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Optimal Sequence Alignment Techniques in Computational Biology

This article explores essential concepts in sequence alignment, a crucial method in computational biology for comparing DNA, RNA, or protein sequences. It discusses pairwise alignments with examples, scoring schemes for matches, mismatches, and gaps, and introduces both global and local alignments. The document outlines algorithms for calculating alignment scores and the significance of affine gap penalties, alongside practical applications in genetic research. Understanding these principles is vital for advancements in bioinformatics and genetic analysis.

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Optimal Sequence Alignment Techniques in Computational Biology

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

  3. Dot Matrix C G G A T C A T Sequence A：CTTAACT Sequence B：CGGATCAT CTTAACT

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

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

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

  7. 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

  8. 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.

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

  10. Initializations C G G A T C A T CTTAACT

  11. S3,5 = ？ C G G A T C A T CTTAACT

  12. S3,5 = 5 C G G A T C A T CTTAACT optimal score

  13. 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

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

  15. Initializations G A A T C T G C CAATTGA

  16. S4,2 = ？ G A A T C T G C CAATTGA

  17. S5,5 = ？ G A A T C T G C CAATTGA

  18. S5,5 = 14 G A A T C T G C CAATTGA optimal score

  19. 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

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

  21. 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.

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

  23. 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)

  24. 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.

  25. 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

  26. 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

  27. 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.

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

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

  30. 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

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

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

  33. 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

  34. 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

  35. 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 a deletion an insertion

  36. 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.

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

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

  39. 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

  40. 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.

  41. Constant gap penalties

  42. 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

  43. Restricted affine gap penalties

  44. 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)

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

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