Representation of nonepistatic selection models and analysis of multilocus Hardy-Weinberg equilibrium configurations

Summary The paper develops conditions for the existence and the stability of central equilibria emanating from selection recombination interaction with generalized nonepistatic selection forms operating in multilocus multiallele systems. The selection structure admits a natural representation as simple sums of Kronecker products based on a common set of marginal selection components. A flexible parametrization of the recombination process is introduced leading to a canonical derivation of the transformation equations connecting gamete frequency states over successive generations. Conditions for the existence and stability of multilocus Hardy-Weinberg (H.W.) type equilibria are elaborated for the classical nonepistatic models (multiplicative and additive viability effects across loci) as well as for generalized nonepistatic selection expressions. It is established that the range of recombination distributions maintaining a stable H.W. polymorphic equilibrium is confined to loose linkage in the pure multiplicative case, but is not restricted in the additive model. In the bisexual case we ascertain for the generalized nonepistatic model the stability conditions of a common H.W. polymorphism.This paper was supported in part by NIH Grant GM 10452-14 and NSF Grant MCS 75-23608.     

A Numerical Simulation of the One-Locus, Multiple-Allele Fertility Model

Numerical simulations were performed to determine the equilibrium behavior of the one-locus fertility model in which fitness is considered as a property of a pair of mating diploids. A series of patterns of "fertility matrices" were considered for a single locus with two to six alleles. From these simulations, 19 different statistics were collected that characterize, at equilibrium, the heterozygosity, the mean fitness and the fate of populations begun at the allele-frequency centroid. For more than one-half of the trajectories produced by random fertility matrices, there was a decrease in the mean fitness at some time on the way to equilibrium. The mean number of alleles maintained at equilibrium increased only slightly with matrix dimension. Despite the potential for fertility models to display multiple stable equilibria, random fertility models maintain fewer distinct stable points than do random one-locus viability models. Pleiotropic models were also considered with fertility and viability selection operating sequentially within each generation. Most of the equilibrium statistics (with the exception of mean fertility) for the pleiotropic model were intermediate between the corresponding random viability and fertility models.     

The generalized multiplicative model for viability selection at multiple loci

Selection due to differential viability is studied in an n-locus two-allele model using a set indexation that allows the simplicity of the one-locus two-allele model to be carried to multi-locus models. The existence condition is analyzed for polymorphic equilibria with linkage equilibrium: Robbins' equilibria. The local stability condition is given for the Robbins' equilibria on the boundaries in the generalized non-epistatic selection regimes of Karlin and Liberman (1979). These generalized non-epistatic regimes include the additive selection model, the multiplicative selection model and the multiplicative interaction model, and their symmetric versions cover all the symmetric viability models.Research supported by grant no. 11-7805 from the Danish Natural Science Research Council, by NIH grant GM 28016, by a fellowship from the Research Foundation of Aarhus University, and by a visiting fellowship from the University of New England, N.S.W.     

Maintenance of multilocus variability under strong stabilizing selection

We present exact conditions for stability of monomorphic equilibria in a general multilocus multiallele system and of specific polymorphic equilibria in general one- and two-locus multiallele systems. We show how these exact results on one- and two-locus systems can be used in approximate analysis of polymorphic equilibria in multilocus systems under selection strong relative to recombination. We determine conditions for existence and stability of polymorphic equilibria in specific models of quadratic stabilizing selection on additive polygenic traits.     

Evolutionary dynamics in frequency-dependent two-phenotype models

General frequency-dependent selection models based on two phenotypic classes are analyzed with underlying one-locus multiallele phenotypic determination systems in diploid populations. It is proved that the mean phenotypic fitnesses tend to equality over discrete generations and genetic mutations if a phenotypic polymorphism is to be maintained. The exact conditions are examined. The present results are valid for a wide class of models whenever random groupings or assortative patterns based on phenotype and affecting fitness, linearly or not, are independent of sex, mating preferences, or kinship. They can also be applied to two-sex haploid models.     

A theoretical and numerical assessment of genetic variability

The equilibrium behavior of one-locus viability selection models is studied numerically. The selection schemes include randomly chosen viabilities, viabilities chosen to measure a hypothetical distance between the alleles making up the genotype and viabilities that obey various allelic dominance relations. From 3 to 8 alleles are considered. Among the key conclusions are (1) equilibria that are most polymorphic do not usually have the highest mean fitness, (2) the more structure there is in the choice of the viability model, the greater is the level of polymorphism at equilibrium, and (3) for the numbers of alleles chosen here, the equilibrium reached by iteration from the centroid of the allele frequency simplex is the best predictor of the equilibrium attainable from randomly chosen starting vectors. Preliminary evidence shows that this is not the case for 16 alleles.     

Evolutionary principles for general frequency-dependent two-phenotype models in sexual populations

The evolutionary dynamics in general two-sex two-phenotype frequency-dependent selection models are studied with respect to underlying multi-allele one-locus genetic systems. Two classes of equilibria come into play: genotypic equilibria, with equilibrium allelic frequencies independent of the phenotype, and phenotypic equilibria, which are characterized by equal mean phenotypic fitnesses. The exact conditions for genotypic equilibria to exist and be stable and for phenotypic equilibria to exist and be evolutionarily attractive are examined. Using adequate definitions of mean fitnesses in general contexts of frequency-dependent selection in dioecious populations, we show that two phenotypes, when they can coexist in the population, tend to balance their fitnesses as far as is allowed by the genetic system as more alleles responsible for phenotype determination are introduced into the population.     

Sex-specific meiotic drive and selection at an imprinted locus

We present a one-locus model that breaks two symmetries of Mendelian genetics. Whereas symmetry of transmission is breached by allowing sex-specific segregation distortion, symmetry of expression is breached by allowing genomic imprinting. Simple conditions for the existence of at least one polymorphic stable equilibrium are provided. In general, population mean fitness is not maximized at polymorphic equilibria. However, mean fitness at a polymorphic equilibrium with segregation distortion may be higher than mean fitness at the corresponding equilibrium with Mendelian segregation if one (or both) of the heterozygote classes has higher fitness than both homozygote classes. In this case, mean fitness is maximized by complete, but opposite, drive in the two sexes. We undertook an extensive numerical analysis of the parameter space, finding, for the first time in this class of models, parameter sets yielding two stable polymorphic equilibria. Multiple equilibria exist both with and without genomic imprinting, although they occurred in a greater proportion of parameter sets with genomic imprinting.     

Deviations of Genotypic Structures from Hardy-Weinberg Proportions under Random Mating and Differential Selection between the Sexes

Population genetic models, such as differential viability selection between the sexes and differential multiplicative fecundity contributions of the sexes, are considered for a single multiallelic locus. These selection models usually produce deviations of the zygotic genotype frequencies from Hardy-Weinberg proportions. The deviations are investigated (with special emphasis put on equilibrium states) to quantify the effect of selective asymmetry in the two sexes. For many selection regimes, the present results demonstrate a strong affinity of zygotic genotype frequencies for Hardy-Weinberg proportions after two generations, at the latest. It is shown that the deviations of genotypic equilibria from the corresponding Hardy-Weinberg proportions can be expressed and estimated by means of selection components of only that sex with the lower selection intensity. This corresponds to the well-known fact that viability selection acting in only one sex yields Hardy-Weinberg equilibria.     

Population models of genomic imprinting. I. Differential viability in the sexes and the analogy with genetic dominance

Many single-locus, two-allele selection models of genomic imprinting have been shown to reduce formally to one-locus Mendelian models with a modified parameter for genetic dominance. One exception is the model where selection at the imprinted locus affects the sexes differently. We present two models of maternal inactivation with differential viability in the sexes, one with complete inactivation, and the other with a partial penetrance for inactivation. We show that, provided dominance relations at the imprintable locus are the same in both sexes, a globally stable polymorphism exists for a range of viabilities that is independent of the penetrance of imprinting. The conditions for a polymorphism are the same as in previous models with differential viability in the sexes but without imprinting and in a model of the paternal X-inactivation system in marsupials. The model with incomplete inactivation is used to illustrate the analogy between imprinting and dominance by comparing equilibrium bifurcation plots for fixed values of dominance and penetrance. We also derive a single expression for the dominance parameter that leaves the frequency and stability of equilibria unchanged for all levels of inactivation. Although an imprinting model with sex differences does not formally reduce to a nonimprinting scheme, close theoretical parallels clearly exist.     

Evolutionarily stable strategies and viability selection in mendelian populations

For various genetical structures, including haploid and diploid, one-locus n-alleles, and n-locus additive viability random mating models, natural selection resulting from intrapopulation conflicts between random individuals leads to exactly those genetical equilibria which determine a mixture of strategies evolutionarily stable according to the game theory definition of Maynard Smith and Price (1973).     

On the maintenance of genetic variation: global analysis of Kimura's continuum-of-alleles model

Methods of functional analysis are applied to provide an exact mathematical analysis of Kimura's continuum-of-alleles model. By an approximate analysis, Kimura obtained the result that the equilibrium distribution of allelic effects determining a quantitative character is Gaussian if fitness decreases quadratically from the optimum and if production of new mutants follows a Gaussian density. Lande extended this model considerably and proposed that high levels of genetic variation can be maintained by mutation even when there is strong stabilizing selection. This hypothesis has been questioned recently by Turelli, who published analyses and computer simulations of some multiallele models, approximating the continuum-of-alleles model, and reviewed relevant data. He found that the Kimura and Lande predictions overestimate the amount of equilibrium variance considerably if selection is not extremely weak or mutation rate not extremely high. The present analysis provides the first proof that in Kimura's model an equilibrium in fact exists and, moreover, that it is globally stable. Finally, using methods from quantum mechanics, estimates of the exact equilibrium variance are derived which are in best accordance with Turelli's results. This shows that continuum-of-alleles models may be excellent approximations to multiallele models, if analysed appropriately.     

Complex dynamical behaviour of frequency-dependent viability selection: an example

The model of viability selection based on a one-locus, two-allele diploid population is considered. Frequency dependence is introduced through the fitnesses of three phenotypes (strategies) exhibited by the population. Both discrete and continuous dynamics are analyzed and contrasted with the classical results of frequency-independent selection and with the more recent results of frequency-dependent selection based on two-phenotype multi-allele systems. Cycling and chaotic behaviour are shown to be easily obtained in the discrete model. Intuitive biological conditions for stability are shown to fail as well as at general equilibria of the continuous model.     

The multiple-locus symmetric fertility model

Selection due to variation in the fecundity among matings of genotypes with respect to many loci each with two alleles is studied. The fitness of a mating depends only on the genotypic distinction between homozygote and heterozygote at each locus in the two individuals, and differences among loci are allowed. This symmetric fertility model is therefore a generalization of the multiple-locus symmetric viability model. The phenomena seen in the two-locus symmetric fertility model generalize—e.g., the possibility of joint stability of equilibria with linkage equilibrium and with linkage disequilibrium, and the existence of different types of totally polymorphic equilibria with the gametic proportions in linkage equilibrium. The central equilibrium with genotypic frequencies in Hardy-Weinberg proportions and gametic frequencies in Robbins proportions exists for all symmetric fertility models. For some symmetric fertility regimes additional equilibria exist with gametic frequencies in linkage equilibrium and with genotypic frequencies in Hardy-Weinberg proportions at all except one locus. These equilibria may exist in the dioecious symmetric viability model, and then they will be locally stable. For free recombination the stable equilibria show linkage equilibrium, but several of these with different numbers of polymorphic loci may be stable simultaneously.     

Analysis of partial assortative mating and sexual selection models for a polygamous species involving a trait based on levels of heterozygosity

The consequences of preferential mating in the presence of partial assortative and sexual selection mechanisms are ascertained for a two-allele one-locus trait involving two phenotype classes C₁ = {all homozygotes} and C₂ = {heterozygotes}. Relevant biological cases may include Burley (1977, Proc. Nat. Acad. Sci. USA74, 3476–3479), Wilbur et al. (1978, Evolution32, 264–270), and Singh and Zouros (1978, Evolution32, 342–353). When the preference rate for the heterozygote exceeds that for homozygotes, it is established that the unique stable state is the central Hardy-Weinberg equilibrium. The rate of approach is faster with sexual selection than for the corresponding model of assortative mating. When the preference rates favor the homozygotes then in this symmetric model of sexual selection two asymmetric Hardy-Weinberg polymorphisms can evolve, and which succeeds depends on initial conditions. The models are also analyzed with natural selection acting on phenotypes superimposed on assortative mating. In this case we can have up to three coexisting stable states involving both fixation alternatives and a central polymorphism. The corresponding model with sexual selection maintains either the central equilibrium as in assortative mating or two asymmetric polymorphic equilibria.     

General analysis of frequency-dependent sexual selection at a multi-allelic locus

A general model is analyzed in which arbitrarily frequency-dependent selection acts on one sex of a diploid population with several alleles at one locus, as a result of viability or mating-success differences. The existence of boundary and polymorphic equilibria is examined, and conditions for local stability, internal and external, are obtained. The status of Hardy-Weinberg approximations in studying stability and approach to equilibria is also considered. The general principles are then applied to two specific models: one where genotypes fall into two phenotypic classes; and one with a hierarchy of dominance where viability and sexual selection are opposed. In the latter case it is found that, of all the equilibria present, there is one and only one which could possibly be stable: the existence of a unique globally stable equilibrium might then be inferred.     

Sexual selection and fertility

Genetic models are analyzed in which sexual selection is combined with fertility selection. In these models, the sexual selection acts on males, the fertility selection on either males, females or both sexes. The phenotypes thus selected may be determined either by dominant and recessive alleles or by each homozygous and heterozygous genotype. Polymorphisms of dominant and recessive phenotypes can be maintained in equilibrium by a balance between sexual and fertility selection. Generally fertility selection has a greater effect than viability selection in determining the point of equilibrium. The dominant phenotype is maintained at a lower frequency when at a fertility disadvantage than when at a viability disadvantage. When about 20% or more of the females mate preferentially, the models show that equilibria will be established at very different frequencies depending on whether fertility selection acts on males, females or both sexes. These results, applied to data of preferential mating of melanic two-spot ladybirds, predict differences in fertility which can be use to test the models. Symmetric models of preferences for each genotype also give rise to polymorphisms if the heterozygotes obtain an overall advantage.     

Some multiallele partial assortative mating systems for a polygamous species

The consequences of preferential mating in the presence of partial assortative and sexual selection mechanisms are ascertained for a two-allele one-locus trait involving two phenotype classes C₁ = {all homozygotes} and C₂ = {heterozygotes}. Relevant biological cases may include Burley (1977, Proc. Nat. Acad. Sci. USA 74, 3476–3479), Wilbur et al. (1978, Evolution 32, 264–270), and Singh and Zouros (1978, Evolution 32, 342–353). When the preference rate for the heterozygote exceeds that for homozygotes, it is established that the unique stable state is the central Hardy-Weinberg equilibrium. The rate of approach is faster with sexual selection than for the corresponding model of assortative mating. When the preference rates favor the homozygotes then in this symmetric model of sexual selection two asymmetric Hardy-Weinberg polymorphisms can evolve, and which succeeds depends on initial conditions. The models are also analyzed with natural selection acting on phenotypes superimposed on assortative mating. In this case we can have up to three coexisting stable states involving both fixation alternatives and a central polymorphism. The corresponding model with sexual selection maintains either the central equilibrium as in assortative mating or two asymmetric polymorphic equilibria.     

One- and multi-locus multi-allele selection models in a random environment

Summary We deduce conditions for stochastic local stability of general perturbed linear stochastic difference equations widely applicable in population genetics. The findings are adapted to evaluate the stability properties of equilibria in classical one- and multi-locus multi-allele selection models influenced by random temporal variation in selection intensities. As an example of some conclusions and biological interpretations we analyse a special one-locus multi-allele model in more detail.This work was supported in part by Stiftung Volkswagenwerk.