Sequence Comparison.pdf

130 7 Local Alignment Statistics
A = E(K 1 ), (7.30)
which is the mean distance between two successive ladder points in the walk. Then
the mean number of ladder points is approximately
A n when n is large (see Section
B.8). Ignoring edge effects, we derive the following asymptotic bounds from
Lemma 7.2 by setting y = y ′ + lnA
λ and m = A n :
{
}
exp − C′ e λ
[
A e−λy′ ≤ Pr M(n) ≤ lnn ]
}
λ + y′ ≤ exp
{− C′
A e−λy′ . (7.31)
Set
K = C′
A . (7.32)
Replacing y ′ by (ln(K)+s)/λ, inequality (7.31) becomes
{
exp −e λ−s} ≤ Pr[M(n) ≤ ln(Kn)+s/λ] ≤ exp { −e −s} ,
or equivalently
exp
{
−e λ−s} ≤ Pr[λM(n) − ln(Kn) ≤ s] ≤ exp { −e −s} . (7.33)
In the BLAST theory, the expression
Y (n)=λM(n) − ln(Kn)
is called the normalized score of the alignment. Hence, the P-value corresponding
to an observed value s of the normalized score is
P-value ≈ 1 − exp { −e −s} . (7.34)
7.2.3 The Number of High-Scoring Segments
By Theorem 7.1 and (7.28), the probability that any maximal-scoring segment has
score s or more is approximately C ′ e −λs . By (7.30), to a close approximation there
are N/A maximal-scoring segments in a fixed alignment of N columns as discussed
in Section 7.2.2. Hence, the expected number of the maximal-scoring segments with
score s or more is approximately
NC ′
A e−λs = NKe −λs , (7.35)