## K

gives the rate at which alleles that originally entered the population as mutations go to fixation per gen-

The necessary assumption is that the substitution process is viewed on a time scale that is long relative to the average time to fixation for an individual mutation. If more than 4Ne generations have elapsed, then it is likely that all the alleles in a population will have descended from one allele due to genetic drift. The probability that the lucky allele that is fixed in the population was a new mutation is ||.

This result is remarkable because it says that the probability that a neutral mutation goes to fixation each generation, or the rate of substitution, is simply equal to the mutation rate. Notice that the predicted substitution rate does not depend on the effective population size. This is because a mutation in a smaller population has a greater chance of fixation but there are fewer new mutations each generation, while a mutation in a larger population has a smaller chance of reaching fixation but there are more mutations introduced each generation. The rate of input of new mutations in a population and the chance of fixation due to genetic drift exactly balance out when N changes. Note that this same result holds in the case of haploid loci since there are a total of N alleles and the probability of fixation of a new mutation is —.

Based on the rate of substitution, the neutral theory also predicts that the substitutions that ultimately cause divergence should occur at a regular average rate. For waiting time processes, the time between events is the reciprocal of the rate of events. Using a clock that chimes on the hour as an example, the rate of chiming is 24 per day (or 24/day). Therefore, the expected time between chiming events is

V24 of a day or 1 hour. Since the rate of neutral substitution is ||, the expected time between neutral substitutions is 1/| generations (see Fig. 8.2). For example, if the mutation rate of a locus is 1 x 10-6 (one nucleotide change per 106 gametes per generation) then the expected time between neutral substitutions is 106 generations on average. This offers one explanation of why different loci diverge at different rates: the different loci simply have distinct mutation rates that lead to variable neutral substitution rates.

### Nearly neutral theory

The nearly neutral theory considers the fate of new mutations if some portion of new mutations are acted on by natural selection of different strengths (Ohta & Kimura 19 71, Ohta 19 72, reviewed in Ohta 1992 and Gillespie 1995). The nearly neutral theory recognizes three categories of new mutations: neutral mutations, mutations acted on strongly by either positive or negative natural selection, and mutations acted on weakly by natural selection relative to the strength of genetic drift. This last category contains mutations that are nearly neutral since neither natural selection nor genetic drift will determine their fate exclusively.

For a new mutation in a finite population that experiences natural selection, the forces of directional selection and genetic drift oppose each other. Recall from Chapter 3 that genetic drift causes heterozygosity to decrease at a rate of- per generation (see

1 2Ng equation 3.51). Thus,-quantifies the "push" on

2Ne a new mutation toward fixation caused by genetic drift. The selection coefficient (s) on a genotype describes the "push" on alleles toward fixation or loss due to natural selection. The force of selection on a new mutation can be quantified using the result from section 5.2 that the chance of fixation is approximately 2 s. Setting these forces equal to each other,

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