Design of Optimal Multiple Spaced Seeds for Homology Search

Jinbo XuSchool of Computer Science, University

of WaterlooJoint work with D. Brown, M. Li and B.

Ma

Overview

Seed-based homology search Optimal multiple spaced seeds LP based randomized algorithm Experimental results Future work

Homology Search

Exhaustive search algorithm e.g. Smith-Waterman algorithm 100% sensitivity infeasible if the database is large

Suffix tree Seed-based algorithm, e.g. BLAST,

PatternHunter

Given: database of DNA sequences, query sequence QTask: extract all homologous sequences of Q from the database.

Region and Seed

S1: AGCTTGCCGTAAACCGS2: ACGTAGCACTGAGCTGRegion model: 1001011001010101

seed: 10010010111: a required match0: “don’t care”seed length M: length of the stringseed weight W: the number of 1 in the seed

Seed-based Hit

ACGCGTGGGAAACC

CAATGTGGGCAATT11011011

00001111101100

seed

region

Given a seed, a query sequence hits another sequence ifand only if the seed hits a region model of both sequences.

Query:

A seed S hits a region R at position i if and only ifR[i+j]=1 for every position j where s[j]=1

Single Seed Based Algorithm

Query: GGAAGCTTGCCGTATGCCATAGS1: CCAGGCTAGCCATAGGCCTTCT

Seed:101110111011011101

Length=18, weight=13

Query: GGAAGCTTGCCGTATGCCATAGS2: CCAGGCATGCAGTAGGCCTTCT

S1 hit, but S2 missed.

Multiple Seeds Based Algorithm

Query: GGAAGCTTGCCGTATGCCATAGS1: CCAGGCTAGCCATAGGCCTTCT

seed1:101110111011011101

Length=18, weight=13

Query: GGAAGCTTGCCGTATGCCATAGS2: CCAGGCATGCAGTAGGCCTTCT

seed2:101101110111011101

Both S1 and S2 are hit

Optimal Multiple Seeds (OMS) Problem

Given: random region R under certain distribution, two integers M and W, and an integer k.Find: set of k seeds with weight W and length no more than M to maximize the hit probability of R.

Mandala (J. Buhler et al.) Hill Climbing, good for small k, no result

reported for k>4 Greedy + Monte Carlo sampling

Greedy Algorithm (M. Li and B. Ma et al.) Given i seeds (i=1,2,…,k-1), search for the

next seed by maximizing the incremental sensitivity

Vector Seeds (B. Brejova et al.)

Related Work

Seed Specific OMS problem: Given a random region R, a set of m seeds , and an integer k, find a set of k seeds out of , to maximize the hit probability of R.

Seed-Region Specific OMS problem: Given a set of m seeds , an integer k and a set of

regions , find a set of k seeds, to maximize the hits of .

Variants of OMS

msss ,...,, 21

msss ,...,, 21

msss ,...,, 21

NRRR ,...,, 21

NRRR ,...,, 21

Main Results:1. Approximation ratio by a greedy algorithm

(D.S. Hochbaum)2. Same approximation ratio by linear programming based

randomized algorithm 3. is tight unless P=NP (U. Feige)

Given a ground set H and its subsets and an integer k, Find k sets out of to cover H as much as possible.

Maximum Coverage (MC) problem

632.01 1 e

mHHH ,...,, 21

11 e

mHHH ,...,, 21

OMS vs. MC ProblemOMS

Region Sampling

Seed Specific OMS

Seed-Region Specific OMS=MC Problem

Seed Enumeration

iS seedby hit regions ofset the:iH

Region Model

3-bits

000 001 010 011 100 101 110 111

p .1426

.0573

.1236

.0660

.0710

.0364

.2335

.2696

1. PH: length 64 and each bit independently set to 1 with probability 0.7 (B. Ma et al.)

2. M3: length 64 and each bit independently set to 1 with probability 0.8 if i%3=1 or 2, 0.5 otherwise (B. Brejova et al.)

3. M8: length 63 and each codon satisfy a certain distribution (B. Brejova et al.)

4. HMM: average length 90, two adjacent codons are not independent (B. Brejova et al.)

Observations

1. PH model: the hit probability of any seed with weight 11 and length 18 is at least 0.30

2. M3 model: the hit probability of any seed with weight 11 and length 18 is at least 0.27

3. HMM model: the hit probability of any seed with weight 11 and length 18 is at least 0.70

Variant of MC Problem

possible. asmuch as cover to

,...,, ofout sets find elements, ||least at contains

whichofeach ,,...,, subsets its and set ground aGiven

21

21

H

HHHkH

HHHH

m

m

Can we have a better approximation ratio?

If the sensitivity of each seed is at least and the optimal linear solution is , then the LP based randomized algorithm guarantees to generate a solution with approximation ratio at least

for the seed-region specific OMS problem.

Better Approximation Ratio

)1)(1()1( 1

)1()1( *

*

*

*

ee

lklkk

lklk

*l

Theoretical Results

Practical Approximation Ratio

)11( 02146.0

)(

)(

)(**

l

Ap

AP

Ap

the optimal seed set for the random region R the best seed set found by the LP based algorithm

*AA

with probability 0.99

If 5000 regions are sampled, then we have

Practical Approximation Ratio (W=10)

Practical Approximation Ratio (W=11)

Test Data

All-against-all comparison between mouse EST sequences and human EST sequences by Smith-Waterman algorithm

3346700 pairs found with local alignment score no less than 16

score

16-20

20-30

30-40

40-50

50-60

60-70

70-80

80-90

>90

ratio 93 6.3 0.26 0.06 0.06 0.02 0.01 0.02 0.28

label

1 2 3 4 5 6 7 8 9

Performance of PH Seeds

Performance of HMM Seeds

4 HMM Seeds vs. 1 HMM Seed

Greedy vs. LP

LP-based algorithm gives a mathematical foundation

LP-based algorithm is also good in practice Time complexity is exponential to . Is there

an approximation algorithm without enumerating seeds?

Better approximation ratio by Greedy algorithm?

Summary and Future Work

WM 2