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Math box 6.2 Equilibrium allele frequency with overdominance

By definition, equilibrium allele frequencies are reached when allele frequencies stop changing from one generation to the next. This means that Ap as expressed by

which was first shown as equation 6.23, should be equal to zero.

Two equilibrium points occur when p = 0 or q = 0, biologically equivalent to situations where there is no genetic variation in a population. When there is genetic variation (both p ^ 0 and q ^ 0), the equilibrium point depends on the fitness differences contained in the numerator. Taking the numerator term in square brackets in equation 6.37 and setting it equal to zero, p(wAA - WAa) + q(wAa - Waa) = 0 (638)

and then solving p or q in terms of relative fitness values, will give allele frequencies where Aq is zero. The first step is to substitute q = 1 - p:

and then expand by multiplying the terms:

The relative fitness values that are multiplied by p can be brought together:

and then subtracted:

Dividing both sides by -(wAA - 2wAa + waa) gives

which expresses p as a function of relative fitness values alone. Substituting the relative fitness values of wAA = 1 - s, wAa = 1, and waa = 1 - t as given in Table 6.4:

and then carrying out the addition and subtraction then gives the equilibrium allele frequency in terms of selection coefficients for the two homozygous genotypes:

selection coefficient is between 1.0 and 0.1%. This illustrates the general principle that stronger natural selection (larger selection coefficients or larger fitness differences) results in a more rapid approach to equilibrium allele frequencies. This conclusion applies to all of the situations given in Table 6.4 and to the process of natural selection in general.

6.3 How natural selection works to increase average fitness

• Natural selection acts to increase mean fitness.

• The fundamental theorem of natural selection.

In the five fitness situations shown in Table 6.4 for natural selection on a diallelic locus, there are always two general outcomes. Directional selection of any type ends in fixation and loss (selection against a dominant phenotype) or nearly fixation and loss (selection against a recessive phenotype). Underdominance too results in fixation or loss (with one exception unlikely to be realized in finite populations). Overdominance is the exception that maintains both alleles in the population indefinitely. So the two outcomes are either fixation and loss or intermediate frequencies for both alleles (sometimes called a balanced polymorphism). The reason why these two general outcomes occur can be understood by examining the average fitness of a population (w) as well as the rate of change in allele frequency (Ap) over the entire range of allele frequencies.

Average fitness and rate of change in allele frequency

The mean fitness (w) over all possible allele frequencies is plotted for each case of natural selection on a diallelic locus in Figs 6.10 and 6.11. For the cases of directional selection in Fig. 6.10, notice that the highest mean fitness corresponds exactly to fixation of the A allele for selection against a recessive phenotype and to loss of the A allele for selection against a dominant phenotype. This same pattern is evident in Fig. 6.11 where the highest mean fitness is found at an intermediate allele frequency for over-dominance or at fixation or loss for underdominance. These plots of mean fitness by allele frequency show that natural selection acts to increase the mean fitness of the population to its maximum. It is the maximum mean fitness in a population that really defines the equilibrium points for genotype and allele frequencies. The plots of w against p reveal the generalization that the process of natural selection acts to increase the population mean fitness every generation if possible and stops when the mean fitness can no longer increase. In this sense, natural selection can be metaphorically likened to a mountain climber who continually works to find the highest point, but who cannot ever go downhill and will only rest at the summit. Keeping with this metaphor, plots of w against p are called fitness surfaces, adaptive landscapes, or adaptive topographies and represent a topographic map of the mountain at any point where our imaginary mountain climber might venture.

Figures 6.10 and 6.11 also show the change in allele frequency over a single generation (Ap) over all possible allele frequencies for each case of natural selection. Plots of Ap against p reveal when allele frequencies are increasing (Ap is positive) or decreasing (Ap is negative) as well as when allele frequencies are changing rapidly (the absolute value of Ap is

Selection against a recessive phenotype

Selection against a dominant phenotype

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