SQUARE45

Let

S_A

and

S_B

be two genomic sequences. An alignment

\mathcal{A}

is a set of pairs of sequences

(S'_A, S'_B)

of equal length

L'

, derived from

S_A

and

S_B

by introducing gaps. The objective is to find the optimal alignment

\mathcal{A}^*

that maximizes the total score

Score(\mathcal{A})

Score(\mathcal{A}) = \sum_{i=1}^{L'} (W(S'_A[i], S'_B[i]) - G(S'_A[i], S'_B[i]))

where

W(\cdot, \cdot)

is the substitution matrix score (e.g., BLOSUM) for matching characters, and

G(\cdot, \cdot)

is the gap penalty function, typically defined as

G(a, b) = \max(g_{open}, g_{extend}) \cdot \delta_{gap}

. This maximization is solved using dynamic programming (e.g., Needleman-Wunsch or Smith-Waterman algorithms).

Comparative Genomics