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Figure 7.7 The expected distribution of allele frequencies for a very large number of replicate finite populations under natural selection where there is overdominance for fitness (wAA = waa = 1 - s and wAa = 1). In an infinite population the expected allele frequency at equilibrium is 0.5. However, in finite populations the equilibrium allele frequency will depend on the balance of natural selection and genetic drift. This balance is determined by the product of the effective population size and the selection coefficient (Nes). Low values of Nes mean that selection is very weak compared to drift and each population reaches fixation or loss. High values of Nes mean that selection is strong compared to drift and most populations reach an equilibrium allele frequency near 0.5. Here forward and backward mutation rates are equal (|i = v = 0.00001).

strong because the population is tiny or the selection coefficient is extremely small and so the populations are evolving in a neutral fashion under drift alone. In either of these cases genetic drift is the dominant process and will eventually result in either fixation or loss in all populations. At the lowest values of Nes in Fig. 7.7, populations are most likely to have allele frequencies near zero or near one as expected under genetic drift alone. Alternatively, Nes can take on large values in two general situations. One is when genetic drift is very weak because the effective population size is very large and there is some natural selection favoring heterozygotes (s can occupy a wide range as long as it is not extremely small). The other is when the selection coefficient is large and the effective population size is at least 10 or so individuals and therefore genetic drift is not extreme. At the largest values of Nes in Fig. 7.7, the probability is greatest that the allele frequency in a population will be near 0.5. Equation 6.35 shows that 0.5 is the expected equilibrium allele frequency under balancing selection when wAA = waa and a population is infinite. Therefore, when Nes is large, selection is stronger than drift and the equilibrium allele frequency is deter mined mostly by natural selection. At intermediate values of Nes, the equilibrium allele frequency for many populations is between the equilibrium allele frequencies expected under genetic drift acting alone or natural selection acting alone.

To summarize the balance of natural selection and genetic drift, Motoo Kimura suggested a simple rule of thumb for a diploid locus (Kimura 1983a, 1983b). If four times the product of the effective population size and the selection coefficient is much less than one (4Nes << 1) then selection is weak relative to sampling, and genetic drift will dictate allele frequencies. Alternatively, if four times the product of the effective population size and the selection coefficient is much greater than one (4Nes >> 1) then selection is strong relative to sampling, and natural selection will dictate allele frequencies. When four times the product of the effective population size and the selection coefficient is approximately one (4Nes ~ 1) then allele frequencies are unpredictable.

Figure 7.8 shows an example of the balance between genetic drift and natural selection in replicated laboratory populations of the fruit fly Drosophila melanogaster (Wright & Kerr 1954). The plot shows allele frequencies at the Bar locus for 108 replicate populations that were each founded from four males and four females every generation (since Bar is hemizygous in males the effective population size is equivalent to six rather than eight diploid individuals). Although the vast majority of populations fix for the wild-type allele due to strong natural selection favoring the recessive phenotype of wild-type homozygotes, three populations fix for the Bar allele by the end of the experiment. The selection coefficient against the Bar homozygote was estimated at 0.63, giving an upper bound estimate of 4Nes ~ 15 in the experiment (Ne may have been less than six in actuality). Even with this value of 4Nes, natural selection is not sufficiently strong to dictate equilibrium allele frequency in all populations.

It is also possible to gain biological insight into Nes by recognizing that it is analogous to the quantity Nem that dictates the balance between genetic drift and gene flow discussed in Chapter 4. Both Nes and Nem represent the net balance of the pressure on allele frequencies toward eventual fixation or loss due to genetic drift and the countervailing force driving allele frequencies toward a specific allele frequency caused by either natural selection or by gene flow. In the case of natural selection the specific allele frequency is dictated by the relative fitness values of genotypes while in the case of gene flow

Figure 7.8 Frequency of the Bar allele in 108 replicate D. melanogaster populations over 10 generations (Wright & Kerr 1954). Each population was founded from four males and four females. The Bar locus is found on the X chromosome and so is hemizygous in males, making the effective population size equivalent to about six individuals. The eyes of D. melanogaster individuals homozygous for the wild-type allele are oval, but heterozygotes and homozygotes for the partially dominant Bar allele have bar-shaped eyes with a reduced number of facets. Females homozygous for the Bar allele produced 37% of the progeny compared to females homozygous or heterozygous for the wild-type allele. Despite this strong natural selection against Bar, three populations fixed for Bar by the end of the experiment. Compare with the similar example in Figure 3.11 where the locus is selectively neutral.

Figure 7.8 Frequency of the Bar allele in 108 replicate D. melanogaster populations over 10 generations (Wright & Kerr 1954). Each population was founded from four males and four females. The Bar locus is found on the X chromosome and so is hemizygous in males, making the effective population size equivalent to about six individuals. The eyes of D. melanogaster individuals homozygous for the wild-type allele are oval, but heterozygotes and homozygotes for the partially dominant Bar allele have bar-shaped eyes with a reduced number of facets. Females homozygous for the Bar allele produced 37% of the progeny compared to females homozygous or heterozygous for the wild-type allele. Despite this strong natural selection against Bar, three populations fixed for Bar by the end of the experiment. Compare with the similar example in Figure 3.11 where the locus is selectively neutral.

Interact box 7.4

The balance of natural selection and genetic drift at a diallelic locus

You can use PopGene.S2 to simulate the simultaneous action of genetic drift and natural selection in many identical finite populations. From the Drift menu select the Drift + Selection + Mutation. In the module dialog, run the model a few times using the default parameters to understand the output. The graph on the left shows the allele frequency trajectories of each replicate population over time while the histogram on the right shows the distribution of allele frequencies for all populations (analogous to a two-dimensional slice through Fig. 7.7 for a single value of Nes). Note that mutation will have almost no effect on the outcome of allele frequencies as long as backward and forward mutation rates are equal and very small (you can test this by setting one very high mutation rate such as 0.1 to see the impact).

Try two sets of simulations, one keeping the selection coefficient constant and varying Ne, and another for comparison keeping Ne constant but varying the selection coefficient. Before each of these runs you should compute the value of 4Nes and make a prediction about the distribution of allele frequencies among the populations. First set wAA = waa = 0.9 and wAa = 1.0 (or s = 0.1) and then run separate simulations for Ne = 2, 20, and 200. Then set Ne = 10 and use selection coefficients of 0.05, 0.5, and 0.95.

the specific allele frequency is the average allele frequency for all demes.

Substituting in the expressions for Aqmutation Aqselection into this equation yields and

The balance between natural selection and mutation

Natural selection takes place at the same time that mutation is working to alter allele frequencies and reintroduce alleles that selection may be driving to loss. Therefore, the process of natural selection may be counteracted to some degree by mutation. If a completely recessive allele is both deleterious when homozygous and also produced by spontaneous mutation, there are contrasting forces acting on its frequency. Mutation pressure will continually rein-troduce the allele into a population while natural selection will continually work to drive the allele to loss. What equilibrium allele frequency is expected when the opposing processes of mutation and natural selection balance out?

Let's assume that there are two alleles at one locus and that the a allele is completely recessive and has a frequency of q. Also assume the case of selection against a recessive as given in Table 6.4. An expression for the change in allele frequency per generation for the specific case of natural selection against a recessive homozygote can be obtained by substituting the fitness values of wAA = wAa = 1 and waa = 1 - s into equation 6.24:

Aq selection pq[q((l - s) - 1) + p(1 - 1)] (1)p2 + (1)2pq + (1 - s)q2

which then rearranges to

Aq selection

Let's further assume that mutation is irreversible and that the probability that an A allele mutates to an a allele is ||. The change in the frequency of the allele due to mutation each generation is then

At equilibrium, the action of natural selection pushing the allele toward fixation and the pressure of mutation increasing the allele frequency exactly balance so that allele frequency does not change. This means that, at equilibrium,

i + Aq, selection

spq2

If we assume that the frequency q of the a allele is low, then q2 is very small and the quantity 1 - sq2 is approximately one. This approximation leads to

which can be solved in terms of genotype frequency:

or in terms of allele frequency:

q equilibrium r V/2 -

vs y

Thus, the expected frequency of a deleterious recessive allele at an equilibrium between natural selection and mutation depends on the ratio of the mutation rate and the selection coefficient. Equation 7.41 shows that even if a recessive homozygous genotype is lethal (s = 1), the expected frequency of the allele is pas and the expected frequency of the lethal genotype is p due to recurrent mutation.

The balance between selection and mutation is illustrated in Fig. 7.9. The Aq due to mutation is always positive while the Aq due to selection in this case is negative. Figure 7.9 uses the absolute value of each Aq to show where the change in allele frequency for the two processes intersects. This intersection is the equilibrium point. The expected equilibrium allele frequency is qequihbrium =

0 0

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