Fundamental Bounds for Sequence Reconstruction from Nanopore Sequencers. BICOB, Honolulu, 2017

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1 Fundamental Bounds for Sequence Reconstruction from Nanopore Sequencers Abram Magner, Jarek Duda, Wojciech Szpankowski, Ananth Grama March 20, 2017 BICOB, Honolulu, 2017

2 Outline 1. Information Theory and Biology: New Symbosis 2. Information Theory of Shotgun Sequencing (D. Tse et al.) 3. Nanopore Sequencing Technology 4. Nanopore Sequencing as an Information Theory Channel 5. Main Results 6. Future Work

3 Information Theory and Biology Information Theory can be used in Biology to derive fundamental bounds on quantities of interest (e.g., is reliable reconstruction of DNA). The final goal, howevers, is use those bounds to design efficient algorithms and eventually develop a useful tool that can be used by the biology community. Examples: 1. Finding altrenative splicing using mutual information (Szpankowski et al.) 2. Shotgun DNA Sequencing using pattern matching (Tse, et al.) 3. De novo RNA-Seq Assembler using channel coding (Tse etal al.) 4 Fundamental of nanopore sequencing using sticky deletion channel (IEEE Trans, Molecular, Biol. & Multi-Scale Commun., 2016). 5. Compression of biological database or biological structures using runlength coding ((Milenkovic, Weissman). 6. Protein statistics thru Boltzmann channel (Magner, Kihara, WS).

4 Outline Update 1. Information Theory and Biology: New Symbosis 2. Information Theory of Shotgun Sequencing (D. Tse et al.) 3. Nanopore Sequencing Technology 4. Nanopore Sequencing as an Information Theory Channel 5. Main Results 6. Future Work

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8 Outline Update 1. Information Theory and Biology: New Symbosis 2. Information Theory of Shotgun Sequencing (D. Tse et al.) 3. Nanopore Sequencing Technology 4. Nanopore Sequencing as an Information Theory Channel 5. Main Results 6. Future Work

9 Nanopore Sequencing technology A sequence of bases is fed through a nanopore channel by a molecular ratchet mechanism. Varying current across the channel encodes identity of bases. Nontrivial machine learning methods transform current signal to a string of bases. Mismatch between ratcheting and current reading rate = insertion/deletion of a base.

10 Outline Update 1. Information Theory and Biology: New Symbosis 2. nformation Theory of Shotgun Sequencing (D. Tse et al.) 3. Nanopore Sequencing Technology 4. Nanopore Sequencing as an Information Theory Channel 5. Main Results 6. Future Work

11 Nanopore as a Sticky Insertion-Deletion Channel Input: TTT AAAA C TTTT... (sequence of blocks of identical bases) AAA...A k identical bases Nanopore Sequencer Reconstruction Techniques AAA...A k identical bases Insertion error when k > k No error when k = k Deletion error when k < k AAA...A Block of k identical characters Insertion Deletion Channel AAA...A Block of k identical characters An insertion deletion channel transforms each block of k identical characters to an output block of k identical characters. This transformation of k to k is drawn from a distribution that is derived from machine characteristics. Input X: Random sequence to be recovered from noisy samples. E.g., X = AAACTTTCCGGGAACG. N: Number of blocks in X. In the previous example, N = 8. Blocks: (B(X),S) whereb is the block size ands is the symbol of the block. Example: (3,A),(1,C),(3,T),(2,C),(2,G),(2,A),(1,C),(1,G). Channel q k,l : Probability that block of length k will become block of length l. But Blocks are neither created nor removed entirely: q k,0 = 0 andq 0,0 = 1. E.g., q k,l = q l k (1 q) for some q < 1.

12 How to Recover Reliably DNA? Main question: How many times do we need to pass an input string through the channel to recover it? Z: A vector of r samples of X from the channel. Example. Assume that the input is Then, we may have with r = 3: X = AAACTTTCCGGGAACG. Z = (ACTCGACG,AACCCCCTTCGGGGACCCCG,ACCTTTCGGAACGG). Y = Y( Z): Estimator of X from samples Z. p e : Probability of error. I.e., Pr[Y X]. Main question becomes how large must r be as a function of N to guarantee p e N 0?

13 Lower Bound on Sample Complexity via Information Theory Goal: Lower bound necessary number of samples r for any estimator Y. Information-theoretic tool: Fano s inequality. p e H(X Z) h(p e ) log supp(x) = H(B(X) B(Y ) h(p e) log supp(x) Consequence: Lower bounding H(X Z) gives a lower bound for p e, which gives a lower bound for r. Analysis reveal that: p e 1 (1 e Θ(r) ) N Ne Θ(r). Theorem 1 (Lower bound). Under certain mild conditions on the q k, at least r Ω(logN) samples are needed for any estimator to recover X exactly. Remark. The constant hidden in the Ω( ) depends on the minimum KLdivergence between any two q k,q k.

14 Upper Bound on Sample Complexity: Decoding Algorithm Upper Bound: Find a decoder for the channel output that estimate X from Y( Z). Since the block structure is preserved it suffices to recover block lengths. If B i (Z j ) is the length of the ith block, then to recovre it we use M i ( Z) = 1 r r B i (Z j ). j=1 Theorem 2 (Sample complexity upper bound). Under mild conditions, e.g., we assume that Var(q k ) = Θ(k γ ), we find p e Ne Θ ( ) 1 r2γ+1. Thus suffices for exact recovery. r = O(log 2γ+1 N)

15 Empirical Examples Performance of estimators for two natural error models: Exponential insertion model: 3 q l,k = q k l (1 q), q (0,1), k l. 2.5 Variance of q l = Θ(1) = Θ(log n) samples are necessary and sufficient for exact recovery. Block size estimator: ˆM = M q, 1 q Normalized L1 error # of samples where M = empirical mean estimator. Figure 1: Number of samples versus normalized L 1 error for the exponential insertion

16 Empirical Examples Independent insertion-deletion model: Symbols in a block are duplicated or deleted independently with probability 1/2. Variance of q l = Θ(l) = O(log 3 n) samples are sufficient and Ω(log n) are necessary for exact recovery. Empirical mean estimator suffices. Normalized L1 error # of samples Figure 2: Number of samples versus normalized L 1 error for the independent insertion-deletion model.

17 Connections to Other Problems Capacity of deletion channels (open information theoretic problem): abbabaaaabbabba = abbaaabbba Upper bound on capacity = Lower bound on our sample complexity Trace reconstruction: Reconstruct a string from random subsequences. How many samples are needed? S = abbaaababbaacd s 1 = abaaad s 2 = baaabaac... This work: Can be viewed as Analysis of performance of repetition codes for an insertion-deletion channel. Trace reconstruction problem with novel distributional assumptions.

18 Outline Update 1. Information Theory and Biology: New Symbosis 2. Information Theory of Shotgun Sequencing (D. Tse et al.) 3. Nanopore Sequencing Technology 4. Nanopore Sequencing as an Information Theory Channel 5. Main Results 6. Future Work

19 Extensions Main drawback (theoretical/practical): Errors can change block structure in real life: S = aaabbababbbaabbbaaab = S = aaabbabaaabbbabaab Several natural channel models for this, depending on how new blocks are created. Ambiguity in source of error makes recovery more challenging: S can result from S in multiple ways. All blocks in S deleted, then S is inserted? aaabbababbbaabbbaaab = = aaabbabaaabbbabaab A single block deleted and another inserted? aaabbababbbaabbbaaab = aaabbabaaabbbaaab = aaabbabaaabbbabaab Model validation (practical): assumptions of the model. Need rigorous statistical verification of Difficult with available data. Learning model parameters from data.

20 Thank you!