Figure 1: Needleman-Wunsch pairwise sequence alignment
Class	Sequence alignment
Worst-case performance	${\displaystyle O(mn)}$
Worst-case space complexity	${\displaystyle O(mn)}$

The Needleman–Wunsch algorithm is an algorithm used in bioinformatics to align protein or nucleotide sequences. It was one of the first applications of dynamic programming to compare biological sequences. The algorithm was developed by Saul B. Needleman and Christian D. Wunsch and published in 1970.^[1] The algorithm essentially divides a large problem (e.g. the full sequence) into a series of smaller problems, and it uses the solutions to the smaller problems to find an optimal solution to the larger problem.^[2][3] It is also sometimes referred to as the optimal matching algorithm and the global alignment technique. The Needleman–Wunsch algorithm is still widely used for optimal global alignment, particularly when the quality of the global alignment is of the utmost importance. The algorithm assigns a score to every possible alignment, and the purpose of the algorithm is to find all possible alignments having the highest score.

Introduction

This algorithm can be used for any two strings. This guide will use two small DNA sequences as examples as shown in Figure 1:

GCATGCG
GATTACA

+−++−−+− −> 1*4 + (−1)*4 = 0

G → CG → GCG → -GCG → T-GCG → AT-GCG → CAT-GCG → GCAT-GCG
A → CA → ACA → TACA → TTACA → ATTACA → -ATTACA → G-ATTACA
             ↓
    (branch) → TGCG → -TGCG → ...
             → TACA → TTACA → ...

Scoring systems

GAAAAAAT
G--A-A-T

GAAAAAAT
GAA----T

Advanced presentation of algorithm

Scores for aligned characters are specified by a similarity matrix. Here, S(a, b) is the similarity of characters a and b. It uses a linear gap penalty, here called d.

For example, if the similarity matrix was

then the alignment:

AGACTAGTTAC
CGA---GACGT

with a gap penalty of −5, would have the following score:

S(A,C) + S(G,G) + S(A,A) + (3 × d) + S(G,G) + S(T,A) + S(T,C) + S(A,G) + S(C,T)

= −3 + 7 + 10 − (3 × 5) + 7 + (−4) + 0 + (−1) + 0 = 1

To find the alignment with the highest score, a two-dimensional array (or matrix) F is allocated. The entry in row i and column j is denoted here by ${\displaystyle F_{ij}}$. There is one row for each character in sequence A, and one column for each character in sequence B. Thus, if aligning sequences of sizes n and m, the amount of memory used is in ${\displaystyle O(nm)}$. Hirschberg's algorithm only holds a subset of the array in memory and uses ${\displaystyle \Theta (\min\{n,m\})}$ space, but is otherwise similar to Needleman-Wunsch (and still requires ${\displaystyle O(nm)}$ time).

As the algorithm progresses, the ${\displaystyle F_{ij}}$ will be assigned to be the optimal score for the alignment of the first ${\displaystyle i=0,\dotsc ,n}$ characters in A and the first ${\displaystyle j=0,\dotsc ,m}$ characters in B. The principle of optimality is then applied as follows:

• Basis:

${\displaystyle F_{0j}=d*j}$

${\displaystyle F_{i0}=d*i}$

• Recursion, based on the principle of optimality:

${\displaystyle F_{ij}=\max(F_{i-1,j-1}+S(A_{i},B_{j}),\;F_{i,j-1}+d,\;F_{i-1,j}+d)}$

The pseudo-code for the algorithm to compute the F matrix therefore looks like this:

d ← Gap penalty score
for i = 0 to length(A)
    F(i,0) ← d * i
for j = 0 to length(B)
    F(0,j) ← d * j
for i = 1 to length(A)
    for j = 1 to length(B)
    {
        Match ← F(i−1, j−1) + S(Ai, Bj)
        Delete ← F(i−1, j) + d
        Insert ← F(i, j−1) + d
        F(i,j) ← max(Match, Insert, Delete)
    }

Once the F matrix is computed, the entry ${\displaystyle F_{nm}}$ gives the maximum score among all possible alignments. To compute an alignment that actually gives this score, you start from the bottom right cell, and compare the value with the three possible sources (Match, Insert, and Delete above) to see which it came from. If Match, then ${\displaystyle A_{i}}$ and ${\displaystyle B_{j}}$ are aligned, if Delete, then ${\displaystyle A_{i}}$ is aligned with a gap, and if Insert, then ${\displaystyle B_{j}}$ is aligned with a gap. (In general, more than one choice may have the same value, leading to alternative optimal alignments.)

AlignmentA ← ""
AlignmentB ← ""
i ← length(A)
j ← length(B)
while (i > 0 or j > 0)
{
    if (i > 0 and j > 0 and F(i, j) == F(i−1, j−1) + S(Ai, Bj))
    {
        AlignmentA ← Ai + AlignmentA
        AlignmentB ← Bj + AlignmentB
        i ← i − 1
        j ← j − 1
    }
    else if (i > 0 and F(i, j) == F(i−1, j) + d)
    {
        AlignmentA ← Ai + AlignmentA
        AlignmentB ← "−" + AlignmentB
        i ← i − 1
    }
    else
    {
        AlignmentA ← "−" + AlignmentA
        AlignmentB ← Bj + AlignmentB
        j ← j − 1
    }
}

Complexity

Computing the score ${\displaystyle F_{ij}}$ for each cell in the table is an ${\displaystyle O(1)}$ operation. Thus the time complexity of the algorithm for two sequences of length ${\displaystyle n}$ and ${\displaystyle m}$ is ${\displaystyle O(mn)}$.^[4] It has been shown that it is possible to improve the running time to ${\displaystyle O(mn/\log n)}$ using the Method of Four Russians.^[4][5] Since the algorithm fills an ${\displaystyle n\times m}$ table the space complexity is ${\displaystyle O(mn).}$^[4]

Historical notes and algorithm development

The original purpose of the algorithm described by Needleman and Wunsch was to find similarities in the amino acid sequences of two proteins.^[1]

Needleman and Wunsch describe their algorithm explicitly for the case when the alignment is penalized solely by the matches and mismatches, and gaps have no penalty (d=0). The original publication from 1970 suggests the recursion ${\displaystyle F_{ij}=\max _{h<i,k<j}\{F_{h,j-1}+S(A_{i},B_{j}),F_{i-1,k}+S(A_{i},B_{j})\}}$.

The corresponding dynamic programming algorithm takes cubic time. The paper also points out that the recursion can accommodate arbitrary gap penalization formulas:

A penalty factor, a number subtracted for every gap made, may be assessed as a barrier to allowing the gap. The penalty factor could be a function of the size and/or direction of the gap. [page 444]

A better dynamic programming algorithm with quadratic running time for the same problem (no gap penalty) was introduced later^[6] by David Sankoff in 1972. Similar quadratic-time algorithms were discovered independently by T. K. Vintsyuk^[7] in 1968 for speech processing ("time warping"), and by Robert A. Wagner and Michael J. Fischer^[8] in 1974 for string matching.

Needleman and Wunsch formulated their problem in terms of maximizing similarity. Another possibility is to minimize the edit distance between sequences, introduced by Vladimir Levenshtein. Peter H. Sellers showed^[9] in 1974 that the two problems are equivalent.

The Needleman–Wunsch algorithm is still widely used for optimal global alignment, particularly when the quality of the global alignment is of the utmost importance. However, the algorithm is expensive with respect to time and space, proportional to the product of the length of two sequences and hence is not suitable for long sequences.

Recent development has focused on improving the time and space cost of the algorithm while maintaining quality. For example, in 2013, a Fast Optimal Global Sequence Alignment Algorithm (FOGSAA),^[10] suggested alignment of nucleotide/protein sequences faster than other optimal global alignment methods, including the Needleman–Wunsch algorithm. The paper claims that when compared to the Needleman–Wunsch algorithm, FOGSAA achieves a time gain of 70–90% for highly similar nucleotide sequences (with > 80% similarity), and 54–70% for sequences having 30–80% similarity.

Applications outside bioinformatics

See also

• Wagner–Fischer algorithm
• Smith–Waterman algorithm
• Sequence mining
• Levenshtein distance
• Dynamic time warping
• Sequence alignment

References

External links

• NW-align: A protein sequence-to-sequence alignment program by Needleman-Wunsch algorithm (online server and source code)
• A live Javascript-based demo of Needleman–Wunsch
• An interactive Javascript-based visual explanation of Needleman-Wunsch Algorithm
• Sequence Alignment Techniques at Technology Blog
• Biostrings R package implementing Needleman–Wunsch algorithm among others