Partial selfing and linkage: the effect of a heterotic locus on a neutral locus

Equilibria are determined for the two-locus model in a partially selfing population when one locus is neutral and the other locus is heterotic. At an equilibrium point, the frequency of heterozygotes at the neutral locus is greater than that expected from one-locus theory, even if the heterotic locus is on a different chromosome. Thus, the neutral locus also appears to be heterotic. The magnitude of this effect is determined for several different proportions of selfing and amounts of recombination.     

Mutator Genes and Selection for the Mutation Rate in Bacteria

Gene frequencies in populations of haploid, asexual organisms are described by linear recurrence equations. Several models in which the mutation rate is controlled by one locus and the fitness is controlled at one or more other loci are developed. It is shown that good approximations can be introduced to give explicit solutions for the course of selection in these models. It is shown that a strong non-equilibrium selection for mutator genes is possible even when the presence of such a gene decreases the fitness of an individual. Experiments that corroborate these conclusions are discussed along with the effects of population size that determine the applicability of these results to natural populations.     

The coalescent process in models with selection, recombination and geographic subdivision

A population genetic model with a single locus at which balancing selection acts and many linked loci at which neutral mutations can occur is analysed using the coalescent approach. The model incorporates geographic subdivision with migration, as well as mutation, recombination, and genetic drift of neutral variation. It is found that geographic subdivision can affect genetic variation even with high rates of migration, providing that selection is strong enough to maintain different allele frequencies at the selected locus. Published sequence data from the alcohol dehydrogenase locus of Drosophila melanogaster are found to fit the proposed model slightly better than a similar model without subdivision.     

A slow selection analysis of two locus, two allele traits.

A deterministic, continuous time model for the dynamics of two locus, two allele Mendelian traits in a large randomly mating diploid population is derived. The model allows for frequency and time dependent birth and death rates. It is analyzed under the assumption that the selective forces acting in the population are small. Slow selection approximations to the system's solution are then constructed. Two particular cases are studied. First, when linkage between loci is tight, the population is shown to rapidly approach Hardy-Weinberg proportions, which then may vary on a (slow) time scale determined by differential fitness. In the case of constant birth and death rates, a measure of the population's fitness is shown to increase on the slow time scale after an initial rapid adjustment. The second case considered is for loose linkage; a population near linkage equilibrium is studied. It is shown that the epistatic parameters cancel and that the results agree with the tight linkage case to leading order. The linkage disequlibrium is described in both cases.     

Multilocus selection in subdivided populations II. Maintenance of polymorphism under weak or strong migration

The potential of maintaining multilocus polymorphism by migration-selection balance is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, selection may vary across demes, and loci are multiallelic. It is shown that if migration and recombination are strong relative to selection, then with weak or no epistasis and intermediate dominance at every locus and in every deme, arbitrarily many alleles can be maintained at arbitrarily many loci at a stable equilibrium. If migration is weak relative to selection and recombination, then with weak or no epistasis and intermediate dominance at every locus and in every deme, as many alleles as there are demes can be maintained at arbitrarily many loci at equilibrium. In both cases open sets of such parameter combinations are constructed, thus the results are robust with respect to small, but arbitrary, perturbations in the parameters. For weak migration, the number of demes is, in fact, a generic upper bound to the number of alleles that can be maintained at any locus. Thus, several scenarios are identified under which multilocus polymorphism can be maintained by migration-selection balance when this is impossible in a panmictic population.      

Mutation and selection in a large population

In this paper we study a large, but finite population, in which mutation and selection occur at a single genetic locus in a diploid organism. We provide theoretical results for the equilibrium allele frequencies, their variances and covariances and their equilibrium distribution, when the population size is larger than the reciprocal of the mean allelic mutation rate. We are also able to infer that the equilibrium distribution of allele frequencies takes the form of a constrained multivariate Gaussian distribution. Our results provide a rapid way of obtaining useful information in the case of complex mutation and selection schemes when the population size is large. We present numerical simulations to test the applicability of our theoretical formulations. The results of these simulations are in very reasonable agreement with the theoretical predictions.     

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.     

Enzyme Polymorphism and Cyclic Parthenogenesis in DAPHNIA MAGNA. I. Selection and Clonal Diversity

Genotype frequencies and fecundities were recorded over a period of two years for three polymorphic enzyme loci (Est, Mdh and Got) in a parthenogenetic natural population of Daphnia magna Straus (Crustacea: Cladocera). There was a large excess of heterozygotes at each locus, and some nonrandom association between loci, although 29 different three-locus genotypes were detected. There were small but significant changes in genotype frequencies that did not follow any clear seasonal cycles or overall trends, and the genotypes often differed significantly in fecundity, although the direction of the difference was not constant. These fitness differences were probably not attributable to the specific loci studied.--Models of balancing selection are of two types: segregation-balanced (e.g., heterosis) and competition-balanced (e.g., frequency dependence). Only the latter type can stabilize diversity in a clonal population. The observed selection was not heterotic, but it is not certain that it was stabilizing either. Clonal competition did not lead to victory by a single, fittest clone; genotypic diversity remained high.     

Density-Dependent Selection Incorporating Intraspecific Competition. II. a Diploid Model

A diploid model is introduced and analyzed in which intraspecific competition is incorporated within the context of density-regulated selection. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when only that type is present. Each genotypic fitness at a single diallelic autosomal locus is a decreasing function of a distinctive effective population size perceived as a result of intraspecific competition. The resulting fitnesses are both density and frequency dependent with selective advantage determined by a balance between genotypic carrying capacity and sensitivity to intraspecific competition. A major finding is that intergenotypic interactions may allow genetic variation to be more easily maintained than in the corresponding model of purely density-dependent selection. In addition, numerical study confirms the possible existence of multiple interior equilibria and that neither overdominance in fitness nor carrying capacity is necessary for stability. The magnitude of the equilibrium population size and optimization principles are also discussed.     

Evolution of dispersal in density regulated populations: A haploid model

The evolution of dispersal is explored in a density-dependent framework. Attention is restricted to haploid populations in which the genotypic fitnesses at a single diallelic locus are decreasing functions of the changing number of individuals in the population. It is shown that migration between two populations in which the genotypic response to density is reversed can maintain both alleles when the intermigration rates are constant or nondecreasing functions of the population densities. There is always a unique symmetric interior equilibrium with equal numbers but opposite gene frequencies in the two populations, provided the system is not degenerate. Numerical examples with exponential and hyperbolic fitnesses suggest that this is the only stable equilibrium state under constant positive migration rates (m) less than . Practically speaking, however, there is only convergence after a reasonable number of generations for relatively small migration rates ( ). A migration-modifying mutant at a second, neutral locus, can successfully enter two populations at a stable migration-selection balance if and only if it reduces the intermigration rates of its carriers at the original equilibrium population size. Moreover, migration modification will always result in a higher equilibrium population size, provided the system approaches another symmetric interior equilibrium. The new equilibrium migration rate will be lower than that at the original equilibrium, even when the modified migration rate is a nondecreasing function of the population sizes. Therefore, as in constant viability models, evolution will lead to reduced dispersal.     

The effect of variable environments on polymorphism at loci with several alleles II. Submultiplicative viabilities

Circumstances assuring a unique stable equilibrium are investigated for a subdivided population with several alleles segregating at a single locus. For a broad class of selection regimes entailing heterozygote viabilities greater than the geometric mean of the corresponding homozygote viabilities, a stable fixation state precludes any other stable equilibria if either total-panmixia or temporal variation is operating. This extends known results for two alleles at a single locus and partially delimits when some of the bizarre behaviour engendered by multiple alleles may occur.Supported in part by National Science Foundation (USA) grant MCS-8002227     

Cumulant dynamics of a population under multiplicative selection, mutation, and drift

We revisit the classical population genetics model of a population evolving under multiplicative selection, mutation, and drift. The number of beneficial alleles in a multilocus system can be considered a trait under exponential selection. Equations of motion are derived for the cumulants of the trait distribution in the diffusion limit and under the assumption of linkage equilibrium. Because of the additive nature of cumulants, this reduces to the problem of determining equations of motion for the expected allele distribution cumulants at each locus. The cumulant equations form an infinite dimensional linear system and in an authored appendix Adam Prügel-Bennett provides a closed form expression for these equations. We derive approximate solutions which are shown to describe the dynamics well for a broad range of parameters. In particular, we introduce two approximate analytical solutions: (1) Perturbation theory is used to solve the dynamics for weak selection and arbitrary mutation rate. The resulting expansion for the system's eigenvalues reduces to the known diffusion theory results for the limiting cases with either mutation or selection absent. (2) For low mutation rates we observe a separation of time-scales between the slowest mode and the rest which allows us to develop an approximate analytical solution for the dominant slow mode. The solution is consistent with the perturbation theory result and provides a good approximation for much stronger selection intensities.     

Population genetic models can be used to study the evolution of the interacting behaviours of parents and their progeny

An example of the evolution of the interacting behaviours of parents and progeny is studied using iterative equations linking the frequencies of the gametes produced by the progeny to the frequencies of the gametes in the parental generation. This population genetics approach shows that a model in which both behaviours are determined by a single locus can lead to a stable equilibrium in which the two behaviours continue to segregate. A model in which the behaviours are determined by genes at two separate loci leads eventually to fixation of the alleles at both loci but this can take many generations of selection. Models of the type described in this paper will be needed to understand the evolution of complex behaviour when genomic or experimental information is available about the genetic determinants of behaviour and the selective values of different genomes.     

Natural selection and density-dependent population growth

Natural selection was studied in the context of density-dependent population growth using a single locus, continuous time model for the rates of change of population size and allele frequency. The maximization principle of density-dependent selection was applied to a class of fitness expressions with explicit recruitment and mortality terms. Three general results were obtained: First, at low population densities, the genetic basis of selection is the difference between the mean recruitment rate and the mean mortality rate. Second, at densities much higher than the equilibrium population size, selection is expected to act to minimize the mean mortality rate. Third, as the population approaches its equilibrium density, selection is predicted to maximize the ratio of the mean recruitment rate to the mean mortality rate.     

Effective size and F-statistics of subdivided populations for sex-linked loci.

For a population subdivided into an arbitrary number (s) of subpopulations, each consisting of different numbers of separate sexes, with arbitrary distributions of family size and variable migration rates by males (dm) and females (df), the recurrence equations for inbreeding coefficient and coancestry between individuals within and among subpopulations for a sex-linked locus are derived and the corresponding expressions for asymptotic effective size are obtained by solving the recurrence equations. The usual assumptions are made which are stable population size and structure, discrete generations, the island migration model, and without mutation and selection. The results show that population structure has an important effect on the inbreeding coefficients in any generation, asymptotic effective size, and F-statistics. Gene exchange among subpopulations inhibits inbreeding in initial generations but increases inbreeding in later generations. The larger the migration rate, the greater the final inbreeding coefficients and the smaller the effective size. Thus if the inbreeding coefficient is to be restricted to a specific value within a given number of generations, the appropriate population structure (the values of s, dm, and df) can be obtained by using the recurrence equations. It is shown that the greater the extent of subdivision (large s, small dm and df), the larger the effective size. For a given subdivided population, the effective size for a sex-linked locus may be larger or smaller than that for an autosomal locus, depending on the sex ratio, variance and covariance of family size, and the extend of subdivision. For the special case of a single unsubdivided population, our recurrence equations for inbreeding coefficient and coancestry and formulas for effective size reduce to the simple expressions derived by previous authors.     

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.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

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.     

Sexual recombination under the joint effects of mutation,selection, and random sampling drift

The advantage or disadvantage of sexual reproduction or recombination for the accumulation of mutant genes in a population is studied under the joint effects of recurrent mutations, selection, and random sampling drift. To obtain the rate at which mutant genes are incorporated three different methods are used; numerical integration of Kolmogorov backward equations, simulation of stochastic difference equations, and Monte Carlo experiments. The first two methods are used in a two-locus system to obtain the fixation probability of double mutants and other related quantities under five different selection models. The third one is conducted for a multiple-locus system and the rate of accumulation of mutant genes per locus is studied. Comparison of the results between sexual and asexual populations shows that the effect of recombination depends on initial linkage disequilibrium, mutation rate v, selection intensity s, and population size N_e. The mode of selection is also an important factor and the large effect of recombination is observed when mutant genes are individually deleterious but collectively favorable. Under a given model of selection, the great advantage or disadvantage of recombination is achieved when a large extent of genetic polymorphism is produced not by mutation but by recombination. Extreme values of N_es and N_ev make the effect insignificant. The results of Monte Carlo experiments also reveal the presence of interaction between selection and sampling drift even when the loci segregate independently and selection is multiplicative. Although this interaction is usually small, there are cases in which one locus theory cannot be used freely. In those cases, the effect of recombination is prominent and one locus theory gives an overestimate of the rate.     

Identifying signatures of sexual selection using genomewide selection components analysis

Sexual selection must affect the genome for it to have an evolutionary impact, yet signatures of selection remain elusive. Here we use an individual‐based model to investigate the utility of genome‐wide selection components analysis, which compares allele frequencies of individuals at different life history stages within a single population to detect selection without requiring a priori knowledge of traits under selection. We modeled a diploid, sexually reproducing population and introduced strong mate choice on a quantitative trait to simulate sexual selection. Genome‐wide allele frequencies in adults and offspring were compared using weighted F_ST values. The average number of outlier peaks (i.e., those with significantly large F_ST values) with a quantitative trait locus in close proximity (“real” peaks) represented correct diagnoses of loci under selection, whereas peaks above the F_ST significance threshold without a quantitative trait locus reflected spurious peaks. We found that, even with moderate sample sizes, signatures of strong sexual selection were detectable, but larger sample sizes improved detection rates. The model was better able to detect selection with more neutral markers, and when quantitative trait loci and neutral markers were distributed across multiple chromosomes. Although environmental variation decreased detection rates, the identification of real peaks nevertheless remained feasible. We also found that detection rates can be improved by sampling multiple populations experiencing similar selection regimes. In short, genome‐wide selection components analysis is a challenging but feasible approach for the identification of regions of the genome under selection.