On the Theory of Random Mating

Abstract

INTRODUCTIONA problem of primary importance in population genetics is that of determining the genotypic frequencies in a large population of sexually reproducing organisms mating at random in nonoverlapping generations with negligible mutation and selection. In populations of disomic organisms where the sexes enjoy equal conditions (i.e. sex-linked inheritance is neglected and both sexes are evenly distributed amongst the various genotypes and possess the same frequencies for the modes of gamete formation), Hardy (1908) showed that gene frequency remains constant and that the genotypic distribution for any Mendelian factor reaches a stationary state in the first generation of random mating. Some years after the discovery of genetic linkage, Jennings (1917) andRobbins (1918) showedthat withtwo linked factors the population approaches an equilibrium state in which the factors are associated at random. They determined the rate of approach to equilibrium and gave the genotypic distribution after any finite number of generations. Geiringer (1945) has given a similar analysis for three factors. With more than three linked factors the problem is much more complex, the basic recurrence relations for the gametic frequencies then forming a system of quadratic instead of linear difference equations. A general but involved method of procedure has been outlined for these cases by Geiringer (1944). In the present paper we develop a general representation of random mating in disomic species as a linear transformation of a set of rational integral functions of the gametic frequencies.The principal components provide an elegant form of solution to the problem. Two other questions are considered from the same standpoint: (i) a modification of the above problem in which allowance is made for possible frequency differences between the sexes, and (ii) random-in-time mating in a population of virus (or haploid) particles.

Decomposing Multilocus Linkage Disequilibrium

“…΅ , tirely consistent with the standard definitions for two loci, as well as its previous extensions to three, four, and (1a) six loci (Geiringer 1944;Bennett 1954;Hastings 1984). Third, we show how this definition is entirely where n i ʦ c means that n i ʦ c is a scalar component consistent with the notion of linkage disequilibrium as of the vector c. Equivalently, the deviation from probabilistic independence.…”

“…eral multilocus linkage disequilibrium, even though special cases have been given by Geiringer (1944), Ben- 1954), andHastings (1984). The only explicit…”

Analysis of some nonrandom mating models

“…We have already seen that this changes with time and the mathematics gets quite complicated when more and more loci are involved even for a simpler sit uation of pure random mating. The treatment of changing relations among three or more loci for pure random mating population has been discussed by Geiringer (1944Geiringer ( , 1948 and by Bennett (1954). These formulas can be usefully employed while studying the changes in the gametic frequencies under the model of mixed selfing and random mating because the frequency of a gamete under such a system can be expressed as a linear combination of the corresponding gametic fre quencies under pure random mating and under pure selfing with appropriate weights.…”

The Dynamics of HIV-1 Adaptation in Early Infection

“…These were calculated following Bennett (1954). In this approach, sets of haplotypes and their compliments that may recombine to produce the haplotype of interest are identified.…”