Lecture 44: Population Genetics VI:
Interaction of Selection with Mutation and Drift

(version 20 July 1999)

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Mutation-Selection balance

In an infinite population, while selection may remove a deleterious allele, new copies can be created by mutation. The net result is a mutation-selection balance.

This occurs when the decrease in the allele frequency by selection is countered by the increase in the allele frequency via mutation.

A common situation is the maintenance of a lethal recessive in the population. It can be shown that if

W_AA = 1; W_Aa = 1; W_aa = 1 -s
and the mutation rate from A to a is u, then
The equilibrium frequency of a is ( u / s )^1/2

In particular, if a is a lethal recessive, then its equilibrium frequency is u^1/2

For many human recessive diseases, u = 10^-6, and in these cases the population frequency of this disease allele is

( 10^-6)^1/2 = 10^-3
In this case, the expected population frequency of the disease is (10^-3)² = 10^-6

General Expression

Let the genotypes AA: Aa: aa have fitnesses 1: 1-h s: 1-s

Here h is a measure of the amount of dominance, with h = 1 (A completely dominant to a) and h = 0 (a completely recessive to A)

Generally, if h is not zero, then the equilibrium frequency is approximately u/(hs).

In particular, for a dominant, the equilibrium frequency is u/s

Thus for a dominant lethal (s=1), the equilibrium frequency of allele a is u.

Hence, if u = 10^-6, then the equilibrium frequency of the allele is 10^-6.
Thus the expected frequency of diseased individuals is 2p(1-p) = 2 x 10^-6.

Derivation

Let p = freq(A) and q = 1-p = freq(a). Then with the above fitness parameterization, the change in the frequency of A following selection is

Provided h is not zero, the q² term is expected to be very small, and we can usually ignore it with little error. Solving gives q = u/(hs).

Interactions of Drift and Selection

In a finite population,

an allele favored by selection can still become lost
an allele selected against can still become fixed.

How often do these respective events occur (i.e., when does drift overpower the effects of selection and vise-versa?)

Kimura's expression

Motto Kimura (1957, 1964) showed that for an allele with the simple fitnesses of 1: 1+ s : 1+2s for the genotypes aa: Aa: AA, that

the probability of fixation, U(p), that allele A is fixed given it starts at allele frequency p, is given by

Of greatest interest is that probability that an allele introduced as a single copy, so that p = 1/(2N). Here, Kimura's expression simplifies

Key points:

Drift dominates selection when 4 N | s | << 1, as here U(p) = p.
Even a selectively-favored allele has a low chance (2s) of becoming fixed

Example Consider an allele with s = 0.01 in three different populations:

For N = 10
- p = 1/20 = 0.05, while U(p) = 0.06
For N = 100
- p = 1/200 = 0.005, while U(p) = 0.02
For N = 1000
- p = 1/20 = 0.0005, while U(p) = 0.02