A clarification of the Hardy-Weinberg law
In 1988, eighty years after the seminal papers of Hardy and Weinberg, C.C. Li made an important contribution to population genetics theory by showing that Hardy-Weinberg proportions (HWP) can be maintained under non-random mating. It will be shown that this was not the first time that this had been recognised, as in fact it had been earlier by this author and perhaps by others. Thus Li refuted the assumption, made by Hardy himself, that random mating was a necessary condition for the maintenance of HWP. The publication of Li's paper in a prestigious periodical, accompanied by editorial comment, should have seen an end to fallacious accounts of the H-W law but they continue to appear in textbooks and innumerable web sites. This paper analyses Li's deceptively simple symmetrical mating model by means of a canonical form. It examines Li's assignment of additive values to genotypes leading to his assertion that the correlation between mates in respect of value is zero. It is shown that additive values and zero correlation apply formally if and only if Li's model is constrained and simplified in a precise way which leads to a model given by the author. A key feature of the H-W law is that HWP are achieved in one generation of random mating. It will be shown that it is possible to reach HWP in a single generation with non-random mating and suggests that the term pseudo-random mating introduced by Li may lead to continuing confusion, since his model describes obviously non-random mating. An important feature of the H-W law is that it explains how a population may conserve its properties from generation to generation. The models of Li and the author leave open the way to more realistic formulations of population genetics theory, that is by abandoning the unrealistic assumption of random mating.