Genome-Wide Scoring of Positive and Negative Epistasis through Decomposition of Quantitative Genetic Interaction Fitness Matrices

Recent technological developments in genetic screening approaches have offered the means to start exploring quantitative genotype-phenotype relationships on a large-scale. What remains unclear is the extent to which the quantitative genetic interaction datasets can distinguish the broad spectrum of interaction classes, as compared to existing information on mutation pairs associated with both positive and negative interactions, and whether the scoring of varying degrees of such epistatic effects could be improved by computational means. To address these questions, we introduce here a computational approach for improving the quantitative discrimination power encoded in the genetic interaction screening data. Our matrix approximation model decomposes the original double-mutant fitness matrix into separate components, representing variability across the array and query mutants, which can be utilized for estimating and correcting the single-mutant fitness effects, respectively. When applied to three large-scale quantitative interaction datasets in yeast, we could improve the accuracy of scoring various interaction classes beyond that obtained with the original fitness data, especially in synthetic genetic array (SGA) and in genetic interaction mapping (GIM) datasets. In addition to the known pairs of interactions used in the evaluation of the computational approach, a number of novel interaction pairs were also predicted, along with underlying biological mechanisms, which remained undetected by the original datasets. It was shown that the optimal choice of the scoring function depends heavily on the screening approach and on the interaction class under analysis. Moreover, a simple preprocessing of the fitness matrix could further enhance the discrimination power of the epistatic miniarray profiling (E-MAP) dataset. These systematic evaluation results provide in-depth information on the optimal analysis of the future, large-scale screening experiments. In general, the modeling framework, enabling accurate identification and classification of genetic interactions, provides a solid basis for completing and mining the genetic interaction networks in yeast and other organisms.     

The generation and maintenance of genetic variation by frequency-dependent selection: constructing polymorphisms under the pairwise interaction model

Frequency-dependent selection remains the most commonly invoked heuristic explanation for the maintenance of genetic variation. For polymorphism to exist, new alleles must be both generated and maintained in the population. Here we use a construction approach to model frequency-dependent selection with mutation under the pairwise interaction model. The pairwise interaction model is a general model of frequency-dependent selection at the genotypic level. We find that frequency-dependent selection is able to generate a large number of alleles at a single locus. The construction process generates multiallelic polymorphisms with a wide range of allele-frequency distributions and genotypic fitness relationships. Levels of polymorphism and mean fitness are uncoupled, so constructed polymorphisms remain permanently invasible to new mutants; thus the model never settles down to an equilibrium state. Analysis of constructed fitness sets reveals signatures of heterozygote advantage and positive frequency dependence.     

Frequency-dependent selection with dominance: a window onto the behavior of the mean fitness

Selection in which fitnesses vary with the changing genetic composition of the population may facilitate the maintenance of genetic diversity in a wide range of organisms. Here, a detailed theoretical investigation is made of a frequency-dependent selection model, in which fitnesses are based on pairwise interactions between the two phenotypes at a diploid, diallelic, autosomal locus with complete dominance. The allele frequency dynamics are fully delimited analytically, along with all possible shapes of the mean fitness function in terms of where it increases or decreases as a function of the current allele frequency in the population. These results in turn allow possibly the first complete characterization of the dynamical behavior by the mean fitness through time under frequency-dependent selection. Here the mean fitness (i) monotonically increases, (ii) monotonically decreases, (iii) initially increases and then decreases, or (iv) initially decreases and then increases as equilibrium is approached. We analytically derive the exact initial and fitness conditions that produce each dynamic and how often each arises. Computer simulations with random initial conditions and fitnesses reveal that the potential decline in mean fitness is not negligible; on average a net decrease occurs 20% of the time and reduces the mean fitness by >17%.     

A simple completion of Fisher’s fundamental theorem of natural selection

Fisher''s fundamental theorem of natural selection shows that the part of the rate of change of mean fitness that is due to natural selection equals the additive genetic variance in fitness. Fisher embedded this result in a model of total fitness, adding terms for deterioration of the environment and density dependence. Here, a quantitative genetic version of this neglected model is derived that relaxes its assumptions that the additive genetic variance in fitness and the rate of deterioration of the environment do not change over time, allows population size to vary, and includes an input of mutational variance. The resulting formula for total rate of change in mean fitness contains two terms more than Fisher''s original, representing the effects of stabilizing selection, on the one hand, and of mutational variance, on the other, making clear for the first time that the fundamental theorem deals only with natural selection that is directional (as opposed to stabilizing) on the underlying traits. In this model, the total (rather than just the additive) genetic variance increases mean fitness. The unstructured population allows an explanation of Fisher''s concept of fitness as simply birth rate minus mortality rate, and building up to the definition in structured populations.     

Maximization principles for frequency-dependent selection II: the one-locus multiallele case

In this article we study a single-locus multiallele version of the pairwise-interaction model (PIM) in discrete and continuous time and a density-dependent version of this model (D-PIM) in continuous time. The PIM assumes that the fitnesses of genotypes are proportional to the average amount of competition resulting from pairwise interactions. Hence, fitness is frequency dependent. Our main aim is to provide necessary and sufficient conditions for the validity of maximization principles analogous to Fisher’s Fundamental Theorem for constant selection. We provide a systematic analysis and illustrate our results by concrete examples. We show that in discrete time the mean fitness is nondecreasing along every trajectory provided the interaction coefficients are nonnegative and symmetric. For asymmetric interactions this is in general not true. However, for what we call pseudo-symmetric interactions a function similar to, but in general not identical to, the mean fitness: the adjusted-mean fitness, is nondecreasing along trajectories. For asymmetric interactions, we also provide sufficient conditions for the mean fitness, and more generally for the adjusted-mean fitness, to be nondecreasing and sufficient conditions when it is not. In continuous time, we provide similar but stronger results. If the interaction coefficients are pseudo-symmetric, the adjusted-mean fitness is nondecreasing in the D-PIM.     

Unexpected positive and negative effects of continuing inbreeding in one of the world's most inbred wild animals

Inbreeding depression, the reduced fitness of offspring of related individuals, is a central theme in evolutionary biology. Inbreeding effects are influenced by the genetic makeup of a population, which is driven by any history of genetic bottlenecks and genetic drift. The Chatham Island black robin represents a case of extreme inbreeding following two severe population bottlenecks. We tested whether inbreeding measured by a 20‐year pedigree predicted variation in fitness among individuals, despite the high mean level of inbreeding and low genetic diversity in this species. We found that paternal and maternal inbreeding reduced fledgling survival and individual inbreeding reduced juvenile survival, indicating that inbreeding depression affects even this highly inbred population. Close inbreeding also reduced survival for fledglings with less‐inbred mothers, but unexpectedly improved survival for fledglings with highly inbred mothers. This counterintuitive interaction could not be explained by various potentially confounding variables. We propose a genetic mechanism, whereby a highly inbred chick with a highly inbred parent inherits a “proven” genotype and thus experiences a fitness advantage, which could explain the interaction. The positive and negative effects we found emphasize that continuing inbreeding can have important effects on individual fitness, even in populations that are already highly inbred.     

Redistribution of gene frequency and changes of genetic variation following a bottleneck in population size

Although the distribution of frequencies of genes influencing quantitative traits is important to our understanding of their genetic basis and their evolution, direct information from laboratory experiments is very limited. In theory, different models of selection and mutation generate different predictions of frequency distributions. When a large population at mutation-selection balance passes through a rapid bottleneck in size, the frequency distribution of genes is dramatically altered, causing changes in observable quantities such as the mean and variance of quantitative traits. We investigate the gene frequency distribution of a population at mutation-selection balance under a joint-effect model of real stabilizing and pleiotropic selection and its redistribution and thus changes of the genetic properties of metric and fitness traits after the population passes a rapid bottleneck and expands in size. If all genes that affect the trait are neutral with respect to fitness, the additive genetic variance (VA) is always reduced by a bottleneck in population size, regardless of their degree of dominance. For genes that have been under selection, VA increases following a bottleneck if they are (partially) recessive, while the dominance variance increases substantially for any degree of dominance. With typical estimates of mutation parameters, the joint-effect model can explain data from laboratory experiments on the effect of bottlenecking on fitness and morphological traits, providing further support for it as a plausible mechanism for maintenance of quantitative genetic variation.     

Niche construction and the environmental term of the price equation: How natural selection changes when organisms alter their environments

Organisms construct their own environments and phenotypes through the adaptive processes of habitat choice, habitat construction, and phenotypic plasticity. We examine how these processes affect the dynamics of mean fitness change through the environmental change term of the Price Equation. This tends to be ignored in evolutionary theory, owing to the emphasis on the first term describing the effect of natural selection on mean fitness (the additive genetic variance for fitness of Fisher's Fundamental Theorem). Using population genetic models and the Price Equation, we show how adaptive niche constructing traits favorably alter the distribution of environments that organisms encounter and thereby increase population mean fitness. Because niche-constructing traits increase the frequency of higher-fitness environments, selection favors their evolution. Furthermore, their alteration of the actual or experienced environmental distribution creates selective feedback between niche constructing traits and other traits, especially those with genotype-by-environment interaction for fitness. By altering the distribution of experienced environments, niche constructing traits can increase the additive genetic variance for such traits. This effect accelerates the process of overall adaption to the niche-constructed environmental distribution and can contribute to the rapid refinement of alternative phenotypic adaptations to different environments. Our findings suggest that evolutionary biologists revisit and reevaluate the environmental term of the Price Equation: owing to adaptive niche construction, it contributes directly to positive change in mean fitness; its magnitude can be comparable to that of natural selection; and, when there is fitness G × E, it increases the additive genetic variance for fitness, the much-celebrated first term.     

Genetic quality of individuals impacts population dynamics

Ample evidence exists that an increase in the inbreeding level of a population reduces the value of fitness components such as fecundity and survival. It does not follow, however, that these decreases in the components of fitness impact population dynamics in a way that increases extinction risk, because virtually all species produce far more offspring than can actually survive. We analyzed the effects of the genetic quality (mean fitness) of individuals on the population growth rate of seven natural populations in each of two species of wolf spider in the genus Rabidosa , statistically controlling for environmental factors. We show that populations of different sizes, and different inbreeding levels, differ in population dynamics for both species. Differences in population growth rates are especially pronounced during stressful environmental conditions (low food availability) and the stressful environment affects smaller populations (<500 individuals) disproportionately. Thus, even in an invertebrate with an extremely high potential growth rate and strong density-dependent mortality rates, genetic factors contribute directly to population dynamics and, therefore, to extinction risk. This is only the second study to demonstrate an impact of the genetic quality of individual genotypes on population dynamics in a wild population and the first to document strong inbreeding–environment interactions for fitness among populations. Endangered species typically exist at sizes of a few hundred individuals and human activities degrade habitats making them innately more stressful (e.g. global climate change). Therefore, the interaction between genetic factors and environmental stress has important implications for efforts aimed at conserving the Earth's biodiversity.     

Mutation-selection balance and the evolutionary advantage of sex and recombination

Mutation-selection balance in a multi-locus system is investigated theoretically, using a modification of Bulmer's infinitesimal model of selection on a normally-distributed quantitative character, taking the number of mutations per individual (n) to represent the character value. The logarithm of the fitness of an individual with n mutations is assumed to be a quadratic, decreasing function of n. The equilibrium properties of infinitely large asexual populations, random-mating populations lacking genetic recombination, and random-mating populations with arbitrary recombination frequencies are investigated. With 'synergistic' epistasis on the scale of log fitness, such that log fitness declines more steeply as n increases, it is shown that equilibrium mean fitness is least for asexual populations. In sexual populations, mean fitness increases with the number of chromosomes and with the map length per chromosome. With 'diminishing returns' epistasis, such that log fitness declines less steeply as n increases, mean fitness behaves in the opposite way. Selection on asexual variants and genes affecting the rate of genetic recombination in random-mating populations was also studied. With synergistic epistasis, zero recombination always appears to be disfavoured, but free recombination is disfavoured when the mutation rate per genome is sufficiently small, leading to evolutionary stability of maps of intermediate length. With synergistic epistasis, an asexual mutant is unlikely to invade a sexual population if the mutation rate per diploid genome greatly exceeds unity. Recombination is selectively disadvantageous when there is diminishing returns epistasis. These results are compared with the results of previous theoretical studies of this problem, and with experimental data.     

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 Pleiotropic Nonadditive Model of Variation in Quantitative Traits

A model of mutation-selection-drift balance incorporating pleiotropic and dominance effects of new mutations on quantitative traits and fitness is investigated and used to predict the amount and nature of genetic variation maintained in segregating populations. The model is based on recent information on the joint distribution of mutant effects on bristle traits and fitness in Drosophila melanogaster from experiments on the accumulation of spontaneous and P element-induced mutations. These experiments suggest a leptokurtic distribution of effects with an intermediate correlation between effects on the trait and fitness. Mutants of large effect tend to be partially recessive while those with smaller effect are on average additive, but apparently with very variable gene action. The model is parameterized with two different sets of information derived from P element insertion and spontaneous mutation data, though the latter are not fully known. They differ in the number of mutations per generation which is assumed to affect the trait. Predictions of the variance maintained for bristle number assuming parameters derived from effects of P element insertions, in which the proportion of mutations with an effect on the trait is small, fit reasonably well with experimental observations. The equilibrium genetic variance is nearly independent of the degree of dominance of new mutations. Heritabilities of between 0.4 and 0.6 are predicted with population sizes from 10(4) to 10(6), and most of the variance for the metric trait in segregating populations is due to a small proportion of mutations (about 1% of the total number) with neutral or nearly neutral effects on fitness and intermediate effects on the trait (0.1-0.5σ(P)). Much of the genetic variance is contributed by recessive or partially recessive mutants, but only a small proportion (about 10%) of the genetic variance is dominance variance. The amount of apparent selection on the trait itself generated by the model is very small. If a model is assumed in which all mutation events have an effect on the quantitative trait, the majority of the genetic variance is contributed by deleterious mutations with tiny effects on the trait. If such a model is assumed for viability, the heritability is about 0.1, independent of the population size.     

Fitness costs associated with low genetic variation are reduced in a harsher environment in amphibian island populations

A basic premise of conservation geneticists is that low levels of genetic variation are associated with fitness costs in terms of reduced survival and fecundity. These fitness costs may frequently vary with environmental factors and should increase under more stressful conditions. However, there is no consensus on how fitness costs associated with low genetic variation change under natural conditions in relation to the stressfulness of the environment. On the Swedish west coast, natterjack toad Bufo calamita populations show a strong population genetic structure and large variation in the amount of within-population genetic variation. We experimentally examined the survival of natterjack larvae from six populations with different genetic variation in three thermal environments corresponding to (a) the mean temperature of natural ponds (stable, laboratory), (b) a high temperature environment occurring in desiccating ponds (stable, laboratory) and (c) an outdoor treatment mimicking the natural, variable thermal conditions (fluctuating, semi-natural). We found that larvae in the outdoor treatment had poorer survival than larvae in the stable environments suggesting that the outdoor treatment was more stressful. Overall, populations with higher genetic variation had higher larval survival. However, a significant interaction between treatments and genetic variation indicated that fitness costs associated with low genetic variation were less severe in the outdoor treatment. Thus, we found no support for the hypothesis that fitness costs associated with low genetic variation increase under more stressful conditions. Our results suggest that natural thermal stress may mask fitness losses associated with low genetic variation in these populations.     

A theory of natural selection incorporating interaction among individuals. VI. Use of non-random synchronized groups

The previous study, (V), in this series approached the problem of accommodating interaction among individuals in a population by considering a life-history model which resulted in the synchronization of fitness components (viability and fecundity) for members within groups of interacting individuals. It was shown that such synchronization resulted in symmetric fitness values, and that selection operating on symmetric fitness values produced optimum short- and long-term results. However, it was also shown that selection operating on the random groups of the model could be inefficient.The present paper demonstrates that use of non-random (related) groups can increase the efficiency of symmetric selection without destroying its short-term balance. The consequences of long-term selection are more complicated and depend on the complexity of the genetic model.     

On the evolution of epistasis I: diploids under selection

One interpretation of recent literature on the evolution of phenotypic modularity is that evolution should act to decrease the degree of interaction between genes that contribute to different phenotypes. This issue is addressed directly here using a fitness scheme determined by two genetic loci and a third locus which modifies a measure of statistical interaction between the fitnesses due to the first two. The equilibrium structure of such an epistasis-modifying locus is studied. It is shown that under well-specified conditions a modifying allele that increases epistasis succeeds. In other words, genetic interactions tend to become stronger. It is speculated that this occurs because the mean fitness in such models is locally increasing as a function of the degree of epistasis.     

Field measurements of genotype by environment interaction for fitness caused by spontaneous mutations in Arabidopsis thaliana

As the ultimate source of genetic diversity, spontaneous mutation is critical to the evolutionary process. The fitness effects of spontaneous mutations are almost always studied under controlled laboratory conditions rather than under the evolutionarily relevant conditions of the field. Of particular interest is the conditionality of new mutations—that is, is a new mutation harmful regardless of the environment in which it is found? In other words, what is the extent of genotype–environment interaction for spontaneous mutations? We studied the fitness effects of 25 generations of accumulated spontaneous mutations in Arabidopsis thaliana in two geographically widely separated field environments, in Michigan and Virginia. At both sites, mean total fitness of mutation accumulation lines exceeded that of the ancestors, contrary to the expected decrease in the mean due to new mutations but in accord with prior work on these MA lines. We observed genotype–environment interactions in the fitness effects of new mutations, such that the effects of mutations in Michigan were a poor predictor of their effects in Virginia and vice versa. In particular, mutational variance for fitness was much larger in Virginia compared to Michigan. This strong genotype–environment interaction would increase the amount of genetic variation maintained by mutation‐selection balance.     

Fundamental theorem of natural selection and frequency-dependent selection: analysis of the matrix game diploid model

Following Ewens' interpretation about Fisher's fundamental theorem of natural selection, the matrix game model for diploid populations undergoing non-overlapping, discrete generations is investigated. The total genetic variance is decomposed and it is shown that the partial change in the mean fitness, which is equal to the additive genetic variance over the mean fitness, can be thought of as a change due only to the partial changes in the phenotypic frequencies.     

When three traits make a line: evolution of phenotypic plasticity and genetic assimilation through linear reaction norms in stochastic environments

Genetic assimilation emerges from selection on phenotypic plasticity. Yet, commonly used quantitative genetics models of linear reaction norms considering intercept and slope as traits do not mimic the full process of genetic assimilation. We argue that intercept–slope reaction norm models are insufficient representations of genetic effects on linear reaction norms and that considering reaction norm intercept as a trait is unfortunate because the definition of this trait relates to a specific environmental value (zero) and confounds genetic effects on reaction norm elevation with genetic effects on environmental perception. Instead, we suggest a model with three traits representing genetic effects that, respectively, (i) are independent of the environment, (ii) alter the sensitivity of the phenotype to the environment and (iii) determine how the organism perceives the environment. The model predicts that, given sufficient additive genetic variation in environmental perception, the environmental value at which reaction norms tend to cross will respond rapidly to selection after an abrupt environmental change, and eventually becomes equal to the new mean environment. This readjustment of the zone of canalization becomes completed without changes in genetic correlations, genetic drift or imposing any fitness costs of maintaining plasticity. The asymptotic evolutionary outcome of this three‐trait linear reaction norm generally entails a lower degree of phenotypic plasticity than the two‐trait model, and maximum expected fitness does not occur at the mean trait values in the population.