Xx

i=l i=i xixiwa

The marginal fitness and mean fitness can be combined with equation 7.8 to give an expression for the change in a gamete frequency under selection and recombination.

To continue with the AB gamete as an example, notice that the marginal fitness w1 is equal to x1w11 + x, w-p + + x4w^4. Making this substitution in equation 7.8, dividing by the mean fitness, and substituting wHfor w14 or w, , gives the change in the AB gamete frequency over one generation of natural selection:

This is exactly like the expression for Ap for a diallelic locus (compare with equation 6.2 3). Using analogous steps for the other three gametes gives the recursion equations for change in gamete frequency after one generation of natural selection and recombination:

Equations 7.11-7.14 show that the change in gamete frequency under natural selection is due to both fitness values and recombination. If there is no recombination (r = 0), then each gamete is analogous to a single allele. The outcome of selection is then like four alleles at a single locus as dictated by the gamete fitness values. The process of recombination may either reinforce or oppose the changes in gamete frequencies due to natural selection. For example, if gametes Ab and aB have the highest fitness values and there is no recombination then Ax, and Ax, would be positive while Ax 1 andAx4wouldbe negative (when not at equilibrium). The gamete frequency changes caused by recombination would amplify the effect of natural selection on gamete frequencies since the rwHD term would increase Ax, and Ax, but decrease Axx and Ax4. In contrast, if gametes AB and ab have the highest fitness and there is recombination, then the rwHD term would decrease Axx and Ax4 but increase Ax, and Ax, in opposition to natural selection.

Examining natural selection on two loci with and without recombination demonstrates that selection and recombination working in opposition can produce counterintuitive equilibrium gamete frequencies. Figure 7.3 shows a fitness surface where

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Frequency of A allele for locus 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Frequency of A allele for locus 1

Figure 7.3 A fitness surface for two loci that each have two alleles where gene action is additive. The blue dots show generation-by-generation allele frequencies based on equations 7.11-7.14 for seven different initial sets of four gamete frequencies. When recombination is a weak force (r = 0.05), equilibrium allele frequencies are dictated by natural selection and all initial gamete frequencies eventually reach the highest mean fitness point (a). In contrast, when recombination is a strong force (r = 0.5) then equilibrium allele frequencies depend on initial gamete frequencies (b). When recombination is strong, equilibrium allele frequencies may not correspond to the highest mean fitness. Relative fitness values are wA ^

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Frequency of A allele for locus 1

Figure 7.3 A fitness surface for two loci that each have two alleles where gene action is additive. The blue dots show generation-by-generation allele frequencies based on equations 7.11-7.14 for seven different initial sets of four gamete frequencies. When recombination is a weak force (r = 0.05), equilibrium allele frequencies are dictated by natural selection and all initial gamete frequencies eventually reach the highest mean fitness point (a). In contrast, when recombination is a strong force (r = 0.5) then equilibrium allele frequencies depend on initial gamete frequencies (b). When recombination is strong, equilibrium allele frequencies may not correspond to the highest mean fitness. Relative fitness values are wA ^

0.4, and waabb = 0.3. The seven initial allele frequency points, shown as open circles, are identical for the two surfaces.

gene action is completely additive. Since the fitness surface is a tilted plane, our earlier experience with one locus selection suggests that the equilibrium under natural selection should be the highest fitness point. When recombination is a weak force relative to selection (Fig. 7.3a), then the change in gamete frequencies follows the slope of the fitness surface and the equilibrium point reached from all initial gamete frequencies is the highest mean fitness. However, when recombination is strong relative to selection (Fig. 7.3b), then equilibrium gamete frequencies depend strongly on initial gamete frequencies. Figure 7.4 shows another example of two-locus selection on a fitness surface with two peaks shaped like a saddle due to dominance andepistasis at the two loci. When recombination is weak (Fig. 7.4a), then the equilibrium points depend on the slope of the fitness surface at the initial gamete frequencies since populations move uphill due to the strong force of selection. However, when recombination is strong relative to selection (Fig. 7.4b) then gamete frequencies will change in opposition to selection and change in directions that decrease mean fitness. When recombination is strong in Figs 7. 3 and 7.4, the gamete frequency trajectories take sharp turns and move downhill on the fitness surfaces due to the force of recombination. This happens because recombination works toward gametic equilibrium (D = 0) whereas selection works toward the highest mean fitness. When one process is much stronger then it will win out over the other process to determine the equilibrium. When the two processes are of approximately equal strength then the result is a compromise that may produce an equilibrium that is neither gametic equilibrium nor maximum mean fitness.

The fitness surface in Fig. 7.4 demonstrates an additional point about the action of natural selection on two loci. Gene action is a key variable in determining the equilibrium reached by natural selection. With additive gene action for two loci, the genotype at one locus has the same fitness value regardless of the genotype at the other locus. This means continual small changes in genotype frequencies that each increase mean fitness will eventually reach the highest mean fitness. In contrast, with non-additive gene action (dominance and epistasis) those same small, generation-by-generation changes in allele frequencies may lead to local maxima because the fitness surface is not a plane. Such peaks and valleys of mean fitness occur when the genotype at one locus has an impact on fitness values at another locus. Fitness surfaces therefore have increasingly com-

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