Sequence Comparison.pdf

134 7 Local Alignment Statistics
where q ij is the target frequency at which we expect to see residue i aligned with
residue j in the MSP, and s ij is the score for aligning i and j. With the value of λ,
the s ij and the background frequencies q i and q j in hand, q ij can be calculated as
q ij ≈ p i p j e λs ij
. (7.50)
Simulation shows that the values calculated from (7.49) are often larger than the empirical
mean lengths of MSPs especially when n 1 and n 2 are in the range from 10 2
to 10 3 (Altschul and Gish, 1996, [6]). Accordingly, the effective lengths defined by
(7.46) might lead to P-value estimates less than the correct values. The current version
of BLAST calculates empirically the mean length of MSPs in database search.
7.3 Gapped Local Alignment Scores
In this section, we concern with the optimal local alignment scores S (in the general
case that gaps are allowed). Although the explicit theory is unknown in this case,
a number of empirical studies strongly suggest that S also has asymptotically an
extreme value distribution (7.1) under certain conditions on the scoring matrix and
gap penalty used for alignment. As we will see, these conditions are satisfied for
most combinations of scoring matrices and gap penalty costs.
7.3.1 Effects of Gap Penalty
Most empirical studies focus on the statistical distribution of the scores of optimal
local alignments with affine gap costs. Each such gap cost has gap opening penalty
o and gap extension penalty e, by which a gap of length k receives a score of −(o +
k × e).
Consider two sequences X ′ and X ′′ that are random with letters generated independently
according to a probabilistic distribution. The optimal local alignment
score S max of X ′ and X ′′ depends on the sequence lengths m and n, and the letter
distribution, the substitution matrix, and affine gap cost (o,e). Although the exact
probabilistic distribution of S max is unclear, S max has either linear or logarithmic
growth as m and n go to infinite.
This phase transition phenomenon for the optimal local alignment score was
studied by Arratia and Waterman [15]. Although a rigorous treatment is far beyond
the scope of this book, an intuitive account is quite straightforward. Consider
two sequences x 1 x 2 ...x n and y 1 y 2 ...y n .LetS(t) denote the score of the optimal
alignment of x 1 x 2 ...x t and y 1 y 2 ...y t . Then,
S t+k ≥ S t + S(x t+1 x t+2 ···x t+k ,y t+1 y t+2 ···y t+k ).