Biomedical Engineering Reference
In-Depth Information
given site and time
t
,
S
1
is characterized by nucleotide
i
and
S
2
by nucleotide
j
.
Assume that
O
.0/ D ˘
,where
˘
denotes a diagonal matrix whose
j
th diagonal
entry is
j
. Then it holds that:
o
ij
.t / D
X
k2
p
ik
.t /
k
p
kj
.t /
8 i; j 2 ; t 0;
or equivalently
.t / D
P
0
.t /˘
O
P
.t /
t
0;
(8.6)
where
P
0
.t /
˘
1
denotes the transpose of
P
.t /
. Premultiplying for
both sides of
(
8.6
)wehave
.t / D ˘
1
e
R
0
t
˘
˘
1
O
.t / D ˘
1
P
0
.t /˘
e
R
t
:
P
ABA
1
/ D
A
A
1
,wehave
Since for any matrix function it holds that
f.
f.
B
/
.t / D
e
˘
1
R
0
t˘
e
R
t
:
˘
1
O
(8.7)
If we assume that the hypothesis of
time-reversibility
holds, i.e.:
R
D
R
0
˘;
˘
˘
1
R
0
t˘
then
and
R
t
are commutative, and (
8.7
) becomes:
.t / D
e
˘
1
R
0
t˘C
R
t
:
˘
1
O
(8.8)
By applying the logarithmic matrix function to both members of (
8.8
) and premul-
tiplying for
˘
, we obtain
R
0
t˘ C ˘
.˘
1
O
R
t D ˘
log
.t //:
As the negative trace of
2t˘
R
represents the expected number of substitution events
per site between
S
1
and
S
2
, at time
t
the evolutionary distance
d
S
1
;S
2
between
S
1
and
S
2
can be computed as:
.˘
1
O
d
S
1
;S
2
D2t
tr
Œ˘
R
D
tr
Œ˘
log
.t //:
(8.9)
Equation (
8.9
) is known as the general time-reversible (GTR) distance [
48
,
63
]and
is the most general formula to quantify the similarity between molecular data using
a time-reversible Markov model of molecular evolution. It is worth noting that if in
one hand the hypothesis of time-reversibility simplifies the formalization of the evo-
lutionary process of a pair of molecular sequences, on the other hand its introduction
gives rises to important consequences. In fact, the hypothesis of time-reversibility
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