PopGen 8: Transient verses equilibrium polymorphism
We will now consider multiple forces (selection, drift, etc.) acting at the same time. In some cases these forces act in opposite directions and an equilibrium state can be reached by the population. This means there might be populations where a polymorphism is not in a transient state (between fixation or loss), but rather represents a stable equilibrium.
Mutationselection equilibrium
Given our separate treatments of selection and mutation as forces for evolution of populations, it is now possible to consider if the combined effects could explain the presence of any of the polymorphisms that are found in natural populations. For the moment we will ignore the effects of genetic drift. First let’s review some of our earlier work. The effectiveness of both selection and mutation as forces for change in allele frequencies is dependent on the frequency of the allele in the population. Let’s call the change in the frequency of the
a
allele (p) from one generation to the next
p. Below are plots of
p under both mutation and selection pressure:
p is not constant; rather it depends on the initial allele frequency. Let’s assume two alleles at a locus,
A
and
a
, and the
a
allele is a deleterious recessive allele. Mutation pressure is a source of increase in the allele frequency (red line above, with +
q), and selection is a force for a decrease in allele frequency (blue line in plot on right, with 
q) If both processes operate for a long enough time, equilibrium will be reached. The equilibrium point will be frequency of q in the population where the increase in allele frequency under mutation pressure (+
q) is
exactly balanced
by the decrease in mutation pressure under selections (
q).
q (mutation pressure) =
q (selection)
Mutation pressure and selection can operate in opposite directions as a force for change in allele frequencies in populations. Note that effectiveness of both depends on the allele frequency.
p is the change in allele frequency from one generation to the next. In this example, mutation and selection are acting in opposite directions as
p is positive under mutation pressure and negative under selection pressure. Note that the values of
p under both forces only become comparable when the allele frequency is low.
F r e q u e n c y o f
a
a l l e l e
= 0.0001
So, we need to determine
p under both forces (mutation and selection) and specify that they are at equilibrium. Remember the following from previous lectures:
Attainment of the equilibrium allele frequency given selection and a variety of different mutation rates. Note that the time to equilibrium varies in addition to the actual equilibrium frequencies.
s
= 0.1 (
W
aa
= 0.9)
= mut rate A
a
= 0.01
= 0.001 = 0.0001 = 0.00001
a Note that fore realistic mutation rates, the equilibrium frequencies are quite low (freq of
a
allele > 0.05). In this example selection pressure is also quite weak (
s
= 0.1). If we assume stronger selection pressure (
s
> 0.1), the equilibrium point will be lower and the rate to equilibrium will be faster.
1. Mutation pressure:
Let
= the mutation rate from
A
a
Let
= the mutation rate from
a
A
Let
p
t
= the frequency of A in the population in generation
t
. Let
q
t
= the frequency of a in the population in generation
t
, with
q
t
= (1 –
p
t
). The change in allele frequency is:
rateat mutation
byA tochangethat
allelesaof freqThe
rateat mutaion
byatochangethat
allelesA of freqThe
vq pq
This is clearly dependent on the frequencies
p
and
q
.
Now we can specify equilibrium:
q (mutation pressure) =
q (selection) p
 q
=
S
q
t 2
(1
q
) / 1
sq
t
2
Yuck, that’s far too complicated to be much help, we need to simplify it. To do this we will make some simplifying assumptions, and use some approximations. First, take my word that equilibrium conditions will only be met for alleles with very low frequencies in the populations. In such cases, q is small (< 0.05) and the rate of “back mutation” (a to A at rate q
), is so small that we can safely ignore it. [Note that frequency of p in such cases will necessarily be quite high (> 0.95) so we can’t ignore mutations from this large body of A alleles to the recessive (p
).] So let’s drop the q
term! p
=
S
q
t 2
(1
q
) / 1
sq
t
2
Now the selection term can’t be simplified. But, we can use a much simpler approximate formula. In short, if the gene frequency q is small enough, the denominator gets very close to unity. The denominator is 1
sq
t
2
; so (1 (very small number))
1. So, let’s just call it unity and use the numerator for the selection term (
S
q
t 2
(1
q
))
p
=
S
q
t 2
(1
q
) (1
q
)
=
S
q
t 2
(1
q
)
=
S
q
t 2
sq
(approx.) It’s a royal pain, but that’s how it is done.
2. Natural selection against a deleterious recessive allele
Remember form our earlier lecture:
q
t
+1
=
q
t

S
q
t 2
/ 1
sq
t
2
So for
q,
q =
q
t
+1

q
t
q = (
q
t

S
q
t 2
/ 1
sq
t
2
) 
q
t
q = 
S
q
t 2
(1
q
) / 1
sq
t
2
When there is partial dominance (i.e., h > 0) we can do similar tricks and get the following approximation.
q
=
/
hs
(approx.) Why the difference in the approximate formulas? The reason is that even with a small amount of partial dominance, selection can “see” the recessive allele in the heterozygotes. Remember that we are looking at equilibrium values where q is small; and when q is small the number of a alleles in the heterozygotes will be much much larger than those found in the aa homozygotes. Hence, selection will be more effective and the equilibrium frequency will be driven much lower. See the figure below for an illustration on the effect of partial dominance on mutationselection equilibrium.
Effect of partial dominance on mutationselection equilibrium. The fitness of genotypes AA, Aa, and aa are assumed to be 1, 1–hs, and 1s respectively.
00.0050.010.0150.020.0250.030.0350.0000010.000010.00010.001
h = 0h = 0.01h = 0.05h = 0.1h = 0.5
E q u i l i b r i u m f r e q u e n c y o f r e c e s s i v e a l l e l e (
a
)
Ratio of mutation rate to selection coefficient against aa (
/s)
The symbol h is the amount of dominance in the heterozygote genotype. Note, that even a small amount of dominance (h = 0.01) reduced the equilibrium frequency of the recessive allele. Hence, dominance has a significant influence on the equilibrium point. The reason is that when q, the freq of the recessive allele is small, the majority of those alleles are in the heterozygote configuration, and even a small amount of selection on the heterozygotes leads to a major reduction in its equilibrium frequency as compared with full dominance. Note that for reasonable values of
, h, and s, the equilibrium frequencies are < 0.01, This means that mutation selection equilibrium is not sufficient to explain low frequency detrimental alleles in populations where those alleles have frequencies > 0.01