Molecular simulation of proteins: How to account for conformational variability when

Diss. ETH No. 20182
Molecular simulation of proteins:
How to account for conformational variability when
calculating relative free energies and 3J-couplings?
A dissertation submitted to the
¨
ETH ZURICH
for the degree of
Doctor of Sciences
presented by
DENISE STEINER
MSc ETH Chemistry, ETH Z¨urich
born 1 April, 1982
citizen of Biberist (SO)
accepted on the recommendation of
Prof. Dr. Wilfred F. van Gunsteren, examiner
Dr. Jane R. Allison, Prof. Dr. Beat H. Meier, and
Prof. Dr. Chris Oostenbrink, co-examiners
2011
dedicated to my family
”Mathematics may be hard, but to make things really disgusting
you need a computer.”
Anonymous
Acknowledgements
First of all I would like to thank my supervisor Wilfred F. van Gunsteren for giving me the
possibility to do my PhD in his group. Because of his excellent leadership skills it was a great
pleasure to work in this environment. Wilfred always managed to see the positive aspects of
my work and after a meeting with him, his new ideas gave motivation for digging further into
the projects for another couple of months. I am grateful for all the things I learned in the last
four years from you, about research and about life in general. Warm thanks also to Jolande for
taking care of Wilfred and for the warm welcoming to their house for the numerous farewell
parties.
I also would like to thank my co-examiners Jane Allison, Beat Meier and Chris Oostenbrink
for agreeing to read my thesis and making my defense happening. Many thanks go to Jane,
I appreciate the various discussions we had about J-values and your critically reading of my
thesis. Which combination of subset 1, 2 and 3 was it again...? Beat Meier introduced me to
the world of NMR in the lectures during my studies at ETH. I consider it an honour to defend
this thesis against such an expert in NMR experiments. Chris Oostenbrink contributed a lot of
ideas to the replica exchange calculations. He accepted me as a scientific guest in his group in
Vienna for seven weeks, which was a great time for me, not only in terms of science, but also
in terms of personal experience. The progress we made in Vienna was an important driving
force for this thesis. Thank you for all of this.
Working is IGC (Informatik-Gest¨utzte Chemie) was a pleasure because of many very nice
and helpful people. The group would not work without the secretaries. Many thanks go
to Daniela for numerous useful advice and for always having a sympathetic ear. Thanks to
Carmen for always being helpful and willing to solve problems. With Ana I had a lot of
interesting discussions about music and more.
Performing simulations is not possible if the machines do not work properly. Therefore I
would like to thank everybody who took or still takes care of our computers: Clara, Jozica
and Katharina for having a look at the Windows machines, Nathan, Halvor, Niels, Stephan,
for maintaining the zoo. A big thanks goes to Alexandra and Moritz for keeping Beaver alive
for me. And of course I thank the whole Ray/Gromos babysitting team. Bettina and Maria
introduced me to the SUN system, Andreas E., Oliwia and Alice shared many (desperate)
hours of debugging with me. I wish Noah much fun joining this group.
A special thank goes to Bojan for introducing me into GROMOS in a very nice and helpful
way during my semester thesis. Thanks to Sereina’s new replica code I could run my last
replica exchange simulations much faster. The fruitful discussions with Zhixiong and Niels
helped me a lot to analyse my free energy calculations. Dongqi was always very helpful, I
am especially thankful for the many tips I got for shell scripting. Daan and Nathan were the
7
8
Acknowledgements
living dictionaries of GROMOS, never too tired to answer questions. Thanks to Phil for many
inspiring ideas during group talks.
Thank you Wei for giving me very interesting insights into the Chinese culture. I had many
nice, motivating conversations with Jozica and Zrinka. Thanks to DJ Lovorka for bringing
good mood to the parties and the lab. Thanks to Pascal for contributing to the good atmosphere
too.
It was fun to get up early on Wednesday morning for jogging with Phil, Halvor, Maria,
and Lovorka. I enjoyed sharing my passion of music and other things with Peter, Monica and
Lorna. Thanks to Dirk, Andreas G. and Halua I never forgot about quantum chemistry. It was
very interesting to hear from Elizabeth about India. Skiing with Hans-Peter in Saas Fee was
a lot of fun and I thank him for providing me nice office mates. With Hiroko I enjoyed many
wonderful hours in Switzerland and Japan. I will not forget the inspiring conversations and
the fun I had with Stefano.
My stay in Vienna was pleasing because of the warm welcome by Chris, Anita, Stephanie
and Sonja, for which I am very thankful.
Thanks a lot to Pitschna for always having a empty chair next to her for me and to Katharina
for listening to all my complaining. Thank you to all my friends inside and outside ETH for
being there for me during the last four years.
Last but not least I am very grateful for all the support I got from my family in the last 30
years for whatever I did.
Thanks to all who carried me to where I am now, and for your patience with me.
Contents
Acknowledgements
7
Summary
11
Zusammenfassung
13
Publications
15
1 Introduction
1.1 Structure refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Experimental approach . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Computational approach . . . . . . . . . . . . . . . . . . . . . .
1.2 Relative free energy calculations . . . . . . . . . . . . . . . . . . . . . .
1.2.1 One-step perturbation . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Thermodynamic integration (TI) . . . . . . . . . . . . . . . . . .
1.2.3 Hamiltonian replica exchange thermodynamic integration (RE-TI)
1.2.4 Slow-growth . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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19
19
20
22
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24
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2 Structural characterisation of Plastocyanin using local-elevation MD
2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Supplementary material . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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25
25
26
30
34
55
56
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins 95
3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.3.1 Generalised Karplus relation . . . . . . . . . . . . . . . . . . . . . . 101
3.3.2 Determination of the parameters of the Karplus relation . . . . . . . . 102
3.3.3 Analysis of the structural and 3 Jαβ -coupling data . . . . . . . . . . . 103
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.4.1 Calculation of 3 Jαβ -values . . . . . . . . . . . . . . . . . . . . . . . 114
9
10
Contents
3.5
3.4.2 Least-squares fitting of Karplus parameters . . . . . . . . . . . . . . 117
exp
-coupling constants . . . . . . . . . . . 129
3.4.3 Reassignment of FKBP 3 Jαβ
Conclusions and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4 Calculation of binding free energies of inhibitors to Plasmepsin II
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Supplementary material . . . . . . . . . . . . . . . . . . . . . . . . . .
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135
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155
5 Calculation of the relative free energy of oxidation of Azurin at pH 5 and
pH 9
171
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.3 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.6 Supplementary material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
6 Outlook
191
Bibliography
193
Curriculum Vitae
203
Summary
Molecular dynamics simulation is a valuable tool to investigate the structure, dynamics and
functionality of biomolecules. On an atomic level, interactions can be examined which may
explain observations from experiment, for which experiments can only give a rough idea about
the origin. However, the accuracy of the results of such a simulation depends on many factors,
e.g. the approximations used for the calculation of the interactions, the calibration of the different model parameters, and the relevance of the sampled part of the conformational space. In
this thesis the main focus is on the relevance of the sampled part of the conformational space.
In Chapter 1 an overview over the different levels of approximation in simulations is given
together with a short description of the effects of conformational variety in protein structure
refinement and the calculation of relative free energies.
In structure refinement, conformational averaging plays an important role, as the measured
observables are averages over time and an ensemble of structures. In Chapter 2 this averaging
is accounted for by applying time-averaged distance restraining on atom-atom distances and
by biasing the calculated (time-averaged) 3 J-coupling constants to obtain agreement with measured 3 J-values from Nuclear Magnetic Resonance (NMR) spectroscopy by adaptive localelevation sampling of conformations of the protein Plastocyanin.
Chapter 3 describes an investigation of the influence of conformational averaging in experiments and in simulations on the relation between the measured side-chain 3 Jαβ -values and
the corresponding dihedral angles θβ of three proteins, Plastocyanin, Lysozyme, and FKBP,
by comparing different definitions and parametrisations of this relationship.
Free energies are driving forces for chemical equilibria such as binding or dissociation processes or conformational changes. In molecular dynamics simulation, the thermodynamic
integration method is often used for the calculation of the relative free energies between two
states. In Chapters 4 and 5 two proteins, Plasmepsin II and Azurin, were studied which are
rather flexible and for which more than one energetically metastable conformation is present
at a specific thermodynamic state. In both cases, the results of the thermodynamic integration suffered from insufficient sampling of the conformational space. Therefore, Hamiltonian
replica exchange thermodynamic integration was applied which may enlarge the part of the
conformational space taken into account for the calculation of the relative free energy at a
particular thermodynamic state. This leads to better converged values for the relative free
energies.
In Chapter 6, possibilities for further improvement of the results presented are discussed.
11
Zusammenfassung
Molekulardynamik-Simulationen sind ein n¨utzliches Werkzeug um die Struktur, Dynamik und
Funktion von Biomolek¨ulen zu erforschen. Auf atomarer Ebene k¨onnen Interaktionen untersucht werden, die experimentelle Beobachtungen erkl¨aren k¨onnten, u¨ ber deren Ursprung
Experimente nur eine grobe Absch¨atzung geben k¨onnen. Allerdings h¨angt die Richtigkeit
der Resultate einer solchen Simulation von vielen Faktoren ab, wie zum Beispiel von den
verwendeten Ann¨aherungen in der Berechnung der Interaktionen, der Kalibrierung der verschiedenen Modell-Parameter, und der Relevanz des abgetasteten Teils des konformationellen
Raums. In der vorliegende Arbeit liegt der Schwerpunkt auf der Relevanz des abgetasteten
¨
Teils des konformationellen Raums eines Proteins. Im Kapitel 1 ist eine Ubersicht
u¨ ber die
unterschiedlichen Grade der N¨aherungen in einer Simulation gegeben, zusammen mit einer
kurzen Beschreibung der Effekte der konformationellen Vielfalt auf Strukturoptimierungsversuche und die Berechnung von relativen freien Energien.
In der Strukturoptimierung spielt die Mittelung u¨ ber mehrere Konformationen eine wichtige
Rolle, da experimentelle Messgr¨ossen Durchschnitte u¨ ber die Zeit und mehrere Strukturen
sind. Im Kapitel 2 wird diese Mittelung ber¨ucksichtigt, indem eine Ann¨aherung der zeitlichen
Durchschnittswerte von Atom-Atom-Abst¨anden im Protein Plastocyanin an experimentelle
Werte aus Kernspinresonanzspektroskopie-Messungen erzwungen wurde und berechneten,
zeitlich gemittelten 3 J-Kopplungskonstanten mit dem “local-elevation” Algorithmus, welcher
den Konformationsraum abtastet, an experimentell gemessene 3 J-Werte angen¨ahert wurden.
Das Kapitel 3 untersucht den Einfluss der Mittelung u¨ ber mehrere Konformationen, welche
sowohl in Experimenten wie auch in Simulationen vorkommen, auf die Relation zwischen
gemessenen 3 Jαβ -Werten von Seitenketten und den zugeh¨origen Torsionswinkeln θβ in drei
Proteinen, Plastocyanin, Lysozyme und FKBP, wobei verschiedene Definitionen und
Parametrisierungen dieser Abh¨angigkeit verglichen werden.
Freie Energien sind treibende Kr¨afte in chemischen Gleichgewichtszust¨anden wie Bindungs¨
oder Dissoziationssprozessen oder konformationellen Anderungen.
In MolekulardynamikSimulationen wird oft die Methode der thermodynamischen Integration verwendet um die
relative freie Energie zwischen zwei Zust¨anden zu berechnen. In den Kapiteln 4 und 5 wurden zwei Proteine, Plasmepsin II und Azurin, betrachtet, die relativ flexibel sind und welche
in einem spezifischen thermodynamischen Zustand in mehr als einer energetisch metastabilen
Konformation vorkommen. In beiden F¨allen wurden die Resultate der thermodynamischen Integration aufgrund des ungen¨ugendem Abtastens des konformationellen Raums beeintr¨achtigt.
Daher wurde eine Verkn¨upfung der “Hamiltonian-replica-exchange”-Method mit der thermodynamischen Integration angewendet, welche den Teil des konformationellen Raums
vergr¨ossern kann, der ber¨ucksichtigt wird in der Berechnung der relativen freien Energie eines
13
14
Zusammenfassung
bestimmten thermodynamischen Zustands. Dies f¨uhrte zu besser konvergierten Werten f¨ur die
relativen freien Energien.
In Kapitel 6 werden die Verbesserungsm¨oglichkeiten der hier pr¨asentierten Resultate diskutiert.
Publications
This thesis led to the following publications:
Chapter 2
Denise Steiner and Wilfred F. van Gunsteren,
”An improved structural characterisation of reduced french bean Plastocyanin based on NMR
data and local-elevation molecular dynamics simulation.”
European Biophysics Journal, 14 (2012), 579-595
Chapter 3
Denise Steiner, Jane R. Allison, Andreas P. Eichenberger, and Wilfred F. van Gunsteren,
”On the calculation of 3 Jαβ -coupling constants for side chains in proteins.”
Journal of Biomolecular NMR, 53 (2012), 223-246
Chapter 4
Denise Steiner, Chris Oostenbrink, Franc¸ois Diederich, Martina Z¨urcher, and Wilfred F. van
Gunsteren,
”Calculation of binding free energies of inhibitors to Plasmepsin II.”
Journal of Computational Chemistry, 32 (2011), 1801-1812
Chapter 5
Denise Steiner, Chris Oostenbrink, and Wilfred F. van Gunsteren,
”Calculation of the relative free energy of oxidation of Azurin at pH 5 and pH 9.”
Journal of Computational Chemistry, 33 (2012), 1467-1477
15
16
Publications
Related publications
Andreas P. Eichenberger, Jane R. Allison, Jo˘zica Dolenc, Daan P. Geerke, Bruno A. C. Horta,
Katharina Meier, Chris Oostenbrink, Nathan Schmid, Denise Steiner, Dongqi Wang, and Wilfred F. van Gunsteren,
”GROMOS++ software for the analysis of biomolecular simulation trajectories.”
Journal of Chemical Theory and Computation, 7 (2011), 3379-3390
Sereina Riniker, Clara D. Christ, Halvor S. Hansen, Philipp H. H¨unenberger, Chris Oostenbrink, Denise Steiner, and Wilfred F. van Gunsteren,
”Calculation of relative free energies for ligand-protein binding, solvation and conformational
transitions using the GROMOS software.”
Journal of Physical Chemistry B, 115 (2011), 13570-13577
1 Introduction
Computer simulation is a promising method complementary to experiments to investigate the
structure and functionality of a biomolecular system, or to be used in drug design or molecular
modeling in industry, where simulations may reduce the number of experiments drastically.
Simulations may replace experiments in the case of very expensive or dangerous experiments,
for example experiments with radioactive material, or in cases where the experiment is impossible as in weather prediction. Molecular dynamics (MD) simulation is capable of providing
insight into mechanisms of biomolecules at the atomic level that are hardly accessible by experiment, because a higher resolution, an atomistic one, in a small system of some nanometers
size is applied, whereas in the experiments often systems of larger scale, e.g. in test tubes of
millimeter size, at lower resolution are investigated. In the ideal case, the simulations reproduce the available experimental data and give insight into the mechanism at the molecular
level. This permits a better understanding of biological processes and quantitative predictions
for new experiments to engineer new molecules with particular properties. Increasing amounts
of computer power are available, making simulations at higher resolutions and of larger and
more complex systems possible. But independent of the progress in computer technology, a
simulation always yields numerical solutions whose accuracy depends on the approximations
made in the molecular model.
A molecular model used in a simulation is defined by four basic choices:
• The degrees of freedom. The elementary particles of a system have to be defined. This
could be a nucleus with electrons around it, an atom where quantum-mechanical particles are treated implicitly, united atoms where hydrogen atoms are implicitly treated,
coarse-grained particles, i.e. one particle per residue of a protein or even a vector field
for fluid dynamics or any hybrid version bridging two models, i.e. implicit treatment of
the solvent and explicit treatment of the solute atoms.
• The interactions. The interactions between the elementary particles can be described
with different levels of approximations. Quantum-mechanically they are described by a
Hamiltonian operator, while a potential energy function is used in molecular mechanics,
and laws of transport dynamics can be applied for fluids. The different levels may be
combined, i.e. in a hybrid quantum-mechanical molecular-mechanical (QM/MM) simulation the interactions of a part of a protein involved in chemical reactions may be
calculated using a Hamiltonian operator, whereas for the remaining part of a protein
further away from the reaction center the interaction may be approximated by a classical potential energy function. In classical mechanics, atomic interactions are calculated
17
18
1 Introduction
considering bonded (through chemical bonds) and nonbonded (through space) interaction types.
• The generation of an ensemble of configurations. The motions of atoms or particles
are governed by statistical mechanics which means that the appropriate configurational
ensemble is to be generated, from which the ensemble-averaged properties can be calculated. This can be done using MD or Monte Carlo (MC) simulation.
• The boundary conditions. These can be of thermodynamic nature, such as a prescribed
temperature or pressure, or of spatial nature, such as a particular wall or interface to the
world outside the simulated system. Often, constant temperature or pressure algorithms
and vacuo or so-called periodic boundary conditions that mimic a periodic system to
avoid surface induced artifacts are used.
Simulation not only allows an analysis of the dynamics of a molecule, it also produces an
ensemble of structures from which an (averaged) experimentally observed quantity Q can be
calculated. MD or MC simulation techniques generate a Boltzmann-weighted ensemble, in
which only the relevant parts of the conformational space are sampled, as the probability of
finding the system in a part of the conformation space with high potential energy is small and
therefore the contribution of these conformations to an observable Q is negligible.
In MC simulations, the ensemble is generated by randomly displacing one (or more) particles and accepting the new conformation depending on the difference between the Hamiltonian
of the old and the new state Ho and Hn , respectively: it is accepted if Hn < Ho , i.e.
1 < e−(Hn −Ho )/kB T
(1.1)
W < e−(Hn −Ho )/kB T
(1.2)
or if
with kB being the Boltzmann constant and T the temperature of the system and W a random
number between 0 and 1. MC simulations lack time-dependent information, as the conformational space is randomly sampled, but may be more efficient in sampling the relevant parts of
a Boltzmann distribution than a search as performed in MD simulations.
In MD simulations, the behaviour of the system is followed by integrating Newton’s equations of motion
∂ V rN
m¨ri = fi = −
,
(1.3)
∂ ri
in which the forces fi acting on the positions
ri of
all the atoms i are given by the partial spatial
derivative of the potential energy V rN . V rN is generally dependent on the positions of all
N particles in the system. This interaction function is called the force field, V phys (rN ). Typically, biomolecular force fields consist of the contribution of the bonded and the nonbonded
interaction functions of the atoms,
V phys (rN ) = V bonded (rN ) +V nonbonded (rN ).
(1.4)
1.1 Structure refinement
19
In the Groningen Molecular Simulation (GROMOS) force field, V bonded (rN ) includes interaction functions that describe the bonds V bond (rN ), bond angles V angles (rN ), and improper
dihedral angles V improp (rN ) using harmonic potential-energy functions and the proper dihedral angles using a function of the form
V
dih
N
(r ) =
Nd
∑ Kϕn (1 + cos(δn) cos(mnϕn)) ,
(1.5)
n=1
where the force constants Kϕn , the phase shifts cos(δn ) and the multiplicities mn are the force
field parameters of the Nd dihedral angles ϕn .
The nonbonded term V nonbonded (rN ) = V LJ (rN ) +V ele (rN ) takes into account the Lennard
Jones interaction between two atoms i and j with distance ri j
#
"
N−1 N
1
C
(i,
j)
12
−C6 (i, j) 6
(1.6)
V LJ (rN ) = ∑ ∑
6
ri j
ri j
i=1 j>i
using the repulsive and attractive van der Waals parameters C12 (i, j) and C6 (i, j) that depend
on the atom types of i and j, and the electrostatic interaction between the charges qn of the
atoms by a Coulombic expression
V
ele
1
2
1 − 21 Cr f
qi q j N−1 N 1
2 Cr f ri j
(r ) =
∑ ∑ ri j − R 3 − R r f ,
4πεo ε1 i=1
j>i
rf
N
(1.7)
with the dielectric permittivity of vacuum ε0 , the relative dielectric permittivity ε1 of the system, and the two reaction field parameters Cr f and Rr f , which define the interactions due to
the reaction field outside a cutoff distance Rr f induced by the charges inside this cutoff.
Special potential energies V special can be added to the force field term V phys , e.g. to restrain
a property to a specific value, a restraining potential energy term V special may be applied as
soon as the deviation of the property from the desired value is too large. The total potential
energy is then given by
V = V phys +V special .
(1.8)
1.1 Structure refinement
There exists a wide interest in investigating the structure and dynamics of biomolecules in
order to understand their function. This can be done experimentally or computationally.
1.1.1 Experimental approach
There are different approaches to investigate the structure of a biomolecule. Two experimental
techniques widely used for structure determination are Nuclear Magnetic Resonance (NMR)
20
1 Introduction
spectroscopy and X-ray diffraction crystallography. As with every method, they each have
advantages and disadvantages.
The technique of X-ray crystallography localises atomic positions by interpreting measured
scattering intensities in terms of electron densities and assigning these to an atom type. As the
measured molecules are in crystallographic form, they are quite static, thus little conformational motion of the molecule is expected to occur. Nowadays structures of good resolution
˚
(lower than 2 Angstr¨
om) can be obtained routinely.
These advantages are at the same time also drawbacks. The crystalline state is most probably different from the aggregation and thermodynamic state in which the biomolecule is found
in nature. The molecule may not be crystallisable in its natural solvent or environment and
shows most probably less flexibility in the crystal than under natural conditions. Crystallographic contacts between different molecules in the crystal, which are not present in a natural
environment, may also influence the conformation. So the measured conformation of the
molecule can be different from the one having the functionality of interest. Besides this, the
accuracy of a structure obtained by X-ray measurement depends very much on the resolution
of the measured scattering intensity. At poor resolution, the signals of different atom types
are similar and improper assignment is more probable. Also, the bigger the molecule, the less
well-resolved are the spectra, as the peaks tend to be smeared out.
Using NMR, there is the possibility to measure the molecule in a more natural environment,
i.e. in solution. Nowadays, even measurements in membranes or in the cell are performed. No
crystalline state is required. For example, chemical shifts of the atoms, indicating atom types
and interactions, Nuclear Overhauser Effects (NOEs), depending on atom-atom distances, and
3 J-coupling constants related to spin-spin interaction and dihedral angles are measured. But
generally, the number of values of observables that can be obtained is smaller than the number
of degrees of freedom in a biomolecule, which means in a structure determination based on
NMR, the structure is underdetermined.
Even in an NMR experiment, the thermodynamic state of the molecule when measured
may differ from the biologically active one. Also, NMR observables are averages over the
many energetically metastable conformations accessible to the multiple copies of the molecule
present in the test tube and over the time taken to carry out the experiment. This complicates
the interpretation of NMR data, as the dependence of a property on a measured average of
an observable Q, e.g. of an atom-atom distance on the NOE, can be complex and non-linear.
The dynamics of the molecules in the NMR test tube may lead to peak broadening. Peak
overlapping may be an issue in large molecules or if repetitive elements are present.
1.1.2 Computational approach
A method that can take into account the averaging and the flexibility of a biomolecule in structure refinement is molecular dynamics simulation. As an ensemble of structures is generated,
averages of observables, not just instantaneous values, can be calculated. Dynamic properties
and flexibility on an atomic level of the molecule can be examined, which may not be directly
accessible by experiment.
1.1 Structure refinement
21
Of course there are also limitations in MD simulations. Approximations are made by only
considering the molecules at the atomic level, neglecting quantum effects. Choices of charges,
masses, bond lengths, etc. have to be made, which are then kept constant in a standard MD
simulation, meaning that reactions cannot be simulated. Also, no adaption of charge upon
conformational change occurs.
For the GROMOS force fields used in this thesis, parametrisation was performed against
small molecule data, which induces uncertainty upon application to systems involving other,
larger molecules. The environment of the system under investigation is not the same as the
natural one, as the system is simulated either in vacuum or in a specific solvent. If necessary,
counterions are added to keep the system neutral, but never all of the co-agents which are
present in nature such as other biomolecules. Inclusion of these would increase the simulation time drastically, and would as well require additional parametrisation work. Besides the
parametrisation issue, the chosen starting structure also plays an important role. If the simulation is not started from a reasonable structure, it may be stuck in a part of the conformational
space that is different from the one the molecule visits during the experiment and therefore
any property calculated from the MD simulation could diverge from the experimental one.
If the experimentally measured and calculated properties do not agree, there may be more
than one possible reason. The question is then how to find out where the disagreement comes
from.
There are methods to enforce the conformational sampling to cover a larger part of the conformational space or to explore a different region of the conformational space if the current
one is not the relevant one. Chapter 2 shows an example where the potential energy surface is
changed by applying an additional, artificial energy term V special to the force field term V phys
to allow the system to overcome energetic barriers. Distances were restrained to upper values
given by NOE distance bounds using a half-harmonic penalty function V dr , which becomes
large for distances outside the measured distance bound. To test the influence of averaging,
two restraining methods were compared, in which either instantaneous or time-averaged distances calculated from the simulation are used. The sampling of the MD simulation was
simultaneously biased towards measured side-chain 3 Jαβ -values by applying a time-averaged
Jres for every dihedral
local-elevation biasing method, where restraining potential
3
functions V
angle build up in the case that the time-averaged Jαβ -values calculated from the simulated dihedral angle values θ do not agree with the measured ones. This approach may help
to overcome energetic barriers and drive the unfavourable dihedral angles to more relevant
values.
On the other hand, the quality of a calculation of a property depends on the accuracy of the
theory behind the calculation, i.e. the reliability of the formula used. This becomes important in the interpretation of experimental data. An unreliable relationship between primary,
i.e. measured data and secondary, i.e. derived data, e.g. structural properties, introduces uncertainty into the secondary data. In Chapter 3 different approaches to define the relation 3 Jαβ (θ )
between measured side-chain 3 Jαβ -values and dihedral angle values θ , are investigated.
22
1 Introduction
1.2 Relative free energy calculations
For many decades, research on the calculation of free energies using MD simulation has been
performed, leading to the introduction of several different approaches. Estimating the free energy of a system by MD can give insight at the atomic level into a thermodynamic process that
is not accessible by experiment and could be used to reduce the number of experimental trials,
e.g. in the selection of inhibitors for enzymes, by preselecting promising inhibitor molecules
from others.
The Helmholtz free energy F of a canonical ensemble at constant volume V , with a constant
number of particles N and constant temperature T is given in statistical mechanics by
Z Z
1
−H (pN ,rN )/kB T
N N
(1.9)
e
dp dr ,
F(N,V, T ) = −kB T ln 3N
h
where H denotes the Hamiltonian of the system, pN the molecular momenta, rN the molecular
coordinates, kB the Boltzmann constant, and h the Planck constant. In an isothermal-isobaric
ensemble, the corresponding Gibbs free energy or free enthalpy G
Z Z Z
1
−(H (pN ,rN )+pV )/kB T
N N
G(N, p, T ) = −kB T ln
e
dp dr dV
(1.10)
V h3N
at pressure p can be defined.
The calculation of the absolute free energy for systems with many particles is often not
possible, as it would involve a numerical solution of a high-dimensional integral. However,
in many cases, one is interested in the relative free energy between two states, e.g. in ligand
binding, in the difference in free energy between the states of the ligand bound to the protein
and of the ligand in solution. Thermodynamic cycles can be specified, see e.g. Fig. 1.1, which
define a series of transformations from one thermodynamic state to another. As the free energy
is a state function, the free energy difference is not dependent on the pathway taken between
two states. The calculation of the binding free energy ∆FAbind between a state where the ligand
ligand A in solution
∆FAbind
/
ligand A bound to protein
bound
∆FBA
f ree
∆FBA
ligand B in solution
∆FBbind
/
ligand B bound to protein
Figure 1.1 Example of a thermodynamic cycle between different thermodynamic states, i.e. ligands A and B in solution and bound to a protein in solution.
1.2 Relative free energy calculations
23
is bound to a protein and the state where it is in solution is not feasible with the computer
power available nowadays, as the transition of a ligand from being free in solution to being
close to and eventually bound to the protein may need very long simulation time. However,
the relative free energy ∆FBA = FB − FA or free enthalpy ∆GBA = GB − GA between different
thermodynamic states A and B can be calculated, e.g. the difference in free energy of a ligand
A bound to a protein and of a ligand B bound to the same site of a protein. By using the
thermodynamic cycles in Fig. 1.1, the calculated difference in the free energy of two ligands
f ree
bound can be subtracted to estimate
A and B in solution ∆FBA and bound to the protein ∆FBA
the difference in binding free energy of the two ligands ∆FBbind − ∆FAbind . There are several
methods available to calculate the relative free energies of two states. Below, those that are
relevant for this thesis will be described.
1.2.1 One-step perturbation
Assuming similar relevant conformational spaces for the two states A and B under investigation, a single MD simulation of a not necessarily physical reference state R can be performed
that is supposed to sample all parts of the relevant conformational spaces. For the states A and
B, the free energy difference to the reference state R can then be calculated from this single
simulation using, e.g. for state A and a canonical ensemble,
D
E
−(HA −HR )/kB T
∆FAR = FA − FR = −kB T ln e
.
(1.11)
R
This so-called one-step perturbation method is simple and fast, as only one simulation is
needed, from which relative free energies with respect to several other states can be calculated. But it is limited by the fact that the conformational space sampled by the reference state
simulation has to contain the relevant parts of the conformational spaces for the states under
investigation. This can result in problems for systems with large differences in the Hamiltonians, e.g. if several atoms are introduced or deleted.
1.2.2 Thermodynamic integration (TI)
In the thermodynamic integration approach, a λ -parameter dependence of the change in Hamiltonian between state A and state B is introduced, i.e. HA = H (λA ) and HB = H (λB ). The
free energy difference between the two states is obtained by integrating the derivative of the
free energy, dF
d λ over the range of λ from λA to λB . Using Eq. 1.9, ∆FBA can be defined as
follows:
Z λB
Z λB dF
∂H
∆FBA = FB − FA =
dλ =
dλ .
(1.12)
∂λ λ
λA d λ
λA
Simulations at different discrete λ -values between
D λEA and λB are performed, while recording the averaged derivative of the Hamiltonian ∂∂H
at each λ -value. The free energy
λ
λ
24
1 Introduction
difference is deduced from the integral over these averages. The simulations at the intermediate λ -values allow the system to adapt to the change in Hamiltonian stepwise and if simulated
long enough, to sample the relevant conformational space for the applied Hamiltonian and to
get converged averages.
This method may be quite time-consuming, as simulations at several intermediate λ -values
have to be executed.
1.2.3 Hamiltonian replica exchange thermodynamic integration
(RE-TI)
Even if a system has the possibility to adapt to an altered Hamiltonian in thermodynamic integration, and therefore adjust the sampled conformational space, it may be stuck in a local
minimum of the potential energy surface and not be able to change to a part of the conformational space more relevant for the current Hamiltonian. To take into account more parts
of the conformational space for a specific Hamiltonian, Hamiltonian replica exchange simulation can be applied. In this method, different simulations at different λ -values are performed
simultaneously and after a certain amount of time, exchange trials between adjacent λ -values
are executed, in which the conformations of two λ -values are exchanged if a Monte Carlo
exchange criterion depending on the potential energy of the system is fulfilled. With this
approach,
D
E the conformations obtained by simulation at similar λ -values are mixed and the
∂H
∂ λ λ at these λ -values, which are now averages over the mixed conformations, may be
better converged. In Chapters 4 and 5, the performance of RE-TI simulations compared to TI
simulations in terms of conformational sampling is investigated.
1.2.4 Slow-growth
Instead of performing simulations at discrete λ -values as in TI to estimate the free energy
difference in Eq. 1.12, one can also continuously change the λ -value from λA to λB during a
simulation by increasing it slightly at every time step in the simulation. This can be considered
as an interpolation of Eq. 1.12 used for TI calculations. With this approach, the system does
not undergo sudden changes in the Hamiltonian, but on the other hand also does not have time
to equilibrate at a specific λ -value, as the value is changed with every step. In Chapter 5 the
slow-growth method is applied, not to calculate differences in free energies, but to generate
starting structures for RE-TI simulations.
2 An improved structural
characterisation of reduced french
bean Plastocyanin based on NMR
data and local-elevation molecular
dynamics simulation
2.1 Summary
Deriving structural information on a protein from NMR experimental data is still a non-trivial
challenge to computational biochemistry. This is due to the low ratio of independent observables and molecular degrees of freedom, the approximations involved in the various relations
between particular observable quantities and molecular conformation, and the averaged character of the experimental data. For example, 3 J-coupling data on a protein is seldom used
in structure refinement due to the multiple-valuedness and limited accuracy of the Karplus
relation linking a 3 J-coupling to a torsional angle. Moreover, sampling of the large conformational space is still problematic. Using the 99-residue protein Plastocyanin as example it is
investigated whether the use of a thermodynamically calibrated force field, inclusion of solvent degrees of freedom, and application of adaptive local-elevation sampling that accounts
for conformational averaging produces a more realistic representation of the ensemble of protein conformations than standard single-structure refinement in non-explicit solvent using restraints that do not account for averaging and are partly based on non-observed data. Yielding
better agreement with the observed experimental data the protein conformational ensemble
is less restricted than when using standard single-structure refinement techniques, which are
likely to yield a too rigid picture of a protein.
25
26
2 Structural characterisation of Plastocyanin using local-elevation MD
2.2 Introduction
Structural information about biomolecules such as proteins, DNA, RNA, carbohydrates and
lipids is essential to an understanding of their role in biomolecular processes in the cell, but is
not very easy to obtain with a high degree of accuracy for any particular biomolecule. This is
due to a variety of reasons: their size, their heterogeneity of composition, the relatively small
free energy differences that characterise different molecular conformations and mixtures, and
the atomic dimensions combined with the great variety of time scales governing their dynamics. X-ray, electron or neutron diffraction techniques are able to produce pictures at the atomic
level of biomolecules in the solid state, while spectroscopic techniques such as NMR, CD,
IR, Raman and fluorescence spectroscopy can be used to obtain, albeit less extensive, structural information under more physiological, i.e. relevant conditions. Such techniques measure
one or more particular observable quantities Q which depend on the molecular coordinates
rN ≡ (r1 , r2 , ..., rN ) and momenta pN of the N atoms of the molecule. Due to the conformational variability that is governed by the laws of statistical mechanics, any observable Q(rN )
that is a function of conformation rN will also show a distribution P(Q(rN )) of Q-values. In
general, experimental techniques only measure an average over space and time, hQiexp , over
this distribution, not the distribution itself.
The challenge of deriving structural information on biomolecules, or in the ideal case deriving biomolecular structure, from experimental data is not a trivial one for the following six
reasons:
1. The function Q(rN ) that yields values of the observable Q as a function of molecular
conformation rN may not be precisely known. For example, an accurate calculation of
NMR chemical shifts as a function of molecular conformation requires sophisticated
quantum-chemical methodology, and still does not reach the precision obtained experimentally. Or, the relation between a 3 J-coupling constant and the corresponding torsional angle θ is generally approximated by the Karplus relation [1, 2] with empirically
derived coefficients a, b, and c, which render this function 3 J(θ ) rather inaccurate. However, for particular observables Q, e.g. X-ray diffraction intensities, the relation Q(rN )
is relatively well known and not too expensive to evaluate.
2. To derive a molecular conformation rN from a measured Q-value one needs the inverse function rN (Q) of the function Q(rN ). Regarding X-ray diffraction, this poses no
problem, because the structure factors are related by Fourier transform to the electron
density. For the inversion of a chemical shift calculation, however, one would need to
invert the quantum-chemical calculation, a clearly impossible task.
3. Even if the inverse function rN (Q) of the function Q(rN ) is known, it may be multiplevalued, i.e. more than one rN -value corresponds to one Q-value. This is e.g. the case
when calculating torsional-angle θ -values from NMR 3 J-coupling constants using the
Karplus relation 3 J(θ ). Its inverse θ (3 J) is multiple-valued.
4. Due to the averaging inherent in the measurement it is usually not possible to determine the Q-distribution P(Q(rN )) or the underlying conformational distribution P(rN )
2.2 Introduction
27
from Q(rN ) exp . If the conformational distribution P(rN ) is characterised by a single
conformation, such as is approximately the case for proteins in crystalline environment,
a single conformation rN (hQi) may serve as a useful approximation to the conformational distribution P(rN ). However,
if different molecular conformations rN contribute
significantly to the average Q(rN ) , as is often the case for observables Q measurable
by NMR, the conformation rN (hQi) derived from the measured averages hQi may be
very unphysical, i.e. may have a negligible Boltzmann weight in the conformational
ensemble, and thus will not be representative for it.
5. The experimentally measured hQi-values possess a finite accuracy, i.e. X-ray or NMR
signal intensities may show a varying accuracy depending on a variety of experimental
parameters.
6. The number NQexp of observable quantities that can be measured for a biomolecular system is generally very much smaller than the number of (atomic) degrees of freedom Ndo f
of the system.
This makes
the problem of determining the conformational distribution
N
from a set of Q(r ) exp -values highly underdetermined. Combining alternative sets of
experimental data pertaining to one system could improve the situation, provided these
data do not represent inconsistent information, e.g. due to measurement under different
thermodynamic conditions or over different time scales [3].
These issues have been discussed in the literature as long as information on protein structure
has been derived from experimental NMR data,
see e.g. [4] and[5]. In 1980, Jardetzky [6]
discussed the difference between the average Q(rN ) and Q( rN ) from quantities Q observable by NMR that possess a non-linear dependence Q(rN ) on rN . The issue of using a conformational Boltzmann weighted ensemble when averaging Q in protein structure refinement
was brought up in 1989 for time-averaged refinement [7] and a few years later for ensembleaveraging refinement based on NMR NOE data [8, 9]. The approximate nature of the Karplus
relation between a 3 J-coupling and the corresponding torsional angle became the focus of
investigations in the 1990s [10–12], while during the past decade approximations involved
in the use of residual dipolar couplings (RDCs) in structure refinement were investigated, as
discussed e.g. in [13] and [14], and the use of chemical shifts in structure determination was
considered again [15, 16].
Some of the six challenges can be met by the use of molecular dynamics (MD) simulation
techniques, which allow for a Boltzmann sampling of conformational space based on a force
field that mimics the atomic interactions at the molecular level. Use of MD simulation allows
for appropriate averaging and enhances the ratio of the number of values of observable quantities over the number of degrees of freedom, because the (bio)molecular force fields are based
on, i.e. are parametrised against, a wide range of experimental data. The use of, be it primitive,
force fields has always been a necessary ingredient of methodology to derive biomolecular
structure from experimental data [17, 18]. Since the 1980s MD simulation has been used to
search conformational space for low-energy conformers, first using a non-physical force field
energy term that represents NMR observables [19], and later one that represents X-ray diffraction intensities [20]. The sampling of conformational space can be biased towards obtaining
28
2 Structural characterisation of Plastocyanin using local-elevation MD
a particular hQiexp -value by restraining the (running) simulated average hQisim -value towards
the given hQiexp -value [7]. In this way an ensemble compatible with the hQiexp -values can be
generated.
Note, however, that hQiexp denotes an observable quantity, that is, a property that can be
measured directly. Such primary experimental data should not be confused with secondary,
non-observed experimental data, QNO , that is, data derived from hQiexp by applying a given
procedure, f , based on a variety of assumptions and approximations: QNO = f (hQiexp ). For
example, peak location and intensity from X-ray diffraction or NMR spectroscopic measurements represent primary, observed data, whereas molecular structures, NMR order parameters,
and so on are secondary, i.e. derived, quantities. Such secondary, non-observed, “experimental” quantities reflect, at least partly, the approximations and assumptions associated with the
conversion procedure f and may in reality carry little experimental information [21]. Clearly,
when coupling or restraining a simulation to a set of hQiexp -values in order to ensure that the
conformational distribution satisfies hQisim = hQiexp , only primary experimental data should
be used. Use of secondary data such as hydrogen-bond or torsional-angle restraints may restrict the sampling artificially and distort the proper Boltzmann weighting of the conformational ensemble. Yet, due to the low ratio of the number of observables to the number of
degrees of freedom in protein structure determination based on NMR data, such secondary,
non-observed, “experimental” data are often used in protein structure refinement, which leads
inevitably to a reduced accuracy of the obtained protein structures.
The use of MD simulation based on atomic level force fields also has its caveats. First, a
force field, no matter how sophisticated or well calibrated using theoretical and experimental
data, has limited accuracy. Second, available computing power still severely limits the extent
of sampling of conformational space for a macromolecule. Yet, the progress made on both
counts over the past decades has made it possible to enhance the accuracy by which protein
structure can be derived significantly.
In the present study we investigate this progress by applying a recently proposed technique for protein structure refinement based on NMR data to the 99-residue protein reduced
french bean Plastocyanin, see Fig. 2.1. Its structure was determinated almost two decades
ago based on NMR data: 1120 NOE intensities, 59 backbone 3 JHN Hα - and 108 side-chain
3 J -couplings [22], see Fig. 2.2. The NOE intensities were represented as NOE atom-atom
αβ
distance bounds. For the determination of the torsional angle restraints, the 3 J-coupling constants were converted to secondary, non-observed, data by specifying allowed ranges for 103
φ - and χ1 -torsional angles. Of the 108 measured 3 Jαβ -couplings, 37 were not used in the
structure determination due to a lack of indication of the preferred χ1 rotamer conformations.
In addition, hydrogen-exchange data were converted to secondary, non-observed, data by specifying 21 backbone-backbone hydrogen-bond restraints. The structure calculations involved
distance geometry calculations to generate a set of structures, which were consecutively refined using molecular dynamics temperature annealing without explicit solvent based on a
modified AMBER force field [23–25]. This resulted in a set of 16 NMR model structures that
largely satisfied the imposed restraints, but did not wholly comply with all measured (primary)
2.2 Introduction
Figure 2.1 Cartoon representation of
the last of the 16 NMR model structures of Plastocyanin [22] with secondary structure (purple: α -helix,
blue: 310 -helix, yellow: β -strand) and
the Cu-ion in orange.
29
Figure 2.2 Tube representation of the backbone
of Plastocyanin. The residues for which 3 Jαβ couplings are used for restraining are shown as
balls, Val in red, Ile in purple, Thr in green, and all
the amino acid residues with two stereospecifically
assigned Hβ in yellow. The Cu-ion is indicated in
orange.
data. This was attributed to inadequate sampling of conformational space in relatively unconstrained regions of the protein, and to the inadequate representation of 3 Jαβ -couplings in the
conformational sampling of χ1 -torsional angles and possible artifacts arising from the force
field used [22].
Due to the ample availability of NMR data and the careful description of the structure determination of Plastocyanin in [22], this molecule offers an appropriate test case to investigate
the accuracy that can be reached by using more recently developed force fields and sampling
methodology:
1. Instead of the AMBER force field [23–25] developed in the 1980’s we use the relatively
recent GROMOS force field parameter sets 45B3 [26] and 53A6 [27] for the vacuum
and water simulations respectively, which were obtained by calibrating against thermodynamic (free energy, enthalpy, density) data for small molecules [27].
2. Instead of structure refinement without explicit solvent, which ignores some solvent
effects, we use explicit water molecules and periodic boundary conditions, which also
allow for constant pressure simulation.
3. Instead of simulated temperature annealing we use local-elevation biasing to enhance
the sampling of side-chain conformations when both the instantaneous and the averaged
30
2 Structural characterisation of Plastocyanin using local-elevation MD
3 J -coupling
αβ
constants calculated from the MD simulation do not match the experimentally measured values.
4. Instead of (instantaneously) applying restraints to every molecular configuration, thereby
ignoring the averaged nature of the measured observables, we use averaged quantities
hQisim in the restraining or biasing.
5. Instead of using, apart from primary NOE data, secondary (non-observed) data such as
hydrogen bonds and torsional angle value ranges in restraints, we only use primary data,
i.e. 957 NOE distance bounds and 62 3 Jαβ -coupling constants in restraining or biasing.
A number of the 1120 NOE bounds involve non-stereospecifically assigned Hβ2 and
Hβ3 atoms with the same value for the NOE bound. These pairs are represented by one
restraint to the pseudo-atom position between Hβ2 and Hβ3 . The 59 3 JHN Hα -couplings
had only been classified as larger than 9 Hz or smaller than 6 Hz [22] and are therefore
not used as restraints. Of the 108 measured 3 Jαβ -values, 46 are not used in the structure
determination because of a lack of stereospecific assignment. The distribution of the
3 J -couplings used as restraints over the protein is shown in Fig. 2.2.
αβ
These differences reflect the development of computational methodology with respect to force
field accuracy and sampling efficiency for structure refinement of a protein and of computing
power to allow for inclusion of solvent degrees of freedom and conformational ensembles.
The focus of the analysis is on the use of 3 Jαβ -couplings in the structure refinement based
on local-elevation sampling [28] of the χ1 dihedral angle degrees of freedom [29]. Recently,
Markwick et al. [30] applied accelerated MD [31], a method based on the same idea as localelevation MD [28], to analyse backbone torsional-angle distributions of the proteins GB3 and
ubiquitin using 3 J-couplings pertaining to the backbone ϕ -angle.
2.3 Method
The simulations were carried out with the GROMOS biomolecular simulation software [32].
For the simulations in vacuo, the 45B3 GROMOS force field [26] was used, and for the simulations in explicit solvent, the 53A6 GROMOS force field [27] was used with the SPC [33]
water model. The lysines and histidines present in the molecule were protonated. The resulting charge of the Cu(I)-Plastocyanin was -8.5 e (with the half charge originating from one
cysteine), thus 8 Na+ counterions were added to the water simulations to get a nearly neutral
solution. The vacuum simulation was performed without any counterions, as in the 45B3 force
field the charged side chains (Glu, Asp, Lys, Cys) and chain termini are neutralised.
As starting structure for the simulations, the last of the 16 NMR model structures described
in [22] was taken from the Protein Data Bank [34, 35] (PDB ID:9PCY). In the vacuum simulations the structure was energy minimised followed by a thermalisation, which involved
position restraining the protein atoms. Initial velocities were generated from a MaxwellBoltzmann distribution. The simulation temperature was raised from 50 K in steps of 50 K
up to 298 K while simultaneously decreasing the position-restraining coupling constant from
2.3 Method
31
25000 kJmol−1 nm−2 to 0 kJmol−1 nm−2 in logarithmic steps, 25000, 2500, 250, 25, 2.5, 0
kJmol−1 nm−2 . For every step simulations of 10 ps were performed. The simulations were
continued for 1 ns at 298 K and the trajectories were used for analysis.
For the water simulations the energy minimised PDB structure was introduced into a truncated octrahedron SPC-water box of 6.3 nm edge length containing 3553 water molecules.
Periodic boundary conditions were used for the simulations in solvent. After another energy
minimisation and thermalisation as described above the starting structure for the unrestrained
MD simulation in water was obtained. The simulations were carried out at a constant temperature of 298 K using the weak coupling method [36] and coupling the solute (protein and
Cu-ion) and solvent degrees of freedom separately to the heat bath with a coupling time τT =
0.1 ps, and at a constant pressure of 1 atm using τP = 0.5 ps and an isothermal compressibility
of 4.575 · 10−4 molnm3 kJ−1 . In both types of simulations a triple-range cutoff scheme was
used for nonbonded interactions where at every time step interactions within a short-range cutoff of 0.8 nm were calculated from a pair list generated every 5th time step. At every 5th time
step interactions between 0.8 and 1.4 nm were updated. A reaction field approach [37, 38]
and a dielectric permittivity of 61 [39] for water were applied for electrostatic interactions
outside a 1.4 nm cutoff distance. The equations of motion were integrated with a step-size of
2 fs applying the leap-frog scheme [40]. The SHAKE algorithm [41] was used for constraining all bonds of the protein and water and the bond angle of the water molecules. Different
restraining functions were used for the NOEs and the 3 Jαβ -couplings.
3 J -couplings depend on torsional angles θ between H -C -C -H via the Karplus relaα α β
αβ
β
tion (Fig. 2.3):
J(θ (t)) = a cos2 θ (t) + b cos θ (t) + c.
(2.1)
Since aliphatic hydrogens are not explicitly represented in the GROMOS force fields, the χ1
torsional angle N-Cα -Cβ -Cγ is used which differs by a phase shift δ from the angle θ [44]:
χ1 = θ + δ .
(2.2)
The value of δ is either -120◦ or 0◦ , depending on whether the hydrogen is Hβ2 or Hβ3 (see
Fig. 2.3).
a MD simulation towards a particular measured value
3 In order3 to0 bias the sampling in restr
J exp = J , a penalty function V
can be added to the physical force field term V phys
for the potential energy:
V rN (t) = V phys rN (t) +V restr rN (t) .
(2.3)
In the case of 3 Jαβ -coupling restraining, a time-averaging and local-elevation biasing method
proposed earlier [29] was applied. The restraining potential energy function VkJres for the k-th
3 J -value related to the torsional angle χ is built up by N (here 36) local-elevation terms
k
le
αβ
32
2 Structural characterisation of Plastocyanin using local-elevation MD
Figure 2.3 Upper panel: Karplus curve for the side-chain dihedral angle θ (Hα − Cα − Cβ − Hβ ),
but given as a function of χ1 (N −Cα −Cβ −Cγ ). The continuous line shows the Karplus curve with
δ = 0◦ for θ (Hα − Cα − Cβ − Hβ3 ), the dashed line is the Karplus curve with a shift δ = −120◦ for
θ (Hα − Cα − Cβ − Hβ2 ). The parameters a, b and c are 9.5, -1.6 and 1.8 Hz [42]. Lower panel:
Karplus curve for the backbone dihedral angle θ (HN − N − Cα − Hα ), but given for the backbone
dihedral angle φ (C − N −Cα −C). The phase shift is δ = −60◦ . The parameters a, b and c are 6.4,
-1.4 and 1.9 Hz [43].
[28]:
VkJres
Nle le
χk r (t) = ∑ Vki χk rN (t) ,
N
(2.4)
i=1
in which the penalty terms are Gaussian functions centered around χki0 :
Vkile
χk r (t) = KkJres ωχki (t)e
N
2
2
−(χk (t)−χki0 ) /2(∆χ 0 )
,
(2.5)
with ∆χ 0 = 360/Nle . KkJres is the overall penalty function force constant (0.005 kJmol−1 Hz−4
here). ωχki (t) is the weight function of the i-th Gaussian penalty function
ωχki (t) = t
−1
Z t
0
δχk (rN (t ′ ))χ 0 V
ki
fb 3
f b N
′
3
J (χk (r (t ))) dt ′ ,
J χk r (t ) V
N
′
which is non-zero if the instantaneous χk rN (t ′ ) -value is in the bin of χki0 :
1
if χki0 − ∆χ 0 /2 < χk rN (t ′ ) < χki0 + ∆χ 0 /2
δχk (rN (t ′ ))χ 0 =
ki
0
otherwise
(2.6)
(2.7)
2.3 Method
33
and both the instantaneous 3 J χk rN (t) and the time-averaged 3 J (χk (rN (t))) deviate more
than ∆J 0 (1 Hz in this study) from the experimental value 3 Jk0 :

3 0
3 J χ rN (t) − 3 J 0 − ∆J 0 2
3 J χ rN (t)

if
> Jk + ∆J 0

k
k
k
3 J χ rN (t) − 3 J 0 + ∆J 0 2
V f b 3 J χk rN (t)
=
if 3 J χk rN (t) < 3 Jk0 − ∆J 0
k
k

 0
otherwise.
(2.8)
3
N
f
b
N
N
3
3
J (χk (r (t))) , J χk r (t) is replaced by J (χk (r (t))), which is the exponenIn V
tially damped temporal average over the course of a MD simulation:
3 J ( χ (rN (t))) =
k
1
τJ (1 − exp (−t/τJ ))
Z t
0
t′ − t
exp
τJ
3
J χk rN (t ′ ) dt ′ ,
(2.9)
with memory relaxation time τJ (here 5 ps). Out of the 108 3 Jαβ -couplings, 62 had been
assigned to Hβ or stereospecifically to Hβ2 or Hβ3 . These values were used for 3 J-restraining
(Table 2.1). The remaining 46 3 Jαβ -couplings (Table 2.2) were only used in the analysis.
For the side-chain 3 Jαβ -couplings the parameters a = 9.5 Hz, b = -1.6 Hz, c = 1.8 Hz [42]
were used in the Karplus relation (Fig. 2.3). The 59 3 JHN Hα -couplings had been categorised
as larger than 9 Hz or smaller than 6 Hz (Table 2.3). These 3 J-couplings were only used in
the analysis. For these backbone 3 JHN Hα -couplings the parameters a = 6.4 Hz, b = -1.4 Hz, c
= 1.9 Hz [43] were used in the Karplus relation (Fig. 2.3).
For distance restraining, NOE data was used. The NOE distance bounds derived [22] from
the measured NOE intensities were used as upper bounds. The distance restraining potential
energy function is attractive half-harmonic:

Ndr

0 2
0
1/2
Kmdr rnn′ − rm
if rnn′ > rm
∑
dr N
V r (t) =
(2.10)
m=1

0
otherwise,
in which the sum is over the Ndr distance restraints and the force constant Kmdr is 1000
kJmol−1 nm−2 . rnn′ is the m-th atom-atom distance restraint between atoms n and n′ with
0 . To take into account the averaged character of the measured
NOE upper distance bound rm
NOE intensity, time-averaged (TAR) restraining was performed using the weighted temporal
average
h
−6
rnn
′ (t)
i−1/6
1
=
τNOE (1 − exp(−t/τNOE ))
Z t
0
t′ − t
exp
τNOE
′
−6
rnn
′ (t)dt
−1/6
(2.11)
instead of rnn′ in Eq. 2.10 with a coupling time τNOE = 5 ps. The NOE violations were
34
2 Structural characterisation of Plastocyanin using local-elevation MD
calculated as
D
−6
rnn
′
E−1/6
0
− rm
.
(2.12)
where h...i denotes an average over the MD ensembles or set of NMR model structures. For
some NOE distance bounds the hydrogen atoms could not be stereospecifically assigned. In
this case a pseudo-atom or averaging correction [45] was added to the bound and a single
pseudo-atom position in between the two or more hydrogen atoms was used in the restraint
[44]. These pseudo-atom positions are denoted in Tables 2.4, 2.5, 2.12 and 2.13 as Q instead of
H. A, B, C, D, E and Z stand for α , β , γ , δ , ε and ζ respectively, indicating the position of the
carbon (C) or hydrogen (H) in the amino acid. This reduced the number of NOE restraints to
957, 414 being “long-range” NOEs between residues separated by at least three other residues
along the polypeptide chain.
Six different MD simulations were performed:
1. UNR VAC: Simulation of the protein in vacuo without restraints.
2. UNR WAT: Simulation of the protein in water without restraints.
3. 3 J LE VAC: Simulation of the protein in vacuo with 3 J-coupling restraining using local
elevation for the 62 3 Jαβ -couplings of Table 2.1.
4. 3 J LE WAT: Simulation of the protein in water with 3 J-coupling restraining using local
elevation for the 62 3 Jαβ -couplings of Table 2.1.
5. 3 J LE NOE WAT: Simulation of the protein in water with 3 J-coupling restraining using local elevation for the 62 3 Jαβ -couplings of Table 2.1 and with instantaneous NOE
distance restraining for the 957 NOE atom pairs of Table 2.13.
3
6. J LE NOE TAR WAT: Simulation of the protein in water with 3 J-coupling restraining
using local elevation for the 62 3 Jαβ -couplings of Table 2.1 and with time-averaged
NOE distance restraining for the 957 NOE atom pairs of Table 2.13.
−1/6
The averaged quantities, NOE atom-atom distances r−6
and 3 J-couplings 3 J , calculated from the trajectories of these simulations were compared to the averages obtained from
the set of 16 NMR model structures. In addition, atom-positional root-mean-square deviations
(RMSD) of the trajectory structures from the initial structure, root-mean-square fluctuations
of atoms and the secondary structure content according to the program dssp [46] were used to
analyse the ensembles.
2.4 Results
Figs. 2.4-2.7 allow a comparison of the 3 J-coupling and NOE data as calculated and averaged
over the six simulated conformational ensembles and over the set of 16 NMR model structures
with the corresponding measured values. In panels a of these figures the results of the MD
simulation of the protein in vacuo without application of any restraints (UNR VAC) are shown.
For the stereospecifically assigned 3 Jαβ -couplings (Fig. 2.4) poor correlation between simul-
2.4 Results
35
Figure 2.4 Comparison of the 62 3 Jαβ -couplings that were stereospecifically assigned and could
be used as restraints calculated from and averaged over each of the 6 different conformational
MD ensembles or the set of 16 NMR model structures with those measured experimentally.
(a) UNR VAC simulation (b) 3 J LE VAC simulation (c) NMR set (d) UNR WAT simulation (e)
3 J LE WAT simulation (f) 3 J LE NOE WAT simulation (g) 3 J LE NOE TAR WAT simulation.
Figure 2.5 Comparison of the 46 3 Jαβ -couplings that were not part of the set of 3 Jαβ -coupling restraints calculated from and averaged over each of the 6 different conformational MD ensembles
or the set of 16 NMR model structures with those measured experimentally. (a) UNR VAC simulation (b) 3 J LE VAC simulation (c) NMR set (d) UNR WAT simulation (e) 3 J LE WAT simulation (f)
3 J LE NOE WAT simulation (g) 3 J LE NOE TAR WAT simulation.
Exp
H
β
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β
β2
β3
β
β2
β3
β
β
β2
β3
β
β
β2
β3
β2
β3
β2
β3
β
β2
β3
β
β
β2
β3
β2
β3
β2
β3
β2
β3
β
β2
3J
exp
10.8
11.7
2.5
5.0
5.4
4.6
9.1
12.1
3.8
11.9
3.2
10.7
3.0
5.5
3.8
5.1
8.5
11.0
11.4
11.8
3.6
10.7
10.4
4.0
11.6
5.7
5.9
8.9
8.4
10.8
5.1
10.9
9.0
3.2
10.6
3.9
8.9
8.0
12.1
3.8
6.6
7.9
2.9
6.6
hχ1 i
184 ± 2
301 ± 3
301 ± 3
53 ± 7
53 ± 7
184 ± 6
184 ± 6
303 ± 2
303 ± 2
282 ± 4
282 ± 4
184 ± 2
65 ± 3
65 ± 3
165 ± 114
223 ± 150
223 ± 150
309 ± 3
179 ± 1
304 ± 2
304 ± 2
185 ± 2
183 ± 4
177 ± 5
177 ± 5
72 ± 72
72 ± 72
198 ± 158
198 ± 158
184 ± 3
175 ± 5
175 ± 5
182 ± 2
54 ± 3
302 ± 2
302 ± 2
232 ± 143
232 ± 143
305 ± 3
305 ± 3
65 ± 101
65 ± 101
45 ± 5
172 ± 6
3 J LE NOE WAT
NMR set
3 3J
hχ1 i
∆3 J
J
12.8 ± 0.0 2.0
189 ± 19 11.6 ± 1.0
12.9 ± 0.1 1.2
297 ± 8 12.7 ± 0.3
3.5 ± 0.3 1.0
297 ± 8
3.2 ± 0.8
2.7 ± 0.6 -2.3
200 ± 81
5.5 ± 2.2
4.3 ± 0.9 -1.1
200 ± 81
4.8 ± 1.9
3.0 ± 0.8 -1.6
169 ± 155
4.8 ± 1.7
12.7 ± 0.2 3.6
169 ± 155
9.4 ± 1.2
12.9 ± 0.0 0.8
296 ± 12 12.5 ± 0.8
3.7 ± 0.3 -0.1
296 ± 12
3.2 ± 1.1
11.9 ± 0.5 -0.0
292 ± 14 12.2 ± 1.0
1.9 ± 0.2 -1.3
292 ± 14
3.0 ± 1.1
12.8 ± 0.1 2.1
196 ± 17 11.4 ± 1.3
4.0 ± 0.4 1.0
42 ± 12
2.2 ± 0.9
2.8 ± 0.3 -2.7
42 ± 12
5.8 ± 1.3
2.9 ± 0.5 -0.9
171 ± 105
3.1 ± 1.2
6.2 ± 3.4 1.1
198 ± 173
5.1 ± 1.3
8.2 ± 0.6 -0.3
198 ± 173
9.4 ± 0.8
12.6 ± 0.2 1.6
310 ± 17 11.7 ± 1.1
12.9 ± 0.0 1.5
188 ± 10 12.4 ± 0.7
12.8 ± 0.0 1.0
292 ± 12 12.3 ± 0.9
3.9 ± 0.3 0.2
292 ± 12
2.8 ± 0.9
12.8 ± 0.1 2.1
197 ± 14 11.5 ± 1.0
12.8 ± 0.1 2.4
197 ± 19 11.1 ± 1.2
3.8 ± 0.6 -0.2
181 ± 15
3.5 ± 1.3
12.8 ± 0.1 1.2
181 ± 15 12.4 ± 1.2
3.1 ± 1.2 -2.6
193 ± 86
5.0 ± 2.3
4.6 ± 1.7 -1.3
193 ± 86
6.0 ± 2.4
6.2 ± 3.3 -2.7
329 ± 56
8.5 ± 1.6
8.6 ± 0.4 0.2
329 ± 56
8.4 ± 1.1
12.8 ± 0.1 2.0
190 ± 19 11.5 ± 1.2
4.1 ± 0.7 -1.0
164 ± 11
5.6 ± 1.2
12.7 ± 0.2 1.8
164 ± 11 11.8 ± 1.0
12.9 ± 0.0 3.9
199 ± 33
9.7 ± 1.4
2.7 ± 0.3 -0.5
43 ± 28
2.7 ± 1.0
12.9 ± 0.0 2.3
202 ± 89 10.4 ± 1.8
3.7 ± 0.2 -0.2
202 ± 89
3.8 ± 1.6
8.0 ± 3.8 -0.9
318 ± 74
8.7 ± 1.9
7.7 ± 0.8 -0.3
318 ± 74
8.2 ± 1.1
12.8 ± 0.1 0.7
294 ± 21 12.4 ± 1.2
4.0 ± 0.4 0.2
294 ± 21
3.1 ± 1.2
2.9 ± 2.6 -3.7
304 ± 111
7.1 ± 1.8
8.0 ± 0.2 0.1
304 ± 111
9.1 ± 0.7
2.1 ± 0.3 -0.8
44 ± 14
2.4 ± 0.9
4.5 ± 0.9 -2.1
164 ± 16
5.8 ± 1.2
Table 2.1 Continued on next page
∆3 J
0.8
1.0
0.7
0.5
-0.6
0.2
0.3
0.4
-0.6
0.3
-0.2
0.7
-0.8
0.3
-0.7
0.0
0.9
0.7
1.0
0.5
-0.8
0.8
0.7
-0.5
0.8
-0.7
0.1
-0.4
0.0
0.7
0.5
0.9
0.7
-0.5
-0.2
-0.1
-0.2
0.2
0.3
-0.7
0.5
1.2
-0.5
-0.8
3J
LE NOE TAR
3 WAT
hχ1 i
J
∆3 J
199 ± 16 11.2 ± 1.2 0.4
296 ± 9 12.6 ± 0.4 0.9
296 ± 9
3.1 ± 1.0 0.6
191 ± 87
5.1 ± 2.0 0.1
191 ± 87
5.0 ± 2.0 -0.4
178 ± 158
4.8 ± 1.4 0.2
178 ± 158
9.4 ± 1.3 0.3
295 ± 17 12.5 ± 0.7 0.4
295 ± 17
3.2 ± 1.1 -0.6
293 ± 28 12.2 ± 1.2 0.3
293 ± 28
3.2 ± 1.3 0.0
193 ± 19 11.5 ± 1.3 0.8
179 ± 85
3.4 ± 1.1 0.4
179 ± 85
6.1 ± 1.3 0.6
218 ± 101
3.0 ± 1.1 -0.8
212 ± 170
5.3 ± 1.5 0.2
212 ± 170
9.4 ± 0.9 0.9
313 ± 21 11.7 ± 1.0 0.7
188 ± 11 12.4 ± 0.8 1.0
297 ± 14 12.3 ± 0.8 0.5
297 ± 14
3.4 ± 1.4 -0.2
197 ± 17 11.3 ± 1.4 0.6
194 ± 23 11.2 ± 1.3 0.8
183 ± 13
3.3 ± 1.2 -0.7
183 ± 13 12.4 ± 0.9 0.8
195 ± 86
5.1 ± 2.2 -0.6
195 ± 86
5.8 ± 2.3 -0.1
329 ± 55
8.5 ± 1.7 -0.4
329 ± 55
8.4 ± 1.1 0.0
191 ± 19 11.7 ± 1.0 0.9
193 ± 106
5.5 ± 1.4 0.4
193 ± 106 10.9 ± 1.6 0.0
174 ± 51
9.3 ± 1.6 0.3
168 ± 65
2.1 ± 0.5 -1.1
162 ± 80 10.0 ± 1.2 -0.6
162 ± 80
4.1 ± 1.5 0.2
336 ± 23
9.0 ± 1.4 0.1
336 ± 23
8.3 ± 0.9 0.3
293 ± 25 12.4 ± 0.9 0.3
293 ± 25
3.1 ± 1.1 -0.7
320 ± 89
7.3 ± 1.6 0.7
320 ± 89
9.1 ± 0.7 1.2
120 ± 80
2.2 ± 0.8 -0.7
158 ± 59
5.8 ± 1.2 -0.8
2 Structural characterisation of Plastocyanin using local-elevation MD
Residue
Name
VAL
LEU
LEU
SER
SER
SER
SER
LEU
LEU
PHE
PHE
VAL
PHE
PHE
VAL
PRO
PRO
ILE
VAL
HISB
HISB
VAL
VAL
ASP
ASP
GLU
GLU
PRO
PRO
VAL
ASP
ASP
VAL
ILE
SER
SER
PRO
PRO
LEU
LEU
PRO
PRO
THR
TYR
36
Nr
3
4
4
7
7
11
11
12
12
14
14
15
19
19
21
22
22
27
28
37
37
39
40
42
42
43
43
47
47
50
51
51
53
55
56
56
58
58
63
63
66
66
69
70
Residue
Name
TYR
VAL
VAL
THR
LEU
LEU
THR
THR
TYR
TYR
CYS
CYS
PRO
PRO
VAL
VAL
THR
VAL
Exp
H
β3
β
β
β
β2
β3
β
β
β2
β3
β2
β3
β2
β3
β
β
β
β
3J
exp
11.2
10.3
5.1
8.6
12.1
3.4
9.7
7.9
12.7
2.1
7.3
10.4
5.8
8.4
4.9
5.6
9.4
11.2
hχ1 i
172 ± 6
188 ± 2
64 ± 2
307 ± 10
304 ± 3
304 ± 3
298 ± 75
309 ± 8
292 ± 3
292 ± 3
164 ± 3
164 ± 3
141 ± 148
141 ± 148
308 ± 2
56 ± 2
319 ± 14
185 ± 2
NMR set
3J
12.6 ± 0.5
12.7 ± 0.1
3.0 ± 0.2
12.5 ± 0.5
12.8 ± 0.1
3.9 ± 0.3
11.4 ± 2.0
12.5 ± 0.7
12.7 ± 0.1
2.6 ± 0.2
5.6 ± 0.4
12.1 ± 0.3
5.0 ± 3.9
8.0 ± 0.2
4.4 ± 0.2
3.9 ± 0.2
11.3 ± 1.2
12.8 ± 0.1
∆3 J
1.4
2.4
-2.1
3.9
0.7
0.5
1.7
4.6
-0.0
0.5
-1.7
1.7
-0.8
-0.4
-0.5
-1.7
1.9
1.6
3 J LE
hχ1 i
164 ± 16
200 ± 17
147 ± 106
334 ± 22
302 ± 11
302 ± 11
296 ± 60
216 ± 98
294 ± 13
294 ± 13
237 ± 103
237 ± 103
217 ± 165
217 ± 165
235 ± 88
45 ± 21
330 ± 21
186 ± 13
NOE WAT
3J
11.7 ± 0.9
11.1 ± 1.0
4.9 ± 1.8
9.3 ± 1.3
12.5 ± 0.7
3.8 ± 1.1
10.3 ± 1.2
8.2 ± 1.3
12.3 ± 1.0
3.1 ± 1.2
7.0 ± 1.6
9.8 ± 1.4
5.5 ± 2.0
9.0 ± 1.5
5.0 ± 1.6
6.0 ± 1.3
10.0 ± 1.1
12.3 ± 0.6
∆3 J
0.5
0.8
-0.2
0.7
0.4
0.4
0.6
0.2
-0.4
1.0
-0.3
-0.6
-0.3
0.6
0.0
0.4
0.6
1.1
3J
LE NOE TAR
3 WAT
hχ1 i
J
∆3 J
158 ± 59 11.3 ± 1.3 0.1
196 ± 18 11.2 ± 1.0 0.9
167 ± 101
4.9 ± 1.6 -0.2
155 ± 82
9.2 ± 1.2 0.6
298 ± 26 12.5 ± 0.8 0.4
298 ± 26
3.6 ± 1.2 0.2
258 ± 90
9.8 ± 1.5 0.1
240 ± 108
8.0 ± 1.6 0.1
293 ± 18 12.4 ± 0.8 -0.3
293 ± 18
2.9 ± 0.9 0.8
234 ± 104
7.0 ± 1.5 -0.3
234 ± 104
9.7 ± 1.4 -0.7
242 ± 159
5.6 ± 1.7 -0.2
242 ± 159
9.2 ± 1.3 0.8
150 ± 114
4.9 ± 1.4 -0.0
166 ± 112
5.6 ± 1.5 -0.0
318 ± 33
9.7 ± 1.4 0.3
188 ± 13 12.2 ± 0.7 1.0
2.4 Results
Nr
70
71
72
73
74
74
76
79
80
80
84
84
86
86
93
96
97
98
Table 2.1 The 62 3 Jαβ -couplings and corresponding side-chain χ1 torsional angles that were selected as restraints. 3 Jexp are the
values from experiment, h...i denotes averaging either over the set of 16
model structures (NMR set) or over the indicated MD
NMR
conformational ensemble. ∆3 J is the difference between the calculated 3 J and the experimental 3 Jexp .
37
38
2 Structural characterisation of Plastocyanin using local-elevation MD
Figure 2.6 Comparison with experimental bounds (smaller than 6 Hz, dotted line; larger than 9 Hz,
dashed line) for the 59 3 JHN Hα -couplings calculated from and averaged over each of the 6 different
conformational MD ensembles or the set of 16 NMR model structures. (a) UNR VAC simulation (b) 3 J LE VAC simulation (c) NMR set (d) UNR WAT simulation (e) 3 J LE WAT simulation (f)
3 J LE NOE WAT simulation (g) 3 J LE NOE TAR WAT simulation. The experimental 3 J
HN Hα -value
was set to the bounds 6 Hz or 9 Hz.
ation and experiment is observed with deviations up to 7 Hz (Table 2.6). For the other 3 Jαβ couplings (Fig. 2.5) almost no correlation is found, again with sizable deviations (Table 2.7).
All but one of the 3 JHN Hα -couplings smaller than 6 Hz are indeed smaller than 6 Hz (Fig.
2.6 and Table 2.8), but only a few of the 3 JHN Hα -couplings that were measured to be larger
than 9 Hz satisfy this lower bound in the simulation. This is not too surprising in view of
the maximum of about 9.7 Hz of the corresponding Karplus curve (Fig. 2.3). Out of the 414
“long-range” NOEs, 32 NOEs show a violation larger than 0.1 nm in the simulation (Fig. 2.7
and Table 2.12). The discrepancies between simulated and experimental data could be due to
force-field deficiencies, insufficient sampling or carrying out the simulation in vacuo.
In panel d of Figs. 2.4-2.7 the results of the MD simulation of the protein in water without application of any restraints (UNR WAT) are shown. For the 3 J-couplings the agreement
between simulation and experiment does not significantly improve by inclusion of the water
degrees of freedom in the simulation, but the NOE distance bound violations are much reduced. Out of the 414 “long-range” NOEs, only 8 NOEs show a violation larger than 0.1
nm in the simulation (Fig. 2.7 and Table 2.13). The discrepancies between the simulated and
experimental 3 J-coupling data can be due to force-field deficiencies or insufficient sampling
of the torsional-angle degrees of freedom that determine the 3 J-couplings. In particular for
the χ1 side-chain torsional angles energetic barriers due to non-bonded
interactions
3 repulsive
hindering side-chain rotations may lead to insufficiently sampled Jαβ -values. The sampling of the χ1 angles that determine the 62 stereospecifically assigned 3 Jαβ -couplings can
be biased towards producing on average the measured 3 J-couplings by using the technique of
2.4 Results
39
Figure 2.7 Difference between the r−6 averaged distances and the NOE distance bounds for
414 pairs of hydrogen atoms that are “long-range” along the residue sequence, in each of
the 6 different conformational MD ensembles or the set of 16 NMR model structures. (a)
UNR VAC (b) 3 J LE VAC (c) NMR set (d) UNR WAT (e) 3 J LE WAT (f) 3 J LE NOE WAT and (g)
3 J LE NOE TAR WAT simulation.
local-elevation biasing based on adaptive 3 J-coupling restraints.
In panels b and e of Figs. 2.4-2.7 the results of the MD simulations of the protein with
application of the 62 3 J-coupling restraints in vacuo (3 J LE VAC) and in water (3 J LE WAT)
are shown, respectively. Since the restraints are applied with a flat-bottom potential energy
restraining function with a flat bottom of 2 Hz, the measured 3 Jαβ -couplings are reproduced
within ±1 Hz (Fig. 2.4 and Table 2.9). For the other 3 J-couplings no improvement of the
deviations between simulations and experiment can be observed (Figs. 2.5, 2.6 and Tables
2.10, 2.11). A comparison of the NOE distance bound violations (Fig. 2.7) shows that, as
observed before without 3 Jαβ -coupling restraints, the inclusion of water in the simulation
reduces the discrepancies with experiment significantly.
In panels c of Figs. 2.4-2.7 the 3 J-couplings and NOE distance bound violations as obtained
by averaging over the set of 16 NMR model structures that were derived using this data as described in the Introduction are shown. The set of 62 measured and stereospecifically assigned
3 J -couplings is roughly reproduced (Fig. 2.4 and Table 2.1) with deviations up to 4 Hz.
αβ
The other measured 3 J-coupling data are reproduced as poorly as in the simulations (Figs.
2.5, 2.6 and Tables 2.2, 2.3). The NOE distance bounds are basically satisfied with only a
few small violations (Fig. 2.4 and Table 2.4). Compared to the simulations in which the 62
3 J -couplings were restrained, the set of 16 NMR model structures shows slightly worse agαβ
Residue
Name
LEU
LEU
LEU
LEU
ASP
ASP
SER
SER
GLU
GLU
LYSH
LYSH
PHE
PHE
ASN
ASN
ASN
ASN
ASN
ASN
ASP
ASP
GLU
GLU
LYSH
LYSH
GLU
GLU
GLU
GLU
GLU
GLU
ASN
ASN
SER
SER
PHE
PHE
SER
SER
HISB
HISB
GLN
GLN
ASN
ASN
Exp
H
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
3J
exp
4.1
10.9
11.6
2.9
4.7
4.1
7.3
6.4
4.6
11.0
8.0
5.3
12.4
2.8
5.3
6.6
4.6
12.4
3.5
4.2
4.0
10.7
5.2
9.9
5.7
9.7
7.1
7.1
4.2
9.1
6.9
8.0
9.6
4.3
6.8
8.3
3.4
6.0
8.8
7.6
13.6
3.5
5.3
9.8
7.3
4.8
hχ1 i
246 ± 54
246 ± 54
294 ± 20
294 ± 20
72 ± 5
72 ± 5
297 ± 10
297 ± 10
241 ± 43
241 ± 43
165 ± 117
165 ± 117
294 ± 1
294 ± 1
236 ± 37
236 ± 37
182 ± 7
182 ± 7
68 ± 6
68 ± 6
216 ± 54
216 ± 54
261 ± 72
261 ± 72
272 ± 67
272 ± 67
223 ± 115
223 ± 115
207 ± 28
207 ± 28
283 ± 62
283 ± 62
126 ± 65
126 ± 65
301 ± 20
301 ± 20
58 ± 2
58 ± 2
180 ± 142
180 ± 142
301 ± 2
301 ± 2
128 ± 56
128 ± 56
165 ± 106
165 ± 106
NMR
set
3J
7.8 ± 4.5
7.2 ± 4.9
12.1 ± 2.5
3.6 ± 1.3
4.9 ± 0.8
2.3 ± 0.4
12.6 ± 0.5
3.2 ± 0.9
6.9 ± 4.6
6.7 ± 4.5
7.0 ± 4.9
4.3 ± 2.6
12.8 ± 0.0
2.7 ± 0.1
6.8 ± 3.1
5.9 ± 4.9
3.3 ± 0.8
12.8 ± 0.1
4.5 ± 0.9
2.6 ± 0.6
6.4 ± 4.3
9.8 ± 4.6
9.4 ± 5.1
5.8 ± 3.4
12.2 ± 1.6
3.0 ± 0.6
9.7 ± 4.6
3.8 ± 1.1
3.6 ± 2.2
10.0 ± 4.1
12.2 ± 2.6
3.5 ± 0.7
3.1 ± 0.7
7.8 ± 4.6
12.1 ± 2.3
4.3 ± 1.0
3.1 ± 0.2
3.6 ± 0.2
5.6 ± 3.7
7.0 ± 2.3
12.9 ± 0.0
3.5 ± 0.2
3.9 ± 1.1
7.6 ± 5.3
6.9 ± 4.4
4.0 ± 3.2
∆3 J
3.7/-3.1
-3.7/3.2
0.5/9.2
0.7/-8.0
0.2/0.8
-1.8/-2.4
5.3/6.2
-3.2/-4.1
2.3/-4.1
-4.3/2.1
-1.0/1.7
-1.0/-3.7
0.4/10.0
-0.1/-9.7
1.5/0.2
-0.7/0.6
-1.3/-9.1
0.3/8.2
1.0/0.3
-1.6/-0.9
2.4/-4.3
-0.9/5.8
4.2/-0.5
-4.1/0.6
6.5/2.5
-6.7/-2.8
2.6/2.6
-3.3/-3.3
-0.6/-5.5
0.9/5.8
5.3/4.2
-4.5/-3.5
-6.5/-1.2
3.5/-1.8
5.3/3.8
-4.0/-2.5
-0.2/-2.9
-2.4/0.2
-3.2/-2.0
-0.6/-1.8
-0.7/9.4
-0.0/-10.1
-1.4/-5.9
-2.2/2.3
-0.4/2.1
-0.8/-3.3
hχ1 i
221 ± 37
221 ± 37
297 ± 13
297 ± 13
56 ± 11
56 ± 11
225 ± 51
225 ± 51
192 ± 21
192 ± 21
270 ± 59
270 ± 59
293 ± 8
293 ± 8
193 ± 13
193 ± 13
194 ± 9
194 ± 9
60 ± 9
60 ± 9
201 ± 27
201 ± 27
284 ± 24
284 ± 24
272 ± 34
272 ± 34
68 ± 41
68 ± 41
259 ± 46
259 ± 46
291 ± 16
291 ± 16
130 ± 84
130 ± 84
242 ± 54
242 ± 54
59 ± 7
59 ± 7
74 ± 11
74 ± 11
317 ± 11
317 ± 11
237 ± 50
237 ± 50
290 ± 24
290 ± 24
3J
LE NOE
3 WAT
J
∆3 J
4.7 ± 3.9
0.6/-6.2
8.8 ± 4.2
-2.1/4.7
12.3 ± 0.7
0.7/9.4
3.3 ± 1.3
0.4/-8.3
3.2 ± 1.1
-1.5/-0.9
4.0 ± 1.3
-0.1/-0.7
6.7 ± 4.4
-0.6/0.3
8.4 ± 5.0
2.0/1.1
3.1 ± 2.0
-1.5/-7.9
12.0 ± 2.2
1.0/7.4
10.8 ± 3.6
2.8/5.5
4.0 ± 3.1
-1.3/-4.0
12.6 ± 0.4
0.2/9.8
2.7 ± 0.7
-0.1/-9.7
2.5 ± 0.9
-2.8/-4.1
11.9 ± 1.4
5.3/6.6
2.2 ± 0.5 -2.3/-10.2
12.1 ± 0.8
-0.3/7.5
3.5 ± 1.0
0.0/-0.7
3.4 ± 1.0
-0.8/-0.1
3.2 ± 2.5
-0.8/-7.5
11.1 ± 2.9
0.4/7.1
11.4 ± 2.5
6.2/1.5
3.1 ± 2.1
-6.8/-2.1
10.3 ± 3.7
4.6/0.6
3.9 ± 3.3
-5.8/-1.8
4.0 ± 2.0
-3.1/-3.1
3.4 ± 1.2
-3.7/-3.7
8.9 ± 4.7
4.7/-0.2
5.7 ± 4.4
-3.4/1.5
12.1 ± 1.4
5.2/4.1
3.0 ± 1.3
-5.0/-4.0
4.4 ± 3.2
-5.2/0.1
6.2 ± 4.3
1.9/-3.4
7.7 ± 4.8
0.9/-0.6
7.4 ± 4.7
-0.9/0.6
3.3 ± 0.8
-0.1/-2.7
3.6 ± 0.8
-2.4/0.2
5.3 ± 1.4
-3.5/-2.3
2.4 ± 0.7
-5.2/-6.4
11.8 ± 1.1
-1.8/8.3
5.7 ± 1.4
2.2/-7.9
7.2 ± 4.6
1.9/-2.6
7.2 ± 4.9
-2.6/1.9
11.7 ± 2.2
4.4/6.9
3.5 ± 2.1
-1.3/-3.8
3 J LE
hχ1 i
276 ± 33
276 ± 33
286 ± 15
286 ± 15
56 ± 12
56 ± 12
121 ± 99
121 ± 99
189 ± 9
189 ± 9
131 ± 110
131 ± 110
289 ± 10
289 ± 10
191 ± 10
191 ± 10
192 ± 13
192 ± 13
59 ± 9
59 ± 9
251 ± 44
251 ± 44
274 ± 31
274 ± 31
267 ± 40
267 ± 40
284 ± 30
284 ± 30
281 ± 36
281 ± 36
274 ± 65
274 ± 65
179 ± 105
179 ± 105
227 ± 57
227 ± 57
64 ± 10
64 ± 10
96 ± 74
96 ± 74
309 ± 9
309 ± 9
275 ± 36
275 ± 36
266 ± 41
266 ± 41
NOE TAR
3 WAT 3
J
∆ J
10.7 ± 3.5
6.6/-0.2
3.8 ± 3.1
-7.1/-0.3
11.7 ± 1.0
0.1/8.8
2.6 ± 1.3
-0.3/-9.0
3.1 ± 1.2
-1.6/-1.0
4.1 ± 1.3
-0.0/-0.6
4.9 ± 3.8
-2.4/-1.5
5.1 ± 3.1
-1.3/-2.2
2.6 ± 0.7
-2.0/-8.4
12.4 ± 0.6
1.4/7.8
6.0 ± 4.3
-2.0/0.7
3.6 ± 1.5
-1.7/-4.4
12.2 ± 0.9
-0.2/9.4
2.5 ± 0.7
-0.3/-9.9
2.5 ± 0.7
-2.8/-4.1
12.3 ± 0.9
5.7/7.0
2.6 ± 0.9
-1.9/-9.8
12.1 ± 1.4
-0.3/7.5
3.4 ± 1.1
-0.1/-0.8
3.6 ± 1.1
-0.6/0.1
8.0 ± 4.8
4.0/-2.7
6.1 ± 4.5
-4.6/2.1
10.4 ± 3.5
5.2/0.5
3.6 ± 3.0
-6.2/-1.6
9.9 ± 4.0
4.2/0.2
4.5 ± 3.9
-5.2/-1.2
11.3 ± 3.0
4.2/4.2
3.7 ± 2.7
-3.4/-3.4
10.9 ± 3.6
6.7/1.8
4.2 ± 3.1
-4.8/0.0
11.5 ± 2.8
4.6/3.5
3.0 ± 1.3
-5.0/-3.9
7.1 ± 4.4
-2.5/2.8
4.3 ± 3.4
0.0/-5.3
6.3 ± 4.6
-0.5/-2.0
9.2 ± 4.4
0.9/2.4
4.0 ± 1.2
0.6/-2.0
3.1 ± 1.0
-2.9/-0.3
5.5 ± 2.9
-3.3/-2.1
2.9 ± 1.5
-4.7/-5.9
12.4 ± 0.6
-1.2/8.9
4.6 ± 1.2
1.1/-9.0
10.7 ± 3.6
5.4/0.9
4.0 ± 3.4
-5.8/-1.3
10.0 ± 3.8
2.7/5.2
4.4 ± 3.9
-0.4/-2.9
2 Structural characterisation of Plastocyanin using local-elevation MD
Nr
1
1
5
5
9
9
23
23
25
25
26
26
29
29
31
31
32
32
38
38
44
44
45
45
54
54
59
59
60
60
61
61
64
64
81
81
82
82
85
85
87
87
88
88
99
99
40
Table 2.2 The 46
3 J -couplings and
αβ
corresponding sidechain χ1 torsional
angles not used
as restraints. 3 Jexp
are the values from
h...i
experiment,
denotes averaging
either over the
set of 16 NMR
model
structures
(NMR set) or over
the indicated MD
conformational
ensemble.
∆3 J
is the difference
between
the calculated 3 J and the
experimental 3 Jexp .
The
alternative
assignments
are
separated by “/” in
the ∆3 J columns.
2.4 Results
Residue
Nr
Name
1
LEU
2
GLU
3
VAL
4
LEU
6
GLY
8
GLY
10
GLY
12
LEU
13
VAL
14
PHE
17
SER
19
PHE
20
SER
22
PRO
25
GLU
26
LYSH
27
ILE
28
VAL
29
PHE
30
LYSH
31
ASN
36
PRO
38
ASN
39
VAL
42
ASP
44
ASP
45
GLU
47
PRO
49
GLY
51
ASP
54
LYSH
55
ILE
58
PRO
60
GLU
61
GLU
62
LEU
63
LEU
64
ASN
67
GLY
68
GLU
69
THR
70
TYR
71
VAL
72
VAL
73
THR
75
ASP
78
GLY
79
THR
80
TYR
81
SER
83
TYR
86
PRO
87
HISB
91
GLY
92
MET
95
LYSH
96
VAL
97
THR
98
VAL
41
Exp
3J
exp
>9
>9
>9
>9
>9
>9
<6
>9
>9
>9
>9
>9
>9
<6
<6
>9
>9
>9
>9
<6
>9
>9
>9
>9
<6
>9
>9
<6
>9
<6
>9
>9
<6
>9
>9
>9
>9
>9
<6
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
<6
<6
>9
>9
>9
>9
>9
hφ i
265 ± 16
253 ± 15
271 ± 6
277 ± 4
262 ± 26
284 ± 3
298 ± 8
227 ± 4
272 ± 5
240 ± 3
244 ± 12
226 ± 7
215 ± 6
308 ± 4
276 ± 15
260 ± 18
242 ± 3
268 ± 12
257 ± 14
294 ± 2
268 ± 7
211 ± 3
226 ± 4
244 ± 3
314 ± 8
253 ± 26
207 ± 3
315 ± 8
279 ± 7
314 ± 2
274 ± 12
278 ± 5
317 ± 9
266 ± 11
206 ± 7
278 ± 6
262 ± 17
266 ± 31
317 ± 9
250 ± 6
250 ± 16
255 ± 13
211 ± 3
235 ± 19
254 ± 11
290 ± 4
250 ± 17
245 ± 8
264 ± 7
222 ± 3
258 ± 2
227 ± 4
301 ± 2
281 ± 1
235 ± 4
224 ± 13
262 ± 4
275 ± 10
245 ± 26
NMR set
3 J
8.0 ± 0.6
8.9 ± 0.9
7.7 ± 0.6
7.2 ± 0.5
7.5 ± 1.0
6.2 ± 0.3
4.5 ± 1.0
9.3 ± 0.2
7.6 ± 0.5
9.7 ± 0.0
9.4 ± 0.3
9.2 ± 0.4
8.4 ± 0.5
3.4 ± 0.4
7.2 ± 1.7
8.4 ± 1.2
9.7 ± 0.0
7.9 ± 0.9
8.8 ± 0.6
5.0 ± 0.3
8.0 ± 0.7
8.1 ± 0.3
9.2 ± 0.2
9.7 ± 0.1
2.9 ± 0.9
8.1 ± 1.2
7.6 ± 0.3
2.8 ± 0.6
6.8 ± 0.7
2.8 ± 0.2
7.3 ± 1.0
7.0 ± 0.6
2.7 ± 0.8
8.1 ± 1.0
7.4 ± 0.8
6.9 ± 0.7
8.2 ± 1.2
6.7 ± 1.0
2.6 ± 0.5
9.4 ± 0.2
9.0 ± 0.4
8.9 ± 0.4
8.0 ± 0.3
8.9 ± 0.7
9.0 ± 0.8
5.5 ± 0.5
8.9 ± 0.6
9.5 ± 0.2
8.5 ± 0.6
9.0 ± 0.2
9.0 ± 0.2
9.3 ± 0.2
4.2 ± 0.3
6.6 ± 0.1
9.6 ± 0.1
8.8 ± 0.6
8.7 ± 0.4
7.3 ± 0.9
8.3 ± 0.9
∆3 J
-4.0
-3.1
-4.3
-4.8
-4.5
-5.8
1.5
-2.7
-4.4
-2.3
-2.6
-2.8
-3.6
0.4
4.2
-3.6
-2.3
-4.1
-3.2
2.0
-4.0
-3.9
-2.8
-2.3
-0.1
-3.9
-4.4
-0.2
-5.2
-0.2
-4.7
-5.0
-0.3
-3.9
-4.6
-5.1
-3.8
-5.3
-0.4
-2.6
-3.0
-3.1
-4.0
-3.1
-3.0
-6.5
-3.1
-2.5
-3.5
-3.0
-3.0
-2.7
1.2
3.6
-2.4
-3.2
-3.3
-4.7
-3.7
3J
LE
hφ i
268 ± 18
255 ± 15
272 ± 13
275 ± 12
254 ± 27
277 ± 24
257 ± 80
230 ± 10
278 ± 9
243 ± 8
243 ± 11
250 ± 12
238 ± 12
305 ± 13
271 ± 15
263 ± 16
257 ± 11
271 ± 14
255 ± 12
292 ± 8
250 ± 15
235 ± 10
239 ± 11
248 ± 9
298 ± 12
269 ± 16
238 ± 19
296 ± 64
250 ± 19
299 ± 12
267 ± 15
250 ± 22
301 ± 10
268 ± 16
251 ± 20
273 ± 14
245 ± 13
278 ± 16
300 ± 13
250 ± 14
238 ± 14
248 ± 13
240 ± 11
256 ± 16
259 ± 15
295 ± 12
252 ± 13
245 ± 10
261 ± 16
236 ± 10
249 ± 11
258 ± 16
297 ± 13
240 ± 23
235 ± 9
244 ± 11
259 ± 11
262 ± 12
256 ± 18
NOE
WAT
3J
7.7 ± 1.7
8.8 ± 1.1
7.5 ± 1.4
7.3 ± 1.3
8.5 ± 1.7
6.6 ± 2.2
6.1 ± 2.1
9.3 ± 0.5
6.9 ± 1.1
9.6 ± 0.2
9.4 ± 0.4
9.2 ± 0.7
9.4 ± 0.4
3.8 ± 1.3
7.6 ± 1.5
8.2 ± 1.4
8.9 ± 0.8
7.6 ± 1.4
8.9 ± 0.8
5.2 ± 1.0
9.1 ± 0.9
9.4 ± 0.3
9.5 ± 0.4
9.4 ± 0.4
4.6 ± 1.4
7.7 ± 1.4
9.0 ± 1.1
3.5 ± 1.2
8.9 ± 1.3
4.5 ± 1.3
8.0 ± 1.4
8.6 ± 1.5
4.2 ± 1.2
7.8 ± 1.5
8.8 ± 1.5
7.4 ± 1.5
9.3 ± 0.6
6.8 ± 1.6
4.3 ± 1.4
9.1 ± 0.8
9.3 ± 0.6
9.3 ± 0.7
9.4 ± 0.4
8.8 ± 1.2
8.6 ± 1.2
4.9 ± 1.3
9.1 ± 0.8
9.4 ± 0.4
8.4 ± 1.4
9.5 ± 0.3
9.3 ± 0.6
8.6 ± 1.3
4.8 ± 1.5
8.7 ± 1.2
9.5 ± 0.3
9.4 ± 0.4
8.8 ± 0.9
8.5 ± 1.0
8.6 ± 1.4
∆3 J
-4.3
-3.2
-4.5
-4.7
-3.5
-5.4
3.1
-2.7
-5.1
-2.4
-2.6
-2.8
-2.6
0.8
4.6
-3.8
-3.1
-4.4
-3.1
2.2
-2.9
-2.6
-2.5
-2.6
1.6
-4.3
-3.0
0.5
-3.1
1.5
-4.0
-3.4
1.2
-4.2
-3.2
-4.6
-2.7
-5.2
1.3
-2.9
-2.7
-2.7
-2.6
-3.2
-3.4
-7.1
-2.9
-2.6
-3.6
-2.5
-2.7
-3.4
1.8
5.7
-2.5
-2.6
-3.2
-3.5
-3.4
3J
LE NOE TAR
3 WAT 3
hφ i
J
∆ J
270 ± 20 7.5 ± 1.9
-4.5
255 ± 13 8.9 ± 0.9
-3.1
272 ± 13 7.6 ± 1.4
-4.4
274 ± 12 7.3 ± 1.3
-4.7
250 ± 22 8.7 ± 1.6
-3.3
264 ± 24 7.7 ± 2.0
-4.3
253 ± 89 5.8 ± 2.2
2.8
236 ± 14 9.3 ± 0.8
-2.7
273 ± 14 7.4 ± 1.4
-4.6
244 ± 9
9.5 ± 0.3
-2.5
250 ± 15 9.1 ± 0.9
-2.9
255 ± 17 8.7 ± 1.2
-3.3
239 ± 11 9.4 ± 0.4
-2.6
299 ± 24 4.2 ± 1.3
1.2
274 ± 16 7.2 ± 1.6
4.2
262 ± 15 8.4 ± 1.3
-3.6
260 ± 12 8.6 ± 1.0
-3.4
270 ± 14 7.7 ± 1.4
-4.3
248 ± 12 9.3 ± 0.5
-2.7
287 ± 9
5.8 ± 1.1
2.8
252 ± 16 8.9 ± 1.0
-3.1
232 ± 13 9.2 ± 0.7
-2.8
243 ± 16 9.2 ± 0.8
-2.8
250 ± 11 9.2 ± 0.7
-2.8
302 ± 12 4.1 ± 1.3
1.1
270 ± 18 7.6 ± 1.6
-4.4
254 ± 30 8.0 ± 2.2
-4.0
291 ± 32 5.0 ± 1.6
2.0
266 ± 19 7.9 ± 1.8
-4.1
309 ± 15 3.5 ± 1.1
0.5
273 ± 16 7.3 ± 1.6
-4.7
259 ± 25 8.0 ± 1.8
-4.0
299 ± 24 4.2 ± 1.3
1.2
266 ± 16 8.0 ± 1.4
-4.0
241 ± 15 9.2 ± 0.9
-2.8
265 ± 16 8.1 ± 1.5
-3.9
252 ± 18 8.8 ± 1.2
-3.2
280 ± 16 6.5 ± 1.8
-5.5
298 ± 15 4.5 ± 1.6
1.5
239 ± 16 9.2 ± 0.8
-2.8
232 ± 14 9.2 ± 0.8
-2.8
251 ± 11 9.2 ± 0.7
-2.8
241 ± 11 9.4 ± 0.4
-2.6
250 ± 14 9.1 ± 0.9
-2.9
249 ± 12 9.2 ± 0.7
-2.8
291 ± 13 5.3 ± 1.5
-6.7
251 ± 14 9.1 ± 0.8
-2.9
258 ± 15 8.7 ± 1.2
-3.3
270 ± 18 7.6 ± 1.7
-4.4
238 ± 13 9.4 ± 0.5
-2.6
252 ± 14 9.1 ± 0.8
-2.9
279 ± 22 6.5 ± 2.1
-5.5
112 ± 97 6.3 ± 1.1
3.3
245 ± 21 8.8 ± 1.3
5.8
236 ± 14 9.3 ± 0.7
-2.7
248 ± 13 9.3 ± 0.7
-2.7
267 ± 13 8.0 ± 1.2
-4.0
264 ± 14 8.2 ± 1.3
-3.8
251 ± 15 9.1 ± 0.9
-2.9
Table 2.3 The 59 3 JHN Hα -couplings and corresponding backbone φ torsional angles. 3 Jexp are the
values from experiment, h...i denotes averaging either over the set of 16 NMR model structures
(NMR set) or over
the indicated MD conformational ensemble. ∆3 J is the difference between the
3
calculated J and the experimental 3 Jexp .
42
NOE
atom 1
HA
H
H
HA
H
HA
HA
H
HZ
HB
HB
H
H
CG
H
HZ
HD2
HE2
H
H
H
H
H
HA
H
H
H
HA
CZ
HH
HH
HH
HH
HH
HA
CZ
H
HB
HB
HB
HD2
HD2
HE1
H
H
H
H
1
28
9
96
16
96
96
19
72
4
4
32
35
31
33
66
35
35
37
82
50
50
54
72
58
63
74
50
50
47
76
77
77
98
93
42
82
84
12
90
86
90
86
87
88
20
77
RES2
LEU
VAL
ASP
VAL
PRO
VAL
VAL
PHE
VAL
LEU
LEU
ASN
PHE
ASN
ALA
PRO
PHE
PHE
HISB
PHE
VAL
VAL
LYSH
VAL
PRO
LEU
LEU
VAL
VAL
PRO
THR
LYSH
LYSH
VAL
VAL
ASP
PHE
CYS
LEU
ALA
PRO
ALA
PRO
HISB
GLN
SER
LYSH
atom 2
QG
H
H
HA
HB
HA
QG2
HB
QG1
HB
HB
HB
CG
HA
HA
HB
HA
QB
HB
HB
H
QG1
H
QG2
HB
HB
HB
QG2
QG1
HB
QG2
HA
H
HB
HA
HB
HB
HB
HB
QB
HB
QB
HB
HA
HA
HA
HA
rexp
0.720
0.350
0.350
0.430
0.400
0.430
0.450
0.350
0.500
0.400
0.400
0.270
0.710
0.710
0.500
0.500
0.300
0.590
0.350
0.500
0.300
0.600
0.430
0.450
0.400
0.400
0.400
0.600
0.610
0.500
0.500
0.500
0.500
0.500
0.270
0.610
0.350
0.500
0.500
0.500
0.500
0.500
0.500
0.430
0.350
0.270
0.430
NMR set
-0.104
-0.004
-0.056
0.001
0.058
-0.028
-0.032
-0.005
-0.038
-0.005
0.048
0.072
-0.147
-0.138
-0.178
0.001
0.007
-0.188
-0.023
0.005
-0.046
-0.028
0.002
-0.146
0.030
0.022
0.020
-0.054
-0.129
0.092
0.015
0.118
0.123
0.048
-0.053
0.033
-0.002
0.093
0.004
-0.005
0.058
-0.101
0.013
-0.121
-0.041
-0.033
0.022
D
r−6
E−1/6
− rexp
3 J LE WAT
3 J LE NOE TAR WAT
-0.127
0.001
-0.030
-0.043
0.066
0.056
0.141
-0.013
0.028
0.061
0.070
0.093
-0.076
-0.003
-0.030
-0.066
0.004
0.029
0.032
0.015
0.007
0.050
0.011
-0.066
0.038
0.012
0.003
0.046
0.084
0.042
-0.033
0.225
0.186
0.151
0.009
0.017
0.021
-0.028
0.014
0.077
0.048
0.014
-0.033
-0.012
0.023
0.046
0.034
0.048
0.021
0.047
0.010
0.030
0.043
0.110
0.010
0.010
0.043
0.067
0.098
0.024
0.115
0.030
0.020
0.043
0.099
0.019
0.028
0.010
0.061
0.018
0.044
0.019
0.021
-0.005
0.064
0.091
-0.008
-0.105
-0.018
0.013
-0.054
0.017
0.016
0.031
-0.000
0.054
0.132
0.032
0.044
-0.013
0.027
0.029
0.047
0.036
2 Structural characterisation of Plastocyanin using local-elevation MD
Table 2.4 NOE distances which have
a violation of larger
than 0.01 nm in the
3 J LE NOE TAR WAT
simulation or the
set of 16 NMR
model
structures.
The
−1/6 deviation
( r−6
− rexp )
from
the
distance bound rexp
is given for the
3 J LE WAT and the
3 J LE NOE TAR WAT
simulation and the
set of 16 NMR
model structures.
3
3
8
18
18
20
20
20
29
32
32
33
33
35
35
35
37
37
38
40
49
52
52
55
61
64
75
76
80
80
80
80
80
80
83
83
83
87
87
87
87
87
87
90
92
97
98
RES1
VAL
VAL
GLY
GLU
GLU
SER
SER
SER
PHE
ASN
ASN
ALA
ALA
PHE
PHE
PHE
HISB
HISB
ASN
VAL
GLY
ALA
ALA
ILE
GLU
ASN
ASP
THR
TYR
TYR
TYR
TYR
TYR
TYR
TYR
TYR
TYR
HISB
HISB
HISB
HISB
HISB
HISB
ALA
MET
THR
VAL
bound
RES1
VAL
GLU
SER
SER
ASN
ASN
ALA
HISB
HISB
HISB
ASN
VAL
ALA
ALA
GLU
ASN
ASP
TYR
TYR
TYR
TYR
TYR
TYR
TYR
TYR
CYS
HISB
HISB
HISB
HISB
THR
VAL
atom 2
H
HB
HA
QG2
HB
HB
HB
HB
HB
QB
HB
HB
QG2
H
HB
HB
HB
HB
QG2
QG2
HA
H
HB
HB
HB
QE
HB
HB
HB
HB
HA
HA
bound
rexp
0.350
0.400
0.430
0.450
0.400
0.400
0.270
0.350
0.350
0.590
0.350
0.500
0.550
0.430
0.400
0.400
0.400
0.500
0.650
0.500
0.500
0.500
0.500
0.610
0.350
0.500
0.500
0.500
0.500
0.500
0.270
0.430
NMR set
-0.004
0.058
-0.028
-0.032
-0.005
0.048
0.072
-0.071
-0.004
-0.188
-0.023
0.005
-0.030
0.002
0.030
0.022
0.020
0.092
-0.183
0.015
0.118
0.123
0.048
0.033
-0.002
-0.051
0.093
0.004
0.058
0.013
-0.033
0.022
−1/6
r−6
− rexp
3 J LE WAT
3 J LE NOE WAT
0.001
0.015
0.066
0.029
0.056
0.011
0.141
0.014
0.061
0.029
0.070
0.051
0.093
0.085
0.057
0.020
0.020
0.015
0.029
0.020
0.032
0.029
0.015
0.017
0.044
0.028
0.011
0.021
0.038
0.024
0.012
0.017
0.003
-0.001
0.042
0.012
-0.141
0.018
-0.033
-0.090
0.225
-0.074
0.186
-0.032
0.151
-0.091
0.017
0.000
0.021
0.019
0.011
0.019
-0.028
-0.013
0.014
0.025
0.048
-0.040
-0.033
-0.015
0.046
0.013
0.034
0.022
2.4 Results
3
18
20
20
32
32
33
37
37
37
38
40
52
52
61
64
75
80
80
80
80
80
80
83
83
84
87
87
87
87
97
98
NOE
atom 1
RES2
H
28 VAL
H
16 PRO
HA
96 VAL
HA
96 VAL
HB
4
LEU
HB
4
LEU
H
32 ASN
HD2
5
LEU
HD2
37 HISB
HE2
35 PHE
H
37 HISB
H
82 PHE
HA
55 ILE
H
54 LYSH
H
58 PRO
H
63 LEU
H
74 LEU
HH
47 PRO
HH
55 ILE
HH
76 THR
HH
77 LYSH
HH
77 LYSH
HH
98 VAL
CZ
42 ASP
H
82 PHE
HB
92 MET
HB
84 CYS
HB
12 LEU
HD2
86 PRO
HE1
86 PRO
H
20 SER
H
77 LYSH
Table 2.5 NOE distances which have a violation of larger than 0.01 nm in the 3 J LE NOE WAT simulation or the set of 16 NMR model
−1/6
structures. The deviation ( r−6
− rexp ) from the distance bound rexp is given for the 3 J LE WAT and the 3 J LE NOE WAT simulation
and the set of 16 NMR model structures.
43
44
2 Structural characterisation of Plastocyanin using local-elevation MD
Figure 2.8 Cα -atom-positional root-mean-square deviation (RMSD) from the initial structure in the
MD simulations. Red: UNR VAC. Green: 3 J LE VAC. Blue: UNR WAT. Yellow: 3 J LE WAT. Black:
3 J LE NOE WAT. Magenta: 3 J LE NOE TAR WAT.
Figure 2.9 Secondary structure analysis [46] of the 3 J LE NOE TAR WAT simulation. Black: 310 helix. Red: α -helix. Cyan: Bend. Magenta: β -Bridge. Blue: β -Strand. Orange: Turn. The right
hand panel shows the root-mean-square fluctuation (RMSF) of the backbone (N, Cα , C) atoms.
The lower panel shows the root-mean-square distance between the instantaneous positions of the
Cα , N and C atoms of the backbone and their positions in the initial structure.
2.4 Results
45
reement with experiment for the 3 J-couplings and better agreement with the NOE distance
bounds. This is no surprise, because the latter were used as restraints in the determination
of the set of NMR model structures, whereas they were not used as such in the simulations
discussed so far. Thus the next step is to consider the MD simulations in which NOE distance
restraints are applied in addition to the 3 Jαβ -coupling restraints.
In panels f and g of Figs. 2.4-2.7 the results of the MD simulations of the protein in water with application of the 62 3 Jαβ -coupling restraints and the 957 NOE distance restraints
either using instantaneous restraining (3 J LE NOE WAT) or using time-averaged restraining
(3 J LE NOE TAR WAT) are shown, respectively. The additional NOE distance restraining
does not affect the agreement of the 3 J-couplings with experiment (Figs. 2.4-2.6 and Tables
2.1-2.3) and slightly improves the agreement with the NOE distance bounds (Fig. 2.7 and
Tables 2.4, 2.5). Since the measurement of observables such as 3 J-couplings and NOE intensities involves averaging over time and space, we consider the simulation that involves timeaveraged restraints instead of instantaneous ones as the better representation of reality. Therefore, we analyse and compare in more detail only the MD simulation 3 J LE NOE TAR WAT
and compare its ensemble of conformations with the set of 16 NMR model structures and with
the experimental 3 Jαβ -coupling data.
Fig. 2.8 shows that all MD simulations except the one using 3 Jαβ -coupling restraints in
vacuo stay reasonably close to the initial structure, one of the 16 NMR model structures.
Not surprisingly, the simulation 3 J LE NOE WAT stays closest to the NMR model structure because its restraints are most similar to the ones used to derive the NMR model structure. The secondary structure analysis shown in Figs. 2.9-2.11 indicates that the β -strands
(Sheet I: residues Leu 1 to Gly 6, Val 13 to Val 15, Glu 25 to Asn 32, and Gly 67 to Leu 74;
Sheet II: residues Ser 17 to Val 21, His 37 to Asp 42, Gly 78 to Cys 84, and Met 92 to Asn 99)
and two short helical elements (residues Asp 51 to Ser 56 and Cys 84 to Gly 91) are preserved
in the 3 J LE NOE WAT and 3 J LE NOE TAR WAT simulations and in the set of 16 NMR
model structures. Thus the different types of restraining do not distort the overall structure of
the protein significantly.
The global comparison of the set of 16 NMR model structures and the MD simulation
3 J LE NOE TAR WAT with the measured NMR data shows that both sets of conformations
agree on average equally well with the experimental data, which is no surprise since these data
were used as restraints in both cases. However, a comparison of individual side-chain χ1 -angle
distributions and the corresponding averaged 3 Jαβ -couplings shows interesting differences.
Below we analyse these for nine different side-chains that serve as example of a particular
type of side-chain behaviour in protein structure refinement.
Figs. 2.12-2.25 show the behaviour of the χ1 torsional angle, the corresponding 3 Jαβ coupling and the biasing local-elevation potential energy Vle (χ1 ) as a function of time during
a simulation together with the resulting χ1 -angle and 3 Jαβ -coupling distributions and localf inal
elevation biasing potential energy function Vle (χ1 ) for the nine side chains used as examples.
46
2 Structural characterisation of Plastocyanin using local-elevation MD
Figure 2.10 Secondary structure analysis [46] of the set of 16 NMR model structures. Black: 310 helix. Red: α -helix. Cyan: Bend. Magenta: β -Bridge. Blue: β -Strand. Orange: Turn. The right
hand panel shows the root-mean-square fluctuation (RMSF) of the backbone (N, Cα , C) atoms.
The lower panel shows the root-mean-square distance between the instantaneous positions of the
Cα , N and C atoms of the backbone and their positions in the initial structure.
Figure 2.11 Secondary structure analysis [46] of the 3 J LE NOE WAT simulation. Black: 310 -helix.
Red: α -helix. Cyan: Bend. Magenta: β -Bridge. Blue: β -Strand. Orange: Turn. The right hand
panel shows the root-mean-square fluctuation (RMSF) of the backbone (N, Cα , C) atoms. The
lower panel shows the root-mean-square distance between the instantaneous positions of the Cα ,
N and C atoms of the backbone and their positions in the initial structure.
2.4 Results
47
Figure 2.12 Properties of the χ1 torsional angle of Phe 14 and the corresponding 3 JHα Hβ in the
2
3 J LE NOE TAR WAT simulation. Upper row: Left: Final built-up local-elevation restraining potential energy Vlef inal (χ1 ). Middle: Distribution of χ angles. Right: Distribution of 3 J-couplings. Lower
row: Left: Local-elevation potential energy Vle (t) acting on χ1 at specific time points. Middle: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation (circles) and the UNR WAT simulation
(triangles). Right: Evolution of 3 J-value in the 3 J LE NOE TAR WAT simulation (circles) and the
UNR WAT simulation (triangles). The dashed line shows the experimental 3 J-value.
Figure 2.13 Properties of the χ1 torsional angle of Val 50 and the corresponding 3 JHα Hβ in the
3 J LE NOE TAR WAT simulation. Upper row: Left: Final built-up local-elevation restraining potential energy Vlef inal (χ1 ). Middle: Distribution of χ1 angles. Right: Distribution of 3 J-couplings. Lower
row: Left: Local-elevation potential energy Vle (t) acting on χ1 at specific time points. Middle: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation (circles) and the UNR WAT simulation
(triangles). Right: Evolution of 3 J-value in the 3 J LE NOE TAR WAT simulation (circles) and the
UNR WAT simulation (triangles). The dashed line shows the experimental 3 J-value.
48
2 Structural characterisation of Plastocyanin using local-elevation MD
In Fig. 2.12, the χ1 -angle of Phe 14 serves as an example of the case in which the initial structure is such that the 3 Jαβ2 -coupling agrees with the measured value of 11.9 Hz.
Thus no local-elevation biasing energy function is built up which means that the simulation 3 J LE NOE TAR WAT (circles) yields the same distribution of χ1 -angles and 3 Jαβ2 couplings as the unrestrained simulation UNR WAT (triangles).
In Fig. 2.13, the χ1 -angle of Val 50 shows, however, different behaviour for these two
simulations. The unrestrained simulation yields an incorrect 3 Jαβ -coupling which can be
easily corrected in the biased simulation by the build-up of a local-elevation energy
function
around χ1 = 290◦ which drives the dihedral angle value to about 190◦ yielding a 3 Jαβ -value
in better agreement with the 3 Jexp -value.
Figs. 2.14 and 2.15 show a case, the χ1 angle of Glu 43, in which averaging over a wide
range of χ1 -angles is needed. For both H atoms, Hβ2 (Fig. 2.14) and Hβ3 (Fig. 2.15) the
16 NMR model structures (squares) also show a considerable spread in 3 Jαβ -couplings, but
reproduce the 3 Jαβ exp less well than the simulation. The averaged χ1 -angle values are quite
different in each case, 195◦ in the simulation and 72◦ in the set of NMR model structures. The
non-linear character of the Karplus relation between 3 Jαβ and χ1 is illustrated by the different
shapes of the respective distributions. For cases such as this, the biasing energy function serves
to enhance the sampling.
Fig. 2.16 shows a case, the χ1 -angle of Val 3, in which the biasing energy function provides a small correction of 15◦ to the χ1 -angle value that is preferred by the force field used.
Compared to the values around 184◦ observed in the set of NMR model
(squares),
3 structures
◦
a slightly larger χ1 -angle of 199 leads to a reduction of 1.6 Hz in the Jαβ -value and better
agreement with experiment.
Fig. 2.17 shows a case, the χ1 -angle of Val 53, in which the set of NMR model structures
also predicts a too large 3 Jαβ -coupling of 12.9 Hz for a χ1 -angle of 182◦ . In this case the
GROMOS force field and the local-elevation biasing not only shift the distribution of χ1 angle values but also induce transitions between two χ1 -angle ranges on either side of 180◦ .
Thus the sampling is enhanced and a slight force-field deficiency is compensated for.
Up till now we have considered examples of side-chain χ1 -angles that were members of
the
3 list of 62 χ1 -angles that feel a biasing local-elevation force when the discrepancy with the
Jαβ exp becomes too large. It comes as no surprise that for these angles the experimental
However, the behaviour of χ1 -angles that could
not be restrained because of a lack of stereospecific assignment also matches the experimental
data better in the local-elevation biasing simulation as the following examples show.
Figs. 2.18 and 2.19 show a case, the χ1 -angle of Lys 54, in which the 3 Jαβ -values cal
culated from the set of NMR model structures show a large deviation from the 3 Jαβ exp that
can be greatly reduced by averaging over different χ1 -angles as observed in the simulation. A
comparison of Figs. 2.18 and 2.19 also gives an indication of a better stereospecific assignment than that chosen in these figures: a choice of 9.7 Hz for Hβ2 and 5.7 Hz for Hβ3 would
improve the agreement between the simulated and experimental data.
3 J -couplings are well reproduced (Fig. 2.4).
αβ
2.4 Results
49
Figure 2.14 Properties of the χ1 torsional angle of Glu 43 and the corresponding 3 JHα Hβ in the
2
LE NOE TAR WAT simulation. Upper row: Left: Final built-up local-elevation restraining potential energy Vlef inal (χ1 ). Middle: Distribution of χ1 angles. Right: Distribution of 3 J-couplings. Lower
row: Left: Local-elevation potential energy Vle (t) acting on χ1 at specific time points. Middle: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation (circles) and the χ1 angles in set of 16
NMR model structures (squares, from down to up structure 1 to 16). Right: Evolution of 3 J-value in
the 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in the set of 16 NMR model structures
(squares, from down to up structure 1 to 16). The dashed line shows the experimental 3 J-value.
3J
Figure 2.15 Properties of the χ1 torsional angle of Glu 43 and the corresponding 3 JHα Hβ in the
3J
3
LE NOE TAR WAT simulation. Upper row: Left: Final built-up local-elevation restraining potential energy Vlef inal (χ1 ). Middle: Distribution of χ1 angles. Right: Distribution of 3 J-couplings. Lower
row: Left: Local-elevation potential energy Vle (t) acting on χ1 at specific time points. Middle: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation (circles) and χ1 angles in the set of 16
NMR model structures (squares, from down to up structure 1 to 16). Right: Evolution of 3 J-value in
the 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in the set of 16 NMR model structures
(squares, from down to up structure 1 to 16). The dashed line shows the experimental 3 J-value.
50
2 Structural characterisation of Plastocyanin using local-elevation MD
Figure 2.16 Properties of the χ1 torsional angle of Val 3 and the corresponding 3 JHα Hβ in the
3 J LE NOE TAR WAT simulation. Upper row: Left: Final built-up local-elevation restraining potential energy Vlef inal (χ1 ). Middle: Distribution of χ1 angles. Right: Distribution of 3 J-couplings. Lower
row: Left: Local-elevation potential energy Vle (t) acting on χ1 at specific time points. Middle: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation (circles) and χ1 angles in the set of 16
NMR model structures (squares, from down to up structure 1 to 16). Right: Evolution of 3 J-value in
the 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in the set of 16 NMR model structures
(squares, from down to up structure 1 to 16). The dashed line shows the experimental 3 J-value.
Figure 2.17 Properties of the χ1 torsional angle of Val 53 and the corresponding 3 JHα Hβ in the
3 J LE NOE TAR WAT simulation. Upper row: Left: Final built-up local-elevation restraining potential energy Vlef inal (χ1 ). Middle: Distribution of χ1 angles. Right: Distribution of 3 J-couplings. Lower
row: Left: Local-elevation potential energy Vle (t) acting on χ1 at specific time points. Middle: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation (circles) and χ1 angles in the set of 16
NMR model structures (squares, from down to up structure 1 to 16). Right: Evolution of 3 J-value
in 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in the set of 16 NMR model structures
(squares, from down to up structure 1 to 16). The dashed line shows the experimental 3 J-value.
2.4 Results
51
Figure 2.18 Properties of the χ1 torsional angle of Lys 54 and the corresponding 3 JHα Hβ in the
2
LE NOE TAR WAT simulation. Upper row: Left: Distribution of χ1 angles. Right: Distribution
of 3 J-couplings. Lower row: Left: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation
(circles) and χ1 angles in set of 16 NMR model structures (squares, from down to up structure 1
to 16). Right: Evolution of 3 J-value in 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in
the set of 16 NMR model structures (squares, from down to up structure 1 to 16). The dashed line
shows the experimental 3 J-value.
3J
Figure 2.19 Properties of the χ1 torsional angle of Lys 54 and the corresponding 3 JHα Hβ in the
3
LE NOE TAR WAT simulation. Upper row: Left: Distribution of χ1 angles. Right: Distribution
of 3 J-couplings. Lower row: Left: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation
(circles) and χ1 angles in set of 16 NMR model structures (squares, from down to up structure 1
to 16). Right: Evolution of 3 J-value in 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in
the set of 16 NMR model structures (squares, from down to up structure 1 to 16). The dashed line
shows the experimental 3 J-value.
3J
52
2 Structural characterisation of Plastocyanin using local-elevation MD
Figure 2.20 Properties of the χ1 torsional angle of Asn 31 and the corresponding 3 JHα Hβ in the
2
3 J LE NOE TAR WAT simulation. Upper row: Left: Distribution of χ angles. Right: Distribution
1
of 3 J-couplings. Lower row: Left: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation
(circles) and χ1 angles in set of 16 NMR model structures (squares, from down to up structure 1
to 16). Right: Evolution of 3 J-value in 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in
the set of 16 NMR model structures (squares, from down to up structure 1 to 16). The dashed line
shows the experimental 3 J-value.
Figure 2.21 Properties of the χ1 torsional angle of Asn 31 and the corresponding 3 JHα Hβ in the
3
LE NOE TAR WAT simulation. Upper row: Left: Distribution of χ1 angles. Right: Distribution
of 3 J-couplings. Lower row: Left: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation
(circles) and χ1 angles in set of 16 NMR model structures (squares, from down to up structure 1
to 16). Right: Evolution of 3 J-value in 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in
the set of 16 NMR model structures (squares, from down to up structure 1 to 16). The dashed line
shows the experimental 3 J-value.
3J
2.4 Results
53
Figure 2.22 Properties of the χ1 torsional angle of Ser 81 and the corresponding 3 JHα Hβ in the
2
LE NOE TAR WAT simulation. Upper row: Left: Distribution of χ1 angles. Right: Distribution
of 3 J-couplings. Lower row: Left: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation
(circles) and χ1 angles in set of 16 NMR model structures (squares, from down to up structure 1
to 16). Right: Evolution of 3 J-value in 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in
the set of 16 NMR model structures (squares, from down to up structure 1 to 16). The dashed line
shows the experimental 3 J-value.
3J
Figure 2.23 Properties of the χ1 torsional angle of Ser 81 and the corresponding 3 JHα Hβ in the
3
LE NOE TAR WAT simulation. Upper row: Left: Distribution of χ1 angles. Right: Distribution
of 3 J-couplings. Lower row: Left: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation
(circles) and χ1 angles in set of 16 NMR model structures (squares, from down to up structure 1
to 16). Right: Evolution of 3 J-value in 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in
the set of 16 NMR model structures (squares, from down to up structure 1 to 16). The dashed line
shows the experimental 3 J-value.
3J
54
2 Structural characterisation of Plastocyanin using local-elevation MD
Figure 2.24 Properties of the χ1 torsional angle of Glu 45 and the corresponding 3 JHα Hβ in the
2
3 J LE NOE TAR WAT simulation. Upper row: Left: Distribution of χ angles. Right: Distribution
1
of 3 J-couplings. Lower row: Left: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation
(circles) and χ1 angles in set of 16 NMR model structures (squares, from down to up structure 1
to 16). Right: Evolution of 3 J-value in 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in
the set of 16 NMR model structures (squares, from down to up structure 1 to 16). The dashed line
shows the experimental 3 J-value.
Figure 2.25 Properties of the χ1 torsional angle of Glu 45 and the corresponding 3 JHα Hβ in the
3
LE NOE TAR WAT simulation. Upper row: Left: Distribution of χ1 angles. Right: Distribution
of 3 J-couplings. Lower row: Left: Evolution of χ1 angle in the 3 J LE NOE TAR WAT simulation
(circles) and χ1 angles in set of 16 NMR model structures (squares, from down to up structure 1
to 16). Right: Evolution of 3 J-value in 3 J LE NOE TAR WAT simulation (circles) and 3 J-values in
the set of 16 NMR model structures (squares, from down to up structure 1 to 16). The dashed line
shows the experimental 3 J-value.
3J
2.5 Conclusions
55
Figs. 2.20 and 2.21 show a case, the χ1 -angle of Asn 31, in which the set of NMR model
structures reproduces the experimental values by
over two ranges of χ1 -values,
averaging
while the simulation yields poor agreement with 3 Jαβ exp because it only samples one range
of χ1 -angle values (Table 2.2). An inversion of the chosen Hβ2 versus Hβ3 assignment would
improve the agreement for the set of NMR model structures while worsening it for the simulation.
Figs. 2.22 and 2.23 show a case, the χ1 -angle of Ser
81,in which the averaging in the
MD simulation leads to a reproduction of the observed 3 Jαβ exp -couplings, while the NMR
model structures fail to do so (Table 2.2).
Finally, the case of Glu 45 in Figs. 2.24 and 2.25 also shows the importance of conformational averaging and the 3 Jαβ -value distributions suggest an inversion of the chosen assignment.
These examples of the various effects of time-averaged local-elevation biasing based on
3 J-coupling constants show that the technique enhances the search for the appropriate rotamer
when needed, extends the sampling when needed, and compensates for force-field deficiencies
when needed based on a comparison of time-averaged with measured 3 J-coupling values.
2.5 Conclusions
Structure refinement of a protein based on NMR data still poses a challenge because of the
low ratio of independent observables and molecular degrees of freedom, the approximations
involved in the various relations between particular observable quantities and molecular conformation, and the averaged character of the experimental data which may even, if stemming
from different measurements, represent different thermodynamic state points. The recent literature and the Protein Data Bank still contain structures obtained from single-structure refinement in non-explicit solvent using non-observed data as geometric restraints in addition
to a low-accuracy force field. Such a procedure may easily result in a conformationally too
restricted set of protein structures, as is illustrated in Fig. 2.26. Application of time-averaged
restraints and the use of enhanced sampling techniques yield a conformationally more diverse
ensemble of protein structures while satisfying the experimentally measured 3 J-couplings and
NOE distance bounds better than the conformationally restricted set of structures resulting
from single-structure refinement.
Regarding the use of 3 J-couplings in structure refinement it is clear that the accuracy of the
parametrisation of the Karplus relation between torsional angle and 3 J-coupling or even the
relation itself needs to be improved. Second, current force fields for proteins appear not yet
accurate enough to predict protein structures without additional restraining or biasing terms
representing data measured for the particular proteins. Third, the barriers for conformational
changes, e.g. side-chain rotation, are often too high to be observed in nanosecond MD simulations, which makes the use of sampling enhancement techniques mandatory. Regarding all
three aspects progress is still expected in the coming decade.
56
2 Structural characterisation of Plastocyanin using local-elevation MD
Figure 2.26 Best-fit superposition of the backbone N, Cα , C and O atoms with respect to the last
structure of the set of 16 NMR model structures. The positions of the N, Cα and C atoms of the
backbone and the Cu-ion of the 16 NMR model structures (left) and 16 structures from the second
half of the 3 J LE NOE TAR WAT simulation (right) are shown.
2.6 Supplementary material
Nr
3
4
4
7
7
11
11
12
12
14
14
15
19
19
21
22
22
27
28
37
37
39
Residue
Name
VAL
LEU
LEU
SER
SER
SER
SER
LEU
LEU
PHE
PHE
VAL
PHE
PHE
VAL
PRO
PRO
ILE
VAL
HISB
HISB
VAL
Exp
H
β
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β
β2
β3
β
β2
β3
β
β
β2
β3
β
3J
exp
10.8
11.7
2.5
5.0
5.4
4.6
9.1
12.1
3.8
11.9
3.2
10.7
3.0
5.5
3.8
5.1
8.5
11.0
11.4
11.8
3.6
10.7
UNR VAC
UNR WAT
3 3 hχ1 i
hχ1 i
J
∆3 J
J
181 ± 8 12.7 ± 0.3
1.9
184 ± 10 12.6 ± 0.5
287 ± 26 11.5 ± 2.8
-0.2
278 ± 31 10.7 ± 3.5
287 ± 26
3.6 ± 2.2
1.1
278 ± 31
3.8 ± 2.9
107 ± 110
5.0 ± 3.4
-0.0
56 ± 11
3.1 ± 1.1
107 ± 110
4.9 ± 2.5
-0.5
56 ± 11
4.1 ± 1.4
179 ± 10
3.7 ± 1.1
-0.9
172 ± 114
7.0 ± 4.8
179 ± 10 12.6 ± 0.4
3.5
172 ± 114
4.3 ± 2.8
291 ± 15 12.2 ± 1.1
0.1
298 ± 12 12.5 ± 1.0
291 ± 15
2.9 ± 1.2
-0.9
298 ± 12
3.4 ± 1.2
298 ± 16 12.2 ± 1.1
0.3
289 ± 9 12.3 ± 0.7
298 ± 16
3.5 ± 1.5
0.3
289 ± 9
2.5 ± 0.7
234 ± 56
7.5 ± 5.1
-3.1
220 ± 50
8.7 ± 4.9
59 ± 8
3.3 ± 0.9
0.3
60 ± 8
3.4 ± 0.9
59 ± 8
3.6 ± 0.9
-1.9
60 ± 8
3.5 ± 0.9
291 ± 9
2.6 ± 0.7
-1.2
197 ± 63
9.0 ± 4.8
255 ± 131
8.4 ± 3.6
3.3
245 ± 138
7.8 ± 3.7
255 ± 131
7.6 ± 1.0
-0.9
245 ± 138
7.8 ± 0.9
305 ± 21 12.4 ± 1.0
1.4
305 ± 35 12.2 ± 1.5
194 ± 32 11.6 ± 3.0
0.2
184 ± 9 12.6 ± 0.4
246 ± 19
6.2 ± 3.1
-5.6
286 ± 8 12.1 ± 0.7
246 ± 19
4.7 ± 2.4
1.1
286 ± 8
2.2 ± 0.5
269 ± 34
3.5 ± 3.4
-7.2
190 ± 25 12.0 ± 2.4
Table 2.6 Continued on next page
∆3 J
1.8
-1.0
1.3
-1.9
-1.3
2.4
-4.8
0.4
-0.4
0.4
-0.7
-2.0
0.4
-2.0
5.2
2.7
-0.7
1.2
1.2
0.3
-1.4
1.3
2.6 Supplementary material
Nr
40
42
42
43
43
47
47
50
51
51
53
55
56
56
58
58
63
63
66
66
69
70
70
71
72
73
74
74
76
79
80
80
84
84
86
86
93
96
97
98
Residue
Name
VAL
ASP
ASP
GLU
GLU
PRO
PRO
VAL
ASP
ASP
VAL
ILE
SER
SER
PRO
PRO
LEU
LEU
PRO
PRO
THR
TYR
TYR
VAL
VAL
THR
LEU
LEU
THR
THR
TYR
TYR
CYS
CYS
PRO
PRO
VAL
VAL
THR
VAL
57
Exp
H
β
β2
β3
β2
β3
β2
β3
β
β2
β3
β
β
β2
β3
β2
β3
β2
β3
β2
β3
β
β2
β3
β
β
β
β2
β3
β
β
β2
β3
β2
β3
β2
β3
β
β
β
β
3J
exp
10.4
4.0
11.6
5.7
5.9
8.9
8.4
10.8
5.1
10.9
9.0
3.2
10.6
3.9
8.9
8.0
12.1
3.8
6.6
7.9
2.9
6.6
11.2
10.3
5.1
8.6
12.1
3.4
9.7
7.9
12.7
2.1
7.3
10.4
5.8
8.4
4.9
5.6
9.4
11.2
hχ1 i
188 ± 11
217 ± 50
217 ± 50
264 ± 62
264 ± 62
206 ± 154
206 ± 154
173 ± 31
176 ± 38
176 ± 38
215 ± 83
51 ± 8
239 ± 55
239 ± 55
228 ± 147
228 ± 147
296 ± 10
296 ± 10
231 ± 144
231 ± 144
56 ± 21
194 ± 10
194 ± 10
186 ± 9
242 ± 58
95 ± 94
286 ± 11
286 ± 11
124 ± 126
61 ± 46
298 ± 9
298 ± 9
177 ± 8
177 ± 8
55 ± 79
55 ± 79
298 ± 8
61 ± 9
94 ± 99
193 ± 16
UNR VAC
3 J
12.3 ± 0.8
5.4 ± 4.3
9.8 ± 4.0
10.2 ± 3.9
3.8 ± 2.8
5.9 ± 3.5
8.1 ± 1.0
11.8 ± 2.6
3.0 ± 1.2
11.5 ± 2.6
6.1 ± 4.6
2.6 ± 0.7
8.0 ± 4.6
7.3 ± 5.0
7.0 ± 3.7
7.9 ± 1.0
12.6 ± 0.4
3.1 ± 1.0
7.5 ± 3.8
7.8 ± 0.9
2.9 ± 0.8
2.3 ± 0.6
12.1 ± 0.9
12.5 ± 0.5
6.3 ± 4.9
4.3 ± 3.4
12.0 ± 1.0
2.4 ± 0.6
4.6 ± 3.5
3.1 ± 1.7
12.6 ± 0.3
3.3 ± 1.0
3.8 ± 1.0
12.7 ± 0.3
2.5 ± 2.2
7.2 ± 0.8
3.3 ± 0.9
3.4 ± 1.0
4.2 ± 3.9
12.1 ± 1.3
∆3 J
1.9
1.4
-1.8
4.5
-2.1
-3.0
-0.3
1.0
-2.1
0.6
-2.9
-0.6
-2.6
3.4
-1.9
-0.1
0.5
-0.7
0.9
-0.1
0.0
-4.3
0.9
2.2
1.2
-4.3
-0.1
-1.0
-5.1
-4.8
-0.1
1.2
-3.5
2.3
-3.3
-1.2
-1.6
-2.2
-5.2
0.9
hχ1 i
209 ± 50
193 ± 9
193 ± 9
60 ± 11
60 ± 11
211 ± 150
211 ± 150
287 ± 15
185 ± 10
185 ± 10
169 ± 35
58 ± 9
291 ± 12
291 ± 12
249 ± 137
249 ± 137
278 ± 38
278 ± 38
221 ± 148
221 ± 148
55 ± 10
180 ± 9
180 ± 9
185 ± 10
121 ± 71
129 ± 121
279 ± 19
279 ± 19
87 ± 88
107 ± 111
280 ± 8
280 ± 8
186 ± 8
186 ± 8
285 ± 108
285 ± 108
291 ± 9
222 ± 94
231 ± 122
189 ± 9
UNR WAT
3 J
10.0 ± 4.4
2.4 ± 0.6
12.2 ± 0.7
3.5 ± 1.3
3.6 ± 1.3
6.9 ± 4.0
7.8 ± 1.0
2.6 ± 1.4
3.0 ± 1.1
12.5 ± 0.5
11.4 ± 3.2
3.3 ± 1.0
12.4 ± 1.2
2.8 ± 1.0
7.7 ± 3.6
7.8 ± 0.9
10.4 ± 4.2
4.7 ± 3.3
7.4 ± 3.9
7.8 ± 0.9
3.1 ± 0.9
3.5 ± 1.0
12.7 ± 0.4
12.5 ± 0.5
5.7 ± 4.6
5.4 ± 4.3
11.0 ± 2.4
2.6 ± 1.4
3.3 ± 3.0
4.5 ± 3.8
11.5 ± 0.9
2.0 ± 0.3
2.8 ± 0.8
12.6 ± 0.4
9.2 ± 3.1
7.4 ± 0.9
2.7 ± 0.8
4.1 ± 3.6
9.2 ± 4.6
12.4 ± 0.6
∆3 J
-0.4
-1.6
0.6
-2.2
-2.3
-2.0
-0.7
-8.2
-2.1
1.6
2.4
0.1
1.8
-1.1
-1.2
-0.2
-1.7
0.9
0.8
-0.1
0.2
-3.1
1.5
2.2
0.6
-3.2
-1.1
-0.8
-6.4
-3.4
-1.2
-0.1
-4.5
2.2
3.4
-1.0
-2.2
-1.5
-0.2
1.2
Table 2.6 The 62 3 Jαβ -couplings and corresponding side-chain χ1 torsional angles that were selected as restraints. 3 Jexp are the values from experiment, h...i denotes averaging over the in3
dicated MD conformational ensembles
3 (the UNR VAC and 3UNR WAT simulations). ∆ J is the
difference between the calculated J and the experimental Jexp .
58
2 Structural characterisation of Plastocyanin using local-elevation MD
Nr
1
1
5
5
9
9
23
23
25
25
26
26
29
29
31
31
32
32
38
38
44
44
45
45
54
54
59
59
60
60
61
61
64
64
81
81
82
82
85
85
87
87
88
88
99
99
Residue
Name
LEU
LEU
LEU
LEU
ASP
ASP
SER
SER
GLU
GLU
LYSH
LYSH
PHE
PHE
ASN
ASN
ASN
ASN
ASN
ASN
ASP
ASP
GLU
GLU
LYSH
LYSH
GLU
GLU
GLU
GLU
GLU
GLU
ASN
ASN
SER
SER
PHE
PHE
SER
SER
HISB
HISB
GLN
GLN
ASN
ASN
Exp
H
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
3J
exp
4.1
10.9
11.6
2.9
4.7
4.1
7.3
6.4
4.6
11.0
8.0
5.3
12.4
2.8
5.3
6.6
4.6
12.4
3.5
4.2
4.0
10.7
5.2
9.9
5.7
9.7
7.1
7.1
4.2
9.1
6.9
8.0
9.6
4.3
6.8
8.3
3.4
6.0
8.8
7.6
13.6
3.5
5.3
9.8
7.3
4.8
hχ1 i
227 ± 40
227 ± 40
283 ± 32
283 ± 32
62 ± 10
62 ± 10
297 ± 12
297 ± 12
235 ± 43
235 ± 43
209 ± 85
209 ± 85
289 ± 9
289 ± 9
190 ± 12
190 ± 12
199 ± 38
199 ± 38
60 ± 8
60 ± 8
262 ± 73
262 ± 73
192 ± 71
192 ± 71
257 ± 83
257 ± 83
287 ± 24
287 ± 24
94 ± 73
94 ± 73
257 ± 44
257 ± 44
49 ± 9
49 ± 9
167 ± 84
167 ± 84
63 ± 8
63 ± 8
67 ± 57
67 ± 57
303 ± 9
303 ± 9
290 ± 16
290 ± 16
170 ± 129
170 ± 129
UNR VAC
3 J
5.3 ± 4.2
8.2 ± 4.4
11.1 ± 2.9
3.8 ± 2.7
3.7 ± 1.1
3.3 ± 1.1
12.5 ± 0.9
3.3 ± 1.1
6.3 ± 4.6
7.4 ± 4.6
6.7 ± 4.7
7.0 ± 4.6
12.3 ± 0.6
2.5 ± 0.7
2.7 ± 1.0
12.3 ± 1.1
4.5 ± 3.2
10.9 ± 3.6
3.5 ± 0.9
3.5 ± 1.0
10.6 ± 3.7
3.8 ± 2.8
5.2 ± 4.0
8.4 ± 4.7
10.6 ± 3.8
3.6 ± 2.2
11.6 ± 2.6
3.3 ± 2.1
5.0 ± 2.8
3.2 ± 1.9
8.7 ± 4.6
5.5 ± 4.4
2.5 ± 0.8
4.8 ± 1.2
4.4 ± 3.6
8.6 ± 4.3
3.8 ± 0.9
3.1 ± 0.8
3.5 ± 2.3
4.3 ± 1.6
12.6 ± 0.6
3.8 ± 1.1
12.1 ± 1.7
2.9 ± 1.4
7.2 ± 4.8
4.6 ± 1.5
∆3 J
1.2/-5.6
-2.7/4.1
-0.5/8.2
0.9/-7.8
-1.0/-0.4
-0.8/-1.4
5.2/6.1
-3.1/-4.0
1.7/-4.7
-3.6/2.8
-1.3/1.4
1.8/-1.0
-0.1/9.5
-0.3/-9.9
-2.6/-3.9
5.7/7.0
-0.1/-7.9
-1.5/6.3
-0.0/-0.7
-0.7/-0.0
6.6/-0.1
-6.9/-0.2
0.0/-4.7
-1.5/3.2
4.9/0.9
-6.0/-2.1
4.5/4.5
-3.8/-3.8
0.8/-4.0
-5.9/-1.0
1.8/0.7
-2.5/-1.4
-7.1/-1.8
0.5/-4.8
-2.4/-3.9
0.3/1.8
0.4/-2.2
-2.9/-0.3
-5.3/-4.1
-3.3/-4.5
-1.0/9.1
0.3/-9.8
6.8/2.3
-6.9/-2.4
-0.1/2.4
-0.2/-2.7
hχ1 i
273 ± 36
273 ± 36
287 ± 13
287 ± 13
91 ± 88
91 ± 88
129 ± 113
129 ± 113
189 ± 9
189 ± 9
219 ± 106
219 ± 106
291 ± 7
291 ± 7
190 ± 8
190 ± 8
190 ± 10
190 ± 10
64 ± 8
64 ± 8
239 ± 48
239 ± 48
284 ± 19
284 ± 19
224 ± 46
224 ± 46
266 ± 41
266 ± 41
278 ± 37
278 ± 37
272 ± 39
272 ± 39
61 ± 22
61 ± 22
258 ± 53
258 ± 53
60 ± 8
60 ± 8
71 ± 12
71 ± 12
308 ± 9
308 ± 9
262 ± 38
262 ± 38
264 ± 48
264 ± 48
UNR WAT
3 J
10.1 ± 3.9
4.2 ± 3.3
11.9 ± 0.9
2.5 ± 1.1
4.4 ± 3.5
4.1 ± 1.3
5.8 ± 4.4
4.0 ± 1.5
2.7 ± 0.7
12.4 ± 0.6
8.9 ± 4.7
4.1 ± 2.8
12.5 ± 0.5
2.6 ± 0.6
2.5 ± 0.6
12.4 ± 0.6
2.6 ± 0.7
12.3 ± 0.9
3.9 ± 0.9
3.1 ± 0.8
7.2 ± 4.8
7.2 ± 4.8
11.5 ± 2.3
2.8 ± 1.6
5.7 ± 4.6
8.8 ± 4.6
9.7 ± 4.4
4.8 ± 4.0
10.8 ± 3.7
4.2 ± 3.2
10.3 ± 4.0
4.4 ± 3.7
3.5 ± 1.3
3.6 ± 1.2
8.7 ± 4.6
6.5 ± 4.5
3.5 ± 1.0
3.5 ± 1.0
4.9 ± 1.5
2.6 ± 0.9
12.5 ± 0.6
4.4 ± 1.2
9.2 ± 4.2
4.6 ± 3.9
9.7 ± 4.3
5.3 ± 4.2
∆3 J
6.0/-0.8
-6.7/0.1
0.3/9.0
-0.4/-9.1
-0.3/0.3
-0.0/-0.6
-1.5/-0.6
-2.4/-3.3
-1.9/-8.3
1.4/7.8
0.9/3.6
-1.2/-3.9
0.1/9.7
-0.2/-9.8
-2.8/-4.1
5.8/7.1
-2.0/-9.8
-0.1/7.7
0.4/-0.3
-1.1/-0.4
3.2/-3.5
-3.5/3.2
6.3/1.6
-7.1/-2.4
0.0/-4.0
-0.9/3.1
2.6/2.6
-2.3/-2.3
6.6/1.7
-4.9/0.0
3.4/2.3
-3.6/-2.5
-6.0/-0.8
-0.7/-6.0
1.9/0.4
-1.8/-0.3
0.1/-2.5
-2.5/0.1
-3.9/-2.7
-5.0/-6.2
-1.1/9.0
0.9/-9.2
3.9/-0.6
-5.2/-0.7
2.4/4.9
0.5/-2.0
Table 2.7 The 46 3 Jαβ -couplings and corresponding side-chain χ1 torsional angles not used as
restraints. 3 Jexp are the values from experiment, h...i denotes averaging over the indicated MD
conformational ensembles
WAT simulations). ∆3 J is the difference be
3 (the UNR VAC and UNR
3
tween the calculated J and the experimental Jexp . The alternative assignments are separated
by “/” in the ∆3 J columns.
2.6 Supplementary material
Residue
Nr
Name
1
LEU
2
GLU
3
VAL
4
LEU
6
GLY
8
GLY
10
GLY
12
LEU
13
VAL
14
PHE
17
SER
19
PHE
20
SER
22
PRO
25
GLU
26
LYSH
27
ILE
28
VAL
29
PHE
30
LYSH
31
ASN
36
PRO
38
ASN
39
VAL
42
ASP
44
ASP
45
GLU
47
PRO
49
GLY
51
ASP
54
LYSH
55
ILE
58
PRO
60
GLU
61
GLU
62
LEU
63
LEU
64
ASN
67
GLY
68
GLU
69
THR
70
TYR
71
VAL
72
VAL
73
THR
75
ASP
78
GLY
79
THR
80
TYR
81
SER
83
TYR
86
PRO
87
HISB
91
GLY
92
MET
95
LYSH
96
VAL
97
THR
98
VAL
59
Exp
3J
exp
>9
>9
>9
>9
>9
>9
<6
>9
>9
>9
>9
>9
>9
<6
<6
>9
>9
>9
>9
<6
>9
>9
>9
>9
<6
>9
>9
<6
>9
<6
>9
>9
<6
>9
>9
>9
>9
>9
<6
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
<6
<6
>9
>9
>9
>9
>9
UNR VAC
3 hφ i
J
271 ± 19 7.5 ± 1.8
254 ± 13 9.0 ± 0.9
277 ± 15 7.0 ± 1.7
285 ± 12 6.0 ± 1.4
259 ± 31 7.6 ± 2.2
255 ± 30 7.7 ± 1.6
287 ± 103 2.9 ± 0.9
282 ± 16 6.3 ± 1.8
286 ± 14 5.8 ± 1.5
238 ± 9 9.5 ± 0.2
251 ± 15 9.0 ± 0.9
246 ± 14 9.3 ± 0.8
239 ± 13 9.4 ± 0.6
306 ± 11 3.7 ± 1.0
286 ± 16 5.9 ± 1.8
254 ± 14 9.0 ± 0.9
258 ± 13 8.8 ± 1.0
271 ± 14 7.6 ± 1.4
264 ± 15 8.2 ± 1.3
288 ± 12 5.7 ± 1.4
245 ± 14 9.2 ± 0.7
241 ± 11 9.5 ± 0.4
250 ± 14 9.1 ± 0.9
259 ± 11 8.8 ± 0.9
189 ± 125 4.8 ± 1.8
242 ± 33 7.8 ± 1.8
246 ± 9 9.4 ± 0.4
272 ± 65 5.7 ± 1.7
281 ± 17 6.5 ± 1.8
309 ± 33 3.2 ± 0.9
254 ± 15 8.9 ± 1.0
239 ± 15 9.3 ± 0.7
207 ± 147 3.7 ± 1.8
275 ± 17 7.0 ± 1.6
251 ± 15 9.0 ± 0.9
279 ± 16 6.7 ± 1.7
263 ± 19 8.1 ± 1.5
295 ± 16 4.8 ± 1.7
207 ± 143 4.1 ± 1.7
275 ± 15 7.2 ± 1.6
262 ± 19 8.2 ± 1.4
265 ± 16 8.0 ± 1.4
243 ± 13 9.4 ± 0.5
272 ± 15 7.5 ± 1.5
273 ± 15 7.4 ± 1.5
240 ± 29 8.2 ± 1.6
257 ± 19 8.5 ± 1.4
235 ± 15 9.2 ± 0.6
261 ± 17 8.4 ± 1.3
262 ± 18 8.2 ± 1.4
261 ± 15 8.5 ± 1.3
232 ± 14 9.2 ± 0.7
249 ± 17 9.2 ± 0.7
294 ± 12 4.9 ± 1.4
261 ± 16 8.4 ± 1.2
246 ± 11 9.4 ± 0.5
271 ± 12 7.6 ± 1.3
268 ± 15 7.9 ± 1.4
274 ± 19 7.1 ± 1.9
∆3 J
-4.5
-3.0
-5.0
-6.0
-4.4
-4.3
-0.1
-5.7
-6.2
-2.5
-3.0
-2.7
-2.6
0.7
2.9
-3.0
-3.2
-4.4
-3.8
2.7
-2.8
-2.5
-2.9
-3.2
1.8
-4.2
-2.6
2.7
-5.5
0.2
-3.1
-2.7
0.7
-5.0
-3.0
-5.3
-3.9
-7.2
1.1
-4.8
-3.8
-4.0
-2.6
-4.5
-4.6
-3.8
-3.5
-2.8
-3.6
-3.8
-3.5
-2.8
6.2
1.9
-3.6
-2.6
-4.4
-4.1
-4.9
hφ i
274 ± 18
253 ± 13
270 ± 14
271 ± 13
285 ± 19
287 ± 37
248 ± 82
236 ± 10
278 ± 10
244 ± 8
247 ± 10
265 ± 17
239 ± 11
301 ± 12
274 ± 16
267 ± 14
260 ± 12
277 ± 11
252 ± 12
288 ± 10
250 ± 16
237 ± 11
240 ± 9
246 ± 9
300 ± 11
275 ± 17
269 ± 28
291 ± 14
260 ± 22
304 ± 11
277 ± 14
266 ± 24
307 ± 39
263 ± 21
242 ± 14
271 ± 16
244 ± 12
273 ± 17
299 ± 15
247 ± 12
239 ± 12
256 ± 13
243 ± 11
257 ± 16
257 ± 17
288 ± 14
263 ± 16
266 ± 16
273 ± 19
236 ± 11
256 ± 11
264 ± 16
285 ± 22
253 ± 22
240 ± 10
256 ± 14
268 ± 14
270 ± 11
254 ± 21
UNR WAT
3 J
7.2 ± 1.7
9.1 ± 1.0
7.7 ± 1.4
7.7 ± 1.3
6.0 ± 2.0
5.3 ± 2.2
6.3 ± 2.1
9.4 ± 0.4
6.9 ± 1.2
9.5 ± 0.3
9.4 ± 0.5
8.1 ± 1.6
9.4 ± 0.4
4.2 ± 1.3
7.2 ± 1.7
8.0 ± 1.4
8.7 ± 0.9
7.0 ± 1.2
9.1 ± 0.7
5.7 ± 1.2
9.0 ± 0.9
9.4 ± 0.4
9.5 ± 0.3
9.5 ± 0.4
4.3 ± 1.2
7.1 ± 1.8
7.0 ± 2.3
5.3 ± 1.6
8.2 ± 1.9
3.9 ± 1.1
7.0 ± 1.5
7.6 ± 2.1
3.3 ± 1.1
8.0 ± 1.7
9.3 ± 0.7
7.5 ± 1.6
9.4 ± 0.5
7.2 ± 1.6
4.5 ± 1.5
9.3 ± 0.6
9.4 ± 0.5
8.9 ± 1.0
9.4 ± 0.4
8.7 ± 1.2
8.6 ± 1.3
5.6 ± 1.6
8.2 ± 1.3
8.0 ± 1.5
7.2 ± 1.8
9.4 ± 0.4
8.9 ± 0.8
8.2 ± 1.4
5.9 ± 2.1
8.4 ± 1.5
9.5 ± 0.4
8.8 ± 1.0
7.9 ± 1.4
7.8 ± 1.1
8.5 ± 1.4
∆3 J
-4.8
-2.9
-4.3
-4.3
-6.0
-6.7
3.3
-2.6
-5.1
-2.5
-2.6
-3.9
-2.6
1.2
4.2
-4.0
-3.3
-5.0
-2.9
2.7
-3.0
-2.6
-2.5
-2.5
1.3
-4.9
-5.0
2.3
-3.8
0.9
-5.0
-4.4
0.3
-4.0
-2.7
-4.5
-2.6
-4.8
1.5
-2.7
-2.6
-3.1
-2.6
-3.3
-3.4
-6.4
-3.8
-4.0
-4.8
-2.6
-3.1
-3.8
2.9
5.4
-2.5
-3.2
-4.1
-4.2
-3.5
Table 2.8 The 59 3 JHN Hα -couplings and corresponding backbone φ torsional angles. 3 Jexp are the
values from experiment, h...i denotes averaging over the indicated MD conformational ensembles
(the UNR VAC and UNR WAT simulations). ∆3 J is the difference between the calculated 3 J and
the experimental 3 Jexp .
60
2 Structural characterisation of Plastocyanin using local-elevation MD
Nr
3
4
4
7
7
11
11
12
12
14
14
15
19
19
21
22
22
27
28
37
37
39
40
42
42
43
43
47
47
50
51
51
53
55
56
56
58
58
63
63
66
66
69
70
70
71
72
73
74
74
76
79
80
80
84
84
86
86
93
96
Residue
Name
VAL
LEU
LEU
SER
SER
SER
SER
LEU
LEU
PHE
PHE
VAL
PHE
PHE
VAL
PRO
PRO
ILE
VAL
HISB
HISB
VAL
VAL
ASP
ASP
GLU
GLU
PRO
PRO
VAL
ASP
ASP
VAL
ILE
SER
SER
PRO
PRO
LEU
LEU
PRO
PRO
THR
TYR
TYR
VAL
VAL
THR
LEU
LEU
THR
THR
TYR
TYR
CYS
CYS
PRO
PRO
VAL
VAL
Exp
H
β
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β
β2
β3
β
β2
β3
β
β
β2
β3
β
β
β2
β3
β2
β3
β2
β3
β
β2
β3
β
β
β2
β3
β2
β3
β2
β3
β2
β3
β
β2
β3
β
β
β
β2
β3
β
β
β2
β3
β2
β3
β2
β3
β
β
3J
exp
10.8
11.7
2.5
5.0
5.4
4.6
9.1
12.1
3.8
11.9
3.2
10.7
3.0
5.5
3.8
5.1
8.5
11.0
11.4
11.8
3.6
10.7
10.4
4.0
11.6
5.7
5.9
8.9
8.4
10.8
5.1
10.9
9.0
3.2
10.6
3.9
8.9
8.0
12.1
3.8
6.6
7.9
2.9
6.6
11.2
10.3
5.1
8.6
12.1
3.4
9.7
7.9
12.7
2.1
7.3
10.4
5.8
8.4
4.9
5.6
3 J LE VAC
3J
3 hχ1 i
hχ1 i
J
∆3 J
181 ± 21 11.7 ± 0.9
0.9
186 ± 20
293 ± 20 12.3 ± 1.1
0.6
295 ± 18
293 ± 20
3.1 ± 1.1
0.6
295 ± 18
207 ± 78
5.2 ± 1.9
0.2
203 ± 82
207 ± 78
5.1 ± 1.9
-0.3
203 ± 82
202 ± 168
5.1 ± 1.3
0.5
182 ± 158
202 ± 168
9.5 ± 0.9
0.4
182 ± 158
293 ± 18 12.4 ± 1.0
0.3
294 ± 15
293 ± 18
3.0 ± 1.1
-0.8
294 ± 15
301 ± 11 12.5 ± 0.7
0.6
295 ± 13
301 ± 11
3.7 ± 1.0
0.5
295 ± 13
188 ± 19 11.7 ± 0.9
1.0
192 ± 19
46 ± 36
2.1 ± 0.6
-0.9
197 ± 74
46 ± 36
6.1 ± 1.2
0.6
197 ± 74
214 ± 98
3.1 ± 1.1
-0.6
208 ± 102
221 ± 170
5.2 ± 1.1
0.1
219 ± 169
221 ± 170
9.6 ± 0.4
1.1
219 ± 169
311 ± 15 11.9 ± 0.9
0.9
312 ± 19
185 ± 15 12.3 ± 0.9
0.9
188 ± 10
293 ± 23 12.2 ± 1.2
0.4
293 ± 12
293 ± 23
3.2 ± 1.3
-0.4
293 ± 12
192 ± 19 11.4 ± 1.1
0.7
197 ± 16
191 ± 24 11.1 ± 1.0
0.7
196 ± 21
185 ± 28
3.4 ± 1.4
-0.6
184 ± 12
185 ± 28 12.1 ± 1.3
0.5
184 ± 12
206 ± 82
5.3 ± 2.3
-0.4
196 ± 85
206 ± 82
5.6 ± 2.2
-0.3
196 ± 85
325 ± 62
8.4 ± 1.7
-0.5
328 ± 57
325 ± 62
8.3 ± 1.4
-0.1
328 ± 57
184 ± 21 11.6 ± 1.2
0.8
183 ± 23
186 ± 113
5.3 ± 1.3
0.2
165 ± 81
186 ± 113 10.9 ± 1.4
0.0
165 ± 81
182 ± 69
9.3 ± 1.2
0.3
176 ± 43
150 ± 79
2.3 ± 0.9
-0.9
42 ± 27
183 ± 88 10.1 ± 1.6
-0.5
155 ± 75
183 ± 88
4.0 ± 1.5
0.1
155 ± 75
331 ± 42
8.9 ± 1.5
-0.0
331 ± 39
331 ± 42
8.2 ± 1.2
0.2
331 ± 39
292 ± 29 12.4 ± 1.0
0.3
293 ± 26
292 ± 29
3.1 ± 1.1
-0.7
293 ± 26
313 ± 99
7.2 ± 1.8
0.7
321 ± 85
313 ± 99
9.0 ± 0.9
1.1
321 ± 85
106 ± 81
2.1 ± 0.6
-0.8
39 ± 12
172 ± 67
5.8 ± 1.4
-0.8
166 ± 41
172 ± 67 11.2 ± 1.5
0.0
166 ± 41
189 ± 27 11.0 ± 1.1
0.7
198 ± 20
197 ± 91
5.2 ± 1.4
0.2
169 ± 105
323 ± 53
9.1 ± 1.3
0.5
332 ± 26
303 ± 9 12.6 ± 0.4
0.5
300 ± 18
303 ± 9
3.9 ± 0.9
0.5
300 ± 18
314 ± 29 10.2 ± 1.2
0.5
315 ± 36
223 ± 111
8.2 ± 1.2
0.3
127 ± 54
294 ± 22 12.4 ± 1.1
-0.3
292 ± 18
294 ± 22
3.1 ± 1.0
1.0
292 ± 18
251 ± 111
6.9 ± 1.5
-0.4
246 ± 105
251 ± 111
9.6 ± 1.3
-0.8
246 ± 105
219 ± 165
5.6 ± 1.8
-0.2
234 ± 161
219 ± 165
9.0 ± 1.6
0.6
234 ± 161
112 ± 100
4.9 ± 1.3
0.0
179 ± 105
96 ± 99
5.9 ± 1.4
0.3
214 ± 119
Table 2.9 Continued on next page
LE WAT
3 J
11.7 ± 0.9
12.5 ± 0.9
3.2 ± 1.2
5.1 ± 1.9
5.0 ± 1.7
4.9 ± 1.7
9.3 ± 1.5
12.5 ± 0.6
3.0 ± 1.0
12.3 ± 0.9
3.2 ± 1.2
11.5 ± 1.2
3.6 ± 1.0
6.2 ± 1.3
3.2 ± 1.2
5.2 ± 1.3
9.5 ± 0.6
11.5 ± 1.0
12.4 ± 0.6
12.3 ± 0.7
3.0 ± 1.1
11.3 ± 1.1
11.2 ± 1.5
3.2 ± 1.0
12.5 ± 0.8
5.4 ± 2.3
5.5 ± 2.2
8.6 ± 1.7
8.3 ± 1.2
11.6 ± 1.3
5.2 ± 1.4
11.3 ± 1.6
9.5 ± 1.4
2.4 ± 0.8
9.9 ± 1.3
4.1 ± 1.2
9.2 ± 1.3
8.2 ± 0.7
12.3 ± 1.1
3.2 ± 1.2
7.4 ± 1.9
8.9 ± 1.1
2.1 ± 0.7
5.9 ± 1.2
11.5 ± 1.1
11.2 ± 1.1
5.0 ± 1.6
9.2 ± 1.3
12.5 ± 0.8
3.8 ± 1.2
10.5 ± 1.3
8.3 ± 1.3
12.3 ± 0.9
3.0 ± 1.1
6.8 ± 1.6
9.8 ± 1.4
5.6 ± 2.0
9.1 ± 1.5
4.8 ± 1.5
5.6 ± 1.4
∆3 J
0.9
0.8
0.7
0.1
-0.5
0.3
0.2
0.4
-0.8
0.4
0.0
0.8
0.6
0.7
-0.6
0.1
1.0
0.5
1.0
0.5
-0.6
0.6
0.8
-0.8
0.9
-0.3
-0.4
-0.3
-0.1
0.8
0.1
0.4
0.5
-0.8
-0.7
0.2
0.2
0.2
0.2
-0.6
0.8
1.0
-0.8
-0.7
0.3
0.8
-0.1
0.6
0.4
0.4
0.8
0.4
-0.4
0.9
-0.5
-0.6
-0.2
0.7
-0.2
0.0
2.6 Supplementary material
Nr
97
98
Residue
Name
THR
VAL
61
Exp
H
β
β
3J
exp
9.4
11.2
3J
hχ1 i
313 ± 55
195 ± 10
LE VAC
3 J
9.9 ± 1.3
11.9 ± 0.9
∆3 J
0.5
0.7
3J
hχ1 i
320 ± 28
189 ± 14
LE WAT
3 J
9.8 ± 1.3
12.1 ± 1.0
∆3 J
0.4
0.9
Table 2.9 The 62 3 Jαβ -couplings and corresponding side-chain χ1 torsional angles that were selected as restraints. 3 Jexp are the values from experiment, h...i denotes averaging over the in3
3
3
dicated MD conformational ensembles
3 (the J LE VAC and 3 J LE WAT simulations). ∆ J is the
difference between the calculated J and the experimental Jexp .
62
2 Structural characterisation of Plastocyanin using local-elevation MD
Nr
1
1
5
5
9
9
23
23
25
25
26
26
29
29
31
31
32
32
38
38
44
44
45
45
54
54
59
59
60
60
61
61
64
64
81
81
82
82
85
85
87
87
88
88
99
99
Residue
Name
LEU
LEU
LEU
LEU
ASP
ASP
SER
SER
GLU
GLU
LYSH
LYSH
PHE
PHE
ASN
ASN
ASN
ASN
ASN
ASN
ASP
ASP
GLU
GLU
LYSH
LYSH
GLU
GLU
GLU
GLU
GLU
GLU
ASN
ASN
SER
SER
PHE
PHE
SER
SER
HISB
HISB
GLN
GLN
ASN
ASN
Exp
H
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
β2
β3
3J
exp
4.1
10.9
11.6
2.9
4.7
4.1
7.3
6.4
4.6
11.0
8.0
5.3
12.4
2.8
5.3
6.6
4.6
12.4
3.5
4.2
4.0
10.7
5.2
9.9
5.7
9.7
7.1
7.1
4.2
9.1
6.9
8.0
9.6
4.3
6.8
8.3
3.4
6.0
8.8
7.6
13.6
3.5
5.3
9.8
7.3
4.8
hχ1 i
202 ± 77
202 ± 77
225 ± 46
225 ± 46
263 ± 55
263 ± 55
287 ± 34
287 ± 34
233 ± 42
233 ± 42
202 ± 108
202 ± 108
288 ± 8
288 ± 8
194 ± 28
194 ± 28
191 ± 80
191 ± 80
58 ± 11
58 ± 11
229 ± 49
229 ± 49
218 ± 48
218 ± 48
258 ± 74
258 ± 74
227 ± 90
227 ± 90
253 ± 79
253 ± 79
103 ± 88
103 ± 88
50 ± 9
50 ± 9
178 ± 77
178 ± 77
66 ± 10
66 ± 10
103 ± 95
103 ± 95
211 ± 97
211 ± 97
251 ± 42
251 ± 42
280 ± 82
280 ± 82
3J
LE VAC
3J
5.5 ± 4.3
7.2 ± 4.4
5.5 ± 4.4
8.8 ± 4.4
10.0 ± 4.0
4.4 ± 3.6
11.2 ± 3.5
4.4 ± 2.8
5.9 ± 4.4
7.6 ± 4.4
7.9 ± 4.7
4.5 ± 3.2
12.3 ± 0.7
2.4 ± 0.6
3.6 ± 2.6
11.6 ± 2.9
5.3 ± 4.0
6.9 ± 4.5
3.3 ± 1.2
3.8 ± 1.2
6.2 ± 4.5
8.3 ± 4.7
5.6 ± 4.2
8.6 ± 4.7
9.8 ± 4.3
4.8 ± 3.5
8.5 ± 4.6
4.6 ± 3.6
10.2 ± 3.7
3.4 ± 2.3
3.6 ± 3.1
6.2 ± 3.3
2.5 ± 0.7
4.8 ± 1.2
4.2 ± 3.6
9.2 ± 4.0
4.2 ± 1.2
2.9 ± 0.9
4.9 ± 3.6
4.0 ± 1.8
7.7 ± 4.6
6.1 ± 4.2
8.2 ± 4.5
5.6 ± 4.6
11.3 ± 3.1
4.6 ± 1.4
∆3 J
1.4/-5.4
-3.7/3.1
-6.1/2.6
5.9/-2.8
5.3/5.9
0.3/-0.3
3.9/4.8
-2.0/-2.9
1.3/-5.1
-3.4/3.0
-0.1/2.6
-0.8/-3.5
-0.1/9.5
-0.4/-10.0
-1.7/-3.0
5.0/6.3
0.7/-7.1
-5.5/2.3
-0.2/-0.9
-0.4/0.3
2.2/-4.5
-2.3/4.3
0.4/-4.3
-1.3/3.4
4.1/0.2
-4.9/-0.9
1.4/1.4
-2.5/-2.5
6.0/1.1
-5.7/-0.8
-3.3/-4.4
-1.8/-0.7
-7.1/-1.8
0.5/-4.8
-2.6/-4.1
0.9/2.4
0.8/-1.8
-3.1/-0.5
-3.9/-2.7
-3.5/-4.8
-5.9/4.2
2.6/-7.5
2.9/-1.6
-4.2/0.3
4.0/6.5
-0.2/-2.7
hχ1 i
205 ± 34
205 ± 34
284 ± 11
284 ± 11
121 ± 94
121 ± 94
230 ± 53
230 ± 53
249 ± 43
249 ± 43
258 ± 82
258 ± 82
292 ± 9
292 ± 9
189 ± 10
189 ± 10
192 ± 10
192 ± 10
59 ± 13
59 ± 13
240 ± 48
240 ± 48
279 ± 28
279 ± 28
285 ± 19
285 ± 19
275 ± 33
275 ± 33
269 ± 43
269 ± 43
280 ± 33
280 ± 33
122 ± 102
122 ± 102
283 ± 37
283 ± 37
56 ± 22
56 ± 22
73 ± 16
73 ± 16
307 ± 31
307 ± 31
255 ± 41
255 ± 41
158 ± 111
158 ± 111
3J
LE WAT
3 J
3.8 ± 3.2
10.7 ± 3.4
11.8 ± 1.0
2.3 ± 0.7
4.6 ± 3.7
5.0 ± 3.3
7.3 ± 4.5
8.0 ± 5.1
8.1 ± 4.2
5.7 ± 4.6
10.9 ± 3.5
3.1 ± 1.4
12.5 ± 0.6
2.7 ± 0.8
2.6 ± 0.7
12.4 ± 0.8
2.5 ± 0.7
12.2 ± 0.9
3.2 ± 1.0
3.8 ± 1.2
7.1 ± 4.9
7.4 ± 4.8
11.0 ± 3.2
3.4 ± 2.8
11.8 ± 2.0
2.8 ± 1.6
10.6 ± 3.6
3.8 ± 3.2
9.9 ± 4.2
5.0 ± 3.9
11.1 ± 3.3
3.8 ± 3.0
5.9 ± 4.0
3.3 ± 1.2
11.0 ± 3.4
4.4 ± 3.2
2.9 ± 1.0
4.2 ± 1.2
5.0 ± 1.5
2.6 ± 1.3
12.1 ± 1.4
4.8 ± 1.5
8.3 ± 4.6
5.5 ± 4.3
6.1 ± 4.3
4.6 ± 3.0
∆3 J
-0.3/-7.1
-0.2/6.6
0.2/8.9
-0.6/-9.3
-0.1/0.5
1.0/0.3
-0.0/0.9
1.6/0.7
3.5/-2.9
-5.3/1.1
2.9/5.6
-2.2/-4.9
0.1/9.7
-0.1/-9.7
-2.6/-3.9
5.8/7.1
-2.1/-9.9
-0.2/7.6
-0.2/-1.0
-0.4/0.3
3.1/-3.6
-3.3/3.4
5.8/1.1
-6.5/-1.8
6.0/2.1
-6.9/-2.9
3.5/3.5
-3.3/-3.3
5.7/0.8
-4.1/0.8
4.2/3.1
-4.2/-3.1
-3.7/1.6
-1.0/-6.3
4.2/2.7
-3.9/-2.4
-0.5/-3.1
-1.8/0.8
-3.8/-2.6
-5.0/-6.2
-1.5/8.6
1.3/-8.8
3.0/-1.5
-4.4/0.2
-1.2/1.3
-0.2/-2.7
Table 2.10 The 46 3 Jαβ -couplings and corresponding side-chain χ1 torsional angles not used as
restraints. 3 Jexp are the values from experiment, h...i denotes averaging over the indicated MD
3
3
3
conformational ensembles
3 (the J LE VAC and 3J LE WAT simulations). ∆ J is the difference between the calculated J and the experimental Jexp . The alternative assignments are separated
by “/” in the ∆3 J columns.
2.6 Supplementary material
Residue
Nr
Name
1
LEU
2
GLU
3
VAL
4
LEU
6
GLY
8
GLY
10
GLY
12
LEU
13
VAL
14
PHE
17
SER
19
PHE
20
SER
22
PRO
25
GLU
26
LYSH
27
ILE
28
VAL
29
PHE
30
LYSH
31
ASN
36
PRO
38
ASN
39
VAL
42
ASP
44
ASP
45
GLU
47
PRO
49
GLY
51
ASP
54
LYSH
55
ILE
58
PRO
60
GLU
61
GLU
62
LEU
63
LEU
64
ASN
67
GLY
68
GLU
69
THR
70
TYR
71
VAL
72
VAL
73
THR
75
ASP
78
GLY
79
THR
80
TYR
81
SER
83
TYR
86
PRO
87
HISB
91
GLY
92
MET
95
LYSH
96
VAL
97
THR
98
VAL
63
Exp
3J
exp
>9
>9
>9
>9
>9
>9
<6
>9
>9
>9
>9
>9
>9
<6
<6
>9
>9
>9
>9
<6
>9
>9
>9
>9
<6
>9
>9
<6
>9
<6
>9
>9
<6
>9
>9
>9
>9
>9
<6
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
>9
<6
<6
>9
>9
>9
>9
>9
3J
hφ i
272 ± 19
253 ± 14
280 ± 13
283 ± 12
256 ± 29
213 ± 53
288 ± 18
265 ± 23
276 ± 17
242 ± 12
251 ± 16
253 ± 16
241 ± 13
307 ± 11
283 ± 18
257 ± 15
255 ± 14
268 ± 16
264 ± 17
291 ± 12
235 ± 18
231 ± 12
255 ± 16
262 ± 14
90 ± 92
281 ± 28
248 ± 11
269 ± 33
272 ± 19
220 ± 131
243 ± 30
264 ± 56
170 ± 148
292 ± 29
286 ± 16
280 ± 15
267 ± 21
292 ± 21
200 ± 149
264 ± 23
255 ± 20
265 ± 18
244 ± 14
271 ± 19
279 ± 13
228 ± 25
250 ± 22
236 ± 12
254 ± 18
257 ± 16
246 ± 14
165 ± 81
230 ± 85
253 ± 69
228 ± 18
248 ± 14
273 ± 15
265 ± 13
289 ± 16
LE VAC
3 J
7.3 ± 1.9
9.0 ± 1.0
6.6 ± 1.5
6.3 ± 1.4
8.0 ± 1.9
8.2 ± 1.6
5.7 ± 1.5
7.8 ± 1.9
7.0 ± 1.7
9.4 ± 0.7
9.0 ± 0.9
8.9 ± 1.1
9.4 ± 0.5
3.6 ± 1.1
6.2 ± 1.9
8.7 ± 1.1
8.9 ± 1.0
7.9 ± 1.4
8.1 ± 1.5
5.4 ± 1.5
9.1 ± 1.0
9.3 ± 0.6
8.8 ± 1.2
8.4 ± 1.2
6.1 ± 1.3
6.2 ± 1.8
9.3 ± 0.6
7.2 ± 1.8
7.3 ± 1.9
4.2 ± 1.7
8.2 ± 1.8
5.7 ± 2.6
4.1 ± 1.8
5.1 ± 2.2
5.9 ± 1.6
6.6 ± 1.6
7.6 ± 1.9
4.9 ± 1.8
3.6 ± 1.4
7.8 ± 1.8
8.5 ± 1.3
8.0 ± 1.6
9.3 ± 0.7
7.3 ± 1.8
6.7 ± 1.5
8.3 ± 1.6
8.6 ± 1.5
9.3 ± 0.5
8.8 ± 1.2
8.7 ± 1.2
9.2 ± 0.7
7.3 ± 2.1
7.3 ± 2.2
7.0 ± 1.7
8.8 ± 1.2
9.2 ± 0.7
7.3 ± 1.5
8.2 ± 1.2
5.6 ± 1.8
∆3 J
-4.7
-3.0
-5.4
-5.7
-4.0
-3.8
2.7
-4.2
-5.0
-2.6
-3.0
-3.1
-2.6
0.6
3.2
-3.3
-3.1
-4.1
-3.9
2.4
-2.9
-2.7
-3.2
-3.6
3.1
-5.8
-2.7
4.2
-4.7
1.2
-3.8
-6.3
1.1
-6.9
-6.1
-5.4
-4.4
-7.1
0.6
-4.2
-3.5
-4.0
-2.7
-4.7
-5.3
-3.7
-3.4
-2.7
-3.2
-3.3
-2.8
-4.7
4.3
4.0
-3.2
-2.8
-4.7
-3.8
-6.4
3J
hφ i
273 ± 16
261 ± 14
278 ± 13
278 ± 13
266 ± 23
262 ± 32
269 ± 62
241 ± 11
279 ± 12
244 ± 8
248 ± 13
259 ± 14
241 ± 12
302 ± 17
271 ± 13
264 ± 13
261 ± 11
278 ± 11
254 ± 11
291 ± 9
250 ± 15
235 ± 10
240 ± 10
253 ± 10
302 ± 12
279 ± 18
267 ± 33
290 ± 17
269 ± 21
307 ± 12
267 ± 17
262 ± 22
304 ± 30
269 ± 17
239 ± 24
264 ± 14
246 ± 14
277 ± 16
289 ± 53
248 ± 17
239 ± 14
253 ± 13
241 ± 11
267 ± 16
260 ± 14
295 ± 13
249 ± 13
253 ± 12
269 ± 18
234 ± 10
245 ± 13
272 ± 28
265 ± 30
247 ± 21
237 ± 12
255 ± 12
263 ± 15
262 ± 13
246 ± 17
LE WAT
3 J
7.3 ± 1.6
8.5 ± 1.1
6.9 ± 1.5
6.8 ± 1.4
7.7 ± 1.9
7.3 ± 2.3
6.0 ± 2.0
9.5 ± 0.5
6.8 ± 1.3
9.5 ± 0.2
9.3 ± 0.8
8.6 ± 1.1
9.4 ± 0.5
4.3 ± 1.6
7.7 ± 1.3
8.2 ± 1.2
8.6 ± 0.9
6.9 ± 1.3
9.1 ± 0.7
5.4 ± 1.1
9.0 ± 0.9
9.4 ± 0.3
9.5 ± 0.4
9.2 ± 0.6
4.1 ± 1.2
6.7 ± 1.9
6.9 ± 2.5
5.5 ± 1.8
7.5 ± 1.9
3.6 ± 1.1
7.8 ± 1.6
8.1 ± 1.9
3.7 ± 1.2
7.7 ± 1.6
8.7 ± 1.4
8.2 ± 1.2
9.3 ± 0.7
6.9 ± 1.6
4.6 ± 1.8
9.1 ± 1.2
9.3 ± 0.6
9.0 ± 0.9
9.4 ± 0.4
8.0 ± 1.5
8.6 ± 1.1
4.9 ± 1.5
9.2 ± 0.7
9.1 ± 0.8
7.6 ± 1.7
9.4 ± 0.4
9.3 ± 0.6
7.2 ± 2.0
7.4 ± 2.3
8.8 ± 1.2
9.4 ± 0.4
9.0 ± 0.9
8.3 ± 1.3
8.4 ± 1.1
9.1 ± 1.0
∆3 J
-4.7
-3.5
-5.1
-5.2
-4.3
-4.7
3.0
-2.5
-5.2
-2.5
-2.7
-3.4
-2.6
1.3
4.7
-3.8
-3.4
-5.1
-2.9
2.4
-3.0
-2.6
-2.5
-2.8
1.1
-5.3
-5.1
2.5
-4.5
0.6
-4.2
-3.9
0.7
-4.3
-3.3
-3.8
-2.7
-5.1
1.6
-2.9
-2.7
-3.0
-2.6
-4.0
-3.4
-7.1
-2.8
-2.9
-4.4
-2.6
-2.7
-4.8
4.4
5.8
-2.6
-3.0
-3.7
-3.6
-2.9
Table 2.11 The 59 3 JHN Hα -couplings and corresponding backbone φ torsional angles. 3 Jexp are the
values from experiment, h...i denotes averaging over the indicated MD conformational ensembles
(the 3 J LE VAC and 3 J LE WAT simulations). ∆3 J is the difference between the calculated 3 J
and the experimental 3 Jexp .
RES2
1LEU
28VAL
15VAL
92MET
13VAL
92MET
13VAL
4LEU
9ASP
33ALA
8GLY
9ASP
11SER
33ALA
92MET
12LEU
12LEU
3VAL
92MET
3VAL
3VAL
39VAL
92MET
18GLU
96VAL
16PRO
96VAL
96VAL
19PHE
73THR
27ILE
70TYR
71VAL
72VAL
69THR
5LEU
68GLU
4LEU
4LEU
7SER
32ASN
32ASN
35PHE
35PHE
atom 2
QG
H
HB
QE
H
QE
QB
QG
H
QB
QA
H
H
QB
QE
HB
HA
QG1
QE
QG1
QG2
QG2
QE
H
HA
HB
HA
QG2
HB
QG2
HB
HB
QG1
QG1
QG2
HA
H
HB
HB
HA
HA
HB
CG
H
3J
UNR VAC
UNR WAT
LE VAC
-0.078
-0.008
-0.086
0.011
0.016
0.018
0.003
-0.052
-0.047
0.671
-0.165
0.029
0.107
-0.091
0.070
0.573
-0.134
-0.022
0.009
-0.076
0.002
-0.277
0.123
-0.188
0.060
-0.032
0.080
-0.152
-0.094
-0.005
-0.003
-0.077
0.031
0.044
-0.071
0.035
0.050
-0.044
0.074
-0.188
0.007
-0.161
0.157
-0.037
0.030
-0.014
-0.002
-0.015
0.086
-0.017
-0.012
0.038
-0.084
-0.032
0.398
-0.230
0.024
0.036
-0.014
-0.023
0.035
-0.085
-0.050
0.082
-0.069
-0.020
0.220
-0.299
-0.237
0.011
-0.038
0.005
0.016
-0.025
-0.007
0.039
0.062
0.051
-0.006
0.032
0.032
0.001
0.106
0.076
0.019
0.013
0.010
0.078
-0.062
-0.031
0.008
0.016
-0.002
-0.082
-0.058
-0.048
0.000
-0.000
-0.004
0.128
0.008
0.033
-0.018
-0.031
0.090
-0.013
-0.021
0.014
-0.009
-0.040
0.013
0.098
0.064
0.094
0.085
0.094
0.114
0.131
-0.222
-0.101
-0.006
0.003
-0.067
0.090
0.095
0.018
-0.125
0.064
-0.031
-0.065
0.004
-0.098
Table 2.12 Continued on next page
3J
−1/6
r−6
− rexp
LE WAT NMR set
-0.127
-0.104
0.001
-0.004
-0.063
-0.070
-0.159
-0.185
-0.036
-0.053
-0.136
-0.198
-0.017
-0.106
0.131
-0.044
-0.030
-0.056
0.233
-0.205
0.025
-0.134
-0.030
-0.070
0.021
-0.022
0.168
-0.170
0.015
-0.075
-0.001
0.002
-0.027
-0.006
-0.076
-0.085
-0.229
-0.241
-0.044
-0.036
-0.104
-0.105
-0.048
-0.103
-0.308
-0.253
-0.042
-0.028
-0.043
0.001
0.066
0.058
0.056
-0.028
0.141
-0.032
-0.013
-0.005
-0.111
-0.154
0.017
-0.008
-0.022
-0.047
-0.039
-0.016
0.028
-0.038
-0.050
-0.053
-0.006
-0.000
-0.041
-0.009
0.061
-0.005
0.070
0.048
-0.241
-0.182
0.003
-0.011
0.093
0.072
-0.076
-0.147
-0.040
-0.089
3J
LE NOE WAT
-0.097
0.015
-0.078
-0.174
-0.055
-0.162
-0.085
-0.026
-0.008
-0.187
-0.078
-0.095
-0.065
-0.100
-0.049
-0.002
-0.017
-0.059
-0.245
-0.020
-0.077
-0.043
-0.303
-0.043
-0.005
0.029
0.011
0.014
-0.001
-0.113
0.007
-0.014
-0.053
0.004
-0.041
-0.011
-0.030
0.029
0.051
-0.226
0.000
0.085
-0.098
-0.053
3J
LE NOE TAR WAT
0.048
0.021
-0.063
-0.148
-0.058
-0.130
-0.071
-0.011
0.047
-0.158
-0.074
-0.105
-0.046
-0.080
-0.069
-0.002
-0.016
-0.035
-0.212
-0.011
-0.059
0.000
-0.314
-0.015
0.010
0.030
0.043
0.110
0.010
0.003
0.003
0.008
-0.041
0.010
-0.001
-0.022
-0.029
0.043
0.067
-0.228
-0.001
0.098
0.024
-0.001
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
3VAL
3VAL
4LEU
5LEU
6GLY
6GLY
7SER
8GLY
8GLY
10GLY
10GLY
10GLY
10GLY
10GLY
12LEU
12LEU
13VAL
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
17SER
18GLU
18GLU
20SER
20SER
20SER
24GLY
28VAL
29PHE
29PHE
29PHE
30LYSH
31ASN
31ASN
32ASN
32ASN
32ASN
33ALA
33ALA
33ALA
34GLY
bound
rexp
0.720
0.350
0.400
0.500
0.430
0.500
0.720
0.720
0.350
0.590
0.520
0.350
0.300
0.500
0.500
0.350
0.240
0.500
0.600
0.500
0.810
0.710
0.710
0.270
0.430
0.400
0.430
0.450
0.350
0.540
0.400
0.490
0.600
0.500
0.400
0.500
0.430
0.400
0.400
0.500
0.350
0.270
0.710
0.350
64
NOE Nr
9
19
29
31
38
39
42
51
55
59
60
61
62
63
70
73
79
82
86
87
89
96
109
139
145
146
178
179
183
203
233
244
279
284
288
293
299
305
306
313
315
316
319
324
NOE
atom 1
HA
H
H
HA
H
H
HA
H
H
QA
H
H
H
H
HA
H
H
HA
HA
HB
CG
CG
CZ
H
HA
H
HA
HA
H
QA
H
QB
H
HZ
HA
HA
H
HB
HB
HD22
H
H
H
H
RES2
31ASN
33ALA
33ALA
66PRO
92MET
5LEU
35PHE
35PHE
37HISB
92MET
12LEU
12LEU
92MET
35PHE
35PHE
35PHE
35PHE
92MET
37HISB
85SER
39VAL
82PHE
46ILE
55ILE
46ILE
46ILE
55ILE
55ILE
56SER
80TYR
55ILE
56SER
56SER
46ILE
52ALA
43GLU
44ASP
52ALA
42ASP
46ILE
50VAL
50VAL
46ILE
50VAL
atom 2
HA
HA
HA
HB
QE
HB
HA
CG
HB
QE
HA
HG
QE
HA
QB
CG
H
QE
HB
H
QG1
HB
QG2
QG2
QD
QG2
QG2
QG1
HA
HB
HB
HA
HG
QG1
QB
HB
H
QB
HB
H
H
HB
QD
QG1
3J
UNR VAC
UNR WAT
LE VAC
0.041
0.151
0.028
-0.149
0.042
-0.130
-0.179
-0.081
-0.163
-0.061
0.038
-0.007
0.519
0.018
0.120
-0.003
0.019
-0.007
0.256
0.073
0.079
0.103
-0.028
-0.042
-0.077
0.007
-0.022
0.414
-0.006
0.056
-0.139
-0.157
0.013
0.009
-0.060
0.026
0.189
-0.127
-0.152
0.176
0.015
-0.020
0.276
0.096
0.046
0.130
-0.017
-0.060
0.125
0.006
-0.042
0.186
-0.106
-0.120
0.066
0.050
0.000
-0.006
-0.032
-0.022
0.010
-0.100
-0.080
0.014
0.019
0.017
0.410
-0.211
0.412
0.006
-0.075
0.300
0.568
-0.232
0.365
0.522
-0.206
0.496
-0.023
-0.118
0.255
-0.206
-0.271
0.014
-0.110
-0.099
0.092
-0.076
0.069
-0.096
-0.210
-0.303
0.015
-0.061
-0.051
0.029
-0.115
-0.151
0.004
0.367
-0.089
0.533
-0.085
-0.138
0.138
-0.094
-0.003
-0.065
-0.013
-0.065
0.039
-0.214
-0.151
0.058
-0.008
-0.063
0.029
0.071
-0.064
0.087
0.069
0.005
0.068
-0.111
0.001
-0.101
0.036
-0.160
-0.096
-0.054
0.050
-0.076
Table 2.12 Continued on next page
3J
−1/6
r−6
− rexp
LE WAT
NMR set
-0.003
-0.138
-0.030
-0.178
-0.187
0.005
-0.066
0.001
0.016
-0.041
0.057
-0.071
0.004
0.007
-0.130
-0.162
0.020
-0.004
0.028
-0.068
-0.066
-0.183
-0.031
-0.006
-0.130
-0.086
-0.080
-0.161
0.029
-0.188
-0.134
-0.212
-0.004
-0.159
-0.075
-0.146
0.032
-0.023
-0.032
0.004
-0.081
-0.096
0.015
0.005
-0.237
-0.239
-0.075
-0.139
-0.168
-0.227
-0.213
-0.221
-0.125
-0.196
-0.193
-0.349
-0.120
-0.119
0.015
-0.009
-0.284
-0.348
-0.081
-0.092
-0.115
-0.196
-0.068
-0.073
-0.118
-0.127
-0.135
0.003
-0.067
-0.050
-0.087
-0.175
-0.060
-0.050
-0.055
-0.026
0.007
-0.046
-0.114
-0.093
-0.206
-0.122
-0.069
-0.053
3J
LE NOE WAT
-0.022
-0.050
-0.175
-0.076
0.001
0.020
0.006
-0.133
0.015
0.002
-0.165
-0.027
-0.113
-0.103
0.020
-0.151
-0.028
-0.094
0.029
-0.028
-0.089
0.017
-0.228
-0.126
-0.121
-0.209
-0.178
-0.242
-0.103
-0.001
-0.299
-0.049
-0.135
-0.073
-0.117
-0.135
-0.069
-0.124
-0.055
-0.039
-0.006
-0.099
-0.212
-0.097
3J
LE NOE TAR WAT
0.115
0.030
-0.138
0.020
0.002
-0.025
0.043
-0.030
0.001
-0.040
-0.136
-0.087
-0.117
-0.021
0.099
-0.057
-0.003
-0.110
0.019
-0.054
-0.086
0.028
-0.252
-0.364
-0.195
-0.229
-0.285
-0.296
-0.094
-0.020
-0.157
-0.026
-0.109
-0.072
-0.108
-0.139
-0.063
-0.095
-0.057
-0.055
0.010
-0.102
-0.200
-0.064
65
RES1
35PHE
35PHE
35PHE
35PHE
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
38ASN
38ASN
40VAL
40VAL
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
43GLU
43GLU
43GLU
43GLU
43GLU
45GLU
45GLU
49GLY
50VAL
51ASP
51ASP
bound
rexp
0.710
0.500
0.500
0.500
0.600
0.350
0.300
0.610
0.350
0.600
0.500
0.500
0.450
0.500
0.590
0.710
0.500
0.600
0.350
0.430
0.600
0.500
0.590
0.740
0.810
0.660
0.760
0.700
0.560
0.560
0.710
0.610
0.400
0.390
0.450
0.350
0.350
0.600
0.500
0.350
0.300
0.350
0.600
0.600
2.6 Supplementary material
NOE NR
325
335
337
339
350
352
355
356
357
359
361
362
365
373
374
375
376
377
384
389
402
411
413
415
418
419
423
424
425
430
439
440
453
464
465
468
470
471
477
480
494
497
499
500
NOE
atom 1
CG
H
HZ
HZ
HB
HD2
HD2
HD2
HD2
HD2
HE1
HE1
HE1
HE2
HE2
HE2
HE2
HE2
H
H
HA
H
QB
QB
CG
CG
CG
CG
CG
CG
CZ
CZ
H
HA
HA
H
H
H
H
H
H
H
HA
HA
RES2
46ILE
55ILE
46ILE
50VAL
54LYSH
72VAL
39VAL
40VAL
40VAL
55ILE
56SER
40VAL
52ALA
40VAL
56SER
40VAL
58PRO
58PRO
57MET
63LEU
63LEU
31ASN
65ALA
30LYSH
57MET
69THR
72VAL
73THR
74LEU
50VAL
98VAL
97THR
46ILE
46ILE
46ILE
47PRO
79THR
96VAL
46ILE
46ILE
55ILE
46ILE
50VAL
50VAL
atom 2
QD
QG2
QD
QG1
H
QG2
QG1
QG1
QG2
H
HB
QG1
QB
QG2
HB
QG2
HB
HB
QE
HB
HB
HD22
QB
HA
QG
HB
QG2
QG2
HB
QG2
HB
QG2
HA
HB
QG2
QD
QG2
QG1
HB
QG2
QG2
HB
QG1
QG2
3J
UNR VAC
UNR WAT
LE VAC
0.431
-0.167
0.356
0.049
0.006
-0.025
0.177
-0.132
0.141
0.015
0.197
0.027
0.038
0.009
0.042
0.149
0.012
0.219
0.100
-0.049
0.070
-0.018
-0.037
0.160
0.046
-0.034
0.228
-0.014
0.009
-0.011
-0.057
0.003
-0.076
-0.140
-0.088
0.193
-0.027
-0.047
0.084
-0.108
-0.189
0.136
0.014
-0.014
-0.112
-0.157
0.134
-0.039
-0.058
-0.097
0.002
0.128
0.050
0.132
-0.025
-0.130
0.100
-0.000
-0.041
-0.013
0.017
0.032
0.022
0.014
-0.033
0.027
0.022
-0.059
-0.002
-0.006
-0.006
0.007
0.038
-0.065
-0.028
0.007
-0.043
0.040
0.039
-0.092
-0.043
0.022
-0.031
-0.036
0.045
0.027
0.040
-0.030
0.072
-0.059
0.027
-0.028
0.021
0.060
-0.030
-0.014
0.468
0.015
0.456
0.471
-0.091
0.437
0.561
-0.112
0.567
0.074
-0.164
0.074
0.016
0.000
-0.072
-0.043
0.012
-0.039
0.280
-0.270
0.187
0.280
-0.264
0.265
0.053
0.009
0.268
0.173
-0.305
0.042
-0.077
0.048
0.107
-0.150
-0.405
0.029
Table 2.12 Continued on next page
3J
−1/6
r−6
− rexp
LE WAT NMR set
-0.157
-0.186
0.044
-0.030
-0.150
-0.153
0.050
-0.028
0.011
0.002
-0.066
-0.146
-0.055
-0.080
-0.063
-0.041
-0.022
-0.055
-0.015
-0.038
-0.114
0.008
-0.138
-0.169
-0.044
-0.127
-0.099
-0.141
-0.091
-0.006
-0.140
-0.158
-0.073
-0.011
0.038
0.030
-0.109
-0.154
-0.005
0.007
0.012
0.022
-0.072
-0.109
-0.052
-0.014
-0.000
-0.035
-0.104
-0.213
-0.026
-0.048
-0.064
-0.174
-0.071
-0.091
0.003
0.020
0.046
-0.054
-0.021
-0.057
-0.060
-0.076
0.015
-0.049
-0.027
-0.091
-0.070
-0.126
-0.153
-0.112
0.021
-0.089
0.023
-0.025
-0.236
-0.310
-0.251
-0.310
-0.059
-0.107
-0.253
-0.250
0.084
-0.129
-0.176
-0.364
3J
LE NOE WAT
-0.153
0.028
-0.152
-0.016
0.021
-0.026
-0.032
-0.112
-0.056
-0.010
-0.088
-0.132
-0.057
-0.160
-0.079
-0.066
-0.074
0.024
-0.079
-0.009
0.017
-0.070
-0.059
-0.005
-0.088
-0.023
-0.083
-0.077
-0.001
-0.024
-0.043
-0.080
-0.012
-0.037
-0.078
-0.143
-0.082
-0.048
-0.227
-0.255
-0.101
-0.195
-0.146
-0.413
3J
LE NOE TAR WAT
-0.166
-0.246
-0.134
0.061
0.018
0.044
-0.017
-0.161
-0.110
-0.002
-0.114
-0.136
-0.006
-0.200
-0.089
-0.005
-0.072
0.019
-0.114
-0.006
0.021
-0.018
-0.060
-0.009
-0.043
-0.004
-0.080
0.005
-0.005
0.064
-0.031
-0.048
-0.001
-0.044
-0.080
-0.168
-0.060
-0.010
-0.254
-0.265
-0.055
-0.256
0.091
-0.217
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
52ALA
52ALA
52ALA
52ALA
52ALA
55ILE
56SER
56SER
56SER
56SER
56SER
56SER
56SER
57MET
57MET
59GLU
59GLU
61GLU
62LEU
63LEU
64ASN
65ALA
68GLU
69THR
70TYR
70TYR
73THR
74LEU
75ASP
76THR
77LYSH
78GLY
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
bound
rexp
0.500
0.550
0.500
0.600
0.430
0.450
0.600
0.500
0.600
0.270
0.350
0.500
0.500
0.600
0.400
0.500
0.400
0.400
0.500
0.350
0.400
0.400
0.500
0.270
0.590
0.350
0.450
0.450
0.400
0.600
0.300
0.600
0.300
0.500
0.450
0.490
0.600
0.400
0.710
0.710
0.760
0.710
0.610
0.810
66
NOE NR
507
509
511
512
518
535
542
546
547
549
553
556
557
562
564
569
573
582
589
607
613
619
632
635
647
666
693
699
708
711
720
730
739
740
741
742
743
747
749
750
752
759
763
764
NOE
atom 1
HA
HA
H
H
H
HA
HA
HB
HB
H
H
HG
HG
H
H
HA
H
H
HA
H
H
H
H
HA
HB
H
H
H
H
HA
HA
H
HA
HA
HA
HA
HA
HB
CG
CG
CG
CZ
CZ
CZ
RES2
55ILE
77LYSH
79THR
47PRO
50VAL
55ILE
75ASP
76THR
76THR
76THR
77LYSH
77LYSH
98VAL
98VAL
41PHE
42ASP
96VAL
96VAL
93VAL
93VAL
88GLN
42ASP
82PHE
39VAL
85SER
92MET
12LEU
90ALA
84CYS
90ALA
12LEU
12LEU
90ALA
86PRO
90ALA
37HISB
37HISB
86PRO
83TYR
87HISB
88GLN
90ALA
87HISB
90ALA
atom 2
QG2
HA
H
HB
QG2
QG2
H
HB
H
QG2
HA
H
HB
QB
HA
HB
HB
QG1
HA
QG2
QB
HB
HB
HA
HA
QE
QG
QB
HB
QB
HB
QG
QB
HB
QB
HA
HE1
HB
CG
H
H
H
HA
H
3J
UNR VAC
UNR WAT
LE VAC
-0.079
-0.081
0.037
-0.029
-0.101
0.004
0.008
-0.007
0.005
-0.045
0.051
0.022
0.005
-0.186
0.181
-0.049
-0.007
0.007
0.090
0.135
-0.010
0.055
0.016
-0.061
0.159
0.133
0.054
-0.120
-0.046
-0.171
0.246
0.207
0.116
0.094
0.138
0.077
0.149
0.200
0.054
0.064
0.131
-0.065
0.003
0.002
0.022
0.088
-0.023
0.077
-0.050
-0.006
-0.038
0.005
-0.215
-0.090
0.001
-0.008
0.039
-0.010
0.011
-0.039
0.036
0.022
0.011
-0.019
0.020
-0.046
0.057
0.025
0.041
0.022
-0.023
-0.016
-0.018
-0.020
-0.014
0.210
0.020
-0.096
-0.139
-0.064
0.024
0.488
-0.074
0.085
0.045
0.041
0.069
0.262
-0.252
-0.177
0.126
0.022
0.061
-0.145
-0.022
-0.068
0.538
0.014
0.080
0.077
-0.032
0.093
0.491
-0.075
0.039
0.021
-0.202
0.030
0.203
-0.081
-0.025
-0.001
-0.051
-0.034
0.124
0.035
0.059
0.014
-0.032
-0.072
0.006
-0.171
-0.082
0.086
-0.053
-0.022
0.405
-0.052
0.058
0.146
-0.013
0.066
Table 2.12 Continued on next page
3J
−1/6
r−6
− rexp
LE WAT
NMR set
-0.199
-0.205
-0.053
-0.148
0.005
-0.002
0.042
0.092
-0.033
-0.190
-0.141
-0.183
-0.189
-0.092
-0.039
-0.062
-0.074
-0.053
-0.033
0.015
0.225
0.118
0.186
0.123
0.151
0.048
0.041
-0.085
0.004
-0.007
-0.036
-0.056
-0.032
-0.090
-0.268
-0.042
0.009
-0.053
-0.147
-0.028
0.099
-0.241
0.017
0.033
0.021
-0.002
-0.023
-0.069
-0.021
0.002
0.011
-0.051
0.008
-0.166
-0.025
-0.126
-0.028
0.093
-0.194
-0.262
0.014
0.004
0.043
-0.142
0.077
-0.005
0.048
0.058
0.014
-0.101
-0.243
-0.048
-0.066
-0.011
-0.033
0.013
0.111
-0.091
-0.017
-0.136
-0.136
-0.163
-0.042
-0.081
-0.012
-0.121
-0.030
-0.043
3J
LE NOE WAT
-0.187
-0.152
0.007
0.012
-0.136
0.018
-0.104
-0.086
-0.079
-0.090
-0.074
-0.032
-0.091
-0.173
-0.003
-0.048
-0.063
-0.064
-0.003
-0.068
-0.123
0.000
0.019
-0.044
-0.021
0.019
-0.153
-0.132
-0.013
-0.270
0.025
-0.115
-0.014
-0.040
-0.078
-0.234
-0.078
-0.015
-0.075
-0.066
-0.145
-0.065
-0.090
-0.005
3J
LE NOE TAR WAT
-0.129
-0.106
-0.002
-0.008
-0.043
-0.064
-0.025
-0.057
-0.039
-0.105
-0.018
0.013
-0.054
-0.117
0.006
-0.035
0.006
-0.149
0.017
-0.079
-0.070
0.016
0.031
-0.029
-0.027
-0.010
-0.037
0.005
-0.000
-0.162
0.054
0.008
0.132
0.032
0.044
-0.180
-0.079
-0.013
-0.164
-0.074
-0.088
-0.040
0.027
0.001
67
RES1
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
82PHE
82PHE
82PHE
82PHE
83TYR
83TYR
83TYR
83TYR
83TYR
84CYS
84CYS
84CYS
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
88GLN
88GLN
89GLY
89GLY
90ALA
91GLY
bound
rexp
0.760
0.710
0.430
0.500
0.500
0.650
0.500
0.350
0.400
0.500
0.500
0.500
0.500
0.620
0.240
0.500
0.500
0.600
0.270
0.600
0.650
0.610
0.350
0.400
0.430
0.500
0.720
0.500
0.500
0.600
0.500
0.620
0.500
0.500
0.500
0.500
0.400
0.500
0.710
0.350
0.430
0.350
0.430
0.300
2.6 Supplementary material
NOE NR
766
771
776
784
787
788
790
791
792
793
794
795
796
797
805
807
838
839
841
844
853
858
865
870
874
876
897
900
901
902
904
905
906
908
909
911
912
913
917
918
924
925
926
929
NOE
atom 1
CZ
CZ
H
HH
HH
HH
HH
HH
HH
HH
HH
HH
HH
HH
HA
HA
HZ
HZ
HA
HA
CG
CZ
H
HA
HA
HB
HA
HA
HB
HB
HB
HB
HB
HD2
HD2
HE1
HE1
HE1
H
H
H
H
H
H
RES1
91GLY
92MET
92MET
96VAL
97THR
97THR
98VAL
98VAL
98VAL
RES2
92MET
88GLN
93VAL
79THR
20SER
96VAL
97THR
77LYSH
78GLY
atom 2
H
HA
QG1
QG2
HA
HB
QG2
HA
H
bound
rexp
0.300
0.350
0.500
0.600
0.270
0.400
0.600
0.430
0.350
UNR VAC
0.029
0.194
0.068
0.064
0.031
0.019
0.013
0.079
-0.001
UNR WAT
-0.022
-0.000
-0.050
-0.028
0.033
-0.074
-0.055
0.040
-0.024
3J
LE VAC
0.002
0.053
0.008
-0.058
0.044
-0.018
-0.051
0.090
0.006
3J
−1/6
r−6
− rexp
LE WAT NMR set
-0.022
-0.051
0.023
-0.041
-0.013
-0.050
0.055
-0.139
0.046
-0.033
-0.101
-0.006
-0.079
-0.041
0.034
0.022
-0.027
-0.019
68
NOE NR
931
932
937
959
971
973
979
981
982
NOE
atom 1
H
H
H
H
H
H
HA
H
H
3J
LE NOE WAT
-0.047
-0.035
-0.029
-0.041
0.013
-0.011
-0.069
0.022
-0.034
3J
LE NOE TAR WAT
-0.034
0.029
0.008
-0.094
0.047
-0.044
-0.073
0.036
-0.026
2 Structural characterisation of Plastocyanin using local-elevation MD
−1/6
Table 2.12 List of the NOE distances, for which the average r−6
is larger than the experimental rexp in at least one of the
−6 −1/6
simulations or in the set of 16 NMR model structures. The difference ( r
− rexp ) is given in nm for all 6 simulations and the set
of 16 NMR model structures.
RES2
1LEU
1LEU
3VAL
28VAL
1LEU
1LEU
1LEU
2GLU
1LEU
3VAL
3VAL
2GLU
2GLU
2GLU
3VAL
3VAL
3VAL
28VAL
28VAL
29PHE
4LEU
30LYSH
3VAL
3VAL
3VAL
4LEU
4LEU
4LEU
15VAL
5LEU
92MET
4LEU
4LEU
5LEU
5LEU
33ALA
5LEU
13VAL
92MET
7SER
7SER
13VAL
13VAL
13VAL
atom 2
QB
QG
QG2
HB
HA
QB
QG
QB
QG
QG1
QG2
HA
QB
QG
HB
QG1
QG2
HB
H
HA
QG
QB
HA
QG1
QG2
HB
HG
QG
HB
QG
QE
HA
HB
HG
QG
QB
HA
H
QE
HB
HB
QB
QB
QB
3J
UNR VAC UNR WAT
LE VAC
-0.182
-0.182
-0.189
-0.269
-0.268
-0.264
-0.124
-0.136
-0.140
-0.034
-0.045
-0.048
-0.089
-0.091
-0.087
-0.111
-0.113
-0.155
-0.163
-0.130
-0.173
-0.200
-0.192
-0.199
-0.078
-0.008
-0.086
-0.155
-0.158
-0.161
-0.160
-0.157
-0.155
-0.060
-0.061
-0.060
-0.067
-0.084
-0.068
-0.082
-0.126
-0.100
-0.104
-0.101
-0.100
-0.057
-0.055
-0.060
-0.172
-0.175
-0.188
-0.025
-0.042
-0.030
0.011
0.016
0.018
-0.103
-0.098
-0.118
-0.173
-0.203
-0.171
-0.220
-0.207
-0.232
-0.028
-0.025
-0.026
-0.152
-0.152
-0.148
-0.115
-0.112
-0.112
-0.067
-0.077
-0.053
-0.097
-0.058
-0.101
-0.316
-0.323
-0.326
0.003
-0.052
-0.047
-0.275
-0.263
-0.270
0.671
-0.165
0.029
-0.089
-0.090
-0.088
-0.070
-0.076
-0.071
-0.150
-0.122
-0.137
-0.328
-0.346
-0.310
-0.228
-0.183
-0.154
-0.018
-0.023
-0.019
0.107
-0.091
0.070
0.573
-0.134
-0.022
-0.066
-0.063
-0.041
-0.045
-0.050
-0.039
0.009
-0.076
0.002
-0.064
-0.150
-0.017
-0.013
-0.008
-0.034
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.184
-0.186
-0.266
-0.276
-0.130
-0.130
-0.048
-0.058
-0.099
-0.075
-0.089
-0.208
-0.162
-0.211
-0.192
-0.164
-0.127
-0.104
-0.161
-0.159
-0.156
-0.150
-0.060
-0.058
-0.076
-0.113
-0.111
-0.091
-0.106
-0.088
-0.061
-0.045
-0.190
-0.170
-0.052
-0.040
0.001
-0.004
-0.092
-0.116
-0.175
-0.183
-0.229
-0.211
-0.027
-0.026
-0.124
-0.144
-0.107
-0.103
-0.053
-0.037
-0.104
-0.095
-0.316
-0.308
-0.063
-0.070
-0.269
-0.282
-0.159
-0.185
-0.088
-0.088
-0.077
-0.085
-0.124
-0.135
-0.347
-0.350
-0.168
-0.089
-0.021
-0.024
-0.036
-0.053
-0.136
-0.198
-0.037
-0.062
-0.034
-0.053
-0.017
-0.106
-0.038
-0.201
-0.100
-0.065
3J
LE NOE WAT
-0.182
-0.269
-0.142
-0.048
-0.097
-0.090
-0.154
-0.196
-0.097
-0.162
-0.153
-0.063
-0.078
-0.126
-0.097
-0.057
-0.194
-0.040
0.015
-0.100
-0.175
-0.223
-0.021
-0.149
-0.110
-0.046
-0.100
-0.310
-0.078
-0.269
-0.174
-0.086
-0.082
-0.157
-0.337
-0.174
-0.019
-0.055
-0.162
-0.037
-0.038
-0.085
-0.119
-0.196
3J
LE NOE TAV WAT
-0.182
-0.273
-0.146
-0.050
-0.092
-0.112
-0.122
-0.188
0.048
-0.173
-0.141
-0.064
-0.095
-0.116
-0.084
-0.053
-0.199
-0.041
0.021
-0.100
-0.176
-0.211
-0.021
-0.143
-0.094
-0.047
-0.092
-0.310
-0.063
-0.260
-0.148
-0.088
-0.075
-0.129
-0.349
-0.200
-0.020
-0.058
-0.130
-0.040
-0.037
-0.071
-0.120
-0.193
69
RES1
1LEU
1LEU
2GLU
2GLU
2GLU
2GLU
2GLU
2GLU
3VAL
3VAL
3VAL
3VAL
3VAL
3VAL
3VAL
3VAL
3VAL
3VAL
3VAL
3VAL
4LEU
4LEU
4LEU
4LEU
4LEU
4LEU
4LEU
4LEU
4LEU
5LEU
5LEU
5LEU
5LEU
5LEU
5LEU
6GLY
6GLY
6GLY
6GLY
7SER
7SER
7SER
7SER
7SER
bound
rexp
0.440
0.620
0.600
0.300
0.300
0.490
0.620
0.440
0.720
0.450
0.450
0.270
0.490
0.490
0.350
0.500
0.500
0.400
0.350
0.430
0.520
0.590
0.240
0.500
0.600
0.400
0.350
0.720
0.400
0.620
0.500
0.300
0.500
0.400
0.720
0.590
0.240
0.430
0.500
0.300
0.300
0.720
0.620
0.620
2.6 Supplementary material
NOE Nr
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
NOE
atom 1
HA
HA
HA
HA
H
H
H
H
HA
HA
HA
H
H
H
H
H
H
H
H
H
HA
HA
H
H
H
H
H
H
H
HA
HA
H
H
H
H
QA
H
H
H
HA
HA
HA
HB
HB
RES2
15VAL
6GLY
7SER
13VAL
33ALA
33ALA
4LEU
7SER
7SER
7SER
9ASP
9ASP
8GLY
9ASP
33ALA
8GLY
9ASP
11SER
33ALA
13VAL
10GLY
11SER
11SER
12LEU
12LEU
92MET
11SER
11SER
12LEU
12LEU
12LEU
12LEU
12LEU
13VAL
12LEU
12LEU
13VAL
3VAL
5LEU
14PHE
15VAL
92MET
3VAL
3VAL
atom 2
QG2
QA
HB
H
QB
QB
QG
HA
HB
H
H
QB
QA
QB
QB
QA
H
H
QB
QB
QA
HB
HB
HB
QG
QE
HA
HB
HB
HB
HG
QG
HB
QB
HA
QG
QB
QG1
HA
HB
QG2
QE
QG1
QG1
3J
UNR VAC
UNR WAT
LE VAC
-0.078
-0.059
-0.147
-0.107
-0.097
-0.091
-0.088
-0.050
-0.139
-0.060
-0.122
-0.111
-0.283
-0.123
-0.124
-0.191
-0.272
-0.175
-0.277
0.123
-0.188
-0.133
-0.009
-0.131
-0.034
-0.105
-0.063
-0.049
-0.168
-0.017
0.060
-0.032
0.080
-0.219
-0.212
-0.191
-0.167
-0.182
-0.189
-0.174
-0.195
-0.188
-0.152
-0.094
-0.005
-0.003
-0.077
0.031
0.044
-0.071
0.035
0.050
-0.044
0.074
-0.188
0.007
-0.161
-0.032
-0.246
-0.133
-0.150
-0.153
-0.158
-0.147
-0.103
-0.104
-0.157
-0.119
-0.109
-0.060
-0.062
-0.065
-0.166
-0.175
-0.171
0.157
-0.037
0.030
-0.032
-0.025
-0.020
-0.004
-0.028
-0.013
-0.014
-0.002
-0.015
-0.124
-0.113
-0.120
-0.112
-0.118
-0.111
-0.327
-0.321
-0.321
-0.090
-0.025
-0.033
-0.241
-0.227
-0.233
0.086
-0.017
-0.012
-0.151
-0.154
-0.161
-0.252
-0.231
-0.248
0.038
-0.084
-0.032
-0.038
-0.038
-0.023
-0.062
-0.063
-0.063
-0.135
-0.129
-0.145
0.398
-0.230
0.024
0.036
-0.014
-0.023
-0.106
-0.202
-0.171
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.142
-0.077
-0.105
-0.113
-0.164
-0.041
-0.097
-0.123
-0.161
-0.104
-0.309
-0.265
0.131
-0.044
-0.066
-0.117
-0.106
-0.154
-0.175
-0.087
-0.030
-0.056
-0.208
-0.213
-0.175
-0.146
-0.205
-0.191
0.233
-0.205
0.025
-0.134
-0.030
-0.070
0.021
-0.022
0.168
-0.170
-0.185
-0.240
-0.171
-0.129
-0.097
-0.131
-0.110
-0.133
-0.063
-0.059
-0.176
-0.175
0.015
-0.075
-0.018
-0.033
-0.030
-0.012
-0.001
0.002
-0.111
-0.104
-0.106
-0.119
-0.313
-0.364
-0.042
-0.019
-0.227
-0.227
-0.027
-0.006
-0.185
-0.149
-0.230
-0.203
-0.076
-0.085
-0.025
-0.055
-0.063
-0.063
-0.147
-0.133
-0.229
-0.241
-0.044
-0.036
-0.213
-0.222
3J
LE NOE WAT
-0.130
-0.110
-0.150
-0.121
-0.191
-0.257
-0.026
-0.123
-0.092
-0.085
-0.008
-0.216
-0.170
-0.191
-0.187
-0.078
-0.095
-0.065
-0.100
-0.174
-0.136
-0.104
-0.121
-0.061
-0.175
-0.049
-0.016
-0.049
-0.002
-0.116
-0.114
-0.316
-0.033
-0.236
-0.017
-0.154
-0.214
-0.059
-0.022
-0.064
-0.155
-0.245
-0.020
-0.186
3J
LE NOE TAV WAT
-0.117
-0.111
-0.146
-0.121
-0.212
-0.254
-0.011
-0.121
-0.093
-0.101
0.047
-0.216
-0.177
-0.184
-0.158
-0.074
-0.105
-0.046
-0.080
-0.191
-0.151
-0.095
-0.104
-0.063
-0.194
-0.069
-0.018
-0.043
-0.002
-0.115
-0.094
-0.358
-0.034
-0.227
-0.016
-0.171
-0.227
-0.035
-0.036
-0.064
-0.153
-0.212
-0.011
-0.175
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
7SER
7SER
7SER
7SER
7SER
8GLY
8GLY
8GLY
8GLY
8GLY
8GLY
9ASP
9ASP
9ASP
10GLY
10GLY
10GLY
10GLY
10GLY
11SER
11SER
11SER
11SER
12LEU
12LEU
12LEU
12LEU
12LEU
12LEU
12LEU
12LEU
12LEU
13VAL
13VAL
13VAL
13VAL
13VAL
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
bound
rexp
0.500
0.360
0.400
0.430
0.600
0.690
0.720
0.350
0.430
0.430
0.350
0.440
0.440
0.490
0.590
0.520
0.350
0.300
0.500
0.620
0.440
0.400
0.400
0.350
0.520
0.500
0.240
0.400
0.350
0.350
0.350
0.720
0.500
0.520
0.240
0.720
0.520
0.500
0.350
0.350
0.600
0.600
0.500
0.660
70
NOE Nr
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
NOE
atom 1
HB
H
H
H
H
QA
H
H
H
H
H
HA
H
H
QA
H
H
H
H
HB
H
H
H
HA
HA
HA
H
H
H
H
H
H
HA
HA
H
H
H
HA
HA
HA
HA
HA
HB
CG
RES2
3VAL
5LEU
5LEU
13VAL
14PHE
29PHE
39VAL
39VAL
82PHE
83TYR
92MET
93VAL
94GLY
3VAL
5LEU
5LEU
39VAL
39VAL
83TYR
92MET
92MET
93VAL
13VAL
13VAL
13VAL
14PHE
14PHE
14PHE
39VAL
39VAL
82PHE
15VAL
15VAL
16PRO
16PRO
14PHE
14PHE
14PHE
15VAL
15VAL
15VAL
16PRO
1LEU
3VAL
atom 2
QG2
HG
QG
HA
HA
CZ
QG1
QG2
CZ
HA
QG
HA
QA
QG1
HG
QG
QG1
QG2
HA
QG
QE
HA
HA
HB
QB
HB
HB
CG
HA
QG1
CG
QG1
QG2
HB
HB
HA
HB
CG
QG2
HA
QG1
QG
QG
QG1
3J
UNR VAC UNR WAT
LE VAC
0.035
-0.085
-0.050
-0.185
-0.275
-0.202
-0.237
-0.231
-0.212
-0.235
-0.209
-0.235
-0.329
-0.341
-0.335
-0.060
-0.194
-0.149
-0.237
-0.113
-0.091
0.082
-0.069
-0.020
-0.328
-0.337
-0.348
-0.093
-0.104
-0.134
-0.071
-0.288
-0.185
-0.172
-0.165
-0.170
-0.289
-0.326
-0.325
-0.121
-0.190
-0.168
-0.272
-0.281
-0.262
-0.307
-0.280
-0.310
-0.407
-0.324
-0.300
-0.074
-0.230
-0.155
-0.196
-0.179
-0.258
-0.076
-0.264
-0.256
0.220
-0.299
-0.237
-0.151
-0.109
-0.130
-0.132
-0.117
-0.127
-0.050
-0.094
-0.090
-0.299
-0.303
-0.337
-0.057
-0.051
-0.053
-0.266
-0.268
-0.264
-0.272
-0.244
-0.266
-0.138
-0.182
-0.105
-0.239
-0.176
-0.149
-0.088
-0.118
-0.114
-0.116
-0.117
-0.110
-0.138
-0.137
-0.158
-0.111
-0.095
-0.104
-0.056
-0.049
-0.051
-0.076
-0.078
-0.082
-0.098
-0.069
-0.091
-0.259
-0.244
-0.259
-0.188
-0.197
-0.168
-0.059
-0.052
-0.054
-0.047
-0.052
-0.064
-0.256
-0.257
-0.256
-0.224
-0.187
-0.283
-0.196
-0.160
-0.175
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.104
-0.105
-0.247
-0.140
-0.205
-0.239
-0.217
-0.208
-0.337
-0.341
-0.153
-0.168
-0.165
-0.156
-0.048
-0.103
-0.331
-0.355
-0.094
-0.158
-0.289
-0.265
-0.163
-0.184
-0.338
-0.314
-0.205
-0.195
-0.266
-0.147
-0.253
-0.282
-0.371
-0.357
-0.186
-0.238
-0.198
-0.240
-0.282
-0.256
-0.308
-0.253
-0.134
-0.152
-0.112
-0.130
-0.108
-0.067
-0.313
-0.317
-0.050
-0.044
-0.264
-0.263
-0.252
-0.227
-0.160
-0.205
-0.216
-0.195
-0.115
-0.123
-0.117
-0.107
-0.148
-0.151
-0.097
-0.065
-0.046
-0.040
-0.078
-0.074
-0.081
-0.082
-0.246
-0.257
-0.185
-0.152
-0.048
-0.029
-0.066
-0.075
-0.257
-0.257
-0.321
-0.309
-0.137
-0.162
3J
LE NOE WAT
-0.077
-0.202
-0.194
-0.203
-0.339
-0.167
-0.160
-0.043
-0.346
-0.104
-0.299
-0.170
-0.324
-0.171
-0.220
-0.252
-0.366
-0.178
-0.199
-0.285
-0.303
-0.135
-0.118
-0.074
-0.303
-0.050
-0.266
-0.245
-0.171
-0.213
-0.124
-0.118
-0.143
-0.095
-0.046
-0.075
-0.082
-0.250
-0.188
-0.049
-0.065
-0.255
-0.266
-0.161
3J
LE NOE TAV WAT
-0.059
-0.260
-0.206
-0.222
-0.339
-0.132
-0.101
0.000
-0.359
-0.101
-0.242
-0.141
-0.342
-0.175
-0.281
-0.255
-0.316
-0.162
-0.227
-0.260
-0.314
-0.123
-0.113
-0.093
-0.302
-0.053
-0.260
-0.251
-0.144
-0.171
-0.117
-0.116
-0.147
-0.106
-0.050
-0.079
-0.095
-0.255
-0.185
-0.052
-0.064
-0.256
-0.098
-0.142
71
RES1
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
14PHE
15VAL
15VAL
15VAL
15VAL
15VAL
15VAL
15VAL
15VAL
16PRO
16PRO
16PRO
16PRO
16PRO
bound
rexp
0.810
0.710
0.780
0.710
0.610
0.820
0.810
0.710
0.820
0.710
0.700
0.710
0.800
0.810
0.710
0.780
0.810
0.660
0.610
0.700
0.710
0.610
0.350
0.350
0.720
0.400
0.500
0.560
0.500
0.600
0.610
0.400
0.450
0.500
0.500
0.300
0.500
0.710
0.500
0.240
0.450
0.490
0.810
0.590
2.6 Supplementary material
NOE Nr
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
NOE
atom 1
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
H
H
H
H
H
H
HZ
HZ
HZ
HA
HA
HA
HA
H
H
H
H
HA
HA
HB
QD
QD
RES2
3VAL
15VAL
14PHE
16PRO
16PRO
16PRO
18GLU
19PHE
18GLU
18GLU
95LYSH
95LYSH
96VAL
16PRO
16PRO
18GLU
19PHE
19PHE
1LEU
1LEU
1LEU
16PRO
16PRO
16PRO
16PRO
18GLU
96VAL
96VAL
96VAL
1LEU
3VAL
3VAL
27ILE
27ILE
27ILE
82PHE
96VAL
96VAL
96VAL
18GLU
18GLU
18GLU
19PHE
96VAL
atom 2
QG2
QG1
HB
HA
HB
HB
H
CG
QB
QG
QB
QG
HA
HB
HB
QB
HB
HB
QG
QG
QG
HB
HB
QD
QG
HA
HA
HB
QG2
QG
QG1
QG2
QG1
QD
QG2
HZ
HA
HB
QG2
HA
QB
QG
CG
HA
3J
UNR VAC
UNR WAT
LE VAC
-0.201
-0.190
-0.197
-0.189
-0.180
-0.222
-0.145
-0.152
-0.148
-0.081
-0.071
-0.076
-0.039
-0.066
-0.053
-0.089
-0.137
-0.112
0.011
-0.038
0.005
-0.064
-0.134
-0.072
-0.183
-0.184
-0.184
-0.207
-0.235
-0.203
-0.128
-0.111
-0.121
-0.086
-0.057
-0.063
0.016
-0.025
-0.007
0.039
0.062
0.051
-0.175
-0.149
-0.169
-0.172
-0.160
-0.179
-0.068
-0.066
-0.077
-0.060
-0.058
-0.049
-0.146
-0.155
-0.254
-0.220
-0.258
-0.272
-0.234
-0.273
-0.300
-0.172
-0.180
-0.155
-0.304
-0.314
-0.283
-0.260
-0.239
-0.226
-0.270
-0.267
-0.239
-0.230
-0.232
-0.247
-0.282
-0.261
-0.293
-0.188
-0.127
-0.194
-0.208
-0.133
-0.165
-0.249
-0.310
-0.291
-0.283
-0.234
-0.278
-0.372
-0.375
-0.363
-0.292
-0.314
-0.301
-0.190
-0.170
-0.170
-0.275
-0.284
-0.277
-0.222
-0.237
-0.279
-0.242
-0.215
-0.259
-0.226
-0.135
-0.225
-0.259
-0.219
-0.211
-0.023
-0.028
-0.023
-0.090
-0.088
-0.078
-0.061
-0.056
-0.062
-0.306
-0.312
-0.330
-0.035
-0.047
-0.043
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.172
-0.180
-0.206
-0.187
-0.206
-0.086
-0.065
-0.076
-0.060
-0.057
-0.140
-0.119
-0.042
-0.028
-0.092
-0.083
-0.184
-0.194
-0.239
-0.216
-0.083
-0.115
-0.035
-0.088
-0.043
0.001
0.066
0.058
-0.150
-0.159
-0.158
-0.158
-0.037
-0.062
-0.036
-0.063
-0.337
-0.170
-0.311
-0.259
-0.384
-0.288
-0.208
-0.199
-0.298
-0.315
-0.189
-0.260
-0.244
-0.305
-0.170
-0.229
-0.185
-0.317
-0.117
-0.204
-0.065
-0.218
-0.379
-0.310
-0.111
-0.178
-0.231
-0.379
-0.262
-0.305
-0.006
-0.230
-0.143
-0.277
-0.184
-0.174
-0.068
-0.263
-0.096
-0.254
-0.053
-0.294
-0.030
-0.021
-0.101
-0.114
-0.060
-0.140
-0.255
-0.273
-0.066
-0.066
3J
LE NOE WAT
-0.180
-0.198
-0.200
-0.057
-0.077
-0.147
-0.043
-0.105
-0.185
-0.237
-0.106
-0.027
-0.005
0.029
-0.179
-0.152
-0.075
-0.048
-0.289
-0.341
-0.339
-0.127
-0.264
-0.229
-0.222
-0.248
-0.286
-0.210
-0.182
-0.300
-0.275
-0.358
-0.276
-0.164
-0.278
-0.306
-0.222
-0.210
-0.200
-0.028
-0.091
-0.047
-0.325
-0.050
3J
LE NOE TAV WAT
-0.110
-0.222
-0.133
-0.071
-0.063
-0.131
-0.015
-0.032
-0.185
-0.238
-0.131
-0.061
0.010
0.030
-0.186
-0.180
-0.044
-0.037
-0.138
-0.218
-0.229
-0.095
-0.229
-0.265
-0.220
-0.187
-0.242
-0.132
-0.124
-0.351
-0.216
-0.333
-0.239
-0.075
-0.227
-0.221
-0.125
-0.071
-0.067
-0.029
-0.082
-0.134
-0.277
-0.054
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
16PRO
16PRO
17SER
17SER
17SER
17SER
17SER
17SER
18GLU
18GLU
18GLU
18GLU
18GLU
18GLU
18GLU
18GLU
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
19PHE
bound
rexp
0.590
0.690
0.590
0.300
0.400
0.400
0.270
0.710
0.440
0.490
0.490
0.590
0.430
0.400
0.500
0.440
0.300
0.300
0.720
0.720
0.780
0.710
0.710
0.800
0.650
0.710
0.710
0.610
0.660
0.830
0.710
0.710
0.800
0.660
0.660
0.610
0.710
0.560
0.660
0.240
0.490
0.490
0.610
0.350
72
NOE Nr
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
NOE
atom 1
QD
QD
QB
H
H
H
H
H
HA
HA
HA
HA
HA
H
H
H
HA
HA
HB
HB
CG
CG
CG
CG
CG
CG
CG
CG
CG
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
H
H
H
H
H
RES2
20SER
96VAL
96VAL
97THR
97THR
19PHE
19PHE
19PHE
19PHE
20SER
21VAL
22PRO
20SER
20SER
21VAL
96VAL
98VAL
98VAL
22PRO
99ASN
21VAL
22PRO
22PRO
22PRO
23SER
98VAL
73THR
23SER
23SER
25GLU
25GLU
21VAL
23SER
23SER
25GLU
26LYSH
26LYSH
25GLU
25GLU
26LYSH
26LYSH
3VAL
27ILE
27ILE
atom 2
QB
HA
QG2
HB
QG2
HA
HB
HB
CG
QB
QB
QD
HA
QB
QB
QG2
HA
QB
QG
QB
QB
HA
HB
HB
QB
QB
QG2
HA
QB
H
QG
QB
HA
QB
QB
QB
QG
HA
QB
QB
QG
QG2
HB
QG1
3J
UNR VAC UNR WAT
LE VAC
-0.131
-0.142
-0.134
-0.006
0.032
0.032
0.001
0.106
0.076
-0.069
-0.131
-0.141
-0.077
-0.117
-0.125
-0.044
-0.042
-0.046
0.019
0.013
0.010
-0.149
-0.156
-0.133
-0.194
-0.199
-0.172
-0.221
-0.220
-0.232
-0.228
-0.255
-0.229
-0.113
-0.114
-0.112
-0.023
-0.023
-0.024
-0.042
-0.074
-0.027
-0.319
-0.276
-0.319
-0.184
-0.079
-0.149
-0.040
-0.038
-0.028
-0.208
-0.195
-0.188
-0.157
-0.153
-0.126
-0.226
-0.214
-0.262
-0.237
-0.273
-0.241
-0.136
-0.126
-0.134
-0.071
-0.083
-0.063
-0.148
-0.175
-0.166
-0.303
-0.295
-0.303
-0.053
-0.022
-0.070
0.078
-0.062
-0.031
-0.025
-0.031
-0.024
-0.150
-0.155
-0.173
-0.106
-0.102
-0.077
-0.230
-0.161
-0.225
-0.130
-0.192
-0.107
-0.061
-0.052
-0.065
-0.042
-0.021
-0.053
-0.145
-0.161
-0.151
-0.193
-0.198
-0.216
-0.219
-0.216
-0.156
-0.063
-0.054
-0.060
-0.209
-0.221
-0.213
-0.192
-0.155
-0.142
-0.300
-0.318
-0.350
-0.085
-0.118
-0.089
-0.066
-0.066
-0.068
-0.224
-0.223
-0.171
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.136
-0.148
0.056
-0.028
0.141
-0.032
-0.131
-0.161
-0.158
-0.132
-0.051
-0.019
-0.013
-0.005
-0.083
-0.195
-0.277
-0.268
-0.235
-0.159
-0.232
-0.222
-0.114
-0.097
-0.022
-0.015
-0.027
-0.108
-0.300
-0.278
-0.076
-0.174
-0.032
-0.036
-0.189
-0.181
-0.128
-0.141
-0.156
-0.220
-0.273
-0.292
-0.130
-0.134
-0.080
-0.059
-0.189
-0.171
-0.335
-0.290
-0.077
-0.057
-0.111
-0.154
-0.031
-0.027
-0.148
-0.149
-0.116
-0.091
-0.184
-0.199
-0.163
-0.205
-0.053
-0.075
-0.011
-0.070
-0.137
-0.125
-0.192
-0.205
-0.227
-0.194
-0.049
-0.055
-0.257
-0.246
-0.150
-0.146
-0.301
-0.332
-0.128
-0.089
-0.068
-0.063
-0.196
-0.218
3J
LE NOE WAT
-0.145
0.011
0.014
-0.124
-0.086
-0.040
-0.001
-0.145
-0.174
-0.216
-0.229
-0.113
-0.022
-0.066
-0.291
-0.180
-0.037
-0.203
-0.127
-0.213
-0.280
-0.130
-0.067
-0.174
-0.338
-0.088
-0.113
-0.028
-0.132
-0.113
-0.163
-0.215
-0.058
-0.020
-0.155
-0.187
-0.228
-0.054
-0.226
-0.163
-0.284
-0.104
-0.066
-0.223
3J
LE NOE TAV WAT
-0.135
0.043
0.110
-0.126
-0.148
-0.052
0.010
-0.088
-0.234
-0.234
-0.230
-0.113
-0.022
-0.026
-0.300
-0.101
-0.024
-0.185
-0.130
-0.203
-0.265
-0.127
-0.077
-0.185
-0.309
-0.058
0.003
-0.029
-0.161
-0.097
-0.161
-0.183
-0.056
-0.027
-0.159
-0.210
-0.190
-0.051
-0.223
-0.139
-0.337
-0.119
-0.068
-0.215
73
RES1
20SER
20SER
20SER
20SER
20SER
20SER
20SER
20SER
20SER
20SER
21VAL
21VAL
21VAL
21VAL
21VAL
21VAL
21VAL
21VAL
22PRO
22PRO
22PRO
23SER
23SER
23SER
23SER
23SER
24GLY
24GLY
24GLY
24GLY
25GLU
25GLU
25GLU
25GLU
25GLU
26LYSH
26LYSH
26LYSH
26LYSH
26LYSH
26LYSH
27ILE
27ILE
27ILE
bound
rexp
0.390
0.430
0.450
0.400
0.600
0.270
0.350
0.400
0.610
0.490
0.520
0.330
0.240
0.440
0.620
0.600
0.350
0.720
0.490
0.590
0.710
0.350
0.400
0.500
0.590
0.620
0.540
0.240
0.520
0.350
0.490
0.720
0.430
0.590
0.390
0.440
0.490
0.270
0.590
0.440
0.590
0.600
0.350
0.490
2.6 Supplementary material
NOE Nr
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
NOE
atom 1
HA
HA
HA
HA
HA
H
H
H
H
H
HA
HA
H
H
H
H
H
H
HA
HA
QD
H
H
H
H
H
QA
H
H
H
HA
H
H
H
H
HA
HA
H
H
H
H
HA
HA
HA
RES2
27ILE
26LYSH
27ILE
27ILE
27ILE
72VAL
73THR
28VAL
71VAL
71VAL
2GLU
27ILE
27ILE
27ILE
28VAL
28VAL
3VAL
3VAL
3VAL
5LEU
27ILE
29PHE
39VAL
70TYR
39VAL
3VAL
3VAL
3VAL
5LEU
5LEU
27ILE
29PHE
39VAL
39VAL
70TYR
70TYR
70TYR
70TYR
71VAL
72VAL
3VAL
3VAL
3VAL
27ILE
atom 2
QG2
HA
HB
QG1
QG2
H
HA
QB
QG1
QG2
HA
HA
HB
QG2
HB
QG2
HB
QG1
QG2
QG
QG2
QB
QG2
HB
QG2
HB
QG1
QG2
HG
QG
HB
HA
QG1
QG2
HB
HB
CG
CZ
HA
QG1
HB
QG1
QG2
HB
3J
UNR VAC
UNR WAT
LE VAC
-0.115
-0.117
-0.113
-0.025
-0.027
-0.028
-0.100
-0.098
-0.098
-0.184
-0.205
-0.182
-0.157
-0.159
-0.164
-0.082
-0.087
-0.102
-0.090
-0.098
-0.075
-0.315
-0.318
-0.316
-0.062
-0.062
-0.067
-0.100
-0.119
-0.101
-0.019
-0.014
-0.023
-0.032
-0.028
-0.027
0.008
0.016
-0.002
-0.160
-0.134
-0.167
-0.151
-0.153
-0.161
-0.186
-0.183
-0.180
-0.122
-0.141
-0.113
-0.134
-0.148
-0.124
-0.103
-0.135
-0.140
-0.257
-0.281
-0.219
-0.102
-0.072
-0.095
-0.185
-0.185
-0.183
-0.236
-0.232
-0.255
-0.082
-0.058
-0.048
-0.336
-0.332
-0.355
-0.241
-0.235
-0.241
-0.159
-0.156
-0.155
-0.274
-0.289
-0.315
-0.252
-0.183
-0.281
-0.318
-0.308
-0.299
-0.203
-0.182
-0.210
-0.344
-0.344
-0.344
-0.403
-0.351
-0.376
-0.311
-0.346
-0.336
-0.287
-0.280
-0.242
-0.136
-0.122
-0.120
-0.251
-0.270
-0.278
-0.193
-0.239
-0.295
-0.114
-0.127
-0.116
-0.056
-0.158
-0.154
-0.279
-0.250
-0.282
-0.173
-0.152
-0.174
-0.161
-0.161
-0.195
-0.180
-0.159
-0.177
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.117
-0.115
-0.026
-0.027
-0.101
-0.083
-0.194
-0.197
-0.160
-0.146
-0.092
-0.114
-0.100
-0.115
-0.315
-0.315
-0.114
-0.076
-0.099
-0.112
-0.020
-0.064
-0.028
-0.024
0.017
-0.008
-0.126
-0.179
-0.154
-0.139
-0.192
-0.148
-0.135
-0.155
-0.158
-0.152
-0.144
-0.138
-0.278
-0.272
-0.063
-0.092
-0.186
-0.188
-0.245
-0.252
-0.022
-0.047
-0.345
-0.352
-0.226
-0.270
-0.157
-0.168
-0.298
-0.315
-0.184
-0.227
-0.300
-0.316
-0.186
-0.193
-0.344
-0.340
-0.379
-0.346
-0.335
-0.353
-0.250
-0.290
-0.106
-0.130
-0.294
-0.292
-0.309
-0.271
-0.133
-0.134
-0.153
-0.175
-0.244
-0.269
-0.150
-0.143
-0.165
-0.194
-0.172
-0.178
3J
LE NOE WAT
-0.118
-0.025
-0.098
-0.206
-0.159
-0.096
-0.098
-0.316
-0.121
-0.103
-0.019
-0.025
0.007
-0.153
-0.150
-0.184
-0.125
-0.132
-0.150
-0.268
-0.061
-0.187
-0.247
-0.014
-0.347
-0.229
-0.142
-0.307
-0.218
-0.311
-0.186
-0.345
-0.354
-0.350
-0.237
-0.083
-0.271
-0.283
-0.154
-0.167
-0.249
-0.144
-0.167
-0.183
3J
LE NOE TAV WAT
-0.121
-0.032
-0.095
-0.208
-0.160
-0.096
-0.104
-0.317
-0.113
-0.099
-0.018
-0.025
0.003
-0.150
-0.152
-0.188
-0.129
-0.115
-0.179
-0.279
-0.055
-0.185
-0.256
0.008
-0.356
-0.231
-0.123
-0.341
-0.165
-0.305
-0.195
-0.346
-0.375
-0.345
-0.203
-0.061
-0.249
-0.268
-0.135
-0.171
-0.261
-0.144
-0.219
-0.174
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
27ILE
27ILE
27ILE
27ILE
27ILE
27ILE
27ILE
28VAL
28VAL
28VAL
28VAL
28VAL
28VAL
28VAL
28VAL
28VAL
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
bound
rexp
0.400
0.240
0.350
0.490
0.600
0.430
0.430
0.570
0.500
0.500
0.350
0.240
0.400
0.500
0.400
0.500
0.400
0.600
0.600
0.720
0.600
0.440
0.590
0.490
0.690
0.610
0.660
0.810
0.710
0.780
0.710
0.610
0.810
0.710
0.710
0.710
0.820
0.920
0.710
0.710
0.710
0.660
0.660
0.610
74
NOE Nr
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
NOE
atom 1
HA
H
H
H
H
H
H
HA
HA
HA
H
H
H
H
H
H
HA
HA
HA
HA
HA
HA
QB
QB
QB
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CZ
CZ
CZ
CZ
RES2
27ILE
27ILE
39VAL
39VAL
72VAL
74LEU
82PHE
82PHE
28VAL
28VAL
29PHE
29PHE
70TYR
71VAL
71VAL
71VAL
3VAL
27ILE
39VAL
72VAL
28VAL
30LYSH
30LYSH
69THR
3VAL
29PHE
30LYSH
30LYSH
5LEU
5LEU
5LEU
30LYSH
31ASN
63LEU
68GLU
69THR
31ASN
4LEU
32ASN
32ASN
4LEU
4LEU
4LEU
5LEU
atom 2
QD
QG2
QG1
QG2
QG1
QG
CG
CZ
HA
QG1
QB
CG
H
HA
QG1
QG2
QG2
QG2
QG1
QG1
QG1
QB
QG
QG2
H
HA
QB
QG
HA
HB
HG
QG
QB
QG
H
HA
QB
QG
HB
HB
HB
HB
QG
HB
3J
UNR VAC UNR WAT
LE VAC
-0.205
-0.163
-0.089
-0.296
-0.287
-0.287
-0.286
-0.336
-0.342
-0.308
-0.377
-0.352
-0.195
-0.319
-0.288
-0.263
-0.154
-0.250
-0.478
-0.470
-0.491
-0.389
-0.435
-0.402
-0.027
-0.026
-0.027
-0.096
-0.095
-0.107
-0.161
-0.157
-0.158
-0.248
-0.252
-0.240
-0.080
-0.096
-0.107
-0.092
-0.092
-0.085
0.000
-0.000
-0.004
-0.081
-0.090
-0.082
-0.080
-0.069
-0.105
-0.115
-0.112
-0.114
-0.135
-0.229
-0.218
0.128
0.008
0.033
-0.099
-0.097
-0.071
-0.186
-0.185
-0.184
-0.177
-0.174
-0.179
-0.018
-0.031
0.090
-0.070
-0.082
-0.068
-0.027
-0.025
-0.026
-0.248
-0.244
-0.243
-0.184
-0.157
-0.205
-0.013
-0.021
0.014
-0.086
-0.092
-0.058
-0.118
-0.087
-0.068
-0.159
-0.162
-0.157
-0.207
-0.202
-0.214
-0.187
-0.280
-0.230
-0.009
-0.040
0.013
-0.066
-0.073
-0.085
-0.133
-0.135
-0.137
-0.076
-0.132
-0.063
-0.107
-0.108
-0.098
-0.069
-0.061
-0.078
0.098
0.064
0.094
0.085
0.094
0.114
-0.148
-0.163
-0.096
-0.049
-0.030
-0.087
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.043
-0.237
-0.289
-0.308
-0.354
-0.343
-0.335
-0.349
-0.301
-0.359
-0.244
-0.221
-0.489
-0.443
-0.419
-0.371
-0.023
-0.024
-0.096
-0.104
-0.156
-0.141
-0.256
-0.241
-0.091
-0.101
-0.097
-0.103
-0.039
-0.016
-0.080
-0.101
-0.073
-0.101
-0.118
-0.124
-0.233
-0.218
0.028
-0.038
-0.096
-0.105
-0.185
-0.184
-0.176
-0.166
-0.050
-0.053
-0.070
-0.076
-0.026
-0.023
-0.238
-0.234
-0.199
-0.144
-0.006
-0.000
-0.085
-0.073
-0.077
-0.104
-0.148
-0.189
-0.211
-0.176
-0.254
-0.202
-0.041
-0.009
-0.077
-0.097
-0.136
-0.138
-0.117
-0.174
-0.107
-0.112
-0.061
-0.061
0.061
-0.005
0.070
0.048
-0.147
-0.112
-0.044
-0.046
3J
LE NOE WAT
-0.189
-0.277
-0.351
-0.363
-0.319
-0.199
-0.445
-0.400
-0.025
-0.118
-0.153
-0.247
-0.096
-0.106
-0.053
-0.087
-0.070
-0.103
-0.240
0.004
-0.101
-0.185
-0.178
-0.041
-0.076
-0.028
-0.240
-0.161
-0.011
-0.086
-0.132
-0.159
-0.210
-0.250
-0.030
-0.080
-0.137
-0.155
-0.105
-0.061
0.029
0.051
-0.155
-0.019
3J
LE NOE TAV WAT
-0.183
-0.268
-0.347
-0.344
-0.309
-0.205
-0.471
-0.404
-0.025
-0.118
-0.156
-0.241
-0.098
-0.102
-0.041
-0.076
-0.131
-0.102
-0.229
0.010
-0.086
-0.185
-0.174
-0.001
-0.075
-0.027
-0.239
-0.153
-0.022
-0.089
-0.075
-0.167
-0.199
-0.265
-0.029
-0.075
-0.135
-0.141
-0.106
-0.061
0.043
0.067
-0.146
-0.027
75
RES1
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
29PHE
30LYSH
30LYSH
30LYSH
30LYSH
30LYSH
30LYSH
30LYSH
30LYSH
31ASN
31ASN
31ASN
31ASN
31ASN
31ASN
31ASN
31ASN
31ASN
32ASN
32ASN
32ASN
32ASN
32ASN
32ASN
32ASN
bound
rexp
0.710
0.660
0.710
0.810
0.810
0.930
0.920
0.820
0.240
0.500
0.440
0.560
0.430
0.430
0.600
0.600
0.600
0.500
0.600
0.500
0.600
0.440
0.440
0.400
0.430
0.240
0.490
0.590
0.500
0.300
0.500
0.490
0.440
0.720
0.430
0.430
0.490
0.720
0.350
0.350
0.400
0.400
0.620
0.400
2.6 Supplementary material
NOE Nr
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
NOE
atom 1
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
H
H
H
H
H
H
H
H
HZ
HZ
HZ
HZ
HA
HA
HA
HA
H
H
H
H
HA
HA
HA
H
H
H
H
H
HD21
HA
HA
HA
HB
HB
HB
H
RES2
5LEU
31ASN
32ASN
4LEU
7SER
33ALA
32ASN
32ASN
32ASN
33ALA
35PHE
37HISB
12LEU
33ALA
33ALA
35PHE
31ASN
31ASN
33ALA
35PHE
36PRO
36PRO
36PRO
64ASN
33ALA
66PRO
33ALA
35PHE
33ALA
66PRO
66PRO
66PRO
35PHE
62LEU
64ASN
37HISB
37HISB
62LEU
86PRO
5LEU
5LEU
92MET
5LEU
5LEU
atom 2
H
HA
HB
QG
HA
QB
HA
HB
HB
QB
CG
HD2
QG
HA
QB
H
HA
QB
HA
H
HA
HB
QD
HA
HA
HB
HA
QB
HA
HA
HB
QG
HA
QG
HA
HB
HB
QG
QG
QG
QG
QE
HB
HB
3J
UNR VAC
UNR WAT
LE VAC
-0.091
-0.115
-0.081
-0.059
-0.060
-0.058
-0.108
-0.155
-0.116
-0.272
-0.259
-0.346
0.131
-0.222
-0.101
-0.093
-0.211
-0.068
-0.006
0.003
-0.067
0.090
0.095
0.018
-0.109
-0.119
-0.141
-0.192
-0.192
-0.228
-0.125
0.064
-0.031
-0.124
-0.048
-0.015
-0.266
-0.336
-0.321
-0.082
-0.126
-0.131
-0.332
-0.241
-0.258
-0.065
0.004
-0.098
0.041
0.151
0.028
-0.045
-0.010
-0.086
-0.213
-0.018
-0.124
-0.354
-0.279
-0.357
-0.200
-0.196
-0.178
-0.129
-0.238
-0.130
-0.197
-0.295
-0.254
-0.207
-0.255
-0.198
-0.302
-0.166
-0.248
-0.188
-0.085
-0.173
-0.149
0.042
-0.130
-0.202
-0.211
-0.204
-0.179
-0.081
-0.163
-0.160
-0.083
-0.075
-0.061
0.038
-0.007
-0.165
-0.110
-0.136
-0.089
-0.090
-0.070
-0.120
-0.119
-0.072
-0.114
-0.132
-0.111
-0.129
-0.147
-0.156
-0.078
-0.061
-0.062
-0.132
-0.132
-0.078
-0.043
-0.116
-0.143
-0.224
-0.235
-0.301
-0.034
-0.122
-0.173
0.519
0.018
0.120
-0.165
-0.078
-0.125
-0.003
0.019
-0.007
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.115
-0.135
-0.059
-0.053
-0.156
-0.140
-0.285
-0.298
-0.241
-0.182
-0.220
-0.210
0.003
-0.011
0.093
0.072
-0.128
-0.146
-0.194
-0.168
-0.076
-0.147
-0.051
-0.051
-0.355
-0.246
-0.120
-0.101
-0.265
-0.296
-0.040
-0.089
-0.003
-0.138
-0.127
-0.298
-0.124
-0.182
-0.330
-0.351
-0.208
-0.242
-0.169
-0.161
-0.194
-0.269
-0.288
-0.273
-0.272
-0.157
-0.207
-0.206
-0.030
-0.178
-0.198
-0.189
-0.187
0.005
-0.196
-0.020
-0.066
0.001
-0.163
-0.084
-0.075
-0.002
-0.133
-0.127
-0.110
-0.145
-0.152
-0.159
-0.061
-0.060
-0.131
-0.125
-0.168
-0.153
-0.243
-0.248
-0.170
-0.179
0.016
-0.041
-0.068
-0.122
0.057
-0.071
3J
LE NOE WAT
-0.118
-0.061
-0.155
-0.306
-0.226
-0.197
0.000
0.085
-0.132
-0.187
-0.098
-0.064
-0.266
-0.122
-0.266
-0.053
-0.022
-0.145
-0.142
-0.339
-0.206
-0.173
-0.194
-0.289
-0.275
-0.215
-0.050
-0.202
-0.175
-0.202
-0.076
-0.157
-0.069
-0.137
-0.112
-0.149
-0.060
-0.188
-0.146
-0.259
-0.177
0.001
-0.069
0.020
3J
LE NOE TAV WAT
-0.119
-0.059
-0.155
-0.270
-0.228
-0.193
-0.001
0.098
-0.111
-0.184
0.024
-0.072
-0.316
-0.127
-0.246
-0.001
0.115
-0.046
-0.043
-0.302
-0.182
-0.202
-0.231
-0.243
-0.209
-0.123
0.030
-0.199
-0.138
-0.133
0.020
-0.132
-0.086
-0.141
-0.121
-0.155
-0.061
-0.163
-0.144
-0.221
-0.143
0.002
-0.111
-0.025
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
32ASN
32ASN
32ASN
32ASN
32ASN
32ASN
33ALA
33ALA
33ALA
33ALA
33ALA
33ALA
34GLY
34GLY
34GLY
34GLY
35PHE
35PHE
35PHE
35PHE
35PHE
35PHE
35PHE
35PHE
35PHE
35PHE
35PHE
35PHE
35PHE
35PHE
35PHE
35PHE
36PRO
36PRO
36PRO
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
bound
rexp
0.430
0.270
0.500
0.720
0.500
0.600
0.350
0.270
0.350
0.500
0.710
0.500
0.710
0.350
0.600
0.350
0.710
0.800
0.610
0.610
0.610
0.710
0.800
0.710
0.610
0.710
0.500
0.490
0.500
0.500
0.500
0.490
0.270
0.720
0.430
0.400
0.350
0.720
0.490
0.620
0.570
0.600
0.400
0.350
76
NOE Nr
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
NOE
atom 1
H
H
HD21
HD22
HD22
HD22
H
H
H
H
H
H
QA
H
H
H
CG
CG
CG
CG
CG
CG
CG
CG
CZ
CZ
H
H
HZ
HZ
HZ
HZ
HA
HA
HA
HA
HA
HA
HA
HB
HB
HB
HD2
HD2
RES2
5LEU
31ASN
35PHE
35PHE
37HISB
37HISB
92MET
6GLY
12LEU
12LEU
12LEU
34GLY
92MET
36PRO
36PRO
36PRO
6GLY
12LEU
33ALA
34GLY
35PHE
35PHE
35PHE
35PHE
92MET
38ASN
39VAL
63LEU
62LEU
37HISB
37HISB
37HISB
38ASN
62LEU
84CYS
84CYS
85SER
40VAL
85SER
59GLU
39VAL
39VAL
38ASN
38ASN
atom 2
QG
HA
HA
CG
HB
HB
QE
QA
HA
HG
QG
QA
QE
HA
HB
HB
QA
QG
H
QA
HA
QB
CG
H
QE
QB
QG2
QG
QG
HA
HB
HB
QB
QG
HA
HB
H
QG2
QB
HA
QG1
QG2
HA
QB
3J
UNR VAC UNR WAT
LE VAC
-0.179
-0.098
-0.166
-0.077
-0.116
-0.201
0.256
0.073
0.079
0.103
-0.028
-0.042
-0.077
0.007
-0.022
-0.110
-0.143
-0.146
0.414
-0.006
0.056
-0.061
-0.190
-0.217
-0.139
-0.157
0.013
0.009
-0.060
0.026
-0.086
-0.206
-0.162
-0.181
-0.148
-0.186
0.189
-0.127
-0.152
-0.017
-0.003
-0.019
-0.041
-0.054
-0.056
-0.180
-0.246
-0.207
-0.276
-0.220
-0.265
-0.111
-0.290
-0.246
-0.017
-0.141
-0.122
-0.191
-0.280
-0.195
0.176
0.015
-0.020
0.276
0.096
0.046
0.130
-0.017
-0.060
0.125
0.006
-0.042
0.186
-0.106
-0.120
-0.219
-0.219
-0.221
-0.149
-0.157
-0.164
-0.274
-0.227
-0.253
-0.150
-0.235
-0.163
-0.048
-0.035
-0.033
-0.102
-0.117
-0.169
0.066
0.050
0.000
-0.253
-0.246
-0.246
-0.106
-0.123
-0.051
-0.112
-0.058
-0.095
-0.051
-0.032
-0.255
-0.006
-0.032
-0.022
-0.187
-0.143
-0.176
-0.244
-0.256
-0.225
-0.086
-0.103
-0.087
-0.180
-0.160
-0.169
-0.101
-0.153
-0.155
-0.037
-0.028
-0.032
-0.084
-0.099
-0.087
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.084
-0.159
-0.092
-0.202
0.004
0.007
-0.130
-0.162
0.020
-0.004
-0.137
-0.150
0.028
-0.068
-0.226
-0.210
-0.066
-0.183
-0.031
-0.006
-0.220
-0.136
-0.063
-0.191
-0.130
-0.086
-0.013
-0.021
-0.056
-0.089
-0.235
-0.212
-0.221
-0.263
-0.332
-0.076
-0.151
-0.178
-0.204
-0.190
-0.080
-0.161
0.029
-0.188
-0.134
-0.212
-0.004
-0.159
-0.075
-0.146
-0.220
-0.225
-0.171
-0.173
-0.250
-0.270
-0.146
-0.161
-0.031
-0.035
-0.131
-0.177
0.032
-0.023
-0.249
-0.225
-0.104
-0.034
-0.043
-0.099
-0.213
-0.050
-0.032
0.004
-0.127
-0.206
-0.263
-0.296
-0.232
-0.215
-0.171
-0.161
-0.146
-0.154
-0.024
-0.016
-0.098
-0.095
3J
LE NOE WAT
-0.089
-0.099
0.006
-0.133
0.015
-0.140
0.002
-0.216
-0.165
-0.027
-0.195
-0.095
-0.113
-0.016
-0.054
-0.231
-0.241
-0.238
-0.146
-0.217
-0.103
0.020
-0.151
-0.028
-0.094
-0.220
-0.176
-0.243
-0.184
-0.036
-0.136
0.029
-0.245
-0.147
-0.043
-0.206
-0.028
-0.135
-0.268
-0.249
-0.171
-0.145
-0.023
-0.099
3J
LE NOE TAV WAT
-0.112
-0.153
0.043
-0.030
0.001
-0.141
-0.040
-0.196
-0.136
-0.087
-0.197
-0.187
-0.117
-0.002
-0.056
-0.247
-0.226
-0.258
-0.151
-0.260
-0.021
0.099
-0.057
-0.003
-0.110
-0.221
-0.176
-0.255
-0.199
-0.032
-0.132
0.019
-0.245
-0.137
-0.100
-0.273
-0.054
-0.155
-0.275
-0.229
-0.170
-0.147
-0.018
-0.106
77
RES1
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
37HISB
38ASN
38ASN
38ASN
38ASN
38ASN
38ASN
38ASN
38ASN
38ASN
38ASN
38ASN
38ASN
38ASN
38ASN
38ASN
39VAL
39VAL
39VAL
39VAL
bound
rexp
0.620
0.500
0.300
0.610
0.350
0.400
0.600
0.590
0.500
0.500
0.570
0.590
0.450
0.240
0.400
0.500
0.590
0.720
0.500
0.590
0.500
0.590
0.710
0.500
0.600
0.440
0.600
0.720
0.710
0.270
0.400
0.350
0.590
0.720
0.430
0.500
0.430
0.500
0.590
0.500
0.450
0.450
0.270
0.390
2.6 Supplementary material
NOE Nr
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
NOE
atom 1
HD2
HD2
HD2
HD2
HD2
HD2
HD2
HE1
HE1
HE1
HE1
HE1
HE1
H
H
H
HE2
HE2
HE2
HE2
HE2
HE2
HE2
HE2
HE2
HA
HA
HA
QB
H
H
H
H
H
H
H
H
HD21
HD21
HD22
HA
HA
H
H
RES2
39VAL
39VAL
39VAL
40VAL
57MET
39VAL
40VAL
40VAL
39VAL
39VAL
39VAL
40VAL
40VAL
40VAL
82PHE
41PHE
46ILE
52ALA
55ILE
39VAL
40VAL
46ILE
46ILE
52ALA
52ALA
55ILE
55ILE
55ILE
56SER
56SER
74LEU
80TYR
80TYR
80TYR
80TYR
82PHE
82PHE
82PHE
82PHE
96VAL
29PHE
39VAL
55ILE
56SER
atom 2
HB
QG1
QG2
QG2
QB
QG1
QG1
QG2
HA
QG1
QG2
HB
QG1
QG2
HB
QB
QG2
QB
QG2
QG1
HA
QD
QG2
HA
QB
QD
QG2
QG1
HA
HG
QG
HA
HB
HB
CG
HA
HB
HB
CG
QG1
HZ
QG1
HB
HA
3J
UNR VAC
UNR WAT
LE VAC
-0.033
-0.093
-0.092
-0.221
-0.156
-0.155
-0.228
-0.169
-0.185
-0.106
-0.123
-0.110
-0.149
-0.191
-0.104
0.010
-0.100
-0.080
-0.166
-0.159
-0.170
-0.200
-0.197
-0.203
-0.024
-0.020
-0.023
-0.030
-0.090
-0.087
-0.079
-0.088
-0.094
-0.150
-0.146
-0.149
-0.058
-0.083
-0.063
-0.194
-0.183
-0.200
0.014
0.019
0.017
-0.235
-0.232
-0.237
0.410
-0.211
0.412
-0.316
-0.221
-0.336
0.006
-0.075
0.300
-0.160
-0.186
-0.262
-0.230
-0.228
-0.231
0.568
-0.232
0.365
0.522
-0.206
0.496
-0.200
-0.182
-0.189
-0.262
-0.154
-0.348
-0.425
-0.416
-0.225
-0.023
-0.118
0.255
-0.206
-0.271
0.014
-0.110
-0.099
0.092
-0.262
-0.259
-0.189
-0.311
-0.295
-0.297
-0.153
-0.094
-0.176
-0.239
-0.137
-0.271
-0.076
0.069
-0.096
-0.314
-0.226
-0.363
-0.296
-0.297
-0.265
-0.364
-0.373
-0.343
-0.176
-0.184
-0.155
-0.372
-0.383
-0.357
-0.256
-0.002
-0.248
-0.339
-0.262
-0.337
-0.209
-0.221
-0.302
-0.210
-0.303
0.015
-0.061
-0.051
0.029
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.077
-0.066
-0.145
-0.132
-0.174
-0.137
-0.132
-0.143
-0.171
-0.238
-0.081
-0.096
-0.170
-0.159
-0.195
-0.205
-0.020
-0.032
-0.100
-0.124
-0.084
-0.104
-0.143
-0.136
-0.057
-0.044
-0.194
-0.156
0.015
0.005
-0.233
-0.236
-0.237
-0.239
-0.178
-0.235
-0.075
-0.139
-0.201
-0.198
-0.207
-0.221
-0.168
-0.227
-0.213
-0.221
-0.117
-0.193
-0.102
-0.180
-0.378
-0.468
-0.125
-0.196
-0.193
-0.349
-0.120
-0.119
-0.203
-0.240
-0.304
-0.336
-0.106
-0.100
-0.171
-0.192
0.015
-0.009
-0.243
-0.260
-0.286
-0.268
-0.386
-0.383
-0.181
-0.190
-0.405
-0.400
-0.074
-0.193
-0.365
-0.327
-0.217
-0.254
-0.284
-0.348
-0.081
-0.092
3J
LE NOE WAT
-0.080
-0.146
-0.175
-0.140
-0.190
-0.089
-0.170
-0.195
-0.022
-0.105
-0.085
-0.137
-0.052
-0.183
0.017
-0.233
-0.228
-0.222
-0.126
-0.175
-0.212
-0.121
-0.209
-0.162
-0.147
-0.418
-0.178
-0.242
-0.103
-0.226
-0.312
-0.109
-0.179
-0.001
-0.240
-0.299
-0.388
-0.186
-0.400
-0.199
-0.316
-0.204
-0.299
-0.049
3J
LE NOE TAV WAT
-0.081
-0.148
-0.184
-0.134
-0.197
-0.086
-0.168
-0.197
-0.021
-0.093
-0.087
-0.141
-0.058
-0.186
0.028
-0.234
-0.252
-0.227
-0.364
-0.202
-0.214
-0.195
-0.229
-0.150
-0.137
-0.337
-0.285
-0.296
-0.094
-0.198
-0.325
-0.123
-0.204
-0.020
-0.278
-0.281
-0.353
-0.170
-0.364
-0.141
-0.320
-0.252
-0.157
-0.026
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
39VAL
39VAL
39VAL
39VAL
39VAL
40VAL
40VAL
40VAL
40VAL
40VAL
40VAL
40VAL
40VAL
40VAL
40VAL
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
bound
rexp
0.350
0.600
0.500
0.600
0.490
0.600
0.450
0.500
0.240
0.500
0.600
0.400
0.500
0.500
0.500
0.490
0.590
0.690
0.740
0.710
0.710
0.810
0.660
0.710
0.810
0.810
0.760
0.700
0.560
0.710
0.930
0.710
0.610
0.560
0.820
0.710
0.710
0.610
0.920
0.810
0.710
0.660
0.710
0.610
78
NOE Nr
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
NOE
atom 1
H
H
H
H
H
HA
HA
HA
H
H
H
H
H
H
H
HA
QB
QB
QB
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CZ
CZ
CZ
CZ
RES2
74LEU
80TYR
80TYR
82PHE
82PHE
82PHE
96VAL
40VAL
40VAL
40VAL
41PHE
41PHE
56SER
29PHE
74LEU
41PHE
41PHE
42ASP
42ASP
82PHE
83TYR
43GLU
43GLU
46ILE
52ALA
42ASP
43GLU
43GLU
43GLU
44ASP
52ALA
44ASP
44ASP
45GLU
45GLU
45GLU
42ASP
44ASP
45GLU
46ILE
46ILE
46ILE
47PRO
47PRO
atom 2
QG
HB
HB
HB
CG
CZ
QG1
HA
QG1
QG2
QB
CG
HG
HZ
QG
HA
QB
HB
HB
HA
CG
HB
HB
QG1
QB
HA
HB
HB
QG
H
QB
QB
QB
H
QB
QG
HB
QB
QB
H
HB
QG2
QD
QG
3J
UNR VAC UNR WAT
LE VAC
-0.331
-0.395
-0.370
-0.156
-0.197
-0.174
-0.038
-0.031
-0.071
-0.372
-0.368
-0.333
-0.288
-0.297
-0.261
-0.263
-0.285
-0.256
-0.197
-0.046
-0.201
-0.023
-0.018
-0.028
-0.126
-0.152
-0.117
-0.094
-0.120
-0.099
-0.205
-0.206
-0.209
-0.310
-0.279
-0.326
-0.115
-0.151
0.004
-0.201
-0.113
-0.196
-0.156
-0.244
-0.213
-0.125
-0.132
-0.131
-0.109
-0.150
-0.088
-0.064
-0.103
-0.043
-0.093
-0.111
-0.092
-0.015
-0.048
-0.040
-0.252
-0.155
-0.247
-0.099
-0.116
-0.099
-0.074
-0.106
-0.097
0.367
-0.089
0.533
-0.085
-0.138
0.138
-0.124
-0.142
-0.134
-0.075
-0.150
-0.069
-0.094
-0.003
-0.065
-0.202
-0.224
-0.113
-0.013
-0.065
0.039
-0.214
-0.151
0.058
-0.238
-0.234
-0.238
-0.161
-0.190
-0.140
-0.015
-0.065
-0.005
-0.194
-0.180
-0.186
-0.171
-0.183
-0.184
-0.008
-0.063
0.029
-0.213
-0.297
-0.164
-0.173
-0.164
-0.196
0.071
-0.064
0.087
-0.076
-0.109
-0.103
-0.117
-0.123
-0.095
-0.090
-0.112
-0.091
-0.140
-0.160
-0.140
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.395
-0.396
-0.195
-0.158
-0.063
-0.050
-0.359
-0.350
-0.308
-0.308
-0.330
-0.290
-0.128
-0.206
-0.019
-0.025
-0.133
-0.170
-0.092
-0.117
-0.207
-0.197
-0.300
-0.278
-0.115
-0.196
-0.216
-0.192
-0.242
-0.214
-0.131
-0.136
-0.142
-0.134
-0.118
-0.103
-0.110
-0.084
-0.044
-0.081
-0.148
-0.149
-0.091
-0.119
-0.093
-0.102
-0.068
-0.073
-0.118
-0.127
-0.143
-0.140
-0.144
-0.138
-0.135
0.003
-0.221
-0.218
-0.067
-0.050
-0.087
-0.175
-0.234
-0.236
-0.186
-0.180
-0.060
-0.054
-0.181
-0.190
-0.174
-0.165
-0.060
-0.050
-0.292
-0.276
-0.169
-0.139
-0.055
-0.026
-0.108
-0.109
-0.124
-0.116
-0.111
-0.084
-0.164
-0.136
3J
LE NOE WAT
-0.407
-0.209
-0.088
-0.380
-0.318
-0.319
-0.243
-0.018
-0.162
-0.102
-0.208
-0.284
-0.135
-0.185
-0.250
-0.130
-0.150
-0.117
-0.102
-0.049
-0.153
-0.091
-0.093
-0.073
-0.117
-0.143
-0.146
-0.135
-0.222
-0.069
-0.124
-0.235
-0.205
-0.065
-0.182
-0.187
-0.055
-0.294
-0.162
-0.039
-0.106
-0.127
-0.099
-0.152
3J
LE NOE TAV WAT
-0.405
-0.212
-0.093
-0.353
-0.277
-0.264
-0.175
-0.017
-0.141
-0.099
-0.206
-0.310
-0.109
-0.188
-0.248
-0.134
-0.143
-0.117
-0.108
-0.042
-0.152
-0.091
-0.093
-0.072
-0.108
-0.141
-0.148
-0.139
-0.223
-0.063
-0.095
-0.233
-0.185
-0.059
-0.181
-0.187
-0.057
-0.296
-0.169
-0.055
-0.106
-0.126
-0.109
-0.160
79
RES1
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
41PHE
42ASP
42ASP
42ASP
42ASP
42ASP
42ASP
43GLU
43GLU
43GLU
43GLU
43GLU
43GLU
43GLU
43GLU
43GLU
43GLU
44ASP
44ASP
44ASP
45GLU
45GLU
45GLU
45GLU
45GLU
45GLU
46ILE
46ILE
46ILE
46ILE
bound
rexp
0.830
0.610
0.560
0.710
0.770
0.920
0.660
0.240
0.500
0.600
0.490
0.610
0.400
0.500
0.620
0.350
0.490
0.350
0.350
0.350
0.710
0.350
0.350
0.390
0.450
0.350
0.400
0.350
0.490
0.350
0.600
0.490
0.440
0.300
0.440
0.440
0.500
0.590
0.440
0.350
0.350
0.400
0.330
0.590
2.6 Supplementary material
NOE Nr
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
NOE
atom 1
CZ
CZ
CZ
CZ
CZ
CZ
CZ
H
H
H
H
H
H
HZ
HZ
H
H
H
H
H
H
HA
HA
HA
HA
H
H
H
H
H
H
HA
H
H
HA
HA
H
H
H
H
HA
HA
HA
HA
RES2
45GLU
46ILE
47PRO
46ILE
47PRO
47PRO
47PRO
48ALA
48ALA
50VAL
50VAL
49GLY
50VAL
50VAL
46ILE
50VAL
51ASP
52ALA
50VAL
50VAL
51ASP
51ASP
46ILE
55ILE
55ILE
55ILE
46ILE
50VAL
51ASP
51ASP
51ASP
52ALA
53VAL
54LYSH
53VAL
53VAL
52ALA
53VAL
53VAL
53VAL
54LYSH
53VAL
54LYSH
53VAL
atom 2
HA
QG1
QD
HB
HA
HB
HB
QB
QB
H
QG2
QA
HB
QG2
QD
QG1
HB
QB
HA
QG1
HB
HB
QD
QD
QG2
QG1
QD
QG1
HA
HB
HB
QB
H
H
QG1
QG2
QB
HB
QG1
QG2
H
QG1
QB
HB
3J
UNR VAC
UNR WAT
LE VAC
-0.132
-0.083
-0.137
-0.121
-0.145
-0.066
-0.224
-0.235
-0.227
-0.014
-0.161
-0.018
-0.059
-0.067
-0.027
-0.207
-0.203
-0.238
-0.153
-0.116
-0.186
-0.129
-0.132
-0.126
-0.071
-0.087
-0.065
0.069
0.005
0.068
-0.110
-0.033
-0.101
-0.170
-0.161
-0.183
-0.111
0.001
-0.101
-0.150
-0.175
-0.162
0.036
-0.160
-0.096
-0.054
0.050
-0.076
-0.071
-0.060
-0.089
-0.120
-0.126
-0.124
-0.087
-0.075
-0.087
-0.040
-0.026
-0.117
-0.157
-0.161
-0.174
-0.124
-0.116
-0.091
0.431
-0.167
0.356
-0.246
-0.220
-0.251
0.049
0.006
-0.025
-0.177
-0.216
-0.162
0.177
-0.132
0.141
0.015
0.197
0.027
-0.132
-0.140
-0.139
-0.109
-0.099
-0.147
-0.149
-0.098
-0.148
-0.134
-0.133
-0.127
-0.042
-0.062
-0.024
0.038
0.009
0.042
-0.158
-0.149
-0.164
-0.136
-0.163
-0.149
-0.175
-0.168
-0.211
-0.047
-0.065
-0.066
-0.189
-0.110
-0.145
-0.174
-0.153
-0.197
-0.003
-0.049
-0.010
-0.111
-0.130
-0.133
-0.140
-0.134
-0.133
-0.106
-0.134
-0.125
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.104
-0.117
-0.147
-0.113
-0.233
-0.232
-0.162
-0.220
-0.079
-0.096
-0.196
-0.115
-0.061
-0.016
-0.135
-0.122
-0.085
-0.097
0.007
-0.046
-0.111
-0.106
-0.160
-0.137
-0.114
-0.093
-0.149
-0.142
-0.206
-0.122
-0.069
-0.053
-0.091
-0.061
-0.121
-0.115
-0.085
-0.086
-0.111
-0.058
-0.176
-0.151
-0.097
-0.087
-0.157
-0.186
-0.238
-0.282
0.044
-0.030
-0.199
-0.244
-0.150
-0.153
0.050
-0.028
-0.133
-0.142
-0.156
-0.076
-0.184
-0.082
-0.132
-0.121
-0.060
-0.062
0.011
0.002
-0.152
-0.150
-0.171
-0.155
-0.171
-0.174
-0.082
-0.043
-0.102
-0.055
-0.155
-0.161
-0.043
-0.026
-0.139
-0.126
-0.131
-0.142
-0.106
-0.101
3J
LE NOE WAT
-0.126
-0.153
-0.237
-0.188
-0.081
-0.196
-0.059
-0.133
-0.128
-0.006
-0.100
-0.129
-0.099
-0.132
-0.212
-0.097
-0.064
-0.124
-0.077
-0.123
-0.178
-0.070
-0.153
-0.234
0.028
-0.209
-0.152
-0.016
-0.137
-0.087
-0.117
-0.130
-0.072
0.021
-0.174
-0.147
-0.149
-0.051
-0.090
-0.179
-0.053
-0.104
-0.131
-0.130
3J
LE NOE TAV WAT
-0.115
-0.154
-0.234
-0.158
-0.077
-0.195
-0.062
-0.133
-0.103
0.010
-0.102
-0.158
-0.102
-0.151
-0.200
-0.064
-0.103
-0.118
-0.082
-0.110
-0.168
-0.089
-0.166
-0.090
-0.246
-0.110
-0.134
0.061
-0.130
-0.175
-0.185
-0.134
-0.054
0.018
-0.153
-0.170
-0.174
-0.078
-0.132
-0.162
-0.039
-0.139
-0.132
-0.118
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
46ILE
46ILE
47PRO
47PRO
48ALA
48ALA
48ALA
48ALA
49GLY
49GLY
50VAL
50VAL
50VAL
50VAL
51ASP
51ASP
51ASP
51ASP
51ASP
51ASP
51ASP
51ASP
52ALA
52ALA
52ALA
52ALA
52ALA
52ALA
52ALA
52ALA
52ALA
52ALA
52ALA
52ALA
53VAL
53VAL
53VAL
53VAL
53VAL
53VAL
53VAL
54LYSH
54LYSH
54LYSH
bound
rexp
0.350
0.390
0.590
0.490
0.300
0.500
0.400
0.400
0.450
0.300
0.400
0.440
0.350
0.450
0.600
0.600
0.350
0.600
0.300
0.450
0.400
0.350
0.500
0.600
0.550
0.490
0.500
0.600
0.350
0.500
0.500
0.400
0.350
0.430
0.450
0.450
0.500
0.300
0.500
0.450
0.300
0.600
0.390
0.400
80
NOE Nr
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
NOE
atom 1
H
H
HA
QD
H
H
H
H
H
H
HA
H
H
H
HA
HA
HA
HA
H
H
H
H
HA
HA
HA
HA
H
H
H
H
H
H
H
H
HA
HA
H
H
H
H
H
HA
HA
H
RES2
54LYSH
54LYSH
55ILE
55ILE
55ILE
55ILE
72VAL
52ALA
54LYSH
55ILE
55ILE
55ILE
55ILE
39VAL
55ILE
56SER
56SER
40VAL
40VAL
53VAL
55ILE
55ILE
55ILE
55ILE
56SER
56SER
56SER
40VAL
52ALA
55ILE
57MET
57MET
58PRO
40VAL
56SER
56SER
57MET
57MET
57MET
57MET
40VAL
59GLU
58PRO
58PRO
atom 2
QB
QG
H
HB
QD
QG2
QG2
HA
QB
HB
QD
QG2
QG1
QG1
QG1
HB
HB
QG1
QG2
HA
H
QD
QG2
QG1
HB
HB
HG
QG1
QB
QG1
QB
QG
QD
QG2
HA
HB
QB
QG
QB
QE
QG2
QB
HA
HB
3J
UNR VAC UNR WAT
LE VAC
-0.113
-0.152
-0.116
-0.218
-0.146
-0.219
-0.032
-0.010
-0.028
-0.052
-0.053
-0.048
-0.038
-0.008
-0.065
-0.326
-0.324
-0.332
0.149
0.012
0.219
-0.032
-0.033
-0.011
-0.165
-0.178
-0.175
-0.146
-0.145
-0.158
-0.172
-0.114
-0.193
-0.135
-0.175
-0.126
-0.247
-0.254
-0.247
0.100
-0.049
0.070
-0.070
-0.117
-0.031
-0.035
-0.048
-0.038
-0.042
-0.009
-0.073
-0.018
-0.037
0.160
0.046
-0.034
0.228
-0.055
-0.078
-0.031
-0.014
0.009
-0.011
-0.094
-0.069
-0.105
-0.075
-0.083
-0.138
-0.052
-0.171
-0.075
-0.057
0.003
-0.076
-0.231
-0.259
-0.176
-0.069
-0.112
-0.041
-0.140
-0.088
0.193
-0.027
-0.047
0.084
-0.102
-0.156
-0.005
-0.256
-0.233
-0.235
-0.125
-0.191
-0.188
-0.114
-0.121
-0.110
-0.108
-0.189
0.136
-0.127
-0.119
-0.117
0.014
-0.014
-0.112
-0.186
-0.183
-0.186
-0.311
-0.299
-0.290
-0.199
-0.172
-0.166
-0.026
-0.063
-0.013
-0.157
0.134
-0.039
-0.181
-0.180
-0.192
-0.029
-0.014
-0.038
-0.170
-0.201
-0.128
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.112
-0.090
-0.200
-0.189
-0.013
-0.054
-0.042
-0.057
-0.018
-0.022
-0.334
-0.321
-0.066
-0.146
-0.037
-0.049
-0.190
-0.176
-0.152
-0.137
-0.143
-0.148
-0.139
-0.155
-0.267
-0.251
-0.055
-0.080
-0.089
-0.141
-0.036
-0.052
-0.069
-0.006
-0.063
-0.041
-0.022
-0.055
-0.086
-0.071
-0.015
-0.038
-0.164
-0.125
-0.085
-0.111
-0.165
-0.216
-0.114
0.008
-0.211
-0.248
-0.082
-0.045
-0.138
-0.169
-0.044
-0.127
-0.060
-0.100
-0.235
-0.235
-0.177
-0.151
-0.117
-0.103
-0.099
-0.141
-0.128
-0.136
-0.091
-0.006
-0.181
-0.186
-0.282
-0.250
-0.170
-0.159
-0.015
-0.096
-0.140
-0.158
-0.180
-0.196
-0.014
-0.038
-0.202
-0.124
3J
LE NOE WAT
-0.126
-0.200
-0.005
-0.044
-0.010
-0.330
-0.026
-0.038
-0.179
-0.155
-0.125
-0.147
-0.278
-0.032
-0.098
-0.039
-0.057
-0.112
-0.056
-0.074
-0.010
-0.068
-0.078
-0.148
-0.088
-0.234
-0.090
-0.132
-0.057
-0.120
-0.235
-0.176
-0.115
-0.160
-0.128
-0.079
-0.181
-0.283
-0.175
-0.061
-0.066
-0.215
-0.012
-0.198
3J
LE NOE TAV WAT
-0.128
-0.190
-0.017
-0.058
-0.169
-0.275
0.044
-0.039
-0.188
-0.258
-0.073
-0.196
-0.215
-0.017
-0.096
-0.037
-0.069
-0.161
-0.110
-0.069
-0.002
-0.002
-0.255
-0.038
-0.114
-0.202
-0.056
-0.136
-0.006
-0.002
-0.233
-0.184
-0.118
-0.200
-0.125
-0.089
-0.189
-0.255
-0.164
-0.055
-0.005
-0.182
-0.014
-0.206
81
RES1
54LYSH
54LYSH
54LYSH
55ILE
55ILE
55ILE
55ILE
55ILE
55ILE
55ILE
55ILE
55ILE
55ILE
56SER
56SER
56SER
56SER
56SER
56SER
56SER
56SER
56SER
56SER
56SER
56SER
56SER
56SER
56SER
56SER
56SER
57MET
57MET
57MET
57MET
57MET
57MET
57MET
57MET
58PRO
58PRO
59GLU
59GLU
59GLU
59GLU
bound
rexp
0.390
0.490
0.300
0.300
0.500
0.600
0.450
0.400
0.490
0.500
0.550
0.500
0.490
0.600
0.590
0.300
0.300
0.500
0.600
0.430
0.270
0.600
0.600
0.490
0.350
0.500
0.400
0.500
0.500
0.540
0.490
0.440
0.330
0.600
0.350
0.400
0.440
0.590
0.580
0.590
0.500
0.440
0.240
0.500
2.6 Supplementary material
NOE Nr
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
NOE
atom 1
H
H
H
HA
HA
HA
HA
H
H
H
H
H
H
HA
HA
HA
HA
HB
HB
H
H
H
H
H
H
H
H
HG
HG
HG
HA
HA
HA
H
H
H
H
H
QD
QD
HA
HA
H
H
RES2
58PRO
59GLU
59GLU
59GLU
59GLU
60GLU
60GLU
61GLU
61GLU
58PRO
58PRO
59GLU
60GLU
61GLU
61GLU
38ASN
57MET
62LEU
62LEU
61GLU
61GLU
61GLU
62LEU
62LEU
62LEU
63LEU
63LEU
63LEU
63LEU
37HISB
38ASN
62LEU
62LEU
62LEU
63LEU
63LEU
63LEU
63LEU
64ASN
63LEU
63LEU
63LEU
64ASN
65ALA
atom 2
HB
QB
QG
HA
H
QB
QG
H
QB
HB
HB
HA
HA
QB
QG
QB
QE
QB
QG
HA
QB
QG
QB
HG
QG
HB
HB
HG
QG
H
HA
HA
QB
QG
HB
HB
HG
QG
QB
HA
HB
QG
QB
H
3J
UNR VAC
UNR WAT
LE VAC
-0.058
-0.097
0.002
-0.087
-0.135
-0.088
-0.308
-0.312
-0.303
-0.006
-0.006
-0.008
-0.076
-0.126
-0.063
-0.200
-0.229
-0.214
-0.177
-0.178
-0.179
-0.033
-0.024
-0.019
-0.186
-0.184
-0.202
0.128
0.050
0.132
-0.104
-0.195
-0.082
-0.025
-0.028
-0.042
-0.034
-0.032
-0.053
-0.178
-0.164
-0.191
-0.142
-0.165
-0.207
-0.140
-0.070
-0.104
-0.025
-0.130
0.100
-0.189
-0.184
-0.189
-0.214
-0.214
-0.222
-0.056
-0.058
-0.058
-0.092
-0.089
-0.118
-0.159
-0.155
-0.192
-0.152
-0.145
-0.169
-0.161
-0.122
-0.164
-0.318
-0.284
-0.325
-0.159
-0.153
-0.160
-0.063
-0.073
-0.067
-0.114
-0.120
-0.119
-0.177
-0.175
-0.172
-0.043
-0.104
-0.073
-0.078
-0.084
-0.079
-0.131
-0.119
-0.136
-0.141
-0.171
-0.107
-0.219
-0.246
-0.195
-0.000
-0.041
-0.013
-0.262
-0.263
-0.265
-0.107
-0.088
-0.118
-0.315
-0.299
-0.328
-0.217
-0.216
-0.217
-0.056
-0.047
-0.056
0.017
0.032
0.022
-0.106
-0.094
-0.086
-0.175
-0.157
-0.182
-0.031
-0.077
-0.004
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.073
-0.011
-0.122
-0.088
-0.312
-0.308
-0.011
-0.043
-0.140
-0.131
-0.235
-0.246
-0.178
-0.064
-0.013
-0.052
-0.184
-0.189
0.038
0.030
-0.214
-0.179
-0.042
-0.033
-0.015
-0.015
-0.163
-0.143
-0.172
-0.201
-0.117
-0.146
-0.109
-0.154
-0.206
-0.209
-0.182
-0.175
-0.050
-0.045
-0.131
-0.183
-0.156
-0.139
-0.129
-0.092
-0.174
-0.099
-0.327
-0.329
-0.159
-0.167
-0.066
-0.063
-0.112
-0.092
-0.179
-0.172
-0.102
-0.109
-0.092
-0.079
-0.124
-0.131
-0.187
-0.185
-0.242
-0.223
-0.005
0.007
-0.264
-0.245
-0.082
-0.120
-0.323
-0.326
-0.209
-0.203
-0.049
-0.051
0.012
0.022
-0.097
-0.067
-0.170
-0.208
-0.078
-0.059
3J
LE NOE WAT
-0.074
-0.100
-0.285
-0.007
-0.144
-0.246
-0.198
-0.022
-0.185
0.024
-0.223
-0.045
-0.019
-0.153
-0.174
-0.102
-0.079
-0.190
-0.208
-0.055
-0.093
-0.109
-0.147
-0.149
-0.303
-0.158
-0.063
-0.127
-0.171
-0.110
-0.074
-0.129
-0.163
-0.237
-0.009
-0.265
-0.099
-0.317
-0.204
-0.052
0.017
-0.095
-0.201
-0.074
3J
LE NOE TAV WAT
-0.072
-0.118
-0.319
-0.009
-0.148
-0.229
-0.191
-0.019
-0.189
0.019
-0.205
-0.037
-0.015
-0.156
-0.194
-0.124
-0.114
-0.195
-0.203
-0.059
-0.097
-0.107
-0.140
-0.110
-0.309
-0.159
-0.068
-0.110
-0.181
-0.100
-0.093
-0.119
-0.175
-0.254
-0.006
-0.263
-0.085
-0.319
-0.200
-0.047
0.021
-0.092
-0.202
-0.066
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
59GLU
59GLU
59GLU
60GLU
60GLU
60GLU
60GLU
60GLU
61GLU
61GLU
61GLU
61GLU
61GLU
61GLU
61GLU
62LEU
62LEU
62LEU
62LEU
62LEU
62LEU
62LEU
62LEU
62LEU
62LEU
63LEU
63LEU
63LEU
63LEU
63LEU
63LEU
63LEU
63LEU
63LEU
63LEU
63LEU
63LEU
63LEU
64ASN
64ASN
64ASN
64ASN
64ASN
64ASN
bound
rexp
0.400
0.390
0.590
0.350
0.430
0.490
0.490
0.300
0.440
0.400
0.500
0.430
0.350
0.440
0.490
0.440
0.500
0.440
0.570
0.270
0.490
0.590
0.440
0.400
0.720
0.400
0.350
0.400
0.520
0.430
0.430
0.350
0.490
0.720
0.350
0.500
0.350
0.720
0.440
0.270
0.400
0.620
0.490
0.300
82
NOE Nr
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
NOE
atom 1
H
H
H
H
H
H
H
H
HA
H
H
H
H
H
H
HA
HA
HA
HA
H
H
H
H
H
H
HA
HA
HA
HA
H
H
H
H
H
H
H
H
H
HA
H
H
H
H
H
RES2
66PRO
31ASN
31ASN
64ASN
65ALA
31ASN
66PRO
65ALA
30LYSH
30LYSH
31ASN
66PRO
68GLU
68GLU
31ASN
65ALA
68GLU
68GLU
30LYSH
30LYSH
63LEU
69THR
69THR
68GLU
68GLU
69THR
69THR
70TYR
71VAL
63LEU
57MET
63LEU
29PHE
39VAL
39VAL
39VAL
70TYR
39VAL
39VAL
39VAL
56SER
57MET
70TYR
72VAL
atom 2
QD
HD21
HD22
QB
QB
QB
HB
QB
QD
QG
QB
HA
H
QG
QB
QB
QB
QG
HA
QB
QG
HB
QG2
HA
QB
HB
QG2
HB
QG2
QG
QG
QG
QB
HB
QG1
QG2
HA
HB
QG1
QG2
HA
QG
HA
HB
3J
UNR VAC UNR WAT
LE VAC
-0.115
-0.113
-0.113
-0.050
-0.052
-0.042
0.014
-0.033
0.027
-0.171
-0.104
-0.177
-0.189
-0.178
-0.189
-0.070
-0.072
-0.113
-0.119
-0.120
-0.116
-0.162
-0.161
-0.171
-0.183
-0.164
-0.193
-0.127
-0.120
-0.117
-0.088
-0.092
-0.127
-0.024
-0.029
-0.019
-0.046
-0.076
-0.039
-0.184
-0.185
-0.182
-0.097
-0.132
-0.106
0.022
-0.059
-0.002
-0.091
-0.133
-0.117
-0.228
-0.204
-0.218
-0.006
-0.006
0.007
-0.135
-0.135
-0.088
-0.031
-0.186
-0.185
-0.098
-0.098
-0.105
-0.164
-0.166
-0.098
-0.058
-0.054
-0.052
-0.041
-0.083
-0.086
-0.158
-0.141
-0.219
-0.182
-0.141
-0.194
-0.063
-0.062
-0.093
-0.145
-0.152
-0.148
-0.138
-0.268
-0.243
0.038
-0.065
-0.028
-0.362
-0.416
-0.393
-0.346
-0.348
-0.369
-0.114
-0.149
-0.206
-0.067
-0.078
-0.116
-0.310
-0.215
-0.290
-0.344
-0.330
-0.312
-0.124
-0.164
-0.295
-0.046
-0.107
-0.197
-0.341
-0.233
-0.341
-0.295
-0.219
-0.223
-0.364
-0.453
-0.189
-0.217
-0.199
-0.164
-0.001
-0.235
-0.171
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.110
-0.098
-0.110
-0.133
-0.072
-0.109
-0.148
-0.152
-0.181
-0.163
-0.034
-0.195
-0.117
-0.132
-0.170
-0.179
-0.164
-0.193
-0.133
-0.142
-0.092
-0.189
-0.028
-0.013
-0.073
-0.037
-0.189
-0.162
-0.150
-0.155
-0.052
-0.014
-0.132
-0.087
-0.197
-0.217
-0.000
-0.035
-0.115
-0.143
-0.148
-0.153
-0.088
-0.090
-0.176
-0.161
-0.057
-0.059
-0.075
-0.082
-0.154
-0.144
-0.124
-0.136
-0.076
-0.064
-0.168
-0.135
-0.250
-0.204
-0.104
-0.213
-0.357
-0.404
-0.361
-0.344
-0.132
-0.180
-0.069
-0.075
-0.200
-0.202
-0.316
-0.332
-0.184
-0.173
-0.129
-0.101
-0.251
-0.189
-0.298
-0.237
-0.414
-0.414
-0.172
-0.203
-0.233
-0.299
3J
LE NOE WAT
-0.110
-0.102
-0.070
-0.155
-0.182
-0.053
-0.117
-0.172
-0.171
-0.144
-0.106
-0.027
-0.070
-0.189
-0.162
-0.059
-0.131
-0.199
-0.005
-0.123
-0.146
-0.091
-0.175
-0.055
-0.075
-0.152
-0.133
-0.066
-0.178
-0.288
-0.088
-0.396
-0.348
-0.115
-0.027
-0.195
-0.319
-0.173
-0.089
-0.244
-0.247
-0.450
-0.186
-0.249
3J
LE NOE TAV WAT
-0.112
-0.041
-0.018
-0.191
-0.181
-0.079
-0.117
-0.171
-0.146
-0.105
-0.087
-0.028
-0.082
-0.190
-0.130
-0.060
-0.127
-0.202
-0.009
-0.135
-0.191
-0.099
-0.146
-0.059
-0.083
-0.187
-0.164
-0.086
-0.175
-0.289
-0.043
-0.379
-0.339
-0.102
-0.030
-0.204
-0.312
-0.173
-0.102
-0.273
-0.218
-0.394
-0.166
-0.227
83
RES1
65ALA
65ALA
65ALA
65ALA
65ALA
66PRO
66PRO
66PRO
67GLY
67GLY
67GLY
67GLY
67GLY
68GLU
68GLU
68GLU
68GLU
68GLU
69THR
69THR
69THR
69THR
69THR
69THR
69THR
69THR
69THR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
bound
rexp
0.330
0.500
0.400
0.490
0.450
0.490
0.400
0.540
0.680
0.530
0.490
0.240
0.350
0.440
0.490
0.500
0.390
0.490
0.270
0.590
0.620
0.350
0.450
0.270
0.440
0.500
0.500
0.350
0.600
0.620
0.590
0.720
0.800
0.610
0.710
0.710
0.610
0.610
0.710
0.810
0.710
0.800
0.710
0.610
2.6 Supplementary material
NOE Nr
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
NOE
atom 1
HA
H
H
H
H
HA
HA
QD
QA
QA
H
H
H
HA
H
H
H
H
HA
HA
HA
HA
HA
H
H
H
H
HA
HA
HB
HB
HB
CG
CG
CG
CG
CG
CZ
CZ
CZ
CZ
CZ
CZ
CZ
RES2
72VAL
29PHE
29PHE
63LEU
69THR
69THR
69THR
70TYR
70TYR
28VAL
28VAL
28VAL
71VAL
71VAL
70TYR
70TYR
71VAL
71VAL
71VAL
71VAL
72VAL
28VAL
71VAL
71VAL
71VAL
72VAL
72VAL
72VAL
72VAL
73THR
73THR
72VAL
72VAL
73THR
73THR
74LEU
74LEU
73THR
73THR
74LEU
74LEU
74LEU
23SER
75ASP
atom 2
QG1
QB
CG
QG
HA
HB
QG2
HB
HB
HA
QG1
QG2
QG1
QG2
HA
CG
HB
QG1
QG2
QG1
QG2
HA
HA
QG1
QG2
HB
QG1
QG1
QG2
HB
QG2
HA
QG2
HB
QG2
HB
QG
HA
QG2
HB
HG
QG
HA
QB
3J
UNR VAC
UNR WAT
LE VAC
-0.226
-0.183
-0.213
-0.213
-0.230
-0.228
-0.165
-0.187
-0.182
-0.216
-0.326
-0.280
-0.025
-0.021
-0.027
0.007
-0.043
0.040
-0.021
-0.050
-0.055
-0.248
-0.252
-0.261
-0.100
-0.059
-0.043
-0.137
-0.136
-0.120
-0.162
-0.134
-0.124
-0.162
-0.173
-0.158
-0.160
-0.157
-0.161
-0.153
-0.153
-0.157
-0.056
-0.055
-0.053
-0.333
-0.350
-0.336
-0.104
-0.102
-0.109
-0.160
-0.156
-0.165
-0.148
-0.129
-0.147
-0.125
-0.131
-0.135
-0.177
-0.212
-0.194
-0.076
-0.061
-0.057
-0.026
-0.027
-0.024
-0.076
-0.086
-0.093
-0.099
-0.104
-0.116
-0.121
-0.128
-0.119
-0.113
-0.177
-0.170
-0.031
-0.088
-0.039
-0.014
-0.065
-0.049
-0.148
-0.141
-0.115
-0.166
-0.169
-0.178
-0.027
-0.025
-0.024
0.039
-0.092
-0.043
-0.088
-0.100
-0.125
-0.269
-0.233
-0.173
-0.100
-0.096
-0.107
-0.187
-0.180
-0.173
-0.017
-0.021
-0.017
0.022
-0.031
-0.036
-0.051
-0.056
-0.049
-0.077
-0.046
-0.122
-0.291
-0.271
-0.325
-0.089
-0.136
-0.063
-0.252
-0.260
-0.236
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.255
-0.189
-0.236
-0.225
-0.193
-0.192
-0.286
-0.287
-0.017
-0.026
-0.026
-0.048
-0.065
-0.050
-0.260
-0.242
-0.039
-0.040
-0.138
-0.144
-0.147
-0.109
-0.154
-0.155
-0.167
-0.161
-0.145
-0.149
-0.054
-0.051
-0.340
-0.358
-0.089
-0.084
-0.156
-0.145
-0.143
-0.128
-0.121
-0.140
-0.201
-0.210
-0.065
-0.050
-0.024
-0.022
-0.092
-0.141
-0.097
-0.119
-0.116
-0.071
-0.182
-0.171
-0.046
-0.093
-0.069
-0.103
-0.117
-0.110
-0.174
-0.159
-0.025
-0.017
-0.064
-0.174
-0.119
-0.126
-0.176
-0.140
-0.110
-0.114
-0.177
-0.181
-0.018
-0.025
-0.071
-0.091
-0.039
-0.030
-0.079
-0.075
-0.290
-0.323
-0.140
-0.077
-0.249
-0.243
3J
LE NOE WAT
-0.257
-0.219
-0.179
-0.302
-0.019
-0.023
-0.060
-0.263
-0.043
-0.134
-0.139
-0.141
-0.169
-0.140
-0.052
-0.366
-0.085
-0.153
-0.139
-0.125
-0.211
-0.058
-0.024
-0.111
-0.102
-0.119
-0.187
-0.073
-0.090
-0.114
-0.176
-0.023
-0.083
-0.115
-0.159
-0.110
-0.177
-0.016
-0.077
-0.038
-0.083
-0.291
-0.130
-0.236
3J
LE NOE TAV WAT
-0.239
-0.211
-0.168
-0.314
-0.021
-0.004
-0.057
-0.254
-0.029
-0.138
-0.148
-0.155
-0.167
-0.145
-0.051
-0.360
-0.094
-0.154
-0.139
-0.125
-0.210
-0.066
-0.023
-0.103
-0.101
-0.122
-0.173
-0.092
-0.083
-0.185
-0.138
-0.020
-0.080
-0.073
-0.291
-0.109
-0.174
-0.023
0.005
-0.038
-0.064
-0.279
-0.118
-0.234
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
70TYR
71VAL
71VAL
71VAL
71VAL
71VAL
71VAL
71VAL
71VAL
71VAL
71VAL
72VAL
72VAL
72VAL
72VAL
72VAL
72VAL
72VAL
72VAL
73THR
73THR
73THR
73THR
73THR
73THR
73THR
73THR
74LEU
74LEU
74LEU
74LEU
74LEU
74LEU
74LEU
75ASP
75ASP
bound
rexp
0.660
0.590
0.710
0.720
0.240
0.350
0.500
0.500
0.350
0.400
0.600
0.600
0.450
0.450
0.270
0.710
0.350
0.600
0.450
0.600
0.500
0.430
0.240
0.450
0.600
0.400
0.500
0.600
0.600
0.400
0.450
0.240
0.450
0.400
0.600
0.350
0.520
0.240
0.450
0.400
0.350
0.720
0.430
0.490
84
NOE Nr
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
NOE
atom 1
CZ
H
H
H
H
H
H
H
H
HA
HA
HA
HA
HA
H
H
H
H
H
HA
HA
H
H
H
H
H
H
HA
HA
HA
HA
H
H
H
H
HA
HA
H
H
H
H
H
HA
HA
RES2
98VAL
74LEU
74LEU
74LEU
74LEU
76THR
50VAL
75ASP
76THR
75ASP
76THR
76THR
98VAL
77LYSH
77LYSH
98VAL
98VAL
76THR
76THR
77LYSH
77LYSH
77LYSH
77LYSH
77LYSH
80TYR
97THR
98VAL
79THR
79THR
97THR
97THR
47PRO
78GLY
79THR
46ILE
46ILE
46ILE
47PRO
79THR
80TYR
80TYR
74LEU
96VAL
96VAL
atom 2
QB
HA
HB
HB
QG
H
QG2
QB
QG2
QB
HB
QG2
QB
QB
QG
HB
QB
HA
QG2
QB
QG
HA
QB
QG
CZ
QG2
HB
HB
QG2
HA
QG2
QG
QA
QG2
HA
HB
QG2
QD
QG2
HB
HB
QG
QG1
QG1
3J
UNR VAC UNR WAT
LE VAC
-0.142
-0.235
-0.111
-0.030
-0.030
-0.024
-0.111
-0.168
-0.106
0.045
0.027
0.040
-0.094
-0.134
-0.065
-0.057
-0.124
-0.034
-0.030
0.072
-0.059
-0.105
-0.101
-0.099
-0.170
-0.168
-0.173
-0.114
-0.133
-0.116
-0.080
-0.095
-0.118
-0.227
-0.273
-0.149
-0.292
-0.199
-0.277
-0.185
-0.181
-0.188
-0.228
-0.231
-0.228
0.027
-0.028
0.021
-0.096
-0.160
-0.109
-0.003
-0.019
-0.006
-0.149
-0.091
-0.196
-0.153
-0.169
-0.151
-0.115
-0.145
-0.123
-0.051
-0.053
-0.051
-0.113
-0.140
-0.121
-0.195
-0.186
-0.161
-0.082
-0.120
-0.024
0.060
-0.030
-0.014
-0.072
-0.104
-0.083
-0.148
-0.142
-0.169
-0.164
-0.172
-0.151
-0.023
-0.038
-0.023
-0.189
-0.168
-0.157
-0.107
-0.191
-0.177
-0.200
-0.197
-0.193
-0.155
-0.141
-0.186
0.468
0.015
0.456
0.471
-0.091
0.437
0.561
-0.112
0.567
0.074
-0.164
0.074
0.016
0.000
-0.072
-0.159
-0.145
-0.157
-0.112
-0.114
-0.115
-0.216
-0.286
-0.156
-0.043
0.012
-0.039
-0.242
-0.169
-0.235
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.272
-0.171
-0.028
-0.029
-0.195
-0.146
0.003
0.020
-0.129
-0.088
-0.131
-0.114
0.046
-0.054
-0.115
-0.123
-0.163
-0.162
-0.171
-0.187
-0.147
-0.132
-0.197
-0.166
-0.205
-0.166
-0.182
-0.198
-0.235
-0.211
-0.021
-0.057
-0.150
-0.204
-0.025
-0.032
-0.154
-0.140
-0.159
-0.156
-0.145
-0.157
-0.063
-0.044
-0.118
-0.183
-0.161
-0.215
-0.105
-0.114
-0.060
-0.076
-0.102
-0.088
-0.186
-0.111
-0.131
-0.163
-0.042
-0.062
-0.184
-0.178
-0.184
-0.048
-0.193
-0.204
-0.198
-0.045
0.015
-0.049
-0.027
-0.091
-0.070
-0.126
-0.153
-0.112
0.021
-0.089
-0.155
-0.154
-0.114
-0.110
-0.269
-0.223
0.023
-0.025
-0.183
-0.233
3J
LE NOE WAT
-0.217
-0.030
-0.200
-0.001
-0.134
-0.117
-0.024
-0.114
-0.155
-0.198
-0.156
-0.239
-0.153
-0.181
-0.231
-0.043
-0.177
-0.024
-0.156
-0.160
-0.161
-0.052
-0.167
-0.184
-0.131
-0.080
-0.090
-0.161
-0.147
-0.040
-0.188
-0.137
-0.195
-0.164
-0.012
-0.037
-0.078
-0.143
-0.082
-0.157
-0.112
-0.244
-0.048
-0.238
3J
LE NOE TAV WAT
-0.220
-0.024
-0.205
-0.005
-0.136
-0.126
0.064
-0.145
-0.151
-0.192
-0.157
-0.269
-0.181
-0.181
-0.233
-0.031
-0.163
-0.025
-0.142
-0.186
-0.124
-0.055
-0.134
-0.247
-0.119
-0.048
-0.105
-0.159
-0.160
-0.038
-0.169
-0.192
-0.195
-0.162
-0.001
-0.044
-0.080
-0.168
-0.060
-0.154
-0.113
-0.245
-0.010
-0.204
85
RES1
75ASP
75ASP
75ASP
75ASP
75ASP
75ASP
76THR
76THR
76THR
76THR
76THR
76THR
76THR
77LYSH
77LYSH
77LYSH
77LYSH
77LYSH
77LYSH
77LYSH
77LYSH
78GLY
78GLY
78GLY
78GLY
78GLY
78GLY
79THR
79THR
79THR
79THR
79THR
79THR
79THR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
bound
rexp
0.720
0.240
0.500
0.400
0.620
0.350
0.600
0.590
0.450
0.490
0.400
0.600
0.720
0.440
0.490
0.300
0.570
0.240
0.500
0.440
0.440
0.270
0.490
0.590
0.710
0.600
0.350
0.400
0.450
0.300
0.600
0.490
0.440
0.500
0.300
0.500
0.450
0.490
0.600
0.400
0.400
0.720
0.400
0.500
2.6 Supplementary material
NOE Nr
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
NOE
atom 1
HA
H
H
H
H
H
HA
HA
HA
H
H
H
H
HA
HA
HA
HA
H
H
H
H
H
H
H
H
H
H
HA
HA
HA
HA
H
H
H
HA
HA
HA
HA
HA
HA
HA
HB
HB
HB
RES2
46ILE
46ILE
47PRO
55ILE
74LEU
79THR
80TYR
96VAL
97THR
98VAL
46ILE
47PRO
47PRO
50VAL
50VAL
50VAL
55ILE
55ILE
74LEU
74LEU
74LEU
76THR
77LYSH
98VAL
98VAL
79THR
79THR
79THR
79THR
80TYR
80TYR
80TYR
96VAL
96VAL
97THR
47PRO
47PRO
47PRO
50VAL
55ILE
74LEU
75ASP
76THR
76THR
atom 2
HB
QG2
QD
QG2
QG
HA
HA
QG1
HA
QB
HB
QD
QG
HB
QG1
QG2
QD
QG2
HB
HB
QG
HB
HA
HB
QB
HA
HB
H
QG2
HB
HB
CG
H
QG1
HA
HB
QD
QG
QG2
QG2
QG
H
HB
H
3J
UNR VAC
UNR WAT
LE VAC
0.280
-0.270
0.187
0.280
-0.264
0.265
-0.269
-0.411
-0.319
0.053
0.009
0.268
-0.491
-0.524
-0.430
-0.227
-0.203
-0.220
-0.335
-0.349
-0.337
-0.270
-0.161
-0.258
-0.086
-0.067
-0.053
-0.251
-0.281
-0.251
0.173
-0.305
0.042
-0.364
-0.400
-0.431
-0.474
-0.390
-0.414
-0.277
-0.080
-0.116
-0.077
0.048
0.107
-0.150
-0.405
0.029
-0.391
-0.331
-0.349
-0.079
-0.081
0.037
-0.213
-0.203
-0.210
-0.074
-0.055
-0.056
-0.418
-0.377
-0.359
-0.294
-0.311
-0.321
-0.029
-0.101
0.004
-0.043
-0.015
0.000
-0.173
-0.137
-0.160
-0.022
-0.026
-0.023
-0.176
-0.135
-0.159
0.008
-0.007
0.005
-0.045
-0.068
-0.118
-0.037
-0.055
-0.038
-0.247
-0.274
-0.246
-0.219
-0.219
-0.213
-0.077
-0.086
-0.087
-0.194
-0.115
-0.189
-0.087
-0.078
-0.074
-0.045
0.051
0.022
-0.189
-0.125
-0.224
-0.157
-0.036
-0.036
0.005
-0.186
0.181
-0.049
-0.007
0.007
-0.216
-0.133
-0.268
0.090
0.135
-0.010
0.055
0.016
-0.061
0.159
0.133
0.054
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.236
-0.310
-0.251
-0.310
-0.450
-0.474
-0.059
-0.107
-0.523
-0.487
-0.221
-0.226
-0.338
-0.330
-0.180
-0.269
-0.086
-0.111
-0.250
-0.276
-0.253
-0.250
-0.441
-0.468
-0.372
-0.474
-0.178
-0.267
0.084
-0.129
-0.176
-0.364
-0.341
-0.306
-0.199
-0.205
-0.331
-0.373
-0.171
-0.225
-0.422
-0.427
-0.349
-0.321
-0.053
-0.148
-0.039
-0.121
-0.185
-0.275
-0.024
-0.032
-0.183
-0.090
0.005
-0.002
-0.017
-0.149
-0.042
-0.036
-0.253
-0.249
-0.219
-0.215
-0.086
-0.117
-0.127
-0.181
-0.088
-0.092
0.042
0.092
-0.178
-0.131
-0.006
-0.087
-0.033
-0.190
-0.141
-0.183
-0.217
-0.286
-0.189
-0.092
-0.039
-0.062
-0.074
-0.053
3J
LE NOE WAT
-0.227
-0.255
-0.457
-0.101
-0.502
-0.240
-0.333
-0.270
-0.109
-0.258
-0.195
-0.433
-0.428
-0.241
-0.146
-0.413
-0.316
-0.187
-0.339
-0.186
-0.422
-0.284
-0.152
-0.097
-0.238
-0.028
-0.158
0.007
-0.104
-0.045
-0.260
-0.221
-0.109
-0.192
-0.082
0.012
-0.130
-0.117
-0.136
0.018
-0.208
-0.104
-0.086
-0.079
3J
LE NOE TAV WAT
-0.254
-0.265
-0.457
-0.055
-0.512
-0.224
-0.336
-0.216
-0.085
-0.256
-0.256
-0.430
-0.383
-0.202
0.091
-0.217
-0.340
-0.129
-0.316
-0.181
-0.423
-0.302
-0.106
-0.071
-0.197
-0.027
-0.141
-0.002
-0.107
-0.047
-0.260
-0.232
-0.094
-0.143
-0.076
-0.008
-0.137
-0.034
-0.043
-0.064
-0.214
-0.025
-0.057
-0.039
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
80TYR
bound
rexp
0.710
0.710
0.800
0.760
0.930
0.710
0.610
0.710
0.710
0.830
0.710
0.800
0.800
0.710
0.610
0.810
0.810
0.760
0.710
0.710
0.830
0.710
0.710
0.560
0.730
0.240
0.500
0.430
0.500
0.400
0.500
0.560
0.430
0.600
0.430
0.500
0.590
0.490
0.500
0.650
0.720
0.500
0.350
0.400
86
NOE Nr
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
NOE
atom 1
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
H
H
H
H
H
H
H
H
H
H
HH
HH
HH
HH
HH
HH
HH
HH
HH
RES2
76THR
77LYSH
77LYSH
98VAL
98VAL
81SER
95LYSH
95LYSH
95LYSH
80TYR
80TYR
81SER
41PHE
41PHE
42ASP
82PHE
82PHE
39VAL
39VAL
39VAL
41PHE
80TYR
80TYR
83TYR
94GLY
95LYSH
96VAL
96VAL
96VAL
14PHE
39VAL
94GLY
95LYSH
96VAL
96VAL
96VAL
96VAL
81SER
82PHE
82PHE
95LYSH
3VAL
29PHE
29PHE
atom 2
QG2
HA
H
HB
QB
QB
HA
QB
QG
HA
HB
QB
HA
QB
HB
HB
HB
HA
QG1
QG2
HA
HB
HB
HA
QA
HA
HB
QG1
QG2
HB
QG1
QA
HA
HA
HB
QG1
QG2
HA
HB
CG
HA
QG1
CZ
HZ
3J
UNR VAC UNR WAT
LE VAC
-0.120
-0.046
-0.171
0.246
0.207
0.116
0.094
0.138
0.077
0.149
0.200
0.054
0.064
0.131
-0.065
-0.196
-0.182
-0.201
-0.054
-0.040
-0.048
-0.046
-0.053
-0.049
-0.116
-0.146
-0.138
-0.021
-0.036
-0.020
-0.082
-0.064
-0.092
-0.172
-0.180
-0.161
0.003
0.002
0.022
-0.111
-0.114
-0.090
0.088
-0.023
0.077
-0.117
-0.118
-0.117
-0.115
-0.109
-0.111
-0.218
-0.162
-0.165
-0.291
-0.308
-0.302
-0.189
-0.195
-0.236
-0.187
-0.199
-0.186
-0.103
-0.134
-0.142
-0.081
-0.052
-0.106
-0.100
-0.118
-0.110
-0.363
-0.346
-0.366
-0.239
-0.230
-0.253
-0.214
-0.084
-0.192
-0.157
-0.080
-0.181
-0.094
-0.277
-0.180
-0.237
-0.205
-0.250
-0.284
-0.272
-0.244
-0.451
-0.445
-0.448
-0.213
-0.231
-0.230
-0.135
-0.136
-0.152
-0.127
-0.054
-0.109
-0.107
-0.226
-0.160
-0.056
-0.215
-0.111
-0.025
-0.022
-0.027
-0.213
-0.184
-0.210
-0.431
-0.419
-0.429
-0.124
-0.104
-0.128
-0.206
-0.145
-0.238
-0.137
-0.209
-0.157
-0.079
-0.166
-0.111
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.033
0.015
0.225
0.118
0.186
0.123
0.151
0.048
0.041
-0.085
-0.182
-0.188
-0.044
-0.064
-0.052
-0.110
-0.157
-0.172
-0.027
-0.022
-0.079
-0.081
-0.157
-0.134
0.004
-0.007
-0.106
-0.110
-0.036
-0.056
-0.120
-0.116
-0.106
-0.108
-0.140
-0.154
-0.313
-0.327
-0.180
-0.184
-0.219
-0.239
-0.146
-0.131
-0.094
-0.098
-0.099
-0.131
-0.352
-0.311
-0.222
-0.291
-0.093
-0.165
-0.098
-0.141
-0.212
-0.053
-0.213
-0.241
-0.277
-0.268
-0.434
-0.438
-0.224
-0.300
-0.126
-0.154
-0.076
-0.144
-0.256
-0.142
-0.213
-0.061
-0.018
-0.018
-0.180
-0.167
-0.424
-0.404
-0.097
-0.140
-0.188
-0.142
-0.180
-0.123
-0.129
-0.057
3J
LE NOE WAT
-0.090
-0.074
-0.032
-0.091
-0.173
-0.182
-0.036
-0.039
-0.173
-0.026
-0.086
-0.177
-0.003
-0.117
-0.048
-0.118
-0.108
-0.141
-0.313
-0.164
-0.212
-0.108
-0.080
-0.112
-0.346
-0.245
-0.151
-0.161
-0.060
-0.217
-0.288
-0.438
-0.242
-0.139
-0.116
-0.162
-0.083
-0.020
-0.180
-0.416
-0.111
-0.151
-0.168
-0.122
3J
LE NOE TAV WAT
-0.105
-0.018
0.013
-0.054
-0.117
-0.184
-0.034
-0.036
-0.169
-0.030
-0.082
-0.184
0.006
-0.103
-0.035
-0.116
-0.112
-0.157
-0.299
-0.188
-0.194
-0.111
-0.076
-0.121
-0.347
-0.248
-0.148
-0.146
-0.273
-0.226
-0.266
-0.434
-0.235
-0.149
-0.066
-0.184
-0.181
-0.019
-0.191
-0.414
-0.113
-0.247
-0.162
-0.108
87
RES1
80TYR
80TYR
80TYR
80TYR
80TYR
81SER
81SER
81SER
81SER
81SER
81SER
81SER
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
82PHE
bound
rexp
0.500
0.500
0.500
0.500
0.620
0.440
0.300
0.490
0.490
0.240
0.500
0.440
0.240
0.590
0.500
0.350
0.350
0.710
0.710
0.810
0.710
0.710
0.710
0.610
0.800
0.710
0.710
0.660
0.810
0.710
0.810
0.800
0.710
0.710
0.560
0.660
0.710
0.240
0.500
0.710
0.430
0.600
0.610
0.500
2.6 Supplementary material
NOE Nr
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
NOE
atom 1
HH
HH
HH
HH
HH
HA
HA
HA
HA
H
H
H
HA
HA
HA
HA
HA
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CG
CZ
CZ
CZ
CZ
CZ
CZ
CZ
CZ
H
H
H
H
HZ
HZ
HZ
RES2
94GLY
96VAL
96VAL
83TYR
93VAL
93VAL
93VAL
93VAL
93VAL
40VAL
40VAL
40VAL
83TYR
85SER
85SER
88GLN
88GLN
88GLN
40VAL
40VAL
40VAL
42ASP
42ASP
85SER
85SER
39VAL
40VAL
82PHE
82PHE
82PHE
82PHE
83TYR
83TYR
39VAL
40VAL
84CYS
84CYS
85SER
92MET
92MET
92MET
92MET
14PHE
14PHE
atom 2
QA
HB
QG1
QB
HA
HB
QG1
QG2
QG1
HB
QG1
QG2
HA
HA
QB
HA
QB
QG
HB
QG1
QG2
HB
HB
HA
QB
QG1
H
HA
HB
HB
CG
QB
CG
HA
QG2
HB
HB
HA
QG
QE
QB
QG
CG
CZ
3J
UNR VAC
UNR WAT
LE VAC
-0.134
-0.128
-0.124
-0.050
-0.006
-0.038
0.005
-0.215
-0.090
-0.218
-0.218
-0.217
0.001
-0.008
0.039
-0.127
-0.061
-0.022
-0.213
-0.194
-0.120
-0.010
0.011
-0.039
-0.213
-0.208
-0.122
-0.344
-0.305
-0.318
-0.278
-0.271
-0.266
-0.241
-0.340
-0.249
-0.266
-0.266
-0.271
-0.119
-0.115
-0.074
-0.197
-0.217
-0.191
-0.077
-0.092
-0.071
0.036
0.022
0.011
-0.286
-0.301
-0.266
-0.250
-0.224
-0.241
-0.310
-0.315
-0.280
-0.157
-0.255
-0.156
-0.019
0.020
-0.046
-0.207
-0.233
-0.246
-0.071
-0.107
-0.007
-0.235
-0.302
-0.215
-0.031
-0.081
-0.108
-0.049
-0.032
-0.040
-0.060
-0.044
-0.060
0.057
0.025
0.041
-0.083
-0.135
-0.106
-0.247
-0.284
-0.262
-0.263
-0.251
-0.253
-0.426
-0.402
-0.422
0.022
-0.023
-0.016
-0.140
-0.173
-0.200
-0.117
-0.115
-0.173
-0.172
-0.164
-0.166
-0.018
-0.020
-0.014
-0.114
-0.183
-0.161
0.210
0.020
-0.096
-0.098
-0.226
-0.164
-0.032
-0.183
-0.125
-0.070
-0.074
-0.064
-0.207
-0.167
-0.197
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.109
-0.130
-0.032
-0.090
-0.268
-0.042
-0.218
-0.214
0.009
-0.053
-0.066
-0.119
-0.158
-0.180
-0.147
-0.028
-0.163
-0.219
-0.341
-0.353
-0.281
-0.280
-0.256
-0.240
-0.267
-0.267
-0.073
-0.152
-0.162
-0.185
-0.027
-0.210
0.099
-0.241
-0.240
-0.347
-0.272
-0.302
-0.325
-0.358
-0.202
-0.216
0.017
0.033
-0.236
-0.215
-0.065
-0.127
-0.244
-0.247
-0.088
-0.154
-0.018
-0.068
-0.051
-0.036
0.021
-0.002
-0.123
-0.176
-0.267
-0.326
-0.256
-0.233
-0.410
-0.391
-0.023
-0.069
-0.208
-0.101
-0.168
-0.118
-0.163
-0.177
-0.021
0.002
-0.169
-0.281
0.011
-0.051
-0.162
-0.138
-0.098
-0.215
-0.034
-0.120
-0.154
-0.217
3J
LE NOE WAT
-0.123
-0.063
-0.064
-0.219
-0.003
-0.079
-0.200
-0.068
-0.185
-0.333
-0.265
-0.243
-0.268
-0.112
-0.180
-0.160
-0.123
-0.359
-0.275
-0.313
-0.194
0.000
-0.244
-0.112
-0.266
-0.091
-0.024
-0.047
0.019
-0.139
-0.283
-0.251
-0.407
-0.044
-0.203
-0.165
-0.164
-0.021
-0.175
0.019
-0.148
-0.115
-0.056
-0.172
3J
LE NOE TAV WAT
-0.114
0.006
-0.149
-0.219
0.017
-0.051
-0.117
-0.079
-0.114
-0.340
-0.284
-0.257
-0.267
-0.118
-0.171
-0.302
-0.070
-0.312
-0.271
-0.317
-0.199
0.016
-0.235
-0.124
-0.260
-0.074
-0.021
-0.052
0.031
-0.125
-0.276
-0.255
-0.405
-0.029
-0.212
-0.168
-0.168
-0.027
-0.151
-0.010
-0.186
-0.073
-0.048
-0.190
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
82PHE
82PHE
82PHE
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
83TYR
84CYS
84CYS
84CYS
84CYS
84CYS
84CYS
84CYS
84CYS
84CYS
84CYS
84CYS
bound
rexp
0.490
0.500
0.600
0.440
0.270
0.500
0.600
0.600
0.540
0.710
0.810
0.810
0.610
0.610
0.800
0.710
0.650
0.800
0.610
0.710
0.710
0.610
0.710
0.610
0.800
0.600
0.350
0.270
0.350
0.400
0.710
0.590
0.710
0.400
0.600
0.400
0.400
0.430
0.590
0.500
0.590
0.590
0.710
0.610
88
NOE Nr
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
NOE
atom 1
HZ
HZ
HZ
HA
HA
HA
HA
HA
QB
CG
CG
CG
CG
CG
CG
CG
CG
CG
CZ
CZ
CZ
CZ
CZ
CZ
CZ
H
H
H
H
H
H
H
H
HA
HA
HA
HA
HA
HB
HB
HB
HB
H
H
RES2
83TYR
83TYR
83TYR
84CYS
84CYS
92MET
85SER
40VAL
86PRO
38ASN
40VAL
83TYR
84CYS
85SER
86PRO
85SER
12LEU
87HISB
87HISB
90ALA
84CYS
90ALA
92MET
12LEU
12LEU
90ALA
92MET
86PRO
90ALA
12LEU
37HISB
37HISB
86PRO
86PRO
87HISB
88GLN
83TYR
87HISB
88GLN
88GLN
83TYR
88GLN
88GLN
88GLN
atom 2
HA
QB
CG
HB
HB
QB
QB
QG2
QD
HD21
QG2
CG
HA
QB
QD
QB
QG
HB
HB
QB
HB
QB
QB
HB
QG
QB
QB
HB
QB
QG
HA
HE1
HB
HB
HB
QB
CG
H
QB
QG
CZ
QG
QB
H
3J
UNR VAC UNR WAT
LE VAC
-0.072
-0.061
-0.060
-0.143
-0.135
-0.148
-0.304
-0.316
-0.311
-0.165
-0.164
-0.140
-0.126
-0.143
-0.085
-0.137
-0.207
-0.268
-0.209
-0.214
-0.207
-0.110
-0.135
-0.105
-0.192
-0.173
-0.210
-0.224
-0.210
-0.185
-0.224
-0.240
-0.272
-0.173
-0.168
-0.183
-0.049
-0.050
-0.051
-0.188
-0.180
-0.196
-0.103
-0.127
-0.108
-0.292
-0.273
-0.310
-0.139
-0.064
0.024
-0.113
-0.109
-0.111
-0.112
-0.110
-0.129
0.488
-0.074
0.085
0.045
0.041
0.069
0.262
-0.252
-0.177
-0.103
-0.196
-0.002
0.126
0.022
0.061
-0.145
-0.022
-0.068
0.538
0.014
0.080
-0.072
-0.249
-0.026
0.077
-0.032
0.093
0.491
-0.075
0.039
-0.078
-0.185
-0.088
0.021
-0.202
0.030
0.203
-0.081
-0.025
-0.001
-0.051
-0.034
-0.143
-0.257
-0.142
-0.228
-0.255
-0.194
-0.133
-0.131
-0.135
0.124
0.035
0.059
0.014
-0.032
-0.072
-0.098
-0.125
-0.136
-0.130
-0.128
-0.134
-0.208
-0.271
-0.135
-0.325
-0.320
-0.332
-0.161
-0.188
-0.199
0.006
-0.171
-0.082
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.056
-0.072
-0.142
-0.141
-0.331
-0.313
-0.160
-0.169
-0.104
-0.100
-0.179
-0.169
-0.213
-0.205
-0.112
-0.117
-0.173
-0.246
-0.210
-0.179
-0.244
-0.228
-0.136
-0.218
-0.046
-0.052
-0.181
-0.182
-0.136
-0.082
-0.273
-0.346
0.008
-0.166
-0.112
-0.109
-0.111
-0.111
-0.025
-0.126
-0.028
0.093
-0.194
-0.262
-0.184
-0.227
0.014
0.004
0.043
-0.142
0.077
-0.005
-0.221
-0.250
0.048
0.058
0.014
-0.101
-0.127
-0.189
-0.243
-0.048
-0.066
-0.011
-0.033
0.013
-0.246
-0.151
-0.252
-0.224
-0.132
-0.159
0.111
-0.091
-0.017
-0.136
-0.127
-0.129
-0.128
-0.125
-0.196
-0.306
-0.323
-0.321
-0.171
-0.172
-0.136
-0.163
3J
LE NOE WAT
-0.052
-0.134
-0.317
-0.169
-0.104
-0.167
-0.214
-0.144
-0.175
-0.205
-0.266
-0.148
-0.042
-0.185
-0.122
-0.275
-0.153
-0.113
-0.110
-0.132
-0.013
-0.270
-0.202
0.025
-0.115
-0.014
-0.243
-0.040
-0.078
-0.171
-0.234
-0.078
-0.015
-0.226
-0.244
-0.134
-0.075
-0.066
-0.141
-0.130
-0.305
-0.326
-0.198
-0.145
3J
LE NOE TAV WAT
-0.057
-0.133
-0.322
-0.167
-0.100
-0.245
-0.209
-0.157
-0.196
-0.201
-0.270
-0.151
-0.047
-0.189
-0.111
-0.296
-0.037
-0.108
-0.109
0.005
-0.000
-0.162
-0.163
0.054
0.008
0.132
-0.212
0.032
0.044
-0.158
-0.180
-0.079
-0.013
-0.254
-0.251
-0.133
-0.164
-0.074
-0.093
-0.128
-0.252
-0.322
-0.228
-0.088
89
RES1
84CYS
84CYS
84CYS
84CYS
84CYS
84CYS
85SER
85SER
85SER
85SER
85SER
85SER
85SER
85SER
85SER
86PRO
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
87HISB
88GLN
88GLN
88GLN
88GLN
88GLN
88GLN
88GLN
89GLY
89GLY
bound
rexp
0.300
0.440
0.710
0.400
0.400
0.590
0.440
0.540
0.580
0.500
0.600
0.710
0.270
0.490
0.390
0.680
0.720
0.350
0.400
0.500
0.500
0.600
0.490
0.500
0.620
0.500
0.490
0.500
0.500
0.620
0.500
0.400
0.500
0.500
0.500
0.390
0.710
0.350
0.390
0.490
0.710
0.590
0.490
0.430
2.6 Supplementary material
NOE Nr
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
NOE
atom 1
H
H
H
H
H
H
HA
QB
QB
H
H
H
H
H
H
QD
HA
HA
HA
HA
HB
HB
HB
HB
HB
HB
HB
HD2
HD2
HE1
HE1
HE1
HE1
H
H
HA
H
H
H
HE21
HE22
HE22
H
H
RES2
90ALA
87HISB
89GLY
90ALA
90ALA
90ALA
92MET
88GLN
90ALA
91GLY
92MET
92MET
93VAL
93VAL
83TYR
93VAL
93VAL
93VAL
93VAL
93VAL
14PHE
82PHE
82PHE
93VAL
93VAL
93VAL
93VAL
18GLU
82PHE
82PHE
94GLY
95LYSH
95LYSH
96VAL
79THR
81SER
82PHE
82PHE
95LYSH
95LYSH
96VAL
96VAL
79THR
97THR
atom 2
H
HA
QA
QB
H
QB
H
HA
QB
QA
QB
QG
QG1
QG2
QB
QG1
QG2
QG1
QG2
QG2
CZ
CG
H
HA
HB
QG1
QG2
HA
CZ
HZ
QA
QB
QG
QG2
QG2
HA
CG
CZ
HA
QB
QG1
QG2
QG2
HB
3J
UNR VAC
UNR WAT
LE VAC
0.086
-0.053
-0.022
0.405
-0.052
0.058
-0.272
-0.224
-0.247
-0.186
-0.183
-0.181
0.146
-0.013
0.066
-0.124
-0.176
-0.103
0.029
-0.022
0.002
0.194
-0.000
0.053
-0.092
-0.090
-0.073
-0.218
-0.238
-0.221
-0.211
-0.243
-0.209
-0.333
-0.164
-0.294
0.068
-0.050
0.008
-0.091
-0.045
-0.109
-0.134
-0.134
-0.125
-0.165
-0.167
-0.159
-0.130
-0.130
-0.182
-0.173
-0.130
-0.166
-0.164
-0.166
-0.124
-0.145
-0.143
-0.168
-0.226
-0.198
-0.210
-0.316
-0.302
-0.291
-0.083
-0.095
-0.079
-0.128
-0.124
-0.128
-0.070
-0.089
-0.046
-0.042
-0.068
-0.080
-0.042
-0.044
-0.122
-0.066
-0.055
-0.059
-0.249
-0.213
-0.252
-0.145
-0.119
-0.147
-0.151
-0.145
-0.145
-0.153
-0.152
-0.147
-0.160
-0.162
-0.161
-0.164
-0.122
-0.148
0.064
-0.028
-0.058
-0.104
-0.071
-0.086
-0.217
-0.200
-0.220
-0.192
-0.223
-0.204
-0.058
-0.057
-0.054
-0.087
-0.078
-0.089
-0.106
-0.062
-0.116
-0.146
-0.300
-0.223
-0.102
-0.137
-0.112
-0.145
-0.127
-0.118
Table 2.13 Continued on next page
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.042
-0.081
-0.012
-0.121
-0.232
-0.218
-0.183
-0.166
-0.030
-0.043
-0.178
-0.106
-0.022
-0.051
0.023
-0.041
-0.045
-0.181
-0.244
-0.208
-0.238
-0.233
-0.161
-0.155
-0.013
-0.050
-0.084
-0.010
-0.119
-0.180
-0.152
-0.151
-0.195
-0.130
-0.178
-0.147
-0.129
-0.142
-0.157
-0.195
-0.243
-0.173
-0.304
-0.278
-0.103
-0.110
-0.129
-0.108
-0.038
-0.120
-0.113
-0.057
-0.133
-0.112
-0.032
-0.093
-0.199
-0.272
-0.098
-0.184
-0.145
-0.150
-0.151
-0.115
-0.170
-0.221
-0.132
-0.154
0.055
-0.139
-0.072
-0.076
-0.194
-0.186
-0.244
-0.207
-0.055
-0.045
-0.079
-0.177
-0.085
-0.087
-0.272
-0.133
-0.081
-0.088
-0.116
-0.112
3J
LE NOE WAT
-0.065
-0.090
-0.219
-0.186
-0.005
-0.178
-0.047
-0.035
-0.107
-0.229
-0.231
-0.155
-0.029
-0.139
-0.121
-0.167
-0.160
-0.154
-0.145
-0.151
-0.218
-0.308
-0.100
-0.120
-0.069
-0.124
-0.120
-0.051
-0.226
-0.134
-0.147
-0.148
-0.175
-0.148
-0.041
-0.078
-0.185
-0.197
-0.052
-0.106
-0.118
-0.151
-0.089
-0.113
3J
LE NOE TAV WAT
-0.040
0.027
-0.221
-0.182
0.001
-0.156
-0.034
0.029
-0.041
-0.226
-0.217
-0.226
0.008
-0.017
-0.127
-0.134
-0.196
-0.194
-0.099
-0.176
-0.233
-0.302
-0.098
-0.131
-0.033
-0.064
-0.135
-0.050
-0.222
-0.123
-0.150
-0.148
-0.205
-0.139
-0.094
-0.072
-0.207
-0.200
-0.054
-0.114
-0.083
-0.279
-0.113
-0.116
2 Structural characterisation of Plastocyanin using local-elevation MD
RES1
89GLY
90ALA
90ALA
90ALA
91GLY
91GLY
91GLY
92MET
92MET
92MET
92MET
92MET
92MET
92MET
93VAL
93VAL
93VAL
93VAL
93VAL
94GLY
94GLY
94GLY
94GLY
94GLY
94GLY
94GLY
94GLY
95LYSH
95LYSH
95LYSH
95LYSH
95LYSH
95LYSH
96VAL
96VAL
96VAL
96VAL
96VAL
96VAL
96VAL
96VAL
96VAL
97THR
97THR
bound
rexp
0.350
0.430
0.520
0.450
0.300
0.500
0.300
0.350
0.600
0.520
0.490
0.590
0.500
0.600
0.490
0.450
0.500
0.500
0.450
0.690
0.710
0.710
0.430
0.350
0.350
0.500
0.500
0.350
0.610
0.500
0.390
0.440
0.590
0.450
0.600
0.430
0.710
0.710
0.270
0.490
0.400
0.600
0.500
0.400
90
NOE Nr
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
NOE
atom 1
H
H
H
H
H
H
H
H
H
H
H
H
H
H
HA
HA
HA
H
H
QA
H
H
H
H
H
H
H
H
H
H
H
H
H
HA
H
H
H
H
H
H
H
H
HA
HA
RES1
97THR
97THR
97THR
97THR
97THR
97THR
97THR
97THR
98VAL
98VAL
98VAL
98VAL
98VAL
98VAL
98VAL
98VAL
98VAL
98VAL
98VAL
98VAL
98VAL
99ASN
99ASN
99ASN
99ASN
99ASN
99ASN
99ASN
99ASN
99ASN
99ASN
99ASN
RES2
97THR
98VAL
20SER
96VAL
96VAL
96VAL
96VAL
97THR
21VAL
96VAL
97THR
98VAL
77LYSH
78GLY
79THR
80TYR
80TYR
97THR
97THR
98VAL
98VAL
21VAL
22PRO
23SER
98VAL
98VAL
99ASN
20SER
97THR
99ASN
20SER
99ASN
atom 2
QG2
QB
HA
HA
HB
QG1
QG2
HB
QB
QG2
QG2
QB
HA
H
HA
CG
CZ
HA
QG2
HB
QB
H
HA
H
HA
QB
QB
QB
QG2
QB
QB
QB
bound
rexp
0.450
0.720
0.270
0.270
0.400
0.600
0.500
0.400
0.720
0.600
0.600
0.570
0.430
0.350
0.430
0.710
0.710
0.270
0.500
0.350
0.570
0.430
0.430
0.430
0.270
0.620
0.590
0.590
0.600
0.590
0.590
0.590
UNR VAC
-0.174
-0.183
0.031
-0.063
0.019
-0.092
-0.126
-0.097
-0.271
-0.126
0.013
-0.321
0.079
-0.001
-0.108
-0.150
-0.069
-0.053
-0.053
-0.094
-0.225
-0.092
-0.093
-0.031
-0.059
-0.191
-0.289
-0.167
-0.172
-0.225
-0.014
-0.311
UNR WAT
-0.168
-0.171
0.033
-0.056
-0.074
-0.170
-0.089
-0.130
-0.290
-0.033
-0.055
-0.319
0.040
-0.024
-0.101
-0.159
-0.038
-0.049
-0.093
-0.097
-0.221
-0.081
-0.090
-0.029
-0.059
-0.204
-0.315
-0.221
-0.223
-0.232
-0.211
-0.330
3J
LE VAC
-0.178
-0.179
0.044
-0.057
-0.018
-0.111
-0.113
-0.127
-0.257
-0.143
-0.051
-0.319
0.090
0.006
-0.131
-0.123
-0.022
-0.048
-0.112
-0.089
-0.223
-0.090
-0.114
-0.043
-0.056
-0.190
-0.294
-0.184
-0.217
-0.223
-0.015
-0.308
3J
−6 −1/6
r
− rexp
LE WAT NMR set
-0.170
-0.167
-0.180
-0.172
0.046
-0.033
-0.058
-0.066
-0.101
-0.006
-0.155
-0.115
-0.096
-0.171
-0.138
-0.130
-0.279
-0.293
-0.047
-0.128
-0.079
-0.041
-0.320
-0.326
0.034
0.022
-0.027
-0.019
-0.116
-0.114
-0.134
-0.187
-0.053
-0.141
-0.051
-0.057
-0.135
-0.101
-0.098
-0.093
-0.215
-0.220
-0.080
-0.114
-0.110
-0.114
-0.049
-0.072
-0.062
-0.059
-0.207
-0.204
-0.297
-0.291
-0.215
-0.286
-0.217
-0.191
-0.234
-0.226
-0.210
-0.220
-0.330
-0.325
3J
LE NOE WAT
-0.176
-0.173
0.013
-0.056
-0.011
-0.103
-0.155
-0.121
-0.273
-0.128
-0.069
-0.321
0.022
-0.034
-0.102
-0.160
-0.114
-0.045
-0.137
-0.100
-0.214
-0.080
-0.113
-0.052
-0.063
-0.211
-0.308
-0.227
-0.239
-0.232
-0.227
-0.328
3J
LE NOE TAV WAT
-0.170
-0.177
0.047
-0.056
-0.044
-0.163
-0.090
-0.141
-0.285
-0.056
-0.073
-0.320
0.036
-0.026
-0.107
-0.149
-0.085
-0.049
-0.119
-0.099
-0.219
-0.068
-0.114
-0.049
-0.057
-0.203
-0.315
-0.114
-0.246
-0.231
-0.174
-0.327
2.6 Supplementary material
NOE Nr
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
NOE
atom 1
HA
HA
H
H
H
H
H
H
HA
HA
HA
HA
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
HD21
HD21
HD21
HD22
HD22
−1/6
−1/6
Table 2.13 List of all NOE distances and the difference ( r−6
− rexp ) between the averaged distance r−6
and the distance
bound rexp in nm for all 6 simulations and for the set of 16 NMR model structures.
91
92
2 Structural characterisation of Plastocyanin using local-elevation MD
100
Residue number
80
60
40
20
0
0
200
400
600
Time [ps]
800
0 0.1 0.2 0.3
RMSF [nm]
Figure 2.27 Secondary structure analysis [46] of the unrestrained simulation UNR VAC. Black:
310 -helix. Red: α -helix. Cyan: Bend. Magenta: β -Bridge. Blue: β -Strand. Orange: Turn. The
right hand panel shows the root-mean-square fluctuation (RMSF) of the backbone (N, Cα , C)
atoms.
Figure 2.28 Secondary structure analysis [46] of the unrestrained simulation UNR WAT. Black:
310 -helix. Red: α -helix. Cyan: Bend. Magenta: β -Bridge. Blue: β -Strand. Orange: Turn. The
right hand panel shows the root-mean-square fluctuation (RMSF) of the backbone (N, Cα , C)
atoms.
2.6 Supplementary material
93
Figure 2.29 Secondary structure analysis [46] of the restrained simulation 3 J LE VAC. Black: 310 helix. Red: α -helix. Cyan: Bend. Magenta: β -Bridge. Blue: β -Strand. Orange: Turn. The right
hand panel shows the root-mean-square fluctuation (RMSF) of the backbone (N, Cα , C) atoms.
Figure 2.30 Secondary structure analysis [46] of the restrained simulation 3 J LE WAT. Black: 310 helix. Red: α -helix. Cyan: Bend. Magenta: β -Bridge. Blue: β -Strand. Orange: Turn. The right
hand panel shows the root-mean-square fluctuation (RMSF) of the backbone (N, Cα , C) atoms.
3 On the calculation of 3Jαβ -coupling
constants for side chains in
proteins
3.1 Summary
Structural knowledge about proteins is mainly derived from values of observables, measurable
in NMR spectroscopic or X-ray diffraction experiments, i.e. absorbed or scattered intensities,
through theoretically derived relationships between structural quantities such as atom positions or torsional angles on the one hand and observable quantities such as squared structure
factor amplitudes, NOE intensities, or 3 J-coupling constants on the other. The standardly used
relation connecting 3 J-couplings to torsional angles is the Karplus relation, which is used in
protein structure refinement as well as in the evaluation of simulated properties of proteins.
The accuracy of the simple and generalised Karplus relations is investigated using side-chain
structural and 3 Jαβ -coupling data for three different proteins, Plastocyanin, Lysozyme, and
FKBP, for which such data are available. The results show that the widely used Karplus relations are only a rough estimate for the relation between 3 Jαβ -couplings and the corresponding
χ1 -angle in proteins.
95
96
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
3.2 Introduction
A precise determination of the structural properties of proteins is still one of the major challenges in molecular biology, although thousands of protein structures in a crystalline environment or in aqueous solution at a particular thermodynamic state point, i.e. temperature, pH,
ionic strength, etc., have been determined through X-ray diffraction or NMR spectroscopic
experiments [34]. The protein structures derived from X-ray diffraction intensities are generally of relatively high precision, because the ratio of the number of observable (independent)
intensities N obs and the number of spatial degrees of freedom of the protein N d f is larger than
one. Moreover, the relation between the intensity of the diffracted beam and the structure of
the protein is simple and well-known: the intensity of a diffraction peak is proportional to the
square of the amplitude of the corresponding spatial Fourier transform of the electron density
[47], which is in turn directly related to the structure of the protein in terms of atom positions.
The precision of a protein structure derived from X-ray diffraction data is mainly determined
by the spatial resolution of the latter, which determines the ratio N obs /N d f .
NMR experiments can deliver measured values for a variety of observable quantities, including intensities of nuclear Overhauser effect (NOE) peaks, residual dipolar couplings
(RDCs), 3 J-coupling constants, or chemical shifts [48]. The precision of protein structures
derived from NMR data is generally much lower than that of protein structures derived from Xray diffraction data. This relatively low precision is caused by various aspects of the methodology used to derive protein structural information from NMR data [14, 49–51], and of the
NMR data itself.
With respect to the NMR data, the first issue is that the number of measured values of observable quantities at a particular thermodynamic state point is much smaller than the number
of protein degrees of freedom. Even if different observables, such as NOEs, RDCs, or 3 Jcouplings, are combined, the ratio N obs /N d f is still lower than one. Moreover, correlation between different data may reduce the number of independent data, the data may originate from
different experiments at different thermodynamic state points at which the protein’s structure
may not be the same, and the quantities observable in an NMR experiment are in general
related to particular subsets of the atoms of a protein.
A second problem is that the relation between an observable quantity Q(rN ) and the structure of a protein represented by the Cartesian coordinates rN ≡ (r1 , r2 , ..., rN ) of its N atoms is
generally not very precisely known for the aforementioned observables measurable by NMR.
An NOE intensity depends not only on the distance between the two atoms involved, but also
on the rotational motion of the protein and on the distances to other protein atoms surrounding the atom pair due to spin diffusion effects. An RDC depends not only on the angle θ
between the vector connecting two atoms and the magnetic field direction for a single protein
structure, but also on the spatial distribution of these vectors, i.e. on the distribution of protein
orientations in a medium that induces a slight deviation from a uniform spherical distribution.
A 3 J-coupling constant depends not only on the dihedral angle θ between the four atoms involved and their types, but also on the substituents at the two central atoms of the dihedral
3.2 Introduction
97
angle. A chemical shift depends not only on the relative position of the atom involved with
respect to its covalently bound neighbour atoms and their types, but also on the distance to
nonbonded neighbour atoms and their types. In principle, the value of an NMR observable Q
can be calculated from rN using quantum-chemical methods, but the accuracy that currently
can be reached is rather low due to the approximations made during the calculations that are
required by the finite computing power available. Therefore, semi-empirical, approximate
functions Q(rN ) are generally used to relate protein structure to observable quantities.
A further difficulty in determing structural properties from NMR data is that the relations
Q(rN ) for NMR observables are highly non-linear: Q depends on r−3 or r−6 for a NOE
distance r between two atoms, it depends on the cosine of the angle θ for an RDC, and on the
cosine of the angle θ and its square for a 3 J-coupling. Together, these aspects make a precise
determination of protein structural properties based on NMR data a challenging task.
Protein structure determination, be it based on X-ray, NMR or other experimental data,
should also account for the motion or conformational variability of a protein, because all but
a few experiments involve averaging over time and over the ensemble of protein structures
in the sample. Due to the crystalline packing and the linear character of the Fourier transform, the neglect of properly accounting for conformational averaging in procedures to derive
protein structure from measured data is much less aggravating when using X-ray diffraction
data than when using NMR spectroscopic data. In particular, for 3 J-coupling data measured
for protein side chains, it is essential to properly account for averaging because of the strong
non-linearity of the function Q(rN ) and the variety of possible side-chain conformations [52].
3 J-coupling data for protein side chains are less often used to determine protein structures than
the corresponding backbone data. Yet, the information they provide about the distribution of
side-chain dihedral angle values is essential for characterising protein structures in view of the
tight spatial packing of side chains in the interior of a protein.
A first approximation of the relation Q(rN ) between a vicinal 3 J-coupling constant
3 J(A , A ) = 3 J(A − A − A − A ) between two atoms A and A that are covalently con1 4
1
2
3
4
1
4
nected through three bonds involving the atoms A2 and A3 and the dihedral angle
θ = A1 −A2 −A3 −A4 is given by the Karplus relation [1, 2]
3
JA1 A4 (θ ) = a cos2 (θ ) + b cos(θ ) + c,
(3.1)
in which the coefficients a, b and c are parameters that depend on the types of the atoms A1
to A4 and in principle on the number and types of substituents at atoms A2 and A3 . Using
Eq. 3.1, the dependence of 3 JA1 A4 on the geometry of the configuration of atoms A1 to A4 and
their substituents is reduced to a simple function of one dihedral angle θ .
In proteins, different types of 3 J-couplings can be observed, which are related to particular
torsional angles, for example for θα = H−N−Cα −Hα , 3 JHN Hα is related to
ϕ = C−N−Cα −C, and for θβ = Hα −Cα −Cβ −Hβ , 3 JHα Hβ = 3 Jαβ is related to
χ1 = N−Cα −Cβ −Cγ /Oγ /Sγ .
The relation between the angles θα and θβ and the angles ϕ and χ1 depends on the config-
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
98
uration of the atoms involved and is generally approximated by
and
θα = ϕ + δα
(3.2)
θβ = χ1 + δβ ,
(3.3)
with δα = −60◦ , δβH = −120◦ , and δβH = 0◦ for an L-amino acid residue and δα = +60◦ ,
2
3
δβH = 0◦ , and δβH = +120◦ for a D-amino acid residue.
2
3
Z
Hα
Cα
C
N
Cβ
Hβ3
Hβ2
Figure 3.1 Fragment H−C−C−H bearing three non-hydrogen substituents as found in the Lamino acids suitable for application of the generalised Karplus relation.
A more complex description of the relation Q rN between measured proton-proton scalar
3J
HH -couplings and the corresponding dihedral angle values is given by the generalised
Karplus equation proposed by Haasnoot et al. [53–55], which takes into account the substituents of atoms A2 and A3 . It applies to 3 JHH -couplings, i.e. A1 = H and A4 = H, for which
A2 = C and A3 = C, and for which the fragment H−A2 −A3 −H bears three non-hydrogen
substituents, see Fig. 3.1 and Method section. This generalised Karplus relation can be used
to calculate 3 Jαβ -couplings that depend on the θβ torsional angle related to the χ1 torsional
angle for 15 of the 20 amino acids naturally occurring in proteins. The exceptions are Ile, Thr
and Val, for which the fragments H−Cα −Cβ −H bear four non-hydrogen substituents, and the
residues Ala and Gly, which do not have χ1 -angles.
The parameters a, b and c for the standard Karplus relation are generally determined empirically. A variety of different sets of Karplus parameters have been determined using different
molecules and methodologies [11, 12, 42, 43, 56–64]. Often, the value of a particular torsional
angle ϕ or χ1 in the X-ray diffraction structure of a molecule in crystal form is assumed to be
related through Eqs. 3.1-3.3 to the corresponding 3 J-coupling measured for the same molecule
in aqueous solution [42, 43, 61]. Using all available 3 J-couplings for one or more molecules,
a set of Karplus parameters can be obtained that minimises the sum of the squared differences
of the measured 3 J-couplings 3 J exp to the ones calculated from the ϕ - or χ1 -angles using Eqs.
3.1-3.3. Such a procedure rests upon the assumption that the value of a torsional angle in the
crystal is a good approximation of the value of the same angle in solution, and that conformational averaging plays a similar role in both environments. The approximate nature of these
assumptions is illustrated by the variation of the Karplus parameters obtained using different
sets of data. For 3 JHN Hα , the parameter ranges found in the literature [11, 12, 43, 59, 61, 63, 64]
are 3.1 Hz for a, 0.64 Hz for b, and 1.6 Hz for c. For 3 Jαβ , the variety of parameter values is
3.2 Introduction
99
Source
Molecule (θβ determination method)
Abraham et al., 1962 [56]
a
b
c
Hydroxy-L-proline (theoretical)
12.1
-1.6
0
Deber et al., 1971 [57]
Cyclo(tri-L-prolyl) and derivatives (simulation)
9.5
-1.0
1.4
Kopple et al., 1973 [58]
Several molecules (X-ray and theoretical)
9.4
-1.4
1.6
De Marco et al., 1978 [42]
χ1 − χ3 dihedral angles of ornityl
9.5
-1.6
1.8
P´erez et al., 2001 [62]
Flavodoxin (self-consistent fitting)
7.23
-1.37
2.22
residues in a cyclohexapeptide (X-ray)
Table 3.1 Karplus relation parameters a, b, and c from the literature. The molecules for which the
were measured and the methodology of θβ determination are indicated. All values
are in Hz.
3 J -couplings
αβ
even larger [42, 56–58, 60, 62], 4.87 Hz for a, 0.6 Hz for b, and 2.2 Hz for c, see Table 3.1,
leading to quite some variation in the resulting Karplus curves (Fig. 3.2).
Figure 3.2 Karplus curves of Eq. 3.1 for 3 Jαβ3 as a function of θβ3 using different values of the
parameters a, b, and c from the literature. The solid line was generated using the Abraham [56]
parameters, the dot-dot-dashed line using the Deber [57] parameters, the dashed line using the
Kopple [58] parameters, the dotted line using the De Marco [42] parameters, and the dot-dashed
´
line using the Perez
[62] parameters.
The precision of a, b and c values may be affected by the fact that the values of the ϕ - or χ1 angles in the crystal structures of the proteins or other molecules used to obtain the Karplus
parameters do not cover the whole 360◦ domain of dihedral angle values. This situation is
illustrated in Fig. 3.3, which shows the stereospecifically assigned 3 Jαβ -values as obtained
from NMR experiments in solution for three different proteins and the corresponding θβ angles in the X-ray crystal or NMR model structures of each protein.
An alternative procedure to obtain Karplus parameters that avoids the use of crystal data,
100
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
Figure 3.3 Stereospecifically assigned 3 Jαβ -coupling constants as measured by NMR as a function of the corresponding θβ -angle values in the X-ray or NMR model structures for three proteins,
plus the Karplus curve using the parameters from De Marco [42] (solid line). The top panel shows
data for the NMR model structures 9PCY of Plastocyanin (circles). Data for the X-ray structures
1AKI (crosses), 193L A (triangles up), 193L B (triangles down), and the NMR model structures
1E8L (circles) of HEWL are shown in the middle panel. In the bottom panel, data for the X-ray
structure 1FKF (crosses) of FKBP is plotted.
which are characterised by low atom mobility and a particular environment, is to use structural
data from molecular dynamics (MD) simulations of proteins in aqueous solution [11, 63, 64].
Least-squares fitting of the calculated 3 J-couplings averaged over an ensemble or trajectory of
structures h3 J calc i to the measured couplings 3 J exp results in values of a, b, and c optimised for
that particular combination of NMR data and protein structures. Given a high accuracy protein force field and sufficient conformational sampling, such a procedure may lead to a more
accurate set of Karplus parameters than the currently available ones. The question remains,
however, as to how robust such fitted parameters are.
Whilst several groups [11, 63, 64] have optimised the Karplus parameters for backbone or
side chain 3 J-couplings using MD simulations, none of them made use of 3 Jαβ -couplings.
One possible reason may be the limited amount of data available, as 15 out of the 20 types of
amino acids naturally occurring in proteins have two Hβ atoms, meaning that stereospecific
assignment is required. Additionally, the measurement of 3 Jαβ -coupling constants becomes
more difficult with increasing molecule size due to resonance overlaps and larger line-widths
in the spectra. However, the 3 JHN Hα -coupling depends on the ϕ -angle, which is in turn correlated to the ψ -angle of the previous residue in a protein via the peptide bond. Since a
variation of the orientation of the peptide plane is easily obtained without changing the spatial fold of the polypeptide backbone as long as the value of the sum ψ + ϕ is constant, the
3J
HN Hα -couplings do not unambiguously determine the fold and are therefore less useful for
protein structure determination. For these reasons, we shall not consider 3 JHN Hα -couplings,
and instead concentrate on 3 Jαβ -couplings, which are related to χ1 -angle value distributions.
3.3 Method
101
calc and experWe investigate whether the agreement between calculated 3 Jαβ -couplings 3 Jαβ
exp
in proteins can be improved either by using the generimentally measured couplings 3 Jαβ
calc i-values to measured 3 J exp -values using
alised Karplus relation or by fitting calculated h3 Jαβ
αβ
conformational ensembles of proteins generated by MD simulation or X-ray or NMR model
structures to find optimal values for the parameters a, b and c of the standard Karplus relation.
We use three proteins, Plastocyanin [22], hen egg white Lysozyme (HEWL) [65], and FK506
binding protein (FKBP) [66], for which measured, stereospecifically assigned 3 Jαβ -couplings
are available, as test proteins and for calibration of the Karplus parameters.
3.3 Method
3.3.1 Generalised Karplus relation
For a fragment H−C−C−H in which each of the C atoms carries three substituents, the generalised Karplus relation takes the form [53–55]
3
JHH (θ ) = a1 cos2 (θ ) + a2 cos(θ ) + a3 +
6
∑
i=1
substituents
∆xi′ a4 + a5 cos2 (ξi θ + a6 ∆xi′ ) , (3.4)
in which θ is the H−C−C−H dihedral angle (IUPAC convention [67]), ∆xi′ are the effective
electronegativity differences between the substituent atoms at the two C-atoms and an H-atom
as given by the expression
∆xi′ = ∆xi − a7
3
∑
∆xk ,
(3.5)
k=1
substituents
in which ∆xi is the electronegativity difference between the substituent i on the C-atom and an
H-atom, and the ∆xk are these quantities for the atoms k bound to the substituent i, representing
the secondary substituent effect. Table II of Huggins [68] gives
∆xH = xH − xH = 2.20 − 2.20 = 0.00
∆xC = xC − xH = ∆xS = xS − xH = 2.60 − 2.20 = 0.40
∆xN = xN − xH = 3.05 − 2.20 = 0.85
∆xO = xO − xH = 3.50 − 2.20 = 1.30.
(3.6)
The quantity ξi depends on the orientation of the substituent i with respect to its geminal
coupled proton. Since in an L-amino acid fragment as in Fig. 3.1 the substituents X = N and
Y = C on the Cα atom are the same for all residues, only substituent Z on the Cβ atom varies,
which is Cγ for Arg, Asn, Asp, Glu, Gln, His, Leu, Lys, Met, Phe, Pro, Trp, and Tyr, Sγ for
Cys, and Oγ for Ser. We have for the pair (Hα , Hβ2 ) the values ξN = +1, ξC = −1, ξZ = +1,
102
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
and δβ2 = −120◦ , and for the pair (Hα , Hβ3 ) the values ξN = +1, ξC = −1, ξZ = −1, and
δβ3 = 0◦ , with Z = Cγ , Oγ , or Sγ and χ1 = N−Cα −Cβ −Z, with δβ defined by Eq. 3.3 and
θ = θβ = Hα −Cα −Cβ −Hβ2 /β3 . The coefficients ai are [55] a1 = 13.22 Hz, a2 = −0.99 Hz,
a3 = 0 Hz, a4 = 0.87 Hz, a5 = −2.46 Hz, a6 = 19.9, and a7 = 0. Because a7 = 0, ∆xi′ = ∆xi
and so
3
JHH (θ ) = a1 cos2 (θ ) + a2 cos(θ ) + ∆xN a4 + a5 cos2 (ξN θ + a6 |∆xN |)
+ ∆xC a4 + a5 cos2 (ξC θ + a6 |∆xC |)
(3.7)
+ ∆xZ a4 + a5 cos2 (ξZ θ + a6 |∆xZ |) .
Considering the β2 /β3 protons and Z = Cγ /Sγ /Oγ , but with the simplification ∆xC = ∆xS , this
yields four different expressions for 3 Jαβ (θβ ). The four corresponding generalised Karplus
curves 3 Jαβ (θβ ), two for Ser (Z = O) and two for the other 14 residues with two β -protons,
are displayed in Fig. 3.4.
Figure 3.4 The four generalised Karplus relations for different substituents Z (O: thick lines, C/S:
thin lines) and different Hβ types (Hβ2 : dashed lines, Hβ3 : solid lines), and the curve obtained using
the standard Karplus relation with the De Marco [42] parameters (dash-dotted).
3.3.2 Determination of the parameters of the Karplus relation
As an alternative to using the generalised Karplus relation of Eq. 3.4 we may optimise the
parameters a, b, and c of the standard Karplus relation in Eq. 3.1 by fitting MD trajectory
calc i to the corresponding experimental values 3 J exp [11, 63, 64].
averaged 3 Jαβ -couplings h3 Jαβ
αβ
Using ensemble-averaged values
hcosθβi i = hcosχi icosδβi − hsinχi isinδβi
(3.8)
3.3 Method
and
103
hcos2 θβi i = hcos2 χi icos2 δβi − 2hcosχi sinχi icosδβi sinδβi + hsin2 χi isin2 δβi
(3.9)
obtained from an MD trajectory of a particular Hα , Hβ torsional angle θβi in the Karplus
calc i can be obtained. By using all or a
relation in Eq. 3.1, ensemble-averaged values of h3 Jαβ
exp
particular subset, e.g. 3 Jαβ , 3 Jαβ2 , or 3 Jαβ3 , of the NJ experimental values 3 Jαβ
measured for a
calc i-values
protein, optimal values for a, b, and c can be obtained by least-squares fitting of h3 Jαβ
exp
to the corresponding 3 Jαβ
-values. In doing so the quantity
Q2 =
2
1 NJ
ahcos2 θβi i + bhcosθβi i + c − 3 Jiexp
∑
NJ i=1
(3.10)
is minimised with respect to variation of the parameters a, b, and c. Their values follow from
the equations ∂ Q/∂ a = ∂ Q/∂ b = ∂ Q/∂ c = 0, or
NJ
∑
i=1
NJ
∑
i=1
ahcos2 θβi i + bhcosθβi i + c − 3 Jiexp hcos2 θβi i = 0
(3.11)
ahcos2 θβi i + bhcosθβi i + c − 3 Jiexp hcosθβi i = 0
(3.12)
ahcos2 θβi i + bhcosθβi i + c − 3 Jiexp = 0,
(3.13)
NJ
∑
i=1
which can be solved using Cramer’s rule.
3.3.3 Analysis of the structural and 3 Jαβ -coupling data
For each of the three proteins we use three different subsets of 3 Jαβ -couplings, i.e.
1. the stereospecifically assigned 3 Jαβ2 and 3 Jαβ3 for the side chains with two stereospecifically assigned Hβ protons,
2. the 3 Jαβ for the side chains with one Hβ proton (Ile, Thr and Val), and
3. the non-stereospecifically assigned 3 Jαβ2 and 3 Jαβ3 .
Two types of structural data sets for the three proteins were used: (i) X-ray or NMR model
structures, and (ii) trajectories of protein structures obtained from MD simulations of the proteins in aqueous solution. The simulations were carried out using the GROMOS [32, 71]
software and different GROMOS biomolecular force fields, namely the force field parameter sets 45A3 of the year 2001 [26], 53A6 of 2004 [27], and 54A7 of 2011 [70], see Table
3.2. The nonbonded interaction parameters of the 45A3 force field were obtained by fitting
the heat of vaporisation, density and solvation free energy in water and in cyclohexane for
a set of compounds representing apolar side chains in proteins. In the 53A6 force field this
104
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
Plastocyanin
HEWL
FKBP/asc
Steiner et al., 2012[69]
Schmid et al., 2011[70]
Allison et al., 2009[52]
99
129
107
(16 9PCY)
1AKI, 193L, (50 1E8L)
1FKF
Number of water molecules
3553
14365/14355/14378
6285
Box type, length [nm]
t, 6.26
r, 7.72
r, 5.94
45B3[26], 53A6[27]
54A7[70], 54B7[70], 53A6[27], 45A3[26]
45A3[26], 45B3[26]
Initial structure
9PCY model 16
1AKI
1FKF
Temperature [K]
300
300
303
1
20
8
Number of 3 Jαβ -couplings total
108
100
94
Number of entries in subset 1
42
46
37+37
Number of entries in subset 2
20
14
20
Number of entries in subset 3
46
40
0
Number of residues
PDB code of X-ray (NMR) structures
Force field
Simulation length [ns]
Table 3.2 Proteins and structure sets investigated. The structure sets are either molecular dynamics simulation (MD) trajectories from a simulation in a vacuum or water environment with a particular force field, or experimental X-ray or NMR model structures from the PDB [34]. For MD simulations in water, a rectangular (r) or truncated octahedron (t) box was used. The number of NMR
model structures is indicated before the PDB code name of the NMR structure set. Subset 1 contains the measured, stereospecifically assigned 3 Jαβ2/3 -values, subset 2 the measured, assigned
3 J -values of amino acids
αβ
assigned 3 Jαβ2/3 -values.
with only one Hβ , and subset 3 the measured, non-stereospecifically
set was extended to compounds representing polar side chains in proteins. The 54A7 force
field contains a slight modification of protein backbone nonbonded and torsional-angle parameters compared to 53A6. The corresponding force field parameter sets for simulations of
proteins in vacuo are denoted as 45B3, 53B6 and 54B7. The X-ray and NMR model structures
were taken from the Protein Data Bank (PDB, [34]): 9PCY [22] (16 NMR model structures)
for Plastocyanin, 1AKI [72][73] and 193L [74] (X-ray structures) and 1E8L [75] (50 NMR
model structures) for HEWL, and 1FKF [66] for FKBP. The setups of the MD simulations are
described in earlier studies of Plastocyanin [69], HEWL [70], and ascomycin bound to FKBP
[52].
For the evaluation of the generalised Karplus relation in Tables 3.3-3.9 simulation trajectories of lengths 1 ns for Plastocyanin, 20 ns for HEWL, and 8 ns for FKBP were used.
calc i-couplings and Q-values, defined as the square-root of the Q2 obtained from Eq. 3.10,
h3 Jαβ
were calculated for the set of stereospecifically assigned 3 Jαβ2 and 3 Jαβ3 -coupling constants
(subset 1). The same trajectories were used to calculate the Q-values for the least-squares
fitted Karplus relations in Tables 3.11-3.13 and 3.16. The values of a, b, and c obtained from
exp
least-squares fitting of subsets 1 and 2 of the 3 Jαβ
for one protein were used to determine Q-
3.3 Method
105
Residue
4 LEU
4 LEU
7 SER
7 SER
11 SER
11 SER
12 LEU
12 LEU
14 PHE
14 PHE
19 PHE
19 PHE
22 PRO
22 PRO
37 HISB
37 HISB
42 ASP
42 ASP
43 GLU
43 GLU
47 PRO
47 PRO
51 ASP
51 ASP
56 SER
56 SER
58 PRO
58 PRO
63 LEU
63 LEU
66 PRO
66 PRO
70 TYR
70 TYR
74 LEU
74 LEU
80 TYR
80 TYR
84 CYS
84 CYS
86 PRO
86 PRO
Proton
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Q:
3 J exp
αβ
11.7
2.5
5.0
5.4
4.6
9.1
12.1
3.8
11.9
3.2
3.0
5.5
5.1
8.5
11.8
3.6
4.0
11.6
5.7
5.9
8.9
8.4
5.1
10.9
10.6
3.9
8.9
8.0
12.1
3.8
6.6
7.9
6.6
11.2
12.1
3.4
12.7
2.1
7.3
10.4
5.8
8.4
hθβ i
158± 31
278± 31
295± 11
55± 11
221± 75
341± 75
178± 12
298± 12
169± 9
289± 9
299± 7
59± 7
226± 24
346± 24
166± 7
286± 7
72± 8
192± 8
299± 11
59± 11
233± 27
353± 27
64± 10
184± 10
171± 11
291± 11
227± 25
347± 25
157± 38
277± 38
228± 25
348± 25
59± 8
179± 8
158± 18
278± 18
159± 7
279± 7
66± 8
186± 8
216± 20
336± 20
calc i
h3 Jαβ
DeMarco
10.7± 3.5
3.8± 2.9
3.1± 1.1
4.1± 1.4
7.0± 4.8
4.3± 2.8
12.5± 1.0
3.4± 1.2
12.3± 0.7
2.5± 0.7
3.4± 0.9
3.5± 0.9
7.8± 3.7
7.8± 0.9
12.1± 0.7
2.2± 0.5
2.4± 0.6
12.2± 0.7
3.5± 1.3
3.6± 1.3
6.9± 4.0
7.8± 1.0
3.0± 1.1
12.5± 0.5
12.4± 1.2
2.8± 1.0
7.7± 3.6
7.8± 0.9
10.4± 4.2
4.7± 3.3
7.4± 3.9
7.8± 0.9
3.5± 1.0
12.7± 0.4
11.0± 2.4
2.6± 1.4
11.5± 0.9
2.0± 0.3
2.8± 0.8
12.6± 0.4
9.2± 3.1
7.4± 0.9
1.8
calc i
h3 Jαβ
genKarplus
9.6± 3.3
3.2± 2.8
1.6± 0.8
3.0± 1.2
5.4± 4.3
3.7± 2.3
11.3± 1.1
2.9± 1.2
11.0± 0.8
2.0± 0.7
2.7± 0.9
3.5± 1.0
7.3± 3.6
7.7± 1.0
10.8± 0.8
1.7± 0.5
2.3± 0.8
11.2± 0.6
2.8± 1.3
3.6± 1.4
6.3± 4.0
7.7± 1.1
3.1± 1.3
11.5± 0.4
10.1± 1.3
3.0± 1.0
7.1± 3.6
7.8± 1.0
9.4± 4.0
4.1± 3.1
6.9± 3.8
7.8± 1.1
3.8± 1.2
11.6± 0.4
9.6± 2.4
2.1± 1.4
10.1± 1.0
1.4± 0.3
2.9± 1.0
11.5± 0.3
8.7± 3.0
7.2± 1.1
2.0
exp
Table 3.3 3 Jαβ -coupling constants of subset 1 measured experimentally 3 Jαβ
[22] and the average
calc calculated using the standard Karplus relation with the De Marco [42] paand rmsd of the 3 Jαβ
rameters and using the generalised Karplus relation from the simulation of Plastocyanin in water
(53A6) and the corresponding averaged dihedral angle value hθβ i. The Q-values quantifying the
exp
calc i are given at the bottom. All 3 J-couplings and
and each set of h3 Jαβ
agreement between 3 Jαβ
Q-values are given in Hz and the hθβ i-angle values are given in degrees.
106
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
Residue
4 LEU
4 LEU
7 SER
7 SER
11 SER
11 SER
12 LEU
12 LEU
14 PHE
14 PHE
19 PHE
19 PHE
22 PRO
22 PRO
37 HISB
37 HISB
42 ASP
42 ASP
43 GLU
43 GLU
47 PRO
47 PRO
51 ASP
51 ASP
56 SER
56 SER
58 PRO
58 PRO
63 LEU
63 LEU
66 PRO
66 PRO
70 TYR
70 TYR
74 LEU
74 LEU
80 TYR
80 TYR
84 CYS
84 CYS
86 PRO
86 PRO
Proton
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Q:
3 J exp
αβ
11.7
2.5
5.0
5.4
4.6
9.1
12.1
3.8
11.9
3.2
3.0
5.5
5.1
8.5
11.8
3.6
4.0
11.6
5.7
5.9
8.9
8.4
5.1
10.9
10.6
3.9
8.9
8.0
12.1
3.8
6.6
7.9
6.6
11.2
12.1
3.4
12.7
2.1
7.3
10.4
5.8
8.4
hθβ i
166± 26
286± 26
269± 45
29± 45
58± 9
178± 9
171± 14
291± 14
177± 15
297± 15
298± 7
58± 7
221± 24
341± 24
125± 19
245± 19
97± 49
217± 49
167± 46
287± 46
240± 25
0± 25
55± 38
175± 38
119± 54
239± 54
232± 26
352± 26
175± 9
295± 9
228± 25
348± 25
73± 9
193± 9
166± 10
286± 10
178± 9
298± 9
57± 7
177± 7
267± 17
27± 17
calc i
h3 Jαβ
DeMarco
11.5± 2.8
3.6± 2.2
5.0± 3.4
4.9± 2.5
3.7± 1.1
12.6± 0.4
12.2± 1.1
2.9± 1.2
12.2± 1.1
3.5± 1.5
3.3± 0.9
3.6± 0.9
8.4± 3.6
7.6± 1.0
6.2± 3.1
4.7± 2.4
5.4± 4.3
9.8± 4.0
10.2± 3.9
3.8± 2.8
5.9± 3.5
8.1± 1.0
3.0± 1.2
11.5± 2.6
8.0± 4.6
7.3± 5.0
7.0± 3.7
7.9± 1.0
12.6± 0.4
3.1± 1.0
7.5± 3.8
7.8± 0.9
2.3± 0.6
12.1± 0.9
12.0± 1.0
2.4± 0.6
12.6± 0.3
3.3± 1.0
3.8± 1.0
12.7± 0.3
2.5± 2.2
7.2± 0.8
2.1
calc i
h3 Jαβ
genKarplus
10.4± 2.7
3.0± 2.2
3.7± 3.3
4.1± 2.5
4.8± 1.3
10.7± 0.4
11.0± 1.2
2.3± 1.2
11.1± 1.2
3.0± 1.5
2.6± 0.9
3.7± 1.0
7.9± 3.5
7.5± 1.2
4.8± 2.9
4.1± 2.4
5.1± 3.9
8.9± 3.8
9.1± 3.7
3.3± 2.7
5.4± 3.5
8.1± 0.9
2.9± 1.4
10.6± 2.3
7.2± 3.1
6.5± 4.0
6.4± 3.7
7.9± 1.1
11.4± 0.5
2.6± 1.1
7.0± 3.8
7.7± 1.1
2.2± 0.8
11.1± 0.8
10.7± 1.1
1.8± 0.7
11.5± 0.4
2.8± 1.1
4.1± 1.1
11.6± 0.3
1.9± 2.2
7.5± 0.8
2.2
exp
Table 3.4 3 Jαβ -coupling constants of subset 1 measured experimentally 3 Jαβ
[22] and the avercalc calculated using the standard Karplus relation with the De Marco [42]
age and rmsd of the 3 Jαβ
parameters and using the generalised Karplus relation from the simulation of Plastocyanin in vacuum (45B3) and the corresponding averaged dihedral angle value hθβ i. The Q-values quantifying
exp
calc i are given at the bottom. All 3 J-couplings and
and each set of h3 Jαβ
the agreement between 3 Jαβ
Q-values are given in Hz and the hθβ i-angle values are given in degrees.
3.3 Method
107
Residue
3 PHE
3 PHE
6 CYS1
6 CYS1
15 HISB
15 HISB
18 ASP
18 ASP
20 TYR
20 TYR
23 TYR
23 TYR
27 ASN
27 ASN
30 CYS1
30 CYS1
34 PHE
34 PHE
39 ASN
39 ASN
46 ASN
46 ASN
48 ASP
48 ASP
52 ASP
52 ASP
53 TYR
53 TYR
59 ASN
59 ASN
61 ARG
61 ARG
66 ASP
66 ASP
75 LEU
75 LEU
87 ASP
87 ASP
94 CYS2
94 CYS2
119 ASP
119 ASP
123 TRP
123 TRP
127 CYS2
127 CYS2
3 J exp
αβ
Proton
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
10.0
3.0
11.5
3.5
11.2
2.6
4.2
11.0
2.3
11.7
10.9
2.7
10.3
2.4
5.3
10.8
10.7
5.0
4.5
10.8
11.2
4.7
2.6
3.7
11.6
3.6
10.4
3.0
5.4
11.3
5.7
10.8
5.1
4.5
12.4
2.1
5.1
11.5
4.0
12.2
4.9
11.7
10.6
2.9
11.6
4.8
Q:
hθβ i
157± 12
277± 12
181± 8
301± 8
165± 9
285± 9
103± 44
223± 44
73± 9
193± 9
167± 11
287± 11
70± 20
190± 20
69± 6
189± 6
172± 8
292± 8
-573± 154
-453± 154
27± 57
147± 57
18± 102
138± 102
174± 14
294± 14
175± 7
295± 7
69± 9
189± 9
151± 34
271± 34
64± 68
184± 68
163± 18
283± 18
168± 22
288± 22
74± 10
194± 10
75± 14
195± 14
179± 8
299± 8
168± 10
288± 10
calc i
h3 Jαβ
DeMarco
11.2± 1.6
2.1± 1.0
12.7± 0.3
3.7± 1.0
12.0± 0.8
2.3± 0.6
5.6± 4.5
8.7± 4.7
2.3± 0.6
12.1± 0.8
12.1± 1.0
2.5± 0.7
3.1± 1.8
12.1± 1.9
2.5± 0.6
12.5± 0.4
12.5± 0.5
2.7± 0.8
5.6± 4.8
8.7± 4.5
3.6± 1.4
10.5± 3.6
5.7± 4.1
4.4± 3.1
12.3± 1.0
3.2± 1.2
12.7± 0.3
3.0± 0.7
2.6± 0.7
12.3± 0.7
10.6± 3.3
3.6± 3.4
3.4± 1.9
7.1± 3.3
11.6± 2.0
2.7± 1.5
12.0± 2.2
3.2± 2.1
2.3± 0.7
11.9± 1.0
2.5± 1.4
12.0± 1.5
12.7± 0.3
3.4± 1.0
12.2± 0.9
2.5± 0.7
3.4
calc i
h3 Jαβ
genKarplus
9.8± 1.6
1.6± 1.0
11.6± 0.4
3.2± 1.0
10.7± 0.9
1.7± 0.6
5.1± 3.9
7.9± 4.5
2.2± 0.8
11.2± 0.8
10.8± 1.1
1.9± 0.8
3.1± 1.7
11.1± 1.8
2.5± 0.7
11.5± 0.4
11.3± 0.6
2.2± 0.8
5.1± 4.3
7.9± 4.3
3.6± 1.7
9.7± 3.1
4.9± 3.8
4.1± 2.9
11.1± 1.1
2.7± 1.3
11.5± 0.4
2.5± 0.8
2.6± 1.0
11.4± 0.6
9.4± 3.0
3.0± 3.3
2.5± 1.7
6.6± 3.0
10.3± 2.0
2.1± 1.5
10.8± 2.0
2.7± 2.0
2.2± 0.9
11.0± 0.9
2.3± 1.3
11.1± 1.4
11.6± 0.4
2.9± 1.0
10.9± 1.1
2.0± 0.8
3.3
exp
Table 3.5 3 Jαβ -coupling constants of subset 1 measured experimentally 3 Jαβ
[65] and the avercalc calculated using the standard Karplus relation with the De Marco [42]
age and rmsd of the 3 Jαβ
parameters and using the generalised Karplus relation from the simulation of HEWL in vacuum
(54B7) and the corresponding averaged dihedral angle value hθβ i. The Q-values quantifying the
exp
calc i are given at the bottom. All 3 J-couplings and
and each set of h3 Jαβ
agreement between 3 Jαβ
Q-values are given in Hz and the hθβ i-angle values are given in degrees.
108
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
Residue
3 PHE
3 PHE
6 CYS1
6 CYS1
15 HISB
15 HISB
18 ASP
18 ASP
20 TYR
20 TYR
23 TYR
23 TYR
27 ASN
27 ASN
30 CYS1
30 CYS1
34 PHE
34 PHE
39 ASN
39 ASN
46 ASN
46 ASN
48 ASP
48 ASP
52 ASP
52 ASP
53 TYR
53 TYR
59 ASN
59 ASN
61 ARG
61 ARG
66 ASP
66 ASP
75 LEU
75 LEU
87 ASP
87 ASP
94 CYS2
94 CYS2
119 ASP
119 ASP
123 TRP
123 TRP
127 CYS2
127 CYS2
Proton
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Q:
3 J exp
αβ
10.0
3.0
11.5
3.5
11.2
2.6
4.2
11.0
2.3
11.7
10.9
2.7
10.3
2.4
5.3
10.8
10.7
5.0
4.5
10.8
11.2
4.7
2.6
3.7
11.6
3.6
10.4
3.0
5.4
11.3
5.7
10.8
5.1
4.5
12.4
2.1
5.1
11.5
4.0
12.2
4.9
11.7
10.6
2.9
11.6
4.8
hθβ i
166± 19
286± 19
173± 10
293± 10
163± 27
283± 27
-37± 52
82± 52
119± 37
239± 37
179± 10
299± 10
180± 11
300± 11
92± 17
212± 17
162± 14
282± 14
72± 16
192± 16
179± 10
299± 10
305± 9
65± 9
177± 10
297± 10
163± 9
283± 9
74± 7
194± 7
104± 35
224± 35
59± 28
179± 28
113± 46
233± 46
69± 25
189± 25
67± 9
187± 9
69± 15
189± 15
162± 13
282± 13
182± 10
302± 10
calc i
h3 Jαβ
DeMarco
11.7± 1.9
2.7± 1.3
12.4± 0.6
2.8± 0.9
11.3± 3.0
3.5± 2.6
3.2± 1.1
5.6± 3.5
6.3± 4.3
6.5± 4.3
12.6± 0.5
3.5± 1.2
12.6± 0.8
3.6± 1.2
2.7± 1.3
9.5± 2.4
11.6± 1.7
2.4± 1.1
2.8± 1.5
11.9± 1.7
12.6± 0.7
3.5± 1.1
4.2± 1.2
3.0± 1.0
12.5± 0.6
3.3± 1.1
11.8± 1.0
2.2± 0.5
2.2± 0.4
12.1± 0.7
4.8± 3.9
8.2± 4.2
3.0± 1.0
11.9± 2.4
6.4± 4.7
7.8± 4.7
3.3± 2.0
11.8± 2.1
2.8± 0.8
12.5± 0.6
2.8± 1.3
12.2± 1.2
11.6± 1.4
2.3± 0.9
12.6± 0.6
3.9± 1.2
2.5
calc i
h3 Jαβ
genKarplus
10.4± 1.9
2.1± 1.3
11.2± 0.7
2.3± 1.0
10.2± 2.8
2.9± 2.5
2.6± 1.2
5.4± 3.2
5.4± 3.9
5.8± 4.2
11.5± 0.6
3.0± 1.2
11.5± 0.8
3.1± 1.2
2.0± 1.2
8.8± 2.3
10.2± 1.7
1.8± 1.1
2.7± 1.5
11.0± 1.6
11.5± 0.7
3.0± 1.2
3.4± 1.2
2.9± 1.2
11.4± 0.7
2.7± 1.1
10.4± 1.1
1.6± 0.5
2.0± 0.6
11.1± 0.6
4.1± 3.5
7.5± 4.0
3.0± 1.1
11.0± 2.2
5.8± 4.2
7.0± 4.5
3.3± 2.0
10.8± 1.9
2.9± 1.0
11.5± 0.6
2.8± 1.4
11.2± 1.1
10.2± 1.4
1.8± 0.9
11.5± 0.6
3.4± 1.2
2.4
exp
Table 3.6 3 Jαβ -coupling constants of subset 1 measured experimentally 3 Jαβ
[65] and the average
calc calculated using the standard Karplus relation with the De Marco [42] paand rmsd of the 3 Jαβ
rameters and using the generalised Karplus relation from the simulation of HEWL in water (45A3)
and the corresponding averaged dihedral angle value hθβ i. The Q-values quantifying the agreeexp
calc i are given at the bottom. All 3 J-couplings and Q-values
and each set of h3 Jαβ
ment between 3 Jαβ
are given in Hz and the hθβ i-angle values are given in degrees.
3.3 Method
109
Residue
3 PHE
3 PHE
6 CYS1
6 CYS1
15 HISB
15 HISB
18 ASP
18 ASP
20 TYR
20 TYR
23 TYR
23 TYR
27 ASN
27 ASN
30 CYS1
30 CYS1
34 PHE
34 PHE
39 ASN
39 ASN
46 ASN
46 ASN
48 ASP
48 ASP
52 ASP
52 ASP
53 TYR
53 TYR
59 ASN
59 ASN
61 ARG
61 ARG
66 ASP
66 ASP
75 LEU
75 LEU
87 ASP
87 ASP
94 CYS2
94 CYS2
119 ASP
119 ASP
123 TRP
123 TRP
127 CYS2
127 CYS2
Proton
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Q:
3 J exp
αβ
10.0
3.0
11.5
3.5
11.2
2.6
4.2
11.0
2.3
11.7
10.9
2.7
10.3
2.4
5.3
10.8
10.7
5.0
4.5
10.8
11.2
4.7
2.6
3.7
11.6
3.6
10.4
3.0
5.4
11.3
5.7
10.8
5.1
4.5
12.4
2.1
5.1
11.5
4.0
12.2
4.9
11.7
10.6
2.9
11.6
4.8
hθβ i
158± 16
278± 16
177± 25
297± 25
167± 10
287± 10
-13± 66
106± 66
107± 40
227± 40
178± 11
298± 11
175± 14
295± 14
106± 50
226± 50
170± 12
290± 12
66± 32
186± 32
181± 14
301± 14
305± 10
65± 10
183± 9
303± 9
166± 10
286± 10
76± 7
196± 7
95± 29
215± 29
66± 10
186± 10
149± 32
269± 32
71± 15
191± 15
74± 12
194± 12
-42± 164
77± 164
171± 12
291± 12
186± 51
306± 51
calc i
h3 Jαβ
DeMarco
10.9± 1.9
2.5± 0.9
12.0± 2.3
3.5± 1.7
12.1± 0.9
2.5± 0.8
2.9± 1.0
7.2± 4.1
5.4± 4.5
8.3± 4.6
12.5± 0.6
3.4± 1.1
12.3± 1.1
3.3± 1.3
6.9± 4.2
8.0± 5.1
12.3± 1.1
2.8± 1.1
2.7± 1.3
11.5± 2.3
12.4± 1.4
3.9± 1.3
4.2± 1.3
3.0± 1.1
12.6± 0.5
3.9± 1.2
12.1± 1.0
2.4± 0.6
2.1± 0.4
11.9± 0.7
3.8± 3.1
9.2± 3.7
2.9± 0.8
12.5± 0.7
10.0± 3.6
3.8± 3.3
2.8± 1.3
11.9± 1.4
2.4± 0.7
11.9± 1.3
2.9± 1.3
12.2± 1.3
12.2± 0.9
2.9± 1.1
10.7± 3.6
3.9± 2.4
2.0
calc i
h3 Jαβ
genKarplus
9.6± 2.0
1.9± 1.0
10.9± 2.2
3.0± 1.7
10.8± 1.0
1.9± 0.8
2.4± 1.0
6.9± 3.6
4.8± 4.0
7.5± 4.5
11.4± 0.7
2.9± 1.2
11.1± 1.2
2.8± 1.3
6.4± 3.6
7.1± 4.9
11.0± 1.1
2.2± 1.1
2.6± 1.4
10.6± 2.1
11.4± 1.4
3.4± 1.4
3.5± 1.3
2.9± 1.3
11.6± 0.4
3.4± 1.2
10.7± 1.1
1.8± 0.7
1.9± 0.6
11.0± 0.7
3.2± 2.8
8.4± 3.5
2.9± 1.0
11.5± 0.6
8.8± 3.3
3.2± 3.2
2.7± 1.4
11.0± 1.3
2.3± 0.9
11.0± 1.2
2.9± 1.4
11.2± 1.2
11.0± 1.1
2.3± 1.1
9.7± 3.4
3.4± 2.3
2.0
exp
Table 3.7 3 Jαβ -coupling constants of subset 1 measured experimentally 3 Jαβ
[65] and the average
calc calculated using the standard Karplus relation with the De Marco [42] paand rmsd of the 3 Jαβ
rameters and using the generalised Karplus relation from the simulation of HEWL in water (53A6)
and the corresponding averaged dihedral angle value hθβ i. The Q-values quantifying the agreeexp
calc i are given at the bottom. All 3 J-couplings and Q-values
and each set of h3 Jαβ
ment between 3 Jαβ
are given in Hz and the hθβ i-angle values are given in degrees.
110
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
Residue
3 PHE
3 PHE
6 CYS1
6 CYS1
15 HISB
15 HISB
18 ASP
18 ASP
20 TYR
20 TYR
23 TYR
23 TYR
27 ASN
27 ASN
30 CYS1
30 CYS1
34 PHE
34 PHE
39 ASN
39 ASN
46 ASN
46 ASN
48 ASP
48 ASP
52 ASP
52 ASP
53 TYR
53 TYR
59 ASN
59 ASN
61 ARG
61 ARG
66 ASP
66 ASP
75 LEU
75 LEU
87 ASP
87 ASP
94 CYS2
94 CYS2
119 ASP
119 ASP
123 TRP
123 TRP
127 CYS2
127 CYS2
Proton
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Q:
3 J exp
αβ
10.0
3.0
11.5
3.5
11.2
2.6
4.2
11.0
2.3
11.7
10.9
2.7
10.3
2.4
5.3
10.8
10.7
5.0
4.5
10.8
11.2
4.7
2.6
3.7
11.6
3.6
10.4
3.0
5.4
11.3
5.7
10.8
5.1
4.5
12.4
2.1
5.1
11.5
4.0
12.2
4.9
11.7
10.6
2.9
11.6
4.8
hθβ i
149± 23
269± 23
165± 17
285± 17
164± 10
284± 10
67± 17
187± 17
92± 30
212± 30
176± 10
296± 10
168± 11
288± 11
58± 13
178± 13
146± 26
266± 26
72± 11
192± 11
114± 39
234± 39
306± 8
66± 8
178± 12
298± 12
168± 9
288± 9
73± 7
193± 7
98± 39
218± 39
65± 20
185± 20
168± 15
288± 15
60± 12
180± 12
69± 8
189± 8
65± 11
185± 11
163± 16
283± 16
66± 155
186± 155
calc i
h3 Jαβ
DeMarco
10.0± 3.1
3.0± 2.5
11.6± 1.5
2.7± 1.5
11.9± 1.1
2.3± 0.7
3.1± 1.3
12.2± 1.5
3.9± 3.0
9.4± 3.9
12.5± 0.5
3.1± 1.0
12.1± 1.1
2.5± 0.9
3.8± 1.3
12.4± 0.9
9.6± 3.3
3.3± 2.8
2.5± 0.9
12.0± 1.0
4.8± 4.5
8.2± 3.1
4.3± 1.1
2.8± 0.9
12.4± 0.8
3.4± 1.2
12.2± 0.7
2.4± 0.6
2.3± 0.5
12.2± 0.6
4.7± 4.0
9.2± 4.3
2.8± 0.9
12.2± 1.6
12.0± 1.7
2.8± 1.2
3.5± 1.3
12.4± 0.7
2.6± 0.7
12.4± 0.5
3.0± 1.1
12.4± 0.8
11.5± 1.6
2.6± 1.3
11.6± 2.5
2.9± 1.2
2.0
calc i
h3 Jαβ
genKarplus
8.6± 2.9
2.4± 2.4
10.3± 1.6
2.2± 1.5
10.5± 1.2
1.7± 0.7
3.1± 1.4
11.2± 1.4
3.3± 2.6
8.6± 3.7
11.4± 0.6
2.6± 1.1
10.8± 1.2
2.0± 0.9
4.0± 1.5
11.3± 0.8
8.3± 3.1
2.7± 2.7
2.4± 1.1
11.1± 0.9
4.0± 4.3
7.5± 3.0
3.5± 1.1
2.7± 1.0
11.3± 0.9
2.9± 1.2
10.9± 0.8
1.9± 0.7
2.2± 0.7
11.3± 0.5
4.2± 3.6
8.4± 4.1
2.8± 1.1
11.2± 1.4
10.7± 1.7
2.2± 1.2
3.7± 1.5
11.3± 0.7
2.6± 0.9
11.4± 0.5
3.1± 1.3
11.4± 0.7
10.1± 1.7
2.0± 1.3
10.4± 2.4
2.4± 1.3
2.0
exp
Table 3.8 3 Jαβ -coupling constants of subset 1 measured experimentally 3 Jαβ
[65] and the average
calc calculated using the standard Karplus relation with the De Marco [42] paand rmsd of the 3 Jαβ
rameters and using the generalised Karplus relation from the simulation of HEWL in water (54A7)
and the corresponding averaged dihedral angle value hθβ i. The Q-values quantifying the agreeexp
calc i are given at the bottom. All 3 J-couplings and Q-values
and each set of h3 Jαβ
ment between 3 Jαβ
are given in Hz and the hθβ i-angle values are given in degrees.
3.3 Method
111
Residue
3 GLN
8 SER
8 SER
11 ASP
11 ASP
13 ARG
13 ARG
15 PHE
15 PHE
17 LYSH
17 LYSH
20 GLN
26 TYR
26 TYR
29 MET
29 MET
30 LEU
30 LEU
32 ASP
32 ASP
34 LYSH
34 LYSH
36 PHE
36 PHE
37 ASP
37 ASP
38 SER
38 SER
39 SER
39 SER
40 ARG
40 ARG
43 ASN
43 ASN
46 PHE
46 PHE
47 LYSH
47 LYSH
48 PHE
48 PHE
49 MET
49 MET
50 LEU
50 LEU
52 LYSH
52 LYSH
59 TRP
61 GLU
61 GLU
67 SER
67 SER
71 ARG
73 LYSH
73 LYSH
74 LEU
74 LEU
77 SER
77 SER
79 ASP
79 ASP
Proton
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ2
Hβ3
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ3
Hβ2
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ2
Hβ3
Hβ3
Hβ2
Hβ2
Hβ3
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ2
Hβ2
Hβ3
Hβ3
Hβ2
Hβ3
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
calc i
hθβ i
h3 Jαβ
DeMarco
9.2
183± 8
12.6± 0.5
2.0
27± 208
5.7± 4.2
4.0
-92± 208
4.6± 2.9
2.0
346± 183
3.9± 3.1
5.5
226± 183
10.8± 3.6
3.0
4± 188
6.3± 4.6
5.1
244± 188
7.8± 4.8
2.0
255± 64
6.3± 4.8
9.0
135± 64
7.9± 4.5
11.0
190± 176
10.2± 4.0
3.2
310± 176
3.8± 2.5
4.1
281± 12
2.3± 0.7
2.0
195± 49
10.6± 3.8
4.0
315± 49
3.0± 1.1
2.0
94± 151
12.2± 1.7
2.0
214± 151
3.1± 1.5
4.2
207± 29
10.4± 3.2
11.6
87± 29
3.3± 3.0
3.0
163± 39
10.9± 3.5
4.0
283± 39
3.8± 2.9
3.0
136± 92
6.3± 4.7
8.1
256± 92
7.6± 4.5
4.0
328± 35
5.4± 1.4
4.0
88± 35
3.3± 3.2
5.0
-20± 49
4.4± 1.2
10.0
99± 49
5.7± 4.6
3.0
66± 9
2.8± 0.9
9.1
186± 9
12.5± 0.6
2.0
241± 178
2.5± 1.5
3.0
121± 178
5.5± 1.5
2.0
255± 52
6.3± 4.6
14.0
135± 52
8.8± 4.7
3.1
-91± 60
3.8± 3.6
5.0
-211± 60
9.3± 3.6
5.0
88± 32
3.7± 3.4
9.1
208± 32
10.4± 3.6
5.0
42± 177
4.9± 3.8
10.1
282± 177
7.5± 4.6
3.0
154± 108
3.1± 1.4
3.0
274± 108
8.6± 4.4
3.0
285± 25
3.3± 2.3
11.0
165± 25
11.6± 2.5
3.0
273± 37
4.2± 3.5
10.0
153± 37
10.6± 3.8
3.0
242± 131
3.9± 2.7
5.0
122± 131
10.5± 3.9
9.1
177± 13
12.5± 1.1
13.0
252± 151
8.1± 4.9
4.1
12± 151
6.2± 4.4
3.0
233± 49
8.5± 4.3
5.1
113± 49
5.7± 4.9
4.1
226± 49
9.0± 4.5
3.0
195± 114
8.2± 4.8
11.1
75± 114
6.3± 4.7
3.0
281± 21
2.9± 1.7
12.2
161± 21
11.1± 2.2
1.0
59± 9
3.6± 1.1
4.0
299± 9
3.4± 1.0
1.0
-84± 688
4.5± 2.6
5.0 -204± 688
4.8± 3.7
Table 3.9 Continued on next page
3 J exp
αβ
calc i
h3 Jαβ
genKarplus
11.6± 0.4
4.6± 3.7
3.5± 2.7
3.3± 2.9
9.7± 3.3
5.7± 4.4
7.0± 4.3
5.7± 4.6
7.2± 3.9
3.4± 2.4
9.2± 3.8
1.7± 0.7
9.4± 3.6
2.5± 1.3
11.1± 1.6
2.6± 1.5
9.6± 3.1
3.0± 2.7
9.8± 3.2
3.2± 2.8
5.8± 4.3
7.0± 4.2
4.8± 1.4
2.9± 3.0
4.0± 1.3
5.2± 4.1
3.7± 1.1
10.5± 0.7
1.8± 1.5
3.6± 1.6
5.6± 4.4
8.0± 4.1
3.2± 3.5
8.1± 3.3
3.3± 3.0
9.5± 3.5
4.4± 3.6
6.7± 4.3
2.7± 1.3
8.0± 4.0
2.7± 2.2
10.4± 2.4
3.6± 3.4
9.5± 3.4
3.4± 2.6
9.5± 3.6
11.4± 1.1
5.5± 4.2
7.3± 4.5
7.1± 3.2
5.1± 3.8
8.2± 4.3
7.4± 4.5
5.8± 4.2
2.3± 1.6
9.8± 2.2
2.6± 0.9
1.7± 0.8
4.3± 2.4
4.1± 3.5
112
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
Residue
80 TYR
80 TYR
82 TYR
82 TYR
97 LEU
97 LEU
99 PHE
99 PHE
100 ASP
100 ASP
106 LEU
106 LEU
107 GLU
107 GLU
3 J exp
αβ
Proton
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ2
Hβ3
Hβ3
Hβ2
Hβ2
Hβ3
3.0
12.0
1.1
13.2
4.0
11.1
2.4
12.1
2.0
10.0
3.0
12.2
3.0
7.0
Q:
Table 3.9 3 Jαβ -coupling constants of
calc calculated
age and rmsd of the 3 Jαβ
hθβ i
181± 37
61± 37
337± 49
217± 49
239± 42
119± 42
256± 20
136± 20
100± 44
220± 44
237± 38
117± 38
259± 81
19± 81
calc i
h3 Jαβ
DeMarco
11.4± 2.9
3.3± 1.9
5.7± 1.8
8.5± 4.8
6.5± 4.8
7.0± 4.3
3.5± 2.3
7.9± 2.9
5.3± 4.4
9.2± 4.5
6.7± 4.5
6.3± 4.1
4.6± 2.4
3.9± 2.8
3.9
calc i
h3 Jαβ
genKarplus
10.5± 2.7
3.3± 1.9
5.4± 1.8
7.8± 4.7
5.8± 4.6
6.2± 3.8
3.0± 2.2
6.5± 2.8
4.9± 3.9
8.3± 4.3
6.0± 4.3
5.4± 3.7
3.8± 2.2
3.7± 2.6
3.6
exp
subset 1 measured experimentally 3 Jαβ
[66] and the aver-
using the standard Karplus relation with the De Marco [42]
parameters and using the generalised Karplus relation from the simulation of FKBP in vacuum
(45B3) and the corresponding averaged dihedral angle value hθβ i. The Q-values quantifying the
exp
calc i are given at the bottom. All 3 J-couplings and
and each set of h3 Jαβ
agreement between 3 Jαβ
Q-values are given in Hz and the hθβ i-angle values are given in degrees.
Residue
3 GLN
8 SER
8 SER
11 ASP
11 ASP
13 ARG
13 ARG
15 PHE
15 PHE
17 LYSH
17 LYSH
20 GLN
26 TYR
26 TYR
29 MET
29 MET
30 LEU
30 LEU
32 ASP
32 ASP
34 LYSH
34 LYSH
36 PHE
36 PHE
37 ASP
37 ASP
38 SER
38 SER
39 SER
39 SER
40 ARG
40 ARG
Proton
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ2
Hβ3
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ3
Hβ2
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ3
Hβ2
Hβ3
Hβ2
calc i
hθβ i
h3 Jαβ
DeMarco
9.2
122± 51
7.5± 4.9
2.0
221± 57
9.7± 4.3
4.0
101± 57
5.7± 4.3
2.0
240± 75
8.9± 4.4
5.5
120± 75
4.9± 4.1
3.0
63± 156
4.6± 3.6
5.1
-56± 156
8.4± 4.6
2.0
277± 35
3.8± 3.2
9.0
157± 35
11.1± 3.2
11.0
184± 46
11.1± 3.3
3.2
304± 46
3.4± 1.9
4.1 217± 140
3.9± 3.2
2.0
309± 6
4.7± 0.9
4.0
69± 6
2.5± 0.5
2.0
205± 61
9.7± 4.3
2.0
325± 61
3.5± 1.9
4.2
257± 40
5.5± 4.1
11.6
137± 40
8.4± 4.9
3.0 198± 105
4.0± 3.2
4.0 318± 105
6.9± 4.2
3.0
69± 132
7.8± 4.8
8.1 189± 132
5.9± 4.3
4.0
305± 9
4.1± 1.2
4.0
65± 9
3.0± 0.9
5.0
78± 10
2.2± 0.5
10.0
198± 10
11.5± 1.3
3.0
57± 23
3.4± 1.1
9.1
177± 23
12.3± 1.7
2.0
66± 7
2.8± 0.7
3.0
306± 7
4.2± 1.0
2.0
288± 11
2.6± 0.9
14.0
168± 11
12.2± 1.0
Table 3.10 Continued on next page
3 J exp
αβ
calc i
h3 Jαβ
genKarplus
6.9± 4.4
8.3± 3.3
5.7± 3.2
8.1± 4.1
4.4± 3.7
4.1± 3.4
7.5± 4.3
3.2± 3.1
10.0± 2.8
2.9± 1.9
10.0± 3.1
3.3± 3.0
4.0± 0.9
2.3± 0.6
8.7± 4.1
3.1± 1.9
4.8± 3.9
7.4± 4.5
3.5± 3.0
6.5± 3.8
7.0± 4.4
5.3± 4.0
3.4± 1.2
2.9± 1.0
1.9± 0.7
10.7± 1.2
4.4± 1.4
10.4± 1.5
1.9± 0.6
2.3± 0.9
2.0± 0.9
10.9± 1.1
3.3 Method
113
Residue
43 ASN
43 ASN
46 PHE
46 PHE
47 LYSH
47 LYSH
48 PHE
48 PHE
49 MET
49 MET
50 LEU
50 LEU
52 LYSH
52 LYSH
59 TRP
61 GLU
61 GLU
67 SER
67 SER
71 ARG
73 LYSH
73 LYSH
74 LEU
74 LEU
77 SER
77 SER
79 ASP
79 ASP
80 TYR
80 TYR
82 TYR
82 TYR
97 LEU
97 LEU
99 PHE
99 PHE
100 ASP
100 ASP
106 LEU
106 LEU
107 GLU
107 GLU
3 J exp
αβ
Proton
Hβ3
Hβ2
Hβ2
Hβ3
Hβ3
Hβ2
Hβ2
Hβ3
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ2
Hβ2
Hβ3
Hβ3
Hβ2
Hβ3
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ3
Hβ2
Hβ2
Hβ3
Hβ3
Hβ2
Hβ2
Hβ3
3.1
5.0
5.0
9.1
5.0
10.1
3.0
3.0
3.0
11.0
3.0
10.0
3.0
5.0
9.1
13.0
4.1
3.0
5.1
4.1
3.0
11.1
3.0
12.2
1.0
4.0
1.0
5.0
3.0
12.0
1.1
13.2
4.0
11.1
2.4
12.1
2.0
10.0
3.0
12.2
3.0
7.0
Q:
hθβ i
46± 159
-73± 159
64± 14
184± 14
25± 166
-94± 166
317± 10
77± 10
212± 37
92± 37
215± 37
95± 37
250± 73
130± 73
124± 19
143± 46
263± 46
56± 15
296± 15
285± 35
212± 43
92± 43
250± 43
130± 43
62± 8
302± 8
51± 10
291± 10
233± 28
113± 28
234± 45
114± 45
292± 15
172± 15
286± 10
166± 10
67± 9
187± 9
273± 40
153± 40
166± 41
286± 41
calc i
h3 Jαβ
DeMarco
4.3± 3.2
9.5± 4.3
3.2± 1.2
12.3± 1.0
6.7± 4.3
7.4± 4.9
5.7± 1.4
2.2± 0.6
9.7± 4.1
4.4± 3.6
9.8± 3.8
4.2± 4.0
7.8± 4.6
5.8± 4.5
6.2± 2.7
9.4± 4.5
5.5± 4.3
3.8± 1.1
3.4± 1.4
4.0± 2.9
9.9± 4.2
5.0± 4.0
5.7± 4.1
7.9± 4.1
3.2± 0.9
3.8± 1.0
4.6± 1.3
2.7± 0.9
6.7± 4.0
5.2± 3.6
7.1± 5.1
6.9± 4.3
3.1± 1.2
12.1± 1.3
2.4± 0.6
12.1± 0.9
2.8± 0.9
12.5± 0.8
4.2± 3.6
10.3± 3.7
10.7± 3.6
3.8± 2.9
3.5
calc i
h3 Jαβ
genKarplus
3.8± 3.1
8.5± 4.0
3.3± 1.5
11.3± 1.0
6.0± 4.1
6.7± 4.5
5.1± 1.5
1.9± 0.7
8.8± 3.9
4.0± 3.2
9.0± 3.6
3.7± 3.6
7.0± 4.3
5.2± 4.1
4.8± 2.4
4.8± 4.1
8.5± 4.1
2.7± 1.0
1.7± 1.2
3.5± 2.8
9.0± 4.0
4.6± 3.5
5.0± 4.0
6.9± 3.8
2.3± 0.8
1.9± 0.8
4.7± 1.5
2.0± 0.9
6.0± 3.9
4.2± 3.1
6.3± 4.9
6.2± 3.6
2.5± 1.3
10.9± 1.4
1.8± 0.7
10.7± 1.0
2.9± 1.1
11.4± 0.7
3.6± 3.4
9.2± 3.3
9.6± 3.3
3.3± 2.8
3.4
exp
Table 3.10 3 Jαβ -coupling constants of subset 1 measured experimentally 3 Jαβ
[66] and the avcalc calculated using the standard Karplus relation with the De Marco
erage and rmsd of the 3 Jαβ
[42] parameters and using the generalised Karplus relation from the simulation of FKBP in water
(45A3) and the corresponding averaged dihedral angle value hθβ i. The Q-values quantifying the
exp
calc i are given at the bottom. All 3 J-couplings and
and each set of h3 Jαβ
agreement between 3 Jαβ
Q-values are given in Hz and the hθβ i-angle values are given in degrees.
exp
values for subsets 1 and 2, or 1, 2, and 3 of the 3 Jαβ
in all other proteins as a jack-knife test.
The stereospecific assignment of subset 3 giving the lowest Q-value was chosen by assigning
exp
calc i-value of
-values given for a residue to the lower calculated h3 Jαβ
the lower of the two 3 Jαβ
this residue. The degree of convergence of the trajectory averages of quantities such as 3 Jαβ ,
θ , cos θ , and cos2 θ was investigated using cumulative ensemble averages h....it . To follow
114
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
the evolution of the ensemble averages h....i over time, the total simulation time was divided
into 10 equal time periods, i.e. 100 ps for Plastocyanin, 2 ns for HEWL and 800 ps for FKBP.
The determination of an average value hθβ i for a dihedral angle θβ will depend on the range
of values considered, e.g. [−180◦ , +180◦ ] or [0◦ , +360◦ ]. Therefore, as well as calculating the
mean, we calculated the probability distribution P(θβ ) for a given range, i.e. [−180◦ , +180◦ ],
from a MD trajectory and used the θβ -value for which P(θβ ) is largest, i.e. the median, rather
than the mean. For the ensemble of NMR structures of Plastocyanin, 9PCY, and HEWL,
1E8L, and the two X-ray structures of HEWL, 193L, the mean dihedral angle was always
used. The root-mean-square fluctuation of θβ was calculated from the P(θ ) obtained from the
MD trajectory in which the θβ -values were not mapped onto a finite range.
3.4 Results
3.4.1 Calculation of 3 Jαβ -values
A diverse range of parameters for the standard Karplus relation have been proposed [42,
56–58, 60, 62], see Fig. 3.2 and Table 3.1, based on different parametrisation methods and
molecules. This creates uncertainty as to how to choose the optimal parameter set for Eq. 3.1
for use in protein structure determination. The parameters of De Marco et al. [42] are most
widely used, and also result in Karplus curves that lie between the curves generated by the
other parameter sets, thus we use these for our initial investigations. The distribution of the
exp
measured 3 Jαβ
-couplings over the protein structures is shown in Fig. 3.5. For all three proteins they are well spread over the residues and throughout the space occupied by the protein.
Figure 3.5 Cartoon pictures of Plastocyanin (NMR model structure 16 of 9PCY [22] with the
coordinated copper ion in orange, left panel), HEWL (X-ray structure 1AKI [72][73], middle panel),
and FKBP (X-ray structure 1FKF [66] with the bound ascomycin in blue, right panel). The amino
acids for which 3 Jαβ -values are available are shown in red (stereospecifically assigned 3 Jαβ2/3 ),
green (residues Val, Ile and Thr with only one Hβ ), and yellow (non-stereospecifically assigned
3J
αβ2/3 ). In the right panel the orange amino acids are the ones for which only one of the two
stereospecifically assigned 3 Jαβ2/3 is available.
exp
Fig. 3.3 shows the measured 3 Jαβ
-coupling constants of subsets 1 and 2 versus the value of
θβ (or hθβ i for a set of NMR model structures) in the X-ray or NMR structures along with
3.4 Results
115
exp
the Karplus relation using the De Marco [42] parameters. Several 3 Jαβ
-values deviate considerably from the value suggested by the curve for the corresponding θβ or hθβ i. Moreover,
the values of θβ or hθβ i for the dihedral angles for which 3 Jαβ -couplings were measured do
not cover the whole dihedral angle value range. For HEWL and FKBP in particular, they are
clustered around the canonical rotamer positions. This is a known issue in the determination
of Karplus parameters.
Figure 3.6 Comparison of the stereospecifically assigned 3 Jαβ -couplings (subset 1) measured
exp
calc i using the parameters of De Marco [42] and the
and calculated h3 Jαβ
experimentally 3 Jαβ
2/3
2/3
standard Karplus relation (black) and using the generalised Karplus relation (red) for Plastocyanin
(top row), HEWL (second and third rows), and FKBP (bottom row). The blue lines indicate a
calc i are calculated from the simulations in vacuum (left panels except
deviation of ± 1 Hz. The h3 Jαβ
second row) and in water (middle panels and left panel second row) and from the experimental
model structures (right panels). In the second row, results from two water simulations (45A3, left
panel, and 53A6, middle panel) and two X-ray structure sets 1AKI (green and blue dots for the
standard and generalised Karplus relation respectively) and 193L (black and red dots) are given.
calc i calculated from the X-ray or NMR model structures
Firstly, the averaged values h3 Jαβ
or from MD simulation trajectories are compared to those measured experimentally for each
calc i-values were computed using the standard Karplus relation
protein, see Fig. 3.6. The h3 Jαβ
with the De Marco parameters, and using the generalised Karplus relation as described in the
116
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
Method section. A deviation of ± 1 Hz is considered to be acceptable, given the uncertainty
calc i calculated from the water simulation
in the Karplus parameters. For Plastocyanin, the h3 Jαβ
using the 53A6 force field using the standard Karplus relation lie slightly closer to the experimental values than those calculated from the vacuum simulation using the 45B3 force field,
giving rise to Q-values of 1.8 Hz (Table 3.3) and 2.1 Hz (Table 3.4) respectively. Many of
calc i calculated from the set of NMR model structures, shown in Fig. 3.6, deviate more
the h3 Jαβ
exp
-values, but the overall agreement is better than for the
than ± 1 Hz from the measured 3 Jαβ
MD simulations (Q = 1.5 Hz). Use of the generalised Karplus relation of Eq. 3.4 results in
Q-values of 2.0 Hz for the water simulation, 2.2 Hz for the vacuum simulation and 1.4 Hz for
the set of NMR model structures. Except for the latter, the deviation from the measured data is
calc i-values calculated with the standard Karplus relation of Eq. 3.1.
even larger than for the h3 Jαβ
calc i-couplings
It is also obvious that with the generalised Karplus relation, the calculated h3 Jαβ
shift to lower values.
For HEWL, Fig. 3.6 shows a similar situation: the vacuum simulation (54B7 force field)
yields the highest Q (3.4 Hz, Table 3.5), the water simulations perform better (Q = 2.5 Hz
for 45A3, Table 3.6, and 2.0 Hz for force fields 53A6 and 54A7, Tables 3.7 and 3.8), and
calc i obtained from the set of NMR model structures (1E8L) agree best with the exthe h3 Jαβ
perimental data (Q = 1.7 Hz). Upon application of the generalised Karplus relation, the same
calc i-values is observed, but the Q-value improves slightly for the
shift downwards of the h3 Jαβ
vacuum simulation (Q = 3.3 Hz) and for the water simulation using the 45A3 force field (Q
= 2.4 Hz), stays the same for the two other water simulations using the force fields 53A6 and
54A7 (Q = 2.0 Hz), and becomes worse for the NMR model structures (Q = 2.0 Hz). The
Q-value for the X-ray structure 1AKI decreases from 1.6 Hz for the standard Karplus relation
to 1.5 Hz for the generalised Karplus relation. The same tendency is observed for the average
over the two X-ray structures from 193L, where the Q-value is 1.4 Hz for the standard Karplus
relation and 1.2 Hz for the generalised Karplus relation.
calc i-values diverge much more from the measured ones than
For FKBP, the calculated h3 Jαβ
for the other two proteins, see Fig. 3.6. This is also evident in the high Q-values obtained using
the standard Karplus relation: 3.9 Hz for the vacuum simulation using the 45B3 force field
(Table 3.9) and 3.5 Hz for the water simulation using the 45A3 force field (Table 3.10). Even
calc -values from the X-ray structure yield a Q-value of 3.0 Hz. Using the generalised
the 3 Jαβ
Karplus relation, the agreement improves slightly, but with Q-values of 3.6, 3.4, and 2.8 Hz for
the vacuum simulation, water simulation and X-ray structure respectively it is still worse than
calc i-couplings
for the other two proteins. As for Plastocyanin and HEWL, the calculated h3 Jαβ
shift towards lower values when the generalised Karplus relation is used. It is noteworthy that
exp
-values are close to integer values, suggesting that
for FKBP, most of the experimental 3 Jαβ
this data may be of limited precision.
For all three proteins, the inclusion of solvent in the MD simulations improves the agreement with experimental data when either the standard or generalised Karplus relation is used.
Despite including substituent effects, use of the generalised Karplus relation does not signi-
3.4 Results
117
exp
ficantly improve the agreement with the measured 3 Jαβ
-couplings. In all cases, however, the
calc i-values calculated from the X-ray and NMR model structures agree better with the
h3 Jαβ
exp
. This may be a consequence of using parameters for the Karplus relations,
measured 3 Jαβ
both standard and generalised, that were derived using rather rigid small molecules.
Indeed, both the standard and the generalised Karplus relation link experimentally measured
3 J exp -couplings to a single angle value, implying a static structure. Because NMR experiments
αβ
take place in solution or are measured from a powder, however, it is expected that the proteins
are mobile on the time-scale of the measurement or that they point in different directions
exp
-couplings are averages over an ensemble of
in the powder, meaning that the measured 3 Jαβ
structures as well as over the timescale of the experiment.
3.4.2 Least-squares fitting of Karplus parameters
Protein
Structure set
Source
Number of structures
(Simulation length)
Least-squares fitted parameters
a
b
c
Q
Plasto-
45B3
MD (vac)
200 (1 ns)
5.66
-1.41
4.51
2.00
cyanin
53A6
MD (wat)
200 (1 ns)
6.61
-0.93
3.96
1.86
9PCY
NMR
16
6.61
-1.07
3.36
1.13
54B7
MD (vac)
4000 (20 ns)
-0.21
-3.72
6.32
2.87
54A7
MD (wat)
4000 (20 ns)
5.00
-2.56
4.10
1.69
53A6
MD (wat)
4000 (20 ns)
4.80
-2.42
4.27
1.98
45A3
MD (wat)
4000 (20 ns)
2.40
-2.67
5.11
2.69
1AKI
X-ray
1
3.99
-2.68
4.42
1.10
193L
X-ray
2
5.87
-1.59
3.50
0.99
1E8L
NMR
50
3.55
-3.38
4.51
1.10
45B3
MD (vac)
1600 (8 ns)
0.60
-2.72
5.60
3.63
45A3
MD (wat)
1600 (8 ns)
-1.14
-4.48
6.25
3.38
1FKF
X-ray
1
5.34
-0.65
3.70
3.15
HEWL
FKBP
Table 3.11 Karplus relation parameters a, b, and c and the corresponding Q-value obtained by
calc i calculated for the indicated structures to the stereospecifically assigned
fitting the values of h3 Jαβ
exp
3
measured Jαβ -coupling constants (subsets 1 and 2). All values are given in Hz. The structure
sets are either MD trajectories from simulations in a vacuum (vac) or water (wat) environment with
a particular force field, or experimental X-ray or NMR model structures from the PDB [34]. The
structure set is denoted either by the PDB entry or by the code of the force field used.
calc i,
One way to overcoming this discrepancy is to employ least-squares fitting of the h3 Jαβ
averaged over MD trajectories or a set of experimental X-ray or NMR model structures, to the
exp
measured 3 Jαβ
-couplings to determine a, b, and c parameters for the standard Karplus relation
118
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
in Eq. 3.1. Table 3.11 lists the parameters a, b, and c obtained in this manner using only subsets
1 and 2 of the 3 Jαβ -couplings for each simulation of each protein and from the experimentally
determined structure(s), along with the Q-values for each fit. For all three proteins, the same
calc i-values from the vacuum simulations resulting in the
result is seen as before, with the h3 Jαβ
calc i-values from the experimental structures yielding a better fit
highest Q-values, and the h3 Jαβ
exp
-values than those from the MD simulation trajectories. The enhanced conformato the 3 Jαβ
tional flexibility in the MD trajectories seems to complicate the fitting to a Karplus relation of
the form of Eq. 3.1. This may be due to the fact that not all of the dihedral angles necessarily undergo the same degree of conformational averaging in an MD simulation, meaning that
calc i-values used for the fitting are averages over a wide distribution and other
some of the h3 Jαβ
arise from dihedral angles that are nearly rigid. This degree of conformational averaging for a
specific dihedral angle may or may not correspond to the one occurring in experiment.
Figure 3.7 Median (black circles) and rms variation (bars) of the dihedral angle values θβ and
calc i-values of subsets 1 and 2 calculated from the MD simulations in water of
corresponding h3 Jαβ
2/3
Plastocyanin (53A6 force field, upper panel), HEWL (54A7 force field, middle panel) and FKBP
(45A3 force field, lower panel) using the standard Karplus relation with the De Marco [42] parameters. The Karplus curves generated using the De Marco parameters are shown as black
lines.
The degree of conformational sampling that the side-chain dihedral angles θβ with stereexp
ospecifically assigned 3 Jαβ
-values (subset 1 and 2) undergo during MD simulation is shown
in Fig. 3.7 for the simulations of Plastocyanin (53A6), HEWL (54A7) and FKBP (45A3) in
calc -values calculated using the De Marco parameters.
water, along with the corresponding 3 Jαβ
calc -values show significant variation, with the
Both the dihedral angle values θβ and the 3 Jαβ
3 J calc varying by up to 10 Hz. Not all dihedral angles undergo the same degree of conforαβ
calc -values depends on where on
mational sampling. Moreover, the range of corresponding 3 Jαβ
the Karplus curve the dihedral angle value θβ lies: variation in dihedral angle values situated
3.4 Results
119
calc i-values of subsets 1
Figure 3.8 Median of the dihedral angle values θβ and the average h3 Jαβ
2/3
and 2 calculated from the MD simulations of Plastocyanin (upper panel), HEWL (middle panel),
and FKBP (lower panel) and the Karplus curves obtained using the parameters optimised by leastcalc i-values to the measured 3 J exp -values of the subsets 1 and 2.
squares fitting of the plotted h3 Jαβ
αβ
2/3
2/3
For Plastocyanin and FKBP, the filled circles and solid lines are for the simulations in water (53A6
or 45A3 force field, respectively) and the open circles and dotted lines are for the simulations in
vacuum (45B3 force field). For HEWL, the filled circles and solid line are for the simulation in water
(54A7 force field), the open circles and dotted line for the simulation in vacuum (54B7 force field),
the crosses and dashed line are for the simulation in water (45A3 force field), and the triangles
and dot-dot-dashed line are for the simulation in water (53A6 force field).
calc -values, whereas a
in flat parts of the Karplus curve has relatively little effect on the 3 Jαβ
small change in a θβ located in a steep part of the Karplus curve results in a comparably large
calc -value. Because of this, the h3 J calc i-values calculated from the paramchange in the 3 Jαβ
αβ
eters obtained in the fitting procedure are often different from the 3 Jαβ -values predicted by
the Karplus relation using the same parameters for the corresponding hθβ i. Together, these
effects cause a large variation in the parameters in Table 3.11 obtained using least-squares
fitting to subsets 1 and 2 for the different simulations of the three proteins studied here and in
the resulting Karplus curves shown in Figs. 3.8 and 3.9. Even for the same protein, fitting the
Karplus parameters to different simulations with different force fields, in water or in vacuum,
yields different parameter sets, most noticeably for HEWL and FKBP, see Figs. 3.8 and 3.9
and Table 3.11. The three curves obtained for FKBP are all rather different, with the curves
obtained from the MD trajectories exhibiting only one maximum. For HEWL, the curve for
the vacuum simulation also has only one maximum. The remainder of the curves for HEWL,
along with those for Plastocyanin, display the expected two maxima, but their heights vary
considerably. For HEWL, the simulations in water using the 54A7 and 53A6 force fields
produce very similar Karplus parameters and curves, but the simulation carried out using the
45A3 force field yields a curve almost without a second maximum. The two different X-ray
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
120
Figure 3.9 Karplus curves generated using the optimised parameters obtained by least-squares
calc i-values calculated from different structure sets to the measured 3 J exp for subsets
fitting of h3 Jαβ
αβ
2/3
2/3
1 and 2 for Plastocyanin (black), HEWL (red), or FKBP (green). For Plastocyanin and FKBP, the
solid lines correspond to the simulation in water (53A6 and 45A3 force field, respectively), the
dotted lines to the simulation in vacuum (45B3 force field) and the dash-dotted lines to the NMR
model structures (9PCY) or X-ray structure (1FKF), respectively. For HEWL, three different Xray or NMR model structures (dot-dashed lines) and MD trajectories (solid lines) in water were
analysed: the 1AKI X-ray structure and the simulation using the 54A7 force field (thin lines), the
1E8L NMR model structures and the simulation using the 53A6 force field (normal lines), and
the 193L X-ray structures and the simulation using the 45A3 force field (thick lines). The Karplus
curve generated using the parameters optimised against the vacuum simulation of HEWL (54B7
force field) is shown as a dotted red line.
structure sets, 1AKI and 193L, and the NMR model structures 1E8L give rise to three quite
different curves in terms of the height of the maximum centred at θβ = 0◦ , and the curves
obtained from fitting to the MD simulation data using the 54A7 and 53A6 force fields lie in
between.
The large variation around θβ = 0◦ between the differently fitted Karplus curves in Fig.
3.9 is due to the lack of dihedral angle values in the range −60◦ < θβ < 60◦ in the X-ray
or NMR structures or MD simulation trajectories, as shown in Figs. 3.3 and 3.7. Chemically
this makes sense, as eclipsed conformations are generally disfavored compared to staggered
conformations. In experimental structure refinement, often only staggered conformations,
the rotamers g+ , g− and t, are considered to be energetically favourable. In contrast, in the
MD simulations, quite a wide range of angle values is sampled outside of −60◦ < θβ < 60◦ ,
although the median, i.e. the most populated dihedral angle values θβ , are concentrated around
the classical rotamer positions θβ = ± 60◦ and ± 180◦ . Ultimately, however, it is angle values
around 0◦ ± 60◦ that determine the shape of the curve, as the minima and maxima of the
Karplus relation are mainly defined by the cos2 function and the b parameter of the cos part of
the Karplus relation determines the shape of the curve around 0◦ .
3.4 Results
121
The conformational motion that takes place during the MD simulations means that the leastsquares fitted Karplus parameters will depend on the length of the simulation used in the fitting
procedure, i.e. on the range of different structures that are included. The dependence of the
Figure 3.10 Karplus parameters a (open circles), b (triangles), and c (filled circles) as a function
calc i-values used in the fitting
of the proportion of the simulation period used to calculate the h3 Jαβ
procedure for Plastocyanin (53A6 force field, 100% = 1 ns), HEWL (54A7 force field, 100% = 20
ns), and FKBP (45A3 force field, 100% = 8 ns).
parameters a, b, and c on the size of the time range considered for fitting is shown in Fig.
3.10 for the three proteins. The a, b, and c values differ between the proteins and vary over
the whole simulation period, even for the 20 ns simulation of HEWL in water. The origin
of this variation can be seen in Figs. 3.11-3.13, where the averages hcos2 θβ i and hcos θβ i
calc i calculated using
over each 10% window of the total simulation time together with the h3 Jαβ
the least-squares fitted parameters for this time window are given for the three proteins. For
Plastocyanin, Fig. 3.11, the side-chain angles θβ of many of the residues, e.g. Ser 11, Val 15,
Val 21, Pro 22, Val 40, Pro 47, Val 53, Pro 58, Leu 63, Val 72, Thr 73, Thr 79, Pro 86 , Val
96, and Thr 97 show considerable motion during the simulation, resulting in quite different
calc i-values for each window. In the case of HEWL, Fig. 3.12, less
hcos2 θβ i, hcos θβ i and h3 Jαβ
motion occurs, but still some residues, e.g. Val 2, Phe 3, Tyr 20, Val 29, Phe 34, Asn 46, Thr
51, Thr 69, Val 92, 99, and 109, Trp 123 and Ile 124 show different values of the averages in
each time window. In the FKBP simulation, nearly half of the residues exhibit quite different
calc i over time as shown in Fig. 3.13.
values of hcos2 θβ i, hcos θβ i, and h3 Jαβ
calc i-values for all three subsets 1, 2 and 3 for all structure sets of all proThe calculated h3 Jαβ
teins calculated using the least-squares fitted Karplus parameters of the corresponding simulat-
122
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
calc i-values calculated from 10 time windows (each 100 ps) of the MD simulation of Plastocyanin
Figure 3.11 hcos2 θβ i, hcosθβ i, and h3 Jαβ
exp
calc i-values were obtained using the
in water (53A6 force field). The measured 3 Jαβ
-values [22] are shown as black squares. h3 Jαβ
exp
3
Karplus parameters a, b, and c from the least-squares fit to the measured Jαβ -values using the averaged hcos2 θβ i and hcosθβ i of the
corresponding time window.
3.4 Results
calc i-values calculated from 10 time windows (each 2 ns) of the MD simulation of HEWL in water
Figure 3.12 hcos2 θβ i, hcosθβ i, and h3 Jαβ
exp
calc i-values were obtained using the Karplus param(54A7 force field). The measured 3 Jαβ
-values [65] are shown as black squares. h3 Jαβ
exp
eters a, b, and c from the least-squares fit to the measured 3 Jαβ -values using the averaged hcos2 θβ i and hcosθβ i of the corresponding
time window.
123
124
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
calc i-values calculated from 10 time windows (each 800 ps) of the MD simulation of FKBP
Figure 3.13 hcos2 θβ i, hcosθβ i, and h3 Jαβ
exp
calc i-values were obtained using the
in water (45A3 force field). The measured 3 Jαβ
-values [66] are shown as black squares. h3 Jαβ
exp
3
Karplus parameters a, b, and c from the least-squares fit to the measured Jαβ -values using the averaged hcos2 θβ i and hcosθβ i of the
corresponding time window.
3.4 Results
125
exp
Figure 3.14 Comparison of all (subsets 1-3) measured 3 Jαβ
-values for Plastocyanin (top row),
HEWL (second and third rows), and FKBP (bottom row) with those calculated using the parameters of [42] (black) or the least-squares fitted parameters (red) optimised for that structure set.
Note that the optimised Karplus parameters were obtained using subsets 1 and 2. The blue lines
calc i are calculated from the simulations in vacuum (left
indicate a deviation of ± 1 Hz. The h3 Jαβ
panels) and in water (middle panels) and from the experimental model structures (right panels). In
the second row, results from two water simulations (45A3, left panel, and 53A6, middle panel) and
two X-ray structure sets 1AKI (green and blue dots for using the de Marco and the least-squares
fitted parameters respectively) and 193L (black and red dots) are given.
exp
tion or the De Marco parameters are compared to the measured 3 Jαβ
in Fig. 3.14. For subset
3, the assignment was chosen to minimise the Q-value, i.e. by assigning the larger of the two
calc i-couplings to the larger of the two 3 J exp -coupling constants.
h3 Jαβ
αβ
The robustness of a given set of parameters may be quantified by conducting jack-knife
tests, in which the parameters obtained from fitting to one particular structure set of one protein
calc i for another structure set, possibly of another protein, and
are used to back-calculate h3 Jαβ
the goodness of fit (Q-value) is compared to the one obtained for the structure set used in
the fitting procedure. Jack-knife tests were carried out for all possible combinations of fitted
Karplus parameters, structure sets and proteins.
exp
The assignment of the unassigned 3 Jαβ
of subset 3 adds some complication to this procedure. Two possible assignment protocols were tested:
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
126
1. Using a particular structure set and Karplus parameters a, b, and c optimised using
the 3 Jαβ -values of subset 1 and 2 and that particular structure set, the assignment of
the 3 Jαβ -values of subset 3 is chosen such that the Q-value is minimal. Subsequently,
this assignment of subset 3 is used for all calculations of Q-values for that particular
structure set using all the different sets of a, b, and c parameters, see Table 3.12.
2. For every combination of structure set and Karplus parameters a, b, and c, the assignment of the 3 Jαβ -values of subset 3 is chosen such that the Q-value is minimal for that
combination, see Table 3.13.
The Q-values obtained using the second procedure are, as expected, lower, but the differences
are mostly small or nonexistent.
Q calculated for
Parameters fitted to
Plastocyanin
HEWL
FKBP
45B3
53A6
9PCY
54B7
54A7
53A6
45A3
1AKI
193L
1E8L
45B3
45A3
1FKF
Plasto-
45B3
1.98
1.98
2.07
2.43
2.14
2.10
2.24
2.18
2.14
2.30
2.56
2.72
2.24
cyanin
53A6
1.84
1.81
1.87
2.31
1.97
1.93
2.09
2.01
1.94
2.14
2.44
2.58
2.07
9PCY
1.55
1.51
1.44
2.09
1.61
1.56
1.74
1.63
1.49
1.81
2.19
2.44
1.65
54B7
2.66
2.66
2.64
2.42
2.69
2.64
2.46
2.59
2.64
2.66
2.49
2.46
2.56
54A7
1.77
1.71
1.63
1.79
1.64
1.63
1.69
1.61
1.63
1.63
1.98
1.80
1.82
53A6
1.95
1.89
1.81
1.94
1.84
1.83
1.86
1.81
1.81
1.84
2.10
1.95
1.95
45A3
2.53
2.49
2.38
2.34
2.46
2.43
2.32
2.37
2.37
2.43
2.39
2.33
2.37
1AKI
1.73
1.71
1.55
1.62
1.68
1.61
1.43
1.52
1.52
1.68
1.70
1.75
1.51
193L
1.65
1.61
1.48
1.58
1.68
1.60
1.44
1.54
1.49
1.71
1.72
1.71
1.45
1E8L
1.63
1.71
1.61
1.52
1.25
1.26
1.30
1.18
1.45
1.13
1.69
1.59
1.83
HEWL
FKBP
45B3
3.92
3.89
3.78
3.69
3.85
3.83
3.69
3.77
3.76
3.82
3.63
3.67
3.68
45A3
3.63
3.61
3.51
3.41
3.52
3.51
3.42
3.46
3.48
3.48
3.42
3.38
3.48
1FKF
3.38
3.34
3.25
3.34
3.41
3.36
3.24
3.32
3.26
3.45
3.28
3.45
3.15
exp
Table 3.12 Q-values in Hz quantifying the similarity between the measured 3 Jαβ
-values and the
calc i of all three subsets for each structure set for each of the three proteins using the
calculated h3 Jαβ
Karplus parameters obtained using the same structure set of that protein (bold) and using each
structure set of all proteins. Q-values obtained using parameters optimised for another structure
set that are lower than the value obtained using the parameters optimised for the same structure
set are in italics. The assignment of the non-stereospecifically assigned couplings (subset 3) for
a given structure set was always the same, namely the one that minimises the Q-value obtained
when back-calculating all three subsets of couplings from that structure set using the fitted Karplus
parameters determined using subsets 1 and 2 and the same structure set (protocol 1).
A robust parameter set might be expected to perform similarly in terms of Q-values for all
structure sets of all proteins, not just for the one it was optimised for. Applying this criterion is
calc i-values calculated from the various structure
complicated, however, by the fact that the h3 Jαβ
3.4 Results
Q calculated for
Parameters fitted to
Plastocyanin
HEWL
FKBP
De Marco [42]
45B3
53A6
9PCY
54B7
54A7
53A6
45A3
1AKI
193L
1E8L
45B3
45A3
1FKF
Plasto-
45B3
1.98
1.98
2.07
2.40
2.14
2.10
2.23
2.17
2.14
2.30
2.55
2.67
2.24
2.23
cyanin
53A6
1.84
1.81
1.87
2.30
1.97
1.93
2.09
2.01
1.94
2.13
2.43
2.57
2.07
2.42
9PCY
1.55
1.51
1.44
2.06
1.60
1.55
1.73
1.62
1.49
1.80
2.18
2.41
1.65
1.93
54B7
2.65
2.63
2.61
2.42
2.69
2.64
2.46
2.58
2.63
2.66
2.49
2.46
2.54
3.23
54A7
1.77
1.71
1.63
1.79
1.64
1.63
1.69
1.61
1.63
1.63
1.98
1.78
1.82
1.95
53A6
1.95
1.89
1.81
1.94
1.84
1.83
1.86
1.81
1.81
1.84
2.09
1.94
1.95
2.10
45A3
2.53
2.48
2.37
2.34
2.45
2.43
2.32
2.37
2.37
2.43
2.39
2.33
2.36
2.77
1AKI
1.73
1.71
1.55
1.61
1.68
1.61
1.43
1.52
1.52
1.68
1.70
1.75
1.51
2.44
193L
1.65
1.61
1.48
1.55
1.68
1.60
1.44
1.54
1.49
1.71
1.71
1.67
1.45
2.39
1E8L
1.51
1.53
1.42
1.52
1.23
1.24
1.30
1.17
1.33
1.13
1.69
1.59
1.68
2.00
45B3
3.92
3.89
3.78
3.69
3.85
3.83
3.69
3.77
3.76
3.82
3.63
3.67
3.68
4.08
45A3
3.63
3.61
3.51
3.41
3.52
3.51
3.42
3.46
3.48
3.48
3.42
3.38
3.48
3.74
1FKF
3.38
3.34
3.25
3.34
3.41
3.36
3.24
3.32
3.26
3.45
3.28
3.45
3.15
3.79
HEWL
FKBP
exp
calc i of all three subsets
Table 3.13 Q-values in Hz quantifying the similarity between the measured 3 Jαβ
-values and the calculated h3 Jαβ
for each structure set for each of the three proteins using the Karplus parameters obtained using the same structure set of that protein
(bold) and using each structure set of all proteins. Q-values obtained using parameters optimised for another structure set that are
lower than the value obtained using the parameters optimised for the same structure set are in italics. For comparison, the Q-values
calc i with the De Marco [42] parameters are also given. The assignment of the non-stereospecifically
obtained when calculating the h3 Jαβ
assigned couplings (subset 3) was individually chosen to optimise the Q-value obtained when back-calculating all three subsets of
couplings for each combination of structure set and Karplus parameters (protocol 2).
127
128
Q calculated for
Parameters fitted to
Plastocyanin
HEWL
FKBP
De Marco [42]
53A6
9PCY
54B7
54A7
53A6
45A3
1AKI
193L
1E8L
45B3
45A4
1FKF
Plasto-
45B3
2.00
2.02
2.13
2.61
2.17
2.14
2.34
2.23
2.19
2.36
2.74
2.95
2.35
2.47
cyanin
53A6
1.88
1.86
1.95
2.57
2.07
2.03
2.26
2.13
2.03
2.29
2.68
2.91
2.18
2.30
9PCY
1.31
1.27
1.13
2.13
1.39
1.33
1.62
1.43
1.19
1.69
2.22
2.58
1.44
1.78
54B7
3.04
3.04
3.03
2.87
3.10
3.05
2.89
3.00
3.04
3.08
2.95
2.89
2.97
3.57
54A7
1.81
1.78
1.76
2.03
1.69
1.70
1.89
1.72
1.76
1.70
2.29
2.03
2.08
1.99
53A6
2.06
2.03
2.02
2.20
1.99
1.98
2.11
2.00
2.03
2.00
2.44
2.24
2.27
2.28
45A3
2.84
2.82
2.75
2.71
2.82
2.79
2.69
2.74
2.75
2.81
2.80
2.73
2.75
3.14
1AKI
1.42
1.41
1.23
1.53
1.17
1.14
1.28
1.10
1.17
1.16
1.86
1.66
1.55
1.79
193L
1.32
1.25
1.02
1.46
1.12
1.08
1.21
1.03
0.99
1.12
1.79
1.53
1.36
1.64
1E8L
1.57
1.64
1.54
1.66
1.18
1.20
1.40
1.16
1.39
1.10
1.94
1.76
1.89
1.91
45B3
3.92
3.89
3.78
3.69
3.85
3.83
3.69
3.77
3.76
3.82
3.63
3.67
3.68
4.08
45A3
3.63
3.61
3.51
3.41
3.52
3.51
3.42
3.46
3.48
3.48
3.42
3.38
3.48
3.74
1FKF
3.38
3.34
3.25
3.34
3.41
3.36
3.24
3.32
3.26
3.45
3.28
3.45
3.15
3.79
HEWL
FKBP
exp
calc i-values of the two
Table 3.14 Q-values in Hz quantifying the similarity between the measured 3 Jαβ
-values and the calculated h3 Jαβ
stereospecifically assigned subsets 1 and 2 for each structure set for each of the three proteins using the Karplus parameters obtained
using the same structure set of that protein (bold) and using each structure set of all proteins. For comparison, the Q-values obtained
calc i with the De Marco [42] parameters are also given.
when calculating the h3 Jαβ
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
45B3
3.4 Results
129
sets of the different proteins do not all match the experimental data equally well, even when the
Karplus parameters optimised for that structure set of that protein are used. This is in fact the
dominant factor determing the magnitude of the Q-values in the majority of cases, see Tables
3.12 and 3.13, that is, there is more variation between the Q-values obtained using a given set
calc i-values from each protein structure set (variation
of Karplus parameters to calculate the h3 Jαβ
within columns) than between those obtained using each different set of Karplus parameters
for a given structure set (variation within rows). Indeed, the Q-values obtained using a given
calc i-values from a protein structure set other than
set of Karplus parameters to calculate the h3 Jαβ
the one to which the parameters were fitted is in several cases better than the Q-value obtained
during the fitting procedure. The main exception to this trend is Plastocyanin, for which the
Karplus parameters obtained from fitting to the Plastocyanin structure sets give quite different
Q-values to the parameters obtained from the FKBP and some of the HEWL structure sets.
A somewhat surprising result is that the Q-value obtained using the parameters optimised
for the same structure set is not the lowest Q-value for that structure set in all cases, although it
is always among the lowest. For instance, the Q-value calculated for the simulation of HEWL
in the 54A7 force field is lower when the Karplus parameters obtained from the NMR model
structures of Plastocyanin (Q = 1.63 Hz), the simulation of HEWL in 53A6 (Q = 1.63 Hz),
the 1AKI (Q = 1.61 Hz) and 193L (Q = 1.63 Hz) X-ray structures or the 1E8L (Q = 1.63 Hz)
NMR model structures are used than when the Karplus parameters obtained from the fit to the
simulation of HEWL in the 54A7 force field are used (Q = 1.64 Hz). Even more intriguing
is the fact that the Karplus parameters obtained from fitting to the X-ray structure of FKBP
calc i-values from the HEWL X-ray structures 1AKI
perform well in the back-calculation of h3 Jαβ
(Q = 1.51 Hz) and 193L (Q = 1.45 Hz). These unexpected results may, however, occur due
to the uncertainty introduced by the unassigned 3 Jαβ -couplings of subset 3. To avoid this
uncertainty, a jack-knife test was carried out for subsets 1 and 2 only to calculate the Q-values
in Table 3.14. With this approach, the Q-value of a specific structure set is always lowest when
the Karplus parameters optimised for that structure set were used. The Q-values are now also
more sensitive to the set of Karplus parameters used, indicating that some of the apparent
dominance of the structure set in the goodness of fit was due to assignment uncertainty.
exp
-coupling constants
3.4.3 Reassignment of FKBP 3 Jαβ
It is obvious from Tables 3.12, 3.13, and 3.14 that for FKBP, none of the parameter sets, even
calc i-values were back-calculated, prothose fitted to the same structure set from which the h3 Jαβ
vides a good fit between the measured and calculated data. This is surprising given that FKBP
exp
is the only one of the three proteins for which all of the 3 Jαβ
-values were stereospecifically
assigned. To check whether any of the couplings had been incorrectly assigned, the assignment of all 3 Jαβ2 - and 3 Jαβ3 -couplings for residues with two Hβ protons and two measured
3 J exp -coupling constants was compared and changed according to the same exchange criterion
αβ
as was described earlier for the fitting procedure. These comparisons were carried out using
130
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
the average hcos2 θβ i and hcos θβ i values calculated from the MD simulation of FKBP in wacalc i-values calculated using either the De Marco Karplus parameters
ter (45A3) and the h3 Jαβ
or the fitted Karplus parameters. The resulting assignment changes are given in Table 3.15.
A new set of “re-fitted” Karplus parameters and corresponding Q-values were then calculated
using the different optimised assignments, see Table 3.16. In Fig. 3.15, the Karplus curves
obtained using these “re-fitted” Karplus parameters are compared.
Residue
DMopt
LSFopt
COMopt
8 SER
c
c
c
11 ASP
c
c
c
34 LYS
c
c
c
39 SER
-
c
c
49 MET
c
c
c
50 LEU
c
c
c
52 LYS
c
c
c
67 SER
c
-
c
73 LYS
c
c
c
77 SER
-
c
c
79 ASP
c
-
c
80 TYR
c
c
c
82 TYR
c
c
c
107 GLU
c
c
c
Table 3.15 Residues of FKBP for which the assignment was changed (c) in order to optimise the
exp
Q-value comparing the measured 3 Jαβ
of the subsets 1-3 to those back-calculated from the MD
simulation of FKBP in water (45A3) using the De Marco [42] parameters (DMopt) or the leastsquares fitted Karplus parameters (LSFopt). The two sets of assignment changes are merged to
form the set “COMopt”.
In all cases, the “re-fitted” parameters lead to a decrease in the Q-values, from 3.38 Hz
for using least-squares fitting with the original assignment (LSF/Xu) to 2.95, 2.94 and 2.95
Hz for LSF-DMopt, LSF-LSFopt and LSF-COMopt respectively, although they are still not
particularly good. The main difference between the fitted and “re-fitted” Karplus parameters
is the change of sign of a, although only for the parameter sets re-fitted using the assignments
optimised for the De Marco parameters or the combined set of assignment changes, LSFDMopt and LSF-COMopt, is this enough to induce a small maximum in the Karplus curve at
θβ = 0◦ . The similar performance of all three “re-fitted” parameter sets is due to their similarity
outside of the region −60◦ < θβ < 60◦ , which is where most of the time-averaged dihedral
angle values lie. To illustrate this, the mean of the dihedral angles hθβ i over the simulations,
not the median, is shown in Fig. 3.15.
3.4 Results
131
Parameter set
Assignment
a
b
c
Q
DM
Xu
9.5
-1.6
1.8
3.74
LSF
Xu
-1.14
-4.48
6.25
3.38
LSF-DMopt
DMopt
2.75
-3.85
4.47
2.95
LSF-LSFopt
LSFopt
0.97
-4.74
5.19
2.94
LSF-COMopt
COMopt
2.29
-4.08
4.66
2.95
Table 3.16 Karplus parameters a, b, and c and the corresponding Q-values obtained by leastcalc i-values of all three subsets after the re-assignment of the measured
squares fitting of the h3 Jαβ
3 J exp -values for FKBP. DM refers to the De Marco [42] Karplus parameters and LSF to the Karplus
αβ
parameters obtained by least-squares fitting to the simulation of FKBP using the 45A3 force field.
Xu refers to the original, published assignment of Xu et al. [66], DMopt to the assignment opticalc i-values calculated using the DM parameters, LSFopt to the assignment opmised using the h3 Jαβ
calc i-values calculated using the LSF parameters and COMopt to the merged
timised using the h3 Jαβ
set of assignment changes in Table 3.15.
calc i-values calculated from the MD simulation of FKBP in waFigure 3.15 Karplus curves and h3 Jαβ
ter (45A3) using the parameters optimised for each different assignment given in Table 3.15 plotted
against the unmapped mean of the corresponding dihedral angle hθβ i in the simulation, shifted
to the value hθβ i − n · 360 in the range [-180◦ , +180◦ ] for integer values of n. The assignments
used in the optimisation and the Karplus parameter sets are LSF-DMopt/DMopt (cyan), LSFcalc i-values
LSFopt/LSFopt (blue), and LSF-COMopt/COMopt (green), see Table 3.16. The h3 Jαβ
calculated using the assignments by Xu [66] and using the De Marco [42] (DM, black) and optimised (LSF, red) parameters and the corresponding Karplus curves are given for reference.
132
3 On the calculation of 3 Jαβ -coupling constants for side chains in proteins
3.5 Conclusions and discussion
The variety of available parameter values for the standard Karplus relation between a 3 Jαβ coupling and the corresponding θβ -angle hints at an insufficient description of this relation in
exp
-values measured experithe form of the standard Karplus equation, Eq. 3.1. Indeed, the 3 Jαβ
mentally for the three proteins studied here, Plastocyanin, HEWL, and FKBP, deviate considcalc i-values predicted using the commonly-used De Marco parameters and
erably from the h3 Jαβ
calc i-values from the X-ray or NMR model structhe standard Karplus relation to calculate h3 Jαβ
tures or from MD simulation trajectories in vacuum or water in various force fields. Therefore,
we explored two avenues for improving the relation 3 Jαβ θβ .
calc i-coupling constants using the generalised
The first approach of calculating the h3 Jαβ
calc i
Karplus relation in Eq. 3.4 yielded at best only slightly better agreement between the h3 Jαβ
exp
-coupling constants than when using the simpler Karplus relation of Eq.
and measured 3 Jαβ
3.1 with the De Marco parameters. Thus accounting for substituent effects is not sufficient
to improve the agreement between the calculated and measured 3 Jαβ -couplings for these proteins. Moreover, use of dihedral angle values θβ from X-ray or NMR model structures to
calc i-values leads to better agreement with the 3 J exp -values than when using simcalculate h3 Jαβ
αβ
calc i-values are averages over a variety of conformations
ulation trajectories, for which the h3 Jαβ
as they are in the NMR experiments. This may be related to the fact that the parameters of the
standard Karplus relation are often fitted assuming a single structure.
To investigate the effect of conformational averaging, the parameters a, b, and c of the stancalc i-coupling constants
dard Karplus relation were obtained by least-squares fitting of the h3 Jαβ
exp
-coupling
averaged over MD trajectories or X-ray or NMR model structures to measured 3 Jαβ
constants. The parameters a, b, and c and the Q-values quantifying the goodness of fit depend
not only on the choice of protein, but on the particular structure set of each protein that is used.
It is noticeable that the shape of the fitted Karplus curves is highly dependent on the values of
calc i in the fitting procedure. A general lack of
the dihedral angles θβ used to calculate the h3 Jαβ
sampling of angle values in the range −60◦ < θβ < 60◦ means that the fitted curves are not
well defined in this region, with some lacking the maximum located here when parameter sets
from the literature are used.
A further factor influencing the performance of the fitting procedure is how well the relative
weights of the different conformations sampled during the MD simulations match the conformational probability density in the NMR experiment. This will depend on both the quality of
the force field and the degree of sampling. Indeed, it was seen that the Karplus parameters obtained from the least-squares fitting procedure are rather sensitive to the part of the simulation,
i.e. subset of the conformational ensemble, to which they are fitted.
exp
It was observed that the goodness of fit between the measured 3 Jαβ
-couplings and the
calc i-values depends as much on the 3 J exp dataset as on the choice of structure
calculated h3 Jαβ
αβ
set or Karplus parameters. In particular, the Q-values calculated for FKBP are always rather
3.5 Conclusions and discussion
133
exp
-couplings and
high. Optimisation of the assignment of the stereospecifically assigned 3 Jαβ
2/3
re-fitting of the Karplus parameters using the optimised assignment improved the Q-values,
but only marginally.
Overall, this study highlights the uncertainty inherent in the parameters of the Karplus relation used to link 3 J-couplings to dihedral angle values and in the relation itself in the case
of side chain θβ -angles. Similar conclusions are expected to hold for other dihedral angles,
although for those that are less mobile, fewer problems are anticipated.
To improve the quality of the function 3 Jαβ θβ , an extended generalised Karplus equation
as suggested by Imai et al. [76] could be applied. This takes into account more substituent effects. Given the minimal improvement seen here for the generalised Karplus relation, however,
it seems unlikely that the extended version will offer significant further improvement. Another
possibility would be to consider asymmetric, amino-acid specific Karplus relations as done by
exp
Schmidt [77]. The experimental data in Fig. 3.3 show larger measured 3 Jαβ
-coupling con◦
◦
stants for dihedral angle values θβ around +60 than around −60 , which would support the
concept of asymmetric relations. Schmidt [77] parametrised asymmetric Karplus relations for
each amino acid type using a self-consistent method [12]. A wide spread in the parameter sets
obtained for 3 Jαβ -couplings for different amino acid types was observed. Other approaches to
calculate different side-chain vicinal coupling constants around χ1 [78] of Valine also showed
the highest deviation from the experimental values when considering 3 Jαβ -couplings, illus
trating the particular difficulty of finding an appropriate 3 Jαβ θβ relation compared to other
types of side chain 3 J(χ1 )-coupling constants.
4 Calculation of binding free energies
of inhibitors to Plasmepsin II
4.1 Summary
An understanding at the atomic level of the driving forces of inhibitor binding to the protein
Plasmepsin (PM) II would be of interest to the development of drugs against malaria. To this
end, three state of the art computational techniques to compute relative free energies - thermodynamic integration (TI), Hamiltonian replica-exchange (H-RE) TI, and comparison of bound
versus unbound ligand energy and entropy - were applied to a protein-ligand system of PM II
and several exo-3-amino-7-azabicyclo[2.2.1]heptanes and the resulting relative free energies
were compared to values derived from experimental IC50 values. For this large and flexible protein-ligand system the simulations could not properly sample the relevant parts of the
conformational space of the bound ligand, resulting in failure to reproduce the experimental
data. Yet, the use of Hamiltonian replica exchange in conjunction with thermodynamic integration resulted in enhanced convergence and computational efficiency compared to standard
thermodynamic integration calculations.
The more approximate method of calculating only energetic and entropic contributions of
the ligand in its bound and unbound states from conventional molecular dynamics (MD) simulations reproduced the major trends in the experimental binding free energies, which could
be rationalised in terms of energetic and entropic characteristics of the different structural and
physico-chemical properties of the protein and ligands.
135
136
4 Calculation of binding free energies of inhibitors to Plasmepsin II
4.2 Introduction
Malaria is a life-threatening disease caused by a parasite called Plasmodium. It is transmitted
into the human body by bites of infected Anopheles mosquitoes. In 2008 nearly one million
people died of it [79]. Various treatments are available, but growing resistance against current
drugs makes a continuing research effort to control malaria necessary.
Figure 4.1 Initial conformation of Plasmepsin II. Flap pocket surface in green, S1/S3 pocket surface in blue, Asp 34 in yellow, Asp 213 in cyan, and ligand L1 in red.
One of the protein classes that are possible targets for drug design against malaria are three
aspartic proteases Plasmepsin (PM) I, II, IV and an histo-aspartic protease (HAP) produced
by the malaria parasite Plasmodium falciparum, which causes the most severe variant of the
disease. Plasmepsins catalyse the hydrolysis of the peptide bond in proteins and the products
of their degradation of human Hemoglobin are a source of nutrition for the parasite. Two aspartic acid residues act as proton donor and acceptor respectively (Fig. 4.1). Plasmepsins vary
in the specificity of their cleavage site and inhibition of these proteases was considered to be
a promising approach in antimalaria drug design [80, 81]. More recent knockout-studies indicate, however, that the activities of the Plasmepsins may be either redundant or, collectively,
not essential [82]. Notwithstanding this loss of immediate practical interest, the Plasmepsins
still constitute, in view of the binding data available, a challenging protein-ligand system
for testing methodology to predict and understand ligand-protein binding forces. For several
malarial Plasmepsin inhibitors different computational techniques to predict binding affinities
have been applied and the linear-interaction energy (LIE) approach was shown to yield good
agreement with experimental binding energies for a particular set of ligands [83].
4.2 Introduction
137
Figure 4.2 The scaffold and sidechains of the inhibitors L1-L7 under investigation. For each ligand
the experimental inhibition concentrations IC50 are given[84].
The compounds under investigation in the current study are PM II (Fig. 4.1) and several exo3-amino-7-azabicyclo[2.2.1]heptanes (Fig. 4.2), non-peptidomimetic inhibitors which were
shown to have inhibition concentrations IC50 ≥ 30 nM against PM II [84]. They were designed
to interact with the catalytic dyad of the aspartic proteases and target the S1/S3 pocket and the
flap pocket of PM II [85]. The compounds vary in the flap vector (L1-L7). The pattern in the
inhibition concentrations seems to follow roughly the 55% rule of Mecozzi and Rebek [86],
which states that inclusion complexes are most favorable if the guest occupies 55% of the
available space within the host.
To investigate the interaction between a flexible host and guest we performed molecular
dynamics (MD) simulations which may explain the trends in binding affinities by considering
the structure and mobility of the protein and ligands. It offers not only the energies, but also
entropic contributions to ligand binding.
Calculation of the binding free energy for large biomolecular compounds by MD is nontrivial, as the sampling of the bound and unbound state of the ligand requires long simulations
and the result depends, in addition, on the accuracy of the force field and parametrisation
chosen. According to the thermodynamic cycle in Fig. 4.3, the following condition should be
fulfilled [87]
f ree
bound
∆FAbinding + ∆FBA
− ∆FBbinding − ∆FBA = 0,
(4.1)
or
f ree
binding
bound
∆FBA
− ∆FBA = ∆FBA
≡ ∆FBbinding − ∆FAbinding .
(4.2)
138
4 Calculation of binding free energies of inhibitors to Plasmepsin II
Figure 4.3 Thermodynamic cycle used to calculate the relative free energies of binding for two
ligands A and B to a target protein.
∆FAbinding is the free energy of binding for ligand A. Thermodynamic integration (TI) [88] can
env between two ligands A and B in a
be used to calculate the difference in free energy ∆FBA
certain environment, i.e. bound to the host or free in solution,
env
∆FBA
= FBenv − FAenv .
(4.3)
From experimental data, e.g. inhibition constants or IC50 values, binding free energy difbinding
ferences ∆FBA
between two ligands A and B can be estimated. So using experimental
binding
data for ∆FBA
we can estimate the accuracy of our simulations by checking whether the
thermodynamic cycle is closed. When more than two compounds are involved, one can additionally define thermodynamic cycles along which the calculated free energy should be zero
by definition. The transitions a - i and cycles I - III for which we calculate the differences in
env in the bound and free environment, are shown in Fig. 4.4.
free energy ∆FBA
TI does not give information about relative entropies ∆S and energies ∆E. As a first approximation these can be estimated from MD simulations in the form of ligand interaction energies
and conformational entropies of the solute. Schlitter [89] suggested a formula to estimate an
upper bound to the absolute conformational entropy of a molecule. This can be used to obtain
an approximate picture of the entropic and energetic contribution to binding by the ligand,
protein and solvent. A theoretically less approximate way of estimating relative entropies of
binding would be from the corresponding free energies calculated at different temperatures.
However, this method has insufficient precision when applied to ligand binding [90]. Free
energy calculations based on TI are time-consuming and computationally expensive, as simulations at many intermediate points along a chosen alchemical pathway between compounds
A and B have to be performed and these sometimes need to be done sequentially along the
pathway. The accuracy of MD simulations depends not only on the force-field accuracy, but
4.2 Introduction
139
Figure 4.4 Thermodynamic cycles for free energy calculations between the ligands L1-L7, free
in solution and when bound to the protein. Using the transitions a-i, three cycles (I, II, III) can be
defined along which the total free energy change is theoretically zero.
also on the characteristics of the particular system under investigation with respect to a proper
sampling of conformational space. Since some protein motions may have long characteristic time scales, they may be insufficiently sampled on a short time scale. A simulation may
get trapped in a local energy minimum leading to insufficient sampling and incorrect (free)
energies.
To overcome this problem, Replica Exchange (RE) simulation [91, 92] can be used. Several
independent simulations (MD or Monte Carlo) are run in parallel, each at a different temperature (T-RE) or using a different Hamiltonian (H-RE). At a chosen frequency pairs of structures
(replicas) are exchanged with a transition probability governed by the detailed balance condition. This results in diffusion of structures obtained at different temperatures or Hamiltonians,
which allows them to cross energy barriers that may be too high to pass with standard MD
or TI. In this way RE enhances conformational sampling, while still producing a canonical
ensemble for each temperature or Hamiltonian.
In this work, we report on MD simulations and TI calculations of PM II in complex with
the inhibitors shown in Fig. 4.2. Free energy calculations were performed using standard TI
MD simulations and by employing a Hamiltonian RE scheme [93]. Results are compared
qualitatively and quantitatively to experimental data. This gives insights into the strengths
and limitations of free energy calculations of a challenging protein-ligand system and into the
peculiarities of ligand PM II binding.
140
4 Calculation of binding free energies of inhibitors to Plasmepsin II
4.3 Method
From the experimental inhibition concentrations IC50 (A) and IC50 (B) of two compounds A
binding
and B the experimental relative free energy of binding ∆FBA
(exp) can be approximated
using
binding
∆FBA
(exp) = ∆FBbinding − ∆FAbinding
Ki (A)
IC50 (A)
= −kB T ln
= −kB T ln
,
Ki (B)
IC50 (B)
(4.4)
where kB is the Boltzmann constant and T the absolute temperature. Here we have used
the Cheng-Prusoff equation [94] and the fact that all measurements were performed at the
same substrate concentration to convert IC50 values into inhibition constants, Ki (A) and Ki (B).
∆Fxbinding is defined as the difference in free energy of the protein-ligand system x between the
bound, ligand in protein, and the free, only ligand in solution, state,
∆Fxbinding = Fxbound − Fxf ree .
(4.5)
To close the thermodynamic cycles in Fig. 4.3, we need to calculate the free energy difference between two ligands in the bound or free environment, respectively. In a TI simulation
we simulate the transition between two states (in this example two ligands) A and B by changing the Hamiltonian H of the system from state A to state B. The Hamiltonian H is coupled
to a parameter λ which runs from 0 (state A) to 1 (state B) and different simulations at different λ -values in between are performed. The free energy difference ∆FBA between two states
A and B, is then [88]:
Z λB ∂H
dλ .
(4.6)
∆FBA = F(λB ) − F(λA ) =
∂λ λ
λA
As TI does not give information about entropies, Schlitter’s formula for the calculation of
configurational entropy was applied to trajectories of the ligand in conventional, i.e. non-TI,
MD simulations. Schlitter’s entropy S is a heuristic formula to get an upper bound of the true
configurational entropy Strue of a system
kB Te2
1
Strue ≤ S = kB ln det 1 + 2 M σ ,
(4.7)
2
h¯
where h¯ is the Planck constant divided by 2π and e Euler’s number. M is the mass matrix,
holding on the diagonal the masses belonging to the Cartesian degrees of freedom, σ is the
covariance matrix of atom-positional fluctuations. The elements of σ are
(4.8)
σi j = (xi − hxi i) x j − hx j i ,
4.3 Method
141
with xi being the Cartesian coordinates of the atoms considered for the entropy calculation
after least-squares fitting of the position of a given subset of atoms of the coordinate trajectories of the MD simulation. Superposition by least-squares fitting was performed for all nonhydrogen atoms of the ligands. Because of the superpositioning, rotational and translational
entropy was excluded from the final entropy calculations, only the internal conformational
entropy of the ligand was obtained. As an estimate for the change in internal entropy of the
f ree
ligand upon binding to the protein, the difference of the entropies Slbound and Sl in the bound
and the free simulations of ligand l respectively, was considered:
f ree
∆Slbinding = Slbound − Sl
.
(4.9)
This use of the expression in Eq. 4.7 suffers from various approximations: (i) only the internal
configurational entropy of the ligand is calculated, no contributions from the protein or solvent
are considered, (ii) Eq. 4.7 is an upper bound and is based on a quasi-harmonic assumption,
which has its limitations [95]. Moreover, due to the large size of the system, sampling is
limited.
As our standard TI simulations appeared to suffer from insufficient sampling of the conformational space of the ligand in the bound environment (see Results), we also applied H-RE
to these systems. Several independent MD simulations, replicas, were run, each having a
different Hamiltonian depending on the λ -coupling parameter that was also used in the TI
simulations before. After a chosen number of MD steps, exchanges of the configurations Xm
and Xn of replicas with adjacent λ -values, i.e. λm and λn defining the Hamiltonians Hm and
Hn respectively, are attempted using the detailed balance condition,
PS′ t(S′ → S′′ ) = PS′′ t(S′′ → S′ ),
(4.10)
where S′ denotes the state before the exchange, i.e. Hm has configuration Xm and Hn has
configuration Xn , and S′′ the state after the exchange, i.e. Hm has configuration Xn and Hn
has configuration Xm . PS denotes the configurational probability of state S and t(S → S′ ) the
transition probability from state S to state S′ . The relative configurational probabilities of
states S′′ and S′ are
e−Hm (Xn )/kB T e−Hn (Xm )/kB T
PS′′
= −H (X )/k T −H (X )/k T .
(4.11)
PS′
e m m B e n n B
Combining Eqs. 5.6 and 5.7 we find for the probability p of exchange of replicas m and n
with
p(m ↔ n) = t(S′ → S′′ ) = min(1, e−∆/kB T )
(4.12)
∆ = [Hm (Xn ) + Hn (Xm )] − [Hm (Xm ) + Hn (Xn )].
(4.13)
142
4 Calculation of binding free energies of inhibitors to Plasmepsin II
4.4 Simulation setup
All simulations were carried out using the GROMOS05 software [71] and the GROMOS force
field 53A6 [27], at constant temperature (310 K) and constant volume under periodic boundary
conditions. The protein consists of 329 amino acid residues and the ligands have a size of 45 to
53 (united) atoms. SPC-water [33] was used as solvent for all simulations. In the simulations
of the unbound ligand the cubic box had an edge length of 4.1 (L2), 4.3 (L1, L3, L4, L6, L7)
or 4.8 (L5) nm (about 2500 water molecules), for the ligand bound to the protein in water the
edge length was about 9.1 nm (about 22500 water molecules). Since the number of atoms in
the different ligands differs by 0, 1, 2, 5 or 7 and the average change in number of atoms in
the cycles I-III of Fig. 4.4 is only 2, the use of constant volume instead of constant pressure is
not problematic for these large systems.
The force-field parameters for the amines in the ligands were deduced from the work of
Oostenbrink et al. [96], the SO2 group parameters were taken to be the same as the ones for
SO2 used by Zagrovic et al. [97]. The parameters for L1 are shown in the Supplementary
material. The force-field parameters for the various aliphatic tails in L2-L7 were taken from
the work of Schuler et al. [26]. The starting structure for the simulation of the bound ligand
L1 was the optimised structure of L1 bound to PM II (taken from the Protein Data Bank [34],
PDB ID 2BJU [98]) from a calculation using the software MOLOC described elsewhere [84].
To obtain initial structures for the other ligands, CHx groups were added or removed with the
GROMOS++ program gca. For L4, an additional MM2 structure calculation of the ligand
using the Chem3D Ultra software [99] was performed to generate a starting structure of the
ring. As initial structure for the transitions of cycle III an unphysical reference ligand LR
consisting of the atoms of ligand L1 and four additional dummy atoms D1-D4 was used. Fig.
4.11 in the Supplementary material shows the sidechain of reference ligand LR. Its coordinates
were taken from the final structure of the conventional MD simulation of ligand L4, while the
missing two atom positions were generated using the GROMOS++ program gca.
The initial structures were energy minimised followed by a thermalisation and equilibration.
Initial velocities were generated from a Maxwell-Boltzmann distribution at 50 K. The atoms
of the solute were positionally restrained with a harmonic force. In six 10 ps simulation
periods the simulation temperature was raised in steps of 50 K from 50 K up to 298 K while
simultaneously decreasing the position-restraining force constant from 25000 kJmol−1 nm−2
to 0 kJmol−1 nm−2 by a factor of 10 in subsequent simulations.
To evaluate the nonbonded interactions a triple-range cutoff scheme was used. Within a
short-range cutoff of 0.8 nm interactions were calculated at every time step from a pair list
generated every 5th time step. At every 5th time step interactions between 0.8 and 1.4 nm
were updated. A reaction field approach [38] and a dielectric permittivity of 61 [39] were
applied to represent electrostatic interactions outside a 1.4 nm cutoff. A step-size of 2 fs was
used for integration of the equations of motion using the leap frog scheme. All bonds were
constrained using the SHAKE algorithm [41]. After equilibration, regular MD simulations of
the ligand in water were performed for 6 ns, while the ligands bound to the protein in water
4.4 Simulation setup
143
were simulated for 3 ns. Trajectory coordinates were saved every 5 ps for analysis.
Various protocols to change from one λ -value to another and different numbers of λ -values,
equilibration and sampling periods were used, depending on the observed convergence of
the h∂ H /∂ λ iλ ensemble averages. The final structures of the standard MD simulations of
ligands L1, L5 and L6 were taken as starting points for the TI transitions a-f in cycles I and
II of Fig. 4.4. For transitions g and h of cycle III the final structure of the standard MD
simulation of the reference ligand LR was used as starting structure. Transition i was started
from the final configuration of transition h. A soft-core atom approach [100, 101] was used
to prevent instabilities during the transition from state A to state B, with αiLJ
j = 0.5 for the
Lennard-Jones interactions, and electrostatic interactions were treated using αiCj = 0.5 nm2 .
21 equidistant λ -values were used to carry out the transition from λ = 0 to λ = 1. For each
transition in cycles I-III (Fig. 4.4) of the unbound, free ligand a simulation at λ = 0 was
performed for 100 ps. The end structure of this run was taken as starting structure for all
other λ -values. At every λ -value at least 400 ps of simulation, 100 ps equilibration and 300
ps for analysis, were performed. If the statistical error [102] estimate of h∂ H /∂ λ iλ was
larger than 2 kJmol−1 after 400 ps, the simulation at that λ -value was prolonged for 400 ps
or 800 ps if needed and the last 600 ps were used for the calculation of h∂ H /∂ λ iλ . If there
was a large difference in h∂ H /∂ λ iλ between two adjacent λ -values, additional simulations
at intermediate λ -values were performed.
In the simulations of the bound ligands, a slightly different protocol was used. At a given
λ -value, first, 100 ps of equilibration was performed followed by 300 ps of simulation. The
last 300 ps were used to calculate an average value of ∂ H /∂ λ and a statistical error estimate
[102]. If the error estimate was larger than 2 kJmol−1 , an additional 400 ps were simulated at
the same λ -value and the last 600 ps were used to calculate the average of ∂ H /∂ λ and its
error estimate. If the error estimate was still over 2 kJmol−1 , another 400 ps were calculated
and again the last 600 ps used for averaging ∂ H /∂ λ . The final structure of the simulation
at one λ -value, either after getting an error estimate lower than 2 kJmol−1 or after 1.2 ns of
simulation time, was used as starting structure for the next higher λ -value, for which the same
procedure was carried out. Using a bound lower than 2 kJmol−1 for the error estimates for the
whole simulation would have led to very long and thus expensive simulations.
The final structures at each λ -value of the TI simulations of each transition were used as
starting structures for the H-RE calculations, which therefore benefitted from the relaxation
that occurred during the TI simulations. So for every H-RE simulation, 21 different starting
structures for the 21 different λ -values were used. Only for transition f, 23 λ -values were
used, i.e. λ = 0.700 and 0.750 were replaced by 0.680, 0.710, 0.740 and 0.770, to enhance
the exchange probability in this λ -range. For the two first λ -values the final structure at λ =
0.700 of the TI simulation was taken as the starting structure, for the latter two λ -values the
TI final structure at λ = 0.750 (see Figs. 4.15 and 4.16 in the Supplementary material). In the
H-RE simulations, exchanges were tried every 2 ps. Transitions a-d and g-i were run for 100
ps, transitions e-f for 200 ps. For analysis of the data the first 20 ps of the H-RE simulations
were not considered.
144
4 Calculation of binding free energies of inhibitors to Plasmepsin II
4.5 Results
First conventional or standard MD simulations, so-called endstate simulations, are considered
for all ligands in water (free) and bound to the protein in water (bound). The latter show that
the ligands stay in the binding pocket. As an example, Fig. 4.1 shows ligand L1 in the protein
after 3 ns of simulation. The surface areas of the amino acid residues forming the flap pocket
(green) and the S1/S3 pocket (blue) according to [85] are marked. This structure was used as
starting structure for TI simulations of transitions a, b, d, g and h.
Table 4.1 presents the free energy differences obtained by thermodynamic integration (TI),
H-RE (RE), and from experimental data.
The TI simulations of the ligand free in water yield cycle closure of thermodynamic cycles
I, II and III, suggesting convergence of the free energy values for these systems, although
the error estimates are in the range of 1-2 kJmol−1 . Except for the transitions involving L5,
the statistical error estimates per transition are lower than 1.6 kJmol−1 . Fig. 4.5 shows for
transitions a-c of the ligands in water well converged averages of ∂ H /∂ λ with statistical
error estimates per λ -value. Fig. 4.12 serves as a warning that cycle closure does not neces-
Figure 4.5 h ∂∂Hλ i as function of λ for transitions a, b and c in the free ligand simulations.
Results from TI. Error bars indicate statistical
error estimates of the ensemble averages.
Figure 4.6 h ∂∂Hλ i as function of λ for transitions a, b and c in the bound ligand simulations. The solid lines show results from HRE, dotted lines the results from TI. Error bars
indicate statistical error estimates of the ensemble averages.
f ree
∆FBA (T I)
-2.1 ± 1.0
-10.1 ± 0.8
-8.4 ± 1.2
-0.3 ± 1.8
-0.5 ± 1.2
-0.8 ± 1.9
-1.7 ± 2.4
-0.4 ± 3.3
1.9 ± 0.7
-1.4 ± 1.6
3.2 ± 1.3
-0.1± 2.2
bound (T I)
∆FBA
3.7 ± 1.2
-5.7 ± 1.1
-7.2 ± 1.4
2.2 ± 2.1
9.8 ± 1.4
4.6 ± 3.4
11.7 ± 3.5
-2.7 ± 5.1
2.8 ± 1.0
-1.2 ± 1.3
4.7 ± 1.4
0.8 ± 2.2
bound (RE)
∆FBA
1.6 ± 1.1
-5.7 ± 0.5
-10.7 ± 1.4
-3.4 ± 1.8
9.3 ± 1.0
5.2 ± 2.5
14.3 ± 3.3
-0.2 ± 4.3
3.0 ± 1.0
-2.7 ± 1.2
5.4 ± 1.7
-0.2 ± 2.3
binding
∆FBA
(T I)
5.7
4.4
1.2
2.5
10.3
5.4
13.3
-2.3
0.9
0.2
1.5
0.9
7.2
binding
binding
(exp)
∆FBA
(RE) ∆FBA
3.6
0.3
4.4
10.2
-2.3
9.9
-3.1
9.8
9.0
6.0
-7.0
16.0
2.0
0.3
1.1
-3.8
-1.3
-3.8
2.2
0.0
-0.2
8.1
0.0
4.5 Results
Ligand
Transition A B
a
L1 L6
b
L1 L7
c
L6 L7
Cycle I
d
L1 L2
e
L5 L1
f
L5 L2
Cycle II
g
L1 L3
h
L1 L4
i
L4 L3
Cycle III
RMS deviation exp.
Table 4.1 Differences in free energy in kJmol−1 . The ligands L1-L7 are defined in Fig. 4.2, the transitions a-i between them in Fig.
f ree
4.4, and the free energy differences in Eqs. 4.2 - 4.5. ∆FBA
(T I) is the data obtained from simulation of the ligands in water using TI.
bound
∆FBA (T I) is the data from the TI simulations of the ligands bound to the protein in water. The results of Hamiltonian RE are indicated
bound (RE). ∆F binding (T I) and ∆F binding (RE) are the difference between the TI and H-RE results respectively of the ligand bound
with ∆FBA
BA
BA
binding
to the protein and the free ligand TI simulations for a given transition. ∆FBA
(exp) is the binding free energy difference obtained from
experimental IC50 values [84]. The bottom line shows the root-mean-square deviation of the calculated from the experimental values
for the 9 transitions.
145
146
4 Calculation of binding free energies of inhibitors to Plasmepsin II
sarily mean convergence: Cycle closure for transitions g-i in the free ligand systems is good
even though the curves of h∂ H /∂ λ i as a function of λ show sizeable jumps between adjacent
λ -values and statistical error estimates per λ -value are large. Fig. 4.14 shows much better
converged results for transitions d, e and f.
In Figs. 4.6, 4.13 and 4.15 the corresponding averages are shown for the TI simulations of
the bound ligand-protein systems. The TI results, dotted lines, show poor convergence, both
in the sense of statistical error estimates per λ -value and in terms of the smoothness of the
curves. This is most likely due to the long relaxation times of motions in the protein, which
are thus poorly sampled in standard TI. As a consequence, cycle closure for cycles I-III for the
TI simulations of the bound ligand systems is generally worse than for the free ligand systems.
Due to the imprecise results of the bound ligand TI simulations, the differences in binding free
binding
energies ∆FBA
(T I) between two ligands of a transition calculated using TI are not very
precise either and show values deviating up to 4.3 kJmol−1 from 0 for cycle closure. Since
the range of experimental binding free energies is only 14 kJmol−1 , it is not very surprising
that the tendencies in the difference in binding free energy derived from experimental values
binding
binding
∆FBA
(exp) are not reproduced. Transition a should yield the lowest ∆FBA
in cycle I
binding
according to the experimental data, but TI yields transition c as the lowest ∆FBA
in this
binding
cycle. For cycle II TI shows transition f to have the largest ∆FBA
, whereas according
to experiment the value for this transition should lie between the ones of transitions d and
binding
e. Only in cycle III transition i shows the largest ∆FBA
according to TI and experiment.
A Spearman rank-order correlation coefficient of only 0.40 (Table 4.2) was obtained for the
results of TI simulation with respect to the experimental data showing its poor performance.
Considering the TI simulation of the bound ligand system for transition a in Fig. 4.6 in
more detail, we observe the biggest change in h∂ H /∂ λ i as function of λ between λ = 0.60
and λ = 0.65. In the upper panel of Fig. 4.7 the evolution of ∂ H /∂ λ over time for λ = 0.65
is displayed as dotted line with the average over the first and last 500 ps as solid line. After
around 550 ps there is a remarkable jump in the ∂ H /∂ λ value. The lower panel of the figure
shows the change in the torsional angle around the bond between the R-group of the ligand
and the phenyl ring. At the moment when the jump in ∂ H /∂ λ occurs, the torsional angle
changes from 0◦ to 180◦ indicating a conformational change in the ligand causing a change in
∂ H /∂ λ . Fig. 4.8 illustrates the change of conformation in the ligand. Note that not only the
R-group of the ligand is moving, but also the naphthalene group is very mobile and changing
its orientation. While the orientation of the naphthalene group is changing continuously, the
lower panel of Fig. 4.7 shows that the R-group is rotating rarely, here only once, and then it
is stuck in the new conformation. In Fig. 4.9 the left hand panels show the change of this
torsional angle for several other λ -values during TI simulation of transition a.
4.5 Results
147
Figure 4.7 ∂∂Hλ as function of λ (upper panel, dotted line) and time series of the torsional angle
around the bond connecting the R-group of the ligand to the phenyl ring (lower panel) during the
TI simulation at λ = 0.65 of transition a. In the upper panel, the average value of ∂∂Hλ over the first
and last 500 ps is shown as solid line.
Figure 4.8 Picture of two structures of the ligand of the TI simulation at λ = 0.65 of transition a.
Grey is at time point 200 ps before the jump, black is at time point 1005 ps.
-T ∆Sbinding ∆F binding ∆F binding (exp) ∆F binding (T I) ∆F binding (RE)
0
0
0
0
0
-16
11
9.0
10.3
9.8
21
28
-3.8
0.9
1.1
43
38
-3.8
0.2
-1.3
43
81
7.0
-5.4
-6.0
-14
12
0.3
5.7
3.6
12
40
10.2
4.4
4.4
29
39
0
6.5
6.4
-0.37
0.24
1.0
0.40
0.55
Table 4.2 Differences in free energy ∆F binding and entropy ∆Sbinding as defined in Eqs. 4.9 and 4.14 between the ligands L1-L7 (Fig.
4.2). The difference in energy ∆E binding consists only of contributions from interactions involving ligand atoms (see Tables 4.4-4.6) and
the entropy S is given as in Schlitter’s entropy formula (Eq. 4.7) for the ligand. Experimental values were derived from IC50 values of
[84] using Eq. 4.4. ∆F binding (T I) and ∆F binding (RE) are the differences in binding free energies calculated from TI and H-RE simulations
respectively. All values are given in kJmol−1 and with respect to ligand L1. The bottom line shows the root-mean-square deviation of
the calculated values from the experimental ones for the 6 ligands L2 to L7.
4 Calculation of binding free energies of inhibitors to Plasmepsin II
∆E binding
0
27
8
-5
38
25
28
20
0.83
148
Ligand
L1
L2
L3
L4
L5
L6
L7
RMS deviation exp.
Rank order correlation
4.5 Results
149
Figure 4.9 Time series and normalised distributions of the torsional angle around the bond between the R-group and the phenyl group of the ligand bound to the protein in water during TI (left)
and H-RE (right) simulations of transition a for the λ -values 0.0, 0.2, 0.4, 0.6, 0.8 and 1.0 (top to
bottom).
Especially the upper three graphs for λ -values 0.0, 0.2 and 0.4 indicate that a rare conformational change may induce a sizeable change in ∂ H /∂ λ , such that the statistical error
estimate of the average becomes large and so the simulations of the first 400 ps had to be
extended for another 400 or 800 ps to get an acceptable error estimate. Because of the poor
conformational sampling of the ligand bound to the protein using standard TI, in a next step
H-RE was applied to the systems.
In Fig. 4.10 the frequency of exchanges in H-RE for transition e of the bound ligand is depicted indicating the migration of the 21 initial structures through the λ -values. For example
the structure at λ = 0.05 (in red) exchanges the Hamiltonians with its neighboring structures
with higher λ -value, moves up to λ = 0.25 within 10 ps then down to λ = 0.00 within 22 ps,
after 100 ps it is at λ = 0.75, after 200 ps of H-RE simulation it is back to λ = 0.10. This
means that using H-RE more different structures with potentially different conformations are
simulated per λ -value. For transition e there is a λ -range between 0.7 and 0.8 where exchanges occur less often than elsewhere. For transition f such a lack of exchanges was even
more prominent, so additional λ -values were added in this region to enhance the transition
probability (see Method section and Fig. 4.16 of the Supplementary material for more detail).
150
4 Calculation of binding free energies of inhibitors to Plasmepsin II
Figure 4.10 Hamiltonian replica-exchange structure exchanges between different λ -values of transition e, shown for 200 ps, 100 exchanges. Different colors correspond to different starting structures at different λ -values. In the right panel the average acceptance ratio for an exchange per
λ -value is shown.
As a consequence the curves of h∂ H /∂ λ i as a function of λ in Figs. 4.6, 4.13 and 4.15
are much smoother for H-RE than they are for TI, and the H-RE values for h∂ H /∂ λ i cover a
smaller range than those of TI. The statistical error estimates are still large, but the total error
over the whole transition is smaller for most of the transitions using H-RE. Only for transitions
c and i it did not improve. On the other hand, cycle closure got worse for cycle I calculated
binding
with H-RE than with TI. In cycle I the ∆FBA
(RE) for transition a is lower than the one
for transition b, as in experiment. But the value for transition c is the lowest one in the cycle,
whereas it lies between the experimental values of transitions a and b. For Cycle II better cycle
binding
closure is achieved using H-RE, but the relative order of the ∆FBA
(RE) is the same as for
TI and not in accordance with the experimental one. For Cycle III the value of ∆F binding for
binding
the cycle closure got closer to 0 for H-RE, the difference of ∆FBA
(RE) between transitions
h and i is close to the experimental value, but the value for transition g changed in the wrong
direction for H-RE compared to TI. For H-RE simulations a Spearman rank-order correlation
coefficient of 0.55 (Table 4.2) with respect to the experimental data was obtained, a still rather
low value, which is however higher than the one obtained from TI.
4.5 Results
Ligand
Transition A B
a
L1 L6
b
L1 L7
c
L6 L7
Cycle I
d
L1 L2
e
L5 L1
f
L5 L2
Cycle II
g
L1 L3
h
L1 L4
i
L4 L3
Cycle III
first 100ps
4.6 ± 10.9
-5.9 ± 8.2
-8.2 ± 13.2
2.2 ± 19.0
10.7 ± 12.7
9.1 ± 22.7
24.2 ± 26.0
4.5 ± 36.8
5.2 ± 10.8
-1.5 ± 16.7
6.8 ± 17.0
0.1 ± 26.2
bound (T I)
∆FBA
4th traj (300-400ps)
3.1 ± 11.5
-5.7 ± 7.8
-6.5 ± 13.3
2.3 ± 19.2
10.2 ± 13.1
2.6 ± 24.3
12.8 ± 27.2
0.1 ± 38.8
3.2 ± 10.5
-3.8 ± 16.3
3.6 ± 17.2
-3.5 ± 25.9
last 100ps
3.5 ± 11.3
-5.3 ± 8.0
-6.6 ± 13.4
2.2 ± 19.3
8.2 ± 13.3
6.6 ± 23.5
12.4 ± 27.9
1.6 ± 38.8
3.0 ± 10.4
-3.1 ± 16.7
5.8 ± 17.3
-0.2 ± 26.2
bound (RE)
∆FBA
first 100ps
last 100ps
1.6 ± 12.2
-5.7 ± 8.7
-10.7 ± 14.9
-3.4 ± 21.1
9.3 ± 14.1
9.3 ± 14.1
6.6 ± 26.6
4.0 ± 26.9
14.6 ± 30.8 14.0 ± 32.1
-1.3 ± 43.1
0.7 ± 44.2
3.0 ± 12.7
-2.7 ± 18.2
5.4 ± 20.4
-0.2 ± 30.1
Table 4.3 Differences in free energy (kJmol−1 ) depending on the time period used for analysis. The ligands L1-L7 are defined in
bound (T I) is the data from the TI
Fig. 4.2, the transitions a-i between them in Fig. 4.4, and the free energy differences in Eq. 4.3. ∆FBA
simulation of the ligand bound to the protein in water. Data from the first 100, from 300-400 ps and from the last 100 ps of simulation
bound (RE) and calculated based on the first and
was taken into account for every λ -value. The results of H-RE are indicated with ∆FBA
last 100 ps, which were the same in the case of transition d, as only 100 ps were simulated for this transition. Errors are given as
root-mean-square fluctuations.
151
152
4 Calculation of binding free energies of inhibitors to Plasmepsin II
bound values for TI for only 100 ps of
To compare simulations of similar lengths, the ∆FBA
simulation time were compared to the values obtained from H-RE simulations in Table 4.3.
For H-RE, only 80 ps of simulation were used for calculating the ∆F values, as the first 20
ps were considered equilibration time. For TI, different ranges of 100 ps were investigated,
first the h∂ H /∂ λ i of the first 100 ps of every λ -value was used to calculate ∆F. Second,
the 100 ps from 300 to 400 ps were analysed for every λ -value. Finally, the last 100 ps at
every λ -value were considered. The results for H-RE for transitions b and d-h lie in the range
between the numbers for TI at the beginning and end of the simulations, which again reflects
the conformational averaging effect of H-RE. For transition i the H-RE result is between the
TI value at the beginning and the one between 300-400 ps, for transitions a and c H-RE yields
a lower value than TI.
As the relative ∆F values calculated using TI and H-RE suffered from insufficient sampling
of the ligand-protein conformational space and thus did not match the measured values, an
alternative methodology to estimate relative binding free energies was applied, in which the
ligand energy and configurational entropy is calculated from conventional MD simulations of
the bound and unbound ligand in water. This approach allows for a separate calculation of
energetic and entropic contributions to binding,
binding
binding
binding
∆FBA
= ∆EBA
− T ∆SBA
.
(4.14)
The values of these quantities for the 7 ligands are given in Table 4.2 with respect to ligand L1, which was also used as starting compound for the TI and H-RE calculations. The
contributions of the different force-field terms are given in Tables 4.4 - 4.6. In view of the
approximative nature of only considering contributions of the ligand in Eq. 4.14 and the still
limited - though longer than in TI or H-RE - sampling of the ligand-protein complex (3 ns simulations), we shall only interpret the largest energy and entropy differences. The convergence
of the ∆S values as a function of time is shown in Fig. 4.17.
binding
The largest ∆EBA
values are found for the transitions from L5 to L4, L1 and L3. Shortening the long alkane tail favours binding energetically. The protein environment favours chains
of length as in L1, L3 and L4. For the still shorter L2 chain this effect is lost. Transitions from
L1, L3 and L4 to L6 and L7 are energetically unfavourable for binding. The introduction of
an oxygen atom in an alkane chain of favourable length weakens binding due to unfavourable
desolvation of oxygen.
binding
The largest −T ∆SBA
values are found for the transitions from L4 and L5 to L2 and L6.
Compared to the other ligands the binding of a ligand with a branching carbon in the tail
(L4, L3) or with a long tail (L5) is entropically unfavourable. Entropically most favourable is
binding of L2 and L6. The short chain of L2 has room to move within the binding pocket. The
positioning of the oxygen atom in the alkane tail has a larger entropic than energetic effect
upon binding.
For this conventional MD simulations calculation of the Spearman rank-order correlation
coefficient yield a value of 0.24 for the ∆F binding and -0.37 for −T ∆Sbinding (Table 4.2). In-
4.6 Discussion
153
terestingly, the Spearman rank-order correlation coefficient becomes 0.83 when ∆E binding is
considered. The overall picture arising from the results obtained is that the major effects of the
various modifications of the ligands upon binding may be rationalised and the mechanisms in
terms of energetic and entropic contributions are rather diverse.
4.6 Discussion
Comparing the curves of h∂ H /∂ λ i as a function of λ for all the transitions of the bound
ligand system, TI shows more erratic behaviour than H-RE. The curves of H-RE are much
smoother, even though the simulation time was only up to 200 ps, whereas TI was prolonged
for 1.2 ns in some cases. The values for h∂ H /∂ λ i obtained from H-RE calculations lie in
between the values of TI due to the enhanced conformational sampling of H-RE per λ -value.
The statistical error estimates are in a similar range, but considering the different simulation
lengths, H-RE does sample better. With root-mean-square deviations of 6-8 kJmol−1 between
the TI and H-RE free energies of binding and the experimental ones (Tables 4.1 and 4.2), the
tendencies observed in the experiments were barely reproduced. For transitions e and f which
involved a change from a long chain of 11 carbon atoms in the sidechain R to shorter chains of
6 and 4 carbon atoms respectively, the big perturbation yields a large statistical error estimate.
Here the simulation time may have been too short for both TI and H-RE to sufficiently sample
the conformational space.
The deviations from experimental values for the transitions in cycle I are difficult to explain,
as not the introduction of an oxygen, but its position along the chain seems to be important.
The transitions of cycle III were expected to be the most difficult ones to calculate, because
of the introduction of branching carbon atoms in the chain, but they yield the best results
concerning cycle closure. This may indicate that the S1/S3 pocket of the protein does not have
to undergo large conformational changes for these ligand modifications.
Experimental data contains uncertainties too, the estimated error for the experimental IC50
values is ± 50%. Using Eq. 4.4 this leads to uncertainties in the relative free energies of
approximately 4 kJmol−1 . Moreover, in this work we assume that all ligands are competitive
inhibitors that will bind in roughly the same orientation (Fig. 4.1). For some of these ligands it
is not unimaginable that the two hydrophobic moieties of the ligand exchange binding pockets
or that multiple binding modes contribute to the overall binding.
It is interesting to note that considering the Spearman rank-order correlation coefficient,
the observed trends in binding in the experimental data is best reproduced with the relative
energies ∆E binding of the ligand.
f ree
The results for ∆FBA in Table 4.1 can be compared to the difference in free energy of
hyd
hydration ∆FBA
for the two ligands of the transition. Data for these free energies of hydration
for different alkanes is given in [26], taken from experiments and from calculations using
hyd
the GROMOS force field 45A3. ∆FBA
is negative for a transition from hexane to butane,
−1
e.g. around -1.7 kJmol experimentally and -2.8 kJmol−1 for the force field 45A3. The TI
154
4 Calculation of binding free energies of inhibitors to Plasmepsin II
simulation of transition d, where the side chain R is changed from a hexyl to a butyl, yields
f ree
∆FBA = -0.5 kJmol−1 , which is only slightly smaller. Similar tendencies can be observed for
transitions e and f which concern a reduction of the length of the alkane chain by 5 or 7 CH2
groups respectively. The corresponding values for both the 45A3 force field and experiment
are -3.9 kJmol−1 (nonane to butane) and -5.2 kJmol−1 (nonane to ethane) [26], which could
f ree
be compared to a ∆FBA of -0.8 kJmol−1 for transition e and of -1.7 kJmol−1 for transition
f. Of course the remaining, non-alkane tail part of the ligand will have its effect upon the
f ree
bound values can be
free energies of solvation. The difference between the ∆FBA and ∆FBA
rationalised by considering the different molecules under investigation, as only the R part of
the ligands are similar to the compounds in [26], and the rest of the ligand will influence
f ree
the free energy differences too. A value of 1.9 kJmol−1 for ∆FBA for transition g is also in
hyd
accordance with an experimental value of 1.3 kJmol−1 for ∆FBA
from butane to isopentane.
−1
The -1.4 kJmol for transition h compares well to an experimental value of -3.2 kJmol−1 for
hyd
∆FBA
from propane to cyclopentane. Also the value of 3.2 kJmol−1 for transition i does not
deviate too much from the experimental value of 4.7 kJmol−1 for a change from cyclopentane
to isobutane. Thus the results of our TI simulations of the unbound ligands show a pattern of
f ree
hyd
∆FBA values similar to that of experimental ∆FBA
values and those from calculations using
the GROMOS force field 45A3 for alkanes in water.
4.7 Conclusion
The Plasmepsin II protein with exo-3-amino-7-azabicyclo[2.2.1]heptane ligands in aqueous
solution poses a very challenging case for the computation of binding free energies, as both the
ligand and the protein are flexible, the ligands are rather different, and many different factors
seem to influence the binding affinities. The length of the ligand plays an important role, but
also the polarity of its atoms, branching of the ligand, the surrounding water molecules and
the character and mobility of the protein, especially the environment of the flap pocket.
All three methods that were used to compute relative free energies of binding, thermodynamic integration (TI), TI with Hamiltonian replica exchange (H-RE) to enhance sampling,
and the calculation of ligand energy and entropy in the bound and unbound states, failed to
reproduce the experimental values derived from IC50 values. The relative free energies of
binding calculated from TI and H-RE simulations deviated by 6-8 kJmol−1 from the measured values. A reason for this is lack of sampling, especially of the conformational ensemble
of the ligand bound to the protein. The use of Hamiltonian replica-exchange MD simulation
enhances the computational efficiency and the convergence of the standard TI results. But,
force-field deficiencies or uncertainties in the measured values could also play a role.
The third method of estimating ligand energy and entropy in the bound and unbound states
is theoretically much more approximative than TI or H-RE. This is reflected in the large,
39 kJmol−1 , root-mean-square deviation of the calculated from the experimental binding free
energies. Yet for the PM II systems studied here it seems to yield relative free energies of
4.8 Supplementary material
155
binding that better match the trend in the experimental data. This may be due to cancellation
of approximation errors or of sampling and force-field errors. Due to the straightforward
separation of energetic and entropic contributions in the third method, the observed differences
in binding can be more easily related to the structural and physico-chemical characteristics of
the protein and the various ligands, which may be useful to ligand design.
4.8 Supplementary material
bonded
Free El−l
L1
121
L2
114
L3
124
L4
171
L5
136
L6
129
L7
120
nonb
El−l
-128
-123
-127
-130
-136
-130
-129
nonb
El−w
-497
-488
-506
-506
-529
-517
-509
E f ree
-505
-497
-510
-466
-529
-518
-518
T S f ree
299
263
327
341
428
292
304
Table 4.4 Energies E f ree of the ligand in the simulations of the unbound ligand in water, its bonded
bonded and non-bonded components E nonb (ligand-ligand interactions), E nonb (ligand-water interEl−l
l−l
l−w
actions), ligand configurational entropies T S f ree (310 K), calculated using conventional MD simulations. All values are given in kJmol−1 .
Bound
L1
L2
L3
L4
L5
L6
L7
bonded E nonb
El−l
l−l
87
-95
82
-89
91
-91
140
-90
107 -102
103
-92
93
-86
nonb
El−p
-758
-761
-793
-832
-759
-800
-767
nonb
El−w
-28
10
3
23
-26
8
-18
E bound
-793
-758
-790
-758
-780
-781
-778
T Sbound
204
183
211
202
290
210
197
Table 4.5 Energies E bound of the ligand of the simulations of the ligand bound to the protein in water,
bonded and non-bonded components E nonb (ligand-ligand interactions), E nonb (ligandits bonded El−l
l−l
l−w
nonb (ligand-protein interactions), ligand configurational entropies T Sbound (310
water interactions), El−p
K), calculated using conventional MD simulations. All values are given in kJmol−1 .
156
4 Calculation of binding free energies of inhibitors to Plasmepsin II
Bound-Free
L1
L2
L3
L4
L5
L6
L7
∆E binding
-288
-261
-280
-293
-250
-263
-260
T ∆Sbinding
-96
-80
-116
-139
-139
-82
-108
Table 4.6 Differences of ligand energies ∆E binding and ligand configurational entropies ∆Sbinding
between the bound and free ligand simulations in kJmol−1 .
Figure 4.11 Sidechain of the unphysical reference ligand LR used for simulations of transitions g-i
in Cycle III.
4.8 Supplementary material
Figure 4.12 h ∂∂Hλ iλ as function of λ for transitions g, h and i in the free ligand simulations. Results from TI. Error bars indicate statistical error estimates on the ensemble averages. For transition h the same procedure as
in the bound ligand TI simulations was used,
for all other transitions the simpler procedure
as in the free ligand TI simulations was applied.
157
Figure 4.13 h ∂∂Hλ iλ as function of λ for transitions g, h and i in the bound ligand simulations. Solid lines show results from H-RE,
dotted lines results from TI with error bars indicating statistical error estimation on the ensemble averages.
158
4 Calculation of binding free energies of inhibitors to Plasmepsin II
Figure 4.14 h ∂∂Hλ iλ as function of λ for transitions d, e and f in the free ligand simulations. Results from TI. Error bars indicate
statistical error estimation on the ensemble
averages.
Figure 4.15 h ∂∂Hλ iλ as function of λ for transitions d, e and f in the bound ligand simulations.
Solid lines show results from H-RE, dotted lines
results from TI with error bars indicating statistical error estimation on the ensemble averages. Dashed line: results for transition f with
more λ -values. H-RE for transition e and H-RE
with more λ -values for transition f with 100 exchange trials, all others with 50.
4.8 Supplementary material
159
Figure 4.16 Hamiltonian replica-exchange structure exchanges between different λ -values of transition f with more λ -values, shown for 200 ps, 100 exchanges. Different colors correspond to
different starting structures at different λ -values.
Figure 4.17 Propagation of ligand configurational entropy S over time, Eqs. 4.7-4.8, for simulations
of free ligand in water (left panel) and ligand bound to protein in water (right panel).
160
4 Calculation of binding free energies of inhibitors to Plasmepsin II
Parameter settings for ligand L1:
TITLE
MAKE_TOP topology, using:
/home/dsteiner/BB/NORBORNAN.dat
/usr/local/gromos/forcefields/official/ifp53a6.dat
Force-field code: 53A6
END
TOPPHYSCON
# FPEPSI: 1.0/(4.0*PI*EPS0) (EPS0 is the permittivity
138.9354
# HBAR: Planck’s constant HBAR = H/(2* PI)
0.0635078
END
RESNAME
# NRAA2: number of residues in a solute molecule
1
# AANM: residue names
NBL1
END
SOLUTEATOM
#
NRP: number of solute atoms
48
# ATNM: atom number
# MRES: residue number
# PANM: atom name of solute atom
#
IAC: integer (van der Waals) atom type code
# MASS: mass of solute atom
#
CG: charge of solute atom
#
CGC: charge group code (0 or 1)
#
INE: number of excluded atoms
# INE14: number of 1-4 interactions
# ATNM MRES PANM IAC
MASS
CG CGC INE
#
INE14
1
1
C1 12 12.01100 -0.10000 0 12
2
9
0
2
1 HC1 20 1.00800 0.10000 1 6
3
0
3
1
C2 12 12.01100 -0.10000 0 8
4
15
0
4
1 HC2 20 1.00800 0.10000 1 4
5
0
5
1
C3 12 12.01100 -0.10000 0 6
6
0
6
1 HC3 20 1.00800 0.10000 1 3
7
0
7
1
C4 12 12.01100 -0.10000 0 7
8
16
0
8
1 HC4 20 1.00800 0.10000 1 3
9
0
9
1
C5 12 12.01100 -0.10000 0 8
10
17
0
10
1 HC5 20 1.00800 0.10000 1 5
11
0
11
1
C6 12 12.01100 -0.10000 0 7
12
18
1
19
12
1 HC6 20 1.00800 0.10000 1 4
13
of vacuum)
3
13
4
14
5
15
6
16
7
17
4
5
13
15
16
5
16
6
7
8
13
6
7
16
7
8
9
15
16
8
15
9
10
11
13
15
15
16
11
18
12
13
15
16
12
15
16
17
13
14
15
16
15
17
18
17
4.8 Supplementary material
13
1
C8
14
1
15
161
0
5
1
4
0
2
0
2
0
2
2
3
2
4
4
1
2
6
6
8
12 12.01100 -0.10000
0
HC8
20
1.00800
0.10000
1
1
C4A
12 12.01100
0.00000
0
16
1
C8A
12 12.01100
0.00000
1
17
1
C7
12 12.01100
0.00000
1
18
1
C9
15 14.02700
0.22000
0
19
1
N10
6 14.00670 -0.88000
0
20
1
HA1
21
1.00800
0.44000
0
21
1
C11
14 13.01900
0.22000
1
22
1
C12
14 13.01900
0.00000
1
23
1
C14
15 14.02700
0.00000
0
24
1
C15
15 14.02700
0.00000
1
25
1
C13
14 13.01900
0.12700
0
26
1
C16
14 13.01900
0.12700
0
27
1
N17
7 14.00670
0.25000
0
28
1
HN1
21
1.00800
0.24800
0
29
1
HN2
21
1.00800
0.24800
1
30
1
S18
23 32.06000
0.72000
0
31
1
OS1
1 15.99940 -0.36000
0
32
1
OS2
1 15.99940 -0.36000
1
33
1
C19
12 12.01100
0.00000
1
0
2
2
1
2
9
34
1
C20
12 12.01100 -0.10000
0
0
8
35
1 HC20
20
0.10000
1
36
1
12 12.01100 -0.10000
0
37
1 HC21
20
0.10000
1
38
1
12 12.01100 -0.10000
0
39
1 HC23
20
0.10000
1
40
1
12 12.01100 -0.10000
0
41
1 HC24
20
1
C21
C23
C24
1.00800
1.00800
1.00800
1.00800
0.10000
5
4
3
3
2
5
3
3
1
2
1
1
0
0
0
9
0
4
0
6
1
3
0
5
1
4
0
3
0
1
0
14
19
15
15
16
17
16
17
18
16
17
17
18
18
20
19
22
20
24
21
22
22
23
23
32
24
24
28
25
28
26
31
27
30
28
30
29
19
21
20
26
21
25
26
24
28
25
33
28
25
29
26
29
27
32
28
18
21
22
27
26
30
25
29
26
26
31
27
27
32
30
29
26
30
27
34
27
40
28
33
29
29
30
33
41
34
35
36
36
42
37
38
39
40
41
43
29
31
38
32
40
32
34
33
34
34
40
33
40
40
35
41
35
42
36
43
37
38
36
37
40
42
37
44
38
38
39
40
42
42
43
39
44
40
40
41
42
43
41
42
43
41
42
43
42
30
33
31
162
4 Calculation of binding free energies of inhibitors to Plasmepsin II
42
1
C22
12 12.01100
0.00000
0
43
1
C25
15 14.02700
0.00000
1
44
1
C26
15 14.02700
0.00000
0
45
1
C27
15 14.02700
0.00000
0
46
1
C28
15 14.02700
0.00000
1
47
1
C29
15 14.02700
0.00000
0
48
1
C30
16 15.03500
0.00000
1
END
BONDTYPE
# NBTY: number of covalent bond types
52
# CB: force constant
# B0: bond length at minimum energy
#
CB
B0
1.57000e+07 1.00000e-01
1.87000e+07 1.00000e-01
1.23000e+07 1.09000e-01
3.70000e+07 1.12000e-01
1.66000e+07 1.23000e-01
1.34000e+07 1.25000e-01
1.20000e+07 1.32000e-01
8.87000e+06 1.33000e-01
1.06000e+07 1.33000e-01
1.18000e+07 1.33000e-01
# 10
1.05000e+07 1.34000e-01
1.17000e+07 1.34000e-01
1.02000e+07 1.36000e-01
1.10000e+07 1.38000e-01
8.66000e+06 1.39000e-01
1.08000e+07 1.39000e-01
8.54000e+06 1.40000e-01
8.18000e+06 1.43000e-01
9.21000e+06 1.43000e-01
6.10000e+06 1.43500e-01
# 20
8.71000e+06 1.47000e-01
5.73000e+06 1.48000e-01
7.64000e+06 1.48000e-01
8.60000e+06 1.48000e-01
8.37000e+06 1.50000e-01
5.43000e+06 1.52000e-01
7.15000e+06 1.53000e-01
4.84000e+06 1.61000e-01
4.72000e+06 1.63000e-01
2.72000e+06 1.78000e-01
# 30
5.94000e+06 1.78000e-01
5.62000e+06 1.83000e-01
3.59000e+06 1.87000e-01
6.40000e+05 1.98000e-01
6.28000e+05 2.00000e-01
5.03000e+06 2.04000e-01
5.40000e+05 2.21000e-01
2.32000e+07 1.00000e-01
1.21000e+07 1.10000e-01
2
1
2
1
2
1
2
1
2
0
1
0
0
0
43
45
44
46
45
47
46
48
47
48
44
45
46
47
48
4.8 Supplementary material
8.12000e+06 1.75800e-01
# 40
8.04000e+06 1.53000e-01
4.95000e+06 1.93799e-01
8.10000e+06 1.76000e-01
1.31000e+07 1.26500e-01
1.03000e+07 1.35000e-01
8.71000e+06 1.63299e-01
2.68000e+06 2.33839e-01
2.98000e+06 2.90283e-01
2.39000e+06 2.79388e-01
2.19000e+06 2.91189e-01
# 50
3.97000e+06 2.07700e-01
3.04000e+06 2.87407e-01
END
BONDH
# NBONH: number of bonds involving H atoms in solute
14
# IBH, JBH: atom sequence numbers of atoms forming a bond
# ICBH: bond type code
#
IBH
JBH ICBH
1
2
3
3
4
3
5
6
3
7
8
3
9
10
3
11
12
3
13
14
3
19
20
2
27
28
2
27
29
2
# 10
34
35
3
36
37
3
38
39
3
40
41
3
END
BOND
# NBON: number of bonds NOT involving H atoms in solute
38
# IB, JB: atom sequence numbers of atoms forming a bond
# ICB: bond type code
#
IB
JB ICB
1
3
16
1
16
16
3
5
16
5
7
16
7
15
16
9
11
16
9
15
16
11
17
16
13
16
16
13
17
16
# 10
15
16
16
17
18
27
18
19
21
19
21
21
21
22
27
21
26
27
22
25
27
22
30
32
163
164
4 Calculation of binding free energies of inhibitors to Plasmepsin II
23
23
24
25
27
27
24
25
26
30
30
30
33
33
34
36
26
27
27
31
32
33
34
40
36
42
27
21
21
25
25
32
16
16
16
16
38
38
42
43
44
45
46
47
40
42
43
44
45
46
47
48
16
16
27
27
27
27
27
27
# 20
# 30
END
BONDANGLETYPE
# NTTY: number of bond angle types
54
# CT: force constant
# T0: bond angle at minimum energy in degrees
#
CT
T0
3.80000e+02 9.00000e+01
4.20000e+02 9.00000e+01
4.05000e+02 9.60000e+01
4.75000e+02 1.00000e+02
4.20000e+02 1.03000e+02
4.90000e+02 1.04000e+02
4.65000e+02 1.08000e+02
2.85000e+02 1.09500e+02
3.20000e+02 1.09500e+02
3.80000e+02 1.09500e+02
# 10
4.25000e+02 1.09500e+02
4.50000e+02 1.09500e+02
5.20000e+02 1.09500e+02
4.50000e+02 1.09600e+02
5.30000e+02 1.11000e+02
5.45000e+02 1.13000e+02
5.00000e+01 1.15000e+02
4.60000e+02 1.15000e+02
6.10000e+02 1.15000e+02
4.65000e+02 1.16000e+02
# 20
6.20000e+02 1.16000e+02
6.35000e+02 1.17000e+02
3.90000e+02 1.20000e+02
4.45000e+02 1.20000e+02
5.05000e+02 1.20000e+02
5.30000e+02 1.20000e+02
5.60000e+02 1.20000e+02
6.70000e+02 1.20000e+02
7.80000e+02 1.20000e+02
6.85000e+02 1.21000e+02
# 30
7.00000e+02 1.22000e+02
4.8 Supplementary material
4.15000e+02 1.23000e+02
7.30000e+02 1.24000e+02
3.75000e+02 1.25000e+02
7.50000e+02 1.25000e+02
5.75000e+02 1.26000e+02
6.40000e+02 1.26000e+02
7.70000e+02 1.26000e+02
7.60000e+02 1.32000e+02
2.21500e+03 1.55000e+02
# 40
9.13500e+04 1.80000e+02
4.34000e+02 1.09500e+02
4.84000e+02 1.07570e+02
6.32000e+02 1.11300e+02
4.69000e+02 9.74000e+01
5.03000e+02 1.06750e+02
4.43000e+02 1.08530e+02
6.18000e+02 1.09500e+02
5.07000e+02 1.07600e+02
4.48000e+02 1.09500e+02
# 50
5.24000e+02 1.10300e+02
5.32000e+02 1.11400e+02
6.36000e+02 1.17200e+02
6.90000e+02 1.21400e+02
END
BONDANGLEH
# NTHEH: number of bond angles involving H atoms in solute
29
# ITH, JTH, KTH: atom sequence numbers
#
of atoms forming a bond angle in solute
# ICTH: bond angle type code
#
ITH
JTH
KTH ICTH
2
1
3
25
2
1
16
25
1
3
4
25
4
3
5
25
3
5
6
25
6
5
7
25
5
7
8
25
8
7
15
25
10
9
11
25
10
9
15
25
# 10
9
11
12
25
12
11
17
25
14
13
16
25
14
13
17
25
18
19
20
11
20
19
21
11
25
27
28
18
25
27
29
18
26
27
28
18
26
27
29
18
# 20
28
27
29
10
33
34
35
25
35
34
36
25
34
36
37
25
37
36
42
25
39
38
40
25
39
38
42
25
33
40
41
25
165
166
4 Calculation of binding free energies of inhibitors to Plasmepsin II
38
40
41
25
END
BONDANGLE
# NTHE: number of bond angles NOT
#
involving H atoms in solute
54
# IT, JT, KT: atom sequence numbers of atoms
#
forming a bond angle
# ICT: bond angle type code
#
IT
JT
KT ICT
3
1
16
27
1
3
5
27
3
5
7
27
5
7
15
27
11
9
15
27
9
11
17
27
16
13
17
27
7
15
9
27
7
15
16
27
9
15
16
27
# 10
1
16
13
27
1
16
15
27
13
16
15
27
11
17
13
27
11
17
18
27
13
17
18
27
17
18
19
13
18
19
21
21
19
21
22
15
19
21
26
15
# 20
22
21
26
5
21
22
25
5
21
22
30
16
25
22
30
16
24
23
25
5
23
24
26
5
22
25
23
8
22
25
27
5
23
25
27
5
21
26
24
8
# 30
21
26
27
5
24
26
27
5
25
27
26
3
22
30
31
14
22
30
32
14
22
30
33
5
31
30
32
29
31
30
33
14
32
30
33
14
30
33
34
27
# 40
30
33
40
27
34
33
40
27
33
34
36
27
34
36
42
27
40
38
42
27
33
40
38
27
36
42
38
27
36
42
43
27
38
42
43
27
4.8 Supplementary material
42
43
44
15
43
44
45
46
44
45
46
47
45
46
47
48
15
15
15
15
# 50
END
IMPDIHEDRALTYPE
# NQTY: number of improper dihedrals
3
# CQ: force constant of improper dihedral per degrees square
# Q0: improper dihedral angle at minimum energy in degrees
#
CQ
Q0
5.10000e-02 0.00000e+00
1.02000e-01 3.52644e+01
2.04000e-01 0.00000e+00
END
IMPDIHEDRALH
# NQHIH: number of improper dihedrals
#
involving H atoms in the solute
11
# IQH,JQH,KQH,LQH: atom sequence numbers
#
of atoms forming an improper dihedral
# ICQH: improper dihedral type code
#
IQH
JQH
KQH
LQH ICQH
1
2
3
16
1
3
1
4
5
1
5
3
6
7
1
7
5
8
15
1
9
10
11
15
1
11
9
12
17
1
13
14
16
17
1
34
33
36
35
1
36
34
42
37
1
38
39
42
40
1
# 10
40
33
41
38
1
END
IMPDIHEDRAL
# NQHI: number of improper dihedrals NOT
#
involving H atoms in solute
25
# IQ,JQ,KQ,LQ: atom sequence numbers of atoms
#
forming an improper dihedral
# ICQ: improper dihedral type code
#
IQ
JQ
KQ
LQ ICQ
1
3
5
7
1
3
1
16
15
1
3
5
7
15
1
5
7
15
16
1
7
15
16
1
1
9
11
17
13
1
9
15
16
13
1
11
9
15
16
1
15
7
9
16
1
15
9
11
17
1
# 10
16
1
3
5
1
16
1
15
13
1
16
13
17
11
1
17
11
18
13
1
17
13
16
15
1
21
25
30
22
2
167
168
4 Calculation of binding free energies of inhibitors to Plasmepsin II
26
33
33
34
19
30
34
33
22
40
36
40
21
34
42
38
2
1
1
1
34
40
40
42
42
36
33
38
36
38
42
34
42
38
40
38
36
36
43
33
1
1
1
1
1
# 20
END
DIHEDRALTYPE
# NPTY: number of dihedral types
41
# CP: force constant
# PD: cosine of the phase shift
# NP: multiplicity
#
CP
PD NP
2.67000 -1.00000
1
3.41000 -1.00000
1
4.97000 -1.00000
1
5.86000 -1.00000
1
9.35000 -1.00000
1
9.45000 -1.00000
1
2.79000
1.00000
1
5.35000
1.00000
1
1.53000 -1.00000
2
5.86000 -1.00000
2
# 10
7.11000 -1.00000
2
16.70000 -1.00000
2
24.00000 -1.00000
2
33.50000 -1.00000
2
41.80000 -1.00000
2
0.00000
1.00000
2
0.41800
1.00000
2
2.09000
1.00000
2
3.14000
1.00000
2
5.09000
1.00000
2
# 20
16.70000
1.00000
2
1.05000
1.00000
3
1.26000
1.00000
3
1.30000
1.00000
3
2.53000
1.00000
3
2.93000
1.00000
3
3.19000
1.00000
3
3.65000
1.00000
3
3.77000
1.00000
3
3.90000
1.00000
3
# 30
4.18000
1.00000
3
4.69000
1.00000
3
5.44000
1.00000
3
5.92000
1.00000
3
7.69000
1.00000
3
8.62000
1.00000
3
9.50000
1.00000
3
0.00000
1.00000
4
1.00000 -1.00000
6
1.00000
1.00000
6
# 40
3.77000
1.00000
6
4.8 Supplementary material
END
DIHEDRALH
# NPHIH: number of dihedrals involving H atoms in solute
0
# IPH, JPH, KPH, LPH: atom sequence numbers
#
of atoms forming a dihedral
# ICPH: dihedral type code
#
IPH
JPH
KPH
LPH ICPH
END
DIHEDRAL
# NPHI: number of dihedrals NOT involving H atoms in solute
18
# IP, JP, KP, LP: atom sequence numbers
#
of atoms forming a dihedral
# ICP: dihedral type code
#
IP
JP
KP
LP ICP
11
17
18
19
40
17
18
19
21
39
18
19
21
26
39
26
21
22
25
39
22
21
26
24
34
21
22
25
23
34
25
22
30
33
26
25
23
24
26
39
24
23
25
22
34
23
24
26
21
34
# 10
23
25
27
26
29
21
26
27
25
29
22
30
33
40
20
38
42
43
44
40
42
43
44
45
34
43
44
45
46
34
44
45
46
47
34
45
46
47
48
34
END
169
5 Calculation of the relative free
energy of oxidation of Azurin at pH
5 and pH 9
5.1 Summary
Free energy calculations are described for the small copper-containing redox protein Azurin
from Pseudomonas aeruginosa. A thermodynamic cycle connecting the reduced and oxidised
states at pH 5 and pH 9 is considered, allowing for an assessment of convergence in terms
of hysteresis and cycle closure. Previously published thermodynamic integration (TI) data
is compared to Hamiltonian replica exchange TI (RE-TI) simulations using different simulation setups. The effects of varying simulation length, initial structure, position restraints on
particular atoms and the strength of temperature coupling are studied. It is found that RE-TI
simulations do stimulate the distribution of conformational changes over the relevant values
of the TI coupling parameter λ . This results in significantly improved values for hysteresis
and cycle closure when compared to regular TI.
171
172
5 Calculation of the relative free energy of oxidation of Azurin at pH 5 and pH 9
5.2 Introduction
The computation of the free enthalpy or Gibbs free energy G of a biomolecular system is
an impossible task, because it would require the evaluation of an integral over a (6N+1)dimensional space with N being the number of atoms in the system,
Z Z Z
1
−(H (pN ,rN )+pV )/kB T
N N
e
dp dr dV ,
(5.1)
G = −kB T ln
V h3N
in which kB is Boltzmann’s constant, T is the temperature, h is the Planck constant and the
Hamiltonian
N
p2
H (pN , rN ) = ∑ i +V rN
(5.2)
i=1 2mi
is the sum of the kinetic energy and the potential energy V rN of the system,
rN ≡ (r1 , r2 , ..., rN ) and pN ≡ (p1 , p2 , ..., pN ) are the Cartesian coordinates and conjugate momenta of the N atoms, mi is the mass of atom i, p is the pressure and V is the volume of the
system. Ignoring solvent degrees of freedom, the number of atoms of a protein easily exceeds
102 , which renders the integral in Eq. 5.1 wholly intractable. Yet, the free enthalpy is a central
thermodynamic quantity which strives for a minimum for any system.
A less ambitious goal is to compute the relative free enthalpy or the free enthalpy difference
of two systems or of two states of a system, because this only requires the calculation of the
ratio of two integrals of the type in Eq. 5.1. This means that only those parts of configuration
N N
space for which the integrand e−(H (p ,r )+pV )/kB T is different for the two systems or states is
to be evaluated. Over the past decades a great variety of methods to sample these differences
have been proposed and tested for a variety of systems and states, see [103, 104] and references
therein. A well established method is called thermodynamic integration (TI) [88] in which the
Hamiltonian is made a function of a coupling parameter λ , H (pN , rN ; λ ), which connects
two systems or states A and B through a variation of λ between the values λA and λB ,
H (pN , rN ; λA ) = HA (pN , rN )
(5.3)
H (pN , rN ; λB ) = HB (pN , rN ).
(5.4)
Differentiation of G(λ ) with respect to λ yields the TI expression for the free enthalpy difference
Z
∆GBA ≡ GB − GA =
λB
λA
h∂ H /∂ λ iλ d λ
(5.5)
in which the ensemble average for a system with H (pN , rN ; λ ) is denoted by h...iλ .
Although Eq. 5.5 is exact, it is often not easy to obtain precise values for ∆GBA because the
convergence of the ensemble average h...iλ depends on the characteristics of the Hamiltonian
H (pN , rN ; λ ):
• The type of change of H from HA to HB , e.g. how many atoms are involved and which
5.2 Introduction
173
type of interactions such as van der Waals or Coulomb interactions are involved.
• The choice of λ -dependence of the Hamiltonian, i.e. the pathway connecting HA with
HB .
• The ease of configurational relaxation of the environment upon the change from HA to
HB , i.e. how fast the equilibrium corresponding to H (pN , rN ; λ ) is reached for each
λ -value.
Generally, it is not straightforward to determine whether the ensemble average h∂ H /∂ λ iλ
has sufficiently converged. A necessary but not sufficient condition is that a change in free
enthalpy along a closed path of λ -values must be zero. Such a closed path is for example the
successive reduction, pH change, oxidation and reverse pH change of a protein in aqueous
solution, processes that have been experimentally studied for the protein Azurin.
Figure 5.1 (a) Cartoon view of the chain A parts in the crystallographic structures of Azurin with
PDB code 4AZU (oxidised, pH 5) in white. For 5AZU (oxidised, pH 9), residues 36 to 38 are shown
in blue and residues 89 to 90 in red. The Cu-ion is shown in orange. (b) Ribbon view of R5 (red),
R9 (blue), O5 (silver), and O9 (yellow) after 1 ns of MD simulation in water starting from the energy
minimised 4AZU structure.
The blue copper protein Azurin is a small, 128-residue electron transfer protein involved
in the redox reaction with Cytochrome c551 in Pseudomonas aeruginosa. The interaction
between these two small proteins, both containing a single metal center, is well studied. The
process is quite remarkable as it has complex redox kinetics, the bimolecular electron transfer
being fast, followed by a slow, mononuclear reaction in Azurin, see [105] and references
therein. The slow reaction was explained by an equilibrium between two conformational
states of the reduced Azurin, the difference between the two states consisting mainly in a flip
of the peptide plane between Pro 36 and Gly 37 [105], see Figure 5.1a. These two residues
174
5 Calculation of the relative free energy of oxidation of Azurin at pH 5 and pH 9
lie in a cleft near the binding site of the copper, but are not coordinated to the metal. The
slow mononuclear reaction was shown to be comparable to the rate of protonation of His 35
[106]. Furthermore a pH dependence of the Azurin peptide plane rotation was demonstrated
[107, 108]. The copper is coordinated by five amino acids, forming a trigonal bipyramidal
coordination geometry and changes its oxidation state from Cu(II) in the oxidised state to
Cu(I) in the reduced state.
R5 (reduced,
pH 5) o
O
R9t5
RtO5
OtR5
/
R5t9
R9 (reduced, pH 9) o
O5 (oxidised,
pH 5)
O
O5t9
O9t5
RtO9
/
O9 (oxidised, pH 9)
OtR9
Figure 5.2 Thermodynamic cycle between the four states of Azurin: R: reduced, O: oxidised, pH
5 or 9. Transitions are indicated with the symbol “t” between the state indicators.
The redox potential or ∆GRO = GR −GO , the free enthalpy of reduction of Azurin, cannot be
determined directly, but the difference in redox potential between two different pH values can
be determined. For Azurin structural and thermodynamic data at pH 5 and pH 9 is available,
see [105, 109, 110]. The relevant thermodynamic cycle is shown in Fig. 5.2.
In terms of testing methodology to calculate relative free enthalpies, this system is of particular interest because the change in free enthalpy associated with each of the four legs of the
thermodynamic cycle can be calculated. This is in contrast to most systems where only part
of the cycle can be calculated, as e.g. for differences in ligand binding free energy, where the
process of the solvated ligand approaching the protein and binding to it is often too demanding
to be simulated by MD on an atomic level.
Since the oxidation process involves a change of the charge at the copper site and the pH
change from 9 to 5 involves the protonation of the side chain of His 35, both free enthalpy
changes are relatively large and not easily determined. In an earlier study of this protein errors
in relative free enthalpies between 1 and 10 kB T were reported [111]. The method used was
TI for Azurin in a relatively small computational box containing only 3208 water molecules.
A random ordering of the λ -values showed jumps between h∂ H /∂ λ iλ of adjacent λ -values
and different results using distinct pathways. This led to the assumption that a better mixing of
the different structures at neighboring λ -values may help to smooth the curve of h∂ H /∂ λ iλ
as a function of λ .
Recently, it has been proposed to combine the multi-copy sampling technique called Hamiltonian replica exchange with TI by exchanging replicas characterised by adjacent λ -values
[93]. In Hamiltonian replica exchange thermodynamic integration (RE-TI) several MD simulations, replicas, that differ in λ -value, are carried out in parallel. After fixed numbers of
5.3 Simulation setup
175
MD time steps, exchanges of the configurations Xm and Xn of replicas with adjacent λ -values,
i.e. λm and λn defining the Hamiltonians Hm and Hn respectively, are attempted using the
detailed balance condition,
PS′ t(S′ → S′′ ) = PS′′ t(S′′ → S′ ),
(5.6)
where S′ denotes the state before the exchange, i.e. Hm has configuration Xm and Hn has
configuration Xn , and S′′ the state after the exchange, i.e. Hm has configuration Xn and Hn
has configuration Xm . PS denotes the configurational probability of state S and t(S → S′ ) the
transition probability from state S to state S′ . The relative configurational probabilities of
states S′′ and S′ are
PS′′
e−Hm (Xn )/kB T e−Hn (Xm )/kB T
= −H (X )/k T −H (X )/k T .
(5.7)
PS′
e m m B e n n B
Combining Eqs. (5.6) and (5.7) we find for the probability p of exchange of replicas m and n
with
p(m ↔ n) = t(S′ → S′′ ) = min(1, e−∆/kB T )
(5.8)
∆ = [Hm (Xn ) + Hn (Xm )] − [Hm (Xm ) + Hn (Xn )].
(5.9)
With RE-TI, the sampling of conformational space at a specific λ -value may be enhanced by
replacing the current conformation by one at a slightly different Hamiltonian, which may lead
to a better mixing of the different structures observed at neighbouring λ -values and hence to
a better convergence of the free energy profile.
Here we investigate the sampling efficiency of this RE-TI method using Azurin as a test
case. Different parameters of the simulations are varied: (i) starting structures of the protein,
(ii) time period over which h...iλ is equilibrated or sampled, (iii) positional restraints applied
to particular atoms to restrict the configurational relaxation, (iv) strength of the coupling of the
temperature to a heat bath. The values of ∆GBA along the four legs of the thermodynamic cycle
are compared, and the degree of cycle closure and conformational relaxation of the protein is
investigated.
The structural reorganisation in a protein involved in electron transfer has been shown to
include many residues in certain cases [112]. In our study the focus is on a conformational
change involving residues His 35 to Gly 37, where different hydrogen-bonding patterns were
found in X-ray structures at pH 5 and pH 9 [105].
5.3 Simulation setup
All simulations were carried out using the GROMOS software [113] and the GROMOS force
field 54A7 [70]. The system consists of the protein, 128 amino acid residues, and a Cu-ion.
Aliphatic CHn groups were treated as united atoms. Simple-point-charge (SPC)-water[33] was
176
5 Calculation of the relative free energy of oxidation of Azurin at pH 5 and pH 9
used as solvent (12383 water molecules). The charges of the atoms in the amino acids around
the Cu-ion which change upon a change of oxidation state of the Cu-ion were taken from the
work of van den Bosch et al. [111] (set I), see Table 5.1.
The initial structure for the simulations was the X-ray structure of oxidised Azurin at pH
5, taken from the Protein Data Bank [34], PDB ID 4AZU [105]. In all simulations, instantaneous distance restraining was applied to the distances between the Cu-ion and the five atoms
residue
Gly 45
His 46
Cys 112
His 117
Met 121
His 35
atom
Cu
Cα
C
O
N
H
Cα
Cβ
Cγ
Nδ 1
Cδ 2
Hδ 2
Cε 1
Hε 1
Nε 2
Hε 2
Cα
Cβ
Sγ
Cα
Cβ
Cγ
Nδ 1
Cδ 2
Hδ 2
Cε 1
Hε 1
Nε 2
Hε 2
Cα
Cβ
Sγ
Sδ
Cε
Cγ
Nδ 1
Hδ 1
Cδ 2
Hδ 2
Cε 1
Hε 1
Nε 2
Hε 2
charge (e)
reduced(I) oxidised(II)
0.215
0.333
0.124
0.147
0.381
0.400
-0.506
-0.526
-0.730
-0.705
0.368
0.384
0.461
0.429
0.094
0.117
0.077
0.079
-0.373
-0.386
0.014
0.055
0.000
0.000
0.350
0.378
0.000
0.000
-0.565
-0.534
0.479
0.494
-0.039
-0.013
0.056
0.153
-0.629
-0.222
-0.035
-0.017
0.214
0.217
-0.003
0.001
-0.378
-0.391
0.142
0.181
0.000
0.000
0.352
0.366
0.000
0.000
-0.489
-0.451
0.409
0.433
-0.110
-0.087
0.168
0.156
0.084
0.131
-0.259
-0.261
0.129
0.141
charge (e)
pH 5
pH 9
-0.05
0.13
0.38
-0.58
0.30
0.00
0.00
0.00
0.00
-0.24
0.26
0.00
0.00
0.31
0.00
0.30
0.19
Table 5.1 Partial charges used for the Cu-ion and the residues Gly 45, His 46, Cys 112, His 117,
and Met 121 coordinated to it for the reduced and oxidised state, and partial charges of His 35 at
pH 5 and pH 9. Values were taken from the work of van den Bosch et al. [111], set I.
5.3 Simulation setup
177
Atom pair
Cu-O Gly 45
Cu-Nδ 1 His 46
Cu-Sγ Cys 112
Cu-Nδ 1 His 117
Cu-Sδ Met 121
distance restraint (nm)
0.2955
0.2064
0.2267
0.1978
0.3164
Table 5.2 Distances used for Cu-ligand atom distance restraining in the simulations. Distances
were taken from the work of van den Bosch et al. [111], set A.
coordinated to it with a force constant of 25000 kJmol−1 nm−2 . The distances shown in Table
5.2 were taken from [111] (set A). Rectangular periodic boundary conditions were applied
to the cubic box with edge lengths of 7.36 nm. The initial structure of oxidised Azurin at
pH 5 was energy minimised, followed by a thermalisation. For this, initial velocities were
generated from a Maxwell-Boltzmann distribution at 50 K. The atoms of the solute were harmonically positionally restrained. In six 10 ps simulation periods the simulation temperature
was raised in steps of 50 K from 50 K up to 300 K while simultaneously decreasing the
position-restraining force constant from 25000 kJmol−1 nm−2 to 0 kJmol−1 nm−2 by a factor
of 10 in the subsequent simulations. Evaluating the non-bonded interactions a triple-range
cutoff scheme was used. Within a short-range cutoff of 0.8 nm interactions were calculated
at every time step from a pair list generated every 5th time step. At every 5th time step interactions between 0.8 and 1.4 nm were updated. A reaction field force [38] with a dielectric
permittivity of 61 [39] was applied to represent electrostatic interactions outside the 1.4 nm
cutoff. A step-size of 2 fs was used for integration of the equations of motion using the leap
frog scheme. All bonds and the bond angles of the water molecules were constrained using
the SHAKE algorithm [41].
After thermalisation, separate equilibrations of all four states of Azurin were performed.
The temperature was maintained at 300 K using the weak coupling method [36]. The solute
(protein and Cu) and solvent degrees of freedom were separately coupled to a heat bath with
a coupling time τT = 0.1 ps, and at a constant pressure of 1 atm using a coupling constant τP
= 0.5 ps and an isothermal compressibility of 4.575 · 10−4 molnm3 kJ−1 . The equilibration
of the four different states of Azurin was performed for 1 ns and trajectory coordinates were
saved every 5 ps for analysis.
For the RE-TI simulations, n = 11 λn -values, from 0 to 1 in steps of 0.1, were used. Trajectory coordinates were saved all 0.5 ps for analysis. Replica-exchange trials were performed
every 2 ps, alternating between the ”odd” pairs of n (X1 ↔ X2 , X3 ↔ X4 , ... ) and the ”even”
pairs (X2 ↔ X3 , X4 ↔ X5 , ... ).
RE-TI simulations were performed using three different sets of initial structures at the 11
λ -values, using three different sets of protein atoms that are positionally restrained in order to
limit the conformational space to be sampled, and using two different coupling strengths for
the temperature coupling. These RE-TI simulation conditions are denoted as follows.
178
5 Calculation of the relative free energy of oxidation of Azurin at pH 5 and pH 9
XO5: The initial structures for all 11 λ -values and all four legs of the thermodynamic cycle in
Fig. 5.2 are identical, i.e. the structure of oxidised Azurin at pH 5 after thermalisation.
XDE: The four different structures, O5, R5, O9 and R9, resulting from the four 1 ns equilibration simulations were used as initial structures, identical for all 11 λ -values, for the
four different legs of the thermodynamic cycle. Thus the RE-TI simulations OtR5 and
O5t9 started from the equilibrated O5 structure, the simulations RtO5 and R5t9 from
the equilibrated R5 structure, the simulations RtO9 and R9t5 from the equilibrated R9
structure, and the simulations OtR9 and O9t5 from the equilibrated O9 structure.
XDS: Initial structures that were different for all λ -values of all four legs of the thermodynamic cycle were generated using so-called slow-growth simulations in which the value
of the coupling parameter is slowly changed during the simulation, by 2 · 10−5 per time
step of 2 fs. Two slow-growth simulations in which the Cα -atoms were positionally
restrained to the initial structure with a force constant of 25000 kJmol−1 nm−2 , each
starting from the equilibrated O5 structure, were performed. One for the OtR5 leg of
the cycle (100 ps) followed by the R5t9 leg (100 ps) and the other for the O5t9 leg
(100 ps) followed by the OtR9 leg (100 ps). The structures at decimal λ -values were
used as initial structures for the RE-TI simulations.
ALL: All protein atoms are positionally restrained to the initial structure with a force constant
of 25000 kJmol−1 nm−2 .
CA: The Cα -atoms are positionally restrained to the initial structure with a force constant of
25000 kJmol−1 nm−2 .
CA-Cu: All Cα -atoms except those of residues Gly 45, His 46, Cys 112, His 117, Met
121, and His 35, for which atoms partial charges are changed upon oxidation or pH
change, are positionally restrained to the initial structure with a force constant of 25000
kJmol−1 nm−2 .
WT: A standard weak coupling time τT = 0.1 ps for the temperature coupling to the heat bath
is used.
TT: A tight coupling of the temperature to the heat bath is executed by using τT = 2 fs.
Note that we use the notation R5t9, etc. in three different ways.
1. It may indicate the free enthalpy difference GR9 − GR5 .
2. It may indicate the direction, from pH = 5 to pH = 9, in which the coupling parameter
λ is changed in a slow-growth simulation, in this case for the reduced form of Azurin.
3. It may indicate the direction in which the two types of Monte Carlo exchanges of Xn
and Xn+1 between the Hamiltonians H (λn ) and H (λn+1 ) are alternating as function
of time. The first type is an exchange of configurations for all odd values of n, and the
second type is an exchange of configurations for all even values of n. R5t9 means that
the series of alternating exchanges starts with type 1 at R5, while for R9t5 it starts with
type 1 at R9. Thus for an even number of λ -values, the free enthalpies will be the same
for R5t9 and R9t5 in the case of identical starting structures as in the XDS simulations,
whereas for an odd number of λ -values, they need not be the same.
5.4 Results
179
The statistical error [102] estimate for the different legs σleg was approximated by calculating
τ f ull
block averages h ∂∂H
λ iτb for nb = 4 blocks of a length of τb = 4 and using the mean value
h ∂∂H
λ iτ f ull over the whole simulation time τ f ull to get an estimate for the variance
2
σleg
∂H
iτ
h
∂λ b
2
1 nb
∂H
∂H
=
∑ h ∂ λ iτb − h ∂ λ iτ f ull .
nb b=1
(5.10)
The error estimate of the cycle closure σcycle of the cycles R5-O5-O9-R9-R5 and R5-R9O9-O5-R5 were obtained using
2
σcycle
=
4
2
.
∑ σleg
n
(5.11)
n=1
5.4 Results
For all four states, reduced Azurin at pH 5 (R5), reduced Azurin at pH 9 (R9), oxidised Azurin
at pH 5 (O5) and oxidised Azurin at pH 9 (O9), standard MD simulations were performed.
Figs. 5.1b and 5.8 show that the secondary structure is well maintained in the four simulations.
Differences between the different states are observed for e.g. residues 40 to 45 or residues 118
to 120, where α -helices are observed, but with different occurrence rates.
4AZU
4AZU
5AZU
O5
O9
R5
R9
0.026
0.090
0.135
0.173
0.155
5AZU
0.045
0.096
0.133
0.176
0.154
O5
0.168
0.166
0.165
0.175
0.176
O9
0.195
0.189
0.211
0.214
0.142
R5
0.231
0.230
0.240
0.271
R9
0.217
0.216
0.241
0.220
0.288
0.239
Table 5.3 Atom positional root-mean-square differences (RMSD) in nm of the backbone atoms C,
O, N and Cα (lower triangle) and of all atoms (upper triangle) between the two X-ray structures,
4AZU (oxidised Azurin at pH 5) and 5AZU (oxidised Azurin at pH 9), energy minimised in water,
and the four final configurations after 1 ns of MD simulations of the four states O5, O9, R5, and
R9.
Table 5.3 compares the four final conformations of these simulations to the two X-ray structures from the PDB, 4AZU in the oxidised state at pH 5 and 5AZU in the oxidised state at pH
9. The simulations, also of the oxidised state at pH 5, show more deviation from the starting
structure 4AZU than the other X-ray structure 5AZU. Also, the final conformation of simulation O9 deviates by a similar amount from both X-ray structures, indicating that the thermal
motion is larger than the positional differences inferred from the X-ray experiments. No transition of the peptide bond dihedral angle ω between Pro 36 and Gly 37 was observed, but
the neighboring ψ of Pro 36 and ϕ of Gly 37 show anti-correlated changes at higher pH, see
Fig. 5.3, which reflects a rotation of the peptide plane between residues 36 and 37. The high
180
5 Calculation of the relative free energy of oxidation of Azurin at pH 5 and pH 9
Figure 5.3 Dihedral angle values as function of time for the dihedral angles in residues 35 to 37 in
the four MD simulations at different pH values and oxidation states. Upper left panel: reduced at
pH 5, upper right panel: reduced at pH 9, lower left panel: oxidised at pH 5, and lower right panel:
oxidised at pH 9. Dihedral angles ϕ of His 35 (red), ψ of His 35 (blue), ϕ of Pro 36 (turquoise),
ψ of Pro 36 (magenta), ω between Pro 36 and Gly 37 (orange), ϕ of Gly 37 (indigo), and the
side-chain angles χ1 (black) and χ2 (green) of His 35 are shown.
Figure 5.4 Distances in nm between particular hydrogen-bond donors and acceptors as a function
of time in the four MD simulations at different pH values and oxidation states. Upper left panel:
reduced at pH 5, upper right panel: reduced at pH 9, lower left panel: oxidised at pH 5, and lower
right panel: oxidised at pH 9. Maroon: Hδ 1 (His 35) - O(Pro 36). Turquoise: Nδ 1 (His 35) - O(Pro
36). Red: Hε 2 (His 35) - O(Met 44). Green: HN (Gly 37) - Nδ 1 (His 35).
5.4 Results
181
energy barrier of 67 kJmol−1 between cis and trans conformers of a peptidic ω dihedral angle
in the force field explains why no transitions of the ω angle are observed.
The hydrogen bonds Hδ 1 (His 35) - O(Pro 36) and Hε 2 (His 35) - O(Met 44) suggested to be
present at low pH by the X-ray study [105] were not observed in simulations O5 and R5, see
Fig. 5.4. At pH 9, the experimentally observed HN (Gly 37) - Nδ 1 (His 35) and Hε 2 (His 35) O(Met 44) hydrogen bonds are formed for part of the time.
As small differences in the conformational states for the oxidised Azurin at pH 5 and pH
9 were observed both in the simulations and experimentally, RE-TI simulations were initially
started using the final conformations of the 1 ns MD simulations. Table 5.4 compares the free
energy differences for the various legs in Fig. 5.2 as well as the cycle closure for the XDE
simulations. For comparison, the TI-values obtained by van den Bosch et al. [111] are also
included. After 140 ps, the XDE TT simulations show large deviations (hysteresis) between
forward and backward simulations along a leg and unreasonably large cycle closures of about
93 kJmol−1 . After 220 exchange trials, 440 ps, the cycle closures are still bad, +62.1 kJmol−1
for cycle R5-O5-O9-R9-R5 and +8.6 kJmol−1 for cycle R5-R9-O9-O5-R5. The largest hysteresis is 29.3 kJmol−1 for RtO5 and OtR5 and error estimates up to 22.0 kJmol−1 are observed. Performing 1200 exchange trials, the large hysteresis for transitions RtO5 and OtR5
(29.3 kJmol−1 ) or O5t9 and O9t5 (22.8 kJmol−1 ) decreases to 19.4 and 2.4 kJmol−1 respectively, but on the other hand the hysteresis increases between transitions R9t5 and R5t9. The
value for the cycle closure of cycle R5-R9-O9-O5-R5 of -2.9 kJmol−1 lies in a reasonable
range, but the value of +17.2 kJmol−1 for cycle R5-O5-O9-R9-R5 is still too high.
To check that these convergence problems did not originate from the tight temperature coupling, the same simulation setup as for the XDE TT simulations, but with a weak temperature
coupling constant was used to perform the XDE WT simulations. After 140 ps of simulation, very similar tendencies as for the XDE TT simulations are observed, see the last column
of Table 5.4. The cycle closure value of 43.5 kJmol−1 for cycle R5-R9-O9-O5-R5 is much
smaller for the XDE WT than for the XDE TT simulation, but still rather large.
Obviously, the structural divergence between the different states observed in the 1 ns MD
simulations is too large to lead to converged free energy estimates using either TI or RE-TI.
To exclude methodological problems of the calculation, and to check for force field consistency, RE-TI simulations using the same starting structure for all transitions were performed.
Considering the high similarity between the experimental structures shown in Table 5.3, this
may even be appropriate.
The structure of oxidised Azurin at pH 5 after thermalisation was taken as starting configuration for RE-TI simulations of 140 ps (XO5 TT and XO5 WT). Table 5.5 shows the resulting
differences in free energy for these simulations. As expected, the hysteresis in these simulations is much reduced, with maximum values of 2.2 kJmol−1 (RtO5 and OtR5 in the XO5 TT)
and 4.3 kJmol−1 (O5t9 and O9t5 in XO5 WT). The cycle closures have also improved significantly, although the value of +9.2 kJmol−1 for the thermodynamic cycle R5-R9-O9-O5-R5
in XO5 TT is still not very satisfying. The error estimates are clearly larger for the transitions
400 ps
140 ps
XDE TT
440 ps
2400 ps
XDE WT
140 ps
78
-88
-111
156
-66
65
136
-106
37
29
123.3 ± 7.5
-86.6 ± 5.1
-186.7 ± 12.1
203.1 ± 10.3
-75.0 ± 3.8
96.3 ± 3.3
221.0 ± 10.8
-179.8 ± 9.2
+92.6 ± 18.3
-93.0 ± 15.1
119.5 ± 4.2
-90.2 ± 3.1
-192.7 ± 9.4
197.6 ± 8.5
-79.7 ± 3.7
93.5 ± 2.6
215.1 ± 22.0
-192.3 ± 10.1
+62.1 ± 24.6
+8.6 ± 13.8
110.9 ± 5.1
-91.5 ± 1.6
-209.1 ± 10.5
197.0 ± 6.5
-84.6 ± 2.8
89.1 ± 3.3
199.9 ± 9.3
-197.5 ± 3.8
+17.2 ± 15.2
-2.9 ± 8.4
124.4 ± 8.3
-84.7 ± 4.4
-187.2 ± 9.3
210.5 ± 11.8
-73.8 ± 6.3
98.4 ± 3.0
225.0 ± 8.2
-180.8 ± 12.5
+88.4 ± 16.2
+43.5 ± 18.0
Table 5.4 Differences in free enthalpy ∆G in kJmol−1 for the different transitions and values for cycle closure for the TI simulations of
the panel A in Fig. 3 in the work of van den Bosch et al. [111], and the RE-TI simulations XDE TT and XDE WT described in the
simulation setup section.
5 Calculation of the relative free energy of oxidation of Azurin at pH 5 and pH 9
Simulation
Lenght
Transition
RtO5
OtR5
R9t5
R5t9
OtR9
RtO9
O5t9
O9t5
cycle R5-O5-O9-R9-R5
cycle R5-R9-O9-O5-R5
RE-TI
182
TI
XO5 TT
140 ps
XO5 WT
140 ps
115.0 ± 3.7
-112.8 ± 1.6
-200.5 ± 12.9
200.7 ± 11.7
-96.4 ± 3.8
98.5 ± 4.3
178.6 ± 10.7
-177.1 ± 7.2
-3.3 ± 17.6
+9.2 ± 14.5
114.0 ± 3.7
-114.7 ± 3.8
-201.6 ± 13.4
201.5 ± 10.8
-99.0 ± 3.3
98.5 ± 2.7
180.9 ± 9.8
-185.2 ± 9.5
-5.7 ± 17.3
+0.1 ± 15.1
XDS WT
140 ps
116.3 ± 2.6
-115.5 ± 2.9
-200.7 ± 13.7
213.1 ± 12.1
-94.2 ± 3.1
97.3 ± 3.6
183.1 ± 7.9
-189.4 ± 14.3
+4.5 ± 16.3
+5.5 ± 19.3
440 ps
116.3 ± 1.6
-113.1 ± 2.6
-206.9 ± 11.6
214.2 ± 10.1
-95.7 ± 3.1
95.3 ± 3.1
184.4 ± 11.1
-193.6 ± 9.8
-1.9 ± 16.4
+2.9 ± 14.6
5.4 Results
Simulation
Lenght
Transition
RtO5
OtR5
R9t5
R5t9
OtR9
RtO9
O5t9
O9t5
cycle R5-O5-O9-R9-R5
cycle R5-R9-O9-O5-R5
Table 5.5 Differences in free enthalpy ∆G in kJmol−1 for the different transitions and values for cycle closure for the RE-TI simulations
XO5 TT, XO5 WT and XDS WT described in the simulation setup section.
Simulation
Lenght
Transition
RtO5
OtR5
R9t5
R5t9
OtR9
RtO9
O5t9
O9t5
cycle R5-O5-O9-R9-R5
cycle R5-R9-O9-O5-R5
XO5 TT ALL
20 ps
XO5 TT CA
140 ps
XO5 TT CA-Cu
140 ps
85.9
-85.4
-289.0
292.6
-57.7
57.6
262.3
-263.6
+1.6
+1.1
104.3 ± 2.8
-104.0 ± 2.3
-201.3 ± 4.2
201.0 ± 6.1
-86.8 ± 2.6
86.6 ± 3.0
181.3 ± 7.0
-183.1 ± 4.7
-2.5 ± 9.0
+0.4 ± 8.6
104.5 ± 2.4
-104.5 ± 1.6
-200.8 ± 6.2
203.5 ± 4.4
-87.5 ± 1.9
86.1 ± 2.4
185.8 ± 4.6
-185.7 ± 4.4
+2.1 ± 8.3
-0.5 ± 6.9
183
Table 5.6 Differences in free enthalpy ∆G in kJmol−1 for the different transitions and values for cycle closure for the RE-TI simulations
XO5 TT ALL, XO5 TT CA and XO5 TT CA-Cu described in the simulation setup section.
184
5 Calculation of the relative free energy of oxidation of Azurin at pH 5 and pH 9
R9t5, R5t9, O5t9 and O9t5 involving a pH change. Fig. 5.5a shows the value of h ∂∂ Hλ iλ as
function of λ for the transitions in the XO5 WT simulations. From this figure it can be seen
that the reduced hysteresis for the transitions involving a pH change is partly due to a fortuitous
cancellation of errors, as the h ∂∂ Hλ iλ -curves are not the same over the complete λ -range.
In a next attempt, RE-TI simulations using the same starting structure for all transitions
were started, but now with harmonic position restraints on selected atoms to prevent the system
from further divergence. Table 5.6 summarises the results for these simulations. In simulation
XO5 TT ALL, all atoms were positionally restrained to their initial positions. As can be
expected, the hysteresis after 20 ps of simulation is reasonably small, but still amounts to
3.6 kJmol−1 between transition R9t5 and R5t9. Cycle closures of +1.6 kJmol−1 for cycle R5O5-O9-R9-R5 and +1.1 kJmol−1 for cycle R5-R9-O9-O5-R5 are well within an uncertainty
range of ± 2 kJmol−1 . This indicates an internal consistency of the methodology and the
definition of the various states and transitions. Deviations from zero for hysteresis and cycle
closures can thus be attributed to divergence in the conformational sampling.
Restraining only the Cα -atoms of the protein in the XO5 TT CA simulations increases the
hysteresis slightly for all transition pairs after 20 ps of simulation, which is then reduced to
maximally 1.8 kJmol−1 for 140 ps of simulation. Cycle closures of -2.5 and +0.4 kJmol−1 are
satisfying for cycle R5-O5-O9-R9-R5 and R5-R9-O9-O5-R5, respectively.
Figure 5.5 h ∂∂ Hλ i as a function of λ for (a) the XO5 WT simulation (140 ps) and for (b) the XDS WT
simulation (440 ps). The black lines show results for the transitions in cycle R5-O5-O9-R9-R5, the
red lines the results for the transitions in cycle R5-R9-O9-O5-R5. Dashed lines belong to transitions RtO5 and OtR5, dot-dashed lines to transitions R9t5 and R5t9, dotted lines to transitions
OtR9 and RtO9, and solid lines to transitions O5t9 and O9t5. The data for transitions OtR5, R9t5,
OtR9, and O9t5 are shown as function of 1-λ and multiplied by -1 for ease of comparison.
5.4 Results
185
In the third set of simulations with position restraints, the XO5 TT CA-Cu simulations,
only the Cα -atoms of the residues undergoing no change of the charge distribution were restrained, yielding very similar ∆G-values as the XO5 TT CA simulations. This is not very
surprising, as most of the residues for which a change in the charge distribution was applied
are still involved in a distance restraint to the Cu-ion, see Table 5.2. The resulting ∆G-values
after 140 ps for both the XO5 TT CA and the XO5 TT CA-Cu simulations fall mostly in the
range of the ∆G-values of the XDE TT simulations, the only exception being transitions O5t9
and O9t5.
The XO5 simulations indicate that it is possible to obtain closure of the thermodynamic
cycle, as long as the protein structure does not change too much. The differences in the conformations resulting after 1 ns of standard MD simulation are obviously too big, such that
using these different starting structures for transitions with the same change in Hamiltonian,
e.g. R9t5 and R5t9, leads to different ∆G-values. However, as some conformational changes
may occur under experimental conditions, position-restraining the atoms of the protein remains an unsatisfactory solution. All setups discussed so far involve instantaneous transitions
of a conformation equilibrated at one state to the interaction parameters (partial charges) of
another state. I.e. the replica at λ = 1 for the O5t9 transition starting from the equilibrated O5
conformation experiences a sudden transition to the Hamiltonian corresponding to state O9 at
the beginning of the simulation. The observation in Fig. 5.5a that the hysteresis may be reduced due to cancellation of deviations for different λ -values indicates that also the XO5 TT
and XO5 WT simulations suffer from an instantaneous change of Hamiltonian.
To avoid sudden changes in pH or in the oxidation state, the XDS WT simulation setup
was used. Starting structures were generated by performing a slow-growth simulation during
which the Hamiltonian of the system was continuously changed from λ -value 0 to 1 in order
to gradually adapt the configurations to the changing Hamiltonian. The structures obtained at
decimal λ -values were used as starting structures for the appropriate λ -values in the RE-TI
simulations. As the replicas now start from (slightly) different initial structures, a cancellation
of errors along the way becomes less likely as is indeed observed by an increased hysteresis
in Table 5.5 and in the free energy profiles in Fig. 5.5b. Nevertheless, overall cycle closures
after 440 ps of -1.9 kJmol−1 for cycle R5-O5-O9-R9-R5 and +2.9 kJmol−1 for cycle R5-R9O9-O5-R5 are in an acceptable range, even though the large error estimates and the significant
hysteresis clearly show that the simulations have not converged yet. The setup of the XDS WT
simulation is probably most comparable to the setup of the TI calculations in [111]. In the
latter, 50 ps of equilibration per λ -value were performed before changing to the next λ -value,
which is similar to running a slow-growth run of λ . At every λ -value 400 ps of continuous
simulation was subsequently performed for data collection. Thus, the TI simulations and the
XDS WT simulations have the same total simulation time, 50 ps equilibration followed by
400 ps production run per λ -value for the former, and 10 ps slow-growth simulation per λ value followed by 440 ps RE-TI for the latter. The main difference in the setup are the replica
exchanges allowing for a more rapid mixing of (diverged) conformations over the λ -values.
Comparing the TI data (Table 5.4) to the XDS WT data after 440 ps (Table 5.5), it is clear that
186
5 Calculation of the relative free energy of oxidation of Azurin at pH 5 and pH 9
Figure 5.6 Dihedral angle values (upper panels), hydrogen-bond distances (middle panels), and
∂ H/∂ λ (lower panels) as a function of time in the RE-TI simulations XDS WT of the transition
R9t5 in Azurin. λ = 0 corresponds to R9 and λ = 1 to R5. The three dihedral angles are ψ (Pro
36) in magenta, ϕ (Gly 37) in indigo, and χ2 (His 35) in green. The four distances are Hδ 1 (His 35) O(Pro 36) in maroon, Nδ 1 (His 35) - O(Pro 36) in turquoise, Hε 2 (His 35) - O(Met 44) in red, HN (Gly
37) - Nδ 1 (His 35) in dark green.
Figure 5.7 Dihedral angle values (upper panels), hydrogen-bond distances (middle panels), and
∂ H/∂ λ (lower panels) as a function of time in the RE-TI simulations XDS WT of the transition
R5t9 in Azurin. λ = 0 corresponds to R5 and λ = 1 to R9. The three dihedral angles are ψ (Pro
36) in magenta, ϕ (Gly 37) in indigo, and χ2 (His 35) in green. The four distances are Hδ 1 (His 35) O(Pro 36) in maroon, Nδ 1 (His 35) - O(Pro 36) in turquoise, Hε 2 (His 35) - O(Met 44) in red, HN (Gly
37) - Nδ 1 (His 35) in dark green.
5.5 Conclusions
187
such mixing does significantly improve both the hysteresis and the cycle closure, even though
the XDS WT simulations have not converged on this timescale.
Figs. 5.6 and 5.7 exemplify the origins of the remaining hysteresis and large error estimates
for transitions R9t5 (Fig. 5.6) and R5t9 (Fig. 5.7). Especially in the R9 state ∂∂ Hλ shows large
fluctuations which seem to correlate with changes in the dihedral angles ψ of residue Pro 36
and ϕ of residue Gly 37. A transition of the peptide bond between Pro 36 and Gly 37 is not
observed, but the anti-correlated changes in the ψ angle of Pro 36 and ϕ of Gly 37 indicate a
rotation of the peptide plane, which leads to a change in hydrogen bonding. This is to a lesser
extent already observed at λ = 0.5 in both simulations. At this λ -value the few occurrences of
alternative backbone conformations do not originate from dynamic barrier crossings, but are
the result of replicas at higher λ -values exchanging their structures to λ = 0.5. The exchanges
for all transitions are available as Fig. 5.9 in the Supplementary material.
The changes in the χ1 side-chain dihedral angle of His 35 correlate with its hydrogen bonding pattern. The hydrogen bond between the proton Hδ 1 and the oxygen of residue Pro 36 as
observed in the X-ray structure is never present in our simulations. Figs. 5.6 and 5.7 show
selected distances over the simulation time for the transitions R9t5 and R5t9. The changes
observed between λ -values 0 and 1 are mostly transferred to λ = 0.5 through the replica exchange approach. The hydrogen bond between HN (Gly 37) and Nδ 1 (His 35) reported to be
weak at pH 9 in the experimental paper is nicely observed in the simulations at λ -values corresponding to higher pH values. The hydrogen bond Hε 2 (His 35) - O(Met 44) is not observed
at low pH, but at higher pH it occurs from time to time. For states at lower pH, i.e. λ = 1 for
transition R9t5 and λ = 0 for transition R5t9, the distance between the Hδ 1 (His 35) and the
O(Pro 36) is always longer than the distance of the Nδ 1 (His 35) to O(Pro 36) indicating that
the orientation of the histidine side chain does not allow for the hydrogen bond between the
Hδ 1 (His 35) and the O(Pro 36). Interestingly, at states of higher pH, i.e. λ = 0 for transition
R9t5 and λ = 1 for transition R5t9 this situation is reversed, although the distances are still
too large to consider the hydrogen bond formed.
5.5 Conclusions
The free enthalpy differences for the protein Azurin between different oxidation states, reduced with Cu(I) and oxidised with Cu(II) and between pH 5 and 9 were calculated. Previously reported thermodynamic integration simulations suffered from poor convergence, large
hysteresis and unsatisfactory cycle closure. In this study different setups for Hamiltonian
replica exchange thermodynamic integration simulations were tested to improve the convergence without significantly increasing the simulation time. Use of different starting structures
obtained after 1 ns MD simulations yielded unfavorable cycle closure and hysteresis values.
Extending the simulation up to 2.4 ns did not improve these values. Using the same starting
structures for all transitions resulted in better cycle closure values after 140 ps, but hysteresis
was still observed. Changing the temperature coupling constant from a tight coupling of τT =
188
5 Calculation of the relative free energy of oxidation of Azurin at pH 5 and pH 9
2 fs to a weaker coupling of τT = 0.1 ps only influenced the results marginally. Positionally
restraining various sets of atoms did lead to small hysteresis and cycle closures, indicating
internal consistency of the simulation settings and force-field parameters.
Allowing the system to adapt to a change in the Hamiltonian in a slow-growth manner
before starting the Hamiltonian replica exchange thermodynamic integration improved the
situation compared to the simulation with different starting structures obtained from MD simulations, but still showed limited conformational sampling around residues His 35, Pro 36 and
Gly 37. These simulations are most comparable to the original thermodynamic integration
simulations and show the favourable effect of the replica exchanges on mixing conformational changes into all relevant λ -values. The application of temperature replica exchange
in addition to Hamiltonian replica exchange would enhance the sampling but at a loss of efficiency because, in contrast to λ -dependent Hamiltonian replica exchange, the systems at
higher temperatures cannot be used to improve the convergence of the ensemble average at
the temperature of interest.
This study demonstrates that small conformational changes may have a significant effect on
relative free energy calculations. Hamiltonian replica exchange thermodynamic integration
crucially depend on the initial structures and in the current setup will not enhance the conformational sampling as compared to regular thermodynamic integration. It does, however,
improve the distribution of conformational changes observed in one replica over the various
λ -values. As such, it may be used to improve convergence of alchemical modifications in
molecular simulation.
5.6 Supplementary material
189
5.6 Supplementary material
Figure 5.8 Secondary structure analysis of the four MD simulations of Azurin at different pH’s and
oxidation states. Upper left panel: reduced pH 5, upper right panel: reduced at pH 9, lower left
panel: oxidised at pH 5, and lower right panel: oxidised at pH 9. 310 -helices are shown in black,
α -helices in red, bends in green, β -bridges in blue, β -strands in yellow and turns in brown, using
the definitions of Kabsch and Sander [46].
190
5 Calculation of the relative free energy of oxidation of Azurin at pH 5 and pH 9
Figure 5.9 First and third column: RE-TI structure exchanges between different λ -values of the
XDS WT simulations, shown for 220 exchanges, 440 ps. Different colors correspond to different
starting structures at different λ -values. Second and fourth columns: the average acceptance ratio
π for an exchange per λ -value is shown. From left to right in the first row data for transition RtO5
and OtR5 is shown, in the second row for transitions R9t5 and R5t9, in the third row for transitions
OtR9 and RtO9 and in the lowest row for transitions O5t9 and O9t5.
6 Outlook
During the past fifty years, the improvement in computer technology in terms of speed, cost
and design allowed the use of more and more sophisticated algorithms and programming. For
the field of (bio)molecular simulation, where the interaction of particles on an atomic level
and their dynamics using classical equations of motions is considered, this development led
to faster computation of the interaction functions, therefore allowing longer simulations of
increased system size. New computational methods were implemented to get access to properties not accessible before or to accelerate the simulations, e.g. by using parallel computing.
Once new methods were developed, they are applied to several (toy) systems to test their
functionality. In this thesis, no new methods were presented, but existing ones were used for
different systems than they were tested on in order to investigate whether they are generally
applicable.
In Chapter 2, methods to calculate NOE distance bounds and 3 J-coupling constants were
investigated. It was shown that considering time-averages, the calculated properties fulfilled
the experimental data better, and using the local-elevation method helped to bias the simulations to bring the calculated properties in better agreement with the measured ones. In a next
step, one could try to find a procedure to extract an ensemble of structures from the trajectories obtained in these simulations that represents relevant conformations of the system under
investigation. Another issue is the length of the simulation. As the local-elevation sampling
method builds up an additional unphysical potential energy term as long as the calculated 3 Jvalue, be it instantaneous or time-averaged, does not match the experimental 3 J-value, it may
lead to very unphysical conformations by continuing to increase the potential energy in cases
where the calculated 3 J-value can never match the experimental value, e.g. because it was
measured wrongly. So the local-elevation sampling procedure should not be applied for a too
long time and in the cases where very high potential energies are built up a careful inspection
of the reason for this is recommended.
A calculated 3 J-value can also deviate from an experimentally observed one because of an
inaccurate definition of the relationship between the 3 J-value and the corresponding dihedral
angle θ . The results in Chapter 3 showed that the commonly used Karplus relation may be a
too rough approximation to get a good agreement between observed and calculated 3 J-values,
especially for side-chain 3 Jαβ -values. Karplus himself stated in 1963 [2] that 3 J-values may
be calculated approximately using the Karplus relation, but that it only gives a zero-order approximation for the numerical values of 3 J-couplings and may only suggest the trends that
are expected on theoretical grounds. This description matches our findings that the agreement
between calculated and measured 3 J-values is not much influenced by changes in the param-
191
192
6 Outlook
eters of the Karplus relation or by adding extra terms to it accounting for substituent effects.
A set of measured data with reasonable agreement to calculated values using one parameter
set of the Karplus relation also showed reasonable agreement using the other parameter sets.
Assuming accurate measured data, this could indicate that the Karplus relation (or a variation
of it) might not be suitable for application to the side chains of amino acids in proteins, which
are rather flexible, and that perhaps another form of the function 3 J(θ ) would lead to more
satisfying results.
Conformational flexibility also showed a considerable influence on the calculation of relative free energies. In the two systems studied in Chapters 4 and 5, the widely used thermodynamic integration method yielded unconverged results because of insufficient conformational
averaging in the ensembles obtained at each Hamiltonian. The averaging was improved by application of Hamiltonian replica exchange thermodynamic integration, where conformations
are exchanged between very similar Hamiltonians according to a particular exchange criterion,
but the results were still not very encouraging considering the limited lengths of simulation
performed. Extension of the simulation might yield slightly improved results, but it appears
that very long simulation times are required to achieve a reasonable degree of averaging. Additionally, the time needed to reach convergence of the results depends on the starting structure
used. If the simulation is not started in a relevant part of the conformational space, it first
needs to equilibrate to a more relevant part. Such conformational changes induce a considerable change in the free energy. If the relevant conformational ensemble consists of several
metastable states at different free energies, many transitions are needed to get a converged relative free energy which may then be compared to an experimental one. Therefore, more effort
is required to find a way to sample the relevant conformational space for a specific system with
the computer power available. Simply extending the simulation length is not sufficient. Using methods like local-elevation sampling may help to enhance the extent of conformational
space sampled, but there remains the question of how to judge the relevance of the sampled
conformations. Further comparison between the results of simulations and experiments will
hopefully lead to advances in computational methodology capable of solving these problems.
Bibliography
[1] M. Karplus. Contact electron-spin coupling of nuclear magnetic moments. J. Chem.
Phys., 30:11–15, 1959.
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Curriculum Vitae
Personal Data
Name
Denise Steiner
Date of birth
Place of birth
Nationality
1 April, 1982
Baden (AG)
Swiss
Education
2007 – 2011
Ph.D. studies in the group of Prof. Wilfred F. van
Gunsteren at the Laboratory of Physical Chemistry at
the ETH Z¨urich, Switzerland
2006 – 2007
Master studies in Chemistry at the ETH Z¨urich
2002 – 2006
Bachelor studies in Chemistry at the ETH Z¨urich
1998 – 2002
Matura at the Kantonsschule Baden, Switzerland
203