Consequences of population structure on genes under balancing selection

This paper describes a new approach to modeling population structure for genes under strong balancing selection of the type seen in plant self-incompatibility systems and the major histocompatibility complex (MHC) system of vertebrates. Simple analytic solutions for the number of alleles maintained at equilibrium and the expected proportion of alleles shared between demes at various levels are derived and checked against simulation results. The theory accurately captures the dynamics of allele number in a subdivided population and identifies important values of m (migration rate) at which allele number and distribution change qualitatively. Starting from a panmictic population, as migration among demes decreases a qualitative change in dynamics is seen at approximately m(crit) approximately equal to the square root of(s/4piNT) where NT is the total population size and s is a measure of the strength of selection. At this point, demes can no longer maintain their panmictic allele number, due to increasing isolation from the total population. Another qualitative change occurs at a migration rate on the same order of magnitude as the mutation rate, mu. At this point, the demes are highly differentiated for allele complement, and the total number of alleles in the population is increased. Because in general u < m<(crit) at intermediate migration rates slightly fewer alleles may be maintained in the total population than are maintained at panmixia. Within this range, total allele number may not be the best indicator of whether a population is effectively panmictic, and some caution should be used when interpreting samples from such populations. The theory presented here can help to analyze data from genes under balancing selection in subdivided populations.     

Population dynamics of sex-determining alleles in honey bees and self-incompatibility alleles in plants

Mathematical theories of the population dynamics of sex-determining alleles in honey bees are developed. It is shown that in an infinitely large population the equilibrium frequency of a sex allele is 1/n, where n is the number of alleles in the population, and the asymptotic rate of approach to this equilibrium is 2/(3n) per generation. Formulae for the distribution of allele frequencies and the effective and actual numbers of alleles that can be maintained in a finite population are derived by taking into account the population size and mutation rate. It is shown that the allele frequencies in a finite population may deviate considerably from 1/n. Using these results, available data on the number of sex alleles in honey bee populations are discussed. It is also shown that the number of self-incompatibility alleles in plants can be studied in a much simpler way by the method used in this paper. A brief discussion about general overdominant selection is presented.     

Stability properties of underdominance in finite subdivided populations

IN ISOLATED populations underdominance leads to bistable evolutionary dynamics: below a certain mutant allele frequency the wildtype succeeds. Above this point, the potentially underdominant mutant allele fixes. In subdivided populations with gene flow there can be stable states with coexistence of wildtypes and mutants: polymorphism can be maintained because of a migration-selection equilibrium, i.e., selection against rare recent immigrant alleles that tend to be heterozygous. We focus on the stochastic evolutionary dynamics of systems where demographic fluctuations in the coupled populations are the main source of internal noise. We discuss the influence of fitness, migration rate, and the relative sizes of two interacting populations on the mean extinction times of a group of potentially underdominant mutant alleles. We classify realistic initial conditions according to their impact on the stochastic extinction process. Even in small populations, where demographic fluctuations are large, stability properties predicted from deterministic dynamics show remarkable robustness. Fixation of the mutant allele becomes unlikely but the time to its extinction can be long.     

Invasion of gene duplication through masking for maladaptive gene flow

Gene duplication can increase an organism's ability to mask the effect of deleterious alleles present in the population, but this is typically a small effect when the source of the genetic variation is mutation. Migration can introduce orders of magnitude more deleterious alleles per generation and may therefore be an important force acting on the structure of genomes. Using formal analytical methods, we study the invasion of haplotypes containing two copies of the resident allele, assuming that a single-locus equilibrium is already established in a continent-island model of migration. Provided that the immigrant allele can be completely masked by multiple functional gene copies, a new duplication will deterministically spread so long as duplicate haplotypes are, on average, fitter than single-copy haplotypes. When fitness depends on the number of immigrant allele copies and their masking ability then the threshold for invasion depends on the rate of immigration and the rate of recombination between the gene copies. Results from several special cases, including formation of protein dimers and Dobzhansky-Muller incompatibilities, suggest that duplications can invade in a wide range of selection regimes. We hypothesize that duplication in response to gene flow may provide an explanation for the high levels of polymorphism in gene copy number observed in natural populations.     

The Effects of Density Dependence and Immigration on Local Adaptation and Niche Evolution in a Black-Hole Sink Environment

We examine the effects of density dependence and immigration on local adaptation in a "black-hole sink" habitat, i.e., a habitat in which isolated populations of a species would tend to extinction but where a population is demographically maintained by recurrent one-way migration from a separate source habitat in which the species persists. Using a diploid, one-locus model of a discrete-generation sink population maintained by immigration from a fixed source population, we show that a locally favored allele will spread when rare in the sink if the absolute fitness (or, in some cases, the geometric-mean absolute fitness) of heterozygotes with the favored allele is above one in the sink habitat. With density dependence, the criterion for spread can depend on the rate of immigration, because immigration affects local densities and, hence, absolute fitness. Given the successful establishment of a locally favored allele, it will be maintained by a migration-selection balance and the resulting polymorphic population will be sustained deterministically with either stable or unstable dynamics. The densities of stable polymorphic populations tend to exceed densities that would be maintained in the absence of the favored allele. With strong density regulation, spread of the favored allele may destabilize population dynamics. Our analyses show that polymorphic populations which form subsequent to the establishment of favorable alleles have the capacity to persist deterministically without immigration. Finally, we examined the probabilistic rate at which new favored alleles arise and become established in a sink population. Our results suggest that favored alleles are established most readily at intermediate levels of immigration.     

Models of Frequency-Dependent Selection with Mutation from Parental Alleles

Frequency-dependent selection (FDS) remains a common heuristic explanation for the maintenance of genetic variation in natural populations. The pairwise-interaction model (PIM) is a well-studied general model of frequency-dependent selection, which assumes that a genotype’s fitness is a function of within-population intergenotypic interactions. Previous theoretical work indicated that this type of model is able to sustain large numbers of alleles at a single locus when it incorporates recurrent mutation. These studies, however, have ignored the impact of the distribution of fitness effects of new mutations on the dynamics and end results of polymorphism construction. We suggest that a natural way to model mutation would be to assume mutant fitness is related to the fitness of the parental allele, i.e., the existing allele from which the mutant arose. Here we examine the numbers and distributions of fitnesses and alleles produced by construction under the PIM with mutation from parental alleles and the impacts on such measures due to different methods of generating mutant fitnesses. We find that, in comparison with previous results, generating mutants from existing alleles lowers the average number of alleles likely to be observed in a system subject to FDS, but produces polymorphisms that are highly stable and have realistic allele-frequency distributions.     

The estimation of selection coefficients in Afrikaners: Huntington disease, porphyria variegata, and lipoid proteinosis.

The effects of mutation, migration, random drift, and selection on the change in frequency of the alleles associated with Huntington disease, porphyria variegata, and lipoid proteinosis have been assessed in the Afrikaner population of South Africa. Although admixture cannot be completely discounted, it was possible to exclude migration and new mutation as major sources of changes in the frequency of these alleles by limiting analyses to pedigrees descendant from founding families. Calculations which overestimated the possible effect of random drift demonstrated that drift did not account for the observed changes in gene frequencies. Therefore these changes must have been caused by natural selection, and a coefficient of selection was estimated for each trait. For the rare, dominant, deleterious allele associated with Huntington disease, the coefficient of selection was estimated to be .34, indicating that this allele has a selective disadvantage, contrary to some recent studies. For the presumed dominant and probably deleterious allele associated with porphyria variegata, the coefficient of selection lies between .07 and .02. The coefficient of selection for the rare, clinically recessive allele associated with lipoid proteinosis was estimated to be .07. Calculations based on a model system indicate that the observed decrease in allele frequency cannot be explained solely on the basis of selection against the homozygote. Thus, this may be an example of a pleiotropic gene which has a dominant effect in terms of selection even though its known clinical effect is recessive.     

Inference of selection based on temporal genetic differentiation in the study of highly polymorphic multigene families

The co-evolutionary arms race between host immune genes and parasite virulence genes is known as Red Queen dynamics. Temporal fluctuations in allele frequencies, or the 'turnover' of alleles at immune genes, are concordant with predictions of the Red Queen hypothesis. Such observations are often taken as evidence of host-parasite co-evolution. Here, we use computer simulations of the Major Histocompatibility Complex (MHC) of guppies (Poecilia reticulata) to study the turnover rate of alleles (temporal genetic differentiation, G'(ST)). Temporal fluctuations in MHC allele frequencies can be ≥≤order of magnitude larger than changes observed at neutral loci. Although such large fluctuations in the MHC are consistent with Red Queen dynamics, simulations show that other demographic and population genetic processes can account for this observation, these include: (1) overdominant selection, (2) fluctuating population size within a metapopulation, and (3) the number of novel MHC alleles introduced by immigrants when there are multiple duplicated genes. Synergy between these forces combined with migration rate and the effective population size can drive the rapid turnover in MHC alleles. We posit that rapid allelic turnover is an inherent property of highly polymorphic multigene families and that it cannot be taken as evidence of Red Queen dynamics. Furthermore, combining temporal samples in spatial F(ST) outlier analysis may obscure the signal of selection.     

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.      

Effects of environmental heterogeneity on victim-exploiter coevolution

We study victim-exploiter coevolution in a spatially heterogeneous island model. In each species, fitness consequences of between-species interactions are controlled by a single haploid diallelic locus. Our emphasis is on the conditions for the maintenance of genetic variation, the dynamic patterns observed, the extent of local adaptation and genetic differentiation between different demes, and on how different parameters (such as the strength and heterogeneity in selection, migration rates, and the number of sites) affect the dynamic and static behavior of the system. We show that under spatially homogeneous selection the maintenance of genetic variation is possible through asynchronous nonlinear dynamics where the allele frequencies in a majority of demes quickly synchronize but the rest do not. Spatially heterogeneous selection can maintain genetic variation even if migration rates are maximal. This happens in an oscillatory way. Genetic variation is most likely to be maintained at high levels if the heterogeneity in selection is large. If there are some restrictions on migration, genetic variation can be maintained at a stable equilibrium. This behavior is most likely at intermediate migration rates. In this case, the system can exhibit high spatial subdivision as measured by F(ST) values but relatively low local adaptation.     

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.     

The effect of deleterious alleles on adaptation in asexual populations

We calculate the fixation probability of a beneficial allele that arises as the result of a unique mutation in an asexual population that is subject to recurrent deleterious mutation at rate U. Our analysis is an extension of previous works, which make a biologically restrictive assumption that selection against deleterious alleles is stronger than that on the beneficial allele of interest. We show that when selection against deleterious alleles is weak, beneficial alleles that confer a selective advantage that is small relative to U have greatly reduced probabilities of fixation. We discuss the consequences of this effect for the distribution of effects of alleles fixed during adaptation. We show that a selective sweep will increase the fixation probabilities of other beneficial mutations arising during some short interval afterward. We use the calculated fixation probabilities to estimate the expected rate of fitness improvement in an asexual population when beneficial alleles arise continually at some low rate proportional to U. We estimate the rate of mutation that is optimal in the sense that it maximizes this rate of fitness improvement. Again, this analysis relaxes the assumption made previously that selection against deleterious alleles is stronger than on beneficial alleles.     

The relative importance of meiotic gene conversion,selection and mutation pressure,in population genetics and evolution

Disparity in the direction of meiotic gene conversion can change allele frequencies, favouring one allele of a pair in heterozygotes. Equilibrium allele frequencies for large diploid populations are examined by means of equations relating them to meiotic gene conversion, selection and mutation for deleterious recessives, deleterious dominants, and deleterious alleles with no dominance. Using observed conversion parameters from various fungi,Zea mays andDrosophila, it is shown that conversion is generally much more important than mutation pressure and may be of greater or lesser importance than selection, depending on dominance and the strength of selection and conversion forces for the alleles involved.     

Indirect selection of implicit mutation protocols

A hypothesis that mutability evolves to facilitate evolutionary adaptation is dismissed by many biologists. Their skepticism is based on a theoretical expectation that natural selection must minimize mutation rates. That view, in turn, is historically grounded in an intuitive presumption that "the vast majority of mutations are harmful." But such skepticism is surely misplaced. Several highly mutagenic genomic patterns, including simple sequence repeats, and transposable elements, are integrated into an unexpectedly large proportion of functional genetic loci. Because alleles arising within such patterns can retain an intrinsic propensity toward a particular style of mutation, natural selection that favors any such allele can indirectly favor the site's mutability as well. By exploiting patterns that have produced beneficial alleles in the past, indirect selection can encourage mutation within constraints that reduce the probability of deleterious effect, thereby shaping implicit "mutation protocols" that effectively promote evolvability.     

DOES MATE LIMITATION IN SELF‐INCOMPATIBLE SPECIES PROMOTE THE EVOLUTION OF SELFING? THE CASE OF LEAVENWORTHIA ALABAMICA

Genetic diversity at the S‐locus controlling self‐incompatibility (SI) is often high because of negative frequency‐dependent selection. In species with highly patchy spatial distributions, genetic drift can overwhelm balancing selection and cause stochastic loss of S‐alleles. Natural selection may favor the breakdown of SI in populations with few S‐alleles because low S‐allele diversity constrains the seed production of self‐incompatible plants. We estimated S‐allele diversity, effective population sizes, and migration rates in Leavenworthia alabamica, a self‐incompatible mustard species restricted to discrete habitat patches in rocky glades. Patterns of polymorphism were investigated at the S‐locus and 15 neutral microsatellites in three large and three small populations with 100‐fold variation in glade size. Populations on larger glades maintained more S‐alleles, but all populations were estimated to harbor at least 20 S‐alleles, and mate availabilities typically exceeded 0.80, which is consistent with little mate limitation in nature. Estimates of the effective size (N_e) in each population ranged from 600 to 1600, and estimated rates of migration (m) ranged from 3 × 10⁻⁴ to nearly 1 × 10⁻³. According to theoretical models, there is limited opportunity for genetic drift to reduce S‐allele diversity in populations with these attributes. Although pollinators or resources limit seed production in small glades, limited S‐allele diversity does not appear to be a factor promoting the incipient breakdown of SI in populations of this species that were studied.     

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.     

Allele frequency distribution under recurrent selective sweeps

The allele frequency of a neutral variant in a population is pushed either upward or downward by directional selection on a linked beneficial mutation ("selective sweeps"). DNA sequences sampled after the fixation of the beneficial allele thus contain an excess of rare neutral alleles. This study investigates the allele frequency distribution under selective sweep models using analytic approximation and simulation. First, given a single selective sweep at a fixed time, I derive an expression for the sampling probabilities of neutral mutants. This solution can be used to estimate the time of the fixation of a beneficial allele from sequence data. Next, I obtain an approximation to mean allele frequencies under recurrent selective sweeps. Under recurrent sweeps, the frequency spectrum is skewed toward rare alleles. However, the excess of high-frequency derived alleles, previously shown to be a signature of single selective sweeps, disappears with recurrent sweeps. It is shown that, using this approximation and multilocus polymorphism data, genomewide parameters of directional selection can be estimated.     

Linking genetic mechanisms of heterozygosity-fitness correlations to footprints of selection at single loci

Investigations of heterozygosity-fitness correlations (HFCs) are central to the understanding how genetic diversity is maintained in natural populations. Advanced genome-wide approaches will enrich the number of functional loci to be tested. We argue that a combined analysis of the genetic mechanisms of HFCs and selection signals at single loci will allow researchers to better understand the micro-evolutionary basis of HFCs. Different dominance relationships among the alleles at the locus can lead to positive, negative or null HFCs depending on the allele frequency distribution. These scenarios differ in the temporal stability of the HFCs and in the patterns of allele frequency changes over time. Here, we describe a simple theoretical framework that links the analyses of heterozygosity-fitness associations (ecological timescale) with tests for selection signals (evolutionary timescale). Different genomic footprints of selection can be expected for the different underlying genetic mechanisms of HFCs, and this information can be independently used for the classification of HFCs. We suggest that in addition to inbreeding and single-locus overdominant effects also loci under directional selection could play a significant role in the development of heterozygosity-fitness effects in large natural populations under recent or fluctuating ecological changes.     

Premeiotic clusters of mutation and the cost of natural selection

Haldane stated that there is a cost of natural selection for new beneficial alleles to be substituted over time. Most of this cost, which leads to "genetic deaths," is in the early generations of the substitution process when the new allele is low in frequency. It depends on the initial frequency and dominance value, but not the selection coefficient, of the advantageous allele. There have been numerous suggestions on how to reduce the cost for preexisting genetic variation that goes from disadvantageous, or neutral, to advantageous with a change in the environment. However, the cost of natural selection for new alleles that arise by mutation is assumed to be high, based on the assumption that new mutant alleles arise in natural populations as single events [1/(2N) of the total alleles]. However, not all mutant alleles arise as single events. Premeiotic mutations occur frequently in individuals (germinal mosaics), giving rise to multiple copies of identical mutant alleles called a "cluster" (C) with an initial allele frequency of C/(2N) instead of 1/(2N). These clusters of new mutant alleles reduce the cost of natural selection in direct proportion to the relative size of the cluster. Hence new advantageous alleles that arise by mutation have the greatest chance of going to fixation if they occur in large clusters in small populations.     

THE MAINTENANCE OF GENETIC VARIATION FOR OVIPOSITION RATE IN TWO‐SPOTTED SPIDER MITES: INFERENCES FROM ARTIFICIAL SELECTION

Despite the directional selection acting on life‐history traits, substantial amounts of standing variation for these traits have frequently been found. This variation may result from balancing selection (e.g., through genetic trade‐offs) or from mutation‐selection balance. These mechanisms affect allele frequencies in different ways: Under balancing selection alleles are maintained at intermediate frequencies, whereas under mutation‐selection balance variation is generated by deleterious mutations and removed by directional selection, which leads to asymmetry in the distribution of allele frequencies. To investigate the importance of these two mechanisms in maintaining heritable variation in oviposition rate of the two‐spotted spider mite, we analyzed the response to artificial selection. In three replicate experiments, we selected for higher and lower oviposition rate, compared to control lines. A response to selection only occurred in the downward direction. Selection for lower oviposition rate did not lead to an increase in any other component of fitness, but led to a decline in female juvenile survival. The results suggest standing variation for oviposition rate in this population consists largely of deleterious alleles, as in a mutation‐selection balance. Consequently, the standing variation for this trait does not appear to be indicative of its adaptive potential.