Sequence Comparison.pdf

7.2 Ungapped Local Alignment Scores 131
where K is defined in (7.32).
7.2.4 Karlin-Altschul Sum Statistic
When two sequences are aligned, insertions and deletions can break a long alignment
into several parts. If this is the case, focusing on the single highest-scoring
segment could lose useful information. As an option, one may consider the scores
of the multiple highest-scoring segments.
Denote the r disjoint highest segment scores as
M (1) = M(n),M (2) ,...,M (r)
in an alignment with n columns. We consider the normalized scores
S (i) = λM (i) − ln(Kn), i = 1,2,...,r. (7.36)
Karlin and Altschul (1993) showed that the limiting joint density f S (x 1 ,x 2 ,···,x r )
of S =(S (1) ,S (2) ,···,S (r) ) is
(
f S (x 1 ,x 2 ,···,x r )=exp −e −x r
−
r
∑
k=1
x k
)
(7.37)
in the domain x 1 ≥ x 2 ≥ ...≥ x r .
Assessing multiple highest-scoring segments is more involved than it might first
appear. Suppose, for example, comparison X reports two highest scores 108 and 88,
whereas comparison Y reports 99 and 90. One can say that Y is not better than X,
because its high score is lower than that of X. But neither is X considered better,
because the second high score of X is lower than that of Y. The natural way to
rank all the possible results is to consider the sum of the normalized scores of the r
highest-scoring segments
S n,r = S (1) + S (2) + ···+ S (r) (7.38)
as suggested by Karlin and Altschul. This sum is now called the Karlin-Altschul
sum statistic.
Theorem 7.2. The limiting density function of Karlin-Altschul sum S n,r is
f n,r (x)=
e−x ∫ ∞ (
y r−2 exp −e (y−x)/r) dy. (7.39)
r!(r − 2)! 0
Integrating f n,r (x) from t to ∞ gives the tail probability that S n,r ≥ t. This resulting
double integral can be easily calculated numerically. Asymptotically, the tail