Large Scale Sequencing By Hybridization. Tel Aviv University

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1 Large Scale Sequencing By Hybridization Ron Shamir Dekel Tsur Tel Aviv University

2 Outline Background: SBH Shotgun SBH Analysis of the errorless case Analysis of error-prone

3 Sequencing By Hybridization (SBH) Hybridize target to array containing a spot for each possible k-mer. TGT TGG TGA TGT CTT CTT GAA GAA TGG CTG CTG GAT GAT TGA CTA CTA GAC GAC

4 Sequencing By Hybridization (SBH) Hybridize target to array containing a spot for each possible k-mer. TGT TGT TGG TGG TGA TGA CTT CTT GAA GAA CTG CTG GAT GAT CTA CTA GAC GAC

5 Sequencing By Hybridization (SBH) Hybridize target to array containing a spot for each possible k-mer. TGT TGT TGG TGG TGA TGA CTT CTT GAA GAA CTG CTG GAT GAT CTA CTA GAC GAC

6 Sequencing By Hybridization The spectrum of a sequence: multi-set of all its k-long substrings (k-mers). Goal: reconstruct the sequence from its spectrum. ACT CTG TGA GAC Pevzner 89: reconstruction is polynomial. But...

7 Reconstruction May Be Non-unique Different sequences can have the same spectrum: ACT, CTA, TAC ACTAC TACTA

8 Non-uniqueness Probability P (N, k): prob. that for a random sequence of length N, another sequence with same k- spectrum (failure probability). Arratia et al (97): asymptotically tight bounds for P (N, k) P N 8 P N 9 PSfrag replacements

9 Resuscitating SBH SBH is currently not competitive for sequencing. How can one make it competitive?

10 Shotgun SBH (Drmanac, Labat, Brukner, Crkvenjakov 89) 1. Fragment target S into overlapping clones; obtain the spectrum of each clone. ACTAGTTACTCTG ACT CTA TAG GTT TTA TAG AGT GTT TTA ACT CTC TTA TAC ACT CTC TCT CTG

11 Shotgun SBH 2. Find the correct clone map (e.g., Mayraz and Shamir, 98).

12 Shotgun SBH 2. Find the correct clone map (e.g., Mayraz and Shamir, 98). 3. The clones endpoints form a partition of the sequence S into subsequences called information fragments (IF). For each IF, compute its spectrum. ACT CTG ACT CTG CTG

13 Shotgun SBH ACT CTG ACT CTG CTG 4. Reconstruct the sequence of each IF.

14 Shotgun SBH ACT CTG ACT CTG CTG 4. Reconstruct the sequence of each IF. 5. Combine the sequences of the IFs.

15 Hybridization Errors Hybridization experiments are error prone. A false negative error: k-mer appears in a clone but does not appear in its measured spectrum. ACT CTG ACT CTG CTG

16 Hybridization Errors Hybridization experiments are error prone. A false negative error: k-mer appears in a clone but does not appear in its measured spectrum. ACT CTG ACT CTG CTG CTG

17 Goal Dramanac et al.: simulation evidence that shotgun SBH works in the absence of errors. Our Goal: Rigorous analysis, also considering the impact of errors.

18 Assumptions Clones positions are known. Equal size IFs (= d). Each k-mer of target appears in at least one clone spectrum. equiprobable bases, inde- Random sequence: pendent positions. False negative probability p independently for each k-mer and for each clone.

19 Hybridization Errors (2) For each k-tuple P in the spectrum, we attribute P to the i-th IF where i is the maximum index of a clone in which P appears CTG CTG CTG CTG

20 Hybridization Errors (2) For each k-tuple P in the spectrum, we attribute P to the i-th IF where i is the maximum index of a clone in which P appears CTG CTG CTG CTG

21 Hybridization Errors (2) For each k-tuple P in the spectrum, we attribute P to the i-th IF where i is the maximum index of a clone in which P appears. 1 2 CTG CTG CTG CTG CTG The computed index is always the true index.

22 Main Result N = sequence length k = probe length d = length of IFs p = false negative probability P (N, k, d, p): failure probability Theorem P (N, k, d, p) ( 1 + c p d ) P (N, k, d, 0).

23 Overview of the Proof Will show: P (N, k, d, 0) = Ω( d3 N ). 42k P (N, k, d, p) P (N, k, d, 0) = O( d2 N 4 2k ).

24 The de-bruijn Graph (Pevzner 89) A = a 1 a n+k 1 : the sequence. A i : the (k 1)-mer a i a i+1 a i+k 2. The de-bruijn graph of A : G A = (V, E) where V = {A i : i = 1,..., n + 1} E = {e i : i = 1,..., n}, e i = (A i, A i+1 ) ACTGCTGCC GCT TGC GCC ACT CTG

25 The de-bruijn Graph ACTGCTGCC GCT TGC GCC ACT CTG Classical SBH: Any solution corresponds to an Euler path in G A.

26 The de-bruijn Graph ACTGCTGCC ACT GCT 1 TGC 2 GCC CTG Shotgun SBH w/o errors: Each edge e i has a label l i = i d = the number of IF containing e i. A solution corresponds to an Euler path in which each e i is in the l i -th IF (i.e. in [(l i 1)d+1, l i d]). 2

27 The de-bruijn Graph GCT 1 TGC 2 GCC ACTGCTGCC ACT 1 CTG Shotgun SBH with errors: l i = number of IF containing e i s sequence. l i = max clone containing e i s sequence. l i l i.

28 The de-bruijn Graph GCT 1 TGC 2 GCC ACTGCTGCC ACT 1 CTG Shotgun SBH with errors: l i = number of IF containing e i s sequence. l i = max clone containing e i s sequence. l i l i. The distribution of l i l i is geometric with parameter p.

29 The de-bruijn Graph ACTGCTGCC ACT Shotgun SBH with errors: GCT 1 TGC 2 GCC CTG l i = number of IF containing e i s sequence. l i = max clone containing e i s sequence. l i l i. The distribution of l i l i is geometric with parameter p. A solution corresponds to an Euler path in which each e i is in an IF with index l i.

30 Definitions Recall: A i - the (k 1)-mer a i a i+1 a i+k 2. A pair (i, j) is a repeat if A i = A j. ACTGCTGCC GCT TGC GCC ACT CTG

31 Definitions Recall: A i - the (k 1)-mer a i a i+1 a i+k 2. A pair (i, j) is a repeat if A i = A j. (i, j) is rightmost repeat if (i + 1, j + 1) is not a repeat. ACTGCTGCC GCT TGC GCC ACT CTG

32 Failure Conditions Interleaved pair of repeats: a rightmost repeat (i, j) and a repeat (i, j ) with i i < j < j. Theorem (Pevzner 95) A sequence A is not uniquely recoverable iff either 1. A contains an interleaved pair of repeats, or 2. A 1 = A n+1. lacements e j e i e i 1 e 1 e j 1 (1) (2)

33 Failure Conditions Interleaved pair of repeats: a rightmost repeat (i, j) and a repeat (i, j ) with i i < j < j. Theorem (Pevzner 95) A sequence A is not uniquely recoverable iff either 1. A contains an interleaved pair of repeats, or 2. A 1 = A n+1. lacements e j e i e i 1 e 1 e j 1 (1) (2)

34 Failure Conditions Interleaved pair of repeats: a rightmost repeat (i, j) and a repeat (i, j ) with i i < j < j. Theorem (Pevzner 95) A sequence A is not uniquely recoverable iff either 1. A contains an interleaved pair of repeats, or 2. A 1 = A n+1. lacements e j e i e i 1 e 1 e j 1 (1) (2)

35 Failure Conditions - Shotgun SBH Theorem A sequence A is not uniquely recoverable iff either 1. A contains an interleaved pair of repeats (i, j)(i, j ) with l i = l j 1, or 2. A 1 = A d+1 = = A cd+1 and A i1 = A i2 = = A ic A 1 for indices i 1, i 2,... i c with l ij = j and i j (j 1)d

36 Failure Probability: The Errorless Case Using the theorem we show that P (N, k, d, 0) = Θ( n ( ) d d k 2) = n Θ(d3 4 ) 2k Arratia et al. Our bounds n k lower upper lower upper Simulation

37 Error-prone Spectra Define event X: the solution is not unique when there are errors, but is unique in the errorless case. If event X happens, then A contains a rightmost repeat (i, j) and a repeat (i, j ) with i < j < j, i / [j, j ], and l i l i and either l i < l j 1, or j 1 = dl i. PSfrag replacements e i e j e i e j

38 Error-prone Spectra Define event X: the solution is not unique when there are errors, but is unique in the errorless case. If event X happens, then A contains a rightmost repeat (i, j) and a repeat (i, j ) with i < j < j, i / [j, j ], and l i l i and either l i < l j 1, or j 1 = dl i. PSfrag replacements e i e j e i e j

39 Error-prone Spectra Define event X: the solution is not unique when there are errors, but is unique in the errorless case. If event X happens, then A contains a rightmost repeat (i, j) and a repeat (i, j ) with i < j < j, i / [j, j ], and l i l i and either l i < l j 1, or j 1 = dl i. PSfrag replacements e i e j e i e j

40 Error-prone Spectra PSfrag replacements e i e j e i e j

41 Error-prone Spectra PSfrag replacements e i 2 e j e i 3 3 e j

42 Error-prone Spectra PSfrag replacements e i 2 e j e i 3 3 e j

43 Error-prone Spectra PSfrag replacements e i 2 e j e i 3 3 e j

44 Error-prone Spectra PSfrag replacements e i 2 e j e i 2 3 e j We need l j+1 2, l j+2 2, and l i 2. The probability for these events is p 3.

45 Error-prone Spectra Theorem If event X happens, then A contains a rightmost repeat (i, j) and a repeat (i, j ) with i < j < j, i / [j, j ], and l i l i and either l i < l j 1, or j 1 = dl i. Furthermore, l r l r (j i) for all j r j 1, and l i l j (j i). PSfrag replacements e i 2 e j e i 2 3 e j

46 Error-prone Spectra Low probability cases: High probability cases:

47 Error-prone Spectra Using the previous theorem, we can bound the probability that event X happen: p Theorem P [X] = O( (1 p) n 4 d d3 4 2k).

48 Simulations Generated data under the assumptions used for the theoretical analysis.

49 The Impact of d n k d P (n,k,d,0) P (n,k,d,0.5) (%) (%)

50 The Impact of Errors n k d P (n,k,d) P (n,k,d,0.5) P (n,k,d,0.5) P

51 Variable Size IFs IF sizes are Poisson distributed with expectation d. Prob(fail) E(# errors) n k d p=0 p=0.5 p=0 p=

52 Real DNA Sequences Prob(error) Avg. # errors n k d p=0 p=0.5 p=0 p= With 9-mer chip, can handle cosmid size target with 99.9% accuracy even with 50% false negative.

53 Summary Full analysis of failure prob. in errorless SBH: improves over Arratia et al. (96) for small k. Main result: Analysis of failure probability in Shotgun SBH, in the presence of errors. Errors have very little effect on failure probability. Simulation show result holds even when some of the assumptions are relaxed.

54 Open Problems Analyze the case of Poisson distribution of clone positions. Analyze the expected no. of errors. Relax independence assumption on errors. In simulation, compute the clones positions from the data. Handle false positives.

55 Open Problems Analyze the case of Poisson distribution of clone positions. Analyze the expected no. of errors. Relax independence assumption on errors. In simulation, compute the clones positions from the data. Handle false positives. The End

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