Bioinformatics, Volume I Data, Sequence Analysis and Evolution

270 Liò and Bishop
the complementary set of small residues, or residues with a different
charge. There is a rate parameter, l, and an equilibrium
frequency parameter, p A (p A + p a = 1), such that the instantaneous
rate of substituting state j for a different state i is equal to lp j .
The matrix of transition probabilities at time t is then:
⎧exp(
− lt) + piexp( 1 − lt)
i = j
Pij () t = ⎨
⎩ piexp( 1 −lt) i ≠ j
This substitution process is reversible, i.e., p i P ij (t) = p j P ji (t).
A further extension is to model correlated change in pairs of
sites. This was first introduced by Pagel (49) for comparative
analysis of discrete characters. Consider a second site with
two states, B and b, with equilibrium frequencies p B and p b
(where p B + p b = 1). Then the matrix of instantaneous transition
rates is:
∑
⎡ − lp
AB B Ab/ pA lp A aB/ pb
0 ⎤
⎢
⎥
⎢lp
B AB/ pA −∑
0 lp
Ab A ab/ pb
⎥
Q = ⎢
⎥
⎢lApAB/
pB 0 −∑
l
aB
Bpab/ pa
⎥
⎢
⎥
⎣
⎢ 0 lApAb/ pb lBpaB/ pa
−∑
ab ⎦
⎥
where Σ ij is the sum of off-diagonal elements for row ij and
l A and l B are the two rate parameters governing substitution
at the two loci, A and B. Rows and columns are ordered as
AB, Ab, aB, ab. The number of free parameters is five: two
rate parameters, and because the p ij sum to one, three independent
values of p ij . There is an extra degree of freedom
that can be represented by the quantity RD = p AB p ab − p Ab p aB ;
this quantity is analogous to the linkage disequilibrium. If
the quantity RD is different from zero, there is some degree
of dependence between the two sites. RD can be negative
or positive and this corresponds to either compensation or
anti-compensation of the residues. Again, the substitution
probabilities for the co-evolving model can be calculated
using P(t) = exp[Qt]. Rather than using this model to construct
a phylogenetic tree (which would be possible in principle),
if there is a given phylogenetic tree, it is possible to
use it to test the evolutionary model based on likelihood
calculations.
As a final target, the understanding of protein evolution
may allow one to distinguish between analogous and homologous
proteins, i.e., detect similarities in those proteins that
have very low sequence homology and have probably diverged
from a common ancestor into the so-called twilight zone.