Affine gaps penalty

To align strings S, T, consider as usual the prefixes $S_{1 \ldots i}$ of S and $T_{1 \ldots j}$ of T. Any alignment of these two prefixes is one of the following three types:

1.

S ------i
T ------j
alignment of $S_{1 \ldots i}$ and $T_{1 \ldots j}$ where characters S(i) and T(j) are aligned opposite each other. This includes both the case that S_i = T_j and that ${S_i \neq T_j}$.

2.

S ------i_ _ _ _ _ _ _
T -------------j

alignment of $S_{1 \ldots i}$ and $T_{1 \ldots j}$ where character S_i is aligned to a character strictly to the left of character T_j. Therefore, the alignment ends with a gap in S.

3.

S --------------i
T ------j_ _ _ _ _ _ _ _

alignment of $S_{1 \ldots i}$ and $T_{1 \ldots j}$ where character S_i is aligned to a character strickly to the right of character T_j. Therefore, the alignment ends with a gap in T.

 

Hence the base conditions are:

$$\begin{eqnarray*}V(i, 0) &=& E(i, 0) = W_g + iW_s\\ V(0, j) &=& F(0, j) = W_g + jW_s \end{eqnarray*}$$

and the recursive computation of V(i, j) will be:

$$V(i, j) = \max \{ E(i, j), F(i, j), G(i, j) \}$$

while

$$\begin{eqnarray*}G(i, j) &=& V(i - 1, j - 1) + \sigma(S_{i}, T_{j})\\ E(i, j) &... ...(i, j) &=& \max \{ F(i - 1, j) + W_s, V(i - 1, j) + W_g + W_s \} \end{eqnarray*}$$

The optimal value alignment is the maximum value in the nth row or m column.

Complexity

• Time complexity - As before O(nm), as we only compute four matrices instead of one.
• Space complexity - There's a need to save four matrices (for E, F, G, and V respectively) during the computation. Hence, O(nm) space is needed, for the trivial implementation.