Fixation probabilities in a spatially heterogeneous environment

 We consider a simple model of a one-locus, two-allele population inhibiting a two-patch system and experiencing spatially heterogeneous viability selection. The populaton size is finite. We use a diffusion approximation and singular perturbation techniques to find the probability of fixation of a mutant allele. We focus on situations in which each allele is advantageous in one patch and deleterious in the other patch. Our theoretical results support the previous conclusions that, under certain conditions, small populations respond faster to selection than do large populations. We emphasize that knowledge of the dependence of migration rates on population size is crucial in evaluating the effects of population size on the rate of evolution.     

On the Theory of Partially Inbreeding Finite Populations. III. Fixation Probabilities under Partial Selfing When Heterozygotes Are Intermediate in Viability

In a previous paper by the senior author, an approximation to the probability of survival was given for a mutant, which is originally present in a single heterozygote, in a population that reproduces partly by selfing and partly by random mating. The population was assumed to be very large, but the result obtained is general with regard to the level of dominance in viability. In this paper two errors which were made in that earlier work are corrected. A general approximate expression is then derived for the probability that an allele A is fixed in a partially self fertilizing population of size N, if its initial frequency is p, selection is weak and heterozygotes with the allele are exactly intermediate in viability compared with genotypes AA and AA. A rigorous proof is given for a special case that is a generalization of the classical binomial sampling model. In this case, but not in general, the approximate fixation probability is independent of the probability of reproduction by selfing. Some implications are discussed.     

The robustness of Hamilton's rule with inbreeding and dominance: kin selection and fixation probabilities under partial sib mating

Assessing the validity of Hamilton's rule when there is both inbreeding and dominance remains difficult. In this article, we provide a general method based on the direct fitness formalism to address this question. We then apply it to the question of the evolution of altruism among diploid full sibs and among haplodiploid sisters under inbreeding resulting from partial sib mating. In both cases, we find that the allele coding for altruism always increases in frequency if a condition of the form rb>c holds, where r depends on the rate of sib mating alpha but not on the frequency of the allele, its phenotypic effects, or the dominance of these effects. In both examples, we derive expressions for the probability of fixation of an allele coding for altruism; comparing these expressions with simulation results allows us to test various approximations often made in kin selection models (weak selection, large population size, large fecundity). Increasing alpha increases the probability of fixation of recessive altruistic alleles (h<1/2), while it can increase or decrease the probability of fixation of dominant altruistic alleles (h>1/2).     

Fixation probability for a beneficial allele and a mutant strategy in a linear game under weak selection in a finite island model

The effect of population structure on the probability of fixation of a newly introduced mutant under weak selection is studied using a coalescent approach. Wright's island model in a framework of a finite number of demes is assumed and two selection regimes are considered: a beneficial allele model and a linear game among offspring. A first-order approximation of the fixation probability for a single mutant with respect to the intensity of selection is deduced. The approximation requires the calculation of expected coalescence times, under neutrality, for lineages starting from two or three sampled individuals. The results are obtained in a general setting without assumptions on the number of demes, the deme size or the migration rate, which allows for simultaneous coalescence or migration events in the genealogy of the sampled individuals. Comparisons are made with limit cases as the deme size or the number of demes goes to infinity or the migration rate goes to zero for which a diffusion approximation approach is possible. Conditions for selection to favor a mutant strategy replacing a resident strategy in the context of a linear game in a finite island population are addressed.     

ON EVOLUTION UNDER SEXUAL AND VIABILITY SELECTION: A TWO-LOCUS DIPLOID MODEL

A two-locus diploid model of sexual selection is presented in which the two loci govern, respectively, a trait limited in expression in one sex (generally male) and the mating preferences of the other sex (generally female). The viability of a male depends on its genotype at the trait locus. In contrast, all females are equally viable and all individuals are equally fertile with respect to the two loci. Near fixation at both loci, evolution at the mating locus is neutral and hence a new mating preference allele will increase only through random genetic drift or through a correlated response to the increase of a new advantageous trait allele. If, however, a polymorphism is already maintained at the trait locus through overdominance in fitness then the increase of a rare preference allele depends only on the recombination rate between the loci and not on the new preference scheme.     

Recombination and hitchhiking of deleterious alleles

When new advantageous alleles arise and spread within a population, deleterious alleles at neighboring loci can hitchhike alongside them and spread to fixation in areas of low recombination, introducing a fixed mutation load. We use branching processes and diffusion equations to calculate the probability that a deleterious allele hitchhikes and fixes alongside an advantageous mutant. As expected, the probability of fixation of a deleterious hitchhiker rises with the selective advantage of the sweeping allele and declines with the selective disadvantage of the deleterious hitchhiker. We then use computer simulations of a genome with an infinite number of loci to investigate the increase in load after an advantageous mutant is introduced. We show that the appearance of advantageous alleles on genetic backgrounds loaded with deleterious alleles has two potential effects: it can fix deleterious alleles, and it can facilitate the persistence of recombinant lineages that happen to occur. The latter is expected to reduce the signals of selection in the surrounding region. We consider these results in light of human genetic data to infer how likely it is that such deleterious hitchhikers have occurred in our recent evolutionary past.     

Hitchhiking of Deleterious Alleles and the Cost of Adaptation in Partially Selfing Species

Self-fertilization is generally seen to be disadvantageous in the long term. It increases genetic drift, which subsequently reduces polymorphism and the efficiency of selection, which also challenges adaptation. However, high selfing rates can increase the fixation probability of recessive beneficial mutations, but existing theory has generally not accounted for the effect of linked sites. Here, we analyze a model for the fixation probability of deleterious mutants that hitchhike with selective sweeps in diploid, partially selfing populations. Approximate analytical solutions show that, conditional on the sweep not being lost by drift, higher inbreeding rates increase the fixation probability of the deleterious allele, due to the resulting reduction in polymorphism and effective recombination. When extending the analysis to consider a distribution of deleterious alleles, as well as the average fitness increase after a sweep, we find that beneficial alleles generally need to be more recessive than the previously assumed dominance threshold (h < 1/2) for selfing to be beneficial from one-locus theory. Our results highlight that recombination aiding the efficiency of selection on multiple loci amplifies the fitness benefits of outcrossing over selfing, compared to results obtained from one-locus theory. This effect additionally increases the parameter range under which obligate outcrossing is beneficial over partial selfing.     

Sexual selection and the evolution of evolvability

Here we show that sexual selection can have an effect on the rate of mutation. We simulated the fate of a genetic modifier of the mutation rate in a sexual population with and without sexual selection (modelled using a female choice mechanism). Female choice for 'good genes' should reduce variability among male subjects, leaving insufficient differences to maintain female preferences. However, female choice can actually increase genetic variability by supporting a higher mutation rate in sexually selected traits. Increasing the mutation rate will be selected against because of the resulting decline in mean fitness. However, it also increases the probability of rare beneficial mutations arising, and mating skew caused by female preferences for male subjects carrying those beneficials with few deleterious mutations ('good genes') can lead to a mutation rate above that expected under natural selection. A choice of two male subjects was sufficient for there to be a twofold increase in the mutation rate as opposed to a decrease found under random mating.     

The influence of local extinctions on the probability of fixation of chromosomal rearrangements

We investigated the influence of local extinctions in a subdivided population on the probability of fixation of an initially rare allele, for different migration rates. The selective regimes considered were strict underdominance, meiotic drive, and underdominance associated with meiotic drive. We show that local extinctions can increase the probability of fixation of initially rare alleles in underdominant loci for relatively high migration rates, even when both homozygotes have the same fitness. This increase is due to drift during founder events. On the contrary, local extinctions decrease the probability of fixation of alleles favoured by meiotic drive. For a locus where both meiotic drive and underdominance act, the effect of local extinctions depends on the relative strength of the two selective regimes and the initial frequency of the rare allele. For parameter values such that the rare allele is initially selected against, local extinctions decrease the probability of fixation for low migration rates while they cause an increase for moderate migration rates. When the parameter values are such that the rare allele should always be favoured by selection, local extinctions always decrease the probability of fixation. In this latter case, we show the existence of an optimal migration rate which maximizes the probability of fixation.     

Selection and drift in subdivided populations: a straightforward method for deriving diffusion approximations and applications involving dominance, selfing and local extinctions

Population structure affects the relative influence of selection and drift on the change in allele frequencies. Several models have been proposed recently, using diffusion approximations to calculate fixation probabilities, fixation times, and equilibrium properties of subdivided populations. We propose here a simple method to construct diffusion approximations in structured populations; it relies on general expressions for the expectation and variance in allele frequency change over one generation, in terms of partial derivatives of a "fitness function" and probabilities of genetic identity evaluated in a neutral model. In the limit of a very large number of demes, these probabilities can be expressed as functions of average allele frequencies in the metapopulation, provided that coalescence occurs on two different timescales, which is the case in the island model. We then use the method to derive expressions for the probability of fixation of new mutations, as a function of their dominance coefficient, the rate of partial selfing, and the rate of deme extinction. We obtain more precise approximations than those derived by recent work, in particular (but not only) when deme sizes are small. Comparisons with simulations show that the method gives good results as long as migration is stronger than selection.     

Sexual selection when the female directly benefits

Why do females of many species mate with males on the basis of traits apparently detrimental to male survival? The answer may lie in the fact that these male traits are correlated with male condition. We consider the argument that high male condition directly benefits female fecundity and/or viability (e.g. through lower transmission of parasites, improved control of resources, or better paternal care). Using a quantitative genetic model we show how female preferences for male traits that indicate condition can evolve, even if the male traits themselves have deleterious effects on both the male and the female's fecundity. So-called ‘arbitrary preferences’ can spread in this way because male traits subject to sexual selection are often under additional selection to become correlated with condition. At equilibrium the positive effects of male condition on a female's fecundity and the negative effects of the male trait on her fecundity are balanced and the female preference is under stabilizing selection. The male trait will often be correlated with viability, but not with fecundity, even though the preference evolved as a result of differences in male fecundity. The mean fecundity of females is not maximized, and can steadily decline as the male trait and female preference evolve. If the male trait has no direct deleterious effects on female fecundity, as may happen in species with no paternal care, female preferences are under continuous directional selection to increase.     

Evolution under relaxed sexual conflict in the bulb mite Rhizoglyphus robini

The experimental evolution under different levels of sexual conflict have been used to demonstrate antagonistic coevolution in muscids, but among other taxa a similar approach has not been employed. Here, we describe the results of 37 generations of evolution under either experimentally enforced monogamy or polygamy in the bulb mite Rhizoglyphus robini. Three replicates were maintained for each treatment. Monogamy makes male and female interests congruent; thus selection is expected to decrease harmfulness of males to their partners. Our results were consistent with this prediction in that females from monogamous lines achieved lower fecundity when housed with males from polygamous lines. Fecundity of polygamous females was not affected by mating system under which their partners evolved, which suggests that they were more resistant to male-induced harm. As predicted by the antagonistic coevolution hypothesis, the decrease in harmfulness of monogamous males was accompanied by a decline in reproductive competitiveness. In contrast, female fecundity and embryonic viability, which were not expected to be correlated with male harmfulness, did not differ between monogamous and polygamous lines. None of the fitness components assayed differed between individuals obtained from crosses between parents from the same line and those obtained from crosses between parents from different lines within the same mating system. This indicates that inbreeding depression did not confound our results. However, interpretation of our results is complicated by the fact that both males and females from monogamous lines evolved smaller body size compared to individuals from polygamous lines. Although a decrease in reproductive performance of males from monogamous lines was still significant when body size was taken into account, we were not able to separate the effects of male body size and mating system in their influence on fecundity of their female partners.     

Separation of time scales, fixation probabilities and convergence to evolutionary stable states under isolation by distance

To a first order of approximation, selection is frequency independent in a wide range of family structured models and in populations following an island model of dispersal, provided the number of families or demes is large and the population is haploid or diploid but allelic effects on phenotype are semidominant. This result underlies the way the evolutionary stability of traits is computed in games with continuous strategy sets. In this paper similar results are derived under isolation by distance. The first-order effect on expected change in allele frequency is given in terms of a measure of local genetic diversity, and of measures of genetic structure which are almost independent of allele frequency in the total population when the number of demes is large. Hence, when the number of demes increases the response to selection becomes of constant sign. This result holds because the relevant neutral measures of population structure converge to equilibrium at a rate faster than the rate of allele frequency changes in the total population. In the same conditions and in the absence of demographic fluctuations, the results also provide a simple way to compute the fixation probability of mutants affecting various ecological traits, such as sex ratio, dispersal, life-history, or cooperation, under isolation by distance. This result is illustrated and tested against simulations for mutants affecting the dispersal probability under a stepping-stone model.     

Separation of time scales, fixation probabilities and convergence to evolutionarily stable states under isolation by distance

To a first order of approximation, selection is frequency independent in a wide range of family structured models and in populations following an island model of dispersal, provided the number of families or demes is large and the population is haploid or diploid but allelic effects on phenotype are semidominant. This result underlies the way the evolutionary stability of traits is computed in games with continuous strategy sets. In this paper similar results are derived under isolation by distance. The first-order effect on expected change in allele frequency is given in terms of a measure of local genetic diversity, and of measures of genetic structure which are almost independent of allele frequency in the total population when the number of demes is large. Hence, when the number of demes increases the response to selection becomes of constant sign. This result holds because the relevant neutral measures of population structure converge to equilibrium at a rate faster than the rate of allele frequency changes in the total population. In the same conditions and in the absence of demographic fluctuations, the results also provide a simple way to compute the fixation probability of mutants affecting various ecological traits, such as sex ratio, dispersal, life-history, or cooperation, under isolation by distance. This result is illustrated and tested against simulations for mutants affecting the dispersal probability under a stepping-stone model.     

Coevolution of self-fertilization and inbreeding depression. I. Mutation-selection balance at one and two loci

Simple theories for the evolution of breeding systems suggest that the fate of an allele that modifies the rate of self-fertilization hinges only on the degree to which selfing reduces opportunities for outcrossing ("pollen discounting") and the extent of inbreeding depression. These theories predict that outcrossing evolves whenever deleterious mutations have a more severe effect in combination than expected from their individual effects. We study the evolutionary dynamics of a modifier of the rate of self-fertilization in populations subject to complete pollen discounting and recurrent mutations which impair viability at a single locus in diploids and at two loci in haploids. Our analysis indicates that genetic associations arising immediately upon the introduction of a rare modifier allele generate substantial quantitative and qualitative departures from expectation. Higher rates of segregation under selfing in our one-locus diploid model generate positive associations between enhancers of selfing and wild-type viability alleles, which in turn favor the evolution of selfing under a wider range of conditions than expected. Greater opportunities for recombination under outcrossing in our two-locus haploid model generate positive associations between enhancers of outcrossing and wild-type viability alleles. These associations favor the evolution of outcrossing under a wider range of conditions, and introduce the possibility of stable mixed mating systems involving both selfing and outcrossing. Our explicit analysis of genetic associations between loci affecting viability and the rate of self-fertilization indicates that modifiers that enhance the production of offspring with very high (and very low) viability by promoting segregation or recombination develop positive associations with high viability. This advantage of producing extremes can compensate for an initial disadvantage in offspring number.     

Effect of epistasis and linkage on fixation probability in three-locus models: an ancestral recombination-selection graph approach

We study the probability of ultimate fixation of a single new mutant arising in an individual chosen at random at a locus linked to two other loci carrying previously arisen mutations. This is done using the Ancestral Recombination-Selection Graph (ARSG) in a finite population in the limit of a large population size, which is also known as the Ancestral Influence Graph (AIG). An analytical expansion of the fixation probability with respect to population-scaled recombination rates and selection intensities is obtained. The coefficients of the expansion are expressed in terms of the initial state of the population and the epistatic interactions among the selected loci. Under the assumption of weak selection at tightly linked loci, the sign of the leading term, which depends on the signs of epistasis and initial linkage disequilibrium, determines whether an increase in recombination rates increases the chance of ultimate fixation of the new mutant. If mutants are advantageous, this is the case when epistasis is positive or null and the initial linkage disequilibrium is negative, which is an expected state in a finite population under directional selection. Moreover, this is also the case for a neutral mutant modifier coding for higher recombination rates if the same conditions hold at the selected loci. Under the same conditions, deleterious mutants are disfavored for ultimate fixation and neutral modifiers for higher recombination rates still favored. The recombination rates between the modifier locus and the selected loci do not come into play in the leading terms of the approximation for the fixation probability, but they do in higher-order terms.     

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.     

PARASITES AND THE EVOLUTION OF SELF-FERTILIZATION

Abstract.— Assuming all else is equal, an allele for selfing should spread when rare in an outcrossing population and rapidly reach fixation. Such an allele will not spread, however, if self‐fertilization results in inbreeding depression so severe that the fitness of selfed offspring is less that half that of outcrossed offspring. Here we consider an ecological force that may also counter the spread of a selfing allele: coevolution with parasites. Computer simulations were conducted for four different genetic models governing the details of infection. Within each of these models, we varied both the level of selfing in the parasite and the level of male‐gamete discounting in the host (i.e., the reduction in outcrossing fitness through male function due to the selfing allele). We then sought the equilibrium level of host selfing under the different conditions. The results show that, over a wide range of conditions, parasites can select for host reproductive strategies in which both selfed and outcrossed progeny are produced (mixed mating). In addition, mixed mating, where it exits, tends to be biased toward selfing.     

INBREEDING DEPRESSION,GENETIC LOAD,AND THE EVOLUTION OF OUTCROSSING RATES IN A MULTILOCUS SYSTEM WITH NO LINKAGE

We studied deterministic models of multilocus systems subject to mutation–selection balance with all loci unlinked, and with multiplicative interactions of the loci affecting fitness, in partially self-fertilizing populations. The aim was to examine the fitnesses of the zygotes produced by outcrossing and by selling, and the magnitude of inbreeding depression, in populations with different levels of inbreeding. The fates of modifiers of the outcrossing rate were also examined. With biologically plausible parameter values, inbreeding depression can be very large in moderately selfing populations, particularly when the mutant alleles are fairly recessive and selection is weak. A modifier allele reducing the selfing rate can be favored under these circumstances. In more inbred populations, inbreeding depression is lower, and selection favors alleles that increase the selfing rate. When inbreeding depression is caused by mutant alleles with strong selective disadvantage, modifiers causing large increases in selfing can often be favored even when the inbreeding depression exceeds one-half, though in these circumstances modifiers increasing selfing by smaller amounts are usually eliminated. Weaker selection appears to be more favorable to the maintenance of outcrossing.     

The Evolution and Consequences of Sex-Specific Reproductive Variance

Natural selection favors alleles that increase the number of offspring produced by their carriers. But in a world that is inherently uncertain within generations, selection also favors alleles that reduce the variance in the number of offspring produced. If previous studies have established this principle, they have largely ignored fundamental aspects of sexual reproduction and therefore how selection on sex-specific reproductive variance operates. To study the evolution and consequences of sex-specific reproductive variance, we present a population-genetic model of phenotypic evolution in a dioecious population that incorporates previously neglected components of reproductive variance. First, we derive the probability of fixation for mutations that affect male and/or female reproductive phenotypes under sex-specific selection. We find that even in the simplest scenarios, the direction of selection is altered when reproductive variance is taken into account. In particular, previously unaccounted for covariances between the reproductive outputs of different individuals are expected to play a significant role in determining the direction of selection. Then, the probability of fixation is used to develop a stochastic model of joint male and female phenotypic evolution. We find that sex-specific reproductive variance can be responsible for changes in the course of long-term evolution. Finally, the model is applied to an example of parental-care evolution. Overall, our model allows for the evolutionary analysis of social traits in finite and dioecious populations, where interactions can occur within and between sexes under a realistic scenario of reproduction.