1. Alignment II Dynamic Programming

2. Pair-wise sequence alignments
Idea: Display one sequence above another with spaces inserted in both to reveal similarity
A: C A T - T C A - C
| | | | |
B: C - T C G C A G C

3. Two types of alignment S = CTGTCGCTGCACG T = TGCCGTG Global alignment
Local alignment
CTGTCG-CTGCACG
-TGC-CG-TG----
CTGTCGCTGCACG--
TGC-CGTG

4. Global alignment: Scoring
CTGTCG-CTGCACG
-TGC-CG-TG----
Reward for matches: 
Mismatch penalty: 
Space penalty: 
score(A) = w – x - y
w = #matches x = #mismatches y = #spaces

5. Global alignment: Scoring
Reward for matches: 10
Mismatch penalty: 2
Space penalty: 5
C T G T C G – C T G C
- T G C – C G – T G -
Total = 11

6. Optimum Alignment
The score of an alignment is a measure of its quality
Optimum alignment problem: Given a pair of sequences X and Y, find an alignment (global or local) with maximum score
The similarity between X and Y, denoted sim(X,Y), is the maximum score of an alignment of X and Y

7. Alignment algorithms Global: Needleman-Wunsch Local: Smith-Waterman
NW and SW use dynamic programming
Variations:
Gap penalty functions
Scoring matrices

8. Global Alignment: Algorithm

9. Theorem. C(i,j) satisfies the following relationships:
Initial conditions:
Recurrence relation: For 1  i  n, 1  j  m:

10. Justification S1 S2 . . . Si-1 Si T1 T2 . . . Tj-1 Tj
C(i-1,j-1) + w(Si,Tj)
C(i-1,j) 
S1 S Si —
T1 T Tj-1 Tj
C(i,j-1) 

11. Example Case 1: Line up Si with Tj i - 1 i S: C A T T C A C
T: C - T T C A G
j -1 j
Case 2: Line up Si with space
i - 1 i
S: C A T T C A - C
T: C - T T C A G - j
Case 3: Line up Tj with space i
S: C A T T C A C -
T: C - T T C A - G
j -1 j

12. Computation Procedure
C(i-1,j)
C(i-1,j-1)
C(i,j-1)
C(i,j)
C(n,m)

13. λ C T C G C A G C λ -5
-10
-15
-20
-25
-30
-35 10 5 C A T T C
We first compute T[i, j] for the smallest possible values of i and j, then for increasing values of i and j
Usually performed with a table of size
(n + 1) X (m + 1) A C
+10 for match, -2 for mismatch, -5 for space

14. λ C T C G C A G C λ -5
-10
-15
-20
-25
-30
-35
-40 10 5 8 3 -2 -7 15 13 -4 20 18 28 23 26 33 * C A T T C A C
Traceback can yield both optimum alignments

15. End-gap free alignment
Gaps at the start or end of alignment are not penalized
Match: +2 Mismatch and space: -1
Best global
Best end-gap free
Score = 1
Score = 9

16. Motivation: Shotgun assembly
Shotgun assembly produces large set of partially overlapping subsequences from many copies of one unknown DNA sequence.
Problem: Use the overlapping sections to ”paste” the subsequences together.
Overlapping pairs will have low global alignment score, but high end-space free score because of overlap.

17. Motivation: Shotgun assembly

18. Algorithm Same as global alignment, except:
Initialize with zeros (free gaps at start)
Locate max in the last row/column (free gaps at end)

19. λ C T C G C A G C λ C A T T C
We first compute T[i, j] for the smallest possible values of i and j, then for increasing values of i and j
Usually performed with a table of size
(n + 1) X (m + 1) A G
+10 for match, -2 for mismatch, -5 for gap

20. Local Alignment: Motivation
Ignoring stretches of non-coding DNA:
Non-coding regions are more likely to be subjected to mutations than coding regions.
Local alignment between two sequences is likely to be between two exons.
Locating protein domains:
Proteins of different kind and of different species often exhibit local similarities
Local similarities may indicate ”functional subunits”.

21. Local alignment: Example
S = g g t c t g a g
T = a a a c g a
Match: +2 Mismatch and space: -1
Best local alignment:
g g t c t g a g
a a a c – g a -
Score = 5

22. Local Alignment: Algorithm
C [i, j] = Score of optimally aligning a
suffix of s with a suffix of t.
Initialize top row and leftmost column to zero.

23. λ C T C G C A G C λ 1 2 C A T T C A C
+1 for a match, -1 for a mismatch, -5 for a space

24. Some Results
Most pairwise sequence alignment problems can be solved in O(mn) time.
Space requirement can be reduced to O(m+n), while keeping run-time fixed [Myers88].
Highly similar sequences can be aligned in O(dn) time, where d measures the distance between the sequences [Landau86].

25. Reducing space requirements
O(mn) tables are often the limiting factor in computing large alignments
There is a linear space technique that only doubles the time required [Hirschberg77]

26. λ C T C G C A G C λ C A T T C
We first compute T[i, j] for the smallest possible values of i and j, then for increasing values of i and j
Usually performed with a table of size
(n + 1) X (m + 1) A G
IDEA: We only need the previous row to calculate the next

27. Linear-space Alignments
mn + ½ mn + ¼ mn + 1/8 mn + 1/16 mn + … = 2 mn

28. Affine Gap Penalty Functions
Gap penalty = h + gk
where
k = length of gap
h = gap opening penalty
g = gap continuation penalty
Can also be solved in O(nm) time using dynamic programming