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.     

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.     

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.     

Convergence of genotypic frequencies for differential selfing and positive assortative mating at a biallelic locus

For a biallelic model of differential self-fertilization and differential positive assortative mating based on genotype, it is shown that the genotypic frequencies converge for all sets of mating system parameters. Overdominance and underdominance with respect to the parameters are necessary but not sufficient conditions for global convergence to a polymorphic equilibrium and local attractiveness of both the fixation states, respectively. There are cases of overdominance and underdominance for which one fixation state is globally attractive. The relationship of the result to those known from the classical viability selection model are briefly discussed. For the multiallelic version, it is shown that after the first generation all of the homozygote frequencies are always in excess of the corresponding Hardy-Weinberg proportions if at least one homozygote rate of self-fertilization or assortment probability is positive.     

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.     

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.     

Regions of Stable Equilibria for Models of Differential Selection in the Two Sexes under Random Mating

The equilibrium structure of models of differential selection in the sexes is investigated. It is shown that opposing additive selection leads to stable polymorphic equilibria for only a restricted set of selection intensities, and that for weak selection the selection intensities must be of approximately the same magnitude in the sexes. General models of opposing directional selection, with arbitrary dominance, are investigated by considering simultaneously the stability properties of the trivial equilibria and the curve along which multiple roots appear. Numerical calculations lead us to infer that the average degree of dominance determines the equilibrium characteristics of models of opposing selection. It appears that if the favored alleles are, on the average, recessive, there may be multiple polymorphic equilibria, whereas only a single polymorphic equilibrium can occur when the favored alleles are, on the average, dominant. The principle that the average degree of dominance controls equilibrium behavior is then extended to models allowing directional selection in one sex with overdominance in the other sex, by showing that polymorphism is maintained if and only if the average fitness in heterozygotes exceeds one.     

EVOLUTION OF THE NUMBER OF SEXES

In sexually reproducing isogamous organisms, gametes (or diploid cells in ciliates) are classified into two or more groups called sexes, and mating occurs only between cells of different sexes. We have studied the evolutionary stability of the number of sexes maintained in a population by examining population-genetic models. For models in which the diploid genome determines the sex of conjugal cells, a one-locus system with three alleles of pecking-order dominance is assumed. Unlike traditional bisexual models, the genetic dynamics then depend on a rule, called mating kinetics, which determines the proportion of matings between each pair of sexes for given proportions of cells of the three sexes. The evolutionary consequences greatly depend on the mating kinetics assumed. Of the four mating kinetics examined, two give a large advantage to rare sexes whose cells quickly find heterosexual partners, which implies an evolutionary increase in the number of sexes. In contrast, the other two mating kinetics, in which gametes wait for suitable mates without being eliminated from the gamete pool during this waiting period, produce neutrally stable dynamics with curves or a surface of equilibria. Then random drift or differential fitness among sexes would result in the loss of sex alleles until only two remain in the population. This suggests a turnover of sexes; a new sex invades and replaces resident sexes after temporary coexistence. Similar results are obtained in models with haploid sex-determination and with autogamy. These two processes, however, may help to maintain many sexes indirectly by preventing the accumulation of recessive lethal mutations on sex chromosomes. The relationship of these models to models of self-sterility factors in plants and sex factors in honeybees is discussed. To summarize, the number of sexes should increase when conjugal cells must find mates during a limited period of time, otherwise a two-sex system should evolve. We conclude that there may be more isogamous species with three or more sexes than are currently known.     

MATING PATTERN AND FITNESS-COMPONENT ANALYSIS ASSOCIATED WITH INVERSION POLYMORPHISM IN A NATURAL POPULATION OF DROSOPHILA BUZZATII

Direct studies of mating success or mating pattern associated with Mendelian factors rarely have been carried out in nature. From the samples taken for the standard analyses of selection components, it is not usually possible to obtain the mating table, and only directional selection for male mating success can be detected. Both processes, mating pattern and differential mating probability, together with other fitness components, have been investigated for the inversion polymorphism of a natural population of the cactophilic species Drosophila buzzatii. Two independent samples of adult flies were collected: nonmating or single individuals (base population) and mating pairs (mating population). All individuals were karyotyped for the second and fourth chromosomes. A sequence of models with increasing simplicity was fitted to the data to test null hypotheses of no selection and random union of gametes and karyotypes. The main results were (1) no deviations from random mating were found; (2) differential mating probability was nonsignificant in both sexes; (3) inversion and karyotypic frequencies did not differ between sexes; and (4) karyotypic frequencies did not depart from Hardy-Weinberg expectations. These results are discussed in light of complementary evidence showing the need for interpreting with caution no-effect hypotheses such as the ones tested here. The use of complementary selective tests in these studies is suggested.     

Selection for Increased Mutation Rates with Fertility Differences between Matings

Previous studies of mutation modification have considered models in which selection is a result of viability differences that are sex symmetric. The results of a numerical study of a model in which selection is a result of fertility differences between mated pairs demonstrate that the type of selection to which a population is subject can have a significant impact on the evolution of various parameters of the genetic system. When the fertility of matings between individuals with different genotypes exceeds the fertility of at least some of the matings between individuals with the same genotype, selection may favor increased rates of mutation, in contrast to the results from all existing constant viability models with random mating and infinite population size. Increased mutation rates are most frequently favored when forward and back mutation occur at approximately equal rates and when the modifying locus is loosely linked to the selected locus. We present one example in which selection favors increased rates of mutation even though the selection scheme is reducible to one of differential viability between the sexes.     

Population trajectories for single locus additive fecundity selection and related selection models

A selection model which comprises models of additive fecundities as well as models of viability, fecundity, or differential mating selection acting only in one sex, is investigated for an autosomal gene locus in a population reproducing in nonoverlapping generations. The recurrence equations and basic properties of the genotypic population trajectories and equilibrium points are formulated for the multiallelic case. For the diallelic case, the trajectory development is discussed in more detail, and it is proven that every population trajectory converges to a Hardy-Weinberg equilibrium point.     

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.     

Epistasis of quantitative trait loci under different gene action models

Modeling and detecting nonallelic (epistatic) effects at multiple quantitative trait loci (QTL) often assume that the study population is in zygotic equilibrium (i.e., genotypic frequencies at different loci are products of corresponding single-locus genotypic frequencies). However, zygotic associations can arise from physical linkages between different loci or from many evolutionary and demographic processes even for unlinked loci. We describe a new model that partitions the two-locus genotypic values in a zygotic disequilibrium population into equilibrium and residual portions. The residual portion is of course due to the presence of zygotic associations. The equilibrium portion has eight components including epistatic effects that can be defined under three commonly used equilibrium models, Cockerham's model, F2-metric, and F(infinity)-metric models. We evaluate our model along with these equilibrium models theoretically and empirically. While all the equilibrium models require zygotic equilibrium, Cockerham's model is the most general, allowing for Hardy-Weinberg disequilibrium and arbitrary gene frequencies at individual loci whereas F2-metric and F(infinity)-metric models require gene frequencies of one-half in a Hardy-Weinberg equilibrium population. In an F2 population with two unlinked loci, Cockerham's model is reduced to the F2-metric model and thus both have a desirable property of orthogonality among the genic effects; the genic effects under the F(infinity)-metric model are not orthogonal but they can be easily translated into those under the F2-metric model through a simple relation. Our model is reduced to these equilibrium models in the absence of zygotic associations. The results from our empirical analysis suggest that the residual genetic variance arising from zygotic associations can be substantial and may be an important source of bias in QTL mapping studies.     

The Evolution of the Y Chromosome with X-Y Recombination

A theoretical population genetic model is developed to explore the consequences of X-Y recombination in the evolution of sex chromosome polymorphism. The model incorporates one sex-determining locus and one locus subject to natural selection. Both loci have two alleles, and the rate of classical meiotic recombination between the loci is r. The alleles at the sex-determining locus specify whether the chromosome is X or Y, and the alleles at the selected locus are arbitrarily labeled A and a. Natural selection is modeled as a process of differential viabilities. The system can be expressed in terms of three recurrence equations, one for the frequency of A on the X-bearing gametes produced by females, one for each of the frequency of A on the X- and Y-bearing gametes produced by males. Several special cases are examined, including X chromosome dominance and symmetric selection. Unusual equilibria are found with the two sexes having very different allele frequencies at the selected locus. A significant finding is that the allowance of recombination results in a much greater opportunity for polymorphism of the Y chromosome. Tighter linkage results in a greater likelihood for equilibria with a large difference between the sex chromosomes in allele frequency.     

A differential viability model for twin-pair blood group data.

Much interest in human genetics studies of blood systems is focussed on the possible incompatibility of discordant twin pairs. Here we discuss a differential viability model which accounts for deviations from the standard multinomial model for twin pair data based on the Hardy-Weinberg frequencies. We discuss related estimation and testing problems, illustrating the techniques with data from the Louisville Twin Study.     

EVOLUTION BY FISHERIAN SEXUAL SELECTION IN DIPLOIDS

Most models of Fisherian sexual selection assume haploidy. However, analytical models that focus on dynamics near fixation boundaries and simulations show that the resulting behavior depends on ploidy. Here we model sexual selection in a diploid to characterize behaviour away from fixation boundaries. The model assumes two di-allelic loci, a male-limited trait locus subject to viability selection, and a preference locus that determines a female's tendency to mate with males based on their genotype at the trait locus. Using a quasi-linkage equilibrium (QLE) approach, we find a general equation for the curves of quasi-neutral equilibria, and the conditions under which they are attracting or repelling. Unlike in the haploid model, the system can move away from the internal curve of equilibria in the diploid model. We show that this is the case when the combined forces of natural and sexual selection induce underdominance at the trait locus.     

A note on the island model with sex dependent migration

Summary A theoretical calculation is presented which extends Wright's island model of drift and migration to differential migration between the two sexes. In this circumstance, local demes no longer have Hardy-Weinberg frequencies. There may be local heterozygote excess or deficiency depending, respectively, on whether migration occurs before or after mating. The magnitude of the local departure from Hardy-Weinberg is directly proportional to the difference between the migration parameters of the two sexes. These results could have important implications for studies where genetic markers are used for inferring population structure. An example from a study of Marmot colonies is cited.     

The Evolution of One- and Two-Locus Systems

Assuming age-independent fertilities and mortalities and random mating, continuous-time models for a monoecious population are investigated for weak selection. A single locus with multiple alleles and two alleles at each of two loci are considered. A slow-selection analysis of diallelic and multiallelic two-locus models with discrete nonoverlapping generations is also presented. The selective differences may be functions of genotypic frequencies, but their rate of change due to their explicit dependence on time (if any) must be at most of the second order in s, (i.e., O( s²)), where s is the intensity of natural selection. Then, after several generations have elapsed, in the continuous time models the time-derivative of the deviations from Hardy-Weinberg proportions is of O(s²), and in the two-locus models the rate of change of the linkage disequilibrium is of O(s²). It follows that, if the rate of change of the genotypic fitnesses is smaller than second order in s (i.e., o(s²)), then to O(s²) the rate of change of the mean fitness of the population is equal to the genic variance. For a fixed value of s, however, no matter how small, the genic variance may occasionally be smaller in absolute value than the (possibly negative) lower order terms in the change in fitness, and hence the mean fitness may decrease. This happens if the allelic frequencies are changing extremely slowly, and hence occurs often very close to equilibrium. Some new expressions are derived for the change in mean fitness. It is shown that, with an error of O( s), the genotypic frequencies evolve as if the population were in Hardy-Weinberg proportions and linkage equilibrium. Thus, at least for the deterministic behavior of one and two loci, deviations from random combination appear to have very little evolutionary significance.     

Population genetics of the HRAS1 minisatellite locus.

Several years ago it was reported that rare HRAS1 VNTR alleles occurred more frequently in U.S. Caucasian cancer patients than in unaffected controls. Such an association, in theory, could be caused by undetected population heterogeneity. Also, in a study clearly relevant to this issue, it was recently reported that significant deviations from Hardy-Weinberg equilibrium exist at this locus in a sample of U.S. Caucasians. These considerations motivate our population genetic analysis of the HRAS1 locus. From published studies of the HRAS1 VNTR locus, which classified alleles into types, we found only small differences in the allele frequency distributions of samples from various European nations, although there were larger differences among ethnic groups (African American, Caucasian, and Oriental). In an analysis of variation of rare-allele frequencies among samples from four European nations, most of the variance was attributable to molecular methodology, and very samples from four European nations, most of the variance was attributable to molecular methodology, and very little of the variance was accounted for by nationality. In addition, we showed that mixture of European subpopulations should result in only minor deviations from expected genotype proportions in a Caucasian database and demonstrated that there was no significant deviation from Hardy-Weinberg equilibrium in our HRAS1 data.     

A Symmetric Two-Locus Fertility Model

A model in which selection is mediated by differential fertilities among the genotypes at two diallelic loci is proposed. Fertility depends only on the number of heterozygous loci participating in the mating. Classes analogous to symmetric equilibria in symmetric viability models are determined explicitly and shown to exhibit stability behavior very different from the viability results. Linkage equilibrium is shown to occur in a relatively asymmetric fashion and to overlap in stability with linkage disequilibrium. In many cases single-locus or two-locus polymorphism is shown to be stable simultaneously with chromosome fixation even under very tight linkage. It is suggested that historical effects may be of great significance in the evolution of systems in which fertility is the primary agent of natural selection.