Page 332 - Applied Probability

15. Diffusion Processes
321
15.3 Wright-Fisher Model
The Wright-Fisher model for the evolution of a deleterious or neutral
gene postulates (a) discrete generations, (b) finite population size, (c) no
immigration, and (d) formation of gametes by random binomial sampling.
In assumption (d), each current population member contributes to an infi-
nite pool of potential gametes in proportion to his or her fitness. Mutation
from the normal allele A 2 to the deleterious allele A 1 takes place at this
stage with mutation rate η; backmutation is not permitted. In the neu-
tral model we neglect mutation and treat the two alleles symmetrically.
Once the pool of potential gametes is formed, actual gametes are sam-
pled randomly. At each generation, the three genotypes occur in the usual
Hardy-Weinberg proportions even though allele frequencies change over
time.
denote the
In Chapter 1 recall that we let w A 1 /A 1 , w A 1 /A 2 , and w A 2 /A 2
average fitnesses of the three genotypes A 1 /A 1 , A 1 /A 2 and A 2 /A 2 of an
autosomally determined trait. Simplifying our previous discussion and no-
= 1 and
tation slightly, for a dominant disease we may suppose w A 2 /A 2
= f< 1. (In the notation of Chapter 1, r ≥ 0 and
w A 1 /A 1 = w A 1 /A 2
= 1, and
1 − s =1/f.) For a neutral trait, w A 1 /A 1 = w A 1 /A 2 = w A 2 /A 2
= f< 1. (In the
for a recessive disease w A 2 /A 2 = w A 1 /A 2 = 1 and w A 1 /A 1
notation of Chapter 1, s = 0 and 1 − r = f for a recessive.) For our pur-
poses, the population size N m at generation m need not be constant. The
primary object of study in the current chapter is the frequency X m of al-
lele A 1 at generation m. This frequency is the ratio of the total number Y m
of A 1 alleles to the total number of genes 2N m. The Wright-Fisher model
specifies that Y m is binomially distributed with 2N m trials and success
probability p(X m−1 ) determined by the proportion p(X m−1 )of A 1 alleles
in the pool of potential gametes for generation m. In passing to a diffusion
approximation, we take one generation as the unit of time and substitute
µ(m, x m )=E(X m+1 − X m | X m = x m ) (15.6)
= p(x m ) − x m
2
σ (m, x m )=Var(X m+1 − X m | X m = x m ) (15.7)
p(x m )[1 − p(x m )]
=
2N m+1
2
for the infinitesimal mean µ(t, x) and variance σ (t, x) of the diffusion
process evaluated at time t = m and position x = x m .
Under neutral evolution, the gamete pool probability p(x)= x. This for-
mula for p(x) entails no systematic tendency for either allele to expand at
the expense of the other allele. For a dominant disease, p(x)= η+fx, which
implies an equilibrium frequency of x ∞ = η/(1 − f) in the corresponding
2
deterministic model. Finally, for a recessive disease, p(x)= η+x−(1−f)x ,

which implies an equilibrium frequency of x ∞ = η/(1 − f). Most popu-