Population Genetics III: Maintenance of Genetic Variation
We have seen that a lot of genetic variation is found in traits on all levels of organismal organization,
from molecular to behavioral. Genetic variation is maintained in a population by pure Medelian
heredity. However, pure mutation, genetic drift (as we will see in the next lecture), and in particular
selection towards an optimal allele will erode it. This leads to one of the most important questions in
evolutionary biology: What maintains genetic variation in a population in the face of selection? There
are two possibilities: various scenarios of balancing selection or mutation-selection balance.
Balancing selection
Previously, we have assumed that one allele of at least two has the highest fitness throughout. As a
consequence, this allele will go to fixation and genetic variation is lost. We will now consider the case
that depending on circumstances, a given allele may either be favored or disfavored. Any such form
of selection is called balancing selection. As we will see, it is possible to maintain genetic variation at
a locus in this case. There are many types of balancing selection. For example:
• Spatial variation: An allele is favored in one part of the population habitat, but disfavored in
another part. E.g. the allele may have a function in predator protection but come at a physiological
cost. It will then increase fitness and be positively selected only in regions of the habitat where
predator density is high.
• Temporal variation: Similarly, alleles may be favored only under certain environmental conditions
(e.g. in a certain temperature range). If the external environment fluctuates, also the selection
pressure on the allele will fluctuate and may switch sign.
• Frequency dependence: The selective advantage or disadvantage of an allele may also depend
on its own frequency in the population. In this case, it is maintained polymorphic in a population if
it is favored as long as it is rare, but disfavored if it is very common. The most famous form of
frequency dependent selection is heterozygote advantage.
Heterozygote advantage (overdominance).
Consider the case where the heterozygote is the fittest. Selection will obviously tend to keep this
heterozygote at high frequency. Intuitively, then, we might expect selection to keep both alleles in the
population. It's easy to check this intuition:
Genotypes
AA
Fitnesses
Aa
wAA = 1-s
wAa = 1
aa
waa = 1-t
Both s and t are selection coefficients (s > 0, t > 0). The general formula for the change of allele
frequencies (derived in the last lecture) reads
Δsp =
pq[p(w AA − w Aa ) + q(wAa − wa a )]
.
w
(1)
Substituting the above fitnesses into this equation, we obtain
Δsp =
pq[qt − ps]
.
w
Notice that Dsp = 0 (allele frequencies stop changing) when the term in brackets = 0. Thus
qt – ps = 0
or
(1-p)t = ps
(2)
Solving this for p gives
pˆ =
t
s+t
and
qˆ =
s
s+t
We have found the "equilibrium allele frequencies" — a point at which selection will no longer change
ˆ , but negative if p > pˆ .
allele frequencies. It is easy to show that ∆sp is positive as long as p < p
Thus, the allele A is favored when it is rare, but disfavored when it is common. The polymorphic
equilibrium is stable; if allele frequencies drift away, selection will push them back. We conclude that,
with heterozygote advantage, selection does not remove one allele from the population. Note that the
frequency of heterozygotes in the population is never at 1 since homozygotes are inevitably created
each generation by random mating.
Example: The classic example in humans is sickle cell anemia. In the homozygous state, aa, the
sickle allele causes distortion of the red blood cells; they can then block capillaries and cause severe
anemia. About 80% of aa individuals die before reproduction. In the heterozygote, the red blood cells
do not normally become distorted, but only under certain conditions. One of these conditions is
infection by Plasmodium falciparum (malaria), meaning that infected cells die, and along with the cell
the parasite. Heterozygotes are therefore better protected against malaria than the wild type
homozygote AA. This leads to heterozygous advantage in malaria infected areas and measured
equilibrium frequencies of the sickle allele of up to 14%.
By comparing observed adult frequencies of the three genotypes with the Hardy-Weinberg
proportions (expected at zygote stage), it is possible to estimate the selection coefficients s and t. In a
study the following genotype frequencies were found in a sample of 12387 individuals from Nigeria:
aa: 29 adults, 0.234%
Aa: 2993 adults, 24.2%
AA: 9365 adults, 75.6%
This results in a frequency q of the a allele of q = (29 + 2993 / 2) / 12387 = 0.123. We can calculate
the expected zygote frequencies (assuming Hardy-Weinberg equilibrium):
aa: 1.51%
Aa: 21.6%
AA: 76.9%
Dividing the adult frequencies by the zygote frequencies, we obtain the fitness measures 0.155 for aa,
1.12 for Aa, and 0.983 for AA. Normalizing the maximum fitness (of Aa) to 1, this gives the estimates:
waa ≈ 0.14
wAa = 1
wAA ≈ 0.88
Mutation-Selection Balance
So far we have only considered one evolutionary force acting on the population at a time. Let us now
analyze the case of mutation and selection acting simultaneously. We can discuss this issue by
asking the question: Why are there disease alleles in populations?
Generally, there is a large number of deleterious alleles in populations although many are not or only
mildly expressed in heterozygous state. In Drosophila, 20-25% of chromosomes carry recessive lethal
alleles (lethal in homozygotes). Things are similar in humans: We all carry, with very high probability,
alleles for heritable deformities. For many of these defects, fitness may look like this:
A1A1
w=1
A1A2
w=0.95
A2A2
w=0.2
Why doesn’t natural selection get rid of the A2 allele that stands for, e.g.:
• Cystic fibrosis,
• Huntington’s chorea,
• Achondroplastic dwarfism?
Answer: Although natural selection is driving bad alleles out of the population, mutation continually
introduces new A 2 alleles in each generation. One can imagine some sort of equilibrium between
natural selection purging bad alleles from the population and mutation continually introducing bad
alleles into the population.
We consider a single locus with two alleles and unidirectional mutation from the wild type allele A1 to
the mutant allele A2:
u
A1
p
A2
q
Back mutations from the mutant allele the wild type are ignored (they are always rare, but even rarer if
the allele A2 is selected against and thus at low frequency). Fitnesses of the genotypes are w11=1,
w12=1-hs, and w22=1-s. The allele frequency change per generation due to mutation alone then is
∆mp = – up
We have calculated the change due to selection alone in the last lecture:
Δsp =
pqs[ ph + q(1 − h )]
w
If both forces act simultaneously, the total change is (approximately) the sum of the two
∆p = ∆sp + ∆mp
The equilibrium point is attained for ∆p = 0, or if the gain in p due to selection equals the loss due to
mutation:
pqs[ ph + q(1 − h )]
w
= up
There is a trivial solution to this equation at p = 0. This is the case where only mutants are present. As
it turns out, this equilibrium is not stable and also vanishes for any arbitrarily small amount of back
mutation. For any other equilibrium, we divide both sides by p and replace ph on the left hand side by
(1-q)h. As long as u is small, there will be only relatively few mutants and we can ignore terms
2
proportional to q and approximate the mean fitness w by 1. Thus
qs[(1-q)h+q(1-h)] ≈ qsh = u
Solving for q,
qˆ =
u
hs
The equilibrium frequency of the mutant allele is thus approximately given by the ratio of the mutation
rate and the selection strength hs on the heterozygote. We note the following points about mutation
selection balance:
1) Mildly deleterious alleles and more recessive alleles occur at higher equilibrium frequencies
as strongly deleterious ones.
2) We can estimate the frequency of carriers at equilibrium: As long as q is small, almost all
mutant alleles will be in heteroyzgotes, thus carrier frequency ≈ 2 x q̂ .
Example: Achondroplasia (Dwarfism) is a genetic decease due to a single dominant nucleotide
substitution on an autosome. A study shows 10 cases of dwarfism among 94075 births. We can
estimate the frequency of the allele as ½ the frequency of these cases (homozygote mutants can be
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ignored), thus q = H/2 = (10 / 94075) / 2 = 5.31 X 10 . A study further shows that heterozygote dwarfs
have on average 0.25 kids as compared to an average of 1.27 kids of their unaffected siblings. This
translates into a relative fitness 1 – hs = 0.25/1.27 = 0.2 of dwarfism, thus hs = 0.8. Since u = qhs, we
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can use this result to estimate the mutation rate as u = qhs = (5.31 X 10 )(0.8) = 4.25 X 10 . This
result has been verified by direct estimates.
Mutation Load
The effect of a deleterious mutation on an individual is given by the reduction in individual fitness, as
measured by the selection coefficient s. Similarly, we can measure the effect of the same mutation on
the level of the population. This is done by the mutation load. The mutation load Lm is defined as
Lm =
wopt − w
wopt
wopt is the optimal (wild type) fitness and w the mean fitness. With the convention wopt = 1 ,
we simply have Lm = 1 − w . We now ask: Given that a deleterious mutation, which occurs at rate u in
where
a population has an individual effect of s, what is its effect L on the population level? We obtain
w = p2 + 2pq(1 – hs) + q2(1 – s) = 1 – 2pq hs – q2 s ≈ 1 – 2q hs ≈ 1 – 2u
and thus L ≈ 2u. The mutation load is independent of the individual selection coefficient. On the
population level, any mildly deleterious mutation is as harmful as a lethal mutation that occurs at the
same rate. The reason, of course, is that individually less harmful mutations are present at a higher
frequency in mutation selection balance. Nevertheless, the finding that on the population level mild
mutations are just as problematic as even lethals is one of the most counterintuitive results of
population genetics.
1) Eugenetics programs to suppress proliferation of severe genetic diseases by preventing
carriers from reproduction (e.g. sterilization) fail to improve the population mean fitness. Also,
it seems arbitrary to single out certain genetic diseases since mildly deleterious mutations
have the same contribution to the load.
2) Better treatment of genetic diseases to increase individual fitness also does not improve the
population mean fitness.
The only way to reduce the mutation load is to reduce the frequency of new mutations. In this context
the mutation load was originally introduced by H.J. Muller. Muller had discovered that x-ray cause
mutations. He was concerned about how much we are lowering the mean fitness of the human
population due to “small” mutations that are introduced by radiation (x-rays or above-ground nuclear
weapons testing).
What maintains genetic variation?
Despite a lot of effort, no final answer to this question has been obtained. Since deleterious mutations
are ubiquitous and unavoidable, some part is certainly due to mutation-selection balance. The crucial
question is whether this is (almost) all or whether some type of balancing selection is also important.
Nucleotide polymorphisms are often observed at high frequencies (i.e. with high heterozygosity),
which is not easily explained by pure mutation-selection balance. Nevertheless, there are good
arguments that heterozygote advantage does not play a major role. However, temporal or special
fluctuations in selection strength could contribute significantly to the genetic variation.