Lecture 44: Population Genetics II:

Mutation, Inbreeding, and Genetic Drift

(version 8 November 2005)

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The next few lectures examine deviations from Hardy-Weinberg that arise when additional evolutionary forces are considered.

Mutations arise, and as a net result alleles change over time (albeit very slowly).

The simplest model assumes two alleles with forward and back mutation

Thus, allele frequencies change each generation, eventually approaching an equilibrium value

Consider the frequency of allele A. A fraction (1-u) of these alleles do not mutate, while a fraction v of the a alleles mutate to A. Hence, the change in allele frequency is

With only mutation acting, the rate of change in allele frequency is very slow, on the order of 1/u generations, typically on the order of millions of generations (in a very large population).

A more realistic model for mutations is the infinite alleles model of Crow and Kimura (1964), which states the each new mutation gives rise to a new allele never seen in the population before. We discuss the features of this model after we examine genetic drift.

Suppose, instead of random mating, related individuals mate. In this case inbreeding occurs, and this causes departures from Hardy-Weinberg.

The key measure of inbreeding is F, the inbreeding coefficient , which can be determined from pedigree information.

F equals the probability that two alleles in an individual are identical by descent (ibd), which means that both alleles can be traced back to a single copy in some distance ancestor.

As an example, consider the offspring in a mating of full sibs

The offspring on the left has both alleles ibd, as they both trace back to the red allele in one parent.

For this locus, the offspring on the right (in the top figure) does not have both alleles ibd. However, other loci are likely ibd. Here, F = 1/4, so 25% of all loci should have both alleles ibd. Alternatively, for any single locus, 25% of all individuals should have both alleles ibd.

Under inbreeding, the genotypic frequencies differ from those under Hardy-Weinberg, with

• Freq(AA) = F*Freq(A) + (1-F)*Freq(A)*Freq(A)

• Freq(Aa) = (1-F)*2Freq(A)*Freq(a)

• Freq(aa) = F*Freq(a) + (1-F)*Freq(a)*Freq(a)

Derivation: Pick a random individual.

• With probability F both alleles are idb, so that only one allele is chosen at random.
  • The probability that such ibd individuals are AA is freq(A).

• With probability (1-F) these individuals have both alleles chosen at random
  • Here, the genotypes are in Hardy-Weinberg frequencies.

Under continued inbreeding (say brother-sister matings or selfing), F approaches one, so that all individuals are homozygotes.

Hence, inbreeding reduces heterozygosity, with F being the fraction by which heterozygosity is reduced.

In a finite population, random mating still results in inbreeding, as there is some small chance that individuals share alleles ibd. This is the basis for Genetic drift.

Viewed another way, if the population size is kept constant, each copy of a particular allele either:

• leaves exactly one descendant
• leaves no descendants
• leaves more than one descendant

The net result is that the population ends up being fixed for only a single allele.

• This repeated sampling results in the fixation of a randomly-chosen allele.

• Drift, by itself, removes genetic variation.

Which allele is fixed is determined entirely by chance, with

• Prob(A is fixed) = initial frequency of A

For example,

• Suppose freq(A) = .5, then 50 percent of the time the population should fix A.
• However, if freq(A) = 0.1, then A is fixed only 10 percent of the time.

In a finite population, all alleles trace back to a common ancestor, which leads to drift often being called a coalescent process

In a population of size N, then if we draw two random alleles (here, two random sequences at a particular locus), then the time back to a common ancestor for these two sequences follows a geometric distribution with per-generation success probability p = 1/(2N). Under a geometric distribution, the probability of no successes in t generations is (1-p)^t, while the probability of a success in generation t is p(1-p)^t-1

• Hence, the probability that both copies come from the same ancestor in the previous generation is 1/(2N)
• Likewise, the expected time back to a common ancestor is 2N generations

If all populations are finite, why do we see variation? Would not drift remove all variation initially present?

The solution is that mutations are occurring all of the time. Mutation introduces new variation, which is removed by drift.

The net result is a dynamic equilibrium wherein

• The population maintains a certain amount of variation
• The alleles making up this variation are changing over time

Here, alleles a and b are initially present, while alleles c and d arise by mutation. At time 1, alleles a and b are segregating; at time 2, b and c are segregating; at time 3 b, c, and d are all segregating. Note that the average level of heterozygosity remains roughly constant over time, but that the actual alleles (and their frequencies) accounting for this heterozygosity changes over time.

Crow and Kimura (1964) introduced the infinite-alleles model of drift and mutation.

• Here, each mutation gives rise to a new allele never before present in the population
• The motivation for this is that treating a long DNA sequence as a gene, each new mutation likely occurs in a different base pair

• the expected heterozygosity H is

Hence,

• when 4Nu is large, lots of variation; with H close to 1 if 4Nu >> 1

• when 4Nu is small, little variation, with H = 4Nu << 1

How many mutations have occurred if we draw two random sequences?

• If the two sequences have a common ancestor t generations ago, the expected number of mutations follows a Poisson distribution (see Problem set 1) with mean 2tu.

If the average time back to a common ancestor for two alleles is t = 2N, the expected number of mutations between two random sequences is 2tu = 4Nu

• If 4Nu > 1, we expect, on average, at least one mutational difference between two random alleles, and we should score this individual as a heterozygote.

One consequence of drift fixing alleles and mutation introducing new alleles is that the DNA sequences of two isolated populations drift apart by substituting different mutations.

The expected substitution rate per generation is given by

• ( Number of new mutations per generation ) * ( Prob of fixation of a single new mutation)

• = ( 2Nu ) (1 / [2N] ) = u

We need to distinguish between the time to fixation and the time between appearance of new mutants destined to become fixed.

If u is the mutation rate,

• The expected time between the initial appearance of new mutants destined to become fixed is u^-1 generations.

• The expected time to fix those mutants that are destined to become fixed is roughly 4N generations.

• The expected time an allele remains in the population following its introduction as a single copy (most are lost) is approximately ln(N) generations.

For example, if u = 10^-7 and N = 10⁵ , then

• the expected time between mutants that are fixed is = 10⁷ generations

• each such mutant taking (on average) about 400,000 generations to become fixed.

• The majority of new mutants (1-1/[2*10⁵] = 0.99999) are lost.

• Each of these persists in the population for an average of ln(10⁵) = 11.5 generations.

Key: the interaction of drift and mutation generates both polymorphism and substitution of alleles.