Bioinformatics Algorithms: Techniques and Applications

12 DYNAMIC PROGRAMMING ALGORITHMS
(Fig. 2.2 b). Each column may consist of two aligned characters, vi and wj (1 ≤ i ≤ m,
1 ≤ j ≤ n), which is called a match (if vi = wj) oramismatch (otherwise), or one
character and one gap, which is called an indel (insertion or deletion). A global alignment
can be evaluated by the sum of the scores of all columns, which are defined
by a similarity matrix between any pair of characters (4 nucleotides for DNAs or
20 amino acids for proteins) for matches and mismatches, and a gap penalty function.
A simple scoring function for the global alignment of two DNA sequences rewards
each match by score +1, and penalizes each mismatch by score −µ and each indel by
score −σ. The alignment of two protein sequences usually involves more complicated
scoring schemes reflecting models of protein evolution, for example, PAM [21] and
BLOSUM [33].
It is useful to map the global alignment problem, that is, to find the global alignment
with the highest score for two given sequences, onto an alignment graph (Fig. 2.2 a).
Given two sequences V and W, the alignment graph is a directed acylic graph G on
(n + 1) × (m + 1) nodes, each labeled with a pair of positions (i, j) ((0 ≤ i ≤ m,
0 ≤ j ≤ n)), with three types of weighted edges: horizontal edges from (i, j)to(i +
1,j) with weight δ(v(i + 1), −), vertical edges from (i, j) to(i, j + 1) with weight
δ(−,w(j + 1)), and diagonal edges from (i, j)to(i + 1,j+ 1) with weight δ(v(i + 1),
w(j + 1)), where δ(vi, −) and δ(−,wj) represent the penalty score for indels, and
δ(vi,wj) represents similarity scores for match/mismatches. Any global alignment
between V and W corresponds to a path in the alignment graph from node (0, 0)
to node (m, n), and the alignment score is equal to the total weight of the path.
Therefore, the global alignment problem can be transformed into the problem of
finding the longest path between two nodes in the alignment graph, thus can be
solved by a dynamic programming algorithm. To compute the optimal alignment
score S(i, j) between two subsequences V = v1...vi and W = w1...wj, that is, the
total weight of the longest path from (0, 0) to node (i, j), one can use the following
(0,0) A T C T i G C
A
C
T
A
A
j
G
(i,j)
ATCT GC
A CTAAGC
C
(6,7)
(a)
(b)
FIGURE 2.2 The alignment graph for the alignment of two DNA sequences, ACCTGC and
ACTAAGC. The optimal global alignment (b) can be represented as a path in the alignment
graph from (0,0) to (6,7) (highlighted in bold).