Alan E. Stark

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.

On S.N. Bernstein's derivation of Mendel's Law and 'rediscovery' of the Hardy-Weinberg distribution

Around 1923 the soon-to-be famous Soviet mathematician and probabilist Sergei N. Bernstein started to construct an axiomatic foundation of a theory of heredity. He began from the premise of stationarity (constancy of type proportions) from the first generation of offspring. This led him to derive the Mendelian coefficients of heredity. It appears that he had no direct influence on the subsequent development of population genetics. A basic assumption of Bernstein was that parents coupled randomly to produce offspring. This paper shows that a simple model of non-random mating, which nevertheless embodies a feature of the Hardy-Weinberg Law, can produce Mendelian coefficients of heredity while maintaining the population distribution. How W. Johannsen’s monograph influenced Bernstein is discussed.

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.

Wilhelm Weinberg’s Early Contribution to Segregation Analysis

Wilhelm Weinberg (1862–1937) is a largely forgotten pioneer of human and medical genetics. His name is linked with that of the English mathematician G. H. Hardy in the Hardy–Weinberg law, pervasive in textbooks on population genetics since it expresses stability over generations of zygote frequencies AA, Aa, aa under random mating. One of Weinberg’s signal contributions, in an article whose centenary we celebrate, was to verify that Mendel’s segregation law still held in the setting of human heredity, contrary to the then-prevailing view of William Bateson (1861–1926), the leading Mendelian geneticist of the time. Specifically, Weinberg verified that the proportion of recessive offspring genotypes aa in human parental crossings Aa × Aa (that is, the segregation ratio for such a setting) was indeed p=14. We focus in a nontechnical way on his procedure, called the simple sib method, and on the heated controversy with Felix Bernstein (1878–1956) in the 1920s and 1930s over work stimulated by Weinberg’s article.

A.N. Kolmogorov's defence of Mendelism

In 1939 N.I. Ermolaeva published the results of an experiment which repeated parts of Mendel’s classical experiments. On the basis of her experiment she concluded that Mendel’s principle that self-pollination of hybrid plants gave rise to segregation proportions 3:1 was false. The great probability theorist A.N. Kolmogorov reviewed Ermolaeva’s data using a test, now referred to as Kolmogorov’s, or Kolmogorov-Smirnov, test, which he had proposed in 1933. He found, contrary to Ermolaeva, that her results clearly confirmed Mendel’s principle. This paper shows that there were methodological flaws in Kolmogorov’s statistical analysis and presents a substantially adjusted approach, which confirms his conclusions. Some historical commentary on the Lysenko-era background is given, to illuminate the relationship of the disciplines of genetics and statistics in the struggle against the prevailing politically-correct pseudoscience in the Soviet Union. There is a Brazilian connection through the person of Th. Dobzhansky.

Hardy-Weinberg Equilibrium as Foundational

The Hardy-Weinberg Principle explains how random mating can produce and maintain a population in equilibrium, that is: with constant genotypic proportions. The Hardy-Weinberg formula is in constant use as a basis for developing population genetics theory. Here we give a complete description of a model which can sustain equilibrium but with a general mating system, thereby giving a much broader basis on which to develop population genetics. It was S. N. Bernstein who first showed how Mendel’s first law could be justified simply on the basis of observations of populations in equilibrium. We show how the model can be applied to exploring the change in incidence of a genetic disorder.

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.

Use of Bayes' Formula to Make a Genetic Risk Estimate

Izabella and her partner sought pre-implantation genetic diagnosis (PGD) because Izabella had retinoblastoma due to a deletion in chromosome 13 and they want to have children not at genetic risk of retinoblastoma. Fortunately, Izabellas tumor was unilateral and was treated successfully and she is well. Izabellas chromosome abnormality is mosaic with 70% of lymphocytes having the deletion. This mosaicism may not be present in Izabellas ovaries. The couple went through PGD on two occasions and 13 embryos were tested. None had the deleted chromosome 13. IVF and PGD failed to produce a pregnancy. The couple wished to know what the experience provides as to the risk to their offspring: in particular, does it indicate a risk low enough to be acceptable if they go ahead with a natural pregnancy instead of another resort to PGD? Also, the couple did not want prenatal diagnosis. The situation therefore requires an estimate of the probability that an embryo will have the deletion. Counseling is problematic because there is no obvious way of selecting a prior probability from which to compute a Bayesian estimate of risk. Two solutions are offered, depending on the amount of information available about genes transmitted from the maternal grandparents.