Spatiotemporal Modeling and Simulation Summer 2015 Prof. Dr. I. F. Sbalzarini TU Dresden & MPICBG, D-01307 Dresden Exercise 5 Release: 26.05.2015 Due: 02.06.2015 The following exercises will be mostly implementation of data structures that are useful for upcoming exercises. Therefore, choose now a programming language that suits you best and stick to it through the rest of the course (I suggest Matlab). Write the code in a modular, re-usable manner. Question 1: Cell and Verlet Lists In the lecture you have been introduced to particle methods and their fundamental difference to ”classical” numerical methods like Finite Differences and Finite Volumes. One key prerequisite to evaluate Particle-Particle (PP) interactions efficiently is obviously fast access to neighboring particles for each particle. Two data structures are presented in the script on pp. 35–36: Cell lists and Verlet lists. a) Implement first a routine that creates cell lists from a set of particles in 1D/2D without symmetry. The function could read like this: %%%%%%%%%%%%%%%%%%%%%%%% % Code f o r E x e r c i s e 5 − C r e a t i o n o f c e l l l i s t s %%%%%%%%%%%%%%%%%%%%%%%% % Input % p a r t i c l e P o s : ( n u m P a r t i c l e s x dim)−Matrix o f p a r t i c l e p o s i t i o n s % lBounds : S c a l a r l o w e r bound on a l l p a r t i c l e p o s i t i o n s % uBounds : S c a l a r upper bound on a l l p a r t i c l e p o s i t i o n s % cellSide : S c a l a r v a l u e o f th e c e l l ’ s s i d e l e n g t h % % Output % p a r t i c l e M a t : ( n u m P a r t i c l e s x dim+1)−Matrix t h a t c o n t a i n s t h e % p a r t i c l e p o s i t i o n s and th e c e l l i n d e x i t b e l o n g s t o % cellList : Matlab c e l l s t r u c t u r e o f l e n g t h number o f o v e r a l l % c e l l s ( prod ( numCells ) ) % numCells : ( dim x 1)− Vector t h a t c o n t a i n s t he number o f c e l l s % p e r dimension f u n c t i o n [ p a r t i c l e M a t , c e l l L i s t , numCells ] = c r e a t e C e l l L i s t ( p a r t i c l e P o s , lBounds , uBounds , c e l l S i d e ) 1 Each cell should get a unique integer ID. Use the built-in functions sub2ind and ind2sub for this unique assignment. The output matrix particleMat consists of the particle positions and the ID of the cell it is contained in. The cell list should be an Octave cell data type where each element is a vector of arbitrary length. b) Given your cell list implementation, it is now possible to tackle Verlet lists. Implement a routine that creates Verlet lists for particle distributions in 1D/2D. The function should read like this: %%%%%%%%%%%%%%%%%%%%%%%% % Code f o r E x e r c i s e 5 − C r e a t i o n o f V e r l e t l i s t s without symmetry %%%%%%%%%%%%%%%%%%%%%%%% % Input % p a r t i c l e M a t : ( n u m P a r t i c l e s x dim+1)−Matrix o f p a r t i c l e p o s i t i o n s % and c e l l i n d e x % cellList : Matlab c e l l s t r u c t u r e o f l e n g t h prod ( numCells ) % ( number o f c e l l s ) % numCells : ( dim x 1)− Vector t h a t c o n t a i n s t he number o f c e l l s % pe r dimension % % cutoff : Scalar distance c u t o f f that d e f i n e s % th e neighborhood . I t s h o u l d be % e q u i v a l e n t t o t he s i d e l e n g t h o f a c e l l % % Output % verletList : Matlab c e l l s t r u c t u r e o f l e n g t h n u m P a r t i c l e s . %Each element c o n t a i n s t h e i n d i c e s o f th e p a r t i c l e s , %i . e . th e row numbers i n t h e p a r t i c l e M a t matrix % function verletList = c r e a t e V e r l e t L i s t ( p a r t i c l e M a t , c e l l L i s t , numCells , c u t o f f ) The cutoff input argument should be equivalent to the cell side length that has been used to create the cell list. The output Verlet list should be a Matlab cell data type where each element is a vector of arbitrary length. c) The next task is dedicated to visually verifying your implementation. Write a program that creates 10,000 particles in 2D uniformly at random in the unit box. Create a cell list and a Verlet list with cell side length (cutoff) 0.05. Create a figure and plot all particles in blue. Choose a random cell index and color all particles that are contained in this cell in green. Color the particles in adjacent cells in yellow. Pick a random particle of the randomly chosen cell and color its Verlet neighbors in red. 2 Question 2: Neighbor lists and Quorum Sensing The phenomenon of Quorum (= critical density) sensing (QS) is a type of decision-making process used by many bacteria. Bacteria use QS to coordinate certain behaviors based on the local density of the bacterial population. The luminescence of the bacterium Vibrio fischeri (see Fig. 2) that lives in symbiosis with deep-sea animals (e.g. squids) is probably the most striking example of QS. PERSPECTIVES such as a cocktail of hydrolytic enzymes positive feedback of than QS and DS. Interestingly, the function for the breakdown of polymeric organic shelters communica of positive feedback might be to speed up material, but only one type of autoinducer (BOX 1). The fewer an the upregulation of cells in the vicinity or might suffice. are, the more they w at the periphery of a cluster, enabling a • Find molecule a parameter setting where all the previous points are fulfilled. The ES concept combines a unifying ence from other aut synchronized response, even for irregularly autoinducer-degrad hypothesis of what cells measure (cell located cells. For simplicity one can startproperties the simulation 0nM concentration ofoften AHL everywhere and other eukaryote density, mass-transfer and spatial with Note that autoinducer sensing is except inside the activated cells. hypothesis potential for cross-t distribution) with a unifying connected with switching from one life of overlapping sets o of why they measure it (to estimate the strategy to another (for example, from rhizosphere35–38,40–44 efficiency of producing extracellular effecnon-virulent to virulent), which involves distribution of strai tors and react accordingly). This function of the autoinducer simultaneously upregulat1.2 Spatial patterns opportunities for cr ES does not, on purpose, stipulate that the ing one set of genes and downregulating the extent that biofi extracellular effectors are produced coopanother86. Some of these genes might be Given a chosen parameterization one the cancase start involved the simulation with and thetherefore different AHL as bu in effector release ES, initial microcolonies eratively because this is not always Figure 1: Microscopy image of glowing V. fischeri colony. whereas others might not. Therefore, theis 0nM holds true for biofil (cellsinside could be alone yet a produce concentration and outside theautoinduccells. A reasonable choice to start with outside ity or other forms o ers and effectors). Nevertheless, cooperative expression of genes that are not involved in the cell, 75nM for activated cells (over the threshold!), and 25nM for all non-activated cells effector release can indirectly depend on ES can spread motile st and coordinated production of effectors threshold!). certain computational domain play e.g. witharound different because of their coupled regulation. and over no emerges linked in manyChoose situations which density neighV. fischeri’s ability to (below glow isthe directly to theainlocal of neighbors. Ifand a bacterium bringing them into bours with the same autoinducer system spatial distributions of cells: senses a critical number of neighbors around itself, it starts glowing. Using this coupling, Clonal microcolo Microcolonies — avoiding problems are present. ES therefore encompasses the bacterium ensures that the cooperative host provides enough situafood for Microcolonies the colony.are clusters of cells. In soil interference but also and non-cooperative • Uniform distribution of that cells a many certain of cooperation by av otherdensity habitats in which bacteria tions.spatial The evolutionary hypothesis ES withand Imagine now that you have given the exact positions of 5,000 bacterialattached cells tobut you interest, as discusse grow predominantly surfaces, evolved because of the2D direct fitness benefits • Inhomogeneous (spatially clustered) with a certain stable spatial structu typically arise be fromdensity the find of optimized effector production the distribution don’t know whether a given bacterium is glowing orfornot. A these firstclusters guess could to als in a microcolony growth and cell division of a progenitor individual and because of group benefits of areas with high density of bacteria. In effector these areas, also theencombacteria will most likely essentially glow. In interactions are with 2); they are therefore cell (FIG. cooperative production • Different degrees of inhomogeneous distribution The more the cells areshows clustered to planktonic bacter passes QS and DSthis concepts. As direct an approximate manner, we can achieve with cell and lists. clonal Fig.45,87–89 2 . schematically together the sooner the threshold for induc- mixing promotes co group benefits work in the same direction, • Different densities, initial conditions and boundary conditions this approach. An exact approach usesevolve Verlet each bacterium, one together can efficiently tion is reached. This, with the usual fact the very reason ES would underlists. broader For conditions diversity of bacteria determine the number of neighbors it has. growth therefore pr a Random b Clustered between clusters an such as interference est. Ironically, altho sensing has been stu mixed liquid cultur studied could not h conditions — DS be sense in bulk liquid evolutionarily stable cheaters are mixed. Conclusions and f QS proposes that ce ing to measure their poses that they mea Figure 4 | Spatial distribution and cell density. Both panels contain 100 cells in a domain of the same size, so the cell density is the same. a | A random spatial distribution of cells (random numbers properties of their e Figure 1:cell Different QSin which cell (dots). distributions (taken [1]). frombacterial a uniform probability distribution, all positions The are equally likely),from is shown. that we have shown that Figure 2: Schematic view of the population bacterium inNotethe positioning cells randomly will result in some clusters, which are probably not clonal. This situation of autoinducer-prod shaded cell, e.g., is isolated andcould willrepresent probably not glow. a snapshot of a mixed liquid culture or cells that disperse after cell division. b | Shows cells can have a stro distribution of clusters that have beenIf growing exponentially at a uniform rate forof a randomly autoinducer concen Report youra random results in a concise manner. possible, produce movies your simulations. chosen time interval (limited to four divisions or 16 cells). Within each cluster, the cells are located experiences than th Comment on the example they give in Box 1 in [1]. Is it possible to re-cover their results a) We have prepared a text filerandomly QSBacterialPos.dat that the represent 2D positions offounded 5,000by we have in a box of 5% of the domain size. Thisstores situation could microcolonies shown that single cells in random locations and at random times. Now consider replacing population-scale cell autoinducer sensing (see Figure 2)? bacteria in the [0,10] box into a matrix particlePos. Write a script (e.g. QS detect.m in density (the density of cells in liquid culture or in a biofilm) by cellular-scale cell density, defined over in autoinducer prod (b). of As a a small file. reference volume, such as the volume elements demarcated by the grid (a) and MATLAB) that loads the data Use a cell side length of 0.5 to create a incell list upregulation of cell consequence, such a small-scale density, for example, the density of 1 cell per unit volume in the green box in (a), does not account for the presence or absence of cells in the neighbouring volume elements, their vicinity. As the three key which clearly affect the 3autoinducer concentration in the focal volume element (BOX 1). Even within a cluster, the peripheral cells sense a different autoinducer concentration than the central cells. autoinducer concen Therefore, autoinducer sensing in clustered cells cannot be simplified as the measurement of cell mass-transfer prope density on the small scale of the cluster volume. tribution of cells) ca the bacterial positions. Find all cell indices that contain at least a critical density of 80 (20 bacteria/cell). Plot all bacteria in blue, and color the bacteria in the high-density cells in red. Create a Verlet list using the constructed cell list and a cutoff of 0.5. Find all bacteria that have at least 50 neighbors within the cutoff. Create a new figure and plot again all bacteria and mark the high-density bacteria red. Compare the two plots. Report also the bacteria that have the most neighbors within the prescribed cutoff. How many neighbors do they have? b) (Optional) Enhance the Verlet list code using intermediate cell lists and symmetry in 3D. 4

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