On Nonrandom Mating Systems for Attaining Hardy-Weinberg Equilibrium

LI (1988) showed that random mating is a sufficient, not a necessary condition for the Hardy-Weinberg principle. A nonrandom mating population that behaves like a random mating population is thus called a ‘pseudo-random mating population’ by him. The pseudo-random mating system studied by him has been focused on those populations in which the parental generation is in Hardy-Weinberg proportions. In other words, the mating type frequency deviations from random mating for each parental genotype add up to zero. In this article these restrictions are relaxed and new pseudo-random mating systems that immediately yield Hardy-Weinberg offspring are also obtained. This is possible because reciprocal crosses have identical segregation probabilities for an autosomal locus, and the manipulation of the combined frequency of reciprocal crosses does not change the gene frequency of the population. A comparison of these new patterns with that of Li is given in the Discussion.     

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.     

A Two-Stage Test for Distinguishing Random,Pseudo-Random and Nonrandom Mating Populations

There is no such an implication that a population in Hardy-Weinberg equilibrium must have undergone random mating. Therefore, it is unequivocal that the usual tests for “Hardy-Weinberg equilibrium” are indeed tests for “random union of gametes” but not for “random mating”. In this paper, utilizing population characteristics expressed in equilibrium state (equilibrium or disequilibrium) and mating behavior (random or nonrandom), a two-stage testing procedure for distinguishing random, pseudo-random and nonrandom mating populations is proposed. At the first stage, a population is tested for Hardy-Weinberg equilibrium. If insignificant result (i.e., in equilibrium) is obtained, then to a second stage the population is further tested for mating behavior. Random mating-pairs data are needed here for analysis instead of random individuals for usual Hardy-Weinberg equilibrium tests. Since distinguishing the three types of mating populations depends on the combined results of two stages, the probability of correct determination of the two-stage tests is discussed by simulation studies.     

Approaches to the Hardy-Weinberg manifold

Let fertilities and death rates be additive, let fertilities be positive, and let mating be random in the Nagylaki-Crow continuous model of selection at a multiallelic locus in a monoecious population. Then polymorphisms are in Hardy-Weinberg proportions. If some fertilities vanish, there is an example of a diallelic polymorphism that is not in Hardy-Weinberg proportions. If the fertilities are larger, in one sense or another, than the difference between any two death rates, then convergence to the Hardy-Weinberg manifold is shown. If, in addition, the Malthusian parameters are constant, and only a finite number of equilibria exist, then global convergence to equilibria is proved.     

Heterozygote deficiencies in small lacustrine populations of brook charr Salvelinus Fontinalis Mitchill (Pisces,Salmonidae): a test of alternative hypotheses

Empirical studies of natural populations have commonly reported departures from Hardy-Weinberg expected proportions of heterozygote individuals. Recent advances in statistical population genetics now offer the potential to exploit individual multilocus genotypic information to test more rigorously for possible sources of heterozygote deficiencies. In a previous study in lacustrine brook charr (Salvelinus fontinalis), we reported stronger deficits in small than in large lakes. In the present paper, we propose a methodology for empirically testing alternative hypotheses to identify the cause of the deficits observed in three of the smallest lakes (85, 109 and 182 ha) analysed. First, as in several salmonid species, brook charr may exhibit a trophic polymorphism in north temperate lakes. If morphs are genetically divergent, indiscriminate sampling of both forms would result in less heterozygote individuals than expected in a randomly mating population (Wahlund effect). Using an individual-based method aiming at detecting cryptic population structure, we can reject this explanation as the sole source of deficits for all three lakes. Secondly, mating among relatives could also be frequent in small lakes and lead to heterozygote deficiencies. Significantly more fish than expected at random had low individual multilocus heterozygosity in two of the lakes, suggesting that inbred fish may have been present. Thirdly, sampling of genetically related fish would also lead to departures from Hardy-Weinberg proportions. In the same two lakes, the distribution of pairwise relatedness coefficients departed from its random expectation, suggesting that non-random sampling of kin may have occurred.     

Hardy-Weinberg equilibria in random mating populations

The structure of multiloci random mating populations is examined. Sufficient conditions for the existence of stable local Hardy-Weinberg equilibria for n loci and an arbitrary number of alleles per locus, are then derived for specified situations under the assumption of multiplicative gene action between loci. It is shown that a stable Hardy-Weinberg equilibrium can not be a local maximum of the mean fitness function with multiplicative gene action between loci. The stability of Hardy-Weinberg type border points and the condition for the increase of newly introduced genes are topics on which some n-loci results are also obtained for an arbitrary number of alleles per locus in systems that allow Hardy-Weinberg equilibria.     

Comparisons of positive assortative mating and sexual selection models

A series of theoretical models of positive assortative mating and sexual selection are contrasted. It is established that for a dominant trait partial positive assortative mating generally implies some fixation, whereas sexual selection exhibits a unique globally stable polymorphism exhibiting Hardy-Weinberg proportions. The effects of monogamy against polygamy do not qualitatively alter the equilibrium outcomes, although the rate of evolutionary change is generally slowed with monogamy vis-à-vis polygamy. For sexual selection the influence of timing of random mating as against preferential mating causes no change in the equilibrium states, although the rates of convergence can be slowed if sexual selection occurs late in the breeding season. Under assortative mating the timing can alter the equilibrium outcomes. The amount of heterozygosity is always deficient in cases of assortative mating, but always exhibits Hardy-Weinberg ratios under a sexual selection mechanism. This suggests that observations consistent with Hardy-Weinberg equilibrium states cannot preclude ipso facto certain forms of selection forces, including mating patterns and some natural selection structures.     

Covariance between relatives for X-chromosomal loci in a population in disequilibrium

Summary According to Hardy-Weinberg, for a single autosomal locus, a population achieves equilibrium in one generation of random mating if allelic frequency is the same in the sexes, or in two generations if the frequency is not. For a single X-chromosomal locus, however, the approach to equilibrium oscillates and is gradual. Covariances between relatives for autosomal and for X-chromosomal loci are in the literature for a random mating population in equilibrium. Although assumption of equilibrium is defensible for an autosomal locus, it is less defensible for an X-chromosomal locus. Covariances between collateral and between lineal relatives are derived for X-chromosomal loci in a random mating population not in equilibrium. Collateral relatives such as sibs are of the same generation, and lineal relatives such as parent-offspring are of different generations. Coefficient of co-ancestry between relatives, based on identity by descent, was used in this development. Results are applicable to crossbreeding in livestock and poultry, and also to haplo-diploid organisms, such as the honeybee, in which the entire genome is equivalent to being X-chromosomal.Supported in part by the Illinois Agricultural Experiment Station, Hatch Project 35-0367     

Cytonuclear Theory for Haplodiploid Species and X-Linked Genes. I. Hardy-Weinberg Dynamics and Continent-Island, Hybrid Zone Models

We develop models that describe the cytonuclear structure for either a cytoplasmic and nuclear marker in a haplodiploid species or a cytoplasmic and X-linked marker in a diploid species. Sex-specific disequilibrium statistics that summarize nonrandom cytonuclear associations in such systems are defined, and their basic Hardy-Weinberg dynamics and admixture formulae are delimited. We focus on the context of hybrid zones and develop continent-island models whereby individuals from two genetically differentiated source populations migrate into and mate within a single zone of admixture. We examine the effects of differential migration of the sexes, assortative mating by pure type females, and census time (relative to mating and migration), as well as special cases of random mating and migration subsumed under the general models. We show that pure type individuals and nonzero cytonuclear disequilibria can be maintained within a hybrid zone if there is continued migration from both source populations, and that females generally have a greater influence over these cytonuclear variables than males. The resulting theoretical framework can be used to estimate the rates of assortative mating and sex-specific gene flow in hybrid zones and other zones of admixture involving haplodiploid or sex-linked cytonuclear data.     

Predicting rates of inbreeding in populations undergoing selection

Tractable forms of predicting rates of inbreeding (DeltaF) in selected populations with general indices, nonrandom mating, and overlapping generations were developed, with the principal results assuming a period of equilibrium in the selection process. An existing theorem concerning the relationship between squared long-term genetic contributions and rates of inbreeding was extended to nonrandom mating and to overlapping generations. DeltaF was shown to be approximately (1)/(4)(1 - omega) times the expected sum of squared lifetime contributions, where omega is the deviation from Hardy-Weinberg proportions. This relationship cannot be used for prediction since it is based upon observed quantities. Therefore, the relationship was further developed to express DeltaF in terms of expected long-term contributions that are conditional on a set of selective advantages that relate the selection processes in two consecutive generations and are predictable quantities. With random mating, if selected family sizes are assumed to be independent Poisson variables then the expected long-term contribution could be substituted for the observed, providing (1)/(4) (since omega = 0) was increased to (1)/(2). Established theory was used to provide a correction term to account for deviations from the Poisson assumptions. The equations were successfully applied, using simple linear models, to the problem of predicting DeltaF with sib indices in discrete generations since previously published solutions had proved complex.     

Microsatellite markers for the Common ringtail possum (Pseudocheirus peregrinus) and their amplification in other Pseudocheirids

Eleven microsatellite markers were developed for the Common ringtail possum, an arboreal marsupial abundant in fragmented forests of south-eastern Australia. Loci were highly polymorphic (4-32 alleles per locus) and heterozygosity ranged from 0.66 to 1. Two loci deviated significantly from Hardy-Weinberg equilibrium proportions, possibly because of low-frequency null alleles. These markers will be informative for examining patterns of gene flow, relatedness and mating systems within fragmented populations of the Common ringtail possum and have potential for use in other Pseudocheirids.     

一个具选择、突变、迁移的群体的遗传差分模型

假设一个群体是由“单位点—双基因”的个体所组成的,在该群体内存在选择、突变、迁移、生死等效应的作用。本文给出了在上述假设下并满足:(1)世代重叠,选择、突变、迁移、生死等效应的作用均在世代遗传之间完成;(2)群体适当大,个体间交配随机,符合孟德尔式遗传;(3)没有任何意外的灾祸等约定的群体遗传的数学模型。通过模型分析,我们能够进一步用数学语言来解释一些生命现象。模型分析指出:虽然某些群体不满足Hardy-Weinberg定律所叙述的条件,但可能具有和Hardy-Weinberg定律的结论相似的结果。该文中还就几个主要参数的变化讨论了群体遗传和进化的某些性质,如平衡等。最后,我们给出了该模型的一个数值例子。     

Pseudo-random mating systems for 3-allele loci

A random mating population attains equilibrium by the Hardy-Weinberg law. By demonstrating some simple examples for 2-allele loci, Li (1988) showed that a nonrandom mating population of certain mating patterns may also attain equilibrium. He called such a type of population a pseudo-random mating population. Tai (1990), then, gave a generalized representation of these pseudo-random mating systems. In this paper the clear patterns of pseudo-random mating behavior for a 3-allele locus are derived. Both autosomal and sex-linked systems are discussed. The study of these mating patterns provides a way to understand the complicated mating system of a population, which usually is only with difficulty realized through sampled individuals from that population.     

Continuous selective models with mutation and migration

The continuous selective model formulated previously for a single locus with multiple alleles in a monoecious population is extended to include mutation and migration. Somatic and germ line genotypic frequencies are distinguished, and the alternative hypotheses of constant mutation rates and age-independent mutation frequencies are analyzed in detail for arbitrary selection and mating schemes. With any mating pattern, if there is no selection, the equilibrium allelic frequencies are shown to be unaffected by the generalizations introduced in this paper. If, in addition, mating is at random, the equilibrium genotypic frequencies are proved to be in Hardy-Weinberg proportions. For both models, the nature of the approach to equilibrium is discussed. Migration is treated in the island model.     

Inbreeding,heterozygote advantage and the evolution of neo-X and neo-Y sex chromosomes

Associations between heterozygosities at different loci are generated by inbreeding. This can cause a fusion or translocation involving a sex chromosome and an autosome to have a selective advantage, when there is selection in favour of heterozygotes. Population genetic models of Y-autosome and X-autosome rearrangments in populations mating by a mixture of full sib-matings and random mating are described, in which the rearrangements cause an autosomal locus with heterozygote advantage to become linked to the true sex chromosomes. Such rearrangements gain a selective advantage under a wide range of conditions. If they can invade, Y-autosome rearrangements always spread to fixation, whereas X-autosome rearrangements may be maintained as stable polymorphisms. The results are discussed in relation to data on breeding systems and karyotypic evolution in termites.     

Continuous selective models

Neglecting age-structure, but taking into account matings with differential fertility in Mendelian reproduction, continuous selective models are formulated for a single locus with an arbitrary number of alleles, with or without distinguishing the sexes, and for two alleles at each of two loci in a monoecious population. In each case, without restricting the mating system, differential equations are derived for the genotypic frequencies, and the validity of the customary Malthusian-parameter differential equations for the gametic frequencies is established. Particular attention is devoted to the conditions for Hardy-Weinberg proportions under random mating. For multiple alleles at a single locus in a monoecious population, exact solutions are obtained for the following three Hardy-Weinberg models: gametic selection, no dominance, and the same selective effect for all alleles but one. The last scheme includes, as special cases, a completely dominant or recessive distinguished allele, and arbitrary selection with only two alleles. Two single-locus assortative mating patterns are analyzed for a monoecious organism using the general formalism. One of these has an arbitrary number of alleles, all the genotypes being distinguishable, while the other involves two alleles, one of which is completely dominant to the other.     

A multi-locus continuous-time selection model

Summary A continuous time selection model is formulated for a diploid monoecious population with multiple alleles at each of an arbitrary number of loci, incorporating differential fertility and mortality as well as arbitrary mating and age structure. The model is simplified in the case of age-independence and for the case of a stable age distribution. The age-independent model is examined in detail for the special case of multiple alleles at each of two loci. This model is analyzed under the assumptions of random mating and additive fertilities, with close attention given to the behavior of the system with respect to Hardy-Weinberg proportions and linkage equilibrium.M. M. was supported by a U.S. Public Health Service training grant (Grant No. GM780).     

Genetic polymorphism of the sixth component of complement (C6) in the rhesus monkey.

With isoelectric focusing, the complement protein C6 has been shown to be genetically polymorphic in the rhesus monkey. Three codominant alleles of a single autosomal locus, Rh C6, have been recognized: C6A, C6B, and C6R, with gene frequencies of 0.592, 0.354, and 0.053 in a random rhesus monkey population. Hardy-Weinberg analysis of the phenotypic frequencies in this population yielded observed values very close to those expected. Both natural mating between individuals carrying the various alleles and artificial combinations of sera of the different C6 types demonstrate patterns consistent with this model. Analysis of several families of monkeys confirmed the Mendelian autosomal codominant inheritance with numbers of offspring very close to expected values and no offspring types inconsistent with the mating pair types.     

The significance of over- and underdominance for the maintenance of genetic polymorphisms. II. Overdominance and instability with random mating

Part I of the present series demonstrates that globally stable polymorphic equilibria may show underdominance in Darwinian fitness. Hence, overdominance in fitness can no longer be conceived of as a necessary condition for the stability of a polymorphism. In the present paper, the question is posed as to whether overdominance is at least sufficient for this stability. A population of randomly mating individuals is considered, where selection operates uniquely through differential fecundities of particular mating types and may generate either a heterozygote excess or deficit relative to Hardy-Weinberg proportions. It turns out that both unstable central overdominance and stable central underdominance are possible and that their occurrence is strongly related to an excess or a deficiency of heterozygotes in the vicinity of the regions of instability or stability. As one consequence, the above suggested sufficiency of heterozygote superiority is not valid, even in random mating populations. Based on the results of both papers of this series, which demonstrate the inadequacy of over- and underdominance as indicators of stability or instability, a modified overdominance principle is discussed. This principle states that a biallelic polymorphism is maintained if the heterozygote is superior in its degree of "heterogamous self-replication" to the degrees of "autogamous self-replication" of the corresponding homozygotes. It is derived with the help of fractional fitnesses, and it is pointed out that certain ratios of these may be more useful for finding evolutionary constants which govern the maintenance of genetic polymorphisms than are ratios of total fitnesses.