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Sequence Alignment

This document presents a comprehensive study on sequence alignment, focusing on the evolutionary relationships and functional correlations of various sequences such as Orz and Crz. It includes detailed examples of dot matrix pairwise alignment, scoring schemes, and optimal alignment through dynamic programming. Additionally, the text delves into global vs. local alignment strategies, as well as affine gap penalties, providing insights into algorithms for maximum-sum intervals. This research is crucial for advancing our understanding of sequence evolution in computational biology. **Relevant

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Sequence Alignment

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  1. Sequence Alignment Kun-Mao Chao (趙坤茂) Department of Computer Science and Information Engineering National Taiwan University, Taiwan WWW: http://www.csie.ntu.edu.tw/~kmchao

  2. GenBank 200.0

  3. orz’s sequence evolution • the origin? • their evolutionary relationships? • their putative functional relationships? • orz (kid) • OTZ (adult) • Orz (big head) • Crz (motorcycle driver) • on_ (soldier) • or2 (bottom up) • oΩ (back high) • STO (the other way around) • Oroz (me)

  4. What? THETR UTHIS MOREI MPORT ANTTH ANTHE FACTS The truth is more important than the facts.

  5. Dot Matrix

  6. Pairwise Alignment Sequence A: CTTAACT Sequence B: CGGATCAT An alignment of A and B: C---TTAACTCGGATCA--T Sequence A Sequence B

  7. Pairwise Alignment Sequence A: CTTAACT Sequence B: CGGATCAT An alignment of A and B: Mismatch Match C---TTAACTCGGATCA--T Deletion gap Insertion gap

  8. Alignment Graph C G G A T C A T Sequence A: CTTAACT Sequence B: CGGATCAT CTTAACT C---TTAACTCGGATCA--T

  9. A simple scoring scheme • Match: +8 (w(x, y) = 8, if x = y) • Mismatch: -5 (w(x, y) = -5, if x ≠ y) • Each gap symbol: -3 (w(-,x)=w(x,-)=-3) C - - - T T A A C TC G G A T C A - - T +8 -3 -3 -3 +8 -5 +8 -3 -3 +8 = +12 Alignment score

  10. An optimal alignment-- the alignment of maximum score • Let A=a1a2…am and B=b1b2…bn . • Si,j: the score of an optimal alignment between a1a2…ai and b1b2…bj • With proper initializations, Si,j can be computedas follows.

  11. ComputingSi,j j w(ai,bj) w(ai,-) i w(-,bj) Sm,n

  12. Match: 8 Mismatch: -5 Gap symbol: -3 Initializations C G G A T C A T CTTAACT

  13. Match: 8 Mismatch: -5 Gap symbol: -3 S3,5 = ？ C G G A T C A T CTTAACT

  14. Match: 8 Mismatch: -5 Gap symbol: -3 S3,5 = 5 C G G A T C A T CTTAACT optimal score

  15. C T T A A C – TC G G A T C A T 8 – 5 –5 +8 -5 +8 -3 +8 = 14 C G G A T C A T CTTAACT

  16. Now try this example in class Sequence A: CAATTGA Sequence B: GAATCTGC Their optimal alignment？

  17. Match: 8 Mismatch: -5 Gap symbol: -3 Initializations G A A T C T G C CAATTGA

  18. Match: 8 Mismatch: -5 Gap symbol: -3 S4,2 = ？ G A A T C T G C CAATTGA

  19. Match: 8 Mismatch: -5 Gap symbol: -3 S5,5 = ？ G A A T C T G C CAATTGA

  20. Match: 8 Mismatch: -5 Gap symbol: -3 S5,5 = 14 G A A T C T G C CAATTGA optimal score

  21. C A A T - T G AG A A T C T G C -5 +8 +8 +8 -3 +8 +8 -5 = 27 G A A T C T G C CAATTGA

  22. Global Alignment vs. Local Alignment • global alignment: • local alignment:

  23. Maximum-sum interval • Given a sequence of real numbers a1a2…an, find a consecutive subsequence with the maximum sum. 9 –3 1 7 –15 2 3 –4 2 –7 6 –2 8 4 -9 For each position, we can compute the maximum-sum interval ending at that position in O(n) time. Therefore, a naive algorithm runs in O(n2) time.

  24. Computing a segment sum in O(1) time? • Input: a sequence of real numbers a1a2…an • Query: the sum of ai ai+1…aj

  25. Computing a segment sum in O(1) time • prefix-sum(i) = a1+a2+…+ai • all n prefix sums are computable in O(n) time. • sum(i, j) = prefix-sum(j) – prefix-sum(i-1) j i prefix-sum(j) prefix-sum(i-1)

  26. ai Maximum-sum interval(The recurrence relation) • Define S(i) to be the maximum sum of the intervals ending at position i. If S(i-1) < 0, concatenating ai with its previous interval gives less sum than ai itself.

  27. Maximum-sum interval(Tabular computation) 9 –3 1 7 –15 2 3 –4 2 –7 6 –2 8 4 -9 S(i) 9 6 7 14 –1 2 5 1 3 –4 6 4 12 16 7 The maximum sum

  28. Maximum-sum interval(Traceback) 9 –3 1 7 –15 2 3 –4 2 –7 6 –2 8 4 -9 S(i) 9 6 7 14 –1 2 5 1 3 –4 6 4 12 16 7 The maximum-sum interval: 6 -2 8 4

  29. An optimal local alignment • Si,j: the score of an optimal local alignment ending at (i, j) between a1a2…ai and b1b2…bj. • With proper initializations, Si,j can be computedas follows.

  30. Match: 8 Mismatch: -5 Gap symbol: -3 local alignment C G G A T C A T CTTAACT

  31. Match: 8 Mismatch: -5 Gap symbol: -3 local alignment C G G A T C A T CTTAACT The best score

  32. A – C - TA T C A T 8-3+8-3+8 = 18 C G G A T C A T CTTAACT The best score

  33. Now try this example in class Sequence A: CAATTGA Sequence B: GAATCTGC Their optimal local alignment？

  34. Did you get it right? G A A T C T G C CAATTGA

  35. A A T – T GA A T C T G 8+8+8-3+8+8 = 37 G A A T C T G C CAATTGA

  36. Osamu Gotoh

  37. Affine gap penalties • Match: +8 (w(a, b) = 8, if a = b) • Mismatch: -5 (w(a, b) = -5, if a ≠ b) • Each gap symbol: -3 (w(-,b) = w(a,-) = -3) • Each gap is charged an extra gap-open penalty: -4. -4 -4 C - - - T T A A C TC G G A T C A - - T +8 -3 -3 -3 +8 -5 +8 -3 -3 +8 = +12 Alignment score: 12 – 4 – 4 = 4

  38. Affine gap panalties • A gap of length k is penalized x + k·y. gap-open penalty Three cases for alignment endings: • ...x...x • ...x...- • ...-...x gap-symbol penalty an aligned pair This is the same as the scoring scheme that penalizes the first symbol x + y and an extended symbol y. a deletion an insertion

  39. Affine gap penalties • Let D(i, j) denote the maximum score of any alignment between a1a2…ai and b1b2…bj endingwith a deletion. • Let I(i, j) denote the maximum score of any alignment between a1a2…ai and b1b2…bj endingwith an insertion. • Let S(i, j) denote the maximum score of any alignment between a1a2…ai and b1b2…bj.

  40. Affine gap penalties (A gap of length k is penalized x + k·y.)

  41. D D D I I I S S S Affine gap penalties -y w(ai,bj) -x-y D -x-y I S -y

  42. Constant gap penalties • Match: +8 (w(a, b) = 8, if a = b) • Mismatch: -5 (w(a, b) = -5, if a ≠ b) • Each gap symbol: 0 (w(-,b) = w(a,-) = 0) • Each gap is charged a constant penalty: -4. -4 -4 C - - - T T A A C TC G G A T C A - - T +8 0 0 0 +8 -5 +8 0 0 +8 = +27 Alignment score: 27 – 4 – 4 = 19

  43. Constant gap penalties • Let D(i, j) denote the maximum score of any alignment between a1a2…ai and b1b2…bj endingwith a deletion. • Let I(i, j) denote the maximum score of any alignment between a1a2…ai and b1b2…bj endingwith an insertion. • Let S(i, j) denote the maximum score of any alignment between a1a2…ai and b1b2…bj.

  44. Constant gap penalties

  45. Restricted affine gap panalties • A gap of length k is penalized x + f(k)·y. where f(k) = k for k <= c and f(k) = c for k > c Five cases for alignment endings: • ...x...x • ...x...- • ...-...x • and 5. for long gaps an aligned pair a deletion an insertion

  46. Restricted affine gap penalties

  47. D(i, j) vs. D’(i, j) • Case 1: the best alignment ending at (i, j) with a deletion at the end has the last deletion gap of length <= c D(i, j) >= D’(i, j) • Case 2: the best alignment ending at (i, j) with a deletion at the end has the last deletion gap of length >= c D(i, j) <= D’(i, j)

  48. Max{S(i,j)-x-ky, S(i,j)-x-cy} S(i,j)-x-cy c k

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