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5000 10,000

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5000 10,000

15,000 20,000 25,000 Generation

30,000 35,000 40,000

Math box 5.1 Equilibrium allele frequency with two-way mutation

To determine the equilibrium allele frequency for a diallelic locus with the possibility of both backward and forward mutation, we take the basic equation that predicts allele frequency over one generation:

and try to express it as

where a and b are constants that depend only on the forward and backward mutation rates ^ and v. Expressing the equation in this way allows us to equate pt+1 with a if the (pt - a)b term goes to zero under certain limiting conditions. Equation 5.24 can be rearranged by adding a to both sides:

and lastly factoring terms containing a to give

Equation 5.23 containing the mutation rates can be put into this same form by expanding to give

which then can be factored to give

Comparing equations 5.27 and 5.29 we see that b = (1 -|-v) (5.30)

Substituting the expression for b into the equation above gives the solution for a:

We can then substitute these values of a and b into equation 5.24 to get a new expression for the change in allele frequency over one generation:

Since the expression for change in allele frequency over any one generation interval is identical and the mutation rates are constant over time, we can recast the equation above in terms of the initial allele frequency p0 and the number of generations that have elapsed:

Pt+1

Notice that as the number of generations grows very large (t ^ the (1 - p. - v)f term approaches zero, making the entire right-hand side of the equation zero. Therefore, when many generations have elapsed, the equilibrium allele frequency is expected to be

v v important to understand that this difference in time scale is just a product of the vastly different rates for the two processes rather than a fundamental difference in the processes themselves. In the figures, the chance that a gamete migrated was one in 10 while the chance of an allele mutated was between one in 1000 and one in 10,000. While these rates are likely to be on the high end of the range of values found in natural populations, gene flow is expected to occur at much higher rates than mutation as a general rule. The conclusion from this comparison is that gene flow is a much more potent force to change allele frequencies at single loci over the short term compared to mutation. Mutation does have an effect, but it is longer term.

The parallel nature of the processes of gene flow and mutation can be used as an advantage to

Interact box 5.4 Simulating irreversible and bi-directional mutation

Both the irreversible and two-way mutation models are available in PopGene.S2 by clicking on the Mutation menu and then selecting Irreversible or Two-way. In the irreversible model only the forward mutation rate can vary while the reverse mutation rate is always 0.0. As an example, compare how rapidly equilibrium is approached with forward mutation rates of 0.01 and 0.001 from an initial allele frequency of 0.9 and the scale set to 2000 generations. For the two-way model, compare approach to equilibrium over 2000 generations when both backward and forward mutation rates are equal (e.g. both 0.001) and when they are unequal (e.g. 0.0015 and 0.0005) starting at an initial allele frequency of 0.9. Note that these mutation rates serve as an illustration only and that biologically realistic mutation rates are usually much lower.

understand more about the process of mutation. In particular, we can learn more about how mutation will impact autozygosity in finite populations where genetic drift is also operating. Recall from Chapter 3 the expression for the level of autozygosity in a finite population caused by genetic drift:

Mutation breaks the chain of descent by changing the state of alleles and therefore reduces the probability that a genotype is composed of two alleles identical by descent (autozygous). Genotypes with no alleles, one allele, or two alleles impacted by mutation each generation have frequencies of (1 - |)2, 2|(1 -1), and |2, respectively. Only the (1 -|)2 genotypes with no mutated alleles can contribute to the pool of alleles that may become identical by descent due to finite sampling. From the opposite perspective, we note that 2| genotypes heterozygous and | 2 genotypes homozygous for a new mutation are expected each generation. Together, these two classes of genotypes with mutations reduce the autozygosity by a factor of 1 - 2| - |2 = (1 - |)2. (This is identical to the reasoning used in Chapter 4 for the case of gene flow.)

Mutation will therefore reduce the autozygosity caused by finite sampling in the present generation

12 (chance of-) by a factor of (1 - |)2. In addition,

2Ne mutation will also reduce any autozygosity from past generations (F(-1) because some alleles that are identical by descent may change to new states via mutation, leaving the proportion (1 - |)2 of genotypes unaffected by mutation and at the same level of autozygosity. Putting these two separate adjustments for the autozygosity together gives

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