Download PopGen 8: Transient verses equilibrium polymorphism Mutation

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.
Mutation-selection 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.
Frequency of a allele
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.
µ = 0.0001
∆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.
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)
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 (Waa = 0.9)
µ = mut rate A → a
a
µ = 0.01
µ = 0.001
µ = 0.0001
µ = 0.00001
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.
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:
1. Mutation pressure:
Let µ = the mutation rate from A ⇒ a
Let ν = the mutation rate from a ⇒ A
Let pt = the frequency of A in the population in generation t.
Let qt = the frequency of a in the population in generation t, with qt = (1 – pt).
The change in allele frequency is:
∆q =
p (µ )
{
The freq of A alleles
that change to a by
mutaion at rate µ
−
q (v )
{
The freq of a alleles
that change to A by
mutation at rate ν
This is clearly dependent on the frequencies p and q.
2. Natural selection against a deleterious recessive allele
Remember form our earlier lecture:
qt+1 = qt - Sqt2 / 1- sqt2
So for ∆q,
∆q = qt+1 - qt
∆q = (qt - Sqt2 / 1- sqt2) - qt
∆q = - Sqt2(1- q) / 1- sqt2
Now we can specify equilibrium:
∆q (mutation pressure) = ∆q (selection)
pµ - qν =
2
S qt
(1- q) / 1- sqt2
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 qν), 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 (pµ).] So let’s drop the qν term!
pµ =
2
S qt
(1- q) / 1- sqt2
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- sqt2; so (1- (very small number)) ≅ 1. So, let’s just call it unity and use the
numerator for the selection term (Sqt2(1- q))
2
S qt
pµ =
(1- q)
2
(1-q)µ =
S qt
µ =
S qt
q= µ
It’s a royal pain, but that’s how it is done.
s
(1- q)
2
(approx.)
When there is partial dominance (i.e., h > 0) we can do similar tricks and get the following
approximation.
q = µ/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 mutation-selection
equilibrium.
Equilibrium frequency of recessive allele (a)
Effect of partial dominance on mutation-selection equilibrium. The
fitness of genotypes AA, Aa, and aa are assumed to be 1, 1–hs, and 1s respectively.
0.035
0.03
h=0
0.025
h = 0.01
h = 0.05
h = 0.1
0.02
h = 0.5
0.015
0.01
0.005
0
0.001
0.0001
0.00001
0.000001
Ratio of mutation rate to selection coefficient
against aa (µ/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 µ, 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
Selection-drift
Directional selection:
Remember the drift affect is dependent on the effective population size. With only drift acting, the
probability of fixation is the frequency q. When selection is acting, the fate of the allele will be
determined by the interaction of drift (depending on effective populations Ne) and selection
(depending on s). The product of Ne and s is used as a guide: if Nes > 1 the beneficial allele is
more likely going to be fixed than under drift alone, and when Nes < 1 selection pressure is not
powerful and the probability of fixing a beneficial allele is close to its initial frequency in the
population.
Then fate of a beneficial recessive allele (A1) is not always predictable under the combined effects of
directional selection and genetic drift. If there is no genetic drift (left: Nes = infinity), the fate of the recessive allele
(A1) is always determined by selection. When there is drift (right: Nes < infinity) the fate of the recessive allele (A1)
is not necessarily determined by selection; hence a deleterious allele can be fixed in a population.
Nes = infinity
Nes = 100
Note that Nes > 1 does not guarantee that an allele is going to be fixed, it simply indicates that (as a long term
average) the frequency that it is fixed will be greater than the frequency under genetic drift alone.
In order for selection to dominate in most cases, the population sizes must be sufficiently large,
and selection pressure sufficiently large; i.e., Nes >> 1, otherwise beneficial alleles will lost
occasionally due to change alone. This means that in real populations, that have finite population
sizes, maladaptive alleles can be fixed. The upper limit of the fitness consequences of such
alleles will be determined by Ne; the smaller the population the more maladaptive the allele that
could be fixed due to drift.
In the above example, the plot on the right indicates that the beneficial allele is lost due to drift
more often than it is fixed. In this example the initial frequency = 0.01 (equivalent to a new
mutant in the population). If there were no selection, the expected frequency of fixation is simply
1 in 100, where Ne = 100 and Nes = 0. However, when Nes = 100 the frequency of fixation
elevated to 1 in 10. So, Nes >1 does not mean that allele is always fixed; it simply means that
probability of fixation is greater than under drift (chance).
Whatever the relationship between directional selection and drift (Nes > 1 or Nes < 1), the ultimate
fate of the allele is fixation or loss. Clearly, in this case polymorphism is transient in the
population.
Over-dominant selection:
In an earlier lecture we examined how opposing selection pressures can result in a stable
equilibrium. Specifically, we looked at a single locus with two alleles, where the heterozygote had
a superior fitness; i.e., W11 < W12 > W22.
The equilibrium that can result from this case is globally stable. That is, the allele frequencies will
converge to the equilibrium value regardless of the initial frequency.
s1 = 0.1
s2 = 0.3
However, in the above figure we modelled over-dominant selection in an otherwise ideal
population. In finite populations the equilibrium will be disturbed each generation by genetic drift.
Equilibrium frequencies under balancing selection (s1 = 0.1 and s2 = 0.3) are less stable under the influence of
genetic drift (right) as compared with an otherwise ideal population (left).
Because drift disturbs the allele frequencies each generation, frequencies in any one generation will not be in equilibrium. However, the
long term average will be the equilibrium frequencies. The polymorphism in both of the above cases is not transient.
In the above case drift was strong enough to disturb the allele frequencies each generation, but
no so strong as to disturb the long term average under s1 = 0.1 and s2 = 0.3. Given the same
selection coefficients but much stranger drift effects the allele frequencies are disturbed more
strongly and the fate of an allele is either fixation or loss.
Again Nes will determine the conditions where drift will be strong enough for the population to go
to fixation for one or the other of the alleles (or selection too weak to avoid this fate). When Nes is
small most populations will be fixed for one allele or the other; whereas when Nes is high many
populations will be polymorphic, but some will nonetheless be fixed.
Polymorphisms under balancing selection (s1 = 0.1 and s2 = 0.3) is
transient if drift effects are strong. (Ne = 50).
This raises an important point: selection acting in opposite directions does NOT guarantee a
stable equilibrium, and proving that a polymorphism in a natural population is the result of
opposing selection pressures is very difficult.
In summary, polymorphism can be either transient or at equilibrium under balancing selection; the
particular outcome depends on the strength of genetic drift in the population; i.e., effective
population size.
Mutation-selection-drift equilibrium
Let’s put them all together; see the plot below.
Combined effects of mutation, selection and drift on polymorphism. When drift is weak polymorphism is not
transient, but when drift is strong the polymorphism is transient, but recurring due to mutation.
Nes = infinity
Nes = 1000
equilibrium
Nes = 100
Nes = 10
When drift is strong relative to directional selection, the population quickly goes to fixation. The
process of mutation occasionally re-introduces the allele back into the population, but at such a
low frequency (1/2Ne) the allele is quickly lost due to drift (see above figure with Nes = 10).
Because mutation pressure is such a weak force as compared with drift (in small populations)
and selection, the dynamics of the system will be dominated by the effects of selection and drift
when population sizes are small. When population sizes are large the long-term equilibrium is set
by selection and mutation.
We may also consider the combination of mutation pressure, overdominant selection and genetic
drift. Here mutation pressure has very small effect on the equilibrium frequencies, even when the
mutation rate is quite high. So in this case the dynamics of the system are dominated by
selection pressure and drift.
Very high mutation rate (0.01) results in only a small shift in the
long term average allele frequency under overdominant selection
drift + mutation + selection
µ = 0.01
drift + selection
Nes = 1000
Inbreeding and selection (does not lead to equilibrium)
We have seen that inbreeding itself cannot alter gene frequencies. However, the combination of
inbreeding and natural selection can yield a powerful force for increasing the frequency of a
homozygous genotype.
Like inbreeding, positive assortative mating in combination with natural selection can be a
powerful force for evolution of populations. An important distinction is that (i) positive assortative
mating only affects those genes that are associated with the phenotypic traits that are influencing
mating frequencies; and (ii) the effect can be stronger when coupled with positive assortative
mating because it is focused on few genes.
Mutation-Drift equilibrium
This interaction was ignored until the late1960’s. The reason it was ignored was: (i) mutation-drift
equilibrium does not lead to a stable polymorphism, and (ii) selection is not involved, so the
process was considered rather uninteresting because it is unrelated to the process of adaptation.
The interaction of mutation and drift is of central importance to molecular evolution. It eventually
leads us to the “neutral theory of molecular evolution”. We will return to this later in the course.
Sources of polymorphism in populations
1. Mutation-selection equilibrium (long-term)
2. Overdominace equilibrium (long-term)
3. Mutation-drift (transient, but important)
• In cases 1 and 2, genetic drift constantly disturbs the equilibrium. The strength of the
disturbance depends on the effective population size. If drift is strong enough, the
disturbances can push the frequency to fixation, thereby terminating the polymorphism.
Hence, we don’t expect to see persistent equilibrium polymorphisms in populations with low
effective size.
• For realistic values of µ, h and s, the equilibrium point is generally very low (p < 0.01). As an
explanation for natural polymorphisms > 0.01, the balance between mutation and selection is
not satisfactory.
• Overdominant selection can explain population polymorphisms with frequencies > 0.01.
However, there is a cost in fitness to an equilibrium involving selection which makes it unlikely
that it can be invoked as a common mechanism for maintaining polymorphisms in natural
populations. We will return the notion of the cost of selection later in this course.