Multi-seed lossless filtration

1. Multi-seed lossless filtration Gregory Kucherov Laurent Noé LORIA/INRIA, Nancy, France Mikhail Roytberg Institute of Mathematical Problems in Biology, Puschino, Russia CPM (Istanbul) July 5-7, 2004

2. potential matches Text filtration: general principle

3. Text filtration: general principle potential matches

4. Text filtration: general principle lossless and lossy filters true match

5. potential similarities Filtration applied to sequence comparison

6. Filtration applied to sequence alignment potential similarities

7. Filtration applied to sequence alignment true similarities

8. Gapless similarities. Hamming distance. • Similarities are defined through Hamming distance GCTACGACTTCGAGCTGC ...CTCAGCTATGACCTCGAGCGGCCTATCTA...

9. Gapless similarities. Hamming distance. • Similarities are defined through Hamming distance

10. m Gapless similarities. Hamming distance. • Similarities are defined through Hamming distance • (m,k)-problem, (m,k)-instances k

11. m Gapless similarities. Hamming distance. • Similarities are defined through Hamming distance • (m,k)-problem, (m,k)-instances • This work: lossless filtering k

12. m=18 #### #########(1) Filtering by contiguous fragment (m,k) • PEX (Navarro&Raffinot 2002) • Searching for a contiguous pattern • PEX with errors • Searching for a contiguouspattern with l possible errors • requires retrieval of alll-variantsin the index. Efficient for • small alphabets (ADN,ARN) • relatively smalll(<= 2) k=3

13. #---#---#---# #---#---#---# #---#---#---# #---#---#---# k+1 Superposition of two filters Pevzner&Waterman 1995 Idea: combine PEX withanother filterbased on a regularly-spaced seed • PEX : • spaced PEX (matches occurring at every k positions). #### #---#---#---#

14. Spaced seeds • Spaced seeds (spaced Q-grams) • proposed byBurkhardt & Kärkkäinen (CPM 2001) for solving (m,k)-problems • Principle • Searching for spaced rather than contiguous patterns • Selectivity • defined by the weightof the seed (numberof #’s) ###-##

15. Example: (18,3)-problem ###-## ###-## ###-## ###-## ###-## ###-##

16. Spaced seeds for sequence comparison • Ma, Tromp, Li 2002 (PatternHunter) • Estimating seed sensitivity: Keich et al 2002, Buhler et al 2003, Brejova et al 2003, Choi&Zhang 2004, Choi et al 2004, Kucherov et al 2004, ... • Extended seed models: BLASTZ 2003, Brejova et al 2003, Chen&Sung 2003, Noé&Kucherov 2004, ...

17. Families of spaced seeds This work:lossless filtration using spaced seed families (extension of Burkhard&Karkkainen 2001) • single filter based on several distinct seeds • each seed detects a part of (m,k)-instances but together they must detect all (m,k)-instances Independent work (lossy seed families for sequence alignment): • Li, Ma, Kisman, Tromp 2004 (PatternHunter II) • Xu, Brown, Li, Ma, this conference • Sun, Buhler, RECOMB 2004 (Mandala)

18. ##-#-#### ###---#--##-# F Example: (18.3)-problem (cont) • every (18,3)-instance contains an occurrence of a seed of F • all seeds of the family have the same weight 7 FamilyFsolves the(18,3)-problem

19. ##-#-#### ###---#--##-# ##-##-##### ###-####--## ###-##---#-### ##----####-### ###---#-#-##-## ###-#-#-#-----### Example: (18.3)-problem (cont) w=7 ###---#--##-# ###---#--##-# w=9 ###-##---#-###

20. ##-#-#### ###---#--##-# ##-##-##### ###-####--## ###-##---#-### ##----####-### ###---#-#-##-## ###-#-#-#-----### Comparative selectivity Selectivity of families onBernoulli similarities (p(match)= 1/4) estimated as the probability for one of the seeds to occur at a given position w=4 ~39.10-4 #### w=5 ~9.810-4 ###-## w=7 ~1.2 10-4 w=9 ~0.23 10-4

21. How far should we go • A trivial extreme solution ... • would be to pick allseeds of weightm - k. • prohibitive cost except for very small problems • We are interested in intermediate solutions: • relatively small number of seeds (< 10) to keep the hash table of a reasonable size, • the seed weight sufficiently large to obtain a good selectivity

22. Results • Computing properties of seed families • Seed design • Seed expansion/contraction • Periodic seeds • Seed optimality • Heuristic seed design • Experiments • Examples of designed seed families • Application to computing specific oligonucleotides • Conclusions

23. Measuringthe efficiency of a family • Optimal threshold (Burkhard&Karkkainen): minimal number of seed occurrences over all (m,k)-instances • A seed family F is lossless iff the optimal threshold TF(m,k)1 • TF(m,k) can be computed by a dynamic programming algorithm in time O(m·k·2(S+1)) and space O(k·2(S+1)), where S is the maximal length of a seed from F • optimizations are possible (see the paper) • the resulting space complexity is the same as in the Burkhard&Karkkainen algorithm

24. Measuringthe efficiency of a family (cont) Using a similar DP technique we can compute, within the same time complexity bound: • the number UF(m,k) of undetected (m,k)-similarities for a (lossy) family F • the contribution of a seed of F, i.e. the number of (m,k)-similarities detected exclusively by this seed [see the paper for details]

25. Design of seed families Pruning exhaustive search tree(Burkhard&Karkkainen) • Construct all solutions of weightwfrom solutions of weightw – 1 • Example: if##--#--#and##-#---#are solutions of weightw-1, considertheir «union» ##-##--#of weightw. • Prohibitive cost: • more than a week for computing all single-seed solutions of the (50,5)-problem • the search space blows up for multi-seed families

26. Seed expansion/contraction Burkhard&Karkkainen: the only two solutions of weight 12 solving the (50,5)-problem: ###-#--###-#--###-# #-#-#---#-----#-#-#---#-----#-#-#---#

27. Seed expansion/contraction Burkhard&Karkkainen: the only two solutions of weight 12 solving the (50,5)-problem: ###-#--###-#--###-# #-#-#---#-----#-#-#---#-----#-#-#---# the only solution of weight 12 of the (25,2)-problem

28. Seed expansion/contraction Burkhard&Karkkainen: the only two solutions of weight 12 solving the (50,5)-problem: ###-#--###-#--###-# #-#-#---#-----#-#-#---#-----#-#-#---# • Letbe thei-regular expansion of F obtained by inserting i-1jokers between successive positions of each seed of F • Example: IfF = {###-# , ##-##}then = {#-#-#---# , #-#---#-#} = {#--#--#-----#, #--#-----#--#} the only solution of weight 12 of the (25,2)-problem

29. ##-#-#### ###---#--##-# ##-#-#### ###---#--##-# #-#---#---#-#-#-# #-#-#-------#-----#-#-# Seed expansion/contraction(cont) Lemma: • If a familyF solvesan (m,k)–problem, thenbothF and solvesthe(i·m, (i+1)·k- 1)–problem • If a familysolvesthe(i·m,k)–problem, then itsi-contraction Fsolvesthe(m, )-problem (18,3) (36,7)

30. Periodic seeds Iterating shortseedswith good properties into longer seeds ###-#-- ###-#--###-#--###-#

31. Cyclic problem --##-# ###-#--#--- Cyclic (11,3)-problem Linear (30,3)-problem ###-#--#---###-#--# Lemma: If a seed Q solves a cyclic (m,k)-problem, then the seed Qi=[Q,- (m-s(Q))]i solves the linear (m·(i+1)+s(Q)-1,k)-problem.

32. ###-#--#---###-#--# #--#---###-#--#---### Extension to multi-seed case ###-#--#--- Cyclic (11,3)-problem Linear (25,3)-problem

33. ###-#--#---###-#--# #--#---###-#--#---### Extension to multi-seed case ###-#--#--- Cyclic (11,3)-problem Linear (25,3)-problem

34. Asymptotic optimality Theorem: Fix a number of errors k. Let w(m) be the maximal weight of a seed solving the linear (m,k)-problem. Then • the fraction of the number of jokers tends to 0 but the convergence speed depends on k • seed expansion cannot provide an asymptotically optimal solution ) (

35. Non-asymptotic optimality • Fix a number of errors k. • For each seed (seed family) Q there exists mQ s.t. mmQ, Q solves the (m,k)-problem • For a class of seeds , Q is an optimal seed in  iff Q realizes the minimal mQ over all seeds of  Lemma: Let n be an integer and r=n/3. For every k2, seed #n-r-#r is optimal among seeds of weight n with one joker.

36. Heuristic seed design: genetic algorithm difficult (m,k)-instances • a population of seed families is evolving by mutating and crossing over • seed families are screened against sets of difficult (m,k)-instances • for a family that detects all difficult instances, the number of undetected similarities is computed by a DP algorithm. A family is kept if it yields a smaller number than currently known families • compute the contribution of each seed of the family. Mutate the less “valuable” seeds. select selectandreorder seed families

37. Example: (25,2)-problem

38. Example: (25,3)-problem

39. Application of lossless filtering: oligo design • Specific oligonucleotides: small DNA molecules (10-50bp) that hybridize with a given target sequence and do not hybridize with the other background sequences (e.g. the rest of the genome) • Formalization: given a sequence, find all windows of length m which do not occur elsewhere within k substitution errors

40. Seed design: (32,5)-problem

41. Experiment • This filter has been applied to the rice EST database (100015 sequences of total size ~42 Mbp) • All 32-windows occurring elsewhere within 5 errors have been computed • The computation took slightly more than 1 hour on a P4 3GHz computer • 87% of the database have been “filtered out”

42. Further questions • Combinatorial structure of optimal seed families • Efficient design algorithm

43. Questions ? ? ?