Note on the computation of critical effective population sizes.

This work extends the work of Whitlock in examining the critical effective population sizes from the fixation of both deleterious and beneficial mutations under drift and selection to prevent mutation breakdown of the population. The validity of approximations for the probability of fixation depends on the nature of the assumed distribution for the fitness effect of both types of mutations. Using no approximation for the probability of fixation and assuming a heavy tailed fitness effect distribution, the current model indicates that the coefficients of variation for the fitness effect distributions of both types of mutations and the fitness effect distribution mean for the beneficial mutations are important predictors of the critical effective population size. The current model further predicts that very small populations can be sustained if the fitness effect variances for both types of mutations and the mean for beneficial mutations are large.     

Population genetics of the cytoplasm and the units of selection on mitochondrial DNA in <Emphasis Type="Italic">Drosophila melanogaster</Emphasis>

Biological variation exists across a nested set of hierarchical levels from nucleotides within genes to populations within species to lineages within the tree of life. How selection acts across this hierarchy is a long-standing question in evolutionary biology. Recent studies have suggested that genome size is influenced largely by the balance of selection, mutation and drift in lineages with different population sizes. Here we use population cage and maternal transmission experiments to identify the relative strength of selection at an individual and cytoplasmic level. No significant trends were observed in the frequency of large (L) and small (S) mtDNAs across 14 generations in population cages. In all replicate cages, new length variants were observed in heteroplasmic states indicating that spontaneous length mutations occurred in these experimental populations. Heteroplasmic flies carrying L genomes were more frequent than those carrying S genomes suggesting an asymmetric mutation dynamic from larger to smaller mtDNAs. Mother-offspring transmission of heteroplasmy showed that the L mtDNA increased in frequency within flies both between and within generations despite sampling drift of the same intensity as occurred in population cages. These results suggest that selection for mtDNA size is stronger at the cytoplasmic than at the organismal level. The fixation of novel mtDNAs within and between species requires a transient intracellular heteroplasmic stage. The balance of population genetic forces at the cytoplasmic and individual levels governs the units of selection on mtDNA, and has implications for evolutionary inference as well as for the effects of mtDNA mutations on fitness, disease and aging.     

Multigeneration maximum-likelihood analysis applied to mutation-accumulation experiments in Caenorhabditis elegans

We develop a maximum-likelihood (ML) approach to estimate genomic mutation rates (U) and average homozygous mutation effects (s) from mutation-accumulation (MA) experiments in which phenotypic assays are carried out in several generations. We use simulations to compare the procedure's performance with the method of moments traditionally used to analyze MA data. Similar precision is obtained if mutation effects are small relative to the environmental standard deviation, but ML can give estimates of mutation parameters that have lower sampling variances than those obtained by the method of moments if mutations with large effects have accumulated. The inclusion of data from intermediate generations may improve the precision. We analyze life-history trait data from two Caenorhabditis elegans MA experiments. Under a model with equal mutation effects, the two experiments provide similar estimates for U of approximately 0.005 per haploid, averaged over traits. Estimates of s are more divergent and average at -0.51 and -0.13 in the two studies. Detailed analysis shows that changes of mean and variance of genetic values of MA lines in both C. elegans experiments are dominated by mutations with large effects, but the analysis does not rule out the presence of a large class of deleterious mutations with very small effects.     

Fitness Is Strongly Influenced by Rare Mutations of Large Effect in a Microbial Mutation Accumulation Experiment

Our understanding of the evolutionary consequences of mutation relies heavily on estimates of the rate and fitness effect of spontaneous mutations generated by mutation accumulation (MA) experiments. We performed a classic MA experiment in which frequent sampling of MA lines was combined with whole genome resequencing to develop a high-resolution picture of the effect of spontaneous mutations in a hypermutator (ΔmutS) strain of the bacterium Pseudomonas aeruginosa. After ∼644 generations of mutation accumulation, MA lines had accumulated an average of 118 mutations, and we found that average fitness across all lines decayed linearly over time. Detailed analyses of the dynamics of fitness change in individual lines revealed that a large fraction of the total decay in fitness (42.3%) was attributable to the fixation of rare, highly deleterious mutations (comprising only 0.5% of fixed mutations). Furthermore, we found that at least 0.64% of mutations were beneficial and probably fixed due to positive selection. The majority of mutations that fixed (82.4%) were base substitutions and we failed to find any signatures of selection on nonsynonymous or intergenic mutations. Short indels made up a much smaller fraction of the mutations that were fixed (17.4%), but we found evidence of strong selection against indels that caused frameshift mutations in coding regions. These results help to quantify the amount of natural selection present in microbial MA experiments and demonstrate that changes in fitness are strongly influenced by rare mutations of large effect.     

Evolution by small steps and rugged landscapes in the RNA virus phi6

Fisher's geometric model of adaptive evolution argues that adaptive evolution should generally result from the substitution of many mutations of small effect because advantageous mutations of small effect should be more common than those of large effect. However, evidence for both evolution by small steps and for Fisher's model has been mixed. Here we report supporting results from a new experimental test of the model. We subjected the bacteriophage phi6 to intensified genetic drift in small populations and caused viral fitness to decline through the accumulation of a deleterious mutation. We then propagated the mutated virus at a range of larger population sizes and allowed fitness to recover by natural selection. Although fitness declined in one large step, it was usually recovered in smaller steps. More importantly, step size during recovery was smaller with decreasing size of the recovery population. These results confirm Fisher's main prediction that advantageous mutations of small effect should be more common. We also show that the advantageous mutations of small effect are compensatory mutations whose advantage is conditional (epistatic) on the presence of the deleterious mutation, in which case the adaptive landscape of phi6 is likely to be very rugged.     

Understanding the evolutionary fate of finite populations: the dynamics of mutational effects

The most consistent result in more than two decades of experimental evolution is that the fitness of populations adapting to a constant environment does not increase indefinitely, but reaches a plateau. Using experimental evolution with bacteriophage, we show here that the converse is also true. In populations small enough such that drift overwhelms selection and causes fitness to decrease, fitness declines down to a plateau. We demonstrate theoretically that both of these phenomena must be due either to changes in the ratio of beneficial to deleterious mutations, the size of mutational effects, or both. We use mutation accumulation experiments and molecular data from experimental evolution to show that the most significant change in mutational effects is a drastic increase in the rate of beneficial mutation as fitness decreases. In contrast, the size of mutational effects changes little even as organismal fitness changes over several orders of magnitude. These findings have significant implications for the dynamics of adaptation.     

A method to infer positive selection from marker dynamics in an asexual population

MOTIVATION: The observation of positive selection acting on a mutant indicates that the corresponding mutation has some form of functional relevance. Determining the fitness effects of mutations thus has relevance to many interesting biological questions. One means of identifying beneficial mutations in an asexual population is to observe changes in the frequency of marked subsets of the population. We here describe a method to estimate the establishment times and fitnesses of beneficial mutations from neutral marker frequency data. RESULTS: The method accurately reproduces complex marker frequency trajectories. In simulations for which positive selection is close to 5% per generation, we obtain correlations upwards of 0.91 between correct and inferred haplotype establishment times. Where mutation selection coefficients are exponentially distributed, the inferred distribution of haplotype fitnesses is close to being correct. Applied to data from a bacterial evolution experiment, our method reproduces an observed correlation between evolvability and initial fitness defect.     

Dynamics of inbreeding depression due to deleterious mutations in small populations: mutation parameters and inbreeding rate

A multilocus stochastic model is developed to simulate the dynamics of mutational load in small populations of various sizes. Old mutations sampled from a large ancestral population at mutation-selection balance and new mutations arising each generation are considered jointly, using biologically plausible lethal and deleterious mutation parameters. The results show that inbreeding depression and the number of lethal equivalents due to partially recessive mutations can be partly purged from the population by inbreeding, and that this purging mainly involves lethals or detrimentals of large effect. However, fitness decreases continuously with inbreeding, due to increased fixation and homozygosity of mildly deleterious mutants, resulting in extinctions of very small populations with low reproductive rates. No optimum inbreeding rate or population size exists for purging with respect to fitness (viability) changes, but there is an optimum inbreeding rate at a given final level of inbreeding for reducing inbreeding depression or the number of lethal equivalents. The interaction between selection against partially recessive mutations and genetic drift in small populations also influences the rate of decay of neutral variation. Weak selection against mutants relative to genetic drift results in apparent overdominance and thus an increase in effective size (Ne) at neutral loci, and strong selection relative to drift leads to a decrease in Ne due to the increased variance in family size. The simulation results and their implications are discussed in the context of biological conservation and tests for purging.     

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.     

Clonal interference and the periodic selection of new beneficial mutations in Escherichia coli

The conventional model of adaptation in asexual populations implies sequential fixation of new beneficial mutations via rare selective sweeps that purge all variation and preserve the clonal genotype. However, in large populations multiple beneficial mutations may co-occur, causing competition among them, a phenomenon called "clonal interference." Clonal interference is thus expected to lead to longer fixation times and larger fitness effects of mutations that ultimately become fixed, as well as to a genetically more diverse population. Here, we study the significance of clonal interference in populations consisting of mixtures of differently marked wild-type and mutator strains of Escherichia coli that adapt to a minimal-glucose environment for 400 generations. We monitored marker frequencies during evolution and measured the competitive fitness of random clones from each marker state after evolution. The results demonstrate the presence of multiple beneficial mutations in these populations and slower and more erratic invasion of mutants than expected by the conventional model, showing the signature of clonal interference. We found that a consequence of clonal interference is that fitness estimates derived from invasion trajectories were less than half the magnitude of direct estimates from competition experiments, thus revealing fundamental problems with this fitness measure. These results force a reevaluation of the conventional model of periodic selection for asexual microbes.     

Mutations of intermediate effect are responsible for adaptation in evolving Pseudomonas fluorescens populations

The fixation of a beneficial mutation represents the first step in adaptation, and the average effect of such mutations is therefore a fundamental property of evolving populations. It is nevertheless poorly characterized because the rarity of beneficial mutations makes it difficult to obtain reliable estimates of fitness. We obtained 68 genotypes each containing a single fixed beneficial mutation from experimental populations of Pseudomonas fluorescens, evolving in medium with serine as the sole carbon source and estimated the selective advantage of each by competition with the ancestor. The distribution of selection coefficients is modal and closely resembles the Weibull distribution. The average selection coefficient (2.1) and beneficial mutation rate (3.8x10(-8)) are high relative to previous studies, possibly because the ancestral population grows poorly in serine-limited medium. Our experiment suggests that the initial stages of adaptation to stressful environments will involve the substitution of mutations with large effect on fitness.     

Sexual recombination under the joint effects of mutation,selection, and random sampling drift

The advantage or disadvantage of sexual reproduction or recombination for the accumulation of mutant genes in a population is studied under the joint effects of recurrent mutations, selection, and random sampling drift. To obtain the rate at which mutant genes are incorporated three different methods are used; numerical integration of Kolmogorov backward equations, simulation of stochastic difference equations, and Monte Carlo experiments. The first two methods are used in a two-locus system to obtain the fixation probability of double mutants and other related quantities under five different selection models. The third one is conducted for a multiple-locus system and the rate of accumulation of mutant genes per locus is studied. Comparison of the results between sexual and asexual populations shows that the effect of recombination depends on initial linkage disequilibrium, mutation rate v, selection intensity s, and population size N_e. The mode of selection is also an important factor and the large effect of recombination is observed when mutant genes are individually deleterious but collectively favorable. Under a given model of selection, the great advantage or disadvantage of recombination is achieved when a large extent of genetic polymorphism is produced not by mutation but by recombination. Extreme values of N_es and N_ev make the effect insignificant. The results of Monte Carlo experiments also reveal the presence of interaction between selection and sampling drift even when the loci segregate independently and selection is multiplicative. Although this interaction is usually small, there are cases in which one locus theory cannot be used freely. In those cases, the effect of recombination is prominent and one locus theory gives an overestimate of the rate.     

The contribution of the mitochondrial genome to sex‐specific fitness variance

Maternal inheritance of mitochondrial DNA (mtDNA) facilitates the evolutionary accumulation of mutations with sex‐biased fitness effects. Whereas maternal inheritance closely aligns mtDNA evolution with natural selection in females, it makes it indifferent to evolutionary changes that exclusively benefit males. The constrained response of mtDNA to selection in males can lead to asymmetries in the relative contributions of mitochondrial genes to female versus male fitness variation. Here, we examine the impact of genetic drift and the distribution of fitness effects (DFE) among mutations—including the correlation of mutant fitness effects between the sexes—on mitochondrial genetic variation for fitness. We show how drift, genetic correlations, and skewness of the DFE determine the relative contributions of mitochondrial genes to male versus female fitness variance. When mutant fitness effects are weakly correlated between the sexes, and the effective population size is large, mitochondrial genes should contribute much more to male than to female fitness variance. In contrast, high fitness correlations and small population sizes tend to equalize the contributions of mitochondrial genes to female versus male variance. We discuss implications of these results for the evolution of mitochondrial genome diversity and the genetic architecture of female and male fitness.     

ADAPTIVE WALKS AND DISTRIBUTION OF BENEFICIAL FITNESS EFFECTS

We study the adaptation dynamics of a maladapted asexual population on rugged fitness landscapes with many local fitness peaks. The distribution of beneficial fitness effects is assumed to belong to one of the three extreme value domains, viz. Weibull, Gumbel, and Fréchet. We work in the strong selection‐weak mutation regime in which beneficial mutations fix sequentially, and the population performs an uphill walk on the fitness landscape until a local fitness peak is reached. A striking prediction of our analysis is that the fitness difference between successive steps follows a pattern of diminishing returns in the Weibull domain and accelerating returns in the Fréchet domain, as the initial fitness of the population is increased. These trends are found to be robust with respect to fitness correlations. We believe that this result can be exploited in experiments to determine the extreme value domain of the distribution of beneficial fitness effects. Our work here differs significantly from the previous ones that assume the selection coefficient to be small. On taking large effect mutations into account, we find that the length of the walk shows different qualitative trends from those derived using small selection coefficient approximation.     

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.     

Heterosis in an isolated, effectively small, and self-fertilizing population of the flowering plant Leavenworthia alabamica

Mildly deleterious mutations are thought to play a major role in the extinction of natural populations, especially those that are small, isolated, or inbred. Self-fertilization should reduce the effective size of populations and simultaneously reduce migration between populations. A history of self-fertilization should therefore cause a population to harbor a substantial "local drift load" caused by the fixation of mildly deleterious mutations. This hypothesis was tested in Leavenworthia alabamica, which contains large, self-incompatible populations and smaller self-compatible populations with adaptations for self-fertilization. The fitness of offspring from within- and between-population crosses was compared to quantify heterosis caused by the masking of deleterious alleles in the heterozygous state. Little heterosis was observed in crosses between five large, self-incompatible populations and two of the three small, self-fertilizing populations of L. alabamica. However, the most geographically isolated and genetically divergent self-fertilizing population (Tuscumbia) exhibited a 110.2% increase in germination and a 73.6% increase in fitness, which is consistent with a sizeable local drift load. The finding of substantial heterosis for fitness supports the idea that small effective size, reproductive isolation, and self-fertilization can make populations particularly vulnerable to mutation accumulation.     

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.     

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.     

Stabilizing selection,mutational bias,and the evolution of sex*

Stabilizing selection around a fixed phenotypic optimum is expected to disfavor sexual reproduction, since asexually reproducing organisms can maintain a higher fitness at equilibrium, while sex disrupts combinations of compensatory mutations. This conclusion rests on the assumption that mutational effects on phenotypic traits are unbiased, that is, mutation does not tend to push phenotypes in any particular direction. In this article, we consider a model of stabilizing selection acting on an arbitrary number of polygenic traits coded by bialellic loci, and show that mutational bias may greatly reduce the mean fitness of asexual populations compared with sexual ones in regimes where mutations have weak to moderate fitness effects. Indeed, mutation and drift tend to push the population mean phenotype away from the optimum, this effect being enhanced by the low effective population size of asexual populations. In a second part, we present results from individual‐based simulations showing that positive rates of sex are favored when mutational bias is present, while the population evolves toward complete asexuality in the absence of bias. We also present analytical (QLE) approximations for the selective forces acting on sex in terms of the effect of sex on the mean and variance in fitness among offspring.     

Synergistic fitness interactions and a high frequency of beneficial changes among mutations accumulated under relaxed selection in Saccharomyces cerevisiae

Spontaneous mutations were accumulated for approximately 4800 generations in 48 lines of yeast protected from effective selection by frequent passage through single-cell bottlenecks. Changes in fitness were evaluated by direct competition with matched parental stocks differing only at a selectively neutral marker locus. Average fitness declined by approximately 5% over the course of the experiment. The rate of change increased sharply in later generations, strongly suggesting synergistic epistasis. Divergence among lines increased rapidly relative to the change in average fitness and also at an accelerating pace. Both results are well matched by a model assuming that fitness cost increases exponentially (approximately second order) with the number of accumulated mutations. This result is consistent with fitness loss due primarily to interactions between specific pairs of gene products. I also estimate that approximately 25% of the mutations with detectable fitness effects were beneficial. This result can be explained by the fact that the effects of most mutations are small relative to the distance from a local fitness optimum.