## The Hardy Weinberg

The basic formula of population genetics is a direct consequence of the binomial theorem. In a randomly interbreeding population segregating for a single gene with two alleles, A and a (i.e., discrete alternative characters), the stable frequency of genotypes is given by p2(AA):2pq(Aa):q2(aa). It can be seen that this expression corresponds to the special case given in row 2 of Pascal's triangle (Table 5.6). This concept was gradually derived over the years by longhand arguments involving the origin of evolutionary changes in populations (see Chapter 17 in A.H. Sturtevant's A History of Genetics), but it was first expressed in simple algebraic terms independently by Hardy and by Weinberg shortly after the rediscovery of Mendel's laws at the beginning of the 1900s. This idea, Figure 5.6 Binomial distribution with n = 20, p — 0.1 and 0.7.

elaborated upon in most textbooks of genetics, is commonly known as the Hardy-Weinberg law or as the Hardy-Weinberg equilibrium.

It is not difficult to see, in retrospect, how the binomial distribution came to be applied to the investigation of variations in Mendelian inheritance if the two primary assumptions implicit in the binomial theorem are fulfilled. The first assumption that two unequivocally distinguishable alternative characters are present in the population of interest is fulfilled by the fact that genes are of a particulate nature and that independent segregation of alleles occurs at meiosis; this mechanism ensures that alternative characters (alleles) are transmitted from parents to offspring. The second assumption that these characters be randomly chosen from the population is in part fulfilled by independent segregation of alleles during the reproductive phase of the cell cycle, but is strictly valid only if mutation, selection, or migration (between populations) is absent, and that mating between interbreeding individuals is random. Many of the further developments in population genetics revolve around the algebraic analysis of the effects of deviations from these stringent requirements. These assumptions are often closely approximated so that the Hardy-Weinberg law is useful in the genetic analysis of human populations.

So far, we have considered the applicability of the Hardy-Weinberg law solely to Mendelian segregation operating in populations that contain different proportions of individuals homozygous and heterozygous for various genes, but the same law is operating within families segregating those genes. It is usually impossible to test for the occurrence of segregation in humans by experimental

 Phenotypic matings Genotypic matingsc Expected frequency of matings