Stages in the evolution of the Hardy-Weinberg law

Abstract

The Hardy-Weinberg law has been used widely for about one hundred years with little question as to the foundations laid down by its originators. The basic assumption of random mating, that is choice of mates by a process akin to that of a lottery, was shown to produce genotypic proportions following the "binomial-square" rule, the so-called Hardy-Weinberg proportions (HWP). It has been assumed by many that random mating was the only way of pairing genes capable of producing HWP. However it has been shown that HWP can be obtained and maintained by non-random mating. The steps along the way to this revelation and some implications are reviewed.Key words: Hardy-Weinberg law, non-random mating. It is a strange fact that the most basic law of population genetics, which is attributed to Hardy (1908) and Weinberg (1908), is poorly understood by many scientists who use it routinely. Support for this apparently extravagant statement can be found in numerous textbooks and web sites. It is unfair to take Hancock (2004), which appears to be an excellent monograph on its topic -plant evolution -to illustrate the point. But, because it is such an admirable book, it does provide a good example. In the section entitled "Random Mating and Hardy-Weinberg Equilibrium" on page 24 is the following statement: "What these two men [Hardy and Weinberg] discovered mathematically is that genotype frequencies will reach an equilibrium in one generation of random mating in the absence of any other evolutionary force. The frequencies of different genotypes will then depend only upon the allele frequencies of the previous generation. If gene frequencies do not accurately predict genotype frequencies, then plants are crossing in a non-random fashion or some other evolutionary force is operating". While some of the quoted text is valid, the overall impression that random mating and Hardy-Weinberg proportions (HWP) are inextricably linked is not correct. Stark (2006, in press) shows that, provided the population has discrete and non-overlapping generations, HWP can be attained in one round of non-random mating and that random mating is a single point in a continuum of such possibilities. Furthermore Li (1988) and Stark (2005) give mating schemes which show that HWP can be maintained by non-random mating.To illustrate the problem further, Hancock (p. 35) writes "While the Hardy-Weinberg formula is very useful in describing the completely randomized situation, it is a rare natural population that is in fact in Hardy-Weinberg equilibrium. More commonly, mating is not random and the populations are subjected to other evolutionary forces, such as migration. genetic drift and natural selection". As noted above, HWP are consistent with (some) non-random mating.

Wilhelm Weinberg’s Early Contribution to Segregation Analysis

“…However, he was in the vanguard of efforts to exploit Mendelism to advance human and medical genetics. For example Weinberg (1908) was a notable contribution to the genetics of twinning in man (Stark 2006). A few writers, such as the late James Crow, did appreciate his work (Crow 1999).…”

“…At about that time he started to produce an impressive list of publications, since his activities and contacts as a medical practitioner gave him access to medical data and provided a stimulus to develop genetic and statistical concepts. His "Hardy-Weinberg law" article, Weinberg (1908), was a study of the genetics of twinning in humans (Stark 2006). Edwards (2008) begins with tribute to Weinberg's comprehensive formulation of the Hardy-Weinberg law.…”

A Reality Check on Hardy–Weinberg

“…Surely something as déclassé as 3:1 ratios was not to be preferred to GALTON's sophisticated and seductive mathematics. (p. 1333) But we leave the last word to Mayo (2008): Li (1988), followed and elaborated by Stark (2006aStark ( , 2006b, showed that panmixia is not the only breeding structure that can yield HW proportions, so that panmixia is a sufficient but not a necessary condition for HWE. However, no natural population is known to manifest the other possible breeding structures so that it appears unlikely that they need to be considered in data collection and analysis.…”

“…In the interests of completing the 'story' , we explain how this is so. As discussed below, Weinberg used the property of stability to explore the question of a possible genetic basis for the occurrence of twins in humans (Stark, 2006a).…”

Estimation of Divergence from Hardy–Weinberg Form

“…Weinberg (1908) used the HW distribution to analyze the inheritance of twinning in man. Stark (2006b) noted that Weinberg's analysis may need to be modified in the light of Li's paper. Stark and Seneta (2012) show that a paper of a Russian mathematician S. N. Bernstein was a fundamental, though largely unrecognized, contribution to genetic analysis by reason of its connection with the HW law.…”