1. Sequence Alignment Using Dynamic Programming
Saurabh Sinha

2. Dynamic Programming Is not a type of programming language
Is a type of algorithm, used to solve many different computational problems
Sequence Alignment is one of these problems
We will see the algorithm in its general sense first

3. Manhattan Tourist Problem1 2 5
source 5 3 10 5 2 1 5 3 5 3 1 2 3 4 5 2
sink
Find most weighted path from source to sink.

4. Manhattan Tourist Problem1 2 5
source 1 3 5 3 10 5 2 1 5 13 3 5 3 1 2 3 4 16 20 5 2
sink 22

5. MTP: Greedy Algorithm Is Not Optimal1 2 5
source 22 5 3 10 5 2 1 5 3 5 3 1 2 3 4
promising start, but leads to bad choices! 5 2
sink 18

6. MTP: Dynamic Programmingj 1
source 1 1 i
S0,1 = 1 5 1 5
S1,0 = 5
Calculate optimal path score for each vertex in the graph
Each vertex’s score is the maximum of the prior vertices score plus the weight of the respective edge in between

7. MTP: Dynamic Programming (cont’d)j 1 2
source 1 2 1 3 i
S0,2 = 3 5 3 -5 1 5 4
S1,1 = 4 3 2 8
S2,0 = 8

8. MTP: Dynamic Programming (cont’d)j 1 2 3
source 1 2 5 1 3 8 i
S3,0 = 8 5 3 10 -5 1 1 5 4 13
S1,2 = 13 3 5 -5 2 8 9
S2,1 = 9 3 8
S3,0 = 8

9. MTP: Dynamic Programming (cont’d)j 1 2 3
source 1 2 5 1 3 8 i 5 3 10 -5 -5 1 -5 1 5 4 13 8
S1,3 = 8 3 5 -3 -5 3 2 8 9 12
S2,2 = 12 3 8 9
S3,1 = 9

10. MTP: Dynamic Programming (cont’d)j 1 2 3
source 1 2 5 1 3 8 i 5 3 10 -5 -5 1 -5 1 5 4 13 8 3 5 -3 2 -5 3 3 2 8 9 12 15
S2,3 = 15 -5 3 8 9 9
S3,2 = 9

11. MTP: Dynamic Programming (cont’d)j 1 2 3
source 1 2 5 1 3 8
Almost Done i 5 3 10 -5 -5 1 -5 1 5 4 13 8 3 5 -3 2 -5 3 3 2 8 9 12 15 -5 1 3 8 9 9 16
S3,3 = 16

12. MTP: Dynamic Programming (cont’d)j 1 2 3
source 1 2 5 1 3 8
Done! i 5 3 10 -5 -5 1 -5 1 5 4 13 8 3 5 -3 2 -5 3 3 2 8 9 12 15 -5 1 3 8 9 9 16
S3,3 = 16

13. MTP Dynamic Programming: Formal Description
Computing the score for a point (i,j) by the recurrence relation:
si, j =
max
si-1, j + weight of the edge between (i-1, j) and (i, j)
si, j-1 + weight of the edge between (i, j-1) and (i, j)

14. Applying Dynamic Programming to Sequence Alignment

15. Representing alignments
Alignment : 2 x k matrix ( k  m, n )
V = ACCTGGTAAA
n = 10 8 2 1
matches
mismatches
deletions
insertions
W = ACATGCGTATA
m = 11 V A C T G W

16. Scoring functions A simple scoring function:
if in an alignment there are nm matches, nmis substitutions and ng gaps, the alignment score is
where wm , wmis ,wg represent match score, mismatch score and gap score (penalty) respectively

17. Sequence Alignment as a MTP-like problem

18. Sequence Alignment as a MTP-like problem
Match = 20
Mismatch = -10
Gap = -20
Score of path = 8 matches + 2 mismatches + 1 gap = 130

19. What alignment is this? V W A C T G A C T G Match = 20 Mismatch = -10
Gap = -20
Score of path = 5 matches + 2 mismatches + 7 gaps = -60

20. Sequence Alignment, formally
Find the best alignment between two strings under a given scoring scheme
Input : Strings v and w and a scoring schema
Output : Alignment of maximum score
Dynamic programming recurrence:
si-1,j-1 + score (vi, wj)
si,j = max si-1,j + gapscore
si,j-1 + gapscore {

21. Sequence Alignment: Example
Calculate and show the Dynamic Programming matrix and an optimal alignment for the DNA sequences GCTTAGC and GCATTGC, scoring +3 for a match, -2 for a mismatch, and -3 for a gap