Lecture 8: Mutation, Migration, with and without selection

Lecture 8: Mutation, Migration, with and without selection¶

Mutation and Evolution¶

Previously we learned about how selection can change the frequencies of alleles and genotypes in populations. Selection typically eliminates variation from within populations. (The general exception to this claim is with the class selection models we have called “balancing” selection where alleles are maintained in the population by overdominance, habitat-specific selection, or frequency dependent selection). If selection removes variation, soon there will be no more variation for selection to act on, and evolution will grind to a halt, right? This might be true if it were not for the reality of mutation which will restore genetic variation eliminated by selection. Thus, mutations are the fundamental raw material of evolution.

The basic model of mutation that we will study is one way mutation. This is when one allele through mutation can turn in to another such that

\[\begin{aligned} A_1 \stackrel{u}\longrightarrow A_2 \end{aligned}\]

$u$ here represents the mutation rate, the probability that a mutation from $A_1$ to $A_2$ occurs during a meiosis.

It should be obvious that mutation will change allele frequencies. This is true because there is a constant flux from $A_1$ to $A_2$ purely as a result of this mutation process. We can study the change in allele frequency due to mutation in a very similar way to how we studied the change in allele frequency due to selection. Consider a population with frequency of the $A_1$ allele $p$. In the next generation, after a round of mutation, each $A_1$ allele must have been $A_1$ in the current generation and it must not have mutated. That is

\[\begin{aligned} p' = p (1-u). \end{aligned}\]

Now lets turn our attention to the change in allele frequency in one generation due to mutation as we did previously for selection

\[\begin{split}\begin{aligned} \Delta_up & = & p' - p \\ & = & p(1-u) -p \\ & = & -up \\ \end{aligned}\end{split}\]

notice that our notation– $\Delta_up$ – emphasizes the source of the change in allele frequency is mutation.

This sort of unidirectional mutation acts to consistently decrease the frequency of the $A_1$ allele from generation to generation. If we instead were to study two way mutation, the mutational flux would depend on the proportional rates to and from $A_1$. So mutation, although it is a random process with respect to target, leads to deterministic effects on allele frequencies. Neat huh?

Mutation rates per generation are very very small. You know this intuitively- think about cloning plants from cuttings. In Drosophila, which is one of the best studied animals from the perspective of rates of spontaneous mutation, the mutation rate per generation per nucleotide is on the order of $10^{-9}$. This means that mutation changes the frequency of alleles are a very slow rate. If we assume that there is sufficiently strong selection against the $A_2$ allele (i.e. $w_{11} >> w_{22}$), then we can further approximate our change in allele frequency due to mutation

\[\begin{split}\begin{aligned} \Delta_up & = & -up \\ & = & -u + qu \\ & \approx & -u, \end{aligned}\end{split}\]

because $q \approx 0$. So in the case of a deleterious $A_2$ allele, we can see that the change in allele frequency due to mutation is independent of allele frequencies. This is our first hint that mutation and selection might combine in interesting and important ways.

Mutation-Selection Balance¶

We can imagine mutation and selection as opposing forces which might come to some equilibrium in terms of the number or frequency of deleterious alleles within a population. To make this concrete think about the human genetic disease cystic fibrosis (CF). CF is a very serious genetic disorder in which a transmembrane protein in lung epithelium cells called CFTR is non-functional. Hundreds, if not thousands of separate mutations in CFTR lead to CF, thus we could imagine that there is a certain, appreciable rate of mutation to CF. If each of these mutations is deleterious (i.e. they cause disease) then over generations they should be selected out of the population. Thus mutation will inject CF mutations into the population, but selection will remove them- can we study this as an equilibrium process?

Our approach will be to study each of our evolutionary forces in isolation, and then combine them to figure out how they interact. Let’s start by considering the change in allele frequency due to selection that we studied in lecture 7, but this time we will approximate it under the assumption that $q \approx 0$

\[\begin{split}\begin{aligned} \Delta_sp & = &\frac{pqs[ph + q(1-h)]}{\bar{w}} \\ & \approx & qhs \end{aligned}\end{split}\]

This approximation goes down the road because when $q \approx 0$, $p \approx 1$, $\bar{w} \approx 1$, and we can ignore all terms of order $q^2$.

Now lets combine the forces of selection and mutation on the change in frequency of $A_1$ using the approximations we have just derived (equations 3 and 5). At equilibrium that change in allele frequency due to the combined actions of mutation and selection must equal zero. That is

\[\begin{split}\begin{aligned} 0 & = & \Delta_up + \Delta_sp \\ & \approx & -u + qhs \end{aligned}\end{split}\]

so the equilibrium frequency of the $A_2$ is

\[\begin{aligned} \hat{q} \approx \frac{u}{hs} \end{aligned}\]

Thus we see that deleterious (e.g. disease) allele frequencies are determined by both the mutation rate to those alleles and their selective effects in heterozygotes. As we saw earlier, new mutations overwhelmingly are found in heterozygous states, so it’s perhaps not surprising that $h$ should dominate the fate of deleterious alleles.

Problem: Go through the same steps of approximations that we just did to find the Mutation-Selection equilibrium value of mutations which are completely recessive (i.e. $h = 0$). This would make a heck of an exam question….

Migration and Gene Flow¶

In population genetics, the term “migration” is really meant to describe Gene flow, defined as the movement of alleles from one area (deme, population, region) to another. Gene flow assumes some form of dispersal or migration (wind pollination, seed dispersal, birds flying, etc.) but dispersal is not gene flow (genes must be transferred, not just their carriers)

We are going to build a model of gene flow in exactly the same way we studied mutation. Consider two populations, a mainland population and an island population. Each of these populations has the $A_1$ allele at frequencies $p_{main}$ and $p_{island}$ respectively. Assume that gene flow is one way, from mainland to island and that the proportion of individuals who become parents in the island population is $m$. Although I’ve said this is a proportion, also notice that we could consider this a probability interchangeably. After a round of migration, in the island population there are then two sources for alleles, they could be from the island population originally with probability $1-m$, or they could have migrated from the mainland with probability $m$. This means that after migration the allele frequency of $A_1$ in the island population is $$\begin{aligned} p_{island}' = p_{island}(1-m) + p_{main}m. \end{aligned}$$

Simple enough right? Now lets to what comes naturally and study the change in allele frequency as a result of mutation. Following what we have done in our other analyses,

\[\begin{split}\begin{aligned} \Delta_{m}p_{island} & = & p_{island}' - p_{island} \\ & = & p_{island}(1-m) + p_{main}m - p_{island} \\ & = & m(p_{main} - p_{island}) \\ \end{aligned}\end{split}\]

Beautiful. Now we have a very simple expression for how allele frequencies in the island population should change due to gene flow from the mainland population. This change in allele frequency makes sense- it only depends on the amount of migration and the differences in allele frequencies between the two populations. It’s simple enough to generalize this to multiple populations, or to populations at different distances away from one another, but we won’t cover that here.

Migration and Selection¶

One of the fundamental observations in biology and natural history is that of local adaptation- populations of organisms adapt to their local surroundings. Think of populations of plants that live in dry environments that might be better able to handle drought than their conspecifics who live in wet environments. Another example might be populations of humans who are better able to handle solar radiation than others. How does this kind of adaptation take place in the face of the homogenizing influence of gene flow? The answer is through the interplay between migration and selection.

We will study this struggle between migration and selection in exactly the same way we have studied the other combinations of forces in this lecture. Consider that some weak allele is wafting over to the other side of the tracks, so to speak, where they do not survive (e.g., fish swimming into New York harbor). There is an evolutionary pressure changing allele frequencies in one direction ( into the harbor), and an opposing evolutionary force eliminating those alleles (sewage killing off genetically intolerant fish). Depending on the relative strengths of these two opposing forces, an equilibrium condition can arise.

Imagine that the fish population outside of the New York Harbor is fixed for the $A_1$ allele (i.e. that $p = 1$), and that in the New York Harbor we get the following array of fitnesses as a result of sewage intolerance

Genotype:	$A_1A_1$	$A_1A_2$	$A_2A_2$
Relative Fitness:	$1 -s$	$1$	$1$

So the $A_1$ allele in this case is completely recessive in the death-by-sewage phenotype.

Let’s first look at the effect of migration on the frequency of the $A_2$ allele in the NY Harbor population. Note that I’m switching attention from $A_1$ to $A_2$ because we are assuming that $A_1$ is fixed outside of the harbor (in the ocean say). The the change in allele frequency, $q$ due to selection in the Harbor population is

\[\begin{aligned} \Delta_mq = -mq \end{aligned}\]

This is just a rearrangement of what we did before, and is left as an exercise to the reader. The next component we need is the change in allele frequency due to selection. We’ve written this down numerous times at this point,

\[\begin{aligned} \Delta_sq = \frac{pqw_{12} + q^2w_{22}}{\bar{w}} - q. \end{aligned}\]

The only thing left is to set up the equilibrium condition. Equilibrium between selection and gene flow occurs when the change in allele frequency due to both forces equals zero,

\[\begin{aligned} 0 = \Delta_sq + \Delta_mq \end{aligned}\]

As an instructive exercise go and iterate this equation for some realistic values of s, h, and m.

Lecture 7: Natural Selection II: Population Genetics

Lecture 9: Genetic Drift

Bio 380-- Evolutionary Biology