The Needleman Wunsch algorithm

Needleman-Wunsch

Dr Avril Coghlan
alc@sanger.ac.uk

Note: this talk contains animations which can only be seen by
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The Needleman-Wunsch algorithm
• Even for relatively short sequences, there are lots of
possible alignments
It will take you (or a computer) a long time to assess each alignment
one-by-one to find the best alignment
• The problem of finding the best possible alignment
for 2 sequences is solved by the Needleman-Wunsch
algorithm
The N-W algorithm was proposed by Christian Wunsch & Saul
Needleman, 1970
• The N-W algorithm is mathematically proven to find
the best alignment of 2 sequences
By the ‘best’ alignment, we mean the alignment that implies the fewest
number of mutations in the 2 sequences

The Needleman-Wunsch algorithm
• The Needleman-Wunsch algorithm saves us the
trouble of assessing all the many possible alignments
to find the best one
• The N-W algorithm takes time proportion to n2 to
find the best alignment of two sequences that are
both n letters long
In contrast, assessing all possible alignments one-by-one
2n
would take time proportional to (n )
2n
n2 is much smaller than ( n ), so N-W is much faster than
assessing all possible alignments one-by-one
eg. for n=11, n2=121, ( 2n )=705432, so N-W is ~5830-fold
n
faster (705432/121) than assessing all alignments

Problem
• How many times faster is it to find the best
alignment for sequences “RQQEP” & “QQESP” using
the Needleman-Wunsch algorithm, compared to
assessing each possible alignment one-by-one?

Answer
alignment for sequences “RQQEP” & “QQESP” using
the Needleman-Wunsch algorithm, compared to
assessing each possible alignment one-by-one?
The sequence length, n, is 5 here
This means it will take time proportional to n2=25 to find the best
alignment using N-W
It will take time proportional to ( 2n ) = 252 to find the best alignment
n
by assessing each possible alignment one-by-one
This means that we can find the best alignment about 10 times
(=252/25) faster by using N-W

Explanation of the N-W algorithm
• In the following explanation, we’ll refer to the ith
letter in sequence S1 as S1(i)
• Similarly, we’ll refer to the jth letter in sequence S2
as S2(j)
eg. for sequences ‘VIVADAVIS’ and ‘VIVALASVEGAS’:
i= 1 2 3 4 5 6 7 8 9 Sequence S1
V I V A D A V I S
j= 1 2 3 4 5 6 7 8 9 10 11 12
V I V A L A S V E G A S Sequence S2
For example, S1(5) = ‘D’, S2(3) = ‘V’

• To use N-W, we must first define:
1 A scoring function (σ): defines the score to give to a substitution
mutation eg. -1 for a match, -1 for mismatch
2 A gap penalty: defines the score to give to an insertion or
deletion mutation, eg. -1
A recurrence relation: defines what actions we repeat at each iteration
3 (step) of the algorithm; for N-W this is:
T(i-1, j-1) + σ(S1(i), S2(j))
T(i, j) = max T(i-1, j) + gap penalty This will be
T(i, j-1) + gap penalty explained later...
• There are 2 parts to computing the best alignment
using the N-W algorithm:
1 Fill up a matrix (table) T using the recurrence relation
2 The traceback step : use the filled-in matrix T to work out the best
alignment

Scoring functions
• We define a scoring function σ(S1(i), S2(j)) for pairs of
amino acids or nucleotides (S1(i), S2(j))
σ(S1(i), S2(j)) is the cost (score) of aligning symbols S1(i) & S2(j)
ie. σ(S1(i), S2(j)) is the cost (score) for a substitution mutation from
S1(i) → S2(j)
• A simple scoring function σ is a score of +1 for
matches, and -1 for mismatches
A
This can be written as: (the symbol means ‘for all’)
a=b
A
a≠b
A
σ(a,b) = +1 and σ(a,b) = -1
• A convenient way of representing many scoring
functions is a substitution matrix
This shows the cost (score) of aligning one letter (nucleotide or amino
acid) with another letter

• Substitution matrix for Letter b function that assigns
a scoring
+1 to matches, and D Cto mismatches: M F P S T W Y V
A R N -1 Q E G H I L K
A 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
σ(a,b) now refers to an entry in the substitution matrix
R -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

N -1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

D -1 -1 -1 1 -1 Letter -1
-1 -1 -1 b -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

C -1 -1 -1 -1 1 -1 -1
A-1 -1 -1
C
-1 -1 -1 -1
G-1 -1 -1
T
-1 -1 -1

Substitution
Letter a
Q -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
A +1 -1 -1-1 -1
matrix σ for DNA
E -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

alignments G -1 Letter a
-1 -1 -1 C -1 -1 -1 -1
1 -1 -1 +1 -1
-1 -1 -1 -1-1 -1 -1 -1
-1 -1 -1

H -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
G -1 -1 +1 -1
I -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
T -1 -1 -1-1 +1
Substitution L -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1

matrix σ for K -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1

protein M -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1

alignments F -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1

P -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1

S -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1

T -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1

W -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1

Y -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1

V -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1

N-W: Initialising table T
• To align 2 sequences S1 & S2 of lengths m & n, N-W starts by building a
table T with m+1 columns & n+1 rows:
eg. for S1=‘TGGTG’ & S2=‘ATCGT’, m=5 and n=5:
T G G T G

A
T
Table T
C
G
T i=0 i=1 i=2 i=3 i=4 i=5
T G G T G
• We number the columns i=0,1,2,....m j=0
We number the rows j=0,1,2,...n j=1 A
j=2 T
j=3 C
j=4 G
j=5 T

• T(i, j) is the cell at the intersection of column i and row j
eg. for S1=‘TGGTG’ & S2=‘ATCGT’, m=5 and n=5:
i=0 i=1 i=2 i=3 i=4 i=5
T G G T G
j=0
j=1 A
Table T j=2 T T(3,2)
j=3 C
j=4 G
j=5 T
• The N-W algorithm starts by initialising (setting the initial value of) T(0,0) to
zero:
T G G T G
0
A
T
C
G
T

• The table T is then filled in using the recurrence relation:
T(i-1, j-1) + σ(S1(i), S2(j)) This will be explained
T(i, j) = max T(i-1, j) + gap penalty in a minute...
T(i, j-1) + gap penalty
• The table is filled in from left to right, and from top to bottom
• The value of T(0,0) is set to zero at the start (initialised to 0)
• We first calculate the values of T(i, j) for row 0 of the table, from left to
right
• We then calculate the values of T(i, j) for row 1 of the table from left to
right, then rows 2, 3, 4 .... row n of the table
T
T G
G G
G T
T G
G
0 x x x x x
A
A x x x x x x
T
T
T x x x x x x
C
C
C
C x x x x x x
G
G
G
G
G x x x x x x
T
T
T
T
T x x x x x x

T(i-1, j-1) + σ(S1(i), S2(j)) 1
T(i, j) = max T(i-1, j) + gap penalty 2
T(i, j-1) + gap penalty 3
• This means that the value in cell T(i, j) is set to be the maximum of the
three possibilities 1 , 2 , 3 , where:
T(i-1, j-1) is the value in the previous column & row
T(i-1, j) is the value in the previous column & same row
T(i, j-1) is the value in the same column & previous row

i=0 i=1 i=2 i=3 i=4 i=5
T G G T G
j=0 T(i, j-1) = T(3,1)
j=1 A T(i-1, j-1) = T(2,1)
Table T j=2 T T(i, j) = T(3,2)
j=3 C T(i-1, j) = T(2,2)
j=4 G
j=5 T

T(i-1, j-1) + σ(S1(i), S2(j)) 1
T(i, j) = max T(i-1, j) + gap penalty 2
T(i, j-1) + gap penalty 3
• This means that the value in cell T(i, j) is set to be the maximum of these
three possibilities 1 , 2 , 3 , where
gap penalty is score that we have decided to use for an insertion or
deletion mutation, for example -1
σ(S1(i), S2(j)) is the cost (score) that we have decided to use for
aligning symbols S1(i) & S2(j), in our substitution matrix σ
eg. using +1 for matches and -1 for mismatches:
Letter b
A C G T
Substitution A +1 -1 -1 -1
matrix σ for
Letter a

C -1 +1 -1 -1
DNA G -1 -1 +1 -1
alignments T -1 -1 -1 +1

• For example, say we decide to use +1 for matches, -1 for mismatches, and
-2 for an insertion/deletion (gap)
• The N-W algorithm starts by initialising (setting the initial value of) T(0,0)
to zero
• We next calculate the value of T(1, 0)
• The value of T(1,0) is set to the maximum of 3 possibilities:
T(i-1, j-1) + σ(S1(i), S2(j)) Not defined here
T(i, j) = max T(i-1, j) + gap penalty = 0 – 2 = -2
T(i, j-1) + gap penalty Not defined here
• We calculate this to be -2, so set T(1, 0) to -2
• We record which previous cell was used to set the value of T(1, 0) :

T G G T G
0 -2
?
A
T
C
G
T

T(i, j) = max T(i-1, j) + gap penalty = -2 – 2 = -4
Not defined here

T G G T G
0 -2 -4
?
A
T
C
G
T

T(i, j) = max T(i-1, j) + gap penalty
= -4 – 2 = -6
Not defined here

T G G T G
0 -2 -4 -6
?
A
T
C
G
T

T(i, j) = max T(i-1, j) + gap penalty
= -6 – 2 = -8
Not defined here

T G G T G
0 -2 -4 -6 -8
?
A
T
C
G
T

T(i, j) = max T(i-1, j) + gap penalty = -8 – 2 = -10
T(i, j-1) + gap penalty Not defined here

T G G T G
0 -2 -4 -6 -8 -10
?
A
T
C
G
T

T(i, j) = max T(i-1, j) + gap penalty Not defined here
T(i, j-1) + gap penalty = 0 – 2 = -2

T G G T G
-10
0 -2 -4 -6 -8
A -2
?
T
T
C
C
G
G
T
T

T(i-1, j-1) + σ(S1(i), S2(j)) = 0 – 1 = -1
T(i, j) = max T(i-1, j) + gap penalty = -2 -2 = -4
T(i, j-1) + gap penalty = -2 -2 = -4

T G G T G
-10
0 -2 -4 -6 -8
A -2 -1
?
T
C
G
T

T(i-1, j-1) + σ(S1(i), S2(j)) = -2 -1 = -3
T(i, j-1) + gap penalty = -4 -2 = -6
• We record which previous cells were used to set the value of T(2, 1) (two
different cells here):

T G G T G
-10
0 -2 -4 -6 -8
A -2 -1 -3
?
T
C
G
T

T(i-1, j-1) + σ(S1(i), S2(j)) = -4 -1 = -5
T(i, j-1) + gap penalty = -6 -2 = -8

T G G T G
-10
0 -2 -4 -6 -8
A -2 -1 -3 -5
?
T
C
G
T

T(i-1, j-1) + σ(S1(i), S2(j)) = -6 -1 = -7
T(i, j-1) + gap penalty = -8 -2 = -10

T G G T G
-10
0 -2 -4 -6 -8
A -2 -1 -3 -5 -7
?
T
C
G
T

T(i-1, j-1) + σ(S1(i), S2(j)) = -8 -1 = -9
T(i, j-1) + gap penalty = -10 -2 = -12

T G G T G
-10
0 -2 -4 -6 -8
A -2 -1 -3 -5 -7 -9
?
T
C
G
T

Problem
• Fill in the next row of matrix T
T G G T G
-10
0 -2 -4 -6 -8
A -2 -1 -3 -5 -7 -9
T ? ? ? ? ? ?
C
G
T

Answer
• Fill in the next row of matrix T
T G G T G
0 -2 -4 -6 -8 -10

A -2 -1 -3 -5 -7 -9
T -4 -1 -2 -4 -4 -6
? ? ? ? ?
C
G
T

N-W: the traceback step
• When we have filled in the whole of matrix T, it looks like:
T G G T G
0 -2 -4 -6 -8 -10

A -2 -1 -3 -5 -7 -9
T -4 -1 -2 -4 -4 -6
C -6 -3 -2 -3 -5 -5
G -8 -5 -2 -1 -3 -4
T -10
-7 -4 -3 0 -2

• In the traceback step we use the filled-in matrix T to work out the best
alignment between the two sequences S1 & S2
• We start at the bottom right cell of matrix T
• We then follow the arrow to the previous cell used to calculate the best
value for that cell
• From there, follow the arrow to the previous cell... and so on..

• The path through matrix T is the traceback (in pink here):
sequence S1
T G G T G
-10
0 -2 -4 -6 -8
A -2 -1 -3 -5 -7 -9 - T G G T G
sequence S2

T -4 -1 -2 -4 -4 -6 | | |
C -6 -3 -2 -3 -5 -5 A T C G T -
G -8 -5 -2 -1 -3 -4
T -10
-7 -4 -3 0 -2

• To work out the best alignment, follow the traceback from top left to
bottom right, & look at the letters aligned in each cell
• Here the 1st cell doesn’t correspond to any letter
• The 2nd cell is ‘A’ in sequence S2 but nothing in sequence S1
• The 3rd cell is ‘T’ in sequence S2 and ‘T’ in sequence S1
• The 4th cell is ‘C’ in sequence S2 and ‘G’ in sequence S1
• The 5th cell is ‘G’ in sequence S2 and ‘G’ in sequence S1
• The 6th cell is ‘T’ in sequence S2 and ‘T’ in sequence S1
• The 7th cell is nothing in sequence S2 and ‘G’ in sequence S1

Problem
• The traceback is shown in pink in the matrix T
below. What is the best alignment?
A C C T
x x x x x
C x x x x x
T x x x x x
G x x x x x

Answer
• The traceback is shown in pink in the matrix T
below. What is the best alignment?
A C C T
x x x x x
C x x x x x
T x x x x x
G x x x x x

• It is: A C C T -
| |
- C - T G

• The Needleman-Wunsch algorithm uses an approach
called dynamic programming (d.p.)
d.p. algorithms solve problems by breaking a large problem into smaller
easy problems of a similar type
The N-W algorithm works by progressively building optimal
alignments of longer and longer subsequences of S1 & S2

• N-W finds the best alignment between 2 sequences
by iteratively (repeatedly):
i. taking the 1st i letters of sequence S1 and the 1st j letters of
sequence S2, for a particular i and j
ii. get the score of the best alignment of the 2 subsequences
This is what we are doing when we are filling matrix T
If S1 is m letters long, & S2 is n letters long, we need to do this for all m×n
possible subsequences of S1 and S2
So N-W takes time proportional to m×n to run (or n2, if m=n)

• During the N-W algorithm we assign scores to
alignments of subsequences of S1 and S2
We store the score for an alignment of the 1st i letters of S1 to the first j
letters of S2 in cell T(i, j)
So, after filling T, the bottom right cell will contain the score for the
best alignment between S1 and S2
This is just the sum of the scores for the matches, mismatches and
gaps in the best alignment:
eg. the best alignment of ‘TGGTG’ and ‘ATCGT’ using a score of +1 for a
match, -1 for a mismatch and -2 for a gap:
T G G T G
0 -2 -4 -6 -8 -10 The best alignment has:
• 3 matches (score +3)
A -2 -1 -3 -5 -7 -9
- T G G T G • 1 mismatch (score -1)
T -4 -1 -2 -4 -4 -6 | | | • 2 gaps (score -4)
C -6 -3 -2 -3 -5 -5 A T C G T - → Score = 3-1-4 = -2
G -8 -5 -2 -1 -3 -4
T -10
-7 -4 -3 0 -2

Software for making alignments
• For Needleman-Wunsch pairwise alignment
pairwiseAlignment() in the “Biostrings” R library
the EMBOSS (emboss.sourceforge.net/) needle program

Problem
alignment for sequences “RQQEPVRSTC” &
“QQESGPVRST” using the Needleman-Wunsch
algorithm, compared to assessing each possible
alignment one-by-one?

Answer
alignment for sequences “RQQEPVRSTC” &
“QQESGPVRST” using the N-W algorithm, compared
to assessing each possible alignment one-by-one?
The sequence length, n, is 10 here
This means it will take time proportional to n2=100 to find the best
alignment using N-W
2n
It will take time proportional to ( n ) = 184,756 to find the best
alignment by assessing each possible alignment one-by-one
We can find the best alignment about 1848 times (=184756/100)
faster by using N-W

Problem
• Find the best alignment between the sequences
“WHAT” and “WHY”, using the Needleman-Wunsch
algorithm, with +1 for a match, -1 for a mismatch,
and -2 for a gap.

Answer
• Find the best alignment between “WHAT” & “WHY”
using N-W with match:+1, mismatch:-1, gap:-2
• Matrix T looks like this, giving 2 possible tracebacks:
W H A T W H A T W H A T
0 -2 -4 -6 -8 0 -2 -4 -6 -8 0 -2 -4 -6 -8
W -2 1 -1 -3 -5 W -2 1 -1 -3 -5 W -2 1 -1 -3 -5
H -4 -1 2 0 -2 H -4 -1 2 0 -2 H -4 -1 2 0 -2
Y -6 -3 0 1 -1 Y -6 -3 0 1 -1 Y -6 -3 0 1 -1

• The two possible tracebacks give two equally good best alignments:
W H A T W H A T
| | | |
W H - Y W H Y -
(Pink traceback) (Orange traceback)

Further Reading
• Chapter 3 in Introduction to Computational Genomics Cristianini & Hahn
• Chapter 6 in Deonier et al Computational Genome Analysis
• Practical on pairwise alignment in R in the Little Book of R for
Bioinformatics:
https://a-little-book-of-r-for-
bioinformatics.readthedocs.org/en/latest/src/chapter4.html

The Needleman Wunsch algorithm

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