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Frequency of the A allele (p)

Figure 6.11 Mean fitness in a population (w) and change in allele frequency over a single generation (Ap) as a function of allele frequency for balancing and disruptive selection. Natural selection changes allele frequencies to increase the average fitness in each generation, eventually reaching an equilibrium when the mean fitness is highest. The change in allele frequencies is faster when average fitness changes more rapidly (the slope of w is steeper). The dashed line in the plots of Ap by p shows where allele frequencies stop changing (Ap = 0) and thus are allele frequency equilibrium points. With underdominance for fitness, Ap is zero when p = 0.5 and so defines an equilibrium point marked by the circle. However, this equilibrium point is unstable since Ap on either side ofp = 0.5 changes allele frequencies away from the equilibrium point (below p = 0.5 Ap is negative leading toward loss and above p = 0.5 Ap is positive leading toward fixation). In contrast, with overdominance Ap on either side ofp = 0.5 changes allele frequencies toward the equilibrium point (below p = 0.5 Ap is positive and above p = 0.5 Ap is negative) and thus p = 0.5

Frequency of the A allele (p)

Figure 6.11 Mean fitness in a population (w) and change in allele frequency over a single generation (Ap) as a function of allele frequency for balancing and disruptive selection. Natural selection changes allele frequencies to increase the average fitness in each generation, eventually reaching an equilibrium when the mean fitness is highest. The change in allele frequencies is faster when average fitness changes more rapidly (the slope of w is steeper). The dashed line in the plots of Ap by p shows where allele frequencies stop changing (Ap = 0) and thus are allele frequency equilibrium points. With underdominance for fitness, Ap is zero when p = 0.5 and so defines an equilibrium point marked by the circle. However, this equilibrium point is unstable since Ap on either side ofp = 0.5 changes allele frequencies away from the equilibrium point (below p = 0.5 Ap is negative leading toward loss and above p = 0.5 Ap is positive leading toward fixation). In contrast, with overdominance Ap on either side ofp = 0.5 changes allele frequencies toward the equilibrium point (below p = 0.5 Ap is positive and above p = 0.5 Ap is negative) and thus p = 0.5

is a stable equilibrium point. Here wAA overdominance.

: 0.7 and wAa = 1.0 for w w aa aa large) or slowly (the absolute value of Ap is small). When allele frequencies are not changing at all (Ap is zero), then an equilibrium allele frequency has been reached. Notice that fixation or loss of the A allele corresponds to Ap = 0 for directional selection. For overdominance and underdominance, Ap = 0 for fixation and loss as well as for the intermediate allele frequency of p = 0.5. These allele frequencies are therefore equilibrium points because natural selection is not causing any change in allele frequency at these specific allele frequencies.

Also compare each plot of Ap against p to the corresponding plot of w against p. There is a striking relationship between Ap and w. Both the magnitude and sign of Ap correspond exactly to the slope of the w line at any value of p. The slope of w is always positive, just as Ap is always positive for selection against a recessive, while the slope of w is always negative, just as Ap is always negative for selection against a dominant (Fig. 6.10). This same pattern is seen in Fig. 6.11 for overdominance and underdominance, where the slope of w is zero at fixation and loss as well as at p = 0.5. The slope of w explains why the polymorphic equilibrium point for overdominance is stable while that for underdominance is unstable. With overdominance, if there is any shift of allele frequencies away from p = 0.5, say by genetic drift or mutation, natural selection will return the population back to the equilibrium of p = 0.5 (Ap is positive for p < 0.5 and negative for p > 0.5). In contrast,

Problem box 6.2 Mean fitness and change in allele frequency

Using equations 6.35 and 6.36 allows us to predict the allele frequencies at equilibrium for selection with overdominance for fitness. We also need to understand why the equilibrium is the point at which genotype frequencies stop changing. Let the fitness values be wAA = 0.9, wAa = 1.0, and waa = 0.8. First, calculate expected frequency of the

A allele at equilibrium or Pequilibrium. Then compute Ap and W at Pequilibrium, P = °.9, and p = 0.2. How do the values of Ap and w at the three allele frequencies compare? Use Ap and W to explain why equilibrium allele frequency is between p = 0.9 and p = 0.2.

with underdominance, if there is any shift of allele frequencies away from p = 0.5, natural selection will change allele frequencies to the maximum mean fitness values found at fixation and loss (Ap is negative for p < 0.5 and positive for p > 0.5).

The fundamental theorem of natural selection

To close this section, let's examine one last generalization about the action of natural selection on a diallelic locus. Sir Ronald Fisher (Fig. 6.12) proposed the impressive sounding fundamental theorem of natural selection (Fisher 1999, originally published in 1930) as a way to summarize and generalize the process. In Fisher's words, the fundamental theorem of natural selection was that "the rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time." A modern restatement of the theorem is that "the rate of increase in the mean fitness of any organism at any time ascribable to natural selection acting through changes in gene frequencies is exactly equal to its genic variance in fitness at that time" (Edwards 1994). The fundamental theorem has also been interpreted as showing that any change in mean fitness caused by natural selection must always be positive. As Crow (2002) and Edwards (2002) recount, this cryptic yet simultaneously insightful statement about natural selection has lead to a great deal of controversy, misunderstanding, and just plain confusion over many years.

Interact box 6.1 Natural selection on one locus with two alleles

Launch Populus. In the Model menu, choose Natural Selection and then Selection on a Diallelic Autosomal Locus. In the options dialog box set view to p vs. t and use fitness coefficients. Set the fitness values as given below. Click on the button for One Initial Frequency and set the initial allele frequency to 0.5. Press the View button to see the simulation results. Clicking on the different radio buttons for Plot Options switches the view among the four different modes of display (p vs. t, Genotypic frequency vs. t, Ap vs. p, and W vs. p). When selection is weak, set the generations to view in the range of 500 to 1000, but when selection is strong 50 to 100 generations will be more appropriate. Using the Six Initial Frequencies option shows the outcome of natural selection for six different initial allele frequencies like Figs 6.4-6.8, but the Genotypic frequency vs. t plot will not be available.

Here are some fitness values to simulate.

Weak selection against recessive: wAA = 1; wAa = 1; waa = 0.9 (h = 0, s = 0.1). Compare with selection against recessive lethal: wAA = 1; wAa = 1; waa = 0.0 (h = 0, s = 1.0) Weak selection with additive gene action: wAA = 1; wAa = 0.95; waa = 0.9 (h = 0.5, s = 0.1). Compare with strong selection with additive gene action: wAA = 1; wAa = 0.7; waa = 0.4 (h = 0.5, s = 0.6)

Weak selection with overdominance: wAA = 0.98; wAa = 1; waa = 0.95. Compare with strong selection with overdominance: wAA = 0.2; wAa = 1; waa = 0.4.

Selection against the heterozygote: wAA = 1; wAa = 0.8; waa = 1. For this case be sure to examine the Genotypic frequency vs. t plot for several different initial allele frequencies such as 0.2, 0.5 and 0.8.

r where k is the total number of haplotypes in the population. Extending the results in Table 6.1 to an arbitrary number of alleles, the frequency of any single haplotype, call it haplotype i, after natural selection is p.w. Fi w

where the prime symbol is used to represent quantities after one generation of natural selection. Based on this haplotype frequency after selection, the average fitness after one generation of selection is then n' = I( P'Wi)

Figure 6.12 Sir Ronald A. Fisher (1890-1963) photographed in 1943, was a pioneer in the theory and practice of statistics. He invented the techniques of analysis of variance and maximum likelihood as well as numerous other statistical tests and methods of experimental design. Fisher's 1930 book The Genetical Theory of Natural Selection established a rigorous mathematical framework that coupled Mendelian inheritance and Darwin's qualitative model of natural selection and is one of the foundation works of modern population genetics. Much of Fisher's work stressed the effectiveness of natural selection in changing gene frequencies in infinite, panmictic populations. Photograph courtesy of the Master and Fellows of Gonville and Caius College, Cambridge.

which when substituting in the expression for p' gives

The change in fitness from one generation to the next standardized by the mean fitness in the initial generation is

which when substituting in the expression for w' from equation 6.48 gives

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