## Interact box Mullers Ratchet

Selecting the Muller's Ratchet module from the Mutation menu of PopGene.S2 opens a simulation of the fate of new deleterious mutations in a finite population of chromosomes that lack recombination. The simulation starts out with a population of haploid, clonal individuals that have no mutations and then lets mutation, genetic drift, and natural selection act. The fitness of each individual determines its chances of contributing progeny to the next generation. The number of progeny produced by each individual is Poisson-distributed with a mean of one. The effective size of the population, the coefficient of selection against deleterious mutations, and the mutation rate can be set in the simulation. The results are given in terms of the proportion of individuals in the population with a given number of mutations.

Initially, run the simulation using the default values. Then try independently increasing the effective population size (or the population size of haploid chromosomes), the selection coefficient against the deleterious mutations, and the chance of a deleterious mutation. Predict the impact of each simulation parameter on the frequency distribution of the number of mutations per genome before you change each parameter.

Genetic drift and natural selection are also acting along with mutation on the frequencies of individuals with different numbers of mutations. The sampling error of genetic drift can result in the stochastic loss of mutation categories with a low frequency in the population. This effect of genetic drift works regardless of the number of mutations. Any category of mutations lost by drift can be reestablished via mutation from individuals with fewer mutations. However, when all the individuals with the lowest number of mutations are lost from the population that fewest-mutation category is gone forever. This is because mutation cannot make wild type-alleles that would reduce the number of deleterious mutations. Also, the fewest-mutation category cannot be reconstituted because there is no recombination. The overall effect of genetic drift is to push the frequency distribution of the number of mutations toward higher numbers. In contrast, natural selection tends to push the distribution of the number of mutations toward lower numbers since individuals with more mutations are increasingly disfavored by natural selection.

If the effective population size is small, Muller's Ratchet also leads to accelerated rates of fixation to a single allele within the category of individuals with fewest mutations. This occurs since the category of fewest mutations is not renewed by mutation. It is also finite and consists of alleles that have identical fitness, so that genetic drift will eventually cause fixation of a single allele within that mutation category. This effect has implications for genomes with low levels of recombination or in diploid populations with mating systems that lead to high levels of homo-

zygosity that effectively nullify recombination. In these situations, fixation may occur at higher rates than for deleterious mutations in genomes with free recombination and the same effective population size (see Charlesworth & Charlesworth 1997).

5.3 Mutation models

• The infinite alleles, k alleles, and stepwise mutation models.

• Understanding the implications of mutation models using the standard genetic distance and RST.

• The infinite sites and finite sites mutation models for DNA sequences.

Mutation acts in diverse ways and can produce a wide range of changes at the level of alleles and DNA sequences. To study the allele frequency consequences of mutation it is helpful to construct some simplifying models of the mutation process itself. Mutation models attempt to capture the essence of the genetic changes caused by mutation while at the same time simplifying the process of mutation into a form that permits generalizations about allele frequency changes. There is no single model of the process of mutation, but rather a series of models that serve to encapsulate different features of the mutation process for different classes of loci and different types of alleles. Often, mutation models are motivated by molecular methods such as allozyme electrophoresis or DNA sequencing used to assay genetic variation in actual populations. This section introduces and describes the major classes of mutation models. Two types of mutation model for discrete alleles are applied in measures of genetic difference between populations to show the role mutation models play in the interpretation of genetic differences. Mutation models for DNA sequences are applied in the last section of the chapter on mutation in genealogical branching models.

### Mutation models for discrete alleles

A repeated theme in earlier chapters was determining expected levels of homozygosity and heterozygosity (autozygosity and allozygosity) under different population genetic processes. A key assumption in many of these expectations is that identity in state can be treated as identity by descent. In other words, alleles identical in state look alike because they descended from a common ancestral allele copy at some point in the past. The infinite alleles model of mutation (see Kimura & Crow 1964) is an assumption used to guarantee that identity in state is equivalent to identity by descent. Under the infinite alleles model, each mutational event creates a new allele unlike any other allele currently in the population. Once a given allelic state is made by mutation the first time it can never be made by mutation ever again. In essence, the allelic state is crossed off the list of possible mutations. The infinite alleles model, serves to avoid the possibility that two alleles are identical in state but not identical by descent, as can occur if the same allele can be made by mutation repeatedly over time. Under the infinite alleles model, mutation produces the original copy of each allele but is not an ongoing process influencing the frequency of any allele already in the population. Processes other than mutation are responsible for allele and genotype frequencies after an allele exists in a population. An additional consequence is that the evolutionary "distance" or number of transition events between all alleles is the same, since all alleles are produced by a single mutational event and alleles can never accumulate multiple mutations. This means that all alleles can be treated as equivalent when estimating heterozygosity or fixation indices.

The infinite alleles model might roughly approximate the mutational process for molecular markers like allozymes since alleles take discrete states (e.g. fast or slow migration on a gel) and allozyme loci are generally observed to have low mutation rates so it is likely that most alleles in a sample are not recently the product of mutation. A length of DNA sequence might also approximate the infinite alleles model. In a sequence of 500 nucleotides there are

4500 = 1.072 x 10301 unique combinations of the nucleotides. If mutation is purely random and mutation changes an existing nucleotide to any other nucleotide with equal probability, many mutations could occur in a population of DNA sequences without producing a duplicate allele since there are a truly staggering number of possible alleles.

Homoplasy The condition where allelic states are identical without the alleles being identical by descent.

Infinite alleles model A model where each mutational event creates a new allele unlike any other allele currently in the population so that identity in state for two or more alleles is always a perfect indication of identity by descent.

k alleles model A mutation model where each allele can mutate to each of the other k - 1 possible allelic states with equal probability.

Stepwise mutation model A mutation model where the allelic states produced by mutation depend on the initial state of an allele. Alleles with a greater difference in state are therefore more likely to be separated by a greater number of past mutational events.

There are several features of the mutational process that the infinite alleles model does not account for, and therefore there are a number of other mutation models. Obviously, there are not an infinite number of alleles possible at actual genetic loci. The k alleles model of mutation is an alternative to the infinite alleles model where k refers to a finite integer representing the number of possible alleles. In this model each allele can mutate with equal probability to each of the other k - 1 possible allelic states. With the k alleles model the same allele can be created by mutation repeatedly, blurring the equivalence of identity in state and identity by descent. As the number of possible alleles or k decreases and as the mutation rate increases, allelic state becomes a poorer and poorer measure of identity by descent since an increasing proportion of alleles with identical states have completely independent histories. The term homoplasy refers to allelic states that are identical in state without being identical by descent.

The infinite alleles and k alleles models both assume that the allelic state produced by mutation is independent of the current state of an allele. With these models each allele has an equal probability of mutating to any of the other allowable allelic states. It is also possible that the state of a new allele produced by mutation is not independent of the initial state of an allele. An example is the common observation that transitions are more common than transversions in diverged DNA sequences. The stepwise mutation model accounts for cases where allelic states are somehow ordered and the allelic states produced by mutation depend on the initial state of an allele (Kimura & Ohta 1978). Mutations by slipped-strand mispairing at microsatellite or simple sequence repeat loci produce new allelic states within one or a few repeats of the initial allelic state much more often than mutations that are many repeats different than the initial allelic state. Microsatellite loci are therefore a prime example of ordered, stepwise mutation where alleles closer in state are more likely to be recently identical by descent than alleles that are very different in state. See evidence for human microsatellite loci in Valdes et al. (1993).

The role of mutation models is illustrated in summary measures that express the genetic similarity or dissimilarity of individuals or populations, called genetic distances. The standard genetic distance or D measure developed by Nei (1972, 19 78) has been widely employed. Given allele frequencies for several populations, D (not to be confused with the measure of gametic disequilibrium) expresses the probability that two alleles each randomly sampled from two different subpopulations will be identical in state relative to the probability that two alleles randomly sampled from the same subpopulation are identical in state. Table 5.4 gives hypothetical allele frequencies at one locus in two subpopulations that can be used to compute D. With random mating, the total probability that two identical alleles are sampled from subpopulation 1 is alleles

and the total probability that two identical alleles are sampled from subpopulation 2 is alleles

where p^ indicates the frequency of allele k in population i. The total probability of sampling an identical allele from subpopulation 1 and subpopulation 2 is alleles

The normalized genetic identity for this locus is then

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