Empirically, we tend to observe that indels occur in consecutive blocks. This makes intuitive sense: it is plausible that a fragment encoding a structural or functional module will be inserted in a single event.
However, the gap function discussed in previous lectures does not prefer blocks of gaps. The cost of an alignment with $i$ indels is the same, regardless of whether the indels are in a single block or scattered through out the alignment.
An alternative is to use a more complex gap function, a function where the cost of adding another indel to an existing gap is lower than the cost of initiating a new gap. There is a broad class of functions that have this property.
The function most often used in practise is the affine gap function:
w(i) = h + i g.Here, h is the gap initiation or gap opening penalty and g is the gap extension penalty. This function performs fairly well in practise and is easily implemented in an O(n2) algorithm. Details are given in Setubal and Meidanis, pp. 64-66, available through electronic reserves.
In class, we worked through an example, aligning ''BAA'' with ''BANANA''. In this example, the matrices are color-coded (a: white, b: orange, c: blue). The color codes are used to indicate from which matrix the last transition was made. For example, if cell i,j in matrix a is orange, then the previous cell was b[i-1,j-1].
Note that using an affine gap function, the optimal alignment is
BANANA
BA___A
With a linear gap function, w(i) = i g, with no gap opening
penalty, other alignments with scattered indels, such as this one:
BANANA
B__A_A
would also have the optimal score.