Sequence Comparison.pdf

7.3 Gapped Local Alignment Scores 137
A(s)= 1
|I s | ∑ i∈I s
(S(i) − s), (7.51)
where S(i) is the score of island i and |I s | the number of islands in I s . Because
the island scores are integral and have no proper common divisors in the cases of
interest, the maximum-likelihood estimate for λ is
By equation (7.45), K is calculated as
λ s = ln(1 + 1 ). (7.52)
A(s)
K s = 1 V ×|I s|×e sλ s
, (7.53)
where V is the size of the search space from which the island scores are collected.
V is equal to n 2 if the islands are obtained from aligning locally two sequences of
length n; itisNn 2 if N such alignments were performed.
Maximum-Likelihood Method for Estimating λ (Altschul et al., 2001, [5])
The island scores S follow asymptotically a geometric-like distribution
Pr[S = x]=De −λx ,
where D is a constant. For a large integer cutoff c,
Pr[S = x|S ≥ c] ≈
De−λx
∑ ∞ j=c De −λ j =(1 − e−λ )e −λ(x−c) .
Let x i denote the ith island scores for i = 1,2,...,M. Then the logarithm of
the probability that all x i shaveavalueofc or greater is
ln(Pr[x 1 ,x 2 ,...,x M |x 1 ≥ c,x 2 ≥ c,...,x M ≥ c])
= −λ
M
∑
j=1
(x j − c)+M ln(1 − e −λ ).
The best value λ ML of λ is the one that maximizes this expression. By equating
the first derivation of this expression to zero, we obtain that
(
)
1
λ ML = ln 1 +
1
M ∑M j=1 (x .
j − c)