The Hardy-Weinberg Law states: In a large, random-mating population that is not affected by the evolutionary processes of mutation, migration, or selection, both the allele frequencies and the genotype frequencies are constant from generation to generation. Furthermore, the genotype frequencies are related to the allele frequencies by the square expansion of those allele frequencies. In other words, the Hardy-Weinberg Law states that under a restrictive set of assumptions, it is possible to calculate the expected frequencies of genotypes in a population if the frequency of the different alleles in a population is known.

The genotype frequencies are calculated using the square expansion of the allele frequencies. To illustrate this concept, assume that at some locus, A, you have two alleles, call them A_{1}, and A_{2}. Assume that the frequency of allele A_{1} is p and the frequency of allele A_{2} is q. We can write this as:

f(A_{1}) = p f(A_{2}) = q

Under Hardy-Weinberg conditions, the expected genotypic proportions in the population are

(p + q)^{2}

The square expansion of allele frequencies when there are two alleles is p^{2} + 2pq + q^{2} meaning that: f(A_{1}A_{1}) = p^{2}, f(A_{1}A_{2}) = 2pq, and f(A_{2}A_{2}) = q^{2}

If there were a third allele, call it A_{3}, and it was present at frequency r, then the expected genotypic proportions would be (p + q + r)^{2}. In other words, the expected genotypic frequencies would be: f(A_{1}A_{1}) = p^{2}, f(A_{2}A_{2}) = q^{2}, f(A_{3}A_{3}) = r^{2} , f(A_{1}A_{2}) = 2pq, f(A_{1}A_{3}) = 2pr, and f(A_{2}A_{3}) = 2qr.

The implications of the Hardy-Weinberg Law are that:

- The population is in a state of equilibrium.
- The frequencies of alleles in a population will remain constant from generation to generation.
- The genotypic frequencies will remain constant from generation to generation.
- The Hardy-Weinberg proportions will be reached in a single generation of random-mating.

As an example, consider a diploid pathogen such as an oomycete that has two alleles at an isozyme locus. If the frequency of the fast allele (F) is 0.40 and the frequency of the slow alleles (S) is 0.60, then the expected frequencies of the genotypes FF, FS, and SS would be 0.16, 0.48, and 0.36, respectively.

The Hardy-Weinberg Law offers a model that is often used as a starting point for studying the population genetics of diploid organisms that meets the basic assumptions of large population size, random-mating, and no migration, mutation, or selection. Unfortunately, the Hardy-Weinberg Law cannot be applied to haploid pathogens. If a population is found not to be in H-W equilibrium, then one of the assumptions in the Law has been violated. That is, there has been non-random mating, or selection or migration have affected the population. At this point, hypotheses are proposed and experiments are conducted to determine why a population is not at equilibrium.

A good example of Hardy-Weinberg testing in a plant pathogen is for *Phytophthora infestans* in- and outside of its putative center of origin (Tooley et. al. 1985).

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