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.     

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.     

Evolution of recombination due to random drift

In finite populations subject to selection, genetic drift generates negative linkage disequilibrium, on average, even if selection acts independently (i.e., multiplicatively) upon all loci. Negative disequilibrium reduces the variance in fitness and hence, by Fisher's (1930) fundamental theorem, slows the rate of increase in mean fitness. Modifiers that increase recombination eliminate the negative disequilibria that impede selection and consequently increase in frequency by "hitchhiking." Thus, stochastic fluctuations in linkage disequilibrium in finite populations favor the evolution of increased rates of recombination, even in the absence of epistatic interactions among loci and even when disequilibrium is initially absent. The method developed within this article allows us to quantify the strength of selection acting on a modifier allele that increases recombination in a finite population. The analysis indicates that stochastically generated linkage disequilibria do select for increased recombination, a result that is confirmed by Monte Carlo simulations. Selection for a modifier that increases recombination is highest when linkage among loci is tight, when beneficial alleles rise from low to high frequency, and when the population size is small.     

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.     

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.     

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.     

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.     

The evolutionary advantage of fitness‐dependent recombination in diploids: A deterministic mutation–selection balance model

Recombination's omnipresence in nature is one of the most intriguing problems in evolutionary biology. The question of why recombination exhibits certain general features is no less interesting than that of why it exists at all. One such feature is recombination's fitness dependence (FD). The so far developed population genetics models have focused on the evolution of FD recombination mainly in haploids, although the empirical evidence for this phenomenon comes mostly from diploids. Using numerical analysis of modifier models for infinite panmictic populations, we show here that FD recombination can be evolutionarily advantageous in diploids subjected to purifying selection. We ascribe this advantage to the differential rate of disruption of lower‐ versus higher‐fitness genotypes, which can be manifested in selected systems with at least three loci. We also show that if the modifier is linked to such selected system, it can additionally benefit from modifying this linkage in a fitness‐dependent manner. The revealed evolutionary advantage of FD recombination appeared robust to crossover interference within the selected system, either positive or negative. Remarkably, FD recombination was often favored in situations where any constant nonzero recombination was evolutionarily disfavored, implying a relaxation of the rather strict constraints on major parameters (e.g., selection intensity and epistasis) required for the evolutionary advantage of nonzero recombination formulated by classical models.     

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.     

Ethnic variation in genetic disease: possible roles of hitchhiking and epistasis.

The high incidence of some genetic diseases in certain ethnic groups is important in planning of medical genetic programs. Simple interaction models predict that at least some lethal recessive alleles will have "hitchhiked" to increased frequencies because of linkage to genes whose alleles have been favored by selection for other reasons in certain populations. In the absence of linkage or epistasis with a gene favored by selection, heterozygote advantage for a recessive lethal may produce the same phenomenon. In the hitchhiking model (linkage), the increase in the gene frequency is temporary, but the length of time that the increased gene frequency is at least double the base frequency may be quite long. Changes in gene frequency for the unlinked epistatic model result in a new equilibrium with a possibly higher gene frequency. The most likely chromosomal regions in which hitchhiked lethal recessives would be found are in the vicinity of genes whose allelic frequencies vary substantially among human racial groups (e.g., Gm, Rh, Duffy, lactose tolerance, or HL-A). There will be a hitchhiking effect if recombination distance is less than the selective advantage. The closer the linkage of two loci, the easier hitchhiking effects will be to detect. Hitchhiking is suggested by nonrandom association of the recessive disease and one of the selected markers, as in the case of Gm and cystic fibrosis. However, there is so far insufficient evidence of linkage between them. More pedigree information is necessary than is now available.     

Lewontin and Kojima Meet Fisher: Linkage in a Symmetric Model of Sex Determination

The effect of linkage and epistasis on the evolution of the sex-ratio is studied in a symmetric two-locus model of autosomal sex determination closely related to the symmetric viability model of R. C. Lewontin and K. Kojima. R. A. Fisher's expectation of an even sex ratio for autosomal sex determination by a single gene governs the dynamics when the loci are tightly linked. However, recombination may preclude optimization of the sex ratio just as occurs in viability selection models. Many of the evolutionary phenomena known for the symmetric viability model also occur here. In addition, we exhibit a series of new phenomena related to the presence of surfaces of even sex ratio.     

Evolution of genetic architecture under directional selection

We investigate the multilinear epistatic model under mutation-limited directional selection. We confirm previous results that only directional epistasis, in which genes on average reinforce or diminish each other's effects, contribute to the initial evolution of mutational effects. Thus, either canalization or decanalization can occur under directional selection, depending on whether positive or negative epistasis is prevalent. We then focus on the evolution of the epistatic coefficients themselves. In the absence of higher-order epistasis, positive pairwise epistasis will tend to weaken relative to additive effects, while negative pairwise epistasis will tend to become strengthened. Positive third-order epistasis will counteract these effects, while negative third-order epistasis will reinforce them. More generally, gene interactions of all orders have an inherent tendency for negative changes under directional selection, which can only be modified by higher-order directional epistasis. We identify three types of nonadditive quasi-equilibrium architectures that, although not strictly stable, can be maintained for an extended time: (1) nondirectional epistatic architectures; (2) canalized architectures with strong epistasis; and (3) near-additive architectures in which additive effects keep increasing relative to epistasis.     

SELECTION FOR RECOMBINATION IN SMALL POPULATIONS

Abstract The reasons that sex and recombination are so widespread remain elusive. One popular hypothesis is that sex and recombination promote adaptation to a changing environment. The strongest evidence that increased recombination may evolve because recombination promotes adaptation comes from artificially selected populations. Recombination rates have been found to increase as a correlated response to selection on traits unrelated to recombination in several artificial selection experiments and in a comparison of domesticated and nondomesticated mammals. There are, however, several alternative explanations for the increase in recombination in such populations, including two different evolutionary explanations. The first is that the form of selection is epistatic, generating linkage disequilibria among selected loci, which can indirectly favor modifier alleles that increase recombination. The second is that random genetic drift in selected populations tends to generate disequilibria such that beneficial alleles are often found in different individuals; modifier alleles that increase recombination can bring together such favorable alleles and thus may be found in individuals with greater fitness. In this paper, we compare the evolutionary forces acting on recombination in finite populations subject to strong selection. To our surprise, we found that drift accounted for the majority of selection for increased recombination observed in simulations of small to moderately large populations, suggesting that, unless selected populations are large, epistasis plays a secondary role in the evolution of recombination.     

QUANTITATIVE GENETIC VARIANCE MAINTAINED BY FLUCTUATING SELECTION WITH OVERLAPPING GENERATIONS: VARIANCE COMPONENTS AND COVARIANCES

The quantitative genetic variance-covariance that can be maintained in a random environment is studied, assuming overlapping generations and Gaussian stabilizing selection with a fluctuating optimum. The phenotype of an individual is assumed to be determined by additive contributions from each locus on paternal and maternal gametes (i.e., no epistasis and no dominance). Recurrent mutation is ignored, but linkage between loci is arbitrary. The genotype distribution in the evolutionarily stable population is generically discrete: only a finite number of polymorphic alleles with distinctly different effects are maintained, even though we allow a continuum of alleles with arbitrary phenotypic contributions to invade. Fluctuating selection maintains nonzero genetic variance in the evolutionarily stable population if the environmental heterogeneity is larger than a certain threshold. Explicit asymptotic expressions for the standing variance-covariance components are derived for the population near the threshold, or for large generational overlap, as a function of environmental variability and genetic parameters (i.e., number of loci, recombination rate, etc.), using the fact that the genotype distribution is discrete. Above the threshold, the population maintains considerable genetic variance in the form of positive linkage disequilibrium and positive gamete covariance (Hardy-Weinberg disequilibrium) as well as allelic variance. The relative proportion of these disequilibrium variances in the total genetic variance increases with the environmental variability.     

Recombination Modification in a Fluctuating Environment

This paper examines the theory of the evolution of increased recombination between two loci subjected to interactive selection in a temporally fluctuating environment. Both cyclical and stochastic environments are considered. It is shown that temporal variation in the linkage disequilibrium coefficient for the pair of selected loci, due to fluctuations in the selective values of the genotypes at these loci, can give rise to selection in favor of modifier genes increasing recombination. The equilibrium level of recombination established in a given population depends on several factors; it is highest for intermediate values of the environmental periodicity or autocorrelation, for cases when the modifier genes are themselves linked to the selected loci, and for high levels of environmental variation. In general, it seems that the rate of modification of recombination values by this process will be low except when the modifiers are tightly linked to the selected loci. The possible evolutionary significance of this process is discussed in relation to observations on genetic systems of plants and animals.     

DELETERIOUS MUTATION AND THE EVOLUTION OF GENETIC LIFE CYCLES

It is often proposed that the ability of diploids to mask deleterious mutations leads to an evolutionary advantage over haploidy. In this paper, we studied the evolution of the relative duration of haploid and diploid phases using a model of recurrent deleterious mutations across the entire genome. We found that a completely diploid life cycle is favored under biologically reasonable conditions, even when prolonging the diploid phase reduces a population's mean fitness. A haploid cycle is favored when there is complete linkage throughout the genome or when mutations are either highly deleterious or partially dominant. These results hold when loci interact multiplicatively and for synergistic epistasis. The strength of selection generated on the life cycle can be substantial because of the cumulative effect of selection against mutations across many loci. We did not find conditions that support cycles that retain both phases, such as those found in some plants and algae. Thus, selection against deleterious mutations may be an important force in the evolution of life cycles but may not be sufficient to explain all the patterns of life cycles seen in nature.     

Detecting linkage for genetically heterogeneous diseases and detecting heterogeneity with linkage data.

Interest in searching for genetic linkage between diseases and marker loci has been greatly increased by the recent introduction of DNA polymorphisms. However, even for the most well-behaved Mendelian disorders, those with clear-cut mode of inheritance, complete penetrance, and no phenocopies, genetic heterogeneity may exist; that is, in the population there may be more than one locus that can determine the disease, and these loci may not be linked. In such cases, two questions arise: (1) What sample size is necessary to detect linkage for a genetically heterogeneous disease? (2) What sample size is necessary to detect heterogeneity given linkage between a disease and a marker locus? We have answered these questions for the most important types of matings under specified conditions: linkage phase known or unknown, number of alleles involved in the cross at the marker locus, and different numbers of affected and unaffected children. In general, the presence of heterogeneity increases the recombination value at which lod scores peak, by an amount that increases with the degree of heterogeneity. There is a corresponding increase in the number of families necessary to establish linkage. For the specific case of backcrosses between disease and marker loci with two alleles, linkage can be detected at recombination fractions up to 20% with reasonable numbers of families, even if only half the families carry the disease locus linked to the marker. The task is easier if more than two informative children are available or if phase is known. For recessive diseases, highly polymorphic markers with four different alleles in the parents greatly reduce the number of families required.     

The role of epistatic gene interactions in the response to selection and the evolution of evolvability

It has been argued that the architecture of the genotype-phenotype map determines evolvability, but few studies have attempted to quantify these effects. In this article we use the multilinear epistatic model to study the effects of different forms of epistasis on the response to directional selection. We derive an analytical prediction for the change in the additive genetic variance, and use individual-based simulations to understand the dynamics of evolvability and the evolution of genetic architecture. This shows that the major determinant for the evolution of the additive variance, and thus the evolvability, is directional epistasis. Positive directional epistasis leads to an acceleration of evolvability, while negative directional epistasis leads to canalization. In contrast, pure non-directional epistasis has little effect on the response to selection. One consequence of this is that the classical epistatic variance components, which do not distinguish directional and non-directional effects, are useless as predictors of evolutionary dynamics. The build-up of linkage disequilibrium also has negligible effects. We argue that directional epistasis is likely to have major effects on evolutionary dynamics and should be the focus of empirical studies of epistasis.     

Coalescent under the evolution of coadaptation

Adaptation to novel environments arises either from new beneficial mutations or by utilizing pre‐existing genetic variation. When standing variation is used as the source of new adaptation, fitness effects of alleles may be altered through an environmental change. Alternatively, changes in epistatic genetic backgrounds may convert formerly neutral mutations into beneficial alleles in the new genetic background. By extending the coalescent theory to describe the genealogical histories of two interacting loci, I here investigated the hitchhiking effect of epistatic selection on the amount and pattern of sequence diversity at the linked neutral regions. Assuming a specific form of epistasis between two new mutations that are independently neutral, but together form a coadapted haplotype, I demonstrate that the footprints of epistatic selection differ markedly between the interacting loci depending on the order and relative timing of the two mutational events, even though both mutations are equally essential for the formation of an adaptive gene combination. Our results imply that even when neutrality tests could detect just a single instance of adaptive substitution, there may, in fact, be numerous other hidden mutations that are left undetected, but still play indispensable roles in the evolution of a new adaptation. We expect that the integration of the coalescent framework into the general theory of polygenic inheritance would clarify the connection between factors driving phenotypic evolution and their consequences on underlying DNA sequence changes, which should further illuminate the evolutionary foundation of coadapted systems.     

Epistasis between deleterious mutations and the evolution of recombination

Epistasis and the evolution of recombination are closely intertwined: epistasis generates linkage disequilibria (i.e. statistical associations between alleles), whereas recombination breaks them up. The mutational deterministic hypothesis (MDH) states that high recombination rates are maintained because the breaking up of linkage disequilibria generated by negative epistasis enables more efficient purging of deleterious mutations. However, recent theoretical and experimental work challenges the MDH. Experimental evidence suggests that negative epistasis, required by the MDH, is relatively uncommon. On the theoretical side, population genetic models suggest that, compared with the combined effects of drift and selection, epistasis generates a negligible amount of linkage disequilibria. Here, we assess these criticisms and discuss to what extent they invalidate the MDH as an explanation for the evolution of recombination.