Sequence Comparison.pdf

6.3 Distance between Non-Overlapping Hits 103
Table 6.1 The values of the upper bound in Theorem 6.6 for different w and p after rounding to
the nearest integer).
❅ w 10 11 12 13 14
p ❅
0.6 49 76 121 196 315
0.7 17 21 26 24 44
0.8 11 12 14 15 17
0.9 10 11 12 13 14
Using the above theorem, the following explicit upper bound on µ π can be proved
for non-uniformly spaced seeds π. Its proof is quite involved and so is omitted.
Theorem 6.5. For any non-uniformly spaced seed π,
µ π ≤
w π
∑
i=1
[
]
(1/p) i +(|π|−w π ) − (q/p) (1/p) (w π −2) − 1
6.3.3 Why Do Spaced Seeds Have More Hits?
Recall that, for the consecutive seed θ of weight w, µ θ = ∑ w i=1 ( 1 p )i . By Theorem 6.5,
we have
Theorem 6.6. Let π be a non-uniformly spaced seed and θ the consecutive seed of
the same weight. If |π| < w π + q p [( 1 p )w π −2 − 1], then, µ π < µ θ .
Non-overlapping hit of a spaced seed π is a recurrent event with the following
convention: If a hit at position i is selected as a non-overlapping hit, then the
next non-overlapping hit is the first hit at or after position i + |π|. By (B.45) in
Section B.8, the expected number of the non-overlapping hits of a spaced seed π
in a random sequence of length N is approximately N µ π
. Therefore, if |π| < w π +
q
p [( 1 p )w π −2 − 1] (see Table 6.1 for the values of this bound for p = 0.6,0.7,0.8,0.9
and 10 ≤ w ≤ 14), Theorem 6.6 implies that π has on average more non-overlapping
hits than θ in a long homologous region with sequence similarity p in the Bernoulli
sequence model. Because overlapping hits can only be extended into one local alignment,
the above fact indicates that a homology search program with a good spaced
seed is usually more sensitive than with the consecutive seed (of the same weight)
especially for genome-genome comparison.