1. Sequence comparison: Dynamic programming
Genome 559: Introduction to Statistical and Computational Genomics
Prof. James H. Thomas

2. Sequence comparison overview
Problem: Find the “best” alignment between a query sequence and a target sequence.
To solve this problem, we need
a method for scoring alignments, and
an algorithm for finding the alignment with the best score.
The alignment score is calculated using
a substitution matrix
gap penalties.
The algorithm for finding the best alignment is dynamic programming (DP).

3. 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

4. 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 possible alignments of two sequences of length n exist?

5. 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?
x102
x105
x1011
x1017
x1023
FYI for two sequences of length m and n, possible alignments number:
2n choose n - the binomial coefficient

6. DP matrix 5 G A T C GA CA j 0 1 2 3 etc. i 1 2 3 4 510 -5 5
The value at (i,j) is the score of the best alignment of the first i characters of one sequence versus the first j characters of the other sequence. GA CA
init. row and column j
etc. G A T C i 1 2 3 4 5

7. A C G T 10 -5
DP matrix
GAA
CA- G A T C 5 1
Moving horizontally in the matrix introduces a gap in the sequence along the left edge.

8. A C G T 10 -5
GA-
CAT
DP matrix G A T C 5 1
Moving vertically in the matrix introduces a gap in the sequence along the top edge.

9. Moving diagonally in the matrix aligns two residuesC G T 10 -5
GAA
CAT
DP matrix G A T C 5
Moving diagonally in the matrix aligns two residues

10. Initialization G A T C Start at top left and move progressively A C G10 -5
Initialization
Start at top left and move progressively G A T C

11. A C G T 10 -5 G -
Introducing a gap G A T C -4

12. A C G T 10 -5 - C
Introducing a gap G A T C -4

13. Complete first row and columnG T 10 -5
Complete first row and column
-----
CATAC G A T C -4 -8
-12
-16
-20

14. Three ways to get to i=1, j=1C G T 10 -5
Three ways to get to i=1, j=1 G- -C j
etc. G A T C -4 -8 i 1 2 3 4 5

15. Three ways to get to i=1, j=1C G T 10 -5 -G C- j
etc. G A T C -4 -8 i 1 2 3 4 5

16. Three ways to get to i=1, j=1C G T 10 -5 G C j
etc. G A T C -5 i 1 2 3 4 5

17. Accept the highest scoring of the three10 -5
Accept the highest scoring of the three G A T C -4 -8
-12
-16
-20 -5
Then simply repeat the same rule progressively across the matrix

18. A C G T 10 -5
DP matrix G A T C -4 -8
-12
-16
-20 -5 ?

19. DP matrix G A T C -4 -8 -12 -16 -20 -5 ? -G CA G- CA --G CA- -4 -9 -1210 -5 -G CA G- CA
--G
CA-
DP matrix -4 -9
-12 G A T C -4 -8
-12
-16
-20 -5 ? -4 -4

20. DP matrix G A T C -4 -8 -12 -16 -20 -5 -G CA G- CA --G CA- -4 -9 -1210 -5 -G CA G- CA
--G
CA-
DP matrix -4 -9
-12 G A T C -4 -8
-12
-16
-20 -5 -4 -4

21. A C G T 10 -5
DP matrix G A T C -4 -8
-12
-16
-20 -5 ?

22. A C G T 10 -5
DP matrix G A T C -4 -8
-12
-16
-20 -5

23. A C G T 10 -5
DP matrix G A T C -4 -8
-12
-16
-20 -5 ?

24. What is the alignment associated with this entry?10 -5
Traceback G A T C -4 -8
-12
-16
-20 -5 -9 5 1 2 -2
What is the alignment associated with this entry?
(just follow the arrows back - this is called the traceback)

25. DP matrix G A T C -4 -8 -12 -16 -20 -5 -9 5 1 2 -2 -G-A CATA A C G T10 -5
DP matrix G A T C -4 -8
-12
-16
-20 -5 -9 5 1 2 -2
-G-A
CATA

26. Continue and find the optimal global alignment, and its score.10 -5
DP matrix G A T C -4 -8
-12
-16
-20 -5 -9 5 1 2 -2 ?
Continue and find the optimal global alignment, and its score.

27. A C G T 10 -5
Best alignment starts at bottom right and follows traceback arrows to top left G A T C -4 -8
-12
-16
-20 -5 -9
-13 -6 5 1 -3 -7 11 7 2 6 -2 17

28. One best traceback GA-ATC CATA-C G A T C -4 -8 -12 -16 -20 -5 -9 -1310 -5
GA-ATC
CATA-C
One best traceback G A T C -4 -8
-12
-16
-20 -5 -9
-13 -6 5 1 -3 -7 11 7 2 6 -2 17

29. Another best tracebackG T 10 -5
GAAT-C
-CATAC
Another best traceback G A T C -4 -8
-12
-16
-20 -5 -9
-13 -6 5 1 -3 -7 11 7 2 6 -2 17

30. GAAT-C -CATAC GA-ATC CATA-C G A T C -4 -8 -12 -16 -20 -5 -9 -13 -6 5 110 -5
GAAT-C
-CATAC
GA-ATC
CATA-C G A T C -4 -8
-12
-16
-20 -5 -9
-13 -6 5 1 -3 -7 11 7 2 6 -2 17

31. Multiple solutions GA-ATC
CATA-C
When a program returns a single sequence alignment, it may not be the only best alignment but it is guaranteed to be one of them.
In our example, all of the alignments at the left have equal scores.
GAAT-C
CA-TAC
GAAT-C
C-ATAC
GAAT-C
-CATAC

32. 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.

33. DP equation graphically
take the max of these three

34. 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.

35. What you should know
Scoring a pairwise alignment requires a substitution matrix and gap penalties.
Dynamic programming (DP) is an efficient algorithm for finding an optimal alignment.
Entry (i,j) in the DP matrix stores the score of the best-scoring alignment up to that position.
DP iteratively fills in the matrix using a simple mathematical rule.
How to use DP to find an alignment.

36. Practice problem: find a best pairwise alignment of GAATC and AATTC10 -5 G A T C
d = -4