1. Pairwise Sequence Alignment
2005/9/22
2005 Autumn / YM / Bioinformatics

2. Outline Motivation A simple alignment problem Scoring alignments
Substition matrices
Gap penalties
Dynamic programming
Global vs. local alignment

3. Reference D.W. Mount / Bioinformatics / Ch.3 Slides from
Prof. C.H.Chang
UW / Genomic Informatics / W.S. Noble

4. Motivation Why align two protein or DNA sequences?
Determine whether they are descended from a common ancestor (homologous).
Infer a common function.
Locate functional elements (motifs or domains).
Infer protein structure, if the structure of one of the sequences is known.

5. Example

6. Brief History
Robinson (1938) described the 1st algorithm for Longest Common Subsequence problem
Levenshtein (1966) introduced “edit distance” between strings
Min. # of edit operations needed to transform from one string to another
No algorithm described
Gibbs & McIntyre (1970): dot matrix analysis
Needleman & Wunsch (1970) “re-discovered” the algorithm for sequence alignment using DP
Smith & Waterman (1981) proposed a clever modification of DP for local alignment

7. Brief History
Knuth (1973) suggested a filtering idea for pattern matching
高德纳 (Donald Knuth)
Dumas & Ninio (1982) first applied the filtering idea for fast sequence comparison
Lipman & Pearson (1985): FASTA
Altschul et al. (1990): BLAST

8. Pairwise Alignment Methods
Dotplot analysis method
Dynamic programming (DP) algorithm
Word / k-tuple methods

9. How dotplot analysis works

10. How dotplot analysis works
Direct Repeat
Reversed Repeat
Aligned
sequence

11. To reduce random noise in dotplots
Specify a window size, w
Examine w residues at a time
Count how many matches are among the w pairs of residues

20. Dotlet

21. A simple alignment problem.
Problem: find the best pairwise alignment of GAATC and CATAC.

22. Scoring alignments GAATC CATAC GAAT-C C-ATAC -GAAT-C C-A-TAC GAATC-
We need a way to measure the quality of a candidate alignment.
Alignment scores consist of two parts: a substition matrix, and a gap penalty.

23. Scoring aligned bases Purine A G Pyrimidine C T Transversion
(low score)
Transition
(high score)

24. Scoring aligned bases GAATC CATAC Purine A G Pyrimidine C T A C G T 10
Transversion
Transition
A hypothetical substitution matrix:
GAATC
CATAC A C G T 10 -5
= 5

25. Scoring aligned bases GAAT-C CA-TAC Purine A G Pyrimidine C T A C G T
Transversion
(expensive)
Transition
(cheap)
A hypothetical substitution matrix:
GAAT-C
CA-TAC A C G T 10 -5
? ? + 10 = ?

26. Scoring gaps Linear gap penalty: every gap receives a score of d.
Affine gap penalty: opening a gap receives a score of d; extending a gap receives a score of e.
GAAT-C d=-4
CA-TAC
= 17
G--AATC d=-4
CATA--C e=-1
= 5

27. A simple alignment problem.
Problem: find the best pairwise alignment of GAATC and CATAC.
Use a linear gap penalty of -4.
Use the following substitution matrix: A C G T 10 -5

28. How many possibilities?
GAATC
CATAC
GAAT-C
C-ATAC
-GAAT-C
C-A-TAC
GAATC-
CA-TAC
GAAT-C
CA-TAC
GA-ATC
CATA-C
How many different alignments of two sequences of length N exist?

29. How many possibilities?
GAATC
CATAC
GAAT-C
C-ATAC
-GAAT-C
C-A-TAC
GAATC-
CA-TAC
GAAT-C
CA-TAC
GA-ATC
CATA-C
How many different alignments of two sequences of length N exist?
Too many to enumerate!

30. -G-
CAT
DP matrix G A T C
The value in position (i,j) is the score of the best alignment of the first i positions of the first sequence versus the first j positions of the second sequence. -8

31. DP matrix -8 G A T C -12 -G-A CAT-
Moving horizontally in the matrix introduces a gap in the sequence along the left edge.

32. DP matrix -8 G A T C -12 -G-- CATA
Moving vertically in the matrix introduces a gap in the sequence along the top edge.

33. Initialization G A T C

34. G -
Introducing a gap G A T C -4

35. - C
DP matrix G A T C -4

36. DP matrix G A T C -4 -8

37. G C
DP matrix G A T C -4 -5

38. -----
CATAC
DP matrix G A T C -4 -8
-12
-16
-20 -5

39. DP matrix G A T C -4 -8
-12
-16
-20 -5 ?

40. DP matrix G A T C -4 -8 -12 -16 -20 -5 -G CA G- CA --G CA- -4 -9 -12-4 -8
-12
-16
-20 -5 -4 -4

41. DP matrix G A T C -4 -8
-12
-16
-20 -5 ?

42. DP matrix G A T C -4 -8
-12
-16
-20 -5

43. DP matrix G A T C -4 -8
-12
-16
-20 -5 ?

44. What is the alignment associated with this entry?
DP matrix G A T C -4 -8
-12
-16
-20 -5 -9 5 1 2 -2
What is the alignment associated with this entry?

45. DP matrix G A T C -4 -8
-12
-16
-20 -5 -9 5 1 2 -2
-G-A
CATA

46. Find the optimal alignment, and its score.
DP matrix G A T C -4 -8
-12
-16
-20 -5 -9 5 1 2 -2 ?
Find the optimal alignment, and its score.

47. DP matrix G A T C -4 -8 -12 -16 -20 -5 -9 -13 -6 5 1 -3 -7 11 7 2 6 -2-4 -8
-12
-16
-20 -5 -9
-13 -6 5 1 -3 -7 11 7 2 6 -2 17

48. DP matrix GA-ATC CATA-C G A T C -4 -8 -12 -16 -20 -5 -9 -13 -6 5 1 -3-4 -8
-12
-16
-20 -5 -9
-13 -6 5 1 -3 -7 11 7 2 6 -2 17

49. DP matrix GAAT-C CA-TAC G A T C -4 -8 -12 -16 -20 -5 -9 -13 -6 5 1 -3-4 -8
-12
-16
-20 -5 -9
-13 -6 5 1 -3 -7 11 7 2 6 -2 17

50. DP matrix GAAT-C C-ATAC G A T C -4 -8 -12 -16 -20 -5 -9 -13 -6 5 1 -3-4 -8
-12
-16
-20 -5 -9
-13 -6 5 1 -3 -7 11 7 2 6 -2 17

51. DP matrix GAAT-C -CATAC G A T C -4 -8 -12 -16 -20 -5 -9 -13 -6 5 1 -3-4 -8
-12
-16
-20 -5 -9
-13 -6 5 1 -3 -7 11 7 2 6 -2 17

52. Multiple solutions
GA-ATC
CATA-C
When a program returns a sequence alignment, it may not be the only best alignment.
GAAT-C
CA-TAC
GAAT-C
C-ATAC
GAAT-C
-CATAC

53. DP in equation form Align sequence x and y.
F is the DP matrix; s is the substitution matrix; d is the linear gap penalty.

54. DP in equation form

55. Dynamic programming Yes, it’s a weird name.
DP is closely related to recursion and to mathematical induction.
We can prove that the resulting score is optimal.

56. Dynamic Programming A method for solving recursive problem
Break a problem into smaller subproblems
Solve subproblems optimally, recursively
Use these optimal solutions to construct an optimal solution for the original problem

57. Summary
Scoring a pairwise alignment requires a substition matrix and gap penalties.
Dynamic programming is an efficient algorithm for finding the optimal alignment.
Entry (i,j) in the DP matrix stores the score of the best-scoring alignment up to those positions.
DP iteratively fills in the matrix using a simple mathematical rule.

58. A simple exercise A G C Find the optimal alignment of AAG and AGC.
Use a gap penalty of d=-5. A C G T 2 -7 -5 A G C

59. A simple exercise A G C Find the optimal alignment of AAG and AGC.
Use a gap penalty of d=-5. A C G T 2 -7 -5 A G C

60. A simple exercise A G -5 -10 -15 C
Find the optimal alignment of AAG and AGC.
Use a gap penalty of d=-5. A C G T 2 -7 -5 A G -5
-10
-15 C

61. A simple exercise A G -5 -10 -15 2 -3 -8 -1 C -6
Find the optimal alignment of AAG and AGC.
Use a gap penalty of d=-5. A C G T 2 -7 -5 A G -5
-10
-15 2 -3 -8 -1 C -6

62. Traceback
Start from the lower right corner and trace back to the upper left.
Each arrow introduces one character at the end of each aligned sequence.
A horizontal move puts a gap in the left sequence.
A vertical move puts a gap in the top sequence.
A diagonal move uses one character from each sequence.

63. A simple example A G -5 2 -3 -1 C -6
Find the optimal alignment of AAG and AGC.
Use a gap penalty of d=-5.
Start from the lower right corner and trace back to the upper left.
Each arrow introduces one character at the end of each aligned sequence.
A horizontal move puts a gap in the left sequence.
A vertical move puts a gap in the top sequence.
A diagonal move uses one character from each sequence. A G -5 2 -3 -1 C -6

64. A simple example A G -5 2 -3 -1 C -6
Find the optimal alignment of AAG and AGC.
Use a gap penalty of d=-5.
Start from the lower right corner and trace back to the upper left.
Each arrow introduces one character at the end of each aligned sequence.
A horizontal move puts a gap in the left sequence.
A vertical move puts a gap in the top sequence.
A diagonal move uses one character from each sequence. A G -5 2 -3 -1 C -6
AAG- AAG-
-AGC A-GC

65. Local alignment
A single-domain protein may be homologous to a region within a multi-domain protein.
Usually, an alignment that spans the complete length of both sequences is not required.

66. BLAST allows local alignments
Global alignment
Local alignment

67. Global alignment DP Align sequence x and y.
F is the DP matrix; s is the substitution matrix; d is the linear gap penalty.

68. Local alignment DP Align sequence x and y.
F is the DP matrix; s is the substitution matrix; d is the linear gap penalty.

69. Local DP in equation form

70. Local alignment Two differences with respect to global alignment:
No score is negative.
Traceback begins at the highest score in the matrix and continues until you reach 0.
Global alignment algorithm: Needleman-Wunsch.
Local alignment algorithm: Smith-Waterman.

71. A simple example
Find the optimal local alignment of AAG and AGC.
Use a gap penalty of d=-5. A C G T 2 -7 -5 A G C

72. A simple example
Find the optimal local alignment of AAG and AGC.
Use a gap penalty of d=-5. A C G T 2 -7 -5 A G C

73. A simple example
Find the optimal local alignment of AAG and AGC.
Use a gap penalty of d=-5. A C G T 2 -7 -5 A G 2 4 C

74. A simple example
Find the optimal local alignment of AAG and AGC.
Use a gap penalty of d=-5. A C G T 2 -7 -5 A G 2 4 C AG

75. Local alignment
Find the optimal local alignment of AAG and GAAGGC.
Use a gap penalty of d=-5. A C G T 2 -7 -5 A G C

76. Local alignment
Find the optimal local alignment of AAG and GAAGGC.
Use a gap penalty of d=-5. A C G T 2 -7 -5 A G 2 4 6 1 C

77. Substitution matrices
Find the optimal local alignment of AAG and GAAGGC.
Use a gap penalty of d=-5. A C G T 2 -7 -5 A G 2 4 6 1 C
Where did this substitution matrix come from?

78. Nucleic acid PAM matrices
PAM = point accepted mutation
1 PAM = 1% probability of mutation at each sequence position.
A uniform PAM1 matrix: A G T C
0.99

79. Transitions and transversions
Transitions (A  G or C  T) are more likely than transversions (A  T or G  C)
Assume that transitions are three times as likely: A G T C
0.99
0.006
0.002

80. Distant relatives
If the probability of a substitution is 2%, simply multiply the probabilities from 1% by themselves.
A PAM N matrix is computed by raising PAM 1 to the Nth power. A G T C

81. Better gap scoring Real gaps are often more than one letter long.
>gi|729942|sp|P40601|LIP1_PHOLU Lipase 1 precursor (Triacylglycerol lipase)
Length = 645
Score = 33.5 bits (75), Expect = 5.9
Identities = 32/180 (17%), Positives = 70/180 (38%), Gaps = 9/180 (5%)
Query: 2038 IYSLYGLYNVPYENLFVEAIASYSDNKIRSKSRRVIATTLETVGYQTANGKYKSESYTGQ 2097
+++ YGL+ Y Y D K +R N G+
Sbjct: 441 VFTAYGLWRY-YDKGWISGDLHYLDMKYEDITRGIVLNDW----LRKENASTSGHQWGGR 495
Query: 2098 LMAGYTYMMPENINLTPLAGLRYSTIKDKGYKETGTTYQNLTVKGKNYNTFDGLLGAKVS 2157
+ AG P KGY+E+G Y++ G LG ++
Sbjct: 496 ITAGWDIPLTSAVTTSPIIQYAWDKSYVKGYRESGNNSTAMHFGEQRYDSQVGTLGWRLD 555
Query: 2158 SNINVNEIVLTPELYAMVDYAFKNKVSAIDARLQGMTAPLPTNSFKQSKTSFDVGVGVTA 2217
+N P F +K I S KQ G+ A
Sbjct: 556 TNFG----YFNPYAEVRFNHQFGDKRYQIRSAINSTQTSFVSESQKQDTHWREYTIGMNA 611
Real gaps are often more than one letter long.

82. Affine gap penalty LETVGY W----L -5 -1 -1 -1
Separate penalties for gap opening and gap extension.
This requires modifying the DP algorithm to store three values in each box.

83. Summary Local alignment finds the best match between subsequences.
Smith-Waterman local alignment algorithm:
No score is negative.
Trace back from the largest score in the matrix.
Substitution matrices represent the probability of mutations.
Affine gap penalties include a large gap opening penalty and small gap extension penalty.