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.     

A population genetics model for multiple quantitative traits exhibiting pleiotropy and epistasis

We study a population genetics model of an organism with a genome of L(tot)loci that determine the values of T quantitative traits. Each trait is controlled by a subset of L loci assigned randomly from the genome. There is an optimum value for each trait, and stabilizing selection acts on the phenotype as a whole to maintain actual trait values close to their optima. The model contains pleiotropic effects (loci can affect more than one trait) and epistasis in fitness. We use adaptive walk simulations to find high-fitness genotypes and to study the way these genotypes are distributed in sequence space. We then simulate the evolution of haploid and diploid populations on these fitness landscapes and show that the genotypes of populations are able to drift through sequence space despite stabilizing selection on the phenotype. We study the way the rate of drift and the extent of the accessible region of sequence space is affected by mutation rate, selection strength, population size, recombination rate, and the parameters L and T that control the landscape shape. There are three regimes of the model. If LTL(tot), there are many small peaks that can be spread over a wide region of sequence space. Compensatory neutral mutations are important in the population dynamics in this case.     

Genetic variation and the fate of beneficial mutations in asexual populations

The fate of a newly arising beneficial mutation depends on many factors, such as the population size and the availability and fitness effects of other mutations that accumulate in the population. It has proved difficult to understand how these factors influence the trajectories of particular mutations, since experiments have primarily focused on characterizing successful clones emerging from a small number of evolving populations. Here, we present the results of a massively parallel experiment designed to measure the full spectrum of possible fates of new beneficial mutations in hundreds of experimental yeast populations, whether these mutations are ultimately successful or not. Using strains in which a particular class of beneficial mutation is detectable by fluorescence, we followed the trajectories of these beneficial mutations across 592 independent populations for 1000 generations. We find that the fitness advantage provided by individual mutations plays a surprisingly small role. Rather, underlying "background" genetic variation is quickly generated in our initially clonal populations and plays a crucial role in determining the fate of each individual beneficial mutation in the evolving population.     

The Genetic Covariance between Characters Maintained by Pleiotropic Mutations

A statistical genetic model of a multivariate phenotype is derived to investigate the covariation of pleiotropic mutations with additive effects under the combined action of phenotypic selection, linkage and the mating system. Equilibrium formulas for large, randomly mating populations demonstrate that, when selection on polygenic variation is much smaller than twice the harmonic mean recombination rate between loci with interacting fitnesses, linkage disequilibrium is negligible and pleiotropy is the main cause of genetic correlations between characters. Under these conditions, approximate expressions for the dynamics of the genetic covariances due to pleiotropic mutations are obtained. Patterns of genetic covariance between characters and their evolution are discussed with reference to data on polygenic mutation, chromosomal organization and morphological integration.     

The multiplicity of divergence mechanisms in a single evolving population

Background

Evolutionary divergence is common within bacterial species and populations, even during a single bacterial infection. We use large-scale genomic and phenotypic analysis to identify the extent of diversification in controlled experimental populations and apply these data to differentiate between several potential mechanisms of evolutionary divergence.

Results

We defined testable differences between five proposed mechanisms and used experimental evolution studies to follow eight glucose-limited Escherichia coli chemostat populations at two growth rates. Simple phenotypic tests identified 11 phenotype combinations evolving under glucose limitation. Each evolved population exhibited 3 to 5 different combinations of the 11 phenotypic clusters. Genome sequencing of a representative of each phenotypic cluster from each population identified 193 mutations in 48 isolates. Only two of the 48 strains had evolved identically. Convergent paths to the same phenotype occurred, but two pleiotropic mutations were unique to slow-growing bacteria, permitting them greater phenotypic variance. Indeed, greater diversity arose in slower-growing, more stressed cultures. Mutation accumulation, hypermutator presence and fitness mechanisms varied between and within populations, with the evolved fitness considerably more uniform with fast growth cultures. Negative frequency-dependent fitness was shown by a subset of isolates.

Conclusions

Evolutionary diversity is unlikely to be explained by any one of the available mechanisms. For a large population as used in this study, our results suggest that multiple mechanisms contribute to the mix of phenotypes and evolved fitness types in a diversifying population. Another major conclusion is that the capacity of a population to diversify is a function of growth rate.     

Selection on a modifier of recombination rate due to linked deleterious mutations

Several models have been suggested to explain the origin and maintenance of recombination. Here I present the results from computer simulations of multilocus haploid and diploid genotypes in small populations. Each chromosome consisted of 1001 loci where deleterious mutations occurred. At "equilibrium" for mutation-selection-genetic drift balance a single recombination variant was introduced to the population in the middle of a chromosome. On average 75,000 replicates for each combination of parameters were followed to fixation or loss of the modifier allele. The results show that, in a small population, increased recombination can be selected, even in the absence of epistasis or beneficial mutations. The effect of the mutation rate for deleterious mutations depends on the ploidy level and the recessiveness of deleterious mutations. A higher deleterious mutation rate is required for an increase in recombination rate to be favored in haploid populations. Increased recombination could not evolve in the case of strong associative overdominance.     

Fitness effects of fixed beneficial mutations in microbial populations

Beneficial mutations are intuitively relevant to understanding adaptation, yet not all beneficial mutations are of consequence to the long-term evolutionary outcome of adaptation. Many beneficial mutations-mostly those of small effect-are lost due either to (1) genetic drift or to (2) competition among clones carrying different beneficial mutations, a phenomenon called the "Hill-Robertson effect" for sexual populations and "clonal interference" for asexual populations. Competition among clones becomes more prevalent with increasing genetic linkage and increasing population size, and it is thus generally characteristic of microbial populations. Together, these two phenomena suggest that only those beneficial mutations of large fitness effect should achieve fixation, despite the fact that most beneficial mutations produced are predicted to have very small fitness effects. Here, we confirm this prediction-both empirically and theoretically-by showing that fitness effects of fixed beneficial mutations follow a distribution whose mode is positive.     

Limits to adaptation in asexual populations

In asexual populations, the rate of adaptation is basically limited by the frequency and properties of spontaneous beneficial mutations. Hence, knowledge of these mutational properties and how they are affected by particular evolutionary conditions is a precondition for understanding the process of adaptation. Here, we address how the rate of adaptation of asexual populations is limited by its population size and mutation rate, as well as by two factors affecting the fraction of mutations that confer a benefit, i.e. the initial adaptedness of the population and the variability of the environment. These factors both influence which mutations are likely to occur, as well as the probability that they will ultimately contribute to adaptation. We attempt to separate the consequences of these basic population features in terms of their effect on the rate of adaptation by using results from evolution experiments with microorganisms.     

The effect of antagonistic pleiotropy on the estimation of the average coefficient of dominance of deleterious mutations

We investigate the impact of antagonistic pleiotropy on the most widely used methods of estimation of the average coefficient of dominance of deleterious mutations from segregating populations. A proportion of the deleterious mutations affecting a given studied fitness component are assumed to have an advantageous effect on another one, generating overdominance on global fitness. Using diffusion approximations and transition matrix methods, we obtain the distribution of gene frequencies for nonpleiotropic and pleiotropic mutations in populations at the mutation-selection-drift balance. From these distributions we build homozygous and heterozygous chromosomes and assess the behavior of the estimators of dominance. A very small number of deleterious mutations with antagonistic pleiotropy produces substantial increases on the estimate of the average degree of dominance of mutations affecting the fitness component under study. For example, estimates are increased three- to fivefold when 2% of segregating loci are over-dominant for fitness. In contrast, strengthening pleiotropy, where pleiotropic effects are assumed to be also deleterious, has little effect on the estimates of the average degree of dominance, supporting previous results. The antagonistic pleiotropy model considered, applied under mutational parameters described in the literature, produces patterns for the distribution of chromosomal viabilities, levels of genetic variance, and homozygous mutation load generally consistent with those observed empirically for viability in Drosophila melanogaster.     

Two steps forward, one step back: the pleiotropic effects of favoured alleles

Pleiotropy is one of the most commonly observed attributes of genes. Yet the extent and influence of pleiotropy have been underexplored in population genetics models. In this paper, I quantify the extent to which pleiotropy inhibits the spread of alleles in response to directional selection on a focal trait. Under the assumption that pleiotropic effects are extensive and deleterious, the fraction of alleles that are beneficial overall is severely limited by pleiotropy and rises nearly linearly with the strength of directional selection on the focal trait. Over a broad class of distribution of pleiotropic effects, the mean selective effect of those alleles that are beneficial overall is halved, on average, by pleiotropy. The fraction of new mutant alleles that are beneficial overall and that succeed in fixing within a population is even more severely limited when directional selection is weak, but it rises quadratically with the strength of directional selection. Finally, the mean selective effect of mutant alleles that are beneficial and succeed in fixing is reduced by one-third, on average, by pleiotropy. These results help to shape our understanding of the evolutionary inertia caused by pleiotropy.     

Fixation of new alleles and the extinction of small populations: drift load, beneficial alleles, and sexual selection

With a small effective population size, random genetic drift is more important than selection in determining the fate of new alleles. Small populations therefore accumulate deleterious mutations. Left unchecked, the effect of these fixed alleles is to reduce the reproductive capacity of a species, eventually to the point of extinction. New beneficial mutations, if fixed by selection, can restore some of this lost fitness. This paper derives the overall change in fitness due to fixation of new deleterious and beneficial alleles, as a function of the distribution of effects of new mutations and the effective population size. There is a critical effective size below which a population will on average decline in fitness, but above which beneficial mutations allow the population to persist. With reasonable estimates of the relevant parameters, this critical effective size is likely to be a few hundred. Furthermore, sexual selection can act to reduce the fixation probability of deleterious new mutations and increase the probability of fixing new beneficial mutations. Sexual selection can therefore reduce the risk of extinction of small populations.     

Population Size Affects Adaptation in Complex Ways: Simulations on Empirical Adaptive Landscapes

Do large populations always outcompete smaller ones? Does increasing the mutation rate have a similar effect to increasing the population size, with respect to the adaptation of a population? How important are substitutions in determining the adaptation rate? In this study, we ask how population size and mutation rate interact to affect adaptation on empirical adaptive landscapes. Using such landscapes, we do not need to make many ad hoc assumption about landscape topography, such as about epistatic interactions among mutations or about the distribution of fitness effects. Moreover, we have a better understanding of all the mutations that occur in a population and their effects on the average fitness of the population than we can know in experimental studies. Our results show that the evolutionary dynamics of a population cannot be fully explained by the population mutation rate \(N\mu\); even at constant \(N\mu\), there can be dramatic differences in the adaptation of populations of different sizes. Moreover, the substitution rate of mutations is not always equivalent to the adaptation rate, because we observed populations adapting to high adaptive peaks without fixing any mutations. Finally, in contrast to some theoretical predictions, even on the most rugged landscapes we study, small population size is never an advantage over larger population size. These result show that complex interactions among multiple factors can affect the evolutionary dynamics of populations, and simple models should be taken with caution.     

SINGLE‐GENE SPECIATION WITH PLEIOTROPY: EFFECTS OF ALLELE DOMINANCE,POPULATION SIZE,AND DELAYED INHERITANCE

Single‐gene speciation is considered to be unlikely, but an excellent example is found in land snails, in which a gene for left‐right reversal has given rise to new species multiple times. This reversal might be facilitated by their small population sizes and maternal effect (i.e., “delayed inheritance,” in which an individual's phenotype is determined by the genotype of its mother). Recent evidence suggests that a pleiotropic effect of the speciation gene on antipredator survival may also promote speciation. Here we theoretically demonstrate that, without a pleiotropic effect, in small populations the fixation probability of a recessive mutant is higher than a dominant mutant, but they are identical for large populations and sufficiently weak selection. With a pleiotropic effect that increases mutant viability, a dominant mutant has a higher fixation probability if the strength of viability selection is sufficiently greater than that of reproductive incompatibility, whereas a recessive mutant has a higher fixation probability otherwise. Delayed inheritance increases the fixation probability of a mutant if viability selection is sufficiently weaker than reproductive incompatibility. Our results clarify the conflicting effects of viability selection and positive frequency‐dependent selection due to reproductive incompatibility and provide a new perspective to single‐gene speciation theory.     

Pleiotropic effects of beneficial mutations in Escherichia coli

Micromutational models of adaptation have placed considerable weight on antagonistic pleiotropy as a mechanism that prevents mutations of large effect from achieving fixation. However, there are few empirical studies of the distribution of pleiotropic effects, and no studies that have examined this distribution for a large number of adaptive mutations. Here we examine the form and extent of pleiotropy associated with beneficial mutations in Escherichia coli. To do so, we used a collection of independently evolved genotypes, each of which contains a beneficial mutation that confers increased fitness in a glucose-limited environment. To determine the pleiotropic effects of these mutations, we examined the fitnesses of the mutants in five novel resource environments. Our results show that the majority of mutations had significant fitness effects in alternative resources, such that pleiotropy was common. The predominant form of this pleiotropy was positive--that is, most mutations that conferred increased fitness in glucose also conferred increased fitness in novel resources. We did detect some deleterious pleiotropic effects, but they were primarily limited to one of the five resources, and within this resource, to only a subset of mutants. Although pleiotropic effects were generally positive, fitness levels were lower and more variable on resources that differed most in their mechanisms of uptake and catabolism from that of glucose. Positive pleiotropic effects were strongly correlated in magnitude with their direct effects, but no such correlation was found among mutants with deleterious pleiotropic effects. Whereas previous studies of populations evolved on glucose for longer periods of time showed consistent declines on some of the resources used here, our results suggest that deleterious pleiotropic effects were limited to only a subset of the beneficial mutations available.     

Efficient simulation under a population genetics model of carcinogenesis

MOTIVATION: Cancer is well known to be the end result of somatic mutations that disrupt normal cell division. The number of such mutations that have to be accumulated in a cell before cancer develops depends on the type of cancer. The waiting time T(m) until the appearance of m mutations in a cell is thus an important quantity in population genetics models of carcinogenesis. Such models are often difficult to analyze theoretically because of the complex interactions of mutation, drift and selection. They are also computationally expensive to simulate because of the large number of cells and the low mutation rate. RESULTS: We develop an efficient algorithm for simulating the waiting time T(m) until m mutations under a population genetics model of cancer development. We use an exact algorithm to simulate evolution of small cell populations and coarse-grained τ-leaping approximation to handle large populations. We compared our hybrid simulation algorithm with the exact algorithm in small populations and with available asymptotic results for large populations. The comparison suggested that our algorithm is accurate and computationally efficient. We used the algorithm to study the waiting time for up to 20 mutations under a Moran model with variable population sizes. Our new algorithm may be useful for studying realistic models of carcinogenesis, which incorporates variable mutation rates and fitness effects.     

Mutation-selection balance accounting for genetic variation for viability in Drosophila melanogaster as deduced from an inbreeding and artificial selection experiment

We carried out an experiment of inbreeding and upward artificial selection for egg-to-adult viability in a recently captured population of Drosophila melanogaster, as well as computer simulations of the experimental design, in order to obtain information on the nature of genetic variation for this important fitness component. The inbreeding depression was linear with a rate of 0.70 +/- 0.11% of the initial mean per 1% increase in inbreeding coefficient, and the realized heritability was 0.06 +/- 0.07. We compared the empirical observations of inbreeding depression and selection response with computer simulations assuming a balance between the occurrence of partially recessive deleterious mutations and their elimination by selection. Our results suggest that a model assuming mutation-selection balance with realistic mutational parameters can explain the genetic variation for viability in the natural population studied. Several mutational models are incompatible with some observations and can be discarded. Mutational models assuming a low rate of mutations of large average effect and highly recessive gene action, and others assuming a high rate of mutations of small average effect and close to additive gene action, are compatible with all the observations.     

Should we expect substitution rate to depend on population size?

The rate of nucleotide substitution is generally believed to be a decreasing function of effective population size, at least for nonsynonymous substitutions. This view was originally based on consideration of slightly deleterious mutations with a fixed distribution of selection coefficients. A realistic model must include the occurrence and fixation of some advantageous mutations that compensate for the loss of fitness due to deleterious substitutions. Some such models, such as so-called "fixed" models, also predict a population size effect on substitution rate. An alternative model, presented here, predicts the near absence of a population size effect on substitution rate. This model is based on concave log-fitness functions and a fixed distribution of mutational effects on the selectively important trait. Simulations of an instance of the model confirm the approximate insensitivity of the substitution rate to population size. Although much experimental evidence has been claimed to support the existence of a population size effect, the body of evidence as a whole is equivocal, and much of the evidence that is supposed to demonstrate such an effect would also suggest that it is very small. Perhaps the proposed model applies well to some genes and not so well to others, and genes therefore vary with regard to the population size effect.     

Evolutionary developmental biology and the problem of variation

Abstract. One of the oldest problems in evolutionary biology remains largely unsolved. Which mutations generate evolutionarily relevant phenotypic variation? What kinds of molecular changes do they entail? What are the phenotypic magnitudes, frequencies of origin, and pleiotropic effects of such mutations? How is the genome constructed to allow the observed abundance of phenotypic diversity? Historically, the neo‐Darwinian synthesizers stressed the predominance of micromutations in evolution, whereas others noted the similarities between some dramatic mutations and evolutionary transitions to argue for macromutationism. Arguments on both sides have been biased by misconceptions of the developmental effects of mutations. For example, the traditional view that mutations of important developmental genes always have large pleiotropic effects can now be seen to be a conclusion drawn from observations of a small class of mutations with dramatic effects. It is possible that some mutations, for example, those in cis‐regulatory DNA, have few or no pleiotropic effects and may be the predominant source of morphological evolution. In contrast, mutations causing dramatic phenotypic effects, although superficially similar to hypothesized evolutionary transitions, are unlikely to fairly represent the true path of evolution. Recent developmental studies of gene function provide a new way of conceptualizing and studying variation that contrasts with the traditional genetic view that was incorporated into neo‐Darwinian theory and population genetics. This new approach in developmental biology is as important for micro‐evolutionary studies as the actual results from recent evolutionary developmental studies. In particular, this approach will assist in the task of identifying the specific mutations generating phenotypic variation and elucidating how they alter gene function. These data will provide the current missing link between molecular and phenotypic variation in natural populations.     

GENETIC SIGNATURE OF ADAPTIVE PEAK SHIFT IN THREESPINE STICKLEBACK

Transition of an evolving population to a new adaptive optimum is predicted to leave a signature in the distribution of effect sizes of fixed mutations. If they affect many traits (are pleiotropic), large effect mutations should contribute more when a population evolves to a farther adaptive peak than to a nearer peak. We tested this prediction in wild threespine stickleback fish (Gasterosteus aculeatus) by comparing the estimated frequency of large effect genetic changes underlying evolution as the same ancestor adapted to two lake types since the end of the ice age. A higher frequency of large effect genetic changes (quantitative trait loci) contributed to adaptive evolution in populations that adapted to lakes representing a more distant optimum than to lakes in which the optimum phenotype was nearer to the ancestral state. Our results also indicate that pleiotropy, not just optimum overshoot, contributes to this difference. These results suggest that a series of adaptive improvements to a new environment leaves a detectable mark in the genome of wild populations. Although not all assumptions of the theory are likely met in natural systems, the prediction may be robust enough to the complexities of natural environments to be useful when forecasting adaptive responses to large environmental changes.     

LIFE AT THE FRONT OF AN EXPANDING POPULATION

Environmental changes have caused episodes of habitat expansions in the evolutionary history of many species. These range changes affect the dynamics of biological evolution in multiple ways. Recent microbial experiments as well as simulations suggest that enhanced genetic drift at the frontier of a two-dimensional range expansion can cause genetic sectoring patterns with fractal domain boundaries. Here, we propose and analyze a simple model of asexual biological evolution at expanding frontiers that explains these neutral patterns and predicts the effect of natural selection. We find that beneficial mutations give rise to sectors with an opening angle that depends sensitively on the selective advantage of the mutants. Deleterious mutations, on the other hand, are not able to establish a sector permanently. They can, however, temporarily "surf" on the population front, and thereby reach unusually high frequencies. As a consequence, expanding frontiers are loaded with a high fraction of mutants at mutation–selection balance. Numerically, we also determine the condition at which the wild type is lost in favor of deleterious mutants (genetic meltdown) at a growing front. Our prediction for this error threshold differs qualitatively from existing well-mixed theories, and sets tight constraints on sustainable mutation rates for populations that undergo frequent range expansions.