Genotypic Variation for a Quantitative Character Maintained under Stabilizing Selection without Mutations: Epistasis

A model of the gene action on a quantitative character is suggested. The model takes into account epistasis by combining multiplicative with the traditional additive approximation of the action of loci. It is demonstrated on the basis of this model that a high level of genotypic variation can be maintained in a population for a quantitative character under stabilizing selection in the absence of mutations, if there is epistasis. It is also shown that a large amount of additive variation as well as high heritability can be "hidden" in such a population and "released" if stabilizing selection is relaxed.     

The Effect of Deleterious Mutations on Neutral Molecular Variation

Selection against deleterious alleles maintained by mutation may cause a reduction in the amount of genetic variability at linked neutral sites. This is because a new neutral variant can only remain in a large population for a long period of time if it is maintained in gametes that are free of deleterious alleles, and hence are not destined for rapid elimination from the population by selection. Approximate formulas are derived for the reduction below classical neutral values resulting from such background selection against deleterious mutations, for the mean times to fixation and loss of new mutations, nucleotide site diversity, and number of segregating sites. These formulas apply to random-mating populations with no genetic recombination, and to populations reproducing exclusively asexually or by self-fertilization. For a given selection regime and mating system, the reduction is an exponential function of the total mutation rate to deleterious mutations for the section of the genome involved. Simulations show that the effect decreases rapidly with increasing recombination frequency or rate of outcrossing. The mean time to loss of new neutral mutations and the total number of segregating neutral sites are less sensitive to background selection than the other statistics, unless the population size is of the order of a hundred thousand or more. The stationary distribution of allele frequencies at the neutral sites is correspondingly skewed in favor of rare alleles, compared with the classical neutral result. Observed reductions in molecular variation in low recombination genomic regions of sufficiently large size, for instance in the centromere-proximal regions of Drosophila autosomes or in highly selfing plant populations, may be partly due to background selection against deleterious mutations.     

Pleiotropic Models of Polygenic Variation, Stabilizing Selection, and Epistasis

We show that in polymorphic populations many polygenic traits pleiotropically related to fitness are expected to be under apparent ``stabilizing selection' independently of the real selection acting on the population. This occurs, for example, if the genetic system is at a stable polymorphic equilibrium determined by selection and the nonadditive contributions of the loci to the trait value either are absent, or are random and independent of those to fitness. Stabilizing selection is also observed if the polygenic system is at an equilibrium determined by a balance between selection and mutation (or migration) when both additive and nonadditive contributions of the loci to the trait value are random and independent of those to fitness. We also compare different viability models that can maintain genetic variability at many loci with respect to their ability to account for the strong stabilizing selection on an additive trait. Let V(m) be the genetic variance supplied by mutation (or migration) each generation, V(g) be the genotypic variance maintained in the population, and n be the number of the loci influencing fitness. We demonstrate that in mutation (migration)-selection balance models the strength of apparent stabilizing selection is order V(m)/V(g). In the overdominant model and in the symmetric viability model the strength of apparent stabilizing selection is approximately 1/(2n) that of total selection on the whole phenotype. We show that a selection system that involves pairwise additive by additive epistasis in maintaining variability can lead to a lower genetic load and genetic variance in fitness (approximately 1/(2n) times) than an equivalent selection system that involves overdominance. We show that, in the epistatic model, the apparent stabilizing selection on an additive trait can be as strong as the total selection on the whole phenotype.     

Pleiotropic Mutations Are Subject to Strong Stabilizing Selection

The assumption that pleiotropic mutations are more deleterious than mutations with more restricted phenotypic effects is an important premise in models of evolution. However, empirical evidence supporting this assumption is limited. Here, we estimated the strength of stabilizing selection on mutations affecting gene expression in male Drosophila serrata. We estimated the mutational variance (V_M) and the standing genetic variance (V_G) from two well-matched panels of inbred lines: a panel of mutation accumulation (MA) lines derived from a single inbred ancestral line and a panel of inbred lines derived from an outbred population. For 855 gene-expression traits, we estimated the strength of stabilizing selection as s = V_M/V_G. Selection was observed to be relatively strong, with 17% of traits having s > 0.02, a magnitude typically associated with life-history traits. Randomly assigning expression traits to five-trait sets, we used factor analytic mixed modeling in the MA data set to identify covarying traits that shared pleiotropic mutations. By assigning traits to the same trait sets in the outbred line data set, we then estimated s for the combination of traits affected by pleiotropic mutation. For these pleiotropic combinations, the median s was three times greater than s acting on the individual component traits, and 46% of the pleiotropic trait combinations had s > 0.02. Although our analytical approach was biased toward detecting mutations with relatively large effects, likely overestimating the average strength of selection, our results provide widespread support for the prediction that stronger selection can act against mutations with pleiotropic effects.THE extent to which new mutations have pleiotropic effects on multiple traits, and ultimately on fitness is central to our understanding of the maintenance of genetic variation and the process of adaptation (Kondrashov and Turelli 1992; Otto 2004; Johnson and Barton 2005; Zhang and Hill 2005). Analyses of Fisher’s (1930) geometric model of adaptation have shown that a mutation with effects on many traits will have a reduced probability of contributing to adaptive evolution (Orr 2000; Welch and Waxman 2003; see also Haygood 2006). For a population close to its optimum under mutation–selection balance, a direct corollary of this is that selection must act more strongly against mutations with wider pleiotropic effects (Zhang 2012).Evidence for the strength of selection increasing with the number of traits that are pleiotropically affected by a mutation is limited. At a phenotypic level, nonlinear (stabilizing) selection is much stronger on combinations of metric traits than on each individual trait contributing to the combination (Blows and Brooks 2003; Walsh and Blows 2009). Given that genetic correlations among such traits are expected to be a consequence of pleiotropic alleles (Lande 1980), stronger selection on trait combinations is consistent with stronger selection on pleiotropic mutations that are likely to underlie the genetic covariance among such traits. There is some evidence that per-trait allelic effects might be greater for alleles with more widespread pleiotropic effects (Wagner et al. 2008; Wang et al. 2010); as mutations with larger phenotypic effects might be more effectively targeted by selection, this also suggests stronger selection against more pleiotropic mutation.Mutation accumulation (MA) breeding designs, in which the opportunity for selection is reduced, allowing new mutations to drift to fixation, provide an opportunity to characterize the strength of selection acting directly against new mutations. Rice and Townsend (2012) proposed an approach for determining the strength of selection acting against mutations at individual loci, combining information from QTL mapping and MA studies. This approach could conceivably be extended to associate the strength of selection with the number of traits a QTL affects. More typically, estimates of selection from MA designs are focused on traits, rather than alleles. Under the assumption that most mutations are deleterious, an assumption supported by MA studies (Halligan and Keightley 2009), the strength of selection acting on mutations affecting quantitative traits can be measured as the ratio of the mutational to the standing genetic variance, s = V_M/V_G, where s is the selection coefficient of the mutation in heterozygous form (Barton 1990; Houle et al. 1996). While estimating s in this way provides a framework for estimating selection on pleiotropic combinations of traits, we are not aware of any studies adopting this approach to directly estimate the strength of selection acting on mutations affecting multiple traits.Within an MA framework, Estes and Phillips (2006) manipulated the opportunity for selection, providing rare direct evidence of stronger selection against mutations with pleiotropic effects. In a DNA repair-deficient strain of Caenorhabditis elegans, Estes and Phillips (2006) observed lower mutational covariance among life-history components when selection was allowed (larger populations) than when the opportunity for selection was limited (small populations). Similarly, McGuigan et al. (2011) compared Drosophila serrata MA lines accumulating mutations in the presence or absence of sexual selection on males, reporting reduced covariance between two fitness components in the selection treatment. These studies reveal that selection can eliminate nonlethal alleles with pleiotropic effects, but whether traits other than life-history components exhibit similar evidence of selection against pleiotropic alleles remains unknown.In parallel to the quantitative genetic predictions that pleiotropic alleles will be under stronger selection, molecular genetic theory predicts that the rate of gene evolution will be negatively correlated with pleiotropy (Pal et al. 2006; Salathe et al. 2006). More highly pleiotropic genes, as identified through the extent of connectivity (the number of interactions) in protein–protein interaction networks (Jeong et al. 2001), or the number of gene ontology (GO) terms (Jovelin and Phillips 2009) are more likely to be essential (i.e., knockout mutations result in lethality), suggesting that selection is stronger against large-effect (knockout) mutations in more highly pleiotropic genes. However, the selection acting against small-effect, nonlethal mutations in pleiotropic genes is less clear (Pal et al. 2006). Several studies have found an association between gene pleiotropy indices, such GO annotation of the number of biological processes or tissue specificity of expression, and the rate of sequence evolution (e.g., Pal et al. 2001; Salathe et al. 2006; Jovelin and Phillips 2009; Su et al. 2010). These pleiotropy indices typically explain little of the variation in sequence evolutionary rates, and it remains unclear whether more highly pleiotropic mutations are typically under stronger selection (Pal et al. 2006; Salathe et al. 2006).Here, we estimate the selection coefficients acting against naturally occurring mutations affecting gene-expression traits in male D. serrata to quantitatively test if selection is stronger on mutations that affect multiple traits. Gene-expression phenotypes are uniquely positioned to enable detailed investigations of pleiotropy: there are many of them, they represent a broad coverage of biological function, they can be analyzed to quantify developmental pleiotropy in the same way as traits traditionally considered in quantitative genetics, and GO information can be used to index molecular genetic pleiotropy. We use multivariate mixed-model analyses of expression traits in a set of inbred lines from a mutation accumulation experiment to estimate the mutational variance in individual expression traits, and the pleiotropic mutational covariance among random sets of five expression traits. Using a second panel of inbred lines, derived from a natural, outbred, population, we estimate the standing genetic variance in the same individual traits and five-trait combinations. From these estimates of mutational and standing genetic variance, we calculate s for each of the individual traits and trait combinations to determine whether selection has typically been stronger on mutations with pleiotropic effects than on other mutations affecting each trait. We complement this quantitative genetic analysis of developmental pleiotropy with an analysis of molecular genetic pleiotropy (Paaby and Rockman 2013), determining whether the strength of selection acting on individual expression traits can be predicted from the number of biological functions that the gene annotates to in the GO database or to the range of tissues in which the gene is expressed.     

Limited Dispersal, Deleterious Mutations and the Evolution of Sex

This study presents a mathematical model that allows for some offspring to be dispersed at random, while others stay close to their mothers. A single genetic locus is assumed to control fertility, and this locus is subject to the occurrence of deleterious mutations. It is shown that, at equilibrium, the frequency of deleterious mutations in the population is inversely related to the rate of dispersal. This is because dispersal of offspring leads to enhanced competition among adults. The results also show that sexual reproduction can lead to a decrease in the equilibrium frequency of deleterious mutations. The reason for this relationship is that sex involves the dispersal of genetic material, and thus, like the dispersal of offspring, sex enhances competition among adults. The model is described using the example of a hermaphroditic plant population. However, the results should apply to animal populations as well.     

Additive Variation Maintained under Stabilizing Selection: A Two-Locus Model of Pleiotropy for Two Quantitative Characters

A model with two diallelic loci controlling two additive quantitative characters is suggested. One of the loci has a similar effect on both characters, whereas the second locus has an antagonistic effect on two characters. Both characters experience direct stabilizing selection. The model yields a stable polymorphic state, with both characters maintaining genetic variation. The genetic correlation between the characters at the equilibrium is zero, in spite of the pleiotropic effects of the loci controlling them.     

Maintenance of Genetic Variability under the Pressure of Neutral and Deleterious Mutations in a Finite Population

In order to assess the effect of deleterious mutations on various measures of genic variation, approximate formulas have been developed for the frequency spectrum, the mean number of alleles in a sample, and the mean homozygosity; in some particular cases, exact formulas have been obtained. The assumptions made are that two classes of mutations exist, neutral and deleterious, and that selection is strong enough to keep deleterious alleles in low frequencies, the mode of selection being either genic or recessive. The main findings are: (1) If the expected value (q) of the sum of the frequencies of deleterious alleles is about 10% or less, then the presence of deleterious alleles causes only a minor reduction in the mean number of neutral alleles in a sample, as compared to the case of q = 0. Also, the low- and intermediate-frequency parts of the frequency spectrum of neutral alleles are little affected by the presence of deleterious alleles, though the high-frequency part may be changed drastically. (2) The contribution of deleterious mutations to the expected total number of alleles in a sample can be quite large even if q is only 1 or 2%. (3) The mean homozygosity is roughly equal to (1--2q)/(1 + theta 1), where theta 1 is twice the number of new neutral mutations occurring in each generation in the total population. Thus, deleterious mutations increase the mean heterozygosity by about 2q/(1 + theta 1). The present results have been applied to study the controversial problem of how deleterious mutations may affect the testing of the neutral mutation hypothesis.     

A Ruby in the Rubbish: Beneficial Mutations, Deleterious Mutations and the Evolution of Sex

This study presents a mathematical model in which a single beneficial mutation arises in a very large population that is subject to frequent deleterious mutations. The results suggest that, if the population is sexual, then the deleterious mutations will have little effect on the ultimate fate of the beneficial mutation. However, if most offspring are produced asexually, then the probability that the beneficial mutation will be lost from the population may be greatly enhanced by the deleterious mutations. Thus, sexual populations may adapt much more quickly than populations where most reproduction is asexual. Some of the results were produced using computer simulation methods, and a technique was developed that allows treatment of arbitrarily large numbers of individuals in a reasonable amount of computer time. This technique may be of prove useful for the analysis of a wide variety of models, though there are some constraints on its applicability. For example, the technique requires that reproduction can be described by Poisson processes.     

The Quantitative Genetic Consequences of Pleiotropy under Stabilizing and Directional Selection

The independence of two phenotypic characters affected by both pleiotropic and nonpleiotropic mutations is investigated using a generalization of M. Slatkin's stepwise mutation model of 1987. The model is used to determine whether predictions of either the multivariate normal model introduced in 1980 by R. Lande or the house-of-cards model introduced in 1985 by M. Turelli can be regarded as typical of models that are intermediate between them. We found that, under stabilizing selection, the variance of one character at equilibrium may depend on the strength of stabilizing selection on the other character (as in the house-of-cards model) or not (as in the multivariate normal model) depending on the types of mutations that can occur. Similarly, under directional selection, the genetic covariance between two characters may increase substantially (as in the house-of-cards model) or not (as in the multivariate normal model) depending on the kinds of mutations that are assumed to occur. Hence, even for the simple model we consider, neither the house-of-cards nor the multivariate normal model can be used to make predictions, making it unlikely that either could be used to draw general conclusions about more complex and realistic models.     

The Maintenance of Genetic Variability in Two-Locus Models of Stabilizing Selection

The maintenance of genetic variability at two diallelic loci under stabilizing selection is investigated. Generations are discrete and nonoverlapping; mating is random; mutation and random genetic drift are absent; selection operates only through viability differences. The determination of the genotypic values is purely additive. The fitness function has its optimum at the value of the double heterozygote and decreases monotonically and symmetrically from its optimum, but is otherwise arbitrary. The resulting fitness scheme is identical to the symmetric viability model. Linkage disequilibrium is neglected, but the results are otherwise exact. Explicit formulas are found for all the equilibria, and explicit conditions are derived fro their existence and stability. A complete classification of the six possible global convergence patterns is presented. In addition to the symmetric equilibrium (with gene frequency 1/2 at both loci), a pair of unsymmetric equilibria may exist; the latter are usually, but not always, unstable. If the ratio of the effect of the major locus to that of the minor one exceeds a critical value, both loci will be stably polymorphic. If selection is weak at the minor locus, the more rapidly the fitness function decreases near the optimum, the lower is this critical value; for rapidly decreasing fitness functions, the critical value is close to one. If the fitness function is smooth at the optimum, then a stable polymorphism exists at both loci only if selection is strong at the major locus.     

The Evolution of Selfing Is Accompanied by Reduced Efficacy of Selection and Purging of Deleterious Mutations

The transition from outcrossing to selfing is predicted to reduce the genome-wide efficacy of selection because of the lower effective population size (N_e) that accompanies this change in mating system. However, strongly recessive deleterious mutations exposed in the homozygous backgrounds of selfers should be under strong purifying selection. Here, we examine estimates of the distribution of fitness effects (DFE) and changes in the magnitude of effective selection coefficients (N_es) acting on mutations during the transition from outcrossing to selfing. Using forward simulations, we investigated the ability of a DFE inference approach to detect the joint influence of mating system and the dominance of deleterious mutations on selection efficacy. We investigated predictions from our simulations in the annual plant Eichhornia paniculata, in which selfing has evolved from outcrossing on multiple occasions. We used range-wide sampling to generate population genomic datasets and identified nonsynonymous and synonymous polymorphisms segregating in outcrossing and selfing populations. We found that the transition to selfing was accompanied by a change in the DFE, with a larger fraction of effectively neutral sites (N_es < 1), a result consistent with the effects of reduced N_e in selfers. Moreover, an increased proportion of sites in selfers were under strong purifying selection (N_es > 100), and simulations suggest that this is due to the exposure of recessive deleterious mutations. We conclude that the transition to selfing has been accompanied by the genome-wide influences of reduced N_e and strong purifying selection against deleterious recessive mutations, an example of purging at the molecular level.     

Deleterious Mutations and the Origin of the Meiotic Ploidy Cycle

A population genetical model is investigated in which the organism either alternates between diploid and haploid states or lives entirely in the haploid state. The behavior of the organism is determined by the genotype at a modifier locus. At an independent locus deleterious mutations occur at a low but constant frequency. It is found that the haploid behavior is always an evolutionarily attainable stable trait, while the ploidy-cyclic behavior is an evolutionarily attainable stable trait only when a certain condition holds. This condition depends on the strength of selection, the degree of "sheltering" given by the heterozygote state, and the degree of linkage between the modifier locus and the locus under selection. The last result leads to the speculation that the eukaryotes are derived from an organism which first developed more than one chromosome before it evolved the ploidy cycle.     

Evolutionary Invasion and Escape in the Presence of Deleterious Mutations

Replicators such as parasites invading a new host species, species invading a new ecological niche, or cancer cells invading a new tissue often must mutate to adapt to a new environment. It is often argued that a higher mutation rate will favor evolutionary invasion and escape from extinction. However, most mutations are deleterious, and even lethal. We study the probability that the lineage will survive and invade successfully as a function of the mutation rate when both the initial strain and an adaptive mutant strain are threatened by lethal mutations. We show that mutations are beneficial, i.e. a non-zero mutation rate increases survival compared to the limit of no mutations, if in the no-mutation limit the survival probability of the initial strain is smaller than the average survival probability of the strains which are one mutation away. The mutation rate that maximizes survival depends on the characteristics of both the initial strain and the adaptive mutant, but if one strain is closer to the threshold governing survival then its properties will have greater influence. These conclusions are robust for more realistic or mechanistic depictions of the fitness landscapes such as a more detailed viral life history, or non-lethal deleterious mutations.     

Maintenance of Genetic Variability under Strong Stabilizing Selection: A Two-Locus Model

We study a two locus model with additive contributions to the phenotype to explore the relationship between stabilizing selection and recombination. We show that if the double heterozygote has the optimum phenotype and the contributions of the loci to the trait are different, then any symmetric stabilizing selection fitness function can maintain genetic variability provided selection is sufficiently strong relative to linkage. We present results of a detailed analysis of the quadratic fitness function which show that selection need not be extremely strong relative to recombination for the polymorphic equilibria to be stable. At these polymorphic equilibria the mean value of the trait, in general, is not equal to the optimum phenotype, there exists a large level of negative linkage disequilibrium which ``hides' additive genetic variance, and different equilibria can be stable simultaneously. We analyze dependence of different characteristics of these equilibria on the location of optimum phenotype, on the difference in allelic effect, and on the strength of selection relative to recombination. Our overall result that stabilizing selection does not necessarily eliminate genetic variability is compatible with some experimental results where the lines subject to strong stabilizing selection did not have significant reductions in genetic variability.     

Epistasis among Deleterious Mutations in the HIV-1 Protease

A central goal in molecular evolution is to understand how genetic interactions between protein mutations shape protein function and fitness. While intergenic epistasis has been extensively explored in eukaryotes, bacteria, and viruses, intragenic epistatic interactions have been insufficiently studied. Here, we employ a model system in which lambda phage fitness correlates with the enzymatic activity of human immunodeficiency virus type 1 (HIV-1) protease to systematically determine the epistatic interactions between intragenic pairs of deleterious protein substitutions. We generated 114 genotypes of the HIV-1 protease, each carrying pairs of nucleotide substitution mutations whose separated and combined deleterious effects on fitness were then determined. A high proportion (39%) of pairs displayed lethality. Several pairs exhibited significant interactions for fitness, including positive and negative epistasis. Significant negative epistatic interactions predominated (15%) over positive interactions (2%). However, the average ± SD epistatic effect, ē = 0.0025 ± 0.1334, was not significantly different from zero (p = 0.8368). Notably, epistatic interactions, regardless of epistatic direction, tend to be more frequent in the context of less deleterious mutations. In the present study, the high frequencies of lethality and negative epistasis indicate that the HIV-1 protease is highly sensitive to the effects of deleterious mutations. Therefore, proteins may not be as robust to mutational change as is usually expected.     

Stabilizing Selection,Purifying Selection,and Mutational Bias in Finite Populations

Genomic traits such as codon usage and the lengths of noncoding sequences may be subject to stabilizing selection rather than purifying selection. Mutations affecting these traits are often biased in one direction. To investigate the potential role of stabilizing selection on genomic traits, the effects of mutational bias on the equilibrium value of a trait under stabilizing selection in a finite population were investigated, using two different mutational models. Numerical results were generated using a matrix method for calculating the probability distribution of variant frequencies at sites affecting the trait, as well as by Monte Carlo simulations. Analytical approximations were also derived, which provided useful insights into the numerical results. A novel conclusion is that the scaled intensity of selection acting on individual variants is nearly independent of the effective population size over a wide range of parameter space and is strongly determined by the logarithm of the mutational bias parameter. This is true even when there is a very small departure of the mean from the optimum, as is usually the case. This implies that studies of the frequency spectra of DNA sequence variants may be unable to distinguish between stabilizing and purifying selection. A similar investigation of purifying selection against deleterious mutations was also carried out. Contrary to previous suggestions, the scaled intensity of purifying selection with synergistic fitness effects is sensitive to population size, which is inconsistent with the general lack of sensitivity of codon usage to effective population size.     

Quantitative Genetic Variation in Body Size of Mice from New Mutations

To measure the amount of new genetic variation in 6-week weight of mice arising each generation from mutation, selection lines derived from an initially inbred strain were maintained for 25 generations. An analysis using an animal model with restricted maximum likelihood was applied to estimate a mutational genetic component of variance for the infinitesimal model of many genes of small effect. Assuming that the inbred base population was at a mutation-drift equilibrium, it is estimated that the heritability for body size has increased by 1.0% per generation, with lower and upper confidence limits of 0.6% and 1.6%, respectively. A model which includes a mutational genetic component of variance fits the data much better than one involving only base population genetic variance. A model with no genetic component fits the data very poorly. An environmental covariance of body size of mother and offspring was included in the model and accounts for 10% of the variance. By using information only from the observed response to selection, the estimated increase in heritability from mutation is 0.3% per generation. These values are higher than published estimates for the increase in variance from spontaneous mutations in bristle traits of Drosophila, for which there are extensive data, but similar to estimates for various skeletal traits in mice.