A Century of Hardy–Weinberg Equilibrium Oliver Mayo CSIRO Livestock Industries, Adelaide, Australia ardy–Weinberg equilibrium (HWE) is the state of essentially stable in genetical terms: a system, like Hthe genotypic frequency of two alleles of one the genome of a population, is at equilibrium in autosomal gene locus after one discrete generation time when no net change occurs or is expected to of random mating in an indefinitely large population: occur from its state at that time. Furthermore, equi- if the alleles are A and a with frequencies p and libria can be stable, meaning that when a small q (=1-p), then the equilibrium gene frequencies are displacement occurs, the system is expected to simply p and q and the equilibrium genotypic fre- return to the equilibrium. 2 2 quencies for AA, Aa and aa are p , 2pq and q . It was Soon after the rediscovery of Mendel’s remarkable independently identified in 1908 by G. H. Hardy and work in 1900, interest arose in the properties of W. Weinberg after earlier attempts by W. E. Castle Mendelian genes in populations; this was the dawn of and K. Pearson. Weinberg, well known for pioneer- population genetics. Castle (1903) and Pearson ing studies of twins, made many important (1903a, 1903b) were among the first to investigate contributions to genetics, especially human genet- these. As Edwards (in press) has pointed out, Castle ics. Existence of this equilibrium provides a did not derive a generalization equivalent to reference point against which the effects of selec- Pearson’s, and will be considered no further in this tion, linkage, mutation, inbreeding and chance can paper. be detected and estimated. Its discovery marked Mendel (1865) had hypothesized that inheritance the initiation of population genetics. of a trait was particulate, that the units of inheritance did not change from generation to generation, that they were contributed equally by an organism’s two Hardy–Weinberg equilibrium (HWE) is the state of parents, and that each parent contributed its unit at the genotypic frequency of two alleles of one gene random from the two it contained. He had shown locus after one generation of random mating in an that a cross between two pure-breeding lines, termed indefinitely large population with discrete genera- A and a in regard to some trait, gave a first genera- tions, in the absence of mutation and selection: if the tion resembling one of the two parents (A) identically alleles are A and a with frequencies p and q (= 1-p), and that crosses among members of this first genera- then the equilibrium gene frequencies are just p and q tion gave a ratio of 3:1 of the two parental types, the and the equilibrium genotypic frequencies for AA, Aa more frequent type being the same as this first genera- and aa are p2, 2pq and q2. Thus, there is equilibrium tion (A). He had described the first generation type as at both the allelic and the genotypic level. Not exces- dominant, the other as recessive. sively fancifully, one could compare this Mendel produced a model of the following kind: Hardy–Weinberg rule with Newton’s first law of the genetic make-up of the two parental lines was AA motion: a physical body will remain at rest, or con- and aa respectively, and their offspring were Aa. tinue to move at a constant velocity, unless an 1 Crossing two Aa gave, by the binomial expansion ( ⁄2A external force acts upon it. If such stability is the rule, 1 1 1 1 1 1 1 + ⁄2a)( ⁄2A + ⁄2a), ⁄4AA, ⁄4Aa, ⁄4aA, ⁄4aa. If Aa = aA then it provides the basis for the detection and estima- (since the units of inheritance are unchanged), and if tion of the effects on the population of ‘the thousand Aa resembles AA exactly (the phenomenon of domi- natural shocks the flesh is heir to’, including natural nance, deduced from the disappearance of a in the and artificial selection, mutation, assortative mating, cross of the two lines), then the proportions of the migration, inbreeding and random sampling (through 3 1 phenotypes A:a will be ⁄4: ⁄4. Mendel made further finite population size). confirmatory crosses, for example showing that two Each word in the topic concept deserves explica- thirds of the A types were Aa and one third were AA. tion: Hardy was a notable pure mathematician, Mendel did not consider directly what would Weinberg was a pioneering human geneticist and happen in a population of an organism, but this was doctor to the poor who made a special contribution to twin studies, and the concept of equilibrium is Received 19 February, 2008; accepted 3 March, 2008. simple and attractive to those in permanent disequi- Address for correspondence: Oliver Mayo, CSIRO Livestock librium, like human beings. The concept made Industries, PO Box 100401 Adelaide BC, SA 5000, Australia. E-mail: immediately clear that human populations were [email protected] Twin Research and Human Genetics Volume 11 Number 3 pp. 249–256 249 Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.234, on 30 Sep 2021 at 04:27:46, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1375/twin.11.3.249 Oliver Mayo essential if evolutionary phenomena were to be the frequencies of the three possible genotypes are as explained or even studied at the level of the hypothesized shown: ‘essential character’, as Mendel called his fundamental A1A1 A1A2 A2A2 particles of inheritance. (The name ‘gene’ was introduced PQR 1 in about 1905 by Wilhelm Johannsen, as an abbrevia- Then the frequencies of the alleles A1 and A2 are P+ ⁄2Q 1 tion of Darwin’s and De Vries’s ‘pangen’.) and ⁄2Q + R respectively. Call these p and q respec- Pearson immediately saw that dominance/ tively. Call this population the parental generation. recessiveness was not essential to the dynamics of the Some very simple algebra shows that the frequencies model, but was rather an additional assumption of of the three genotypes in the offspring produced by Mendel’s. He generalized the model by removing this random mating among this parental generation will be assumption, and also began the analysis of multiple A1A1 A1A2 A2A2 1 2 1 1 1 2 independent genes. (P + ⁄2Q) 2(P + ⁄2Q)( ⁄2Q + R)(⁄2Q + R) On his model, the first two generations described = p2 = 2pq = q2 above would be It is also simple to show that these genotypic frequen- (AA´) × (aa´) = (A+A´)(a+a´) = cies, which are also those chosen by pairwise sampling (Aa) + (Aa´) + (A´a) + (A´a´), of gametes at random in the population, will be the same in the next generation. (At this equilibrium, representing the parents, the gametes and the offspring Q2 =4PR.) The process of pairwise sampling is simple in turn. If the gametes identified by the prime ´ are binomial sampling with replacement, justifiable actually identical, that is A = A´ and so on, then this because the populations of gametes can be regarded as second generation is identical, Aa. This can then be indefinitely large. For a careful and complete mathe- extended to multiple independent genes. Pearson matical account of HWE, see Edwards (2000). (1903b) wrote: The English pure mathematician G. H. Hardy If these hybrids now breed at random and are equally (1908), notable for contributions to number theory fertile among themselves, segregation takes place. If and analysis, simply showed that the relationship the process of random mating with equal fertility be given above would hold; he had been asked what continued generation by generation, what further would happen to gene and genotype frequencies in a changes, if any, take place, and what are the laws of population mating at random, and gave the answer. inheritance within such a population? (p. 506) He participated no further in population genetics. His conclusion was ‘that when the members of this seg- Diaconis (2002) has speculated that Hardy had ‘a true regating population cross at random the population antipathy to the subject’ of probability, which could accurately reproduces itself, and supposing no artificial, explain his failure to contribute further, but it could natural or reproductive selection takes place, a stable equally well be explained by his love of pure mathe- population or ‘race’ is created, which is permanent and matics and total lack of interest in applications. shows a permanent proportional frequency for each Hardy’s place in mathematical history is secure; that sub-class of the population’. From this important con- in genetical history minor but significant. For an clusion, Pearson went on to calculate parent-offspring accessible portrait, see Snow (1967) and Hardy’s own correlations, rOP, and other attributes of the quantita- memoir (1940), and for detailed comments on Hardy’s tive inheritance which he was developing. (1908) paper, see Edwards (in press). Unfortunately, having pointed out that he did not need Weinberg (1908), who was a human geneticist of 1 Mendel’s hypothesis of dominance, he calculated rOP = ⁄3 the first rank, though widely regarded in his own and noted that this was not in agreement with empirical country as an Armenarzt (a doctor employed by a observations which lay round 0.5. For this and other local authority to treat the indigent, an honorable reasons he abandoned particulate inheritance of the calling, perhaps, but hardly a sign of success in his Mendelian kind. Had he assumed that the heterozygote career), did much more work on the topic. 1 was intermediate, he would have obtained rOP = ⁄2. After the publication of Hardy’s note, Pearson (1909a, Weinberg’s Contribution 1909b) obtained correct results, without referring to his Wilhelm Weinberg was born in Stuttgart in 1862, was earlier errors as such.