1. Design of Optimal Multiple Spaced Seeds for Homology Search Jinbo Xu School of Computer Science, University of Waterloo Joint work with D. Brown, M. Li and B. Ma

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

3. 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 Q Task: extract all homologous sequences of Q from the database.

4. Region and Seed S1: AGCTTGCCGTAAACCG S2: ACGTAGCACTGAGCTG Region model: 1001011001010101 seed: 1001001011 1: a required match 0: “ don ’ t care ” seed length M: length of the string seed weight W: the number of 1 in the seed

5. Seed-based Hit ACGCGTGGGAAACC CAATGTGGGCAATT 11011011 00001111101100 seed region Given a seed, a query sequence hits another sequence if and 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 if R[i+j]=1 for every position j where s[j]=1

6. Single Seed Based Algorithm Query: GGAAGCTTGCCGTATGCCATAG S1: CCAGGCTAGCCATAGGCCTTCT Seed: 101110111011011101 Length=18, weight=13 Query: GGAAGCTTGCCGTATGCCATAG S2: CCAGGCATGCAGTAGGCCTTCT S1 hit, but S2 missed.

7. Multiple Seeds Based Algorithm Query: GGAAGCTTGCCGTATGCCATAG S1: CCAGGCTAGCCATAGGCCTTCT seed1: 101110111011011101 Length=18, weight=13 Query: GGAAGCTTGCCGTATGCCATAG S2: CCAGGCATGCAGTAGGCCTTCT seed2: 101101110111011101 Both S1 and S2 are hit

8. 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.

9. 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

10. 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

11. 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

12. OMS vs. MC Problem OMS Region Sampling Seed Specific OMS Seed-Region Specific OMS=MC Problem Seed Enumeration

13. Region Model 3-bits000001010011100101110111 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.)

14. 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

15. Variant of MC Problem Can we have a better approximation ratio?

16. 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

17. Theoretical Results

18. Practical Approximation Ratio the optimal seed set for the random region R the best seed set found by the LP based algorithm with probability 0.99 If 5000 regions are sampled, then we have

19. Practical Approximation Ratio (W=10)

20. Practical Approximation Ratio (W=11)

21. 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 score16-2020-3030-4040-5050-6060-7070-8080-90>90 ratio936.30.260.06 0.020.010.020.28 label123456789

22. Performance of PH Seeds

23. Performance of HMM Seeds

24. 4 HMM Seeds vs. 1 HMM Seed

25. Greedy vs. LP

26. 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