Page 226 - Applied Probability

10. Molecular Phylogeny
212
at time t given that it starts in A at time 0. We will derive a system of
coupled ordinary differential equations obeyed by q AY (t) and p AG (t). The
entry p AG(t)of P(t) is the probability that the chain is in G at time t given
that it starts in A at time 0. By the same reasoning that led to the forward
equation (10.5), we have
q AY (t + h)= q AY (t)(1 − δh − κh)+ p AG (t)(γ + λ)h
+[1 − q AY (t) − p AG(t)](γ + λ)h + o(h),
where 1 − q AY (t) − p AG (t) equals the probability p AA (t) of being in A at
time t. Forming the obvious difference quotient and letting h → 0 yields
the differential equation
q AY (t) = −c 1 q AY (t)+ c 2 ,
where

c 1 = δ + κ + γ + λ
c 2 = γ + λ.
This equation can be solved by multiplying by the integrating factor e c 1t
and isolating the terms [q AY (t)e c 1 t
] involving q AY (t) on the left side of the
equation. These manipulations yield the solution
c 2 −c 1t
q AY (t) = (1 − e ) (10.12)
c 1
satisfying the initial condition q AY (0) = 0.
To solve for p AG (t), write the forward approximation
p AG(t + h)= p AG(t)(1 − h − γh − λh)+ q AY (t)κh
+[1 − q AY (t) − p AG (t)]αh + o(h).

This leads to the differential equation
p (t) = −c 3 p AG (t)+ c 4 q AY (t)+ α,
AG
where

c 3 = + α + γ + λ
c 4 = κ − α.

Substituting the solution (10.12) for q AY (t), one can straightforwardly ver-
ify that this last differential equation has solution

c 2 c 4 + αc 1 c 2 c 4 −c 1t
p AG(t) = − e
c 1 c 3 c 1 (c 3 − c 1 )
c 2 c 4 − α(c 3 − c 1 ) −c 3 t
+ e (10.13)
c 3 (c 3 − c 1 )