The generation and maintenance of genetic variation by frequency-dependent selection: constructing polymorphisms under the pairwise interaction model

Frequency-dependent selection remains the most commonly invoked heuristic explanation for the maintenance of genetic variation. For polymorphism to exist, new alleles must be both generated and maintained in the population. Here we use a construction approach to model frequency-dependent selection with mutation under the pairwise interaction model. The pairwise interaction model is a general model of frequency-dependent selection at the genotypic level. We find that frequency-dependent selection is able to generate a large number of alleles at a single locus. The construction process generates multiallelic polymorphisms with a wide range of allele-frequency distributions and genotypic fitness relationships. Levels of polymorphism and mean fitness are uncoupled, so constructed polymorphisms remain permanently invasible to new mutants; thus the model never settles down to an equilibrium state. Analysis of constructed fitness sets reveals signatures of heterozygote advantage and positive frequency dependence.     

Multivariate stabilizing selection and pleiotropy in the maintenance of quantitative genetic variation

Abstract. We investigate maintenance of quantitative genetic variation at mutation-selection balance for multiple traits. The intrinsic strength of real stabilizing selection on one of these traits denoted the "target trait" and the observed strength of apparent stabilizing selection on the target trait can be quite different: the latter, which is estimable, is much smaller (i.e., implying stronger selection) than the former. Distinguishing them may enable the mutation load to be relaxed when considering multivariate stabilizing selection. It is shown that both correlations among mutational effects and among strengths of real stabilizing selection on the traits are not important unless they are high. The analysis for independent situations thus provides a good approximation to the case where mutant and stabilizing selection effects are correlated. Multivariate stabilizing selection can be regarded as a combination of stabilizing selection on the target trait and the pleiotropic direct selection on fitness that is solely due to the effects of real stabilizing selection on the hidden traits. As the overall fitness approaches a constant value as the number of traits increases, multivariate stabilizing selection can maintain abundant genetic variance only under quite weak selection. The common observations of high polygenic variance and strong stabilizing selection thus imply that if the mutation-selection balance is the true mechanism of maintenance of genetic variation, the apparent stabilizing selection cannot arise solely by real stabilizing selection simultaneously on many metric traits.     

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

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

The maintenance of phenotypic and genetic variation in threshold traits by frequency-dependent selection

Many traits are phenotypically dimorphic but determined by the action of many loci, the phenotype being a result of a threshold of sensitivity. Quantitative genetic analysis has shown that generally there is considerable additive genetic variation for the trait, the average heritability being 0.52. In numerous cases threshold traits have been shown, or are assumed, to be under frequency-dependent selection; examples include satellite-territorial behaviour, sex-determination, wing dimorphism and trophic dimorphism. In this paper I investigate the potential for frequency-dependent selection to maintain both phenotypic and additive genetic variation in threshold traits. The qualitative results are robust to the particular form of the frequency-dependent selection function. The equilibrium proportion is more or less independent of population size but the heritability increases with population size, typically approaching its maximal value at a population size of 5000, when the mutation rate is 10^?4. A tenfold decrease in the mutation rate requires an approximate doubling of the population size before an asymptotic value is approached. Thus frequency-dependent selection can account for both the existence of two morphs in a population and the observed levels of heritability. It is also shown, both via simulation and theory, that the quantitative genetic model and a simple phenotypic analysis predict the same equilibrium morph proportion.     

Joint effects of pleiotropic selection and stabilizing selection on the maintenance of quantitative genetic variation at mutation-selection balance

In quantitative genetics, there are two basic "conflicting" observations: abundant polygenic variation and strong stabilizing selection that should rapidly deplete that variation. This conflict, although having attracted much theoretical attention, still stands open. Two classes of model have been proposed: real stabilizing selection directly on the metric trait under study and apparent stabilizing selection caused solely by the deleterious pleiotropic side effects of mutations on fitness. Here these models are combined and the total stabilizing selection observed is assumed to derive simultaneously through these two different mechanisms. Mutations have effects on a metric trait and on fitness, and both effects vary continuously. The genetic variance (V(G)) and the observed strength of total stabilizing selection (V(s,t)) are analyzed with a rare-alleles model. Both kinds of selection reduce V(G) but their roles in depleting it are not independent: The magnitude of pleiotropic selection depends on real stabilizing selection and such dependence is subject to the shape of the distributions of mutational effects. The genetic variation maintained thus depends on the kurtosis as well as the variance of mutational effects: All else being equal, V(G) increases with increasing leptokurtosis of mutational effects on fitness, while for a given distribution of mutational effects on fitness, V(G) decreases with increasing leptokurtosis of mutational effects on the trait. The V(G) and V(s,t) are determined primarily by real stabilizing selection while pleiotropic effects, which can be large, have only a limited impact. This finding provides some promise that a high heritability can be explained under strong total stabilizing selection for what are regarded as typical values of mutation and selection parameters.     

Frequency-dependent variation in reproductive success in Narcissus: implications for the maintenance of stigma-height dimorphism

Negative frequency-dependent selection is a major selective force maintaining sexual polymorphisms. However, empirical demonstrations of frequency-dependent reproductive success are rare, particularly in plants. We investigate this problem by manipulating the frequencies of style morphs in a natural population of Narcissus assoanus, a self-incompatible herb with style-length dimorphism and intra-morph compatibility. We predicted that the reproductive success of morphs would vary negatively with their frequency because of the effects of morph-specific differences in sex-organ position on patterns of pollen transfer. This prediction was generally supported. The fruit and seed set of the two morphs did not differ significantly in plots with 1 : 1 morph ratios. However, short-styled plants produced significantly fewer seeds than long-styled plants in monomorphic plots, and significantly more seeds than long-styled plants in plots with 'long-biased' morph ratios. These patterns indicate that in the absence of physiological barriers to intra-morph mating, negative frequency-dependent selection contributes to the maintenance of stylar polymorphism through inter-morph pollen transfer. Our experimental results also provide insights into the mechanisms governing the biased style-morph ratios in populations of Narcissus species.     

The maintenance of mitochondrial genetic variation by negative frequency‐dependent selection

Mitochondrial genes generally show high levels of standing genetic variation, which is puzzling given the accumulating evidence for phenotypic effects of mitochondrial genetic variation. Negative frequency‐dependent selection, where the relative fitness of a genotype is inversely related to its frequency in a population, provides a potent and potentially general process that can maintain mitochondrial polymorphism. We assessed the change in mitochondrial haplotype frequencies over 10 generations of experimental evolution in 180 seed beetle populations in the laboratory, where haplotypes competed for propagation to subsequent generations. We found that haplotypes consistently increased in frequency when they were initially rare and decreased in frequency when initially common. Our results have important implications for the use of mtDNA haplotype frequency data to infer population level processes and they revive the general hypothesis that negative frequency‐dependent selection, presumably caused by habitat heterogeneity, may commonly promote polymorphism in ecologically relevant life history genes.     

Pleiotropic model of maintenance of quantitative genetic variation at mutation-selection balance

A pleiotropic model of maintenance of quantitative genetic variation at mutation-selection balance is investigated. Mutations have effects on a metric trait and deleterious effects on fitness, for which a bivariate gamma distribution is assumed. Equations for calculating the strength of apparent stabilizing selection (V(s)) and the genetic variance maintained in segregating populations (V(G)) were derived. A large population can hold a high genetic variance but the apparent stabilizing selection may or may not be relatively strong, depending on other properties such as the distribution of mutation effects. If the distribution of mutation effects on fitness is continuous such that there are few nearly neutral mutants, or a minimum fitness effect is assumed if most mutations are nearly neutral, V(G) increases to an asymptote as the population size increases. Both V(G) and V(s) are strongly affected by the shape of the distribution of mutation effects. Compared with mutants of equal effect, allowing their effects on fitness to vary across loci can produce a much higher V(G) but also a high V(s) (V(s) in phenotypic standard deviation units, which is always larger than the ratio V(P)/V(m)), implying weak apparent stabilizing selection. If the mutational variance V(m) is approximately 10(-3)V(e) (V(e), environmental variance), the model can explain typical values of heritability and also apparent stabilizing selection, provided the latter is quite weak as suggested by a recent review.     

Frequency-dependent selection and competition: empirical approaches

When Darwin and Wallace first formulated the theory of evolution by natural selection, they were greatly influenced by the idea that populations tend to increase geometrically and rapidly outgrow the resources available to them. They argued that the ensuing competition among individuals would be a major agent of natural selection. Since their day, competition has become almost synonymous with the idea of natural selection or survival of the fittest. In this paper we examine the relation between competition and selection by using simple competition models, consider the interaction of density and frequency in determining competitive outcome, and review the literature on frequency-dependent competitive interactions among genotypes within populations.     

Fluctuating selection and the determinants of genetic variation

Recent studies of cosmopolitan Drosophila populations have found hundreds to thousands of genetic loci with seasonally fluctuating allele frequencies, bringing temporally fluctuating selection to the forefront of the historical debate surrounding the maintenance of genetic variation in natural populations. Numerous mechanisms have been explored in this longstanding area of research, but these exciting empirical findings have prompted several recent theoretical and experimental studies that seek to better understand the drivers, dynamics, and genome-wide influence of fluctuating selection. In this review, we evaluate the latest evidence for multilocus fluctuating selection in Drosophila and other taxa, highlighting the role of potential genetic and ecological mechanisms in maintaining these loci and their impacts on neutral genetic variation.     

Frequency-dependent host selection by parasitic mites: a model and a case study

Previous studies on frequency-dependent food selection (changing food preferences in response to changes in relative food abundance) have focused on predators and parasitoids. These organisms utilize several victims during their lifetime. We introduce the case of parasites which, having accepted a host, do not change it. We propose two alternative models to explain the biased occurrence of parasites on different host types: (1) through the option of rejecting less-preferred hosts prior to accepting one of them; (2) by differential parasite survival on different host types. These models predict that host rejection, but not differential survival, can create frequency-dependent parasitism (FDP). Unlike previously described factors responsible for frequency dependence of food selection, which act through changing the foraging behaviour of individual predators or parasitoids, FDP involves no adjustment of parasite foraging strategy according to previous feeding experience. The mite Hemisarcoptes coccophagus is an obligate parasite of armoured scale insects (Homptera: Diaspididae). Our field data show that H. coccophagus is found more frequently on ovipositing than on young host females. Our model, combining the effects of host rejection and differential survival, is used to estimate the relative contribution of these factors to parasite biased occurrence on different hosts. The contribution of differential survival was dominant in H. coccophagus, and overode any effect of host rejection. Nevertheless, our prediction that FDP may be found in parasites is supported by literature data about a parasitic water mite.     

Frequency-dependent predation and maintenance of prey polymorphism

In positive frequency-dependent predation, predation risk of an individual prey correlates positively with the frequency of that prey type. In a number of small-scale experiments individual predators have shown frequency-dependent behaviour, often leading to the conclusion that a population of such predators could maintain prey polymorphism. Using simulations, I studied the dynamics of frequency-dependent predation and prey polymorphism. The model suggests that persistence of prey polymorphism decreases with increasing number of predators that show frequency-dependent behaviour, questioning conclusions about polymorphism based on experiments with few predators. In addition, prey population size, prey crypsis, difference in crypsis between prey morphs and the way the behaviour was adjusted affected the persistence of polymorphism. Under some circumstances prey population remained polymorphic for a shorter time under frequency-dependent than under frequency-independent predation. This suggests that although positive frequency-dependent predator behaviour may maintain prey polymorphism, it is not a sufficient condition for persistent prey polymorphism.     

Frequency-dependent selection and the evolution of assortative mating

A long-standing goal in evolutionary biology is to identify the conditions that promote the evolution of reproductive isolation and speciation. The factors promoting sympatric speciation have been of particular interest, both because it is notoriously difficult to prove empirically and because theoretical models have generated conflicting results, depending on the assumptions made. Here, we analyze the conditions under which selection favors the evolution of assortative mating, thereby reducing gene flow between sympatric groups, using a general model of selection, which allows fitness to be frequency dependent. Our analytical results are based on a two-locus diploid model, with one locus altering the trait under selection and the other locus controlling the strength of assortment (a "one-allele" model). Examining both equilibrium and nonequilibrium scenarios, we demonstrate that whenever heterozygotes are less fit, on average, than homozygotes at the trait locus, indirect selection for assortative mating is generated. While costs of assortative mating hinder the evolution of reproductive isolation, they do not prevent it unless they are sufficiently great. Assortative mating that arises because individuals mate within groups (formed in time or space) is most conducive to the evolution of complete assortative mating from random mating. Assortative mating based on female preferences is more restrictive, because the resulting sexual selection can lead to loss of the trait polymorphism and cause the relative fitness of heterozygotes to rise above homozygotes, eliminating the force favoring assortment. When assortative mating is already prevalent, however, sexual selection can itself cause low heterozygous fitness, promoting the evolution of complete reproductive isolation (akin to "reinforcement") regardless of the form of natural selection.     

Complex dynamics occur in a single-locus,multiallelic model of general frequency-dependent selection

We examine the characteristics of non-equilibrium dynamics produced by a simple well-known model of frequency-dependent selection at a single diploid locus. An examination of the parameter space of this “pairwise-interaction model” (PIM) revealed non-equilibrium dynamics for polymorphisms of 3, 4 and 5 alleles; both allele-frequency cycling and aperiodic trajectories were detected. We measured the number, cycle length and domains of attraction of the various attractors produced by the model. The domains of attraction tended to be smaller, and the cycles longer, for systems with larger number of alleles. Fitnesses that parametrized negative frequency-dependent selection were more likely to allow cycling, and these cycles also had larger domains of attraction. Aperiodic trajectories were detected only in cases with 4 or 5 alleles. The genetic cycles produced by the model do not have periods as short as those predicted in ecological models with cycling (such as predator–prey population cycles, etc.). Consequently, in a real-world system, PIM allele-frequency cycling is likely to be indistinguishable from stable equilibria when observed over short time scales.     

Complex dynamics occur in a single-locus, multiallelic model of general frequency-dependent selection

We examine the characteristics of non-equilibrium dynamics produced by a simple well-known model of frequency-dependent selection at a single diploid locus. An examination of the parameter space of this “pairwise-interaction model” (PIM) revealed non-equilibrium dynamics for polymorphisms of 3, 4 and 5 alleles; both allele-frequency cycling and aperiodic trajectories were detected. We measured the number, cycle length and domains of attraction of the various attractors produced by the model. The domains of attraction tended to be smaller, and the cycles longer, for systems with larger number of alleles. Fitnesses that parametrized negative frequency-dependent selection were more likely to allow cycling, and these cycles also had larger domains of attraction. Aperiodic trajectories were detected only in cases with 4 or 5 alleles. The genetic cycles produced by the model do not have periods as short as those predicted in ecological models with cycling (such as predator–prey population cycles, etc.). Consequently, in a real-world system, PIM allele-frequency cycling is likely to be indistinguishable from stable equilibria when observed over short time scales.     

Joint evolution of interspecific mutualism and regulation of variation of interaction under directional selection in trait space

The present study theoretically examines the process by which interspecific mutualism is established with trait matching. The mathematical model includes joint evolution of the mutualistic relationship between two species and regulation of variation of interaction in one-dimensional trait space, assuming abiotic directional selection. The model considers three types of regulation: homeostasis against environmental variation, developmental stability, and acceptability of dissimilar mutualism partners (mutualism kernel). Mainly focusing on the developmental stability, the analysis indicates that the mutualism can evolve when (1) higher levels of developmental stability are more intensively degenerated by deleterious mutations, (2) the basal rate of deleterious mutation is low, (3) trait expression is less influenced by environmental factors, and (4) the specificity of mutualism is high. It also shows that the evolution of developmental stability can promote the evolution of mutualism with trait matching when the deleterious mutation bias disappears at a certain level of developmental instability. Evolution of homeostasis and mutualism kernel can be discussed in the similar way because of formal similarities in the model. In plant–pollinator interactions, it has recently been proposed that evolutionary increments of developmental stability in mutualistic traits might promote plant diversification. The present results partly support this hypothesis with respect to the evolutionary relationship between mutualism and developmental stability.     

Patterns of multiallelic polymorphism maintained by migration and selection

Evolution at a multiallelic locus under the joint action of migration and viability selection is investigated. Generations are discrete and nonoverlapping. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. The forward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). Several examples of globally attracting multiallelic equilibria are presented. Migration can cause global fixation even if, without migration, there is a globally attracting multiallelic equilibrium in every colony. Migration can also cause the global fixation of an allele that, without migration, is eliminated in every colony. Without dominance, generically, the number of alleles present at equilibrium cannot exceed the number of colonies. Some general properties and examples of the Levene model are studied in detail. If in each colony there is either no dominance or, without migration, a globally attracting internal equilibrium, then there exists a globally attracting equilibrium with migration. Therefore, if an internal equilibrium exists, it is the global attractor.     

Sexual conflict and the maintenance of genetic variation in natural populations

Understanding the factors that maintain genetic variation in natural populations is a foundational goal of evolutionary biology. To this end, population geneticists have developed a variety of models that can produce stable polymorphisms. In one of the earliest models, Owen ( 1953 ) demonstrated that differences in selection pressures acting on males and females could maintain multiple alleles of a gene at a stable equilibrium. If the selection pressures act in opposite directions in males and females, we refer to this as (inter‐) sexual conflict or sexual antagonism (Arnqvist & Rowe, 2005 ). Testing if sexual conflict maintains genetic variation in natural populations is a tremendous challenge—it requires both identifying loci that harbor sexually antagonistic alleles and determining whether those alleles are maintained as stable polymorphisms (Mank, 2017 ). Doing so genome‐wide is even harder because it is not tractable to identify sexually antagonistic alleles and test for stable polymorphisms at all loci. Dutoit et al. ( 2018 ) confront this challenge in a paper published in this issue of Molecular Ecology. Using gene expression and population genomic data from the collared flycatcher, Dutoit et al. ( 2018 ) identify associations and correlations between genomic signatures of balanced polymorphisms and sexual conflict.     

Evolutionary rates for multivariate traits: the role of selection and genetic variation

A fundamental question in evolutionary biology is the relative importance of selection and genetic architecture in determining evolutionary rates. Adaptive evolution can be described by the multivariate breeders'' equation (), which predicts evolutionary change for a suite of phenotypic traits () as a product of directional selection acting on them (β) and the genetic variance–covariance matrix for those traits (G). Despite being empirically challenging to estimate, there are enough published estimates of G and β to allow for synthesis of general patterns across species. We use published estimates to test the hypotheses that there are systematic differences in the rate of evolution among trait types, and that these differences are, in part, due to genetic architecture. We find some evidence that sexually selected traits exhibit faster rates of evolution compared with life-history or morphological traits. This difference does not appear to be related to stronger selection on sexually selected traits. Using numerous proposed approaches to quantifying the shape, size and structure of G, we examine how these parameters relate to one another, and how they vary among taxonomic and trait groupings. Despite considerable variation, they do not explain the observed differences in evolutionary rates.