Sequence Comparison.pdf

132 7 Local Alignment Statistics
probability is well approximated by the formula
Pr[S n,r ≥ t]=
e−t t r−1
r!(r − 1)! . (7.40)
especially when t > r(r − 1). This is the formula used in the BLAST program for
calculating p-value when multiple highest-scoring segments are reported.
7.2.5 Local Ungapped Alignment
The statistic theory of Sections 7.2.1 and 7.2.2 considered the maximal scoring segments
in a fixed ungapped alignment. In practice, the objective of database search is
to find all good matches between a query sequence and the sequences in a database.
Here, we consider a general problem of calculating the statistical significance of
a local ungapped alignment between two sequences. The sequences in a highestscoring
local ungapped alignment between two sequences is called the maximalscoring
segment pairs (MSP).
Consider two sequences of length n 1 and n 2 . To find the MSPs of the sequences,
we have to consider all n 1 + n 2 − 1 possible ungapped alignments between the sequences.
Each such alignment yields a random work as that studied in Section 7.2.1.
Because the n 1 +n 2 −1 corresponding random walks are not independent, it is much
more involved to estimate the mean value of the maximum segment score. The theory
developed by Dembo et al. [56, 57] for this general case is too advanced to be
discussed in this book. Here we simply state the relevant results.
To some extent, the key formulas in Sections 7.2.2 can be taken over to the general
case, with n being simply replaced by n 1 n 2 . Consider the sequences x 1 x 2 ...x n1
and y 1 y 2 ...y n2 , where x i and y j are residues. We use s(x i ,y j ) to denote the score for
aligning residues x i and y j . The optimal local ungapped alignment score S max from
the comparison of the sequences is
S max =
max
∆≤min{n 1 ,n 2 }
max
i≤n 1 −∆
j≤n 2 −∆
∆
∑
l=1
s(x i+l ,x j+l ).
Suppose the sequences are random and independent: x i and y j follows the same
distribution. The random variable S max has the following tail probability distribution:
Pr[S max > 1 λ log(n 1n 2 )+y] ≈ 1 − e −Ke−λy , (7.41)
where λ is given by equation (7.5) and K is given by equation (7.32) with n replaced
by n 1 n 2 . The mean value of S max is approximately
1
λ log(Kn 1n 2 ). (7.42)