Sequence Comparison.pdf

8.7 Gap Cost in Gapped Alignments 163
ble 8.3), whereas in the alignment of two DNA sequences that are diverged at 96 (or
120 × 0.8) PAMs, every three residues (a codon) carry only about 0.62-bit information.
In other words, at this evolutionary distance, as much as 37% of the information
available in protein comparison will be lost in DNA sequence comparison.
8.7 Gap Cost in Gapped Alignments
There is no general theory available for guiding the choice of gap costs. The most
straightforward scheme is to charge a fixed penalty for each indel. Over the years,
it has been observed that the optimal alignments produced by this scheme usually
contain a large number of short gaps and are often not biologically meaningful (see
Section 1.3 for the definition of gaps).
To capture the idea that a single mutational event might insert or delete a sequence
of residues, Waterman and Smith (1981, [180]) introduced the affine gap
penalty model. Under this model, the penalty o+e×k is charged for a gap of length
k, where o is a large penalty for opening a gap and e a smaller penalty for extending
it. The current version of BLASTP uses, by default, 11 for gap opening and 1 for
gap extension, together with BLOSUM62, for aligning protein sequences.
The affine gap cost is based on the hypothesis that gap length has an exponential
distribution, that is, the probability of a gap of length k is α(1 − β)β k for some
constant α and β. Under this hypothesis, an affine gap cost is derived by charging
log ( α(1 − β)β k) for a gap of length k. But, this hypothesis might not be true in
general. For instance, the study of Benner, Cohen, and Gonnet (1993, [26]) suggests
that the frequency of a length k is accurately described by mk −1.7 for some constant
m.
A generalized affine gap cost is introduced by Altschul (1998, [2]). A generalized
gap consists of a consecutive sequence of indels in which spaces can be in either
row. A generalized gap of length 10 may contain 10 insertions; it may also contain
4 insertions and 6 deletions. To reflect the structural property of a generalized gap, a
generalized affine gap cost has three parameters a,b,c. The score −a is introduced
for the opening of a gap; −b is for each residue inserted or deleted; and −c is
for each pair of residues left unaligned. A generalized gap with k insertions and l
deletions scores −(a + |k − l|b + cmin{k,l}).
Generalized affine gap costs can be used for aligning locally or globally protein
sequences. The dynamic programming algorithm carries over to this generalized
affine gap cost in a straightforward manner; and it still has quadratic time complexity.
For local alignment, the distribution of optimal alignment scores also follows
approximately an extreme value distribution (7.1). The empirical study of Zachariah
et al. (2005, [213]) shows that this generalized affine gap cost model improves significantly
the accuracy of protein alignment.