Page 31 - Applied Probability

1. Basic Principles of Population Genetics
14
5. Moran [12] has proposed a model for the approach of allele frequencies
to Hardy-Weinberg equilibrium that permits generations to overlap.
Let u(t), v(t), and w(t) be the relative proportions of the genotypes
A 1 /A 1 , A 1 /A 2 , and A 2 /A 2 at time t. Assume that in the small time
interval (t, t+dt) a proportion dt of the population dies and is replaced
by the offspring of random matings from the residue of the population.
In effect, members of the population have independent, exponentially
distributed lifetimes of mean 1. The other assumptions for Hardy-
Weinberg equilibrium remain in force.
(a) Show that for small dt
1
2
u(t + dt) = u(t)(1 − dt)+ u(t)+ v(t) dt + o(dt).
2
Hence,
1
2

u (t)= −u(t)+ u(t)+ v(t) .
2
(b) Similarly derive the differential equations
1 1

v (t)= −v(t)+ 2 u(t)+ v(t) v(t)+ w(t)
2 2
1
2

w (t)= −w(t)+ v(t)+ w(t) .
2
1
(c) Let p(t)= u(t)+ v(t) be the allele frequency of A 1 . Verify that
2
p (t) = 0 and that p(t)= p 0 is constant.

(d) Show that
2
2
[u(t) − p ] = −[u(t) − p ],
0
0
and so
−t
2
u(t) − p 2 0 =[u(0) − p ]e .
0
(e) Similarly prove
v(t) − 2p 0(1 − p 0 ) = [v(0) − 2p 0 (1 − p 0 )]e −t
−t
2
w(t) − (1 − p 0 ) 2 =[w(0) − (1 − p 0 ) ]e .
(f) If time is measured in generations, then how many generations
does it take for the departure from Hardy-Weinberg equilibrium
to be halved?
6. Consider an X-linked version of the Moran model in the previous
problem. Again let u(t), v(t), and w(t) be the frequencies of the three
female genotypes A 1 /A 1 , A 1 /A 2 , and A 2 /A 2 , respectively. Let r(t)
and s(t) be the frequencies of the male genotypes A 1 and A 2 .