Page 227 - Applied Probability

10. Molecular Phylogeny
213
satisfying p AG(0) = 0.
This analysis has produced the probabilities p AA (t), p AG (t), and q AY (t)
of being in A, G, or either pyrimidine at time t starting from A at time
0. To decompose q AY (t) into its two constituent probabilities p AC (t) and
p AT (t), define q UU (t) to be the probability that the chain is in either purine
at time t given that it starts in either purine at time 0. Likewise, define
q UC (t) to be the probability that the chain is in the pyrimidine C at time t
given that it starts in either purine at time 0. Because of the symmetry of
the transition rates, q UU (t) makes sense, and q UC (t)= p AC (t)= p GC (t).
From q AY (t) and p AC (t), we calculate p AT (t)= q AY (t) − p AC (t).
To derive a differential equation for q UU (t), note the approximation
q UU (t + h)= q UU (t)(1 − γh − λh)+ q UC (t)(δ + κ)h
+[1 − q UU (t) − q UC (t)](δ + κ)h + o(h),
where 1 − q UU (t) − q UC (t) is the probability of being in T at time t. This
approximation leads to
q (t) = −c 1 q UU (t)+ c 5 ,
UU
where c 1 was defined previously and
= δ + κ.
c 5
Again, the solution
c 5 +(c 1 − c 5 )e −c 1 t
q UU (t)= (10.14)
c 1
satisfying q UU (0) = 1 follows directly.
The approximation for q UC (t),
q UC (t + h)= q UC (t)(1 − δh − κh − βh)+ q UU (t)γh
+[1 − q UU (t) − q UC (t)]σh + o(h),
yields the differential equation

q UC (t) = −c 6 q UC (t)+ c 7 q UU (t)+ σ,

where

c 6 = δ + κ + σ + β
c 7 = γ − σ.
In view of equation (10.14) and the initial condition q UC (0) = 0, the solu-
tion for q UC (t)is

c 5 c 7 + σc 1 (c 1 − c 5 )c 7 −c 1 t
q UC (t) = + e
c 1 c 6 c 1 (c 6 − c 1 )
c 7 (c 6 − c 5 )+ σ(c 6 − c 1 ) −c 6 t
− e . (10.15)
c 6 (c 6 − c 1 )