Fluctuations of Fitness Distributions and the Rate of Muller's Ratchet

The accumulation of deleterious mutations is driven by rare fluctuations that lead to the loss of all mutation free individuals, a process known as Muller's ratchet. Even though Muller's ratchet is a paradigmatic process in population genetics, a quantitative understanding of its rate is still lacking. The difficulty lies in the nontrivial nature of fluctuations in the fitness distribution, which control the rate of extinction of the fittest genotype. We address this problem using the simple but classic model of mutation selection balance with deleterious mutations all having the same effect on fitness. We show analytically how fluctuations among the fittest individuals propagate to individuals of lower fitness and have dramatically amplified effects on the bulk of the population at a later time. If a reduction in the size of the fittest class reduces the mean fitness only after a delay, selection opposing this reduction is also delayed. This delayed restoring force speeds up Muller's ratchet. We show how the delayed response can be accounted for using a path-integral formulation of the stochastic dynamics and provide an expression for the rate of the ratchet that is accurate across a broad range of parameters.     

Temporal variation in selection accelerates mutational decay by Muller's ratchet

Asexual species accumulate deleterious mutations through an irreversible process known as Muller's ratchet. Attempts to quantify the rate of the ratchet have ignored the role of temporal environmental heterogeneity even though it is common in nature and has the potential to affect overall ratchet rate. Here we examine Muller's ratchet in the context of conditional neutrality (i.e., mutations that are deleterious in some environmental conditions but neutral in others) as well as more subtle changes in the strength (but not sign) of selection. We find that temporal variation increases the rate of the ratchet (mutation accumulation) and the rate of fitness decline over that of populations experiencing constant selection of equivalent average strength. Temporal autocorrelation magnifies the effects of temporal heterogeneity and can allow the ratchet to operate at large population sizes in which it would be halted under constant selection. Classic studies of Muller's ratchet show that the rate of fitness decline is maximized when selection is of a low but intermediate strength. This relationship changes quantitatively with all forms of temporal heterogeneity studied and changes qualitatively when there is temporal autocorrelation in selection. In particular, the rate of fitness decline can increase indefinitely with the strength of selection with some forms of temporal heterogeneity. Our finding that temporal autocorrelation in selection dramatically increases ratchet rate and rate of fitness decline may help to explain the paucity of asexual taxa.     

Muller''s Ratchet, Epistasis and Mutation Effects

In this study, computer simulation is used to show that despite synergistic epistasis for fitness, Muller's ratchet can lead to lethal fitness loss in a population of asexuals through the accumulation of deleterious mutations. This result contradicts previous work that indicated that epistasis will halt the ratchet. The present results show that epistasis will not halt the ratchet provided that rather than a single deleterious mutation effect, there is a distribution of deleterious mutation effects with sufficient density near zero. In addition to epistasis and mutation distribution, the ability of Muller's ratchet to lead to the extinction of an asexual population under epistasis for fitness depends strongly on the expected number of offspring that survive to reproductive age. This strong dependence is not present in the nonepistatic model and suggests that interpreting the population growth parameter as fecundity is inadequate. Because a continuous distribution of mutation effects is used in this model, an emphasis is placed on the dynamics of the mutation effect distribution rather than on the dynamics of the number of least mutation loaded individuals. This perspective suggests that current models of gene interaction are too simple to apply directly to long-term prediction for populations undergoing the ratchet.     

Dynamic mutation-selection balance as an evolutionary attractor

The vast majority of mutations are deleterious and are eliminated by purifying selection. Yet in finite asexual populations, purifying selection cannot completely prevent the accumulation of deleterious mutations due to Muller's ratchet: once lost by stochastic drift, the most-fit class of genotypes is lost forever. If deleterious mutations are weakly selected, Muller's ratchet can lead to a rapid degradation of population fitness. Evidently, the long-term stability of an asexual population requires an influx of beneficial mutations that continuously compensate for the accumulation of the weakly deleterious ones. Hence any stable evolutionary state of a population in a static environment must involve a dynamic mutation-selection balance, where accumulation of deleterious mutations is on average offset by the influx of beneficial mutations. We argue that such a state can exist for any population size N and mutation rate U and calculate the fraction of beneficial mutations, ε, that maintains the balanced state. We find that a surprisingly low ε suffices to achieve stability, even in small populations in the face of high mutation rates and weak selection, maintaining a well-adapted population in spite of Muller's ratchet. This may explain the maintenance of mitochondria and other asexual genomes.     

Muller's ratchet and the pattern of variation at a neutral locus

The levels and patterns of variation at a neutral locus are analyzed in a haploid asexual population undergoing accumulation of deleterious mutations due to Muller's ratchet. We find that the movement of Muller's ratchet can be associated with a considerable reduction in genetic diversity below classical neutral expectation. The extent to which variability is reduced is a function of the deleterious mutation rate, the fitness effects of the mutations, and the population size. Approximate analytical expressions for the expected genetic diversity are compared with simulation results under two different models of deleterious mutations: a model where all deleterious mutations have equal effects and a model where there are two classes of deleterious mutations. We also find that Muller's ratchet can produce a considerable distortion in the neutral frequency spectrum toward an excess of rare variants.     

Quantifying the genomic decay paradox due to Muller's ratchet in human mitochondrial DNA

The observation of high mitochondrial mutation rates in human pedigrees has led to the question of how such an asexual genetic system can survive the accumulation of slightly deleterious mutations caused by Muller's ratchet. I define a null model to quantify in unprecedented detail the threat from extinction caused by Muller's ratchet. This model is general enough to explore the biological significance of Muller's ratchet in various species where its operation has been suspected. For increased precision over a wide range of parameter space I employ individual-based simulations run by evolution@home, the first global computing system for evolutionary biology. After compiling realistic values for the key parameters in human mitochondrial DNA (mtDNA) I find that a surprisingly large range of biologically realistic parameter combinations would lead to the extinction of the human line over a period of 20 million years - if accepted wisdom about mtDNA and Muller's ratchet is correct. The resulting genomic decay paradox complements a similar threat from extinction due to mutation accumulation in nuclear DNA and suggests evaluation of unconventional explanations for long-term persistence. A substantial list of potential solutions is given, including compensatory back mutations, mutation rate heterogeneity and occasional recombination in mtDNA. Future work will have to explore which of these actually solves the paradox. Nonetheless, the results presented here provide yet another reason to minimize anthropogenic increase of mutation rates.     

The degeneration of asexual haploid populations and the speed of Muller's ratchet

The accumulation of deleterious mutations due to the process known as Muller's ratchet can lead to the degeneration of nonrecombining populations. We present an analytical approximation for the rate at which this process is expected to occur in a haploid population. The approximation is based on a diffusion equation and is valid when N exp(-u/s) > 1, where N is the population size, u is the rate at which deleterious mutations occur, and s is the effect of each mutation on fitness. Simulation results are presented to show that the approximation estimates the rate of the process better than previous approximations for values of mutation rates and selection coefficients that are compatible with the biological data. Under certain conditions, the ratchet can turn at a biologically significant rate when the deterministic equilibrium number of individuals free of mutations is substantially >100. The relevance of this process for the degeneration of Y or neo-Y chromosomes is discussed.     

The speed of Muller's ratchet with background selection, and the degeneration of Y chromosomes.

The rate of accumulation of deleterious mutations by Muller's ratchet is investigated in large asexual haploid populations, for a range of parameters with potential biological relevance. The rate of this process is studied by considering a very simple model in which mutations can have two types of effect: either strongly deleterious or mildly deleterious. It is shown that the rate of accumulation of mildly deleterious mutations can be greatly increased by the presence of strongly deleterious mutations, and that this can be predicted from the associated reduction in effective population size (the background selection effect). We also examine the rate of the ratchet when there are two classes of mutation of similar but unequal effects on fitness. The accuracy of analytical approximations for the rate of this process is analysed. Its possible role in causing the degeneration of Y and neo-Y chromosomes is discussed in the light of our present knowledge of deleterious mutation rates and selection coefficients.     

Muller''s Ratchet under Epistatic Selection

In a finite asexual population mean fitness may decrease by a process known as Muller's ratchet, which proceeds if all individuals with the minimum number of deleterious alleles are randomly lost. If these alleles have independent effects on fitness, previous analysis suggested that the rate of this decrease either remains constant or, if accumulation of mutations leads to the decline of the population size, grows. Here I show that this conclusion is quite sensitive to the assumption of independence. If deleterious alleles have synergistic fitness effects, then, as the ratchet advances, the frequency of the best available genotype will necessarily increase, making its loss less and less probable. As a result, sufficiently strong synergistic epistasis can effectively halt the action of Muller's ratchet. Instead of being driven extinct, a finite asexual population could then survive practically indefinitely, although with lower mean fitness than without random drift.     

Fitness declines in Tobacco etch virus upon serial bottleneck transfers

It has been well established that populations of RNA viruses transmitted throughout serial bottlenecks suffer from significant fitness declines as a consequence of the accumulation of deleterious mutations by the onset of Muller's ratchet. Bottlenecks are unavoidably linked to different steps of the infectious cycle of most plant RNA viruses, such as vector-mediated transmissions and systemic colonization of new leaves. Here we report evidence for fitness declines by the accumulation of deleterious mutations in the potyvirus Tobacco etch virus (TEV). TEV was inoculated into the nonsystemic host Chenopodium quinoa, and local lesions were isolated and used to initiate 20 independent mutation accumulation lineages. Weekly, a random lesion from each lineage was isolated and used to inoculate the next set of plants. At each transfer, the Malthusian growth rate was estimated. After 11 consecutive transfers, all lineages suffered significant fitness losses, and one even became extinct. The average rate of fitness decline was 5% per day. The average pattern of fitness decline was consistent with antagonistic epistasis between deleterious mutations, as postulated for antiredundant genomes. Temporal fitness fluctuations were not explained by random noise but reflected more complex underlying processes related to emergence and self-organization phenomena.     

Molecular basis of fitness loss and fitness recovery in vesicular stomatitis virus

Viral populations subjected to repeated genetic bottleneck accumulate deleterious mutations in a process known as Muller's ratchet. Asexual viruses, such as vesicular stomatitis virus (VSV) can recover from Muller's ratchet by replication with large effective population sizes. However, mutants with a history of bottleneck transmissions often show decreased adaptability when compared to non-bottlenecked populations. We have generated a collection of bottlenecked mutants and allowed them to recover by large population passages. We have characterized fitness changes and the complete genomes of these strains. Mutations accumulated during the operation of Muller's ratchet led to the identification of two potential mutational hot spots in the VSV genome. As in other viral systems, transitions were more common than transversions. Both back mutation and compensatory mutations contributed to recovery, although a significant level of fitness increase was observed in nine of the 13 bottlenecked strains with no obvious changes in the consensus sequence. Additional replication of three strains resulted in the fixation of single point mutations. Only two mutations previously found in non-bottlenecked, high-fitness populations that had been adapting to the same environment were identified in the recovered strains.     

The accumulation of mutations in asexual populations and the structure of genealogical trees in the presence of selection

We study the accumulation of unfavourable mutations in asexual populations by the process of Muller's ratchet, and the consequent inevitable decrease in fitness of the population. Simulations show that it is mutations with only moderate unfavourable effect that lead to the most rapid decrease in fitness. We measure the number of fixations as a function of time and show that the fixation rate must be equal to the ratchet rate once a steady state is reached. Large bursts of fixations are observed to occur simultaneously. We relate this to the structure of the genealogical tree. We derive equations relating the rate of the ratchet to the moments of the distribution of the number of mutations k per individual. These equations interpolate between the deterministic limit (an infinite population with selection present) and the neutral limit (a finite size population with no selection). Both these limits are exactly soluble. In the neutral case, the distribution of k is shown to be non-self-averaging, i.e. the fluctuations remain very large even for very large populations. We also consider the strong-selection limit in which only individuals in the fittest surviving class have offspring. This limit is again exactly soluble. We investigate the structure of the genealogical tree relating individuals in the same population, and consider the probability (T) that two individuals had their latest common ancestor T generations in the past. The function (T) is exactly calculable in the neutral limit and the strong-selection limit, and we obtain an empirical solution for intermediate selection strengths.     

Subclonal components of consensus fitness in an RNA virus clone.

Most RNA virus populations exhibit extremely high mutation frequencies which generate complex, genetically heterogeneous populations referred to as quasi-species. Previous work has shown that when a large spectrum of the quasi-species is transferred, natural selection operates, leading to elimination of noncompetitive (inferior) genomes and rapid gains in fitness. However, whenever the population is repeatedly reduced to a single virion, variable declines in fitness occur as predicted by the Muller's ratchet hypothesis. Here, we quantitated the fitness of 98 subclones isolated from an RNA virus clonal population. We found a normal distribution around a lower fitness, with the average subclone being less fit than the parental clonal population. This finding demonstrates the phenotypic diversity in RNA virus populations and shows that, as expected, a large fraction of mutations generated during virus replication is deleterious. This clarifies the operation of Muller's ratchet and illustrates why a large number of virions must be transferred for rapid fitness gains to occur. We also found that repeated genetic bottleneck passages can cause irregular stochastic declines in fitness, emphasizing again the phenotypic heterogeneity present in RNA virus populations. Finally, we found that following only 60 h of selection (15 passages in which virus yields were harvested after 4 h), RNA virus populations can undergo a 250% average increase in fitness, even on a host cell type to which they were already well adapted. This is a remarkable ability; in population biology, even a much lower fitness gain (e.g., 1 to 2%) can represent a highly significant reproductive advantage. We discuss the biological implications of these findings for the natural transmission and pathogenesis of RNA viruses.     

Adaptive evolution of asexual populations under Muller's ratchet

We study the population genetics of adaptation in nonequilibrium haploid asexual populations. We find that the accumulation of deleterious mutations, due to the operation of Muller's ratchet, can considerably reduce the rate of fixation of advantageous alleles. Such reduction can be approximated reasonably well by a reduction in the effective population size. In the absence of Muller's ratchet, a beneficial mutation can only become fixed if it creates the best possible genotype; if Muller's ratchet operates, however, mutations initially arising in a nonoptimal genotype can also become fixed in the population, since the loss of the least-loaded class implies that an initially nonoptimal background can become optimal. We show that, while the rate at which adaptive mutations become fixed is reduced, the rate of fixation of deleterious mutations due to the ratchet is not changed by the presence of beneficial mutations as long as the rate of their occurrence is low and the deleterious effects of mutations (s(d)) are higher than the beneficial effects (s(a)). When s(a) > s(d), the advantage of a beneficial mutation can outweigh the deleterious effects of associated mutations. Under these conditions, a beneficial allele can drag to fixation deleterious mutations initially associated with it at a higher rate than in the absence of advantageous alleles. We propose analytical approximations for the rates of accumulation of deleterious and beneficial mutations. Furthermore, when allowing for the possible occurrence of interference between beneficial alleles, we find that the presence of deleterious mutations of either very weak or very strong effect can marginally increase the rate of accumulation of beneficial mutations over that observed in the absence of such deleterious mutations.     

A stochastic model for a single click of Muller's ratchet

This work presents a new approach to Muller's ratchet, where Haigh's model is approximately mapped into a simpler model that describes the behaviour of a population after a click of the ratchet, i.e., after loss of what was the fittest class. This new model predicts the distribution of times to the next click of the ratchet and is equivalent to a Wright-Fisher model for a population of haploid asexual individuals with one locus and two alleles. Within this model, the fittest members of a population correspond to carriers of one allele, while all other individuals have suboptimal fitness and are represented as carriers of the other allele. In this way, all suboptimal fitness individuals are amalgamated into a single “mutant” class.The approach presented here has some limitations and the potential for improvement. However, it does lead to results for the rate of the ratchet that, over a wide range of parameters, are accurate within one order of magnitude of simulation results. This contrasts with existing approaches, which are designed for only one or other of the two different parameter regimes known for the ratchet and are more accurate only in the parameter regime they were designed for.Numerical results are presented for the mean time between clicks of the ratchet for (i) the Wright-Fisher model, (ii) a diffusion approximation of this model and (iii) individually based simulations of a full model. The diffusion approximation is validated over a wide range of parameters by its close agreement with the Wright-Fisher model.The present work predicts that: (a) the time between clicks of the ratchet is insensitive to the value of the selection coefficient when the genomic mutation rate is large compared with the selection coefficient against a deleterious mutation, (b) the time interval between clicks of the ratchet has, approximately, an exponential distribution (or its discrete analogue). It is thus possible to determine the variance in times between clicks, given the expected time between clicks. Evidence for both (a) and (b) is seen in simulations.     

Drastic fitness loss in human immunodeficiency virus type 1 upon serial bottleneck events

Muller's ratchet predicts fitness losses in small populations of asexual organisms because of the irreversible accumulation of deleterious mutations and genetic drift. This effect should be enhanced if population bottlenecks intervene and fixation of mutations is not compensated by recombination. To study whether Muller's ratchet could operate in a retrovirus, 10 biological clones were derived from a human immunodeficiency virus type 1 (HIV-1) field isolate by MT-4 plaque assay. Each clone was subjected to 15 plaque-to-plaque passages. Surprisingly, genetic deterioration of viral clones was very drastic, and only 4 of the 10 initial clones were able to produce viable progeny after the serial plaque transfers. Two of the initial clones stopped forming plaques at passage 7, two others stopped at passage 13, and only four of the remaining six clones yielded infectious virus. Of these four, three displayed important fitness losses. Thus, despite virions carrying two copies of genomic RNA and the system displaying frequent recombination, HIV-1 manifested a drastic fitness loss as a result of an accentuation of Muller's ratchet effect.     

Muller's ratchet and the accumulation of favourable mutations

Under the influence of recurrent deleterious mutation and selection, asexual and sexual populations reach a deterministic equilibrium with individuals carrying 0,1,2,. . . harmful mutations. When a favourable mutation (aA) occurs in an asexual population it will usually occur in an individual who has one or more (k) deleterious mutations. Muller's ratchet then applies as A will thereafter never occur in an individual with less than k mutations. If the selective advantage of A is less than the selective disadvantage of k harmful mutations then A will not spread. If it is greater it may spread carrying k deleterious mutations to fixation. Sexual populations are not affected in this way. A will spread through the population experiencing genomes with 0,1,2,. . . deleterious mutations in accordance with the deterministic equilibrium.     

Deleterious mutations can surf to high densities on the wave front of an expanding population

There is an increasing recognition that evolutionary processes play a key role in determining the dynamics of range expansion. Recent work demonstrates that neutral mutations arising near the edge of a range expansion sometimes surf on the expanding front leading them rather than that leads to reach much greater spatial distribution and frequency than expected in stationary populations. Here, we extend this work and examine the surfing behavior of nonneutral mutations. Using an individual-based coupled-map lattice model, we confirm that, regardless of its fitness effects, the probability of survival of a new mutation depends strongly upon where it arises in relation to the expanding wave front. We demonstrate that the surfing effect can lead to deleterious mutations reaching high densities at an expanding front, even when they have substantial negative effects on fitness. Additionally, we highlight that this surfing phenomenon can occur for mutations that impact reproductive rate (i.e., number of offspring produced) as well as mutations that modify juvenile competitive ability. We suggest that these effects are likely to have important consequences for rates of spread and the evolution of spatially expanding populations.