The Impossible Dream - Panmixia.

This study starts with a simple model by which Hardy-Weinberg proportions are attained in a single generation while maintaining gene frequencies. The question of differentiating between random and non-random mating is explored by simulation. Sample mating proportions are generated using the model as base. The difficulty of differentiating between random and non-random mating is illustrated.

C. C. Li and Quasi-Random Mating.

A simple model by which Hardy-Weinberg proportions are attained in a single generation while maintaining gene frequencies is stated and illustrated. The title 'Quasi-random mating' is proposed. Confusion about the Hardy-Weinberg principle can be avoided only if there is clear separation between the basic deterministic model and factors influencing a population's structure. Eighty years passed before C. C. Li coined the term 'pseudo-random mating'. The lesson taught by Li has not been taken on board.

Stable populations and Hardy-Weinberg equilibrium.

The conditions on the mating matrix associated with a stable equilibrium are specified for an autosomal locus with five alleles. This points the way to the maintenance of Hardy-Weinberg proportions with non-random mating. The myth of random mating is exposed.

A Markov Chain Model for the Evolution of Sex Ratio.

A model in the form of a Markov chain is constructed to mimic variations in the human sex ratio. It is illustrated by simulation. The equilibrium distribution is shown to be a simple modification of the binomial distribution. This enables an easy calculation of the variation in sex ratio which could be expected in small populations.

Pseudo-Random Mating with Multiple Alleles.

The conditions on the mating matrix associated with a stable equilibrium are specified for an autosomal locus with four alleles. An example illustrates how Hardy-Weinberg proportions are maintained with nonrandom mating. The ABO blood group provides an illustration.

A Misconception About the Hardy-Weinberg Law.

The Hardy-Weinberg law of population genetics is usually associated with the notion of random mating of parents. A numerical example for a triallelic autosomal locus shows that an uncountable set of mating combinations can maintain Hardy-Weinberg proportions. Therefore, one cannot infer random mating in a population from the observation of Hardy-Weinberg equilibrium. The mating system which ensures that the genotypic distribution of offspring is the same as that of the parents is specified.

On Random and Systematic Variation in the Prevalence of Defective Color Vision.

Defective color vision comes in various forms and its frequency varies from population to population. This article is concerned with only the sex-linked form of essential hereditary color blindness. A model of a 'small' population is constructed to explore the dynamics of occurrence of color blindness. Different mutation rates are introduced for eggs and sperm. Birth and death rates of affected individuals are assumed to be the same as those in the unaffected. Simulation demonstrates that large changes in frequency occur randomly from the combined effects of mutation, transmission of genes from generation to generation and births and deaths. A reference is made to the hypothesis that observed differences in rates are due to selection in the transition from hunter-gatherer to farmer.

Reminiscence of a Teacher of Medical Genetics - E. M. Nicholls.

A brief account is given by E. M. Nicholls, M.D., of the formation and demise of the School of Human Genetics of the University of New South Wales.

Hemophilia B - Mutation Rates and Incidence - A Simulation Study.

The dynamics of rare X-linked recessive traits is explored by simulation. The model follows the prevalence of affected males and carrier females as separate but correlated variables. Different mutation rates and selection coefficients are introduced for males and females. A virtual population based on a published study of hemophilia B in the west of Scotland is followed at weekly intervals over many years. Speculative values of critical parameters to mimic the real population are proposed.

Mixed organic and inorganic tapwater exposures and potential effects in greater Chicago area, USA.

Safe drinking water at the point of use (tapwater, TW) is a public-health priority. TW exposures and potential human-health concerns of 540 organics and 35 inorganics were assessed in 45 Chicago-area United States (US) homes in 2017. No US Environmental Protection Agency (EPA) enforceable Maximum Contaminant Level(s) (MCL) were exceeded in any residential or water treatment plant (WTP) pre-distribution TW sample. Ninety percent (90%) of organic analytes were not detected in treated TW, emphasizing the high quality of the Lake Michigan drinking-water source and the efficacy of the drinking-water treatment and monitoring. Sixteen (16) organics were detected in >25% of TW samples, with about 50 detected at least once. Low-level TW exposures to unregulated disinfection byproducts (DBP) of emerging concern, per/polyfluoroalkyl substances (PFAS), and three pesticides were ubiquitous. Common exceedances of non-enforceable EPA MCL Goal(s) (MCLG) of zero for arsenic [As], lead [Pb], uranium [U], bromodichloromethane, and tribromomethane suggest potential human-health concerns and emphasize the continuing need for improved understanding of cumulative effects of low-concentration mixtures on vulnerable sub-populations. Because DBP dominated TW organics, residential-TW concentrations are potentially predictable with expanded pre-distribution DBP monitoring. However, several TW chemicals, notably Pb and several infrequently detected organic compounds, were not readily explained by pre-distribution samples, illustrating the need for continued broad inorganic/organic TW characterization to support consumer assessment of acceptable risk and point-of-use treatment options.

Variation in Prevalence of Rare Heritable Traits - A Simulation Study - Illustrated by Neurofibromatosis Type 1.

The epidemiology of heritable traits whose prevalence is determined by a balance between mutation and selection is often explored through deterministic models. Here, the properties are explored by simulation of a model population followed through a sequence of closely spaced time points. Mutation and birth and death occur randomly. The condition neurofibromatosis type 1 (NF1) is used as a point of reference. Critical parameters, such as mutation rates and selection forces, are not known precisely for NF1 so speculative values based on published data from Finland and other studies are proposed.

The X-Linked Analog of the Hardy-Weinberg Law.

The system of mating that maintains a general genotypic distribution among females with respect to an X-linked locus is defined. In particular, it is shown that Hardy-Weinberg proportions can be maintained with non-random mating.

Edward Maxwell (Max) Nicholls (1927-2011), a Key Player in the Development of the Two-Hit Model of Tumor Formation.

E. M. Nicholls (1927-2011) was a humanist, medical practitioner, human biologist, geneticist and, above all, a teacher, as well as a husband and father. He believed that he had made a fundamental contribution to the two-hit model of cancer formation. This hypothesis is associated with retinoblastoma, in particular. Nicholls presented it through his observations on neurofibromatosis. He received little credit for what he believed was his most original contribution to medical science. This note attempts to redress the balance in his favor.

Estimation of divergence from Hardy-Weinberg form.

The Hardy­Weinberg (HW) principle explains how random mating (RM) can produce and maintain a population in equilibrium, that is, with constant genotypic proportions. When proportions diverge from HW form, it is of interest to estimate the fixation index F, which reflects the degree of divergence. Starting from a sample of genotypic counts, a mixed procedure gives first the orthodox estimate of gene frequency q and then a Bayesian estimate of F, based on a credible prior distribution of F, which is described here.

A reality check on Hardy-Weinberg.

G. H. Hardy (1877-1947) and Wilhelm Weinberg (1862-1937) had very different lives, but in the minds of geneticists, the two are inextricably linked through the ownership of an apparently simple law called the Hardy-Weinberg law. We demonstrate that the simplicity is more apparent than real. Hardy derived the well-known trio of frequencies {q 2, 2pq, p 2} with a concise demonstration, whereas for Weinberg it was the prelude to an ingenious examination of the inheritance of twinning in man. Hardy's recourse to an identity relating to the distribution of types among offspring following random mating, rather than an identity relating to the mating matrix, may be the reason why he did not realize that Hardy-Weinberg equilibrium can be reached and sustained with non-random mating. The phrase 'random mating' always comes up in reference to the law. The elusive nature of this phrase is part of the reason for the misunderstandings that occur but also because, as we explain, mathematicians are able to use it in a different way from the man-in-the-street. We question the unthinking appeal to random mating as a model and explanation of the distribution of genotypes even when they are close to Hardy-Weinberg proportions. Such sustained proportions are possible under non-random mating.

Mating and offspring frequencies under partial outcrossing in a structured population

This paper gives a model of a structured population with respect to an autosomal locus with two alleles. The population reproduces in discrete and non-overlapping generations. The population is assumed to be in equilibrium in that exactly the same distribution of genotypic proportions is reproduced in each generation. The population is subdivided into 'localities' which are characterized by the local gene frequencies. Within each locality the genotypic proportions may depart from Hardy-Weinberg proportions and the same fixation index applies to all localities. The system departs from reality by assuming that the frequency of the first allele follows the beta distribution. However, this enables a convenient way to derive the mating frequencies of parents so that equilibrium is maintained. Wright's F-statistics are applied to characterize the population as a whole. The system is extended to permit an arbitrary level of outbreeding.

On extending the Hardy-Weinberg law

This paper gives a general mating system for an autosomal locus with two alleles. The population reproduces in discrete and non-overlapping generations. The parental population, the same in both sexes, is arbitrary as is that of the offspring and the gene frequencies of the parents are maintained in the offspring. The system encompasses a number of special cases including the random mating model of Weinberg and Hardy. Thus it demonstrates, in the most general way possible, how genetic variation can be conserved in an indefinitely large population without invoking random mating or balancing selection. An important feature is that it provides a mating system which identifies when mating does and does not produce Hardy-Weinberg proportions among offspring.

A clarification of the Hardy-Weinberg law.

C. C. Li showed that Hardy-Weinberg proportions (HWP) can be maintained in a large population by nonrandom mating as well as random mating. In particular he gave the mating matrix for the symmetric case in the most general form possible. Thus Li showed that, once HWP are attained, the same proportions can be maintained by what he called pseudorandom mating. This article shows that, starting from any genotypic distribution at a single locus with two alleles, the same in each sex, HWP can be reached in one round of nonrandom mating with no change in allele frequency. In the model that demonstrates this fact, random mating is represented by a single point in a continuum of nonrandom possibilities.

Stages in the evolution of the Hardy-Weinberg law

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.

The Hardy-Weinberg principle

Hardy-Weinberg genotypic proportions can be maintained in a population under non-random mating. A compact formula gives the proportions of mating pair types. These are illustrated by some simple examples.