On the evolution of epistasis II: a generalized Wright-Kimura framework

The evolution of fitness interactions between genes at two major loci is studied where the alleles at a third locus modify the epistatic interaction between the two major loci. The epistasis is defined by a parameter epsilon and a matrix structure that specifies the nature of the interactions. When epsilon=0 the two major loci have additive fitnesses, and when these are symmetric the interaction matrices studied here produce symmetric viabilities of the Wright [1952. The genetics of quantitative variability. In: Reeve, E.C.R., Waddington, C.H. (Eds.), Quantitative Inheritance. Her Majesty's Stationary Office, London]-Kimura [1956. A model of a genetic system which leads to closer linkage by natural selection. Evolution 10, 278-281] form. Two such interaction matrices are studied, for one of which epistasis as measured by |epsilon| always increases, and for the other it increases when the linkage between the major loci is tight enough and there is initial linkage disequilibrium. Increase of epistasis does not necessarily coincide with increase in equilibrium mean fitness.     

A gene's eye view of epistasis,selection and speciation

In this mini‐review, I discuss the effects of gene interaction or epistasis from a `gene's eye view.' By a `gene's eye view' of epistasis, I mean that I will consider a single, bi‐allelic locus, A , whose effects on fitness result only from its interactions with alleles of another, unknown locus, X . I will show how changes in the frequencies of alleles at the background locus affect the relationship of alleles at the A ‐locus to fitness. Changing the genetic background changes the fundamental characteristics of the A ‐locus, such as the magnitude and sign of allelic effects on fitness, and, consequently, it changes the strength and pattern of selection. I consider each of the four kinds of pair–wise interactions between two loci and show that some kinds of epistasis are more sensitive than others to population genetic subdivision. Lastly, I show that some kinds of epistasis are more likely than others to affect the process of speciation and contribute to or be responsible for general genetic features of interspecific hybrids, such as Haldane's rule.     

The effect of epistasis on sexually antagonistic genetic variation

There is increasing evidence of segregating sexually antagonistic (SA) genetic variation for fitness in laboratory and wild populations, yet the conditions for the maintenance of such variation can be restrictive. Epistatic interactions between genes can contribute to the maintenance of genetic variance in fitness and we suggest that epistasis between SA genes should be pervasive. Here, we explore its effect on SA genetic variation in fitness using a two locus model with negative epistasis. Our results demonstrate that epistasis often increases the parameter space showing polymorphism for SA loci. This is because selection in one locus is affected by allele frequencies at the other, which can act to balance net selection in males and females. Increased linkage between SA loci had more marginal effects. We also show that under some conditions, large portions of the parameter space evolve to a state where male benefit alleles are fixed at one locus and female benefit alleles at the other. This novel effect of epistasis on SA loci, which we term the ‘equity effect’, may have important effects on population differentiation and may contribute to speciation. More generally, these results support the suggestion that epistasis contributes to population divergence.     

A simple model of co-evolutionary dynamics caused by epistatic selection

Epistasis is the dependency of the effect of a mutation on the genetic background in which it occurs. Epistasis has been widely documented and implicated in the evolution of species barriers and the evolution of genetic architecture. Here we propose a simple model to formalize the idea that epistasis can also lead to co-evolutionary patterns in molecular evolution of interacting genes. This model epistasis is represented by the influence of one gene substitution on the fitness rank of the resident allele at another locus. We assume that increasing or decreasing fitness rank occur equally likely. In simulations we show that this form of epistasis leads to co-evolution in the sense that the length of an adaptive walk between interacting loci is highly correlated. This effect is caused by episodes of elevated rate of evolution in both loci simultaneously. We find that the influence of epistasis on these measures of co-evolutionary dynamics is relatively robust to the details of the model. The main factor influencing the correlation in evolutionary rates is the probability that a substitution will have an epistatic effect, but the strength of epistasis or the asymmetry of the initial fitness ranks of the alleles have only a minor effect. We suggest that covariance in rates of evolution among loci could be used to detect epistasis among loci.     

Impact of epistasis and pleiotropy on evolutionary adaptation

Evolutionary adaptation is often likened to climbing a hill or peak. While this process is simple for fitness landscapes where mutations are independent, the interaction between mutations (epistasis) as well as mutations at loci that affect more than one trait (pleiotropy) are crucial in complex and realistic fitness landscapes. We investigate the impact of epistasis and pleiotropy on adaptive evolution by studying the evolution of a population of asexual haploid organisms (haplotypes) in a model of N interacting loci, where each locus interacts with K other loci. We use a quantitative measure of the magnitude of epistatic interactions between substitutions, and find that it is an increasing function of K. When haplotypes adapt at high mutation rates, more epistatic pairs of substitutions are observed on the line of descent than expected. The highest fitness is attained in landscapes with an intermediate amount of ruggedness that balance the higher fitness potential of interacting genes with their concomitant decreased evolvability. Our findings imply that the synergism between loci that interact epistatically is crucial for evolving genetic modules with high fitness, while too much ruggedness stalls the adaptive process.     

Cluster partitions and fitness landscapes of the Drosophila fly microbiome

Journal of Mathematical Biology - The concept of genetic epistasis defines an interaction between two genetic loci as the degree of non-additivity in their phenotypes. A fitness landscape describes...     

Epistasis and Its Contribution to Genetic Variance Components

We present a new parameterization of physiological epistasis that allows the measurement of epistasis separate from its effects on the interaction (epistatic) genetic variance component. Epistasis is the deviation of two-locus genotypic values from the sum of the contributing single-locus genotypic values. This parameterization leads to statistical tests for epistasis given estimates of two-locus genotypic values such as can be obtained from quantitative trait locus studies. The contributions of epistasis to the additive, dominance and interaction genetic variances are specified. Epistasis can make substantial contributions to each of these variance components. This parameterization of epistasis allows general consideration of the role of epistasis in evolution by defining its contribution to the additive genetic variance.     

Epistasis and its relationship to canalization in the RNA virus phi 6

Although deleterious mutations are believed to play a critical role in evolution, assessing their realized effect has been difficult. A key parameter governing the effect of deleterious mutations is the nature of epistasis, the interaction between the mutations. RNA viruses should provide one of the best systems for investigating the nature of epistasis because the high mutation rate allows a thorough investigation of mutational effects and interactions. Nonetheless, previous investigations of RNA viruses by S. Crotty and co-workers and by S. F. Elena have been unable to detect a significant effect of epistasis. Here we provide evidence that positive epistasis is characteristic of deleterious mutations in the RNA bacteriophage phi 6. We estimated the effects of deleterious mutations by performing mutation-accumulation experiments on five viral genotypes of decreasing fitness. We inferred positive epistasis because viral genotypes with low fitness were found to be less sensitive to deleterious mutations. We further examined environmental sensitivity in these genotypes and found that low-fitness genotypes were also less sensitive to environmental perturbations. Our results suggest that even random mutations impact the degree of canalization, the buffering of a phenotype against genetic and environmental perturbations. In addition, our results suggest that genetic and environmental canalization have the same developmental basis and finally that an understanding of the nature of epistasis may first require an understanding of the nature of canalization.     

Sexual and parental antagonism shape genomic architecture

Populations with two sexes are vulnerable to a pair of genetic conflicts: sexual antagonism that can arise when alleles have opposing fitness effects on females and males; and parental antagonism that arises when alleles have opposing fitness effects when maternally and paternally inherited. This paper extends previous theoretical work that found stable linkage disequilibrium (LD) between sexually antagonistic loci. We find that LD is also generated between parentally antagonistic loci, and between sexually and parentally antagonistic loci, without any requirement of epistasis. We contend that the LD in these models arises from the admixture of gene pools subject to different selective histories. We also find that polymorphism maintained by parental antagonism at one locus expands the opportunity for polymorphism at a linked locus experiencing parental or sexual antagonism. Taken together, our results predict the chromosomal clustering of loci that segregate for sexually and parentally antagonistic alleles. Thus, genetic conflict may play a role in the evolution of genomic architecture.     

Perspective: Sign epistasis and genetic constraint on evolutionary trajectories

Epistasis for fitness means that the selective effect of a mutation is conditional on the genetic background in which it appears. Although epistasis is widely observed in nature, our understanding of its consequences for evolution by natural selection remains incomplete. In particular, much attention focuses only on its influence on the instantaneous rate of changes in frequency of selected alleles via epistatic contribution to the additive genetic variance for fitness. Thus, in this framework epistasis only has evolutionary importance if the interacting loci are simultaneously segregating in the population. However, the selective accessibility of mutational trajectories to high fitness genotypes may depend on the genetic background in which novel mutations appear, and this effect is independent of population polymorphism at other loci. Here we explore this second influence of epistasis on evolution by natural selection. We show that it is the consequence of a particular form of epistasis, which we designate sign epistasis. Sign epistasis means that the sign of the fitness effect of a mutation is under epistatic control; thus, such a mutation is beneficial on some genetic backgrounds and deleterious on others. Recent experimental innovations in microbial systems now permit assessment of the fitness effects of individual mutations on multiple genetic backgrounds. We review this literature and identify many examples of sign epistasis, and we suggest that the implications of these results may generalize to other organisms. These theoretical and empirical considerations imply that strong genetic constraint on the selective accessibility of trajectories to high fitness genotypes may exist and suggest specific areas of investigation for future research.     

CYTONUCLEAR INTERACTIONS CAN FAVOR THE EVOLUTION OF GENOMIC IMPRINTING

Interactions between cytoplasmic (generally organelle) and nuclear genomes may be relatively common and could potentially have major fitness consequences. As in the case of within-genome epistasis, this cytonuclear epistasis can favor the evolutionary coadaptation of high-fitness combinations of nuclear and cytoplasmic alleles. Because cytoplasmic factors are generally uniparentally inherited, the cytoplasmic genome is inherited along with only one of the nuclear haplotypes, and therefore, coadaptation is expected to evolve through the interaction of these coinherited (usually maternally inherited) genomes. Here I show that, as a result of this coinheritance of the two genomes, cytonuclear epistasis can favor the evolution of genomic imprinting such that, when the cytoplasmic factor is maternally inherited, selection favors maternal expression of the nuclear locus and when the factor is paternally inherited selection favors paternal expression. Genomic imprinting evolves in this model because it leads to a pattern of gene expression in the nuclear haplotype that is coadapted with (i.e., adaptively coordinated with) gene expression in the coinherited cytoplasmic genome.     

Magnitude and sign epistasis among deleterious mutations in a positive-sense plant RNA virus

How epistatic interactions between mutations determine the genetic architecture of fitness is of central importance in evolution. The study of epistasis is particularly interesting for RNA viruses because of their genomic compactness, lack of genetic redundancy, and apparent low complexity. Moreover, interactions between mutations in viral genomes determine traits such as resistance to antiviral drugs, virulence and host range. In this study we generated 53 Tobacco etch potyvirus genotypes carrying pairs of single-nucleotide substitutions and measured their separated and combined deleterious fitness effects. We found that up to 38% of pairs had significant epistasis for fitness, including both positive and negative deviations from the null hypothesis of multiplicative effects. Interestingly, the sign of epistasis was correlated with viral protein-protein interactions in a model network, being predominantly positive between linked pairs of proteins and negative between unlinked ones. Furthermore, 55% of significant interactions were cases of reciprocal sign epistasis (RSE), indicating that adaptive landscapes for RNA viruses maybe highly rugged. Finally, we found that the magnitude of epistasis correlated negatively with the average effect of mutations. Overall, our results are in good agreement to those previously reported for other viruses and further consolidate the view that positive epistasis is the norm for small and compact genomes that lack genetic robustness.     

Naturally segregating loci exhibit epistasis for fitness

The extent to which gene interaction or epistasis contributes to fitness variation within populations remains poorly understood, despite its importance to a myriad of evolutionary questions. Here, we report a multi-year field study estimating fitness of Mimulus guttatus genetic lines in which pairs of naturally segregating loci exist in an otherwise uniform background. An allele at QTL x5b—a locus originally mapped for its effect on flower size—positively affects survival if combined with one genotype at quantitative trait locus x10a (aa) but has negative effects when combined with the other genotypes (Aa and AA). The viability differences between genotypes parallel phenotypic differences for the time and node at which a plant flowers. Viability is negatively correlated with fecundity across genotypes, indicating antagonistic pleiotropy for fitness components. This trade-off reduces the genetic variance for total fitness relative to the individual fitness components and thus may serve to maintain variation. Additionally, we find that the effects of each locus and their interaction often vary with the environment.     

The rank ordering of genotypic fitness values predicts genetic constraint on natural selection on landscapes lacking sign epistasis

Sewall Wright's genotypic fitness landscape makes explicit one mechanism by which epistasis for fitness can constrain evolution by natural selection. Wright distinguished between landscapes possessing multiple fitness peaks and those with only a single peak and emphasized that the former class imposes substantially greater constraint on natural selection. Here I present novel formalism that more finely partitions the universe of possible fitness landscapes on the basis of the rank ordering of their genotypic fitness values. In this report I focus on fitness landscapes lacking sign epistasis (i.e., landscapes that lack mutations the sign of whose fitness effect varies epistatically), which constitute a subset of Wright's single peaked landscapes. More than one fitness rank ordering lacking sign epistasis exists for L > 2 (where L is the number of interacting loci), and I find that a highly statistically significant effect exists between landscape membership in fitness rank-ordering partition and two different proxies for genetic constraint, even within this subset of landscapes. This statistical association is robust to population size, permitting general inferences about some of the characteristics of fitness rank orderings responsible for genetic constraint on natural selection.     

The causes of epistasis

Since Bateson's discovery that genes can suppress the phenotypic effects of other genes, gene interactions-called epistasis-have been the topic of a vast research effort. Systems and developmental biologists study epistasis to understand the genotype-phenotype map, whereas evolutionary biologists recognize the fundamental importance of epistasis for evolution. Depending on its form, epistasis may lead to divergence and speciation, provide evolutionary benefits to sex and affect the robustness and evolvability of organisms. That epistasis can itself be shaped by evolution has only recently been realized. Here, we review the empirical pattern of epistasis, and some of the factors that may affect the form and extent of epistasis. Based on their divergent consequences, we distinguish between interactions with or without mean effect, and those affecting the magnitude of fitness effects or their sign. Empirical work has begun to quantify epistasis in multiple dimensions in the context of metabolic and fitness landscape models. We discuss possible proximate causes (such as protein function and metabolic networks) and ultimate factors (including mutation, recombination, and the importance of natural selection and genetic drift). We conclude that, in general, pleiotropy is an important prerequisite for epistasis, and that epistasis may evolve as an adaptive or intrinsic consequence of changes in genetic robustness and evolvability.     

Gene interactions from maternal effects

Theoretical analyses have demonstrated a potential role for epistasis in many of the most important processes in evolution. These analyses generally assume that an individual's genes map directly to its phenotype and epistasis results from interactions among loci that contribute to the same biochemical or developmental pathways (termed physiological, or within-genotype, epistasis). For many characters, particularly those expressed early in life, an individual's phenotype may also be affected by genes expressed by its parents. The presence of these parental effects allows for interactions between the genes present in the parental and offspring genomes. When the phenotypic effect of a locus in the offspring depends on the alleles possessed by its parents, genotype-by-genotype, or among-genotype, epistasis occurs. The among-genotype epistasis resulting from parental effects may contribute to ruggedness of adaptive landscapes because early mortality often accounts for much of the variance in fitness in populations. To demonstrate how parent-offspring interactions can result in among-genotype epistasis, I use a two-locus model, with one maternal effect locus and one direct effect locus, each with two alleles. Dynamical equations are presented for the two-locus model and are directly contrasted with the dynamical equations derived for a model for physiological epistasis. The relationship between the evolutionary dynamics resulting from these two forms of epistasis is discussed. Three scenarios are presented to illustrate systems in which maternal-offspring, genotype-by-genotype epistasis may occur. The implications of maternal-offspring epistasis for quantitative-trait-loci studies are also discussed.     

Reciprocal sign epistasis between frequently experimentally evolved adaptive mutations causes a rugged fitness landscape

The fitness landscape captures the relationship between genotype and evolutionary fitness and is a pervasive metaphor used to describe the possible evolutionary trajectories of adaptation. However, little is known about the actual shape of fitness landscapes, including whether valleys of low fitness create local fitness optima, acting as barriers to adaptive change. Here we provide evidence of a rugged molecular fitness landscape arising during an evolution experiment in an asexual population of Saccharomyces cerevisiae. We identify the mutations that arose during the evolution using whole-genome sequencing and use competitive fitness assays to describe the mutations individually responsible for adaptation. In addition, we find that a fitness valley between two adaptive mutations in the genes MTH1 and HXT6/HXT7 is caused by reciprocal sign epistasis, where the fitness cost of the double mutant prohibits the two mutations from being selected in the same genetic background. The constraint enforced by reciprocal sign epistasis causes the mutations to remain mutually exclusive during the experiment, even though adaptive mutations in these two genes occur several times in independent lineages during the experiment. Our results show that epistasis plays a key role during adaptation and that inter-genic interactions can act as barriers between adaptive solutions. These results also provide a new interpretation on the classic Dobzhansky-Muller model of reproductive isolation and display some surprising parallels with mutations in genes often associated with tumors.     

Detecting epistasis from an ensemble of adapting populations

The role that epistasis plays during adaptation remains an outstanding problem, which has received considerable attention in recent years. Most of the recent empirical studies are based on ensembles of replicate populations that adapt in a fixed, laboratory controlled condition. Researchers often seek to infer the presence and form of epistasis in the fitness landscape from the time evolution of various statistics averaged across the ensemble of populations. Here, we provide a rigorous analysis of what quantities, drawn from time series of such ensembles, can be used to infer epistasis for populations evolving under weak mutation on finite‐site fitness landscapes. First, we analyze the mean fitness trajectory—that is, the time course of the ensemble average fitness. We show that for any epistatic fitness landscape and starting genotype, there always exists a non‐epistatic fitness landscape that produces the exact same mean fitness trajectory. Thus, the presence of epistasis is not identifiable from the mean fitness trajectory. By contrast, we show that two other ensemble statistics—the time evolution of the fitness variance across populations, and the time evolution of the mean number of substitutions—can detect certain forms of epistasis in the underlying fitness landscape.     

On the evolution of epistasis III: the haploid case with mutation

Whether interaction between genes is better represented by synergistic or antagonistic epistasis has been a focus of experimental research in bacterial population genetics. Our previous research on evolution of modifiers of epistasis in diploid systems has indicated that the strength of positive or negative epistasis should increase provided linkage disequilibrium is maintained. Here we study a modifier of epistasis in fitness between two loci in a haploid system. Epistasis is modified in the neighborhood of a mutation-selection balance. We show that when linkage in the three-locus system is tight, an increase in the frequency of a modifier allele that induces either more negative or more positive epistasis is possible. Epistasis here can be measured on either an additive or multiplicative scale.     

GxG epistasis in growth and condition and the maintenance of genetic polymorphism in Gambusia holbrooki

Theory on indirect genetic effects (IGEs) indicates that variation in the genetic composition of social groups can generate GxG epistasis that may promote the evolution of stable polymorphisms. Using a livebearing fish with a genetic polymorphism in coloration and associated behavioral differences, we tested whether genotypes of social partners interacted with focal individual genotypes to influence growth and condition over 16 weeks of development. We found that IGEs had a significant influence on patterns of feeding, regardless of focal fish genotype. There was no influence of social environment on juvenile length, but there was significant GxG epistasis for body condition. Each focal juvenile was in better condition when its own genotype was not present in adult social partners. These data are consistent with negative frequency‐dependent selection in which each morph performs better when it is rare. Neither variation in feeding nor activity‐related behaviors explained variation in body condition, suggesting that GxG epistasis for condition was caused by physiological differences between the two genotypes. These findings indicate that GxG epistasis in a given polymorphism can generate fitness landscapes that contribute to the maintenance of that polymorphism and to maintenance of genetic variation for additional fitness‐related traits.