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Selection is defined as any act that restricts the random mating of individuals in a population, sometimes just called non-random mating. Consider a locus with two alleles with frequencies, p and q. The population is in Hardy-Weinberg equilibrium. Now suppose that a fraction, u, of the A1A1 genotypes, a fraction, v, of the A1A2 genotypes, and a fraction, s, of the A2A2 genotypes have been culled from this population. The resulting parent population can be summarized as follows:
Assuming that culling has been applied equally to both sexes, then
the new frequency of the A1 allele in the parent generation is
Culling could have been applied differentially to males and females, so that the allele frequencies in the remaining males and females could be different. Their progeny would not be in Hardy-Weinberg equilibrium.
The genotypic variance in the initial population is given by
1. Assortative Mating
Consider an initial population in Hardy-Weinberg equilibrium, as before. Below is a mating diagram in which the frequency of the A1allele can be different in the male and female population. The frequencies in the table are the expected frequencies of each type of mating - in a random mating population.
Non-random mating means that the frequencies in the above table are altered from their expected values. An example is assortative mating, which would give a mating diagram as follows:
Note that the resulting frequencies do not sum to one.
pm=pf to simplify the discussion, then
the sum of the non-zero elements is
The progeny genotypic frequencies would be as follows:
The frequency of the A1 allele in the progeny is
To illustrate the above, let p=.4 and let the genotypic values
be a=2 and d=1. The genotypic mean and variance in the
original random mating population would be
Note that S=.3856. The progeny figures would be
The new genetic mean and variance in the progeny generation is
2. Disassortative Mating
Disassortative mating is the mating of unlike genotypes. The mating diagram would look like
Note that the mating of
A1A2 genotypes appears in both
assortative and disassortative mating. Another possibility
in both cases would be to remove these matings as well.
The sum of the frequencies of the matings that were made is
The new frequency of A1 is 0.5, regardless of the starting frequency, and the frequency will remain at 0.5 in future generations whether doing disassortative matings of the same type or random matings. The frequency of homozygous genotypes will diminish in future generations of disassortative matings.
Both types of selection most likely occur simultaneously in the real world, plus other types of selection. Some animals are culled, perhaps for reasons other than their genotype or phenotype, but the reasons may be associated with the genotypes resulting in a change in the parental allele frequencies. After culling, matings may not be random, which will alter the progeny allele frequencies. If these processes occur each generation, then it is unlikely that Hardy-Weinberg equilibrium is ever achieved. Also, genotypic variation could be altered by many different factors. If selection is on phenotypes (because genotypes are masked), then the effects of selection may be reduced if the residual variation is large relative to genotypic variation. However, the effects of selection will still influence the genotypic variance.
Suppose the bottom 25% are culled, and that this affects males and females equally. Let p=0.4 as before, then the new frequencies of parental genotypes would be
The new progeny distribution after random mating of the selected parents would be
If another 25% were culled from the bottom, then the new frequencies of the parent genotypes would be
All of the A2A2 genotypes would be culled and part of the heterozygotes. Eventually fixation to the A1 allele would occur with continued selection.
This LaTeX document is available as postscript or asAdobe PDF.Larry Schaeffer