Figure 7.2 Fitness surfaces for the A, S, and C alleles at the human hemoglobin |3 gene when malaria is common. The surface in (a) corresponds to the top set of fitness values in Table 7.1 and (b) shows the surface for the bottom set of values. The tracks of circles represent generation-by-generation allele frequency trajectories due to natural selection over 50 generations calculated with equation 7.3. In (a), when the initial frequency of the C allele is relatively high, the equilibrium of natural selection is the fixation of the CC genotype. In contrast, when the C allele is initially rare (a frequency of less than about 7%) selec tion reaches an equilibrium with only the A and S alleles segregating and the C allele going to loss. In (b), selection will eventually fix the CC genotype from any initial frequency of the C allele. However, when the C allele is at low frequencies, the increase in the C allele each generation is extremely small so that selection would take hundreds of generations to fix the CC genotype. The six initial allele frequency points, shown as open circles, are identical for the two surfaces.

in Fig. 7.2, we need to extend the viability model of natural selection to three alleles at one locus. We can compute the mean fitness of the population:

w = WaaP2 + wBBr + wccr2 + wAB2pq + wAC2pr+ wEC2qr where p, q, and r represent the frequencies of the three alleles A, B, and C. We can also use the marginal fitness of the genotypes that contain each of the alleles to compute whether an allele will increase or decrease in frequency due to the average fitness of all the geno

where the frequencies of the heterozygous genotypes are multiplied by lh since they carry one copy of the A allele. The marginal fitness is a way to compare the ratio ofp in the current generation with frequency of p in the next generation that will result from natural selection changing genotype frequencies. Allele frequencies change each generation due to differences between the marginal fitness of each allele and the average fitness of the entire population. The change in the frequency of the A allele is

Allele frequency after one generation of selection is then simply pt+1 = p + Ap. Similar expressions are obtained easily for the B and C alleles. Also note that this approach can be extended to an arbitrary number of alleles at one locus as long as genotypes are in Hardy-Weinberg frequencies at the start of each generation before the action of selection.

Returning to the fitness surfaces, Fig. 7.2a is an interesting case because it has two stable equilibrium points. One of the equilibrium points matches our intuition after inspecting Table 7.1 that the CC genotype should be fixed by selection. When the initial frequency of the C allele is relatively high, all three trajectories of allele frequencies over 10 generations of selection calculated with equation 7. 3 are clearly headed for fixation of CC. In contrast, when the initial frequency of the C allele is low the trajectories of allele frequencies show that the C allele will be lost from the population. This is counterintuitive given that the CC genotype has the highest relative fitness. This result is a consequence of the fitness surface. When C is at low frequency its marginal fitness is actually less than the mean fitness. In other words, the fitness surface is going down in elevation toward higher frequencies of the C allele. Since natural selection only works to increase mean fitness, the C allele is reduced in frequency to loss.

To see one possible consequence of this fitness surface, imagine that the A and S alleles are older in human populations than the C allele and the A and S alleles have reached equilibrium frequencies.

(b) Frequency of Cállele

Frequency of A allele

Frequency of S allele

(b) Frequency of Cállele

Frequency of A allele

Frequency of S allele

Using equation 6.35 and Table 7.1, the equilibrium frequency of the A allele would be t/(s + t) = 0.8/ (0.11 + 0.8) = 0.88 and therefore the equilibrium frequency of S would be 1 - 0.88 = 0.11. Next imagine that the C allele occurs in the population at a later time due to mutation. Since mutation rates are low, the resulting frequency of the C allele will also be low and most C alleles would occur in AC and SC heterozygotes. All heterozygotes have overdominance (AS) or underdominance (SC and AC) for fitness. In particular, the SC heterozygote has a lower relative fitness than AA and AS genotypes and so its marginal fitness would be negative when C is at a low frequency. Thus, to get from a mean fitness state where C is infrequent we would have to go through a dip of lower mean fitness as C initially increases. At higher initial frequencies of C, however, mean fitness increases steadily until CC fixes. So if A and S alleles were ancestral, natural selection alone would drive a newly introduced C allele to loss despite the high relative fitness of the CC homozygote.

For the fitness surface in Fig. 7.2b, natural selection will eventually fix the CC genotype from any initial frequency of the C allele. However, when the C allele is at low frequency selection increases the frequency of the C allele very slowly. This is because the marginal fitness of the C allele is only very slightly greater than the mean fitness below a frequency of about 15% when the frequency of the A allele is also high. This can be seen on the fitness surface by noting the

Problem box 7.1 Marginal fitness and Ap for the Hb C allele

Compute the mean fitness, marginal fitness of the C allele, and the change in the C allele using the two sets of initial allele frequencies given below and relative fitness values from the top of Table 7.1. Use Ap along with Fig. 7.2a to predict the equilibrium that will be reached by natural selection for both initial allele frequencies.

Initial allele frequencies set 1: p = 0.75, q = 0.20, r = 0.05

Initial allele frequencies set 2: p = 0.70, q = 0.20, r = 0.10

wide spacing between contour lines toward the left vertex. Widely spaced contour lines indicate areas with little slope. These are areas where the mean fitness of the population is either constant or nearly constant for a range of genotype frequencies. Such flat areas on fitness surfaces can be stable or unstable equilibrium points and are regions where selection is a weak process because the marginal fitnesses are very close in value to the mean fitness.

Determining which of the different hemoglobin P genotype fitness values best describe actual

Interact box 7.1 Natural selection on one locus with three or more alleles

Direct simulation of selection on one locus with three alleles is an easy way to see that equilibrium points depend strongly on over- and underdominance for fitness. Populus has the ability to simulate selection on a locus with three or more alleles. Launch Populus and, in the Model menu, choose Natural Selection and then Selection on a Multi-allelic Locus. Click on each of the radio buttons for the display options to see how the results can be displayed. Note that when using the default fitness values the P3 allele goes to fixation. The options dialog box can be made larger by dragging the tab at the bottom right, making it easier to see the parameter fields. Then try some different fitness values:

Additivity: w11 = 0.6, w12 = 0.7, w13 = 0.8; w21 = 0.7, w22 = 0.8, w23 = 0.9; w31 = 0.8, w32 = 0.9, w33 = 1.0

Overdominance: w11 = w22 = w33 = 0.3; w12 = w13 = w21 = w23 = w13 = w31 = 1.0 Underdominance: w11 = w22 = w33 = 1.0; w12 = w13 = w21 = w23 = w13 = w31 = 0.3

Also be sure to vary the allele frequencies for each set of fitness values. You might try frequencies of all alleles equal at 1/3, and then one allele more common, with 0.67, 0.12, and 0.21 (the default values).

populations is not the main point of this example. Rather, the hemoglobin P gene serves to illustrate that dominance for fitness, the order of appearance of alleles in a population and the relative fitness values may all interact to determining the outcome of natural selection with three alleles.

Natural selection on two diallelic loci

Since phenotypes, and therefore fitness, may be caused by more than one locus, a logical step is to extend the model of natural selection to two loci. Biologically there is strong motivation to consider selection at more than one locus since many phenotypes are known to show variation caused by multiple loci (see Chapter 9). The fate of two mutations could also be considered as two-locus selection. Natural selection on two loci is inherently more complicated than at a single locus because of gametic disequilibrium. As covered in Chapter 2, both linkage and natural selection itself produce gametic disequilibrium that must be accounted for in a two-locus model of natural selection. Because natural selection on two loci is considerably more complex than on just one locus, the goal of this section is to provide a general introduction to two-locus models. It is important to recognize at the outset that there is no easily summarized set of equilibria for two-locus selection as there are for selection on a diallelic locus. The outcome of two-locus selection depends on the balance between natural selection and recombination between loci as well as the initial genotype frequencies in the population.

Two-locus natural selection is commonly approached from the perspective of gametes because gametic disequilibrium is expressed in terms of gamete frequencies. With two diallelic loci there are 16 possible genotypes that result from the union of four possible gametes. Let the frequencies of the gametes AB, Ab, aB, and ab be X1, %2, X3, shows the relative fitness values for all possible combinations of four gametes. There are only 10 unique fitness values if the same gamete inherited from either parent has the same fitness in a progeny genotype. For example, if an Ab gamete from either a male or female parent has the same fitness in an AB/Ab progeny genotype, then w12 = w21 in the fitness matrix. Table 7.3 then gives the expected frequencies of each progeny genotype under the assumption of random mating (compare with Table 2.12). The frequencies of each gamete in the next generation can be obtained by summing each of the columns in Table 7.3 while also weighting each expected frequency by the relative fitness of each genotype. For example, the

Table 7.2 Matrix of fitness values for all combinations of the four gametes formed at two diallelic loci (top). If the same gamete inherited from either parent has the same fitness in a progeny genotype (e.g. w12 = w21), then there are 10 gamete fitness values shown outside the shaded triangle. These 10 fitness values can be summarized by a genotype fitness matrix (bottom) under the assumption that double heterozygotes have equal fitness (w14 = w23) and representing their fitness value by wH. The double heterozygote genotypes are of special interest since they can produce recombinant gametes.

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