Multilocus Population Genetics with Weak Epistasis. II. Equilibrium Properties of Multilocus Models: What Is the Unit of Selection?

Using perturbation techniques, I study the equilibrium of deterministic discrete time multilocus models with weak epistasis. The most important results are on the relationship between epistasis and disequilibrium. Disequilibrium involving a particular set of loci reflects only epistasis simultaneously involving those loci. Moreover, all the disequilibria of all orders vary approximately as the inverse of the probability of at least one recombination event among the loci involved. Finally, higher order disequilibria among loci will be lower than lower order ones, even if the level of epistasis is the same at all orders. In this sense, the unit of selection is small. However, given the larger number of higher order disequilibria, these higher order disequilibria may play an important role in the computation of gametic frequencies from allelic frequencies in models with a large number of loci. Finally, I show that epistasis between blocks of loci will be averages of epistatic effects, not additions of epistatic effects. Thus, failure to find significant epistasis on a chromosomal basis does not rule out the importance of epistatic effects.     

Effect of varying epistasis on the evolution of recombination

Whether recombination decelerates or accelerates a population's response to selection depends, at least in part, on how fitness-determining loci interact. Realistically, all genomes likely contain fitness interactions both with positive and with negative epistasis. Therefore, it is crucial to determine the conditions under which the potential beneficial effects of recombination with negative epistasis prevail over the detrimental effects of recombination with positive epistasis. Here, we examine the simultaneous effects of diverse epistatic interactions with different strengths and signs in a simplified model system with independent pairs of interacting loci and selection acting only on the haploid phase. We find that the average form of epistasis does not predict the average amount of linkage disequilibrium generated or the impact on a recombination modifier when compared to results using the entire distribution of epistatic effects and associated single-mutant effects. Moreover, we show that epistatic interactions of a given strength can produce very different effects, having the greatest impact when selection is weak. In summary, we observe that the evolution of recombination at mutation-selection balance might be driven by a small number of interactions with weak selection rather than by the average epistasis of all interactions. We illustrate this effect with an analysis of published data of Saccharomyces cerevisiae. Thus to draw conclusions on the evolution of recombination from experimental data, it is necessary to consider the distribution of epistatic interactions together with the associated selection coefficients.     

Limits to the Relationship among Recombination, Disequilibrium and Epistasis in Two-Locus Models

I determine limits to the equilibrium relationship among epistasis, recombination and disequilibrium in two-locus, two-allele models using linear programming techniques. I show that when allele frequencies are one-half at each locus, the symmetric model is the fitness pattern that generates the most disequilibrium for the smallest level of epistasis. When allele frequencies deviate from one-half much larger levels of epistasis are required to generate similar levels of disequilibrium. I determine the level of epistasis required to generate observed significant levels of disequilibrium in natural populations. The overall implication is that disequilibrium will be large at equilibrium only between strongly interacting, closely linked loci.     

Linkage Disequilibrium and Genetic Variances under Mutation-Selection Balance

I determine the contribution of linkage disequilibrium to genetic variances using results for two loci and for induced or marginal systems. The analysis allows epistasis and dominance, but assumes that mutation is weak relative to selection. The linkage disequilibrium component of genetic variance is shown to be unimportant for unlinked loci if the gametic mutation rate divided by the harmonic mean of the pairwise recombination rates is much less than one. For tightly linked loci, linkage disequilibrium is unimportant if the gametic mutation rate divided by the (induced) per locus selection is much less than one.     

The evolution of recombination in a heterogeneous environment

Most models describing the evolution of recombination have focused on the case of a single population, implicitly assuming that all individuals are equally likely to mate and that spatial heterogeneity in selection is absent. In these models, the evolution of recombination is driven by linkage disequilibria generated either by epistatic selection or drift. Models based on epistatic selection show that recombination can be favored if epistasis is negative and weak compared to directional selection and if the recombination modifier locus is tightly linked to the selected loci. In this article, we examine the joint effects of spatial heterogeneity in selection and epistasis on the evolution of recombination. In a model with two patches, each subject to different selection regimes, we consider the cases of mutation-selection and migration-selection balance as well as the spread of beneficial alleles. We find that including spatial heterogeneity extends the range of epistasis over which recombination can be favored. Indeed, recombination can be favored without epistasis, with negative and even with positive epistasis depending on environmental circumstances. The selection pressure acting on recombination-modifier loci is often much stronger with spatial heterogeneity, and even loosely linked modifiers and free linkage may evolve. In each case, predicting whether recombination is favored requires knowledge of both the type of environmental heterogeneity and epistasis, as none of these factors alone is sufficient to predict the outcome.     

Recombination and the evolution of coordinated phenotypic expression in a frequency-dependent game

A long standing question in evolutionary biology concerns the maintenance of adaptive combinations of traits in the presence of recombination. This problem may be solved if positive epistasis selects for reducing the rate of recombination between such traits, but this requires sufficiently strong epistasis. Here we use a model that we developed previously to analyze a frequency-dependent strategy game in asexual populations, to study how adaptive combinations of traits may be maintained in the presence of recombination when epistasis is too weak to select for genetic linkage. Previously, in the asexual case, our model demonstrated the evolution of adaptive associations between social foraging strategies and learning rules. We verify that these adaptive associations, which are represented by different two-locus haplotypes, can easily be broken by genetic recombination. We also confirm that a modifier allele that reduces the rate of recombination fails to evolve (due to weak epistasis). However, we find that under the same conditions of weak epistasis, there is an alternative mechanism that allows an association between traits to evolve. This is based on a genetic switch that responds to the presence of one social foraging allele by activating one of the two alternative learning alleles that are carried by all individuals. We suggest that such coordinated phenotypic expression by genetic switches offers a general and robust mechanism for the evolution of adaptive combinations of traits in the presence of recombination.     

Disequilibrium in Two-Locus Mutation-Selection Balance Models

Equilibrium behavior of two-locus mutation-selection balance models is analyzed using perturbation techniques. The classical result of Haldane for one locus is shown to carry over to two loci, if fitnesses are replaced by marginal fitnesses. If the fitness of the double heterozygote is smaller than would be produced by a multiplicative model, as in additive or quantitative fitness models, the disequilibrium is negative--an excess of gametes with one rare allele. In this case the disequilibrium can be as large as one-half its maximum value possible, if the recombination rate is small, not greater than the strength of selection. If the fitness of the double heterozygote is larger than would be produced by a multiplicative model, the disequilibrium is positive, and is very small relative to its maximum value possible, even if the recombination rate is zero.     

Hill–Robertson interference maintained by Red Queen dynamics favours the evolution of sex

Although it is well established theoretically that selective interference among mutations (Hill–Robertson interference) favours meiotic recombination, genomewide mean rates of mutation and strengths of selection appear too low to support this as the mechanism favouring recombination in nature. A possible solution to this discrepancy between theory and observation is that selection is at least intermittently very strong due to the antagonistic coevolution between a host and its parasites. The Red Queen theory posits that such coevolution generates fitness epistasis among loci, which generates negative linkage disequilibrium among beneficial mutations, which in turn favours recombination. This theory has received only limited support. However, Red Queen dynamics without epistasis may provide the ecological conditions that maintain strong and frequent selective interference in finite populations that indirectly selects for recombination. This hypothesis is developed here through the simulation of Red Queen dynamics. This approach required the development of a method to calculate the exact frequencies of multilocus haplotypes after recombination. Simulations show that recombination is favoured by the moderately weak selection of many loci involved in the interaction between a host and its parasites, which results in substitution rates that are compatible with empirical estimates. The model also reproduces the previously reported rapid increase in the rate of outcrossing in Caenorhabditis elegans coevolving with a bacterial pathogen.     

Self-fertilization and the evolution of recombination

In this article, we study the effect of self-fertilization on the evolution of a modifier allele that alters the recombination rate between two selected loci. We consider two different life cycles: under gametophytic selfing, a given proportion of fertilizations involves gametes produced by the same haploid individual, while under sporophytic selfing, a proportion of fertilizations involves gametes produced by the same diploid individual. Under both life cycles, we derive approximations for the change in frequency of the recombination modifier when selection is weak relative to recombination, so that the population reaches a state of quasi-linkage equilibrium. We find that gametophytic selfing increases the range of epistasis under which increased recombination is favored; however, this effect is substantial only for high selfing rates. Moreover, gametophytic selfing affects the relative influence of different components of epistasis (additive x additive, additive x dominance, dominance x dominance) on the evolution of the modifier. Sporophytic selfing has much stronger effects: even a small selfing rate greatly increases the parameter range under which recombination is favored, when there is negative dominance x dominance epistasis. This effect is due to the fact that selfing generates a correlation in homozygosity at linked loci, which is reduced by recombination.     

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.      

Selection for recombination in structured populations

In finite populations, linkage disequilibria generated by the interaction of drift and directional selection (Hill-Robertson effect) can select for sex and recombination, even in the absence of epistasis. Previous models of this process predict very little advantage to recombination in large panmictic populations. In this article we demonstrate that substantial levels of linkage disequilibria can accumulate by drift in the presence of selection in populations of any size, provided that the population is subdivided. We quantify (i) the linkage disequilibrium produced by the interaction of drift and selection during the selective sweep of beneficial alleles at two loci in a subdivided population and (ii) the selection for recombination generated by these disequilibria. We show that, in a population subdivided into n demes of large size N, both the disequilibrium and the selection for recombination are equivalent to that expected in a single population of a size intermediate between the size of each deme (N) and the total size (nN), depending on the rate of migration among demes, m. We also show by simulations that, with small demes, the selection for recombination is stronger than both that expected in an unstructured population (m = 1 - 1/n) and that expected in a set of isolated demes (m = 0). Indeed, migration maintains polymorphisms that would otherwise be lost rapidly from small demes, while population structure maintains enough local stochasticity to generate linkage disequilibria. These effects are also strong enough to overcome the twofold cost of sex under strong selection when sex is initially rare. Overall, our results show that the stochastic theories of the evolution of sex apply to a much broader range of conditions than previously expected.     

Fixation probability in a two-locus model by the ancestral recombination-selection graph

We use the ancestral influence graph (AIG) for a two-locus, two-allele selection model in the limit of a large population size to obtain an analytic approximation for the probability of ultimate fixation of a single mutant allele A. We assume that this new mutant is introduced at a given locus into a finite population in which a previous mutant allele B is already segregating with a wild type at another linked locus. We deduce that the fixation probability increases as the recombination rate increases if allele A is either in positive epistatic interaction with B and allele B is beneficial or in no epistatic interaction with B and then allele A itself is beneficial. This holds at least as long as the recombination fraction and the selection intensity are small enough and the population size is large enough. In particular this confirms the Hill-Robertson effect, which predicts that recombination renders more likely the ultimate fixation of beneficial mutants at different loci in a population in the presence of random genetic drift even in the absence of epistasis. More importantly, we show that this is true from weak negative epistasis to positive epistasis, at least under weak selection. In the case of deleterious mutants, the fixation probability decreases as the recombination rate increases. This supports Muller's ratchet mechanism to explain the accumulation of deleterious mutants in a population lacking recombination.     

Effect on linkage disequilibrium of selection for a quantitative character with epistasis

Summary Selection for a character controlled by additive genes induces linkage disequilibrium which reduces the additive genetic variance usable for further selective gains. Additive x additive epistasis contributes to selection response through development of linkage disequilibrium between interacting loci. To investigate the relative importance of the two effects of linkage disequilibrium, formulae are presented and results are reported of simulations using models involving additive, additive x additive and dominance components. The results suggest that so long as epistatic effects are not large relative to additive effects, and the proportion of pairs of loci which show epistasis is not very high, the predominant effect of linkage disequilibrium will be to reduce the rate of selection response.     

Linkage Disequilibrium in Subdivided Populations

The linkage disequilibrium in a subdivided populaton is shown to be equal to the sum of the average linkage disequilibrium for all subpopulations and the covariance between gene frequencies of the loci concerned. Thus, in a subdivided population the linkage disequilibrium may not be 0 even if the linkage disequilibrium in each subpopulation is 0. If a population is divided into two subpopulations between which migration occurs, the asymptotic rate of approach to linkage equilibrium is equal to either r or 2(m(1) + m(2)) - (m(1) + m(2))(2), whichever is smaller, where r is the recombination value and m(1) and m(2) are the proportions of immigrants in subpopulations 1 and 2, respectively. Thus, if migration rate is high compared with recombination value, the change of linkage disequilibrium in subdivided populations is similar to that of a single random mating population. On the other hand, if migration rate is low, the approach to lnkage equilibrium may be retarded in subdivided populations. If isolated populations begin to exchange genes by migration, linkage disequilibrium may increase temporarily even for neutral loci. If overdominant selection operates and the equilibrium gene frequencies are different in the two subpopulations, a permanent linkage disequilibrium may be produced without epistasis in each subpopulation.     

The evolution of recombination in changing environments

Recombination generates under-represented genotypes by breaking down linkage disequilibrium between genes. Recent analyses have specified the conditions under which recombination is favored. These conditions are surprisingly sensitive to the form of selection and environmental change. This quantification makes it possible to use empirical measurements of critical parameters such as the form of epistasis, the rate of mutation, and the frequency of beneficial sweeps to assess different hypotheses for the evolution of recombination.     

The evolution of sex and recombination in response to abiotic or coevolutionary fluctuations in epistasis

Evolutionary biologists have identified several factors that could explain the widespread phenomena of sex and recombination. One hypothesis is that host-parasite interactions favor sex and recombination because they favor the production of rare genotypes. A problem with many of the early models of this so-called Red Queen hypothesis is that several factors are acting together: directional selection, fluctuating epistasis, and drift. It is thus difficult to identify what exactly is selecting for sex in these models. Is one factor more important than the others or is it the synergistic action of these different factors that really matters? Here we focus on the analysis of a simple model with a single mechanism that might select for sex: fluctuating epistasis. We first analyze the evolution of sex and recombination when the temporal fluctuations are driven by the abiotic environment. We then analyze the evolution of sex and recombination in a two-species coevolutionary model, where directional selection is absent (allele frequencies remain fixed) and temporal variation in epistasis is induced by coevolution with the antagonist species. In both cases we contrast situations with weak and strong selection and derive the evolutionarily stable (ES) recombination rate. The ES recombination rate is most sensitive to the period of the cycles, which in turn depends on the strength of epistasis. In particular, more virulent parasites cause more rapid cycles and consequently increase the ES recombination rate of the host. Although the ES strategy is maximized at an intermediate period, some recombination is favored even when fluctuations are very slow. By contrast, the amplitude of the cycles has no effect on the ES level of sex and recombination, unless sex and recombination are costly, in which case higher-amplitude cycles allow the evolution of higher rates of sex and recombination. In the coevolutionary model, the amount of recombination in the interacting species also has a large effect on the ES, with evolution favoring higher rates of sex and recombination than in the interacting species. In general, the ES recombination rate is less than or equal to the recombination rate that would maximize mean fitness. We also discuss the effect of migration when sex and recombination evolve in a metapopulation. We find that intermediate parasite migration rates maximize the degree of local adaptation of the parasite and lead to a higher ES recombination rate in the host.     

Inferring Selective History from Multilocus Frequency Data: Wright Meets the Hamiltonian

We describe a method to study characteristics of the dynamics of multilocus population genetic models without specifying the form of selection a priori. Our approach consists of specifying initial and final genotypic frequencies (either completely or partially) and then determining the minimum time to go from the initial condition to the final condition according to a continuous time genetic model, with arbitrary constraints on the strength and possibly the form of selection. In analyzing a two-locus, two-allele model with this approach, we show that--so long as r is not much larger than s--substantial linkage disequilibrium can be generated from an initial state of linkage equilibrium in a few hundred generations. We also show that unless recombination is much larger than selection, there is only weak dependence on r of the minimum time to reach a specified state. Thus, similar strengths of selection can lead to similar levels of disequilibrium over a fixed time and a range of small recombination rates. This implies that, within the level of a single gene, selection cannot in general be assumed to lead to any particular relationship between recombination rate and levels of disequilibrium. We indicate a number of other ways in which our method can be useful in asking theoretical questions and in interpreting data.     

Selection on a modifier of recombination rate due to linked deleterious mutations

Several models have been suggested to explain the origin and maintenance of recombination. Here I present the results from computer simulations of multilocus haploid and diploid genotypes in small populations. Each chromosome consisted of 1001 loci where deleterious mutations occurred. At "equilibrium" for mutation-selection-genetic drift balance a single recombination variant was introduced to the population in the middle of a chromosome. On average 75,000 replicates for each combination of parameters were followed to fixation or loss of the modifier allele. The results show that, in a small population, increased recombination can be selected, even in the absence of epistasis or beneficial mutations. The effect of the mutation rate for deleterious mutations depends on the ploidy level and the recessiveness of deleterious mutations. A higher deleterious mutation rate is required for an increase in recombination rate to be favored in haploid populations. Increased recombination could not evolve in the case of strong associative overdominance.     

Meiosis and the evolution of recombination at low mutation rates

The classical understanding of recombination is that in large asexual populations with multiplicative fitness, linkage disequilibrium is negligible, and thus there is no selective agent driving an allele for recombination. This has led researchers to recognize the importance of synergistic epistatic selection in generating negative linkage disequilibrium that thereby renders an advantage to recombination. Yet data on such selection is equivocal, and various works have shown that synergistic epistasis per se, when left unquantified in its magnitude or operation, is not sufficient to drive the evolution of recombination. Here we show that neither it, nor any mechanism generating negative linkage disequilibrium among fitness-related loci, is necessary. We demonstrate that a neutral gene for recombination can increase in frequency in a large population under a low mutation rate and strict multiplicative fitness. We work in a parameter range where individuals have, on average, less than one mutation each, yet recombination can still evolve. We demonstrate this in two ways: first, by examining the consequences of recombination correlated with misrepaired DNA damage and, second, by increasing the probability of recombination with declining fitness. Interestingly, the allele spreads without repairing even a single DNA mutation.     

A general multilocus two-allele model for epistatic selection

Observations show that evolutionary processes often relate to multilocus epistatic selection. Here we develop further the approach suggested earlier by Zhivotovsky and Gavrilets to admit arbitrary multilocus epistasis. The obtained dynamic equations for allelic frequencies and gametic disequilibria are represented in a simple form. If selection is weak, this result extends Wright’s evolutionary equation to the case of cis-trans effects and sex differences in both recombination rates and genotypic fitnesses. Additionally to Wright’s equations for allelic frequencies, we derive equations for the gametic disequilibrium terms. We also give a general expression for the gametic disequilibria in a quasi-linkage state.