A stochastic model for the mutation-selection balance in an infinite asexual population with a genome of fixed size

A stochastic model is presented which describes the evolution of a genome of a haploid species in an infinite population. The genome is a finite set of elements. The elements are divided into different classes according to their effect on the fitness of the organism. Repeated mutations of the genome elements are permitted, in particular positive mutations are introduced. The distribution of the deleterious elements in the genome with respect to the impact on the fitness is given after the replication step. The steady state is fully described including the distribution and the fitness.     

The dynamical way to mutation-selection balance of an infinite population evolving on a truncated fitness landscape

This paper presents the exact analytical solution, valid for all generations and initial conditions, for the frequency distribution of haploids with infinite-sites genome carrying a given number of mutations in a population evolving deterministically on a truncated fitness landscape. This landscape is a generalization of the single sharp peak one, widely used in quasispecies theory, although here there are no reverse mutations.     

The effect of population bottlenecks on mutation rate evolution in asexual populations

In the absence of recombination, a mutator allele can spread through a population by hitchhiking with beneficial mutations that appear in its genetic background. Theoretical studies over the past decade have shown that the survival and fixation probability of beneficial mutations can be severely reduced by population size bottlenecks. Here, we use computational modelling and evolution experiments with the yeast S. cerevisiae to examine whether population bottlenecks can affect mutator dynamics in adapting asexual populations. In simulation, we show that population bottlenecks can inhibit mutator hitchhiking with beneficial mutations and are most effective at lower beneficial mutation supply rates. We then subjected experimental populations of yeast propagated at the same effective population size to three different bottleneck regimes and observed that the speed of mutator hitchhiking was significantly slower at smaller bottlenecks, consistent with our theoretical expectations. Our results, thus, suggest that bottlenecks can be an important factor in mutation rate evolution and can in certain circumstances act to stabilize or, at least, delay the progressive elevation of mutation rates in asexual populations. Additionally, our findings provide the first experimental support for the theoretically postulated effect of population bottlenecks on beneficial mutations and demonstrate the usefulness of studying mutator frequency dynamics for understanding the underlying dynamics of fitness‐affecting mutations.     

The influence of hitchhiking and deleterious mutation upon asexual mutation rates

The question of how natural selection affects asexual mutation rates has been considered since the 1930s, yet our understanding continues to deepen. The distribution of mutation rates observed in natural bacteria remains unexplained. It is well known that environmental constancy can favor minimal mutation rates. In contrast, environmental fluctuation (e.g., at period T) can create indirect selective pressure for stronger mutators: genes modifying mutation rate may "hitchhike" to greater frequency along with environmentally favored mutations they produce. This article extends a well-known model of Leigh to consider fitness genes with multiple mutable sites (call the number of such sites alpha). The phenotypic effect of such a gene is enabled if all sites are in a certain state and disabled otherwise. The effects of multiple deleterious loci are also included (call the number of such loci gamma). The analysis calculates the indirect selective effects experienced by a gene inducing various mutation rates for given values of alpha, gamma, and T. Finite-population simulations validate these results and let us examine the interaction of drift with hitchhiking selection. We close by commenting on the importance of other factors, such as spatiotemporal variation, and on the origin of variation in mutation rates.     

The evolution of mutation rate in finite asexual populations

In this article, we model analytically the evolution of mutation rate in asexual organisms. Three selective forces are present. First, everything else being equal, individuals with higher mutation rate have a larger fitness, thanks to the energy and time saved by not replicating DNA accurately. Second, as a flip side, the genome of these individuals is replicated with errors that may negatively affect fitness. Third, and conversely, replication errors have a potential benefit if beneficial mutations are to be generated. Our model describes the fate of modifiers of mutation rate under the three forces and allows us to predict the long-term evolutionary trajectory of mutation rate. We obtain three major results. First, in asexuals, the needs for both adaptation and genome preservation are not evolutionary forces that can stabilize mutation rate at an intermediate optimum. When adaptation has a significant role, it primarily destabilizes mutation rate and yields the emergence of strong-effect mutators. Second, in contrast to what is usually believed, the appearance of modifiers with large mutation rate is more likely when the fitness cost of each deleterious mutation is weak, because the cost of replication errors is then paid after a delay. Third, in small populations, and even if adaptations are needed, mutation rate is always blocked at the minimum attainable level, because the rate of adaptation is too slow to play a significant role. Only populations whose size is above a critical mass see their mutation rate affected by the need for adaptation.     

Analysis of the biases in the estimation of deleterious mutation parameters from natural populations at mutation-selection balance

Indirect estimates of the genomic rate of deleterious mutations (lambda), their average homozygous effect (s) and their degree of dominance (h) can be obtained from genetic parameters of natural populations, assuming that the frequencies of the loci controlling a given fitness trait are at mutation-selection equilibrium. In 1996, H.-W. Deng and M. Lynch developed a general methodology for obtaining these estimates from inbreeding/outbreeding experiments. The prediction of the sign and magnitude of the biases incurred by these estimators is essential for a correct interpretation of the empirical results. However, the assessment of these biases has been tested so far under a rather limited model of the distribution of dominance effects. In this paper, the application of this method to outbred populations is evaluated, focusing on the level of variation in h values (sigma(h)(2) and the magnitude of the negative correlation (rs,h) between s and h. It is shown that the method produces upwardly biased estimates of lambda and downwardly biased estimates of the average s in the reference situation where rs,h=0, particularly for large values of sigma(h)(2), and biases of different sign depending on the magnitude of the correlation. A modification of the method, substituting the estimates of the average h for alternative ones, allows estimates to be obtained with little or no bias for the case of rs,h=0, but is otherwise biased. Information on rs,h and sigma(h)(2), gathered from mutation-accumulation experiments, suggests that sigma(h)(2) may be rather large and rs,h is usually negative but not higher than about -0.2, although the data are scarce and noisy, and should be used with caution.     

The neutral theory in an infinite population

Selective substitutions at one locus induce stochastic dynamics at a linked neutral locus that resemble genetic drift even when the population size is infinite. This new stochastic force, which is called genetic draft, causes genetic variation at the neutral locus to decrease with population size and the rate of deleterious substitution to increase with population size. The fact that heterozygosities in natural populations are only weakly dependent on population size suggests that genetic draft may be a much more important stochastic force than genetic drift in natural populations. Some of the mathematical properties of genetic draft are explored.     

Effect of chromosomal instability on the mutation-selection balance in unicellular populations

This paper develops a mathematical model describing the evolutionary dynamics of a unicellular, asexually replicating population exhibiting chromosomal instability. Chromosomal instability is a form of genetic instability characterized by the gain or loss of entire chromosomes during cell division. We assume that the cellular genome is divided into several homologous groups of chromosomes, and that a single functional chromosome per homologous group is required for the cell to have the wild-type fitness. If the fitness is unaffected by the total number of chromosomes in the cell, our model is analytically solvable, and yields a mean fitness at mutation-selection balance that is identical to the mean fitness when there is no chromosomal instability. If this assumption is relaxed and the total number of chromosomes in the cell is not allowed to increase without bound, then chromosomal instability leads to a reduction in mean fitness. The results of this paper provide a useful baseline that can inform both future theoretial and experimental studies of chromosomal instability.     

Selection, mutation and sexual reproduction in an infinite haploid population with a genome of finite length

We present a model which describes mutation, selection and sexual reproduction in an infinite haploid population with a finite genome. Each generation is described using an approximation which assures a certain persistent form of the distribution of the number of deleterious elements. The steady state exists and is determined. In addition, we conclude that the introduction of sexual reproduction increases the mean fitness in the equilibrium.     

Mathematical analysis of a model describing evolution of an asexual population in a changing environment

We investigate a mathematical model for an asexual population with non-overlapping (discrete) generations, that exists in a changing environment. Sexual populations are also briefly discussed at the end of the paper. It is assumed that selection occurs on the value of a single polygenic trait, which is controlled by a finite number of loci with discrete-effect alleles. The environmental change results in a moving fitness optimum, causing the trait to be subject to a combination of stabilising and directional selection.This model is different from that investigated by Waxman and Peck [Genetics 153 (1999) 1041] where overlapping generations and continuous effect alleles were considered. In this paper, we consider non-overlapping generations and discrete effect alleles. However in [Genetics 153 (1999) 1041] and the present work, there is the same pattern of environmental change, namely a constant rate of change of the optimum.From [Genetics 153 (1999) 1041], no rigorous theoretical conclusion can be drawn about the form of the solutions as t grows large. Numerical work carried out in [Genetics 153 (1999) 1041] suggests that the solution is a lagged travelling wave solution, but no mathematical proof exists for the continuous model. Only partial results, regarding existence of travelling wave solutions and perturbed solutions, have been established (see [Nonlin. Anal. 53 (2003) 683; An integral equation describing an asexual population in a changing environment, Preprint]). For the discrete case of this paper, under the assumption that the ratio between the unit of genotypic value and the speed of environment change is a rational number, we are able to give rigorous proof of the following conclusion: the population follows the environmental change with a small lag behind, moreover, the lag is represented using a calculable quantity.     

Beneficial mutations, hitchhiking and the evolution of mutation rates in sexual populations

Natural selection acts in three ways on heritable variation for mutation rates. A modifier allele that increases the mutation rate is (i) disfavored due to association with deleterious mutations, but is also favored due to (ii) association with beneficial mutations and (iii) the reduced costs of lower fidelity replication. When a unique beneficial mutation arises and sweeps to fixation, genetic hitchhiking may cause a substantial change in the frequency of a modifier of mutation rate. In previous studies of the evolution of mutation rates in sexual populations, this effect has been underestimated. This article models the long-term effect of a series of such hitchhiking events and determines the resulting strength of indirect selection on the modifier. This is compared to the indirect selection due to deleterious mutations, when both types of mutations are randomly scattered over a given genetic map. Relative to an asexual population, increased levels of recombination reduce the effects of beneficial mutations more rapidly than those of deleterious mutations. However, the role of beneficial mutations in determining the evolutionarily stable mutation rate may still be significant if the function describing the cost of high-fidelity replication has a shallow gradient.     

The balance between mutators and nonmutators in asexual populations

Mutator alleles, which elevate an individual's mutation rate from 10 to 10,000-fold, have been found at high frequencies in many natural and experimental populations. Mutators are continually produced from nonmutators, often due to mutations in mismatch-repair genes. These mutators gradually accumulate deleterious mutations, limiting their spread. However, they can occasionally hitchhike to high frequencies with beneficial mutations. We study the interplay between these effects. We first analyze the dynamics of the balance between the production of mutator alleles and their elimination due to deleterious mutations. We find that when deleterious mutation rates are high in mutators, there will often be many "young," recently produced mutators in the population, and the fact that deleterious mutations only gradually eliminate individuals from a population is important. We then consider how this mutator-nonmutator balance can be disrupted by beneficial mutations and analyze the circumstances in which fixation of mutator alleles is likely. We find that dynamics is crucial: even in situations where selection on average acts against mutators, so they cannot stably invade, the mutators can still occasionally generate beneficial mutations and hence be important to the evolution of the population.     

Experimental estimation of mutation rates in a wheat population with a gene genealogy approach

Microsatellite markers are extensively used to evaluate genetic diversity in natural or experimental evolving populations. Their high degree of polymorphism reflects their high mutation rates. Estimates of the mutation rates are therefore necessary when characterizing diversity in populations. As a complement to the classical experimental designs, we propose to use experimental populations, where the initial state is entirely known and some intermediate states have been thoroughly surveyed, thus providing a short timescale estimation together with a large number of cumulated meioses. In this article, we derived four original gene genealogy-based methods to assess mutation rates with limited bias due to relevant model assumptions incorporating the initial state, the number of new alleles, and the genetic effective population size. We studied the evolution of genetic diversity at 21 microsatellite markers, after 15 generations in an experimental wheat population. Compared to the parents, 23 new alleles were found in generation 15 at 9 of the 21 loci studied. We provide evidence that they arose by mutation. Corresponding estimates of the mutation rates ranged from 0 to 4.97 x 10(-3) per generation (i.e., year). Sequences of several alleles revealed that length polymorphism was only due to variation in the core of the microsatellite. Among different microsatellite characteristics, both the motif repeat number and an independent estimation of the Nei diversity were correlated with the novel diversity. Despite a reduced genetic effective size, global diversity at microsatellite markers increased in this population, suggesting that microsatellite diversity should be used with caution as an indicator in biodiversity conservation issues.     

Inbreeding depression in an asexual population of Mimulus guttatus

The reproductive mechanism, that is whether an organism outcrosses, selfs or asexually reproduces, has a substantial impact on the amount and pattern of genetic variation. In this study, we estimate genetic variation and genetic load for a predominately asexual population of Mimulus guttatus, and then compare our results to other studies of predominately sexually reproducing (outcrossing and selfing) populations of M. guttatus. The asexual population had low levels of heterozygosity (H_e = 0.03) and low (but significantly non‐zero) inbreeding load, especially when compared with other M. guttatus populations. This differs greatly from the sexual populations of Mimulus that display substantial inbreeding depression. We discuss a variety of reasons why we see such low load in this study and suggest future research projects to further explore the questions.     

The evolution of mutation rates: separating causes from consequences

Natural selection can adjust the rate of mutation in a population by acting on allelic variation affecting processes of DNA replication and repair. Because mutation is the ultimate source of the genetic variation required for adaptation, it can be appealing to suppose that the genomic mutation rate is adjusted to a level that best promotes adaptation. Most mutations with phenotypic effects are harmful, however, and thus there is relentless selection within populations for lower genomic mutation rates. Selection on beneficial mutations can counter this effect by favoring alleles that raise the mutation rate, but the effect of beneficial mutations on the genomic mutation rate is extremely sensitive to recombination and is unlikely to be important in sexual populations. In contrast, high genomic mutation rates can evolve in asexual populations under the influence of beneficial mutations, but this phenomenon is probably of limited adaptive significance and represents, at best, a temporary reprieve from the continual selection pressure to reduce mutation. The physiological cost of reducing mutation below the low level observed in most populations may be the most important factor in setting the genomic mutation rate in sexual and asexual systems, regardless of the benefits of mutation in producing new adaptive variation. Maintenance of mutation rates higher than the minimum set by this "cost of fidelity" is likely only under special circumstances.     

Meiosis and the evolution of recombination at low mutation rates

The classical understanding of recombination is that in large asexual populations with multiplicative fitness, linkage disequilibrium is negligible, and thus there is no selective agent driving an allele for recombination. This has led researchers to recognize the importance of synergistic epistatic selection in generating negative linkage disequilibrium that thereby renders an advantage to recombination. Yet data on such selection is equivocal, and various works have shown that synergistic epistasis per se, when left unquantified in its magnitude or operation, is not sufficient to drive the evolution of recombination. Here we show that neither it, nor any mechanism generating negative linkage disequilibrium among fitness-related loci, is necessary. We demonstrate that a neutral gene for recombination can increase in frequency in a large population under a low mutation rate and strict multiplicative fitness. We work in a parameter range where individuals have, on average, less than one mutation each, yet recombination can still evolve. We demonstrate this in two ways: first, by examining the consequences of recombination correlated with misrepaired DNA damage and, second, by increasing the probability of recombination with declining fitness. Interestingly, the allele spreads without repairing even a single DNA mutation.