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Z2
AA Aa aa shown in Table 7.4. X, Y, and Z are the frequencies of the genotypes AA, Aa, and aa, respectively. This is the same notation used for the proof of HardyWeinberg in Chapter 2.
This fecundity selection model is more complex than a viability model because the equations used to solve for genotype frequencies after one generation are functions of genotype frequencies in the parental generation. The mean number of offspring of each genotype after one generation of fecundity selection are found by summing the offspring frequencies (columns in Table 7.4) each weighted by its fecundity. For the progeny with the AA genotype the average fecundity, or X, is
which simplifies to fXt+1 = fnX2 + (fi2 + /2l)V2YX + f221/4Y2 (7.16)
Using similar steps, the equations for the average fecundities of the Aa and aa genotypes are f Yt+1 = (f12 + f21)1/2XY + (f13 + f31)XZ
and fZ+i = f33Z2 + (f32 + f23)1/2 YZ + f221/4Y2 (7.18)
The total average fecundity ( ) is the sum of the average fecundity for each genotype, so that fXt+1, fYt+1, and f Zt+1 give the proportion of the total number of offspring composed of any genotype after one bout of reproduction. Compare these equations for the average fecundity as functions of genotype frequencies with equations 6.21 and 6.22 that are in terms of allele frequencies. Since random mating does not occur by definition when there is fecundity selection, general equilibrium points cannot be found for arbitrary sets of the nine fecundity values shown in Table 7.4. Rather, the change in genotype frequencies caused by fecundity selection model must be understood by considering special cases of fecundity values.
One special case of fecundity selection occurs when the total fecundity of a mating is always the sum of the fecundity value of each genotype for each sex, a situation called additive fecundities, analogous to additive gene action (Penrose 1949). Let the fecundity values of females be fAA, fAa, and faa and the fecundity values of males be mAA, mAa, and maa. With an additive fecundity model, the fecundity values given in Table 7.4 would be fu = fAA + m^and^ = fAA + ^ as two examples. With additive fecundities, higher fecundities for heterozygotes result in both alleles being maintained in a population at equilibrium, as is true for overdominace in the viability selection model. A second special case is when fecundities are multiplicative (Bodmer 1965). For example, f11 = fAAmAA and f12 = fAAmAa. Depending on fecundity values for the three genotypes, there can be equilibrium points where both alleles are maintained in the population. A third special case that has been examined extensively is when there are four fecundity parameters that correspond to the degree of heterozygosity of each mating pair (Hadeler & Liberman 1975; Feldman et al. 1983). In these cases, depending on the specific fecundity values used, it is also possible that fecundity selection can maintain both alleles in the population, since equilibrium points are reached when all three genotypes have nonzero frequencies. Nevertheless, the fecundity selection model does not result in the maintenance of genetic variation for arbitrary fitness values more often than the basic viability model of selection (Clark & Feldman 1986). This means that fecundity models predict that natural selection frequently results in fixation or loss of alleles at equilibrium, just as the viability model does for directional selection.
Pollak (1978) has shown that mean fecundity does not necessarily increase with fecundity selection. This means that the mean fecundity is not necessarily maximized at equilibrium genotype frequencies for fecundity selection, in contrast to the way natural selection maximizes mean fitness in the viability model for one locus with two alleles.
Hybridization between genetically modified and wild sunflowers provides an example of how a simple fecundity selection model can be used to understand changes in allele frequencies. Using transgenic biotechnology, it is now routine practice to permanently incorporate foreign genes into crop plants. There is the possibility that such transgenes can escape into the wild through the hybridization that occurs between some crop plants and wild relatives that are often weeds (reviewed by Snow & Palma 1997). In the case of sunflowers, mating between pure crop genotypes and wild plants produces hybrids with seed production that is only 2% of that shown by wild plants but hybrids and wild plants have identical survival rates.
Cummings et al. (2002) established three experimental populations with half cropwildplant F1 hybrids and half wild plants. In these populations the initial frequency of cropspecific alleles was 25%. The frequencies of the cropspecific alleles at three allozyme loci dropped to about 5% in the next generation. The cropspecific allele frequency in the next generation best matched the allele frequency predicted by a fecundity selection model with additive fecundities.
Natural selection with frequencydependent fitness
In the basic viability model we considered fitness values as invariant properties of the genotypes. Another way to say this is that a fitness value wxx is constant regardless of conditions or the frequencies of any of the genotypes. It seems intuitive to expect that the fitness of a genotype may depend on its frequency in a population and there is direct evidence for frequencydependent fitness in natural populations. For example, mating success of males with different chromosomal inversion genotypes in Drosophila depends on chromosome inversion frequencies in the population (see AlvarezCastro & Alvarez 2005). In plants, the frequency of different flower colors in a population may impact the frequency of visits by pollinators and thereby cause frequencydependent mating success (e.g. Gigord et al. 2001; Jones & Reithel 2001). A whimsical example of frequencydependent fitness values comes from the left and right curvature of the mouths of Lake Tanganyikan cichlid fish Perissodus microlepis that pluck and eat scales from other fish. There appears to be an advantage to the rarer phenotype, presumably since the fish that are attacked anticipate approach of the cichlid from the side of the more frequent mouth phenotype (Hori 1993).
We can also construct a simple selection model where fitness (as genotypespecific viability) depends on genotype frequency and therefore changes as genotype frequencies change. The key concept in frequencydependent selection models is creating a measure of fitness that changes. Suppose that the fitness of a genotype decreases as that genotype becomes more common in the population, called negative frequency dependence. The relative fitness values are:
WAA = 

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