Aa = XY + 2XZ + 2Y2/4 + YZ = 2(XY/2 + XZ + Y2/4 + YZ/2) = 2(X + Y/2)(Z + Y/2) = 2pq

So we have proved that progeny genotype and allele frequencies are identical to parental genotype and allele frequencies over one generation or that f(A)( = f(A)i+1. The major conclusion here is that genotype frequencies remain constant over generations as long as the assumptions of Hardy-Weinberg are met. In fact, we have just proved that under Mendelian heredity genotype and allele frequencies should not change over time unless one or more of our assumptions is not met. This simple model of expected genotype frequencies has profound conclusions! In fact, Hardy-Weinberg expected genotype frequencies serve as one of the most basic tools to test for the action of biological processes that alter genotype and allele frequencies.

You might wonder whether Hardy-Weinberg applies to loci with more than two alleles. For the last point in this section let's explore that question. With three alleles at one locus (allele frequencies symbolized by p, q, and r), Hardy-Weinberg expected genotype frequencies are p2 + q2 + r2 + 2pq + 2pr + 2qr = 1. These genotype frequencies are obtained by expanding (p + q + r)2, a method that can be applied to any number of alleles at one locus. In general, expanding the squared sum of the allele frequencies will show:

• the frequency of any homozygous genotype is the squared frequency of the single allele that composes the genotype ([allele frequency]2);

• the frequency of any heterozygous genotype is twice the product of the two allele frequencies that comprise the genotype (2[allele 1 frequency] x [allele 2 frequency]); and

• there are as many homozygous genotypes as there are alleles and NiN^-1 heterozygoses where N is the number of alleles.

Do you think it would be possible to prove Hardy-Weinberg for more than two alleles at one locus? The answer is absolutely yes. This would just require constructing larger versions of the parental genotype mating table and expected offspring frequency table as we did for two alleles at one locus.

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