Adaptive Topography in Fertility-Viability Selection Models: The Haplodiploid Case

A linear combination of partial changes of mean fitnesses from one generation to the next one is shown to be approximately equal to the additive genetic variance in fitness after enough generations and away from equilibrium in random mating haplodiploid populations under arbitrary weak frequency-dependent selection on sex-differentiated viability of individuals and sex-differentiated fertility of matings controlled at a single multiallelic locus. The result can be applied to X-linked locus models in diploid populations. The result is used to deduce approximate adaptive topographies far frequency-independent selection models in the cases of nonsex-differentiated fertilities and multiplicative sex-differentiated fertilities and for kin selection models in family-structured populations under the assumptions of single insemination and multiple insemination of females. Multiple insemination creates frequency-dependent selection regimes.     

Mutation modification with multiplicative fertility selection

Two diallelic loci in an infinite panmictic population of diploid individuals are modelled. The A/a locus is subject to unidirectional mutation and either multiplicative fertility selection or, equivalently, sex-asymmetric viability selection. The M/m locus acts as a selectively neutral modifier of the mutation rate at A/a. The loci recombine at rate R. If the M/m locus is initially monomorphic, and the A/a locus has reached equilibrium, the fate of a new modifier allele is found to depend not just on its relative effect on mutation but also upon the linkage, R. Each initial equilibrium may be characterized by a critical value of the recombination rate, R*. If 0 less than R* less than 0.5, a sufficiently small "down" modifier of the mutation rate will invade the population when R less than R* whereas a sufficiently small "up" modifier will succeed when R greater than R*. If R* less than 0 or R* greater than 0.5, only mutation reduction may occur. Numerical analysis of 56,000 sample equilibria indicates that mutation rates may be increased, but only when the selection regime is such that the A/a locus would remain polymorphic in the absence of mutation.     

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.     

Fertility and viability at the Sod locus in Drosophila melanogaster: non-additive and asymmetric selection

Experiments were designed to test in Drosophila melanogaster the effect of mating type at the Sod locus on fertility and viability. The experiments show that fertility is neither additive (or multiplicative) nor symmetric, i.e. that the fertility of a mating type cannot be predicted from the average fertility of the two genotypes involved in the mating. There is no significant male x female interaction with respect or progeny viability; but the interaction is significant for productivity, i.e. when fertility and viability are jointly taken into account. There is overdominance with respect to female fertility, but not with respect to male fertility or to viability. There also is alloprocoptic selection with respect to fertility and with respect to productivity, i.e. mating between like homozygotes are less fertile and productive than matings between dissimilar homozygotes. Selection at the Sod locus yields stable polymorphic equilibria, with the frequency of the F allele predicted at P = 0.641 or 0.695, respectively for low and high larval density.     

Mutation-Selection Balance under Genomic Imprinting at an Autosomal Locus

I model the effect of genomic imprinting on the equilibrium allele frequencies at an autosomal diallelic locus subject to viability selection and mutation. The population size is assumed to be very large; male and female mutation rates may be unequal. Different models examine cases of the inactivation of one gene (with both complete and partial penetrance) and of differential expression of genes according to the parent of origin. In the simplest cases the frequency of the deleterious allele is approximately twice that of a dominant nonimprinting mutant, but considerably less than that of a recessive nonimprinting mutant. Under imprinting, selection and unequal mutation rates interact: other things being equal, male-biased mutation leads to lower mutant frequencies under maternal imprinting and higher frequencies under paternal imprinting. I also model cases where just one allele is imprintable (and the other not). These models allow us to predict the frequency of a failure to imprint in a normally imprinting system, as well as the frequency of imprinting at a standard nonimprinting locus.     

Use of restriction fragment length polymorphisms for genetic counseling: Population genetic considerations

Two-locus population genetic models are analyzed to evaluate the utility of restriction fragment length polymorphisms for purposes of genetic counseling. It is shown that the linkage disequilibrium between a neutral marker and a tightly linked overdominant mutant will increase rapidly as the mutant moves to its polymorphic equilibrium. The linkage disequilibrium decays for deleterious recessive mutants. Two measures involving the linkage disequilibrium are investigated to determine how much information the transmission of the neutral marker provides about the transmission of the selected gene. In certain kinds of matings, where the parental two-locus genotypes and linkage phases are known, it is possible to determine whether or not a progeny is homozygous for the selected gene on the basis of the fetal genotype at the marker locus. A quantity of primary interest is the fraction of matings between individuals heterozygous for the selected gene in which exact diagnosis can be made in this way. The expected proportion of such matings, taken over all two-locus matings involving heterozygotes at the selected locus, is calculated as a function of the gene frequencies at the two loci and the linkage disequilibrium between them. This expected value is maximized when the linkage disequilibrium is at its maximum in absolute value. Fewer than half of all matings are informative if the linkage disequilibrium is small in magnitude or if the gene frequencies at the two loci are quite different. Consideration is also given to various conditional measures of association that may be useful when the parental two-locus genotypes are unknown. The results suggest that the utility of tightly linked neutral marker genes in predicting the transmission of a selected gene is generally less when selection acts against a recessive gene than for overdominant selection.     

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.     

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.     

A symmetric two locus model with viability and fertility selection

A two locus deterministic population genetic model is analysed. One locus is under viability selection, the other under fertility selection with both forms of selection completely symmetric. It is shown that linkage equilibrium may occur at two different equilibrium points. For a two-locus polymorphism to be stable, it is necessary that the viability locus be overdominant but not necessary that the fertility locus, considered separately, be able to support a stable polymorphism. The overlaps in stability are not as complex as under two locus symmetric fertilities, but considerably more complex than with symmetric viabilities. Extensions of the analysis for the central linkage equilibrium point with multiple viability and fertility loci are indicated.Research supported in part by NIH grants GM 28106 and GM 10452     

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 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.     

A MODEL FOR THE EVOLUTION OF ASYMMETRICAL MALE HYBRID STERILITY AND ITS IMPLICATIONS FOR SPECIATION

Chromosomal analysis of several cases of asymmetrical male hybrid sterility in Drosophila has implicated the X- or the Y-chromosome and one or more autosomes. Here, I develop a model for the evolution of this phenomenon. An autosomal locus is assumed to affect viability and to interact with a Y-linked or an X-linked locus to determine male fertility. In a new environment, selection for viability favors a new allele at the autosomal locus, but incompatibility of this new allele with the sex-chromosome-linked gene generates male sterility. The incompatibility can be resolved if a new allele at the sex-linked locus invades the population. This results in nonreciprocal male hybrid sterility, the direction of the nonreciprocity being determined by the dominance or recessiveness of the new autosomal gene in its effect on fertility. It is shown that stable polymorphism for the autosomal locus is possible and that, if the equilibrium frequency of the new allele is above a critical value, the population will be constantly at the verge of speciation, “waiting” for the sex-linked mutation to occur. The appearance of this mutation causes a runaway process leading to rapid fixation of the new autosomal and sex-linked alleles. If the equilibrium frequency of the new autosomal allele is less than the critical value, deterministic speciation is impossible, but random drift may increase the frequency above the critical value and predispose the population to the invasion of the new sex-linked allele. Thus, both deterministic and stochastic modes of speciation are possible. Because deterministic speciation requires large selection coefficients, which impose a severe genetic load on the population, and because stochastic speciation requires repeated population bottlenecks, it is concluded that relative to the number of successful speciation events there will be many more events of deme extinction.     

Kin selection is implicated in partial sib-mating populations with constant viability differences before mating

The change in the frequency of a rare mutant allele under constant sex-differentiated viability selection in an infinite, partial full-sib mating population is studied. The diplo-diploid and haplo-diploid polygynous models are considered with a Poisson distribution for the number of offspring produced by every mated female. Reproduction is followed by weak selection among the offspring and then mating to form the next generation. It is shown that the rate of change with respect to the frequency of the mutant allele and the intensity of selection can be expressed in terms of costs or benefits of substituting the mutant type for the wild type, which correspond to average excesses in viability in females and males, multiplied by coefficients of relatedness to the individuals affected by such a substitution and reproductive values associated to the sexes of these individuals. This reveals hidden interactions between mated individuals and between males for mating, the former having positive effects on the reproductive success of related individuals and the latter having negative effects. Such interactions are the result of reproductive constraints when a fixed proportion of females must mate with a male sib and all females are fertilized as long as one mate is available. However, they affect the change in allele frequency because there is inbreeding or relatedness between mates and more generally relatedness between interacting individuals. Surprisingly, the effects of these interactions cancel out in a diploid population when the number of offspring is large enough so that the possibility for a female to have no male sib to mate with can be neglected and the viability differences are the same in both sexes.     

Conditions for the initial increase of new alleles with frequency dependent selection

The paper presents the derivation of conditions for the initial increase of a new allele in a fairly general model of a one locus genetic system. Incorporated within the model are frequency dependent selection coefficients which allow the study of viability differences, fertility differences, segregation distortion and certain forms of non-random mating. Since in many models it is necessary to go beyond linear terms, quadratic, cubic and quartic terms are also derived explicitly and details given of their extension to higher degrees.     

DYNAMICS OF SEXUAL SELECTION IN DIPLOID POPULATIONS

We describe results for a diploid, two-locus model for the evolution of a female mating preference directed at an attractive male trait that is subject to viability and/or fertility selection. Using computer simulation, we studied a large, random sample of parameter values, assuming additivity of alleles at the preference locus and partial dominance at the trait locus. Simulation results were classifiable into nine types of parameter sets, each differing in equilibria, evolutionary trajectories, and rates of evolution. For many parameters, evolutionary trajectories converged on curves within the allelic frequency plane and subsequently evolved along the curves toward fixation. Neutrally stable curves of equilibria did not occur in Fisherian models that assume only viability and sexual selection unless there is complete dominance at the trait locus. The Fisherian models also exhibited oscillation of allelic frequencies and unique polymorphic equilibria. “Sexy son” models in which attractive males had reduced fertility were much less likely to lead to increase in traits and preferences than were the Fisherian models. However, if less fertile males had increased viability, trait polymorphisms and fixation of rare “sexy” alleles occurred. In general, the behavior of the diploid model was much more complex than that of analogous haploid or polygenic models.     

Why men matter: mating patterns drive evolution of human lifespan

Evolutionary theory predicts that senescence, a decline in survival rates with age, is the consequence of stronger selection on alleles that affect fertility or mortality earlier rather than later in life. Hamilton quantified this argument by showing that a rare mutation reducing survival is opposed by a selective force that declines with age over reproductive life. He used a female-only demographic model, predicting that female menopause at age ca. 50 yrs should be followed by a sharp increase in mortality, a "wall of death." Human lives obviously do not display such a wall. Explanations of the evolution of lifespan beyond the age of female menopause have proven difficult to describe as explicit genetic models. Here we argue that the inclusion of males and mating patterns extends Hamilton's theory and predicts the pattern of human senescence. We analyze a general two-sex model to show that selection favors survival for as long as men reproduce. Male fertility can only result from matings with fertile females, and we present a range of data showing that males much older than 50 yrs have substantial realized fertility through matings with younger females, a pattern that was likely typical among early humans. Thus old-age male fertility provides a selective force against autosomal deleterious mutations at ages far past female menopause with no sharp upper age limit, eliminating the wall of death. Our findings illustrate the evolutionary importance of males and mating preferences, and show that one-sex demographic models are insufficient to describe the forces that shape human senescence.     

Imperfect Genes, Fisherian Mutation and the Evolution of Sex

In this paper we present a mathematical model of mutation and selection that allows for the coexistence of multiple alleles at a locus with very small selective differences between alleles. The model also allows for the determination of fitness by multiple loci. Models of this sort are biologically plausible. However, some previous attempts to construct similar models have assumed that all mutations produce a decrease in fitness, and this has led to a tendency for the average fitness of population members to decline when population numbers are finite. In our model we incorporate some of the ideas of R. A. FISHER, so that both deleterious and beneficial mutations are possible. As a result, average fitness tends to approach a stationary distribution. We have used computer simulation methods to apply the Fisherian mutation model to the problem of the evolution of sex and recombination. The results suggest that sex and recombination can provide very large benefits in terms of average fitness. The results also suggest that obligately sexual species will win ecological competitions with species that produce a substantial fraction of their offspring asexually, so long as the number of sites under selection within the genomes of the competing species is not too small and the population sizes are not too large. Our model focuses on fertility selection in an hermaphroditic plant. However, the results are likely to generalize to a wide variety of other situations as well.     

The role of population size in molecular evolution.

The results of a computer simulation study of the role of population size in population genetical models of molecular evolution are presented. If the mutation rate and strength of selection are held fixed and the population size increased, the eight models examined fall into three domains based on their rates of substitution. In the Ohta domain, the rate of substitution decreases with increasing population size; in the Kimura domain, the rate of substitution remains close to the mutation rate; in the Darwin domain, the rate of substitution increases without bound. In the Kimura and Darwin domains, the rate of substitution is much less sensitive to the population size than suggested by two-allele theories. Remarkably, the overdominance model converges to the neutral model with increasing N. The variation at a neutral locus linked to a selected locus is found to be insensitive to the population size for certain models of selection. A selected locus can actually cause the rate of substitution of deleterious alleles at a linked locus to increase with increasing population size. These unexpected results illustrate that intuition based on two-allele theory is often misleading.     

Diffusion approximations for one-locus multi-allele kin selection,mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics

Diffusion approximations are ascertained from a two-time-scale argument in the case of a group-structured diploid population with scaled viability parameters depending on the individual genotype and the group type at a single multi-allelic locus under recurrent mutation, and applied to the case of random pairwise interactions within groups. The main step consists in proving global and uniform convergence of the distribution of the group types in an infinite population in the absence of selection and mutation, using a coalescent approach. An inclusive fitness formulation with coefficient of relatedness between a focal individual J affecting the reproductive success of an individual I, defined as the expected fraction of genes in I that are identical by descent to one or more genes in J in a neutral infinite population, given that J is allozygous or autozygous, yields the correct selection drift functions. These are analogous to the selection drift functions obtained with pure viability selection in a population with inbreeding. They give the changes of the allele frequencies in an infinite population without mutation that correspond to the replicator equation with fitness matrix expressed as a linear combination of a symmetric matrix for allozygous individuals and a rank-one matrix for autozygous individuals. In the case of no inbreeding, the mean inclusive fitness is a strict Lyapunov function with respect to this deterministic dynamics. Connections are made between dispersal with exact replacement (proportional dispersal), uniform dispersal, and local extinction and recolonization. The timing of dispersal (before or after selection, before or after mating) is shown to have an effect on group competition and the effective population size. In memory of Sam Karlin.