High Throughput Sequencing: Microscope in the Big Data Era Sreeram Kannan and David Tse Tutorial ISIT 2014 Research supported by NSF Center for Science of Information. DNA sequencing …ACGTGACTGAGGACCGTG CGACTGAGACTGACTGGGT CTAGCTAGACTACGTTTTA TATATATATACGTCGTCGT ACTGATGACTAGATTACAG ACTGATTTAGATACCTGAC TGATTTTAAAAAAATATT… High throughput sequencing revolution tech. driver for communications Shotgun sequencing read Technologies Sequence Sanger r 3730xl 454 GS Mechanis m Dideoxy chain terminatio n Read length Ion Torrent SOLiDv4 Illumina Pac Bio HiSeq 2000 Pyrose Detection quencin of g hydrogen ion Ligation and twobase coding Reversi Single ble molecule Nucleoti real time des 400-900 bp 700 bp ~400 bp 50 + 50 bp 100 bp PE 1000~1000 0 bp Error Rate 0.001% 0.1% 2% 0.1% 2% 10-15% Output data (per run) 100 KB 1 GB 100 GB 100 GB 1 TB 10 GB High throughput sequencing: Microscope in the big data era Genomic variations, 3-D structures, transcription, translation, protein interaction, etc. The quantities measured can be dynamic and vary spatially. Example: RNA expression is different in different tissues and at different times. Computational problems for high throughput data measure data • Assembly (de Novo) • Variant calling (reference-based assembly) manage data • Compression utilize data • Genome wide association studies • Privacy • Phylogenetic tree reconstruction • Pathogen detection Scope of this tutorial Assembly: three points of view • Software engineering • Computational complexity theoretic • Information theoretic Assembly as a software engineering problem • A single sequencing experiment can generate 100’s of millions of reads, 10’s to 100’s gigabytes of data. • Primary concerns are to minimize time and memory requirements. • No guarantee on optimality of assembly quality and in fact no optimality criterion at all. Computational complexity view • Formulate the assembly problem as a combinatorial optimization problem: – Shortest common superstring (Kececioglu-Myers 95) – Maximum likelihood (Medvedev-Brudno 09) – Hamiltonian path on overlap graph (Nagarajan-Pop 09) • Typically NP-hard and even hard to approximate. • Does not address the question of when the solution reconstructs the ground truth. Information theoretic view Basic question: What is the quality and quantity of read data needed to reliably reconstruct? Tutorial outline I. De Novo DNA assembly. II. Reference-based DNA assembly. III. De Novo RNA assembly Themes • Interplay between information and computational complexity. • Role of empirical data in driving theory and algorithm development. Part I: De Novo DNA Assembly Shotgun sequencing model Basic model : uniformly sampled reads. Assembly problem: reconstruct the genome given the reads. A Gigantic Jigsaw Puzzle Challenges Read errors Long repeats log(# of `-repeats) 16 15 16.5 16 14 15.5 12 15 10 10 14.5 8 14 13.5 6 5 13 4 12.5 2 12 0 11.5 0 0 20 2 40 60 100080 6 500 4 1001500 8 120 140 10 2000 160 12 Human Chr 22 repeat length histogram ` Illumina read error profile Two-step approach • First, we assume the reads are noiseless • Derive fundamental limits and near-optimal assembly algorithms. • Then, we add noise and see how things change. Repeat statistics harder jigsaw puzzle easier jigsaw puzzle How exactly do the fundamental limits depend on repeat statistics? Lower bound: coverage • Introduced by Lander-Waterman in 1988. • What is the number of reads needed to cover the entire DNA sequence with probability 1-²? • NLW only provides a lower bound on the number of reads needed for reconstruction. • NLW does not depend on the DNA repeat statistics! Simple model: I.I.D. DNA, G ! 1 (Motahari, Bresler & Tse 12) normalized # of reads reconstructable by greedy algorithm coverage 1 many repeats of length L no repeats of length L no coverage read length L What about for finite real DNA? I.I.D. DNA vs real DNA (Bresler, Bresler & Tse 12) Example: human chromosome 22 (build GRCh37, G = 35M) log(# of `-repeats) 16 15 16.5 16 16 14 15 15.5 12 15 10 14 10 14.5 8 1314 13.5 6 12 5 13 4 11 12.5 2 12 10 0 11.5 0 0 i.i.d. fit 20 2 2 40 4 4 500 data 60 80 6 10006 8 100 8 120 10 1500 140 10 122000 160 12 14 ` Can we derive performance bounds on an individual sequence basis? Individual sequence performance bounds (Bresler, Bresler, Tse BMC Bioinformatics 13) Given a genome s log(# of `-repeats) GREEDY DEBRUIJN ML lower bound Lcritical repeat length Human Chr 19 Build 37 SIMPLEBRIDGING MULTIBRIDGING Lander-Waterman coverage GAGE Benchmark Datasets http://gage.cbcb.umd.edu/ Rhodobacter sphaeroides lower bound Staphylococcus aureus Human Chromosome14 G = 2,903,081 G = 88,289,540 G = 4,603,060 MULTIBRIDGING lower bound MULTIBRIDGING lower bound MULTIBRIDGING Lower bound: Interleaved repeats Necessary condition: all interleaved repeats are bridged. L m n m n In particular: L > longest interleaved repeat length (Ukkonen) Lower bound: Triple repeats Necessary condition: all triple repeats are bridged L In particular: L > longest triple repeat length (Ukkonen) Individual sequence performance bounds (Bresler, Bresler, T. BMC Bioinformatics 13) log(# of `-repeats) lower bound length Human Chr 19 Build 37 Lander-Waterman coverage Greedy algorithm (TIGR Assembler, phrap, CAP3...) Input: the set of N reads of length L 1. Set the initial set of contigs as the reads 2. Find two contigs with largest overlap and merge them into a new contig 3. Repeat step 2 until only one contig remains Greedy algorithm: first error at overlap repeat contigs bridging read already merged A sufficient condition for reconstruction: all repeats are bridged L Back to chromosome 19 lower bound greedy algorithm log(# of `-repeats) 15 10 longest interleaved repeats at length 2248 non-interleaved repeats are resolvable! 5 0 0 1000 2000 GRCh37 Chr 19 (G = 55M) 3000 4000 longest repeat at Dense Read Model • As the number of reads N increases, one can recover exactly the L-spectrum of the genome. • If there is at least one non-repeating L-mer on the genome, this is equivalent information to having a read at every starting position on the genome. • Key question: What is the minimum read length L for which the genome is uniquely reconstructable from its Lspectrum? de Bruijn graph CCCT CCTA GCCC (L = 5) ATAGACCCTAGACGAT AGCC CTAG TAGA ATAG AGAC AGCG GCGA CGAT 1. Add a node for each (L-1)-mer on the genome. 2. Add k edges between two (L-1)-mers if their overlap has length L-2 and the corresponding L-mer appears k times in genome. Eulerian path CCCT CCTA GCCC (L = 5) ATAGACCCTAGACGAT AGCC CTAG TAGA ATAG AGAC AGCG GCGA CGAT Theorem (Pevzner 95) : If L > max(linterleaved, ltriple) , then the de Bruijn graph has a unique Eulerian path which is the original genome. Resolving non-interleaved repeats Condensed sequence graph non-interleaved repeat Unique Eulerian path. From dense reads to shotgun reads [Idury-Waterman 95] [Pevzner et al 01] Idea: mimic the dense read scenario by looking at K-mers of the length L reads Construct the K-mer graph and find an Eulerian path. Success if we have K-coverage of the genome and K > Lcritical De Bruijn algorithm: performance Loss of info. from the reads! GREEDY DEBRUIJN lower bound length Human Chr 19 Build 37 Lander-Waterman coverage Resolving bridged interleaved repeats bridging read interleaved repeat Bridging read resolves one repeat and the unique Eulerian path resolves the other. Simple bridging: performance GREEDY DEBRUIJN lower bound SIMPLEBRIDGING length Human Chr 19 Build 37 Lander-Waterman coverage Resolving triple repeats all copies bridged neighborhood of triple repeat triple repeat all copies bridged resolve repeat locally Triple Repeats: subtleties Multibridging De-Brujin Theorem: (Bresler,Bresler, Tse 13) Original sequence is reconstructable if: 1. triple repeats are all-bridged 2. interleaved repeats are (single) bridged 3. coverage Necessary conditions for ANY algorithm: 1. triple repeats are (single) bridged 1. interleaved repeats are (single) bridged. 2. coverage. Multibridging: near optimality for Chr 19 GREEDY DEBRUIJN lower bound SIMPLEBRIDGING length MULTIBRIDGING Human Chr 19 Build 37 Lander-Waterman coverage GAGE Benchmark Datasets http://gage.cbcb.umd.edu/ Rhodobacter sphaeroides Staphylococcus aureus Human Chromosome14 G = 2,903,081 G = 88,289,540 G = 4,603,060 Lcritical = length of the longest triple or interleaved repeat. Lcritical Lcritical Lcritical lower bound MULTIBRIDGING lower bound MULTIBRIDGING lower bound MULTIBRIDGING Gap Sulfolobus islandicus. G = 2,655,198 triple repeat lower bound interleaved repeat lower bound MULTIBRIDGING algorithm Complexity: Computational vs Informational • Complexity of MULTIBRIDGING – For a G length genome, O(G2) • Alternate formulations of Assembly – Shortest Common Superstring: NP-Hard – Greedy is O(G), but only a 4-approximation to SCS in the worst case – Maximum Likelihood: NP-Hard • Key differences – We are concerned only with instances when reads are informationally sufficient to reconstruct the genome. – Individual sequence formulation lets us focus on issues arising only in real genomes. Confidence • When the algorithm obtains an answer, can it be sure? • Under the dense read model, we can guarantee that when there is a unique Eulerian cycle, the reconstructed answer is correct. – This happens whenever L > max(linterleaved, ltriple) • Conversely, when L > max(linterleaved, ltriple), there are multiple reconstructions that are consistent with the observed data. • Under the shotgun read model, there is ambiguity in some scenarios. Read Errors ACGTCCTATGCGTATGCGTAATGCCACATATTGCTATGCGTAATGCGT TATA CTTA Error rate and nature depends on sequencing technology: Examples: • Illumina: 0.1 – 2% substitution errors • PacBio: 10 – 15% indel errors We will focus on a simple substitution noise model with noise parameter p. Consistency Basic question: What is the impact of noise on Lcritical? This question is equivalent to whether the L-spectrum is exactly recoverable as the number of noisy reads N -> 1. Theorem (C.C. Wang 13): Yes, for all p except p = ¾. What about coverage depth? Theorem (Motahari, Ramchandran,Tse, Ma 13): Assume i.i.d. genome model. If read error rate p is less than a threshold, then Lander-Waterman coverage is sufficient for L > Lcritical For uniform distr. on {A,G,C,T}, threshold is 19%. A separation architecture is optimal: error correction assembly Why? noise averaging M • Coverage means most positions are covered by many reads. • Multiple aligning overlapping noisy reads is possible if • Assembly using noiseless reads is possible if From theory to practice Two issues: 1) Multiple alignment is performed by testing joint typicality of M sequences, computationally too expensive. Solution: use the technique of finger printing. 2) Real genomes are not i.i.d. Solution: replace greedy by multibridging. Lam, Khalak, T. Recomb-Seq 14 X-phased multibridging Prochlorococcus marinus Lcritical Substitution errors of rate 1.5 % More results Prochlorococcus marinus Lcritical Methanococcus maripaludis Lcritical Helicobacter pylori Lcritical Mycoplasma agalactiae Lcritical A more careful look Mycoplasma agalactiae Lcritical-approx Lcritical Approximate repeat example: Yersinia pestis exact triple repeat, length 1662 5608 approximate triple repeat length Application: finishing tool for PacBio reads raw_reads.fasta PacBio Assembler HGAP raw_reads.fasta contigs.fasta Our finishingTool contigs.fasta improved_contigs.fasta https://github.com/kakitone/finishingTool Experimental results Before After Escherichia coli Meiothermus ruber Pedobacter heparinus More detail of the result Species Before [Ncontigs] After [Ncontigs] % Match with reference Time Size Escherichia coli (MG 1655) 21 7 [finisherSC] 99.60 < 3 mins (laptop) ~ 4.6M Meiothermus ruber (DSM 1279) 3 1 [finisherSC] 99.99 < 1 min (laptop) ~ 3.0M Pedobacter heparinus (DSM 2366) 18 5 [finisherSC] 99.89 < 3 mins (laptop) ~ 5.1M S_cerivisea (fungus) 252 78 [finisherSC] 95.46 < 3 hours (laptop) ~ 12.4M S_cerivisea (fungus) 252 55 [Greedy] 53.91 < 3 hours (laptop) ~ 12.4M Part II: Reference-Based DNA Assembly (Mohajer, Kannan, Tse ‘14) Many genomes to sequence… 100 million species (e.g. phylogeny) 7 billion individuals (SNP, personal genomics) 1013 cells in a human (e.g. somatic mutations such as HIV, cancer) … but not all independent courtesy: Batzoglou Reference Based Assembly: Formulation Reference ACGTCCCATGCGTATGCATAATGCCACATATGGCTATGCGTAATGAGT ACC Target ACGTCCTATGCGTATGCGTAATGCCACATATTGCTATGCGTAATGCGT ACC Assembler Side Information Types of Variations Substitutions (Single Nucleotide Polymorphisms: SNP) Reference ACGTCCCATGCGTATGCATAATGCCACATATGGCTATGCGTAATGAGT ACC Target ACGTCCTATGCGTATGCGTAATGCCACATATTGCTATGCGTAATGCGT ACC Types of Variations Small Indels (Insertions and Deletions) Reference ACGTCC___ATGCGTATGC_TAATGCCACATATTGAGCTATGCGTAATGCTG ACGTCCATGCGTATGCTAATGCCACATATTGAGCTATGCGTAATGCTGTAC TACC C Target ACGTCCTAGATGCGTATGCGTAATGCCACATAT___GCTATGCGTAATG__ ACGTCCTAGATGCGTATGCGTAATGCCACATATGCTATGCGTAATGGTACC GTACC Types of Variations Structural Variation Reference Inversion Duplication Duplication (dispersed) Copy Number Variation Mathematical Formulation Focus on SNP version Define SNP rate Noiseless reads What is Lcritical for this problem? Dense Reads from target t r (Reference DNA) Algorithm SNP Rate Want exact reconstruction Estimate of Target DNA Mathematical Formulation For any given reference DNA and SNP rate, what is the read length required for reconstruction? In the worst case among target DNA sequences Dense Reads from target t r (Reference DNA) Algorithm SNP Rate Lcritical is a function of r, SNP rate Estimate of Target DNA Necessary Conditions Let the reference DNA have a repeat of size lrep > 2L r lrep lrep Consider two possible target DNA sequences t1 and t2 L L t1 t2 Since L < lrep /2, the two targets D1 and D2 indistinguishable from reads Sanity check: interleaved repeat of length lrep /2 in D1 and D2 Necessary Conditions Let the reference DNA have an approximate repeat of size lrep,app > 2L r Can create r’ close to r but having exact repeat of size lrep,app r’ t1 t2 If L < lrep,app / 2: the two possible targets t1 and t2 indistinguishable Tolerance for approximate repeat depends on SNP rate Algorithm lrep,app r Let L > lrep,app / 2 t Map reads to r Keep only uniquely mapped reads Estimate t r ť lrep,app Condition for Success Loci covered by uniquely mapped reads are correctly called. Algorithm fails at a particular locus => None of the (L-1) possible reads uniquely mapped Case 1 Case 2 2L 2L r Second case more typical in real genome => 2L length approximate repeat in r L > lrep,app / 2 => The algorithm succeeds. Assembly Vs. Alignment: I Necessary condition L ≥ lrep,app (r) / 2 Sufficient condition L > lrep,app (r) / 2 (subject to the assumption) => Alignment near optimal and Lref = lrep,app (r) / 2. De Novo algorithm achieves Lcrit (t) = max {linterleaved(t), ltriple(t) } In terms of r, for worst case t Lde-novo = max {linterleaved,app (r), ltriple,app (r)} Assembly Vs. Alignment: II 1. Clearly Lde-novo ≥ Lref since Lref is necessary. 2. Lde-novo = max {linterleaved,app (r), ltriple,app (r)} ≤ lrep,app(r) = 2 Lref Thus gain from reference is at-most a factor of 2 in the read length. The maximal gain happens when linterleaved,app (r) = lrep,app (r), i.e., when the largest approximate repeat is an interleaved repeat. This happens for example, when the DNA is an i.i.d. sequence Reference based Assembly: Reprise • Complexity of alignment – Very fast aligners using fingerprinting available when SNP rate small • Better than alignment ? – Theory shows alignment near optimal – But alignment is what everyone uses anyway – Nothing better is possible? • The limitations of the worst case formulation! • If we adopt a individual sequence analysis for both reference and target, better solution possible. Part III: RNA (Transcriptome) Assembly Kannan, Pachter, Tse Genome Informatics ‘13 RNA: The RAM in Cells DNA transcription RNA translation • The instructions from DNA are copied to mRNA transcripts by transcription – RNA transcripts captures dynamics of cell • RNA Sequencing: Importance – – – – Clinical purposes Research: Discovery of novel functions Understanding gene regulation Most popular *-Seq Protein Alternative splicing DNA ATAC Exon AC GAAT TGAA AGC CAAT TCAG Intron 1000’s to 10,000’s symbols long ATAC CAAT TCAG GAAT RNA Transcript 1 Alternative splicing yields different isoforms. TCAG RNA Transcript 2 RNA-Seq (Mortazavi et al, Nature Methods 08) Reads ATAC CAAT TCAG TCA ATAC CAAT TCAG GAAT TCAG Assembler reconstructs ATT GAAT TCAG GAAT TCAG • Existing Assemblers – Genome guided: Cufflinks, Scripture, Isolasso,.. – De novo: Trinity, Oasis, TransAbyss,… GAA RNA Sequencing: Bottleneck Popular assemblers diverge significantly when fed the same input 24243 448216 6457 IsoLasso 9741 7553 59647 Cufflinks 5588 Scripture Is the bottleneck informational or computational or neither? Source: Wei Li et al, JCB 2011, Data from ENCODE project 78 Informational Limits • Lcritical for transcriptome assembly No algo. can reconstruct Lcritical Proposed algo. can reconstruct in linear time Read Length, L 0 On many examples, these two bounds match, establishing Lcritical ! • Mouse transcriptome: Lcritical = 4077 revealing complex transcriptome structure • What can we do at practical values of L? 79 Near-Optimality at Practical L Fraction of Transcripts Reconstructable Read Length Read Length 80 Near-Optimality at Practical L Fraction of Transcripts Reconstructable Upper bound without Upper bound on any algorithm Upper Bound abundance Read Length Read Length 81 Near-Optimality at Practical L Fraction of Transcripts Reconstructable Proposed Algorithm Read Length Read Length 82 Necessity of Abundance Information Fraction of Transcripts Reconstructable Upper bound without abundance Upper bound without abundance diversity Read Length Read Length 83 Transcriptome Assembly: Formulation • M transcripts s1,..,sM with relative abundances α1,..,αM which are generic (rationally independent). – Dense read model: Look at Lcrit – Get all substrings of length L along with their relative weights s1 α1 s2 α2 α1+α2 αM αM sM . . . What is Lcritical for transcriptome? • Lcritical is lower bounded by the length of the longest interleaved repeat in any transcript • It can potentially be much larger due to inter-transcript repeats of exons across isoforms. ATAC CAAT TCAG GAAT TCAG The Information Bottleneck s1 s3 s4 s2 s3 s5 86 The Information Bottleneck s1 s3 s4 s1 s3 s4 s2 s3 s5 s2 s3 s5 87 The Information Bottleneck s1 s3 s4 s1 s3 s5 s2 s3 s5 s2 s3 s4 Unless L > s3 these two transcriptomes are confused 88 The Information Bottleneck s1 s3 s4 s1 s3 s5 s2 s3 s5 s2 s3 s4 Sparsity can help rule out this four transcript alternative But first two possibilities still confusable unless L > s3 89 How to Distinguish the Two s1 s3 s4 s1 s3 s5 s2 s3 s5 s2 s3 s4 90 Abundance diversity lymphoblastoid cell line Geuvadis dataset Abundance Diversity s1 s3 s4 s1 s3 s5 92 Abundance Diversity s1 s3 s5 s1 s3 s4 s1 s3 s4 s1 s3 s5 This transcriptome is not a viable alternative (non-uniform coverage) Even if L < s3 these transcriptomes are distinguishable. 93 Fooling Set under Abundance Diversity a s1 s2 s3 s1 s2 s4 s5 s2 s3 b c These two transcriptomes are still confusable if L < s2 94 Achievability: Algorithm • From the reads – we construct a transcript graph Reads 0.1 ATTCG GATTC ATCCA 0.3 0.3 TCCAT 0.3 0.3 CCATT CATTC • Weight edges based on relative frequencies 95 Achievability: Algorithm • From the reads, we construct a transcript graph Reads 0.1 ATCCA ATTCG GATTC 0.3 0.3 TCCAT 0.3 0.3 CCATT CATTC • Weight edges based on relative frequencies 96 Achievability: Algorithm • From the reads, we construct a transcript graph Reads 0.1 ATC GAT TCG 0.3 0.3 CAT • Weight edges based on relative frequencies 97 Transcripts from Graph • Paths correspond to transcripts GAT TCG GAT 0.1 ATC TCG 0.3 CAT 0.3 ATC CAT TCG • Naïve Algorithm: Output all paths from the graph 98 Utility of Abundance • Consider the following splice-graph – Not all paths are transcripts – Node frequencies give abundance information 0.12 s1 0.12 0.12 s4 s1 s3 s2 0.88 s3 s4 0.88 0.88 s5 s2 s3 s5 – First idea: Use continuity of copy counts 99 Utility of Abundance: Beyond Continuity 5 • More complex splice graphs: 5 12 s0 s4 7 9 7 9 s1 s3 s5 6 15 s2 s6 6 In general, we are given values on nodes /edges. Need to find sparsest flow (on fewest paths). 100 General Splice graphs • Principle for general splice graphs: – Find the smallest set of paths that corresponds to the node / edge copy counts • Network routing, snooping, societal networks • How to split a flow? – Edge-flow: Flow value on each edge (satisfying conservation) – Path-flow: Flow value on each path – Given a edge-flow, find the sparsest path flow 0.12 0.12 s1 0.12 Start 0.12 s4 0.12 End s3 0.88 s2 0.88 0.88 0.88 s5 0.88 101 Sparsest Flow Decomposition • Problem is NP-Hard. [Vatinlen et al’ 08, Hartman et al ’12] – Closer look at hard instances: most paths have same flow – Equivalent to: Most transcripts have same abundance (!) – This is not characteristic of the biological problem • Our Result: – Assume that abundances are generic – Propose a provably correct algorithm that reconstructs when: L > Lsuff – Algorithm is linear time under this condition • Approximately satisfied by biological data ! 102 Iterative Algorithm • The algorithm locally resolves paths using abundance diversity – Error propagation? • Decompose a node only when sure • If unsure, decompose other nodes before coming back to this node • The algorithm solves paths like a sudoku puzzle – Solving one node can help uniquely resolve other nodes! – Can analyze conditions for correct recovery • L > Lsuff 103 Algorithm: Example Run a a 1 a+b 1 b a a+b 2 b 5 46 a a a 47 c b+c a 1 b 3 2 b 2 c 35 47 3 2 34 6 34 7 1 46 b c b+c 5 a 7 c b+c 5 1 6 4 3 7 c b+c a 4 3 2 6 5 a 1346 b 2347 c 235 104 Practical Implementation Multibridging to construct transcript graph Condensation and intratranscript repeat resolution Identify and discard sequencing errors Aggregate abundance estimation Node-wise copy count estimates Smoothing CC estimates using min-cost network flow Transcripts as paths Sparsest decomposition of edge-flow into paths Deals with inter-transcript repeats Practical Performance • Simulated reads from human chromosome 15, Gencode transcriptome • Hard test case • • • • • 1700 transcripts chosen randomly from Chr 15 Abundance generated from log-uniform distribution Read length=100, 1 Million reads 1% error rates Single-end reads / stranded protocol 106 Practical Performance Fraction of Transcripts Missed False Positives 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 Trinity 0.5 0.4 Our 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 1 to 10 10 to 25 25 to 50 50 to 100 Coverage Depth of Transcripts 100+ Trinity Our 107 Complexity • Sparsest flow problem known to be NP-Hard – Can show using similar reduction that RNA-Seq problem under dense reads is also NP-Hard, assuming arbitrary abundances • Reasons why our formulation leads to poly-time algorithm: – Our assumption that abundances are generic – Only worry about instances where there is enough information – Individual sequence formulation lets us focus on issues arising only in real genomes. Confidence • Can we be sure when the produced solution is correct? – Assume dense read model – We are finding the sparsest set of transcripts that satisfy the given L spectrum • Under the assumption of genericity – Theorem: If the sparsest solution is unique, then it is the only generic solution satisfying the L-spectrum (!) s1 0.12 0.12 s4 0.88 s5 s3 s2 0.88 Summary • An approach to assembly design based on principles of information theory. • Driven by and tested on genomics and transcriptomics data. • Ultimate goal is to build robust, scalable software with performance guarantees. Problem Landscape measure data • Assembly (de Novo) • Noisy reads • RNA: Finite N • Variant calling (reference-based assembly) • Indels • Large variants • Metagenomic assembly manage data utilize data • Compression • Compress memory? • Genome wide association studies • Information bounds • Privacy • Information theoretic methods? • Phylogenetic tree reconstruction • Pathogen detection Acknowledgements DNA Assembly RNA Assembly Abolfazl Motahari Sharif Soheil Mohajer Guy Bresler MIT Lior Pachter Berkeley Ma’ayan Bresler Berkeley Eren Sasoglu Ka Kit Lam Berkeley Asif Khalak Pacific Biosciences Joseph Hui Berkeley Kayvon Mazooji Berkeley

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