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
Derivation: Pick a random individual.
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:
The net result is that the population ends up being fixed for only a single allele.
Which allele is fixed is determined entirely by chance, with
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
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
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.
How many mutations have occurred if we draw two random sequences?
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
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
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,
For example, if u = 10-7 and N = 105 , then
Key: the interaction of drift and mutation generates both polymorphism and substitution of alleles.