# The Hardy – Weinberg Equilibrium

If certain conditions are met, the genetic structure of a population may not change over time. The necessary conditions for such an equilibrium were deduced independently in 1908 by the British mathematician Godfrey Hardy and the German physician Wilhelm Weinberg. Hardy wrote his equations in response to a question posed to him by the geneticist Reginald C. Punnett (the inventor of the Punnett square) at the Cambridge University faculty club. Punnett wondered at the fact that even though the allele for short, stubby fingers (a condition called *brachydactyly*) was dominant and the allele for normal-length fingers was recessive, most people in Britain have normal-length fingers. Hardy’s equations explain why dominant alleles do not necessarily replace recessive alleles in populations, as well as other features of the genetic structure of populations. The **Hardy–Weinberg equilibrium**applies to sexually reproducing organisms. The particular example we will illustrate here assumes that the organism in question is diploid, its generations do not overlap, the gene under consideration has two alleles, and allele frequencies are identical in males and females. The Hardy–Weinberg equilibrium also applies if the gene has more than two alleles and generations overlap but in those cases the mathematics is more complicated. Several conditions must be met for a population to be at Hardy–Weinberg equilibrium:

- Mating is random

- Population size is very large

- There is no migration between populations

- There is no mutation

- Natural selection does not affect the alleles under consideration

If these conditions hold, two major consequences follow. First, the frequencies of alleles at a locus will remain constant from generation to generation. And second, after one generation of random mating, the genotype frequencies will remain in the following proportions: Genotype *AA Aa aa* Frequency *p*2 2*pq q*2 Stated another way, the equation for Hardy–Weinberg equilibrium is

*p*2 + 2*pq *+ *q*2 = 1

To see why, consider population 1 in Figure 23.6, in which the frequency of *A *alleles (*p*) is 0.55. Because we assume that individuals select mates at random, without regard to their genotype, gametes carrying *A *or *a *combine at random—that is, as predicted by the frequencies *p *and *q*. The probability that a particular sperm or egg in this example will bear an *A* allele rather than an *a *allele is 0.55. In other words, 55 out of 100 randomly sampled sperm or eggs will bear an *A *allele. Because *q *= 1 – *p*, the probability that a sperm or egg will bear an *a *allele is 1 – 0.55 = 0.45. To obtain the probability of two *A*-bearing gametes coming together at fertilization, we multiply the two independent probabilities of their occurring separately: *p **p *= *p*2 = (0.55)2 = 0.3025 Therefore, 0.3025, or 30.25 percent, of the offspring in the next generation will have the *AA *genotype. Similarly, the probability of bringing together two *a*-bearing gametes is

*q **q *= *q*2 = (0.45)2 = 0.2025

Thus, 20.25 percent of the next generation will have the *aa *genotype.Figure also shows that there are two ways of producinga heterozygote: An *A *sperm may combine with an *a *egg,the probability of which is *p **q*; or an *a *sperm may combinewith an *A *egg, the probability of which is *q **p*. Consequently,the overall probability of obtaining a heterozygoteis 2*pq*.It is now easy to show that the allele frequencies *p *and *q *remain constant for each generation. If the frequency of *A *allelesin a randomly mating population is *p*2 + *pq*, this frequencybecomes *p*2 + *p*(1 – *p*) = *p*2 + *p *– *p*2 = *p*, the original allelefrequencies are unchanged, and the population is atHardy–Weinberg equilibrium.

If some agent, such as emigration, were to alter the allele frequencies, the genotype frequencies would automatically settle into a predictable new set in the next generation. For instance, if only *AA *and *Aa *individuals left the population, *p *and *q *would change, but there would still be *aa *individuals in the population.

**Why is the Hardy–Weinberg equilibrium important?**

You may already have realized that populations in nature rarely meet the stringent conditions necessary to maintain them at Hardy–Weinberg equilibrium. Why, then, is the Hardy-Weinberg equilibrium considered so important for the study of evolution? The answer is that without it, we cannot tell whether or not evolutionary agents are operating. The most important message of the Hardy–Weinberg equilibrium is that *allele frequencies remain the same from generation to generation unless some agent acts to change them.* In order to ascertain that evolutionary agents are in play, we must estimate the actual allele or genotype frequencies present in a population and then compare them with the frequencies that would be expected at Hardy–Weinberg equilibrium. The pattern of deviation from the Hardy–Weinberg expectations tells us which assumptions are violated. Thus, we can identify the agents of evolutionary change on which we should concentrate our attention.