Bioinformatics, Volume I Data, Sequence Analysis and Evolution

2.3. Modeling
the Process at a
Single Site in Two
Sequences
Phylogenetic Model Evaluation 337
where p ij (t) is the probability that the nucleotide is j at time t,
given that it was i at time 0. Assuming a homogeneous Markov
process, let r ij be the instantaneous rate of change from nucleotide
i to nucleotide j, and let R be the matrix of these rates of
change. Then, representing p ij (t) in matrix notation as P(t), we
can write equation [1] as:
P t I Rt 1
Rt 2!
1
Rt
3!
∞
k
( Rt
)
= ∑ k!
()= + + ( ) + ( ) +
= e
k=
0
Rt
2 3
where R is a time-independent rate matrix satisfying three
conditions:
1. r > 0 for i ≠ j;
ij
2. rii = –S r , implying that R1 = 0, where 1 j¹i ij T = (1, 1, 1, 1)
and 0T = (0, 0, 0, 0)—this condition is needed to ensure that
P(t) is a valid transition matrix for t ≥ 0;
3. pT R = 0T , where pT = (p , p , p , p ) is the stationary distri-
1 2 3 4
bution, 0 < pj < 1, and S 4
j=1 p = 1. j
In addition, if f denotes the frequency of the jth nucleotide in
0j
the ancestral sequence, then the Markov process governing the
evolution of a site along a single edge is:
1. Stationary, if Pr(X(t)= j)= f = p , for j = 1, 2, 3, 4, where p
0j j
is the stationary distribution, and
2. Reversible, if the balance equation p r = p r is met for 1 ≤ i,
i ij j ji
j ≤ 4, where p is the stationary distribution.
In the context of modeling the accumulation of point mutations
at a single site of a nucleotide sequence, R is an essential
component of the Markov models that are used to do so. Each
element of R has a role in determining what state the site will
be in at time t, so it is useful to understand the implications of
changing the values of the elements in R. For this reason, it is
unwise to consider the rate of evolution as a single variable when,
in fact, it is a matrix of rates of evolution.
Consider a site in a pair of nucleotide sequences that evolve from
a common ancestor by independent Markov processes. Let X and
Y denote the Markov processes operating at the site, one along
each edge, and let P X (t) and P Y (t) be the transition functions
that describe the Markov processes X(t) and Y(t). The joint probability
that the sequences contain nucleotide i and j, respectively,
is then given by:
...
[2]
f ij (t) = Pr[X(t) = i, Y(t) = j|X(0) = Y(0)], [3]