Page 29 - Applied Probability

1. Basic Principles of Population Genetics
12
1.6 Balance Between Mutation and Selection
Mutations furnish the raw material of evolutionary change. In practice,
most mutations are either neutral or deleterious. We now briefly discuss
the balance between deleterious mutations and selection. Consider first the
case of a dominant disease. In the notation of the last section, let A 2 be
the normal allele and A 1 the disease allele, and define the fitnesses of the
three genotypes by r ≥ 0 and s< 0. If the mutation rate from A 2 to A 1
is µ, then equilibrium is achieved between the opposing forces of mutation
and selection when
p ∞ q ∞ +(1 − s)q 2
= ∞ (1 − µ).
q ∞ 2 2
1 − rp − sq
∞ ∞
If we multiply this equation by 1 − rp 2 − sq 2 and divide it by q ∞ , we get
∞ ∞
1 − rp 2 − sq 2 =(1 − sq ∞ )(1 − µ).
∞ ∞
2
Dropping the negligible term rp , we find that this quadratic has the
∞
approximate solution

1 − µ 4µ
≈ 1+ 1+
q ∞ 2
2 s(1 − µ)
1 − µ
2µ
≈ 1+1+ 2
2 s(1 − µ)
µ(1 − s)
≈ 1+ ,
s
µ(1−s)
which yields p ∞ =1 − q ∞ ≈ . The corresponding equilibrium fre-
−s
2µ(1−s)
quency of affecteds is 2p ∞q ∞ ≈ .
−s
For a recessive disease (r> 0 and s = 0), the balance equation becomes
p ∞ q ∞ + q 2
= ∞ (1 − µ)
q ∞
¯ w ∞
= q ∞ (1 − µ).
1 − rp 2
∞
In other words, 1 − rp 2 =1 − µ, which has solution p ∞ = µ/r. The
∞ µ
frequency of affecteds at equilibrium is now p 2 = . Thus given equal mu-
∞ r
tation rates, dominant and recessive diseases will afflict comparable num-
bers of people. In contrast, the underlying allele frequencies and rates of
approach to equilibrium vary dramatically. Indeed, it is debatable whether
any human population has existed long enough for the alleles at a recessive
disease locus to achieve a balance between mutation and selection. Ran-
dom sampling of gametes (genetic drift) and small initial population sizes
(founder effect) play a much larger role in determining the frequency of
recessive diseases in modern human populations.