1. Sequence alignment with Needleman-Wunsch
Usman Roshan

2. Pairwise alignment X: ACA, Y: GACAT Match=8, mismatch=2, gap-5
ACA-- -ACA- --ACA ACA----
GACAT GACAT GACAT G—ACAT
Score =

3. Traceback
We can compute an alignment of DNA (or protein or RNA) sequences X and Y with a traceback matrix T.
Sequence X is aligned along the rows and Y along the columns.
Each entry of the matrix T contains D, L, or U specifying diagonal, left or upper

4. Traceback
X: ACA, Y=TACAG T A C G L U D

5. Traceback
X: ACA, Y=TACAG T A C G L U D

6. Traceback code aligned_seq1 = "" aligned_seq2 = "" i = len(seq1)
j = len(seq2)
while(i !=0 or j != 0):
if(T[i][j] == “L”):
aligned_seq1 = “-” + aligned_seq1
aligned_seq2 = seq2[j-1] + aligned_seq2
j = j - 1
elif(T[i][j] == "U"):
aligned_seq2 = "-" + aligned_seq2
aligned_seq1 = seq1[i-1] + aligned_seq1
i = i - 1
else:

7. Optimal alignment
An alignment can be specified by the traceback matrix.
How do we determine the traceback for the highest scoring alignment?
Needleman-Wunsch algorithm for global alignment
First proposed in 1970
Widely used in genomics/bioinformatics
Dynamic programming algorithm

8. Needleman-Wunsch (NW)
Input:
X = x1x2…xn, Y=y1y2…ym
(X is seq1 and Y is seq2)
Notation:
X1..i = x1x2…xi
Score(X1..i,Y1..j) = Optimal alignment score of sequences X1..i and Y1..j.
Suppose we know the optimal alignment scores of
X1…i-1 and Y1…j-1
X1…i and Y1...j-1
X1...i-1 and Y1…j

9. Needleman-Wunsch (NW)
Then the optimal alignment score of X1…i and Y1…j is the maximum of
Score(X1…i-1,Y1…j-1) + match/mismatch
Score(X1…i,Y1…j-1) + gap
Score(X1…i-1,Y1…j) + gap
We build on this observation to compute Score(Xn,Ym)

10. Needleman-Wunsch
Define V to be a two dimensional matrix with len(X)+1 rows and len(Y)+1 columns
Let V[i][j] be the score of the optimal alignment of X1…i and Y1…j.
Let m be the match cost, mm be mismatch, and g be the gap cost.

11. NW pseudocode Initialization:
for i = 1 to length of seq1 { V[i][0] = i*g; }
For i = 1 to length of seq2 { V[0][i] = i*g; }
Recurrence:
for i = 1 to length of seq1{
for j = 1 to length of seq2{
V[i-1][j-1] + m(or mm)
V[i][j] = max { V[i-1][j] + g
V[i][j-1] + g
if(maximum is V[i-1][j-1] + m(or mm)) then T[i][j] = ‘D’
else if (maximum is V[i-1][j] + g) then T[i][j] = ‘U’
else then T[i][j] = ‘L’ }

12. Example V G A C A T A C T Input: seq1: ACA seq2: GACAT m = 5 mm = -4
gap = -20
seq1 is lined along the rows
and seq2 is along the columns
G A C A T
-20
-40
-60
-80
-100 -4
-15
-35
-55
-75
-24 -8
-10
-30
-50
-44
-19
-12 -5
-25 A C T L U D