Dynamic Programming for Pairwise Alignment 2

1. Dynamic Programmingfor Pairwise Alignment 2 Dr Alexei Drummond Department of Computer Science alexei@cs.auckland.ac.nz Semester 2, 2006

2. Review Dynamic programming algorithm for global alignment (Needleman & Wunsch) Given sequences: F(i,j) = score of best alignment between and

3. Principle of Optimality Optimal alignment

4. Principle of Optimality Optimal alignment Looks like ……

5. Principle of Optimality Optimal alignment Looks like …… or ……………

6. Principle of Optimality Optimal alignment Looks like …… or …………… or ……………

7. Principle of Optimality Optimal alignment Looks like …… or …………… or …………… so ……………

8. Basis

9. Filling up table Y F matrix 0 1 2 n 0 1 2 Optimal alignment score X m

10. Constructing alignment Y F matrix 0 1 2 n 0 1 2 Optimal alignment score X m

11. Example Y F matrix 0 1 2 n 0 1 2 Optimal alignment score X m Y Alignment Alignment X

12. Time and space 0 1 2 n 0 1 2 F matrix m table entries space Each entry computed in constant time time

13. Smith & Waterman algorithm Computes local alignment. i.e. look for best alignment of subsequences of X and Y, ignoring scores of regions on either side Y X Best subsequence alignment

14. Recurrences Basis:

15. Example

16. Example Y Alignment X

17. Repeated (local) matches Long sequences - interested in alllocal alignments with significant score, > threshold T. e.g. copies of repeated domain or motif in a protein. X = sequence containing motif Y = target sequence Y Matching parts of X Method is asymmetric

18. Principle of Optimality Given sequences DefineF(i,j)(i ≥ 1) = best sum of match scores in and assuming is in a matched region and match ends in or

19. Ends of matches best sum of completed match scores to is notin a matched region assuming that Row 0 therefore marks unmatched regions and ends of matches in Y.

20. General recurrence Start of new match Extension of previous match

21. Filling up table Y F matrix 0 1 2 n 0 1 2 X m

22. Filling up table Y F matrix 0 1 2 n 0 1 2 X m

23. Filling up table Y F matrix 0 1 2 n 0 1 2 X m

24. Filling up table Y F matrix 0 1 2 n 0 1 2 X m

25. Filling up table Y F matrix 0 1 2 n 0 1 2 X m

26. Filling up table Y F matrix 0 1 2 n 0 1 2 X m

27. Filling up table Y F matrix 0 1 2 n 0 1 2 X m

28. Filling up table Y F matrix 0 1 2 n 0 1 2 X m

29. Filling up table Y F matrix 0 1 2 n 0 1 2 X m

30. Filling up table Y F matrix 0 1 2 n 0 1 2 X m

31. Filling up table Y F matrix 0 1 2 n 0 1 2 Optimal Sum of alignment scores X m

32. Example Extra cell for final total score

33. Example Extra cell for final total score Y Alignment X

34. Overlap matches Y Y X X Y Y X X Don’t penalize overhanging ends i.e. set F(i,0) = F(0,j) = 0 Otherwise

35. Example

36. Example Y Alignment X

37. Affine gap penalities • Affine score:g(g) = -d- (g-1)e gap-open penality gap-extension penalty • Different penalties associated with extending alignment with gap symbol Y = C C T W P X = C S T W - Y = C C T W P X = C S T - - different from

38. General recurrence Extend by matching Extend by matching suffix of Y to gap of length i-k Extend by matching suffix of X to gap of length j-k Problem: Procedure runs in worst-case time

39. version Extra variables

40. Recurrences aligned to start of gap aligned to continuation of gap aligned to start of gap aligned to continuation of gap Procedure runs in worst-case time

41. Linear space alignment Y F matrix 0 1 2 n 0 1 2 X m

42. Linear space alignment Y F matrix 0 1 2 n 0 1 2 X m

43. Linear space alignment Y F matrix 0 1 2 n 0 1 2 X m

44. Linear space alignment Y F matrix 0 1 2 n 0 1 2 X m

45. Linear space alignment Y F matrix 0 1 2 n 0 1 2 X m

46. Linear space alignment Y F matrix 0 1 2 n 0 1 2 X m

47. Linear space alignment Y F matrix 0 1 2 n 0 1 2 X m

48. Linear space alignment Y F matrix 0 1 2 n 0 1 2 X m

49. Linear space alignment Y F matrix 0 1 2 n 0 1 2 X m

50. Linear space alignment Y F matrix 0 1 2 n 0 1 2 X m