The document describes algorithms for sequence alignment, including the Needleman-Wunsch algorithm for global alignment and the Smith-Waterman algorithm for local alignment. It provides examples of filling in a scoring matrix according to the algorithms' recurrence relations and tracing back to find the optimal alignment between two sequences.

1. Sequence Alignment

2. Where we left off
• Output from Sequencing:
AACTGACCTA…
CCGTTGGCAT…
TTTGCGGTCA…
…
…

3. Where we left off
• I have hundreds of millions of short
• Most applications need to figure out where they
came from
• Given a reference genome sequence and millions of
short sequences, how do I figure out where each of
the short sequences came from?

4. What is an algorithm?
• A process or set of rules to be followed in
calculations or other problem-solving operations,
esp. by a computer
• X = 2 * n
• Make a burrito
• Unwrap burrito
• Place on plate in microwave
• Turn microwave on for 2 minutes

5. What makes a good algorithmic
solution?
• Speed
• Memory
• Optimality of answer

6. The alignment problem
• Input
• Two sequences s and t of length n and m
• Output
• An alignment between the two sequences with gaps
inserted appropriately
• Objective Function
• A scoring function that weights particular character to
character alignments

7. How fast do I have to eat?
• Input
• # of sandwiches
• Output
• Sandwiches / minute
• Objective function
• Minimize the number of sandwiches I have to eat per
minute such that I finish all sandwiches in an hour

8. Scoring Function
• Which alignment of ACCTG and ACTTG is better?
• AATAC AATA-C
• ATATC -ATATC
• How did you decide?
• Example scoring function:
• +1 for matches
• -1 for gaps and mismatches

9. Kinds of Alignment
http://3.bp.blogspot.com/_OcdkdnkXwIg/SnRLqde_8PI/AAAAAAAAAZs/_ETauY69JiM/s320/glob
al-local-alignment.png

10. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
T
T
G
C

11. Rules
• Fill in top row and left-most column according to
scoring function
• Start in the upper left-most corner of unfilled squares
• Move left to right filling in the result of the scoring
function
• Break ties arbitrarily
• Trace back from bottom right corner to upper left
corner

12. Needleman-Wunsch Recurrence
𝑆𝑆 𝑖𝑖, 𝑗𝑗 = max �
𝑆𝑆 𝑖𝑖 − 1, 𝑗𝑗 − 1 + 𝛿𝛿 𝑠𝑠𝑖𝑖, 𝑡𝑡𝑗𝑗
𝑆𝑆 𝑖𝑖 − 1, 𝑗𝑗 + 𝛿𝛿 −
𝑆𝑆 𝑖𝑖, 𝑗𝑗 − 1 + 𝛿𝛿 −

13. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
T
T
G
C

14. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0
T
T
G
C

15. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1
T
T
G
C

16. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2
T
T
G
C

17. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3
T
T
G
C

18. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T
T
G
C

19. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1
T
G
C

20. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1
T -2
G
C

21. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1
T -2
G -3
C

22. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1
T -2
G -3
C -4

23. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1
T -2
G -3
C -4

24. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0
T -2
G -3
C -4

25. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1
T -2
G -3
C -4

26. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2
G -3
C -4

27. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2
G -3
C -4

28. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0
G -3
C -4

29. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1
G -3
C -4

30. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3
C -4

31. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3
C -4

32. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1
C -4

33. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0
C -4

34. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0 2
C -4

35. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0 2
C -4 -4

36. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0 2
C -4 -4 -2

37. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0 2
C -4 -4 -2 -1

38. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0 2
C -4 -4 -2 -1 1

39. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0 2
C -4 -4 -2 -1 1

40. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0 2
C -4 -4 -2 -1 1
-
C

41. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0 2
C -4 -4 -2 -1 1
G -
G C

42. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0 2
C -4 -4 -2 -1 1
T G -
T G C

43. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0 2
C -4 -4 -2 -1 1
T T G -
T T G C

44. Global Alignment:
Needleman-Wunsch Algorithm
A T T G
0 -1 -2 -3 -4
T -1 -1 0 -1 -2
T -2 -2 0 1 0
G -3 -3 -1 0 2
C -4 -4 -2 -1 1
A T T G -
- T T G C

45. Rules
• Fill in top row and left-most column according to
scoring function
• Start in the upper left-most corner of unfilled squares
• Move left to right filling in the result of the scoring
function
• Break ties arbitrarily
• Trace back from the max element in the matrix to the
first STOP

46. Rules
• Fill in top row and left-most column according to
scoring function
• Start in the upper left-most corner of unfilled squares
• Move left to right filling in the result of the scoring
function
• Break ties arbitrarily
• Trace back from the max element in the matrix to the
first STOP

47. Local Alignment:
Smith-Waterman Algorithm
𝑆𝑆 𝑖𝑖, 𝑗𝑗 = max
0
𝑆𝑆 𝑖𝑖 − 1, 𝑗𝑗 − 1 + 𝛿𝛿 𝑠𝑠𝑖𝑖, 𝑡𝑡𝑗𝑗
𝑆𝑆 𝑖𝑖 − 1, 𝑗𝑗 + 𝛿𝛿 −
𝑆𝑆 𝑖𝑖, 𝑗𝑗 − 1 + 𝛿𝛿 −

48. Local Alignment:
Smith-Waterman Algorithm
𝑆𝑆 𝑖𝑖, 𝑗𝑗 = max
0
𝑆𝑆 𝑖𝑖 − 1, 𝑗𝑗 − 1 + 𝛿𝛿 𝑠𝑠𝑖𝑖, 𝑡𝑡𝑗𝑗
𝑆𝑆 𝑖𝑖 − 1, 𝑗𝑗 + 𝛿𝛿 −
𝑆𝑆 𝑖𝑖, 𝑗𝑗 − 1 + 𝛿𝛿 −

49. Local Alignment:
Smith-Waterman Algorithm
A T T G
T
T
G
C

50. Local Alignment:
Smith-Waterman Algorithm
A T T G
0
T
T
G
C

51. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0
T
T
G
C

52. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0
T
T
G
C

53. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0
T
T
G
C

54. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T
T
G
C

55. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0
T
G
C

56. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0
T 0
G
C

57. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0
T 0
G 0
C

58. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0
T 0
G 0
C 0

59. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0
T 0
G 0
C 0

60. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1
T 0
G 0
C 0

61. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1
T 0
G 0
C 0

62. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0
G 0
C 0

63. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0
G 0
C 0

64. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1
G 0
C 0

65. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2
G 0
C 0

66. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0
C 0

67. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0
C 0

68. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0 0
C 0

69. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0 0 1
C 0

70. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0 0 1 3
C 0

71. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0 0 1 3
C 0 0

72. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0 0 1 3
C 0 0 0

73. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0 0 1 3
C 0 0 0 0

74. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0 0 1 3
C 0 0 0 0 2

75. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0 0 1 3
C 0 0 0 0 2

76. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0 0 1 3
C 0 0 0 0 2
G
G

77. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0 0 1 3
C 0 0 0 0 2
T G
T G

78. Local Alignment:
Smith-Waterman Algorithm
A T T G
0 0 0 0 0
T 0 0 1 1 0
T 0 0 1 2 1
G 0 0 0 1 3
C 0 0 0 0 2
T T G
T T G

79. How to evaluate statistical
significance?
• Everyone pick a number between 1 and 10 (keep it
to yourself!)

80. The problem with databases
• Query is: ACCT
• Is a match significant?
• Database A:
• ACCT
• CAGG
• AAAA
• Database B:
• ACCT
• ACCT
• ACCT

81. Alignment Projects
• Research BWA
• Research Bowtie
• Research MAQ
• Code a program in the language of your choice that
performs Needleman-Wunsch or Smith Waterman