Biomedical Engineering Reference
In-Depth Information
Let
p
ij
.t /
be the probability that nucleotide
i
undergoes to a substitution to
nucleotide
j
at finite time
t
. Then, if the superposition principle holds, at
t C
d
t
such probability can be written as:
t/ D
X
k2
p
ij
.t C
d
p
ik
.t /p
kj
.
t/
8 i; j
2 :
d
(8.2)
By subtracting
p
ij
.t /
in both sides of (
8.2
) and dividing for d
t
we obtain:
P
k2; k¤j
p
ik
.t /p
kj
.
d
t/
p
ij
.t C
d
t/ p
ij
.t /
d
D
t
d
t
Cp
ij
.t /
p
jj
.
d
t/ 1
d
8 i; j 2 ;
t
i.e.,
P
k2; k¤j
p
ik
.t /p
kj
.
d
t/
p
ij
.t C
d
t/ p
ij
.t /
d
D
t
d
t
Cp
ij
.t /
1
P
k2; k¤j
p
kj
.
d
t/ 1
8 i; j 2 :
d
t
Hence, we have
X
p
ij
.t / D
p
ik
.t /r
kj
C p
ij
.t /r
jj
8 i; j 2 :
(8.3)
k2; k¤j
When expressing (
8.3
) in matrix form, the Chapman-Kolmogorov master equation
arises
P
.t / D
P
.t /
R
D
RP
.t /;
whose integral
X
R
n
t
n
nŠ
.t / D
e
R
t
P
D
(8.4)
nD0
is known as the time homogeneous Markov (THM) model of DNA sequence evolu-
tion [
48
,
63
]. The THM model is a generalization of the Markov models described
in Jukes and Cantor [
44
], Kimura [
46
], Hasegawa et al. [
37
], Tamura and Nei [
78
],
and can be easily adapted to RNA, amino acid and codon sequences as shown in
Felsenstein [
29
] and Schadt and Lange [
71
,
72
]. In the next section, we shall inves-
tigate the dynamics of the THM model in order to derive a commonly used formula
to quantify the similarity between molecular data.
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