Heritable genetic variation via mutation-selection balance: Lerch's zeta meets the abdominal bristle

The anomalous effects of biased mutation revisited: mean-optimum deviation and apparent directional selection under stabilizing selection

Empirical evidence indicates that the distribution of the effects of mutations on quantitative traits is not symmetric about zero. Under stabilizing selection in infinite populations with normally distributed mutant effects having a nonzero mean, Waxman and Peck showed that the deviation of the population mean from the optimum is expected to be small. We show by simulation that genetic drift, leptokurtosis of mutational effects, and pleiotropy can increase the mean-optimum deviation greatly, however, and that the apparent directional selection thereby caused can be substantial.

Effects of spontaneous mutation accumulation on sex ratio traits in a parasitoid wasp

Sex allocation theory has proved extremely successful at predicting when individuals should adjust the sex of their offspring in response to environmental conditions. However, we know rather little about the underlying genetics of sex ratio or how genetic architecture might constrain adaptive sex-ratio behavior. We examined how mutation influenced genetic variation in the sex ratios produced by the parasitoid wasp Nasonia vitripennis. In a mutation accumulation experiment, we determined the mutability of sex ratio, and compared this with the amount of genetic variation observed in natural populations. We found that the mutability (h(2)(m)) ranges from 0.001 to 0.002, similar to estimates for life-history traits in other organisms. These estimates suggest one mutation every 5-60 generations, which shift the sex ratio by approximately 0.01 (proportion males). In this and other studies, the genetic variation in N. vitripennis sex ratio ranged from 0.02 to 0.17 (broad-sense heritability, H(2)). If sex ratio is maintained by mutation-selection balance, a higher genetic variance would be expected given our mutational parameters. Instead, the observed genetic variance perhaps suggests additional selection against sex-ratio mutations with deleterious effects on other fitness traits as well as sex ratio (i.e., pleiotropy), as has been argued to be the case more generally.

Testing the rare-alleles model of quantitative variation by artificial selection

The rare-alleles model of quantitative variation posits that a common allele (the 'wild-type') and one or more rare alleles segregate at each locus affecting a quantitative trait; a scenario predicted by several distinct evolutionary hypotheses. Single locus arguments suggest that artificial selection should substantially increase the genetic variance (Vg) if the rare-alleles model is accurate. This paper tests the 'DeltaVg prediction' using a large artificial selection experiment on flower size of Mimulus guttatus. Vg for flower size does evolve, increasing with selection for larger flower while decreasing in the other direction. These data are consistent with a model in which flower size variation is caused by rare, partially dominant alleles. However, this explanation becomes increasingly tenuous when considered with other data (correlated responses to selection and the effects of inbreeding). A combination of modern (marker-based mapping) and classical (biometric) techniques will likely to be required to determine the distribution of allele frequencies at loci influencing quantitative traits.

Adaptation to slow environmental change, with apparent anticipation of selection

We investigate a genetic model of a large population of sexual organisms in a changing environment. The organisms are subject to stabilising selection on a quantitative trait, with environmental change causing the fitness optimum to move. When the fitness optimum moves slowly, adaptation to the changing environment occurs by means of reasonably well-separated substitutions at the loci controlling the trait. In this way, the trait generally tracks the moving optimum, but in such a case, the population may exhibit periods of time where the mean trait value overshoots the moving optimal trait value, thereby exhibiting an apparent anticipation of selection. The mechanism underlying this phenomenon is determined from consideration of a simpler model that correctly captures the observed dynamical behaviour. We note that very slow rates of changes of traits are seen in the fossil record and the present work may be relevant to this topic.

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309-338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579-612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka-Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.

The mutation matrix and the evolution of evolvability

Evolvability is a key characteristic of any evolving system, and the concept of evolvability serves as a unifying theme in a wide range of disciplines related to evolutionary theory. The field of quantitative genetics provides a framework for the exploration of evolvability with the promise to produce insights of global importance. With respect to the quantitative genetics of biological systems, the parameters most relevant to evolvability are the G-matrix, which describes the standing additive genetic variances and covariances for a suite of traits, and the M-matrix, which describes the effects of new mutations on genetic variances and covariances. A population's immediate response to selection is governed by the G-matrix. However, evolvability is also concerned with the ability of mutational processes to produce adaptive variants, and consequently the M-matrix is a crucial quantitative genetic parameter. Here, we explore the evolution of evolvability by using analytical theory and simulation-based models to examine the evolution of the mutational correlation, r(mu), the key parameter determining the nature of genetic constraints imposed by M. The model uses a diploid, sexually reproducing population of finite size experiencing stabilizing selection on a two-trait phenotype. We assume that the mutational correlation is a third quantitative trait determined by multiple additive loci. An individual's value of the mutational correlation trait determines the correlation between pleiotropic effects of new alleles when they arise in that individual. Our results show that the mutational correlation, despite the fact that it is not involved directly in the specification of an individual's fitness, does evolve in response to selection on the bivariate phenotype. The mutational variance exhibits a weak tendency to evolve to produce alignment of the M-matrix with the adaptive landscape, but is prone to erratic fluctuations as a consequence of genetic drift. The interpretation of this result is that the evolvability of the population is capable of a response to selection, and whether this response results in an increase or decrease in evolvability depends on the way in which the bivariate phenotypic optimum is expected to move. Interestingly, both analytical and simulation results show that the mutational correlation experiences disruptive selection, with local fitness maxima at -1 and +1. Genetic drift counteracts the tendency for the mutational correlation to persist at these extreme values, however. Our results also show that an evolving M-matrix tends to increase stability of the G-matrix under most circumstances. Previous studies of G-matrix stability, which assume nonevolving M-matrices, consequently may overestimate the level of instability of G relative to what might be expected in natural systems. Overall, our results indicate that evolvability can evolve in natural systems in a way that tends to result in alignment of the G-matrix, the M-matrix, and the adaptive landscape, and that such evolution tends to stabilize the G-matrix over evolutionary time.

COMPETITION CAN MAINTAIN GENETIC BUT NOT ENVIRONMENTAL VARIANCE IN THE PRESENCE OF STABILIZING SELECTION

A population in which there is stabilizing selection acting on quantitative traits toward an intermediate optimum becomes monomorphic in the absence of mutation. Further, genotypes that show least environmental variation are also favored, such that selection is likely to reduce both genetic and environmental components of phenotypic variance. In contrast, intraspecific competition for resources is more severe between phenotypically similar individuals, such that those deviating from prevailing phenotypes have a selective advantage. It has been shown previously that polymorphism and phenotypic variance can be maintained if competition between individuals is "effectively" stronger than stabilizing selection. Environmental variance is generally observed in quantitative traits, so mechanisms to explain its maintenance are sought, but the impact of competition on its magnitude has not previously been studied. Here we assume that a quantitative trait is subject to selection for an optimal value and to selection due to competition. Further, we assume that both the mean and variance of the phenotypic value depend on genotype, such that both may be affected by selection. Theoretical analysis and numerical simulations reveal that environmental variance can be maintained only when the genetic variance (in mean phenotypic value) is constrained to a very low level. Environmental variance will be replaced entirely by genotypic variance if a range of genotypes that vary widely in mean phenotype are present or become so by mutation. The distribution of mean phenotypic values is discrete when competition is strong relative to stabilizing selection; but more genotypes segregate and the distribution can approach continuity as competition becomes extremely strong. If the magnitude of the environmental variance is not under genetic control, there is a complementary relationship between the levels of environmental and genetic variance such that the level of phenotypic variance is little affected.

The build up of mutation-selection- drift balance in laboratory Drosophila populations

The build up of an equilibrium between mutation, selection, and drift in populations of moderate size is an important evolutionary issue, and can be critical in the conservation of endangered populations. We studied this process in two Drosophila melanogaster populations initially lacking genetic variability (C1 and C2) that were subsequently maintained during 431 or 165 generations with effective population size N(e) approximately 500 (estimated by lethal complementation analysis). Each population originated synchronously to a companion set of full-sib mutation accumulation (MA) lines, C1 and MA1 were derived from an isogenic origin and C2 and MA2 from a single MA1 line at generation 265. The results suggest that both C1 and C2 populations were close to the mutation-selection-drift balance for viability and bristle traits, and are consistent with a 2.5-fold increase of the mutation rate in C2 and MA2. Despite this increase, the average panmictic viability in C2 was only slightly below that of C1, indicating that the expressed loads due to segregating deleterious mutation were small, in agreement with the low deleterious mutation rate (0.015-0.045) previously reported for the MA1 lines. In C1, the nonlethal inbreeding depression rate for viability was 30% of that usually estimated in segregating populations. The genetic variance for bristles regenerated in C1 and C2 was moderately smaller than the average value reported for natural populations, implying that they have accumulated a substantial adaptive potential. In light of neutral and selective predictions, these results suggest that bristle additive variance was predominantly due to segregation of mutations with deleterious effects of the order of 10(-3), and is consistent with relatively weak causal stabilizing selection (V(s) approximately 30).

Mutational bias for body size in rhabditid nematodes

Mutational bias is a potentially important agent of evolution, but it is difficult to disentangle the effects of mutation from those of natural selection. Mutation-accumulation experiments, in which mutations are allowed to accumulate at very small population size, thus minimizing the efficiency of natural selection, are the best way to separate the effects of mutation from those of selection. Body size varies greatly among species of nematode in the family rhabditidae; mutational biases are both a potential cause and a consequence of that variation. We report data on the cumulative effects of mutations that affect body size in three species of rhabditid nematode that vary fivefold in adult size. Results are very consistent with previous studies of mutations underlying fitness in the same strains: two strains of Caenorhabditis briggsae decline in body size about twice as fast as two strains of C. elegans, with a concomitant higher point estimate of the genomic mutation rate; the confamilial Oscheius myriophila is intermediate. There is an overall mutational bias, such that mutations reduce size on average, but the bias appears consistent between species. The genetic correlation between mutations that affect size and those underlying fitness is large and positive, on average.

Most rare missense alleles are deleterious in humans: implications for complex disease and association studies

The accumulation of mildly deleterious missense mutations in individual human genomes has been proposed to be a genetic basis for complex diseases. The plausibility of this hypothesis depends on quantitative estimates of the prevalence of mildly deleterious de novo mutations and polymorphic variants in humans and on the intensity of selective pressure against them. We combined analysis of mutations causing human Mendelian diseases, of human-chimpanzee divergence, and of systematic data on human genetic variation and found that ~20% of new missense mutations in humans result in a loss of function, whereas ~27% are effectively neutral. Thus, the remaining 53% of new missense mutations have mildly deleterious effects. These mutations give rise to many low-frequency deleterious allelic variants in the human population, as is evident from a new data set of 37 genes sequenced in >1,500 individual human chromosomes. Surprisingly, up to 70% of low-frequency missense alleles are mildly deleterious and are associated with a heterozygous fitness loss in the range 0.001-0.003. Thus, the low allele frequency of an amino acid variant can, by itself, serve as a predictor of its functional significance. Several recent studies have reported a significant excess of rare missense variants in candidate genes or pathways in individuals with extreme values of quantitative phenotypes. These studies would be unlikely to yield results if most rare variants were neutral or if rare variants were not a significant contributor to the genetic component of phenotypic inheritance. Our results provide a justification for these types of candidate-gene (pathway) association studies and imply that mutation-selection balance may be a feasible evolutionary mechanism underlying some common diseases.

To eject or to abandon? Life history traits of hosts and parasites interact to influence the fitness payoffs of alternative anti-parasite strategies

Hosts either tolerate avian brood parasitism or reject it by ejecting parasitic eggs, as seen in most rejecter hosts of common cuckoos, Cuculus canorus, or by abandoning parasitized clutches, as seen in most rejecter hosts of brown-headed cowbirds, Molothrus ater. What explains consistent variation between alternative rejection behaviours of hosts within the same species and across species when exposed to different types of parasites? Life history theory predicts that when parasites decrease the fitness of host offspring, but not the future reproductive success of host adults, optimal clutch size should decrease. Consistent with this prediction, evolutionarily old cowbird hosts, but not cuckoo hosts, have lower clutch sizes than related rarely- or newly parasitized species. We constructed a mathematical model to calculate the fitness payoffs of egg ejector vs. nest abandoner hosts to determine if various aspects of host life history traits and brood parasites' virulence on adult and young host fitness differentially influence the payoffs of alternative host defences. These calculations showed that in general egg ejection was a superior anti-parasite strategy to nest abandonment. Yet, increasing parasitism rates and increasing fitness values of hosts' eggs in both currently parasitized and future replacement nests led to switch points in fitness payoffs in favour of nest abandonment. Nonetheless, nest abandonment became selectively more favourable only at lower clutch sizes and only when hosts faced parasitism by a cowbird- rather than a cuckoo-type brood parasite. We suggest that, in addition to evolutionary lag and gape-size limitation, our estimated fitness differences based on life history trait variation provide new insights for the consistent differences observed in the anti-parasite rejection strategies between many cuckoo- and cowbird-hosts.

Regional heterogeneity and gene flow maintain variance in a quantitative trait within populations of lodgepole pine

Genetic variation is of fundamental importance to biological evolution, yet we still know very little about how it is maintained in nature. Because many species inhabit heterogeneous environments and have pronounced local adaptations, gene flow between differently adapted populations may be a persistent source of genetic variation within populations. If this migration-selection balance is biologically important then there should be strong correlations between genetic variance within populations and the amount of heterogeneity in the environment surrounding them. Here, we use data from a long-term study of 142 populations of lodgepole pine (Pinus contorta) to compare levels of genetic variation in growth response with measures of climatic heterogeneity in the surrounding region. We find that regional heterogeneity explains at least 20% of the variation in genetic variance, suggesting that gene flow and heterogeneous selection may play an important role in maintaining the high levels of genetic variation found within natural populations.

Additive genetic variation and the distribution of QTN effects among sites

Early quantitative genetic theories emphasize the distribution of gene effects at individual loci or the distribution of mutant effects in maintaining quantitative genetic variation, but infrequently stress the distribution of gene effects among loci. In this study, we analyse the effects of the distribution of quantitative trait nucleotides (QTN) effects among sites under artificial and stabilizing selection. Wright's formula is applied to describing the density distribution of allele frequencies of multiple diallelic QTN at the equilibrium of mutation-drift-selection. Our results demonstrate that the distribution of QTN effects among sites can affect additive genetic variation in terms of total additive variance, average gene diversity, per-class contribution of QTN effects and per-QTN contribution. When the distribution of QTN effects among sites is changed from L-shaped to bell-shaped or to be a flatter, both the total additive variance and the average gene diversity are changed. Per-class and per-QTN contributions exhibit different distribution patterns. The L-shaped distribution indicates the predominant role of the aggregative effects from the QTN of small finite effects. The bell-shaped or flatter distributions indicate the predominance of the QTN of intermediate and large effects. These predictions highlight the significance of the distribution of QTN effects among sites in interpreting the maintenance of quantitative genetic variation at the fine genome scale.

The frequency of the perfect genotype in a population subject to pleiotropic mutation

We consider a large population of asexual organisms characterised by a number of quantitative traits that are subject to stabilising selection. Mutation is taken to act pleiotropically, with every mutation generally changing all of the traits under selection. We focus on the equilibrium distribution of the population, where mutation and selection are in balance. It has been previously established that the equilibrium distribution of genotypic effects may be anomalous, as it may contain a singular spike--a Dirac delta function--corresponding to a non-zero proportion of the population having exactly optimal genotypic values. In the present work, we present exact results for the case where three traits are under selection. These results give the equilibrium genetic variance of the population, and the proportion of the population that have the optimal genotype. This is achieved for two different spherically symmetric distributions of mutant effects. Additionally, a simple and robust numerical approach is also presented that allows the treatment of some other mutation distributions, where there are an arbitrary number of selected traits.

Geographical variation in selection, from phenotypes to molecules

Molecular technologies now allow researchers to isolate quantitative trait loci (QTLs) and measure patterns of gene sequence variation within chromosomal regions containing important polymorphisms. I develop a simulation model to investigate gene sequence evolution within genomic regions that harbor QTLs. The QTLs influence a trait experiencing geographical variation in selection, which is common in nature and produces obvious differentiation at the phenotypic level. Counter to expectations, the simulations suggest that selection can substantially affect quantitative genetic variation without altering the amount and pattern of molecular variation at sites closely linked to the QTLs. Even with large samples of gene sequences, the likelihood of rejecting neutrality is often low. The exception is situations where strong selection is combined with low migration among demes, conditions that may be common in many plant species. The results have implications for gene sequence surveys and, perhaps more generally, for interpreting the apparently weak connection between levels of molecular and quantitative trait variation within species.

Near-periodic substitution and the genetic variance induced by environmental change

We investigate a model that describes the evolution of a diploid sexual population in a changing environment. Individuals have discrete generations and are subject to selection on the phenotypic value of a quantitative trait, which is controlled by a finite number of bialleic loci. Environmental change is taken to lead to a uniformly changing optimal phenotypic value. The population continually adapts to the changing environment, by allelic substitution, at the loci controlling the trait. We investigate the detailed interrelation between the process of allelic substitution and the adaptation and variation of the population, via infinite population calculations and finite population simulations. We find a simple relation between the substitution rate and the rate of change of the optimal phenotypic value.

Altruism, sex, and inbreeding when the genotype–phenotype map is additive

Recently published theoretical results suggest that, in a sexual population, when genotypes code for phenotypes in a complex manner, it is possible for altruistic genotypes to spread through a metapopulation (i.e. through a collection of subpopulations). This spread tends to occur during periods when the environment deteriorates throughout the metapopulation. By contrast, under asexual reproduction, non-altruistic genotypes seem to be favoured, at least when subpopulations are substantial in size. The most relevant previous study makes use of Kauffman and Levin's "NK model" as a way to relate genotypes to fitness. Unfortunately, there are both conceptual and technical problems with the application of the NK model to populations that contain many different genotypes (e.g. polymorphic diploid populations with more than a few loci under selection). The present study presents a more tractable and biologically plausible model to study the causal relationship between sexual reproduction and altruism. In particular, phenotypes are determined by additive interactions among alleles at different loci in a diploid genome, with up to 200 loci under selection. In addition, subpopulations are substantially larger than those considered in the most relevant previous work. The results show that, so long as there are multiple "fitness peaks" in "phenotype space", the additive genotype-phenotype map leads to results that are similar to those from the NK model. Various parameters are manipulated in an effort to discover the determinants of altruistic and non-altruistic outcomes. The findings should facilitate further investigations, and they should help to establish the plausibility of the suggested relationship between sexual reproduction and altruism. The results also suggest that inbreeding can lead to a similar result as asexuality. That is, inbreeding seems to enhance the probability that altruistic phenotypes will be eliminated.

Gene action of new mutations in Arabidopsis thaliana

For a newly arising mutation affecting a trait under selection, its degree of dominance relative to the preexisting allele(s) strongly influences its evolutionary impact. We have estimated dominance parameters for spontaneous mutations in a subset of lines derived from a highly inbred founder of Arabidopsis thaliana by at least 17 generations of mutation accumulation (MA). The labor-intensive nature of the crosses and the anticipated subtlety of effects limited the number of MA lines included in this study to 8. Each MA line was selfed and reciprocally crossed to plants representing the founder genotype, and progeny were assayed in the greenhouse. Significant mutational effects on reproductive fitness included a recessive fitness-enhancing effect in one line and fitness-reducing effects, one additive and the other slightly recessive. Mutations conferring earlier phenology or smaller leaves were significantly recessive. For effects increasing leaf number and reducing height at flowering, additive gene action accounted for the expression of the traits. The sole example of a significantly dominant mutational effect delayed phenology. Our findings of recessive action of a fitness-enhancing mutational effect and additive action of a deleterious effect counter a common expectation of (partial) dominance of alleles that increase fitness, but the frequency of occurrence of such mutations is unknown.

A finite locus effect diffusion model for the evolution of a quantitative trait

A diffusion model is constructed for the joint distribution of absolute locus effect sizes and allele frequencies for loci contributing to an additive quantitative trait under selection in a haploid, panmictic population. The model is designed to approximate a discrete model exactly in the limit as both population size and the number of loci affecting the trait tend to infinity. For the case when all loci have the same absolute effect size, formal multiple-timescale asymptotics are used to predict the long-time response of the population trait mean to selection. For the case where loci can take on either of two distinct effect sizes, not necessarily with equal probability, numerical solutions of the system indicate that response to selection of a quantitative trait is insensitive to the variability of the distribution of effect sizes when mutation is negligible.

Constrained evolution of a quantitative character by pleiotropic mutation

The long-term response to directional selection and its selection limit are derived for a quantitative character that is controlled by pleiotropic mutations with direct deleterious effect on fitness. Directional selection is assumed to be weaker than the selection acting directly on mutations via deleterious effects (purging selection), which renders all mutations to eventual elimination. The analysis embedding this restrictive assumption indicates that the evolutionary response of the character starting from an equilibrium state, in which mutation and purging selection balance but no directional selection is operating, decreases monotonically with time at an exponential rate. And the fading rate of responses is mostly determined by the direct deleterious effect. Contrary to the expectation by the standard selection limit theory based on fixation of extant genetic variation, the present model predicts that the selection limit depends on the intensity of directional selection, the limit being proportional to the ratio of the directional selection intensity to the direct deleterious effect. A slightly larger genetic variance is maintained at the selection limit than would be without directional selection.

Theoretical models of selection and mutation on quantitative traits

Empirical studies of quantitative genetic variation have revealed robust patterns that are observed both across traits and across species. However, these patterns have no compelling explanation, and some of the observations even appear to be mutually incompatible. We review and extend a major class of theoretical models, 'mutation-selection models', that have been proposed to explain quantitative genetic variation. We also briefly review an alternative class of 'balancing selection models'. We consider to what extent the models are compatible with the general observations, and argue that a key issue is understanding and modelling pleiotropy. We discuss some of the thorny issues that arise when formulating models that describe many traits simultaneously.

Genetic variability under mutation selection balance

A fundamental problem in evolutionary genetics is understanding how high levels of genetic variation in quantitative traits are maintained in natural populations. Variation is removed by the natural selection of individuals with optimal phenotypes and is recovered by mutation; however, previous analyses had indicated that a mutation-selection balance was insufficient to maintain observed levels of genetic variation in these traits. Using more general models, however, it has recently been shown that it is indeed a sufficient mechanism. These models can be used to explore other phenomena in evolutionary biology.

The population genetic theory of hidden variation and genetic robustness

One of the most solid generalizations of transmission genetics is that the phenotypic variance of populations carrying a major mutation is increased relative to the wild type. At least some part of this higher variance is genetic and due to release of previously hidden variation. Similarly, stressful environments also lead to the expression of hidden variation. These two observations have been considered as evidence that the wild type has evolved robustness against genetic variation, i.e., genetic canalization. In this article we present a general model for the interaction of a major mutation or a novel environment with the additive genetic basis of a quantitative character under stabilizing selection. We introduce an approximation to the genetic variance in mutation-selection-drift balance that includes the previously used stochastic Gaussian and house-of-cards approximations as limiting cases. We then show that the release of hidden genetic variation is a generic property of models with epistasis or genotype-environment interaction, regardless of whether the wild-type genotype is canalized or not. As a consequence, the additive genetic variance increases upon a change in the environment or the genetic background even if the mutant character state is as robust as the wild-type character. Estimates show that this predicted increase can be considerable, in particular in large populations and if there are conditionally neutral alleles at the loci underlying the trait. A brief review of the relevant literature suggests that the assumptions of this model are likely to be generic for polygenic traits. We conclude that the release of hidden genetic variance due to a major mutation or environmental stress does not demonstrate canalization of the wild-type genotype.

Polygenic mutation in Drosophila melanogaster: Mapping spontaneous mutations affecting sensory bristle number

Our ability to predict long-term responses to artificial and natural selection, and understand the mechanisms by which naturally occurring variation for quantitative traits is maintained, depends on detailed knowledge of the properties of spontaneous polygenic mutations, including the quantitative trait loci (QTL) at which mutations occur, mutation rates, and mutational effects. These parameters can be estimated by mapping QTL that cause divergence between mutation-accumulation lines that have been established from an inbred base population and selected for high and low trait values. Here, we have utilized quantitative complementation to deficiencies to map QTL at which spontaneous mutations affecting Drosophila abdominal and sternopleural bristle number have occurred in 11 replicate lines during 206 generations of divergent selection. Estimates of the numbers of mutations were consistent with diploid per-character mutation rates for bristle traits of 0.03. The ratio of the per-character mutation rate to total mutation rate (0.023) implies that >2% of the genome could affect just one bristle trait and that there must be extensive pleiotropy for quantitative phenotypes. The estimated mutational effects were not, however, additive and exhibited dependency on genetic background consistent with diminishing epistasis. However, these inferences must be tempered by the potential for epistatic interactions between spontaneous mutations and QTL affecting bristle number on the deficiency-bearing chromosomes, which could lead to overestimates in numbers of QTL and inaccurate inference of gene action.

Comparative evolutionary genetics of spontaneous mutations affecting fitness in rhabditid nematodes

Deleterious mutations are of fundamental importance to all aspects of organismal biology. Evolutionary geneticists have expended tremendous effort to estimate the genome-wide rate of mutation and the effects of new mutations on fitness, but the degree to which genomic mutational properties vary within and between taxa is largely unknown, particularly in multicellular organisms. Beginning with two highly inbred strains from each of three species in the nematode family Rhabditidae (Caenorhabditis briggsae, Caenorhabditis elegans, and Oscheius myriophila), we allowed mutations to accumulate in the relative absence of natural selection for 200 generations. We document significant variation in the rate of decay of fitness because of new mutations between strains and between species. Estimates of the per-generation mutational decay of fitness were very consistent within strains between assays 100 generations apart. Rate of mutational decay in fitness was positively associated with genomic mutation rate and negatively associated with average mutational effect. These results provide unambiguous experimental evidence for substantial variation in genome-wide properties of mutation both within and between species and reinforce conclusions from previous experiments that the cumulative effects on fitness of new mutations can differ markedly among related taxa.

Predictions of patterns of response to artificial selection in lines derived from natural populations

The pattern of response to artificial selection on quantitative traits in laboratory populations can tell us something of the genetic architecture in the natural population from which they were derived. We modeled artificial selection in samples drawn from natural populations in which variation had been maintained by recurrent mutation, with genes having an effect on the trait, which was subject to real stabilizing selection, and a pleitropic effect on fitness (the joint-effect model). Natural selection leads to an inverse correlation between effects and frequencies of genes, such that the frequency distribution of genes increasing the trait has an extreme U-shape. In contrast to the classical infinitesimal model, an early accelerated response and a larger variance of response among replicates were predicted. However, these are reduced if the base population has been maintained in the laboratory for some generations by random sampling prior to artificial selection. When multiple loci and linkage are also taken into account, the gametic disequilibria generated by the Bulmer and Hill-Robertson effects are such that little or no increase in variance and acceleration of response in early generations of artificial selection are predicted; further, the patterns of predicted responses for the joint-effect model now become close to those of the infinitesimal model. Comparison with data from laboratory selection experiments shows that, overall, the analysis did not provide clear support for the joint-effect model or a clear case for rejection.

A one locus, biased mutation model and its equivalence to an unbiased model

Experimental data suggests that for some continuously-varying characters under stabilising selection, mutation may cause a mean change in the value of the character. A one locus, mathematical model of a continuously-varying biological character with this property of biased mutation is investigated. Via a mathematical transformation, the equilibrium equation describing a large population of individuals is reduced to the equilibrium equation describing a mutationally unbiased problem. Knowledge of an unbiased problem is thus sufficient to determine all equilibrium properties of the corresponding biased problem. In the biased mutation problem, the dependence of the mean equilibrium value of the character, as a function of the mutational bias, is non-monotonic and remains small, for all levels of mutational bias. The analysis presented in this work sheds new light on Turelli's House of Cards Approximation.

Mutation accumulation in populations of varying size: the distribution of mutational effects for fitness correlates in Caenorhabditis elegans

The consequences of mutation for population-genetic and evolutionary processes depend on the rate and, especially, the frequency distribution of mutational effects on fitness. We sought to approximate the form of the distribution of mutational effects by conducting divergence experiments in which lines of a DNA repair-deficient strain of Caenorhabditis elegans, msh-2, were maintained at a range of population sizes. Assays of these lines conducted in parallel with the ancestral control suggest that the mutational variance is dominated by contributions from highly detrimental mutations. This was evidenced by the ability of all but the smallest population-size treatments to maintain relatively high levels of mean fitness even under the 100-fold increase in mutational pressure caused by knocking out the msh-2 gene. However, we show that the mean fitness decline experienced by larger populations is actually greater than expected on the basis of our estimates of mutational parameters, which could be consistent with the existence of a common class of mutations with small individual effects. Further, comparison of the total mutation rate estimated from direct sequencing of DNA to that detected from phenotypic analyses implies the existence of a large class of evolutionarily relevant mutations with no measurable effect on laboratory fitness.

Polygenic variation maintained by balancing selection: pleiotropy, sex-dependent allelic effects and G x E interactions

We investigate three alternative selection-based scenarios proposed to maintain polygenic variation: pleiotropic balancing selection, G x E interactions (with spatial or temporal variation in allelic effects), and sex-dependent allelic effects. Each analysis assumes an additive polygenic trait with n diallelic loci under stabilizing selection. We allow loci to have different effects and consider equilibria at which the population mean departs from the stabilizing-selection optimum. Under weak selection, each model produces essentially identical, approximate allele-frequency dynamics. Variation is maintained under pleiotropic balancing selection only at loci for which the strength of balancing selection exceeds the effective strength of stabilizing selection. In addition, for all models, polymorphism requires that the population mean be close enough to the optimum that directional selection does not overwhelm balancing selection. This balance allows many simultaneously stable equilibria, and we explore their properties numerically. Both spatial and temporal G x E can maintain variation at loci for which the coefficient of variation (across environments) of the effect of a substitution exceeds a critical value greater than one. The critical value depends on the correlation between substitution effects at different loci. For large positive correlations (e.g., rho(ij)2>3/4), even extreme fluctuations in allelic effects cannot maintain variation. Surprisingly, this constraint on correlations implies that sex-dependent allelic effects cannot maintain polygenic variation. We present numerical results that support our analytical approximations and discuss our results in connection to relevant data and alternative variance-maintaining mechanisms.

Influence of dominance, leptokurtosis and pleiotropy of deleterious mutations on quantitative genetic variation at mutation-selection balance

In models of maintenance of genetic variance (V (G)) it has often been assumed that mutant alleles act additively. However, experimental data show that the dominance coefficient varies among mutant alleles and those of large effect tend to be recessive. On the basis of empirical knowledge of mutations, a joint-effect model of pleiotropic and real stabilizing selection that includes dominance is constructed and analyzed. It is shown that dominance can dramatically alter the prediction of equilibrium V (G). Analysis indicates that for the situations where mutations are more recessive for fitness than for a quantitative trait, as supported by the available data, the joint-effect model predicts a significantly higher V (G) than does an additive model. Importantly, for what seem to be realistic distributions of mutational effects (i.e., many mutants may not affect the quantitative trait substantially but are likely to affect fitness), the observed high levels of genetic variation in the quantitative trait under strong apparent stabilizing selection can be generated. This investigation supports the hypothesis that most V (G) comes from the alleles nearly neutral for fitness in heterozygotes while apparent stabilizing selection is contributed mainly by the alleles of large effect on the quantitative trait. Thus considerations of dominance coefficients of mutations lend further support to our previous conclusion that mutation-selection balance is a plausible mechanism of the maintenance of the genetic variance in natural populations.

Orientation of the genetic variance-covariance matrix and the fitness surface for multiple male sexually selected traits

Stabilizing selection has been predicted to change genetic variances and covariances so that the orientation of the genetic variance-covariance matrix (G) becomes aligned with the orientation of the fitness surface, but it is less clear how directional selection may change G. Here we develop statistical approaches to the comparison of G with vectors of linear and nonlinear selection. We apply these approaches to a set of male sexually selected cuticular hydrocarbons (CHCs) of Drosophila serrata. Even though male CHCs displayed substantial additive genetic variance, more than 99% of the genetic variance was orientated 74.9 degrees away from the vector of linear sexual selection, suggesting that open-ended female preferences may greatly reduce genetic variation in male display traits. Although the orientation of G and the fitness surface were found to differ significantly, the similarity present in eigenstructure was a consequence of traits under weak linear selection and strong nonlinear (convex) selection. Associating the eigenstructure of G with vectors of linear and nonlinear selection may provide a way of determining what long-term changes in G may be generated by the processes of natural and sexual selection.

Polygenic Variation Maintained by Balancing Selection: Pleiotropy, Sex-Dependent Allelic Effects and G × E Interactions

We investigate three alternative selection-based scenarios proposed to maintain polygenic variation: pleiotropic balancing selection, G × E interactions (with spatial or temporal variation in allelic effects), and sex-dependent allelic effects. Each analysis assumes an additive polygenic trait with n diallelic loci under stabilizing selection. We allow loci to have different effects and consider equilibria at which the population mean departs from the stabilizing-selection optimum. Under weak selection, each model produces essentially identical, approximate allele-frequency dynamics. Variation is maintained under pleiotropic balancing selection only at loci for which the strength of balancing selection exceeds the effective strength of stabilizing selection. In addition, for all models, polymorphism requires that the population mean be close enough to the optimum that directional selection does not overwhelm balancing selection. This balance allows many simultaneously stable equilibria, and we explore their properties numerically. Both spatial and temporal G × E can maintain variation at loci for which the coefficient of variation (across environments) of the effect of a substitution exceeds a critical value greater than one. The critical value depends on the correlation between substitution effects at different loci. For large positive correlations (e.g., ρij2>3∕4), even extreme fluctuations in allelic effects cannot maintain variation. Surprisingly, this constraint on correlations implies that sex-dependent allelic effects cannot maintain polygenic variation. We present numerical results that support our analytical approximations and discuss our results in connection to relevant data and alternative variance-maintaining mechanisms.

Multivariate stabilizing selection and pleiotropy in the maintenance of quantitative genetic variation

We investigate maintenance of quantitative genetic variation at mutation-selection balance for multiple traits. The intrinsic strength of real stabilizing selection on one of these traits denoted the "target trait" and the observed strength of apparent stabilizing selection on the target trait can be quite different: the latter, which is estimable, is much smaller (i.e., implying stronger selection) than the former. Distinguishing them may enable the mutation load to be relaxed when considering multivariate stabilizing selection. It is shown that both correlations among mutational effects and among strengths of real stabilizing selection on the traits are not important unless they are high. The analysis for independent situations thus provides a good approximation to the case where mutant and stabilizing selection effects are correlated. Multivariate stabilizing selection can be regarded as a combination of stabilizing selection on the target trait and the pleiotropic direct selection on fitness that is solely due to the effects of real stabilizing selection on the hidden traits. As the overall fitness approaches a constant value as the number of traits increases, multivariate stabilizing selection can maintain abundant genetic variance only under quite weak selection. The common observations of high polygenic variance and strong stabilizing selection thus imply that if the mutation-selection balance is the true mechanism of maintenance of genetic variation, the apparent stabilizing selection cannot arise solely by real stabilizing selection simultaneously on many metric traits.