Evolutionary limits to the frequency of aggression between related or unrelated conspecifics in diploid species with simple mendelian inheritance

Game theory has been used by some authors to analyse evolutionary limits to the expression of aggression in theoretical haploid parthenogenetic species. Others have examined frequency dependent selection, of which aggression may be a case, by applying population genetic models to diploid species. A model is presented which attempts to combine these two approaches. Game theory is used to determine evolutionarily stable strategies and corresponding stable polymorphisms for a two-strategy game played by members of a diploid sexual species, when choice of strategy is determined by two alleles at a single locus. Results are given for dominant, co-dominant and recessive determination of choice of the more aggressive of two strategies, for two levels of relationship: unrelated players and sibs. It is found that for a range of models of single locus inheritance the evolutionarily stable strategy (ESS) determined for haploid species remains the stable population strategy for diploid sexual species, when players are unrelated. In sibling contestants aggression is reduced. The mixed strategy haploid ESS underestimates, but the pure strategy haploid ESS provides a good indication of the degree to which relatedness lessens aggression in diploid species. For both haploid and diploid species there may be a considerable advantage to confining conflicts to kin.     

Evolution of dominance under frequency-dependent intraspecific competition

A population-genetic analysis is performed of a two-locus two-allele model, in which the primary locus has a major effect on a quantitative trait that is under frequency-dependent disruptive selection caused by intraspecific competition for a continuum of resources. The modifier locus determines the degree of dominance at the trait level. We establish the conditions when a modifier allele can invade and when it becomes fixed if sufficiently frequent. In general, these are not equivalent because an unstable internal equilibrium may exist and the condition for successful invasion of the modifier is more restrictive than that for eventual fixation from already high frequency. However, successful invasion implies global fixation, i.e., fixation from any initial condition. Modifiers of large effect can become fixed, and also invade, in a wider parameter range than modifiers of small effect. We also study modifiers with a direct, frequency-independent deleterious fitness effect. We show that they can invade if they induce a sufficiently high level of dominance and if disruptive selection on the ecological trait is strong enough. For deleterious modifiers, successful invasion no longer implies global fixation because they can become stuck at an intermediate frequency due to a stable internal equilibrium. Although the conditions for invasion and for fixation if sufficiently frequent are independent of the linkage relation between the two loci, the rate of spread depends strongly on it. The present study provides further support to the view that evolution of dominance may be an efficient mechanism to remove unfit heterozygotes that are maintained by balancing selection. It also demonstrates that an invasion analysis of mutants of very small effect is insufficient to obtain a full understanding of the evolutionary dynamics under frequency-dependent selection.     

PLOIDY AND EVOLUTION BY SEXUAL SELECTION: A COMPARISON OF HAPLOID AND DIPLOID FEMALE CHOICE MODELS NEAR FIXATION EQUILIBRIA

We compare the stability properties of haploid and diploid models of Fisherian sexual selection (with male contribution limited to sperm) by examining both models at equilibria for which a male trait is fixed or absent. Haploid and diploid two locus diallelic models share the property that the stability of such fixation equilibria is determined by the relationship between the harmonic mean of relative preference values for the common male trait, weighted by the frequency of the preferences, and the relative viability associated with the common male trait. When diploid females with heterozygotic-based preferences express preference strengths intermediate between homozygote-based preferences, then boundary equilibria of haploid and diploid models share many stability properties. However, even with intermediate heterozygote preferences, haploid and diploid models do differ: (1) for a particular frequency of the preference allele, both fixation boundaries can be stable for the diploid model, and (2) with over- or underdominance at the preference locus (a possibility precluded in the haploid model), a fixation boundary in the diploid model may show two switches in its stability state for increasing frequencies of one of the preference alleles. These differences are due not just to the impossibility of dominance in haploid models, but also to the larger number of diploid genotypes.     

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.     

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.     

On the evolutionary costs of self-incompatibility: incomplete reproductive compensation due to pollen limitation

Pollen limitation affects plants with diverse reproductive systems and ecologies. In self-incompatible (SI) species, pollen limitation may preclude full reproductive compensation for prezygotic rejection of pollen. We present a model designed to explore the effects of incomplete reproductive compensation on evolutionary changes at a modifier locus that regulates the level of SI expression. Our results indicate that incomplete reproductive compensation greatly increases the evolutionary costs of SI, particularly in populations with low S-allele diversity. The evolutionary fate of modifiers of SI expression depends on the rate at which they are transmitted to future generations as well as the effects of SI on offspring number and quality. Partial SI expression can represent a stable condition rather than an evolutionarily transient state between full expression and full suppression. This unanticipated result provides the first theoretical support for the evolutionary stability of such mixed mating systems, the existence of which has recently been documented.     

Quantifying male attractiveness

Genetic models of sexual selection are concerned with a dynamic process in which female preference and male trait values coevolve. We present a rigorous method for characterizing evolutionary endpoints of this process in phenotypic terms. In our phenotypic characterization the mate-choice strategy of female population members determines how attractive females should find each male, and a population is evolutionarily stable if population members are actually behaving in this way. This provides a justification of phenotypic explanations of sexual selection and the insights into sexual selection that they provide. Furthermore, the phenotypic approach also has enormous advantages over a genetic approach when computing evolutionarily stable mate-choice strategies, especially when strategies are allowed to be complex time-dependent preference rules. For simplicity and clarity our analysis deals with haploid mate-choice genetics and a male trait that is inherited phenotypically, for example by vertical cultural transmission. The method is, however, easily extendible to other cases. An example illustrates that the sexy son phenomenon can occur when there is phenotypic inheritance of the male trait.     

Verification of marker–trait associations in biparental winter barley (<Emphasis Type="Italic">Hordeum vulgare</Emphasis> L.) DH populations

In previous genome-wide association studies, marker–trait associations for grain yield and additional traits of agronomic importance were identified in the German winter barley (Hordeum vulgare L.) breeding gene pool. In the present study, seven doubled haploid populations segregating for the relevant alleles at the associated loci were used to get information whether these marker–trait associations can be verified in biparental populations and reliably used in applied barley breeding. The doubled haploid populations were phenotyped in field trials at two to five locations each in 1 year and genotyped by 40 trait-associated single nucleotide polymorphisms using an Illumina VeraCode GoldenGate assay. Large phenotypic variation was observed for all traits within at least one doubled haploid population. For 19 out of 58 marker–trait associations tested, the phenotypic means of both marker classes were significantly (p ≤ 0.005) different, thus confirming the association of the respective marker and the quantitative trait locus detected. For example, doubled haploid lines derived from a cross of ‘Malta’ × ‘Goldmine’ carrying different marker alleles differed by 0.41 t/ha in mean grain yield. The 19 (out of 58) marker–trait associations verified correspond to 10 (out of 27) genomic regions. Markers that were verified to be associated with a quantitative trait locus can be implemented directly in winter barley breeding for the selection of parental lines and marker-assisted pedigree selection.     

Evolution of assortative mating in a population expressing dominance

In this article, we study the influence of dominance on the evolution of assortative mating. We perform a population-genetic analysis of a two-locus two-allele model. We consider a quantitative trait that is under a mixture of frequency-independent stabilizing selection and density- and frequency-dependent selection caused by intraspecific competition for a continuum of resources. The trait is determined by a single (ecological) locus and expresses intermediate dominance. The second (modifier) locus determines the degree of assortative mating, which is expressed in females only. Assortative mating is based on similarities in the quantitative trait ('magic trait' model). Analytical conditions for the invasion of assortment modifiers are derived in the limit of weak selection and weak assortment. For the full model, extensive numerical iterations are performed to study the global dynamics. This allows us to gain a better understanding of the interaction of the different selective forces. Remarkably, depending on the size of modifier effects, dominance can have different effects on the evolution of assortment. We show that dominance hinders the evolution of assortment if modifier effects are small, but promotes it if modifier effects are large. These findings differ from those in previous work based on adaptive dynamics.     

Coevolution of self-fertilization and inbreeding depression. II. Symmetric overdominance in viability

We describe the evolutionary dynamics of a modifier of selfing coevolving with a locus subject to symmetric overdominance in viability under general levels of reduction in pollination success as a consequence of self-fertilization (pollen discounting). Simple models of the evolution of breeding systems that represent inbreeding depression as a constant parameter do not admit the possibility of stable mixed mating systems involving both inbreeding and random mating. Contrary to this expectation, we find that coevolution between a modifier of selfing and a single overdominant locus situated anywhere in the genome can generate evolutionarily attracting mixed mating systems. Two forms of association between the modifier locus and the viability locus promote the evolution of outcrossing. The favored heterozygous genotype at the viability locus develops positive associations with modifier alleles that enhance outcrossing and with the heterozygous genotype at the modifier locus. Associations between outcrossing and high viability evolve immediately upon the introduction of a rare modifier allele, even in the absence of linkage.     

Host–parasite interactions and the evolution of nonrandom mating

Some species mate nonrandomly with respect to alleles underlying immunity. One hypothesis proposes that this is advantageous because nonrandom mating can lead to offspring with superior parasite resistance. We investigate this hypothesis, generalizing previous models in four ways: First, rather than only examining invasibility of modifiers of nonrandom mating, we identify evolutionarily stable strategies. Second, we study coevolution of both haploid and diploid hosts and parasites. Third, we allow for maternal parasite transmission. Fourth, we allow for many alleles at the interaction locus. We find that evolutionarily stable rates of assortative or disassortative mating are usually near zero or one. However, for one case, in which assumptions most closely match the major histocompatibility complex (MHC) system, intermediate rates of disassortative mating can evolve. Across all cases, with haploid hosts, evolution proceeds toward complete disassortative mating, whereas with diploid hosts either assortative or disassortative mating can evolve. Evolution of nonrandom mating is much less affected by the ploidy of parasites. For the MHC case, maternal transmission of parasites, because it creates an advantage to producing offspring that differ from their parents, leads to higher evolutionarily stable rates of disassortative mating. Lastly, with more alleles at the interaction locus, disassortative mating evolves to higher levels.     

Analysis of disruptive selection in subdivided populations

Background

Analytical methods have been proposed to determine whether there are evolutionarily stable strategies (ESS) for a trait of ecological significance, or whether there is disruptive selection in a population approaching a candidate ESS. These criteria do not take into account all consequences of small patch size in populations with limited dispersal.

Results

We derive local stability conditions which account for the consequences of small and constant patch size. All results are derived from considering R_m, the overall production of successful emigrants from a patch initially colonized by a single mutant immigrant. Further, the results are interpreted in term of concepts of inclusive fitness theory. The condition for convergence to an evolutionarily stable strategy is proportional to some previous expressions for inclusive fitness. The condition for evolutionary stability stricto sensu takes into account effects of selection on relatedness, which cannot be neglected. It is function of the relatedness between pairs of genes in a neutral model and also of a three-genes relationship. Based on these results, I analyze basic models of dispersal and of competition for resources. In the latter scenario there are cases of global instability despite local stability. The results are developed for haploid island models with constant patch size, but the techniques demonstrated here would apply to more general scenarios with an island mode of dispersal.

Conclusions

The results allow to identity and to analyze the relative importance of the different selective pressures involved. They bridge the gap between the modelling frameworks that have led to the R_m concept and to inclusive fitness.

     

Truncation family selection and nonsystematic inbreeding leads to a rapid fixation of a pattern of mobile genetic elements in a computer model

A computer simulation model of the population dynamics of a polygenic system and a pattern of mobile genetic elements (MGEs) under directional truncation selection for a quantitative trait was developed. Modifier MGEs were shown to be rapidly and adaptively fixed (or lost) together with the modified polygenes. Marker MGEs and independent MGE copies were fixed and lost just as rapidly but in a random manner. Using specific marking of initial haploid genomes and direct computing of the mean proportion of identical encounters at each locus in each generation, it was shown that the mean nonselective inbreeding coefficient F(n) dramatically increases in the course of selection, reaching values 0.7-0.9 in 15-20 generations. As a result, adaptive homozygotization of polygenes and modifier MGEs and random homozygotization of marker MGEs, independent MGE copies, and all other genes of the genome occurs. These results confirm the hypothesis on the "champion" polygene pattern advanced earlier to explain the data of selection experiments.     

EVOLUTION OF DOMINANCE UNDER FREQUENCY-DEPENDENT INTRASPECIFIC COMPETITION IN AN ASSORTATIVELY MATING POPULATION

We study the evolution of higher levels of dominance as a response to negative frequency-dependent selection. In contrast to previous studies, we focus on the effect of assortative mating on the evolution of dominance under frequency-dependent intraspecific competition. We analyze a two-locus two-allele model, in which the primary locus has a major effect on a quantitative trait that is under a mixture of frequency-independent stabilizing selection, density-dependent selection, and frequency-dependent selection caused by intraspecific competition for a continuum of resources. The second (modifier) locus determines the degree of dominance at the trait level. Additionally, the population mates assortatively with respect to similarities in the ecological trait. Our analysis shows that the parameter region in which dominance can be established decreases if small levels of assortment are introduced. In addition, the degree of dominance that can be established also decreases. In contrast, if assortment is intermediate, sexual selection for extreme types can be established, which leads to evolution of higher levels of dominance than under random mating. For modifiers with large effects, intermediate levels of assortative mating are most favorable for the evolution of dominance. For large modifiers, the speed of fixation can even be higher for intermediate levels of assortative mating than for random mating.     

Evolution of migration in a periodically changing environment

The ability to migrate can evolve in response to various forces. In particular, when selection is heterogeneous in space but constant in time, local adaptation induces a fitness cost on immigrants and selects against migration. The evolutionary outcome, however, is less clear when selection also varies temporally. Here, we present a two-locus model analyzing the effects of spatial and temporal variability in selection on the evolution of migration. The first locus is under temporally varying selection (various periodic functions are considered, but a general nonparametric framework is used), and the second locus is a modifier controlling migration ability. First, we study the dynamics of local adaptation and derive the migration rate that maximizes local adaptation as a function of the speed and geometry of the fluctuations in the environment. Second, we derive an analytical expression for the evolutionarily stable migration rate. When there is no cost of migration, we show that higher migration rates are favored when selection changes fast. When migration is costly, however, the evolutionarily stable migration rate is maximal for an intermediate speed of the variation of selection. This model may help in understanding the evolution of migration in a broad range of scenarios and, in particular, in host-parasite systems, where selection is thought to vary quickly in both space and time.     

Evolutionarily stable mutation rate in a periodically changing environment

Evolution of mutation rate controlled by a neutral modifier is studied for a locus with two alleles under temporally fluctuating selection pressure. A general formula is derived to calculate the evolutionarily stable mutation rate μ(ess) in an infinitely large haploid population, and following results are obtained. (I) For any fluctuation, periodic or random: (1) if the recombination rate r per generation between the modifier and the main locus is 0, μ(ess) is the same as the optimal mutation rate μ(op) which maximizes the long-term geometric average of population fitness; and (2) for any r, if the strength s of selection per generation is very large, μ(ess) is equal to the reciprocal of the average number T of generations (duration time) during which one allele is persistently favored than the other. (II) For a periodic fluctuation in the limit of small s and r, μ(ess)T is a function of sT and rT with properties: (1) for a given sT, μ(ess)T decreases with increasing rT; (2) for sT </= 1, μ(ess)T is almost independent of sT, and depends on rT as μ(ess)T & 1.6 for rT << 1 and μ(ess)T & 6/rT for rT >> 1; and (3) for sT >/= 1, and for a given rT, μ(ess)T decreases with increasing sT to a certain minimum less than 1, and then increases to 1 asymptotically in the limit of large sT. (III) For a fluctuation consisting of multiple Fourier components (i.e., sine wave components), the component with the longest period is the most effective in determining μ(ess) (low pass filter effect). (IV) When the cost c of preventing mutation is positive, the modifier is nonneutral, and μ(ess) becomes larger than in the case of neutral modifier under the same selection pressure acting at the main locus. The value of c which makes μ(ess) equal to μ(op) of the neutral modifier case is calculated. It is argued that this value gives a critical cost such that, so long as the actual cost exceeds this value, the evolution rate at the main locus must be smaller than its mutation rate μ(ess).     

Mutating away from your enemies: The evolution of mutation rate in a host–parasite system

The rate at which mutations occur in nature is itself under natural selection. While a general reduction of mutation rates is advantageous for species inhabiting constant environments, higher mutation rates can be advantageous for those inhabiting fluctuating environments that impose on-going directional selection. Analogously, species involved in antagonistic co-evolutionary arms races, such as hosts and parasites, can also benefit from higher mutation rates. We use modifier theory, combined with simulations, to investigate the evolution of mutation rate in such a host–parasite system. We derive an expression for the evolutionary stable mutation rate between two alleles, each of whose fitness depends on the current genetic composition of the other species. Recombination has been shown to weaken the strength of selection acting on mutation modifiers, and accordingly, we find that the evolutionarily attracting mutation rate is lower when recombination between the selected and the modifier locus is high. Cyclical dynamics are potentially commonplace for loci governing antagonistic species interactions. We characterize the parameter space where such cyclical dynamics occur and show that the evolution of large mutation rates tends to inhibit cycling and thus eliminates further selection on modifiers of the mutation rate. We then find using computer simulations that stochastic fluctuations in finite populations can increase the size of the region where cycles occur, creating selection for higher mutation rates. We finally use simulations to investigate the model behaviour when there are more than two alleles, finding that the region where cycling occurs becomes smaller and the evolutionarily attracting mutation rate lower when there are more alleles.     

Optimal Recombination Rate in Fluctuating Environments

The optimal recombination rate which maximizes the long-term geometric average of the population fitness is studied for a two-locus haploid model, assuming that the fitnesses of genotypes AB, Ab, aB and ab are 1 + s(t), 1 - s(t), 1 - s(t), and 1 + s(t), respectively, where s(t) follows various stationary stochastic processes with the average zero. With positive recombination, the polymorphism is stably maintained at both loci. After an initial transient phase, the dynamics are reduced to one dimension, and are analyzed for weak selection limit, strong selection limit, and selection with two state Markovian jump. Results are: (1) If the environmental fluctuation has a predominant periodic component, ropt is approximately inversely proportional to the period irrespective of selection intensity. (2) If the fluctuation is a superposition of many periodic components, the one with the longest period is the most effective in determining ropt because the genetic dynamics cannot track very quick fluctuations (low pass filter effect). (3) If the power spectrum density is decreasing with the frequency, as in pink, or 1/f noises, ropt is small when selection is weak, and increases with the selection intensity. Numerical calculation of the genetic dynamics of a recombination modifier supports all these predictions for the evolutionarily stable recombination rate.     

Sex-specific viability, sex linkage and dominance in genomic imprinting

Genomic imprinting is a phenomenon by which the expression of an allele at a locus depends on the parent of origin. Two different two-locus evolutionary models are presented in which a second locus modifies the imprinting status of the primary locus, which is under differential selection in males and females. In the first model, a modifier allele that imprints the primary locus invades the population when the average dominance coefficient among females and males is >12 and selection is weak. The condition for invasion is always heavily contingent upon the extent of dominance. Imprinting is more likely in the sex experiencing weaker selection only under some parameter regimes, whereas imprinting by either sex is equally likely under other regimes. The second model shows that a modifier allele that induces imprinting will increase when imprinting has a direct selective advantage. The results are not qualitatively dependent on whether the modifier locus is autosomal or X linked.     

Polymorphism in the two-locus Levene model with nonepistatic directional selection

For the Levene model with soft selection in two demes, the maintenance of polymorphism at two diallelic loci is studied. Selection is nonepistatic and dominance is intermediate. Thus, there is directional selection in every deme and at every locus. We assume that selection is in opposite directions in the two demes because otherwise no polymorphism is possible. If at one locus there is no dominance, then a complete analysis of the dynamical and equilibrium properties is performed. In particular, a simple necessary and sufficient condition for the existence of an internal equilibrium and sufficient conditions for global asymptotic stability are obtained. These results are extended to deme-independent degree of dominance at one locus. A perturbation analysis establishes structural stability within the full parameter space. In the absence of genotype-environment interaction, which requires deme-independent dominance at both loci, nongeneric equilibrium behavior occurs, and the introduction of arbitrarily small genotype-environment interaction changes the equilibrium structure and may destroy stable polymorphism. The volume of the parameter space for which a (stable) two-locus polymorphism is maintained is computed numerically. It is investigated how this volume depends on the strength of selection and on the dominance relations. If the favorable allele is (partially) dominant in its deme, more than 20% of all parameter combinations lead to a globally asymptotically stable, fully polymorphic equilibrium.