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Alignment II Dynamic Programming

Alignment II Dynamic Programming. Pair-wise sequence alignments. A: C A T - T C A - C | | | | | B: C - T C G C A G C. Idea: Display one sequence above another with spaces inserted in both to reveal similarity. Two types of alignment. S = CTGTCGCTGCACG T = TGCCGTG.

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Alignment II Dynamic Programming

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  1. Alignment IIDynamic Programming

  2. Pair-wise sequence alignments A:C A T - T C A - C | | | | | B:C - T C G C A G C Idea: Display one sequence above another with spaces inserted in both to reveal similarity

  3. Two types of alignment S =CTGTCGCTGCACG T =TGCCGTG Global alignment Local alignment CTGTCG-CTGCACG -TGC-CG-TG---- CTGTCGCTGCACG-- -------TGC-CGTG

  4. Global alignment: Scoring CTGTCG-CTGCACG -TGC-CG-TG---- Reward for matches:  Mismatch penalty:  Space penalty:  score(A) = w – x - y w = #matchesx = #mismatches y = #spaces

  5. Global alignment: Scoring Reward for matches: 10 Mismatch penalty: 2 Space penalty: 5 C T G T C G – C T G C - T G C – C G – T G - -5 10 10-2 -5 -2 -5 -5 10 10 -5 Total = 11

  6. Optimum Alignment • The score of an alignment is a measure of its quality • Optimum alignment problem: Given a pair of sequences X and Y, find an alignment (global or local) with maximum score • The similarity between X and Y, denoted sim(X,Y), is the maximum score of an alignment of X and Y

  7. Alignment algorithms • Global: Needleman-Wunsch • Local: Smith-Waterman • NW and SW use dynamic programming • Variations: • Gap penalty functions • Scoring matrices

  8. Global Alignment: Algorithm

  9. Theorem.C(i,j) satisfies the following relationships: Initial conditions: Recurrence relation: For 1  i n, 1  j m:

  10. S1 S2 . . . Si-1 Si S1 S2 . . . Si-1 Si T1 T2 . . . Tj-1 Tj T1 T2 . . . Tj — C(i-1,j-1) + w(Si,Tj) C(i-1,j)  S1 S2 . . . Si — T1 T2 . . . Tj-1 Tj C(i,j-1)  Justification

  11. Example Case 1: Line up Si with Tj i i - 1 S: C A T T C A C T: C - T T C A G j j -1 Case 2: Line up Si with space i - 1 i S: C A T T C A - C T: C - T T C A G - j Case 3: Line up Tj with space i S: C A T T C A C - T: C - T T C A - G j -1 j

  12. C(i-1,j-1) C(i-1,j) C(i,j-1) Computation Procedure C(0,0) C(i,j) C(n,m)

  13. -5 -10 -15 -20 -25 -30 -35 λ C T C G C A G C 0 -5 -10 -15 -20 -25 -30 -35 -40 λ 10 5 C A T T C A C +10 for match, -2 for mismatch, -5 for space

  14. * * λ C T C G C A G C λ C A T T C A C Traceback can yield both optimum alignments

  15. End-gap free alignment • Gaps at the start or end of alignment are not penalized Match: +2 Mismatch and space: -1 Best global Best end-gap free Score = 1 Score = 9

  16. Motivation: Shotgun assembly • Shotgun assembly produces large set of partially overlapping subsequences from many copies of one unknown DNA sequence. • Problem: Use the overlapping sections to ”paste” the subsequences together. • Overlapping pairs will have low global alignmentscore, but high end-space free score because of overlap.

  17. Motivation: Shotgun assembly

  18. Algorithm • Same as global alignment, except: • Initialize with zeros (free gaps at start) • Locate max in the last row/column (free gaps at end)

  19. 0 0 0 0 0 0 0 0 0 0 0 5 8 5 8 5 20 15 10 0 0 15 10 5 6 15 18 13 0 -2 10 13 8 3 10 13 16 0 10 5 20 15 18 13 8 23 5 8 15 18 13 28 23 18 0 0 0 3 10 25 20 23 38 33 λ C T C G C A G C λ 10 5 10 5 10 5 0 10 C A T T C A G +10 for match, -2 for mismatch, -5 for gap

  20. Local Alignment: Motivation • Ignoring stretches of non-coding DNA: • Non-coding regions are more likely to be subjected to mutations than coding regions. • Local alignment between two sequencesis likely to be between two exons. • Locating protein domains: • Proteins of different kind and of different species often exhibit local similarities • Local similarities may indicate ”functional subunits”.

  21. Local alignment: Example S =g g t c t g a g T =a a a c g a Match: +2 Mismatch and space: -1 Best local alignment: g g tc t g ag a a ac – g a - Score = 5

  22. Local Alignment: Algorithm C [i, j] = Score of optimally aligning a suffix of s with a suffix of t. Initialize top row and leftmost column to zero.

  23. λ C T C G C A G C λ C A T T C A C +1 for a match, -1 for a mismatch, -5 for a space

  24. Some Results • Most pairwise sequence alignment problems can be solved in O(mn) time. • Space requirement can be reduced to O(m+n), while keeping run-time fixed [Myers88]. • Highly similar sequences can be aligned in O(dn) time, where d measures the distance between the sequences [Landau86].

  25. Reducing space requirements • O(mn) tables are often the limiting factor in computing large alignments • There is a linear space technique that only doubles the time required [Hirschberg77]

  26. 0 5 8 5 8 5 20 15 10 λ C T C G C A G C 0 0 0 0 0 0 0 0 0 λ 0 10 5 10 5 10 5 0 10 C A T T C A G IDEA: We only need the previous row to calculate the next

  27. Linear-space Alignments mn + ½ mn + ¼ mn + 1/8 mn + 1/16 mn + … = 2 mn

  28. Affine Gap Penalty Functions Gap penalty = h + gk where k = length of gap h = gap opening penalty g = gap continuation penalty Can also be solved in O(nm) time using dynamic programming

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