Page 28 - Applied Probability

1. Basic Principles of Population Genetics
r ≥ 0 and s< 0 of selection against a dominant. Then p n → 0 and the
approximation
(1 − r)p + p n q n
=
p n+1
2
1 − rp − sq
n 2 n 2 n 11
p n
≈
(1 − s)
for p n ≈ 0 makes it clear that p n approaches 0 locally at geometric rate
1
1−s .
If r> 0 and s = 0, then p n → 0 still holds, but convergence no longer
occurs at a geometric rate. Indeed, the equality
p n (1 − rp n )
p n+1 =
1 − rp 2 n
entails
2
1 1 1 1 − rp n
− = − 1
p n+1 p n p n 1 − rp n
r(1 − p n )
=
1 − rp n
≈ r.
It follows that for p 0 close to 0
1 1 n−1 1 1

− = −
p n p 0 p i+1 p i
i=0
≈ nr.
This approximation implies the slow convergence

1
p n ≈ 1
nr +
p 0
for selection against a pure recessive.
Heterozygote advantage (r and s both positive) is the most inter-
esting situation covered by this classic selection model. Geneticists have
suggested that several recessive diseases are maintained at high frequencies
by the mechanism of heterozygote advantage. The best evidence favoring
this hypothesis exists for sickle cell anemia [2]. A single dose of the sickle
cell gene appears to confer protection against malaria. The evidence is
much weaker for a heterozygote advantage in Tay-Sachs disease and cystic
fibrosis. Geneticists have conjectured that these genes may protect carriers
from tuberculosis and cholera, respectively [14].