The Hardy-Weinberg theorem characterizes the distributions of genotype frequencies in populations that are not evolving and are thus the fundamental null model for population genetics.AaAaAa

## The Hardy-Weinberg Equilibrium

The Hardy-Weinberg Theorem deals with Mendelian genetics in the context of populations of diploid, sexually reproducing individuals. Given a set of assumptions (discussed below), this theorem states that:

- allele frequencies in a population will not change from generation to generation.
- if the allele frequencies in a population with two alleles at a locus are
*p*and*q*, then the expected genotype frequencies are*p*^{2}, 2*pq*, and*q*^{2}. This frequency distribution will not change from generation to generation once a population is in Hardy-Weinberg equilibrium. For example, if the frequency of allele*A*in the population is*p*and the frequency of allele*a*in the population is*q*, then the frequency of genotype*AA*=*p*^{2}, the frequency of genotype*Aa*= 2*pq*, and the frequency of genotype*aa*=*q*^{2}. If there are only two alleles at a locus, then*p*+*q*, by mathematical necessity, equals one. The Hardy-Weinberg genotype frequencies,*p*^{2}+ 2*pq*+*q*^{2}, represent the binomial expansion of (*p*+*q*)^{2}, and also sum to one (as must the frequencies of all genotypes in any population, whether it is in Hardy-Weinberg equilibrium). It is possible to apply the Hardy-Weinberg Theorem to loci with more than two alleles, in which case the expected genotype frequencies are given by the multinomial expansion for all*k*alleles segregating in the population: (*p*_{1}+*p*_{2}+*p*_{3}+ . . . +*p*k)^{2}.

The conclusions of the Hardy-Weinberg Theorem apply only when the population conforms to the following assumptions:

- Natural selection is not acting on the locus in question (i.e., there are no consistent differences in probabilities of survival or reproduction among genotypes).
- Neither mutation (the origin of new alleles) nor migration (the movement of individuals and their genes into or out of the population) is introducing new alleles into the population.
- Population size is infinite, which means that genetic drift is not causing random changes in allele frequencies due to sampling error from one generation to the next. Of course, all-natural populations are finite and thus subject to drift, but we expect the effects of drift to be more pronounced in small than in large populations.
- Individuals in the population mate randomly with respect to the locus in question. Although nonrandom mating does not change allele frequencies from one generation to the next if the other assumptions hold, it can generate deviations from expected genotype frequencies, and it can set the stage for natural selection to cause evolutionary change.

Given these conditions, it is easy to derive the expected Hardy-Weinberg genotype frequencies if we think about random mating in terms of the probability of producing each genotype via the random union of gametes into zygotes (Table 1). If each allele occurs at the same frequencies in sperm and eggs, and gametes unite at random to produce zygotes, then the probability that any two alleles will combine to form a particular genotype equals the product of the allele frequencies. Since there are two ways of generating the heterozygous genotype (*A* egg and *a* sperm, or *an* egg and *A* sperm), we sum the probabilities of those two types of union to arrive at the expected Hardy-Weinberg frequency of the heterozygous genotype (2*pq*).

Table 1: A Punnett square depicting the probabilities of generating all possible genotypes at a diallelic Mendelian locus in a population that conforms to Hardy-Weinberg assumptions.

Reference: This article is taken from the link below for study purposes:

https://www.nature.com/scitable/knowledge/library/the-hardy-weinberg-principle-13235724/