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equilibrium 4N n + 1

This result also depends on each mutation giving rise to a new allele that is not present in the population or the infinite alleles model. Since the allozygosity or expected heterozygosity is just one minus the autozygosity,

Hequilibrium Fequilibrium 4N ^ + 1 4N [I + 1

4Nen 4Nen +1

This is the expected heterozygosity in a finite population where the "push" on allele frequencies toward fixation and loss by genetic drift and the "push" on allele frequencies away from fixation and loss by mutation has reached a net balance.

The quantity 4Ne| has a ready biological interpretation when Ne is large and | is small. In a population of 2Ne alleles, the expected number of mutated alleles each generation is 2Ne|. In a sample of two alleles that compose a diploid genotype, the chance that both alleles have experienced mutation and are therefore not identical by descent is 4Ne| . For example, in a population of 2Ne = 100 alleles with a mutation rate of one in 10,000 alleles per generation (|| = 0.0001), the expected number of mutations is 0.01 and the chance that a sample contains two alleles that are not autozygous is 0.02. The quantity 4Ne| is frequently symbolized by 0 (pronounced "theta"). Under the infinite alleles model, 0 is the probability that two alleles sampled at random from a population at drift-mutation equilibrium will be allozygous. With 0 = 0.02, the expected heterozygosity at driftmutation equilibrium is 0.0099. It is important to note that equilibrium heterozygosity will be lower than that predicted by 0 if the infinite alleles or infinite sites model is not met. This is the case because with a finite number of allelic states not all mutation events will produce a novel allele that forms an allozygous pair, or a heterozygote, when sampled with an existing allele in the population. In fact, mutations that make additional copies of existing alleles actually increase the perceived homozygosity due to homoplasy.

Figure 5.11 shows the expected probability of auto-zygosity and allozygosity at mutation-genetic drift equilibrium. At small values of 4Ne| there will be an intermediate equilibrium level of autozygosity due to the balance of mutation introducing new alleles and genetic drift moving allele frequencies toward fixation or loss. As 4Ne| gets large there is either

4 5 6 4Nem

Figure 5.11 Expected homozygosity (F or autozygosity, solid line) and heterozygosity (H or allozygosity, dashed line) at equilibrium in a population where the processes of both genetic drift and mutation are operating. The chance that two alleles sampled randomly from the population are identical in state depends on the net balance of genetic drift working toward fixation of a single allele in the population and mutation changing existing alleles in the population to new states. A critical assumption is the infinite alleles model, which guarantees that each mutation results in a unique allele and thereby maximizes the allozygosity due to mutations.

4 5 6 4Nem

Figure 5.11 Expected homozygosity (F or autozygosity, solid line) and heterozygosity (H or allozygosity, dashed line) at equilibrium in a population where the processes of both genetic drift and mutation are operating. The chance that two alleles sampled randomly from the population are identical in state depends on the net balance of genetic drift working toward fixation of a single allele in the population and mutation changing existing alleles in the population to new states. A critical assumption is the infinite alleles model, which guarantees that each mutation results in a unique allele and thereby maximizes the allozygosity due to mutations.

little drift or lots of mutation so there will be almost complete heterozygosity (no autozygosity). In the other direction, 4Ne| near zero indicates very strong genetic drift or very infrequent mutation resulting in high levels of autozygosity and very low hetero-zygosity. Bear in mind that reaching the expected equilibrium autozygosity or heterozygosity will take many, many generations because mutation rates are low, making mutation a very slow process. If heterozygosity is perturbed from its mutation-drift equilibrium point, a population will take a very long time to return to that equilibrium.

5.5 The coalescent model with mutation

• Adding the process of mutation to coalescence.

• Longer genealogical branches experience more mutations.

• Genealogies under the infinite alleles and infinite sites models of mutation.

The genealogical branching model was introduced in Chapter 3 for a single finite population and then extended in Chapter 4 to account for branching patterns expected with population subdivision. The goal of those sections was to predict genealogical branching patterns without reference to the identity of the lineages represented by the branches. Those branching models need to be extended to account for the possibility that mutation occurs. Mutations will alter the genes or DNA sequences that are represented by each lineage or branch in the genealogical tree. Therefore, accounting for mutation will be a critical step in developing a coalescent model that explains differences among a sample of lineages in the present. This section will focus on the action of mutation in the coalescent model along with the state of each lineage in a genealogy. This is accomplished by coupling the process of coalescence and the process of mutation while moving back in time toward the most recent common ancestor. The ultimate goal is to build a genealogical branching model that can be used to predict the numbers and types of alleles that might be expected in a sample of lineages taken from an actual population. For example, one prediction might be the number of alleles expected in a single finite population for a given mutation rate. In this way, the combination of the coalescent process and the mutation process is used to form quantitative expectations about patterns of genetic variation produced by various population genetic processes.

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