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

Muller's ratchet in a near-critical regime: Tournament versus fitness proportional selection

Muller's ratchet, in its prototype version, models a haploid, asexual population whose size N is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. The classical variant considers fitness proportional selection, but other fitness schemes are conceivable as well. Inspired by the work of Etheridge et al. (2009) we propose a parameter scaling which fits well to the "near-critical" regime that was in the focus of Etheridge et al. (2009) (and in which the mutation-selection ratio diverges logarithmically as N→∞). Using a Moran model, we investigate the"rule of thumb" given in Etheridge et al. (2009) for the click rate of the "classical ratchet" by putting it into the context of new results on the long-time evolution of the size of the best class of the ratchet with (binary) tournament selection. This variant of Muller's ratchet was introduced in González Casanova et al. (2023), and was analysed there in a subcritical parameter regime. Other than that of the classical ratchet, the size of the best class of the tournament ratchet follows an autonomous dynamics up to the time of its extinction. It turns out that, under a suitable correspondence of the model parameters, this dynamics coincides with the so called Poisson profile approximation of the dynamics of the best class of the classical ratchet.

Mutational meltdown in asexual populations doomed to extinction

Asexual populations are expected to accumulate deleterious mutations through a process known as Muller's ratchet. Lynch and colleagues proposed that the ratchet eventually results in a vicious cycle of mutation accumulation and population decline that drives populations to extinction. They called this phenomenon mutational meltdown. Here, we analyze mutational meltdown using a multi-type branching process model where, in the presence of mutation, populations are doomed to extinction. We analyse the change in size and composition of the population and the time of extinction under this model.

Extinction scenarios in evolutionary processes: a multinomial Wright-Fisher approach

We study a discrete-time multi-type Wright-Fisher population process. The mean-field dynamics of the stochastic process is induced by a general replicator difference equation. We prove several results regarding the asymptotic behavior of the model, focusing on the impact of the mean-field dynamics on it. One of the results is a limit theorem that describes sufficient conditions for an almost certain path to extinction, first eliminating the type which is the least fit at the mean-field equilibrium. The effect is explained by the metastability of the stochastic system, which under the conditions of the theorem spends almost all time before the extinction event in a neighborhood of the equilibrium. In addition to the limit theorems, we propose a maximization principle for a general deterministic replicator dynamics and study its implications for the stochastic model.

Diffusion approximations in population genetics and the rate of Muller's ratchet

The Wright-Fisher binomial model of allele frequency change is often approximated by a scaling limit in which selection, mutation and drift all decrease at the same 1/N rate. This construction restricts the applicability of the resulting 'Wright-Fisher diffusion equation' to the weak selection, weak mutation regime of evolution. We argue that diffusion approximations of the Wright-Fisher model can be used more generally, for instance in cases where genetic drift is much weaker than selection. One important example of this regime is Muller's ratchet phenomenon, whereby deleterious mutations slowly but irreversibly accumulate through rare stochastic fluctuations. Using a modified diffusion equation we derive improved analytical estimates for the mean click time of the ratchet.

Muller's ratchet of the Y chromosome with gene conversion

Muller's ratchet is a process in which deleterious mutations are fixed irreversibly in the absence of recombination. The degeneration of the Y chromosome, and the gradual loss of its genes, can be explained by Muller's ratchet. However, most theories consider single-copy genes, and may not be applicable to Y chromosomes, which have a number of duplicated genes in many species, which are probably undergoing concerted evolution by gene conversion. We developed a model of Muller's ratchet to explore the evolution of the Y chromosome. The model assumes a nonrecombining chromosome with both single-copy and duplicated genes. We used analytical and simulation approaches to obtain the rate of gene loss in this model, with special attention to the role of gene conversion. Homogenization by gene conversion makes both duplicated copies either mutated or intact. The former promotes the ratchet, and the latter retards, and we ask which of these counteracting forces dominates under which conditions. We found that the effect of gene conversion is complex, and depends upon the fitness effect of gene duplication. When duplication has no effect on fitness, gene conversion accelerates the ratchet of both single-copy and duplicated genes. If duplication has an additive fitness effect, the ratchet of single-copy genes is accelerated by gene duplication, regardless of the gene conversion rate, whereas gene conversion slows the degeneration of duplicated genes. Our results suggest that the evolution of the Y chromosome involves several parameters, including the fitness effect of gene duplication by increasing dosage and gene conversion rate.

An informational transition in conditioned Markov chains: Applied to genetics and evolution

In this work we assume that we have some knowledge about the state of a population at two known times, when the dynamics is governed by a Markov chain such as a Wright-Fisher model. Such knowledge could be obtained, for example, from observations made on ancient and contemporary DNA, or during laboratory experiments involving long term evolution. A natural assumption is that the behaviour of the population, between observations, is related to (or constrained by) what was actually observed. The present work shows that this assumption has limited validity. When the time interval between observations is larger than a characteristic value, which is a property of the population under consideration, there is a range of intermediate times where the behaviour of the population has reduced or no dependence on what was observed and an equilibrium-like distribution applies. Thus, for example, if the frequency of an allele is observed at two different times, then for a large enough time interval between observations, the population has reduced or no dependence on the two observed frequencies for a range of intermediate times. Given observations of a population at two times, we provide a general theoretical analysis of the behaviour of the population at all intermediate times, and determine an expression for the characteristic time interval, beyond which the observations do not constrain the population's behaviour over a range of intermediate times. The findings of this work relate to what can be meaningfully inferred about a population at intermediate times, given knowledge of terminal states.

Effect of drift, selection and recombination on the equilibrium frequency of deleterious mutations

We study the stationary state of a population evolving under the action of random genetic drift, selection and recombination in which both deleterious and reverse beneficial mutations can occur. We find that the equilibrium fraction of deleterious mutations decreases as the population size is increased. We calculate exactly the steady state frequency in a nonrecombining population when population size is infinite and for a neutral finite population, and obtain bounds on the fraction of deleterious mutations. We also find that for small and very large populations, the number of deleterious mutations depends weakly on recombination, but for moderately large populations, recombination alleviates the effect of deleterious mutations. An analytical argument shows that recombination decreases disadvantageous mutations appreciably when beneficial mutations are rare as is the case in adapting microbial populations, whereas it has a moderate effect on codon bias where the mutation rates between the preferred and unpreferred codons are comparable.

Clonal interference and Muller's ratchet in spatial habitats

Competition between independently arising beneficial mutations is enhanced in spatial populations due to the linear rather than exponential growth of clones. Recent theoretical studies have pointed out that the resulting fitness dynamics is analogous to a surface growth process, where new layers nucleate and spread stochastically, leading to the build up of scale-invariant roughness. This scenario differs qualitatively from the standard view of adaptation in that the speed of adaptation becomes independent of population size while the fitness variance does not. Here we exploit recent progress in the understanding of surface growth processes to obtain precise predictions for the universal, non-Gaussian shape of the fitness distribution for one-dimensional habitats, which are verified by simulations. When the mutations are deleterious rather than beneficial the problem becomes a spatial version of Muller's ratchet. In contrast to the case of well-mixed populations, the rate of fitness decline remains finite even in the limit of an infinite habitat, provided the ratio [Formula: see text] between the deleterious mutation rate and the square of the (negative) selection coefficient is sufficiently large. Using, again, an analogy to surface growth models we show that the transition between the stationary and the moving state of the ratchet is governed by directed percolation.

Distribution of the fittest individuals and the rate of Muller's ratchet in a model with overlapping generations

Muller's ratchet is a paradigmatic model for the accumulation of deleterious mutations in a population of finite size. A click of the ratchet occurs when all individuals with the least number of deleterious mutations are lost irreversibly due to a stochastic fluctuation. In spite of the simplicity of the model, a quantitative understanding of the process remains an open challenge. In contrast to previous works, we here study a Moran model of the ratchet with overlapping generations. Employing an approximation which describes the fittest individuals as one class and the rest as a second class, we obtain closed analytical expressions of the ratchet rate in the rare clicking regime. As a click in this regime is caused by a rare, large fluctuation from a metastable state, we do not resort to a diffusion approximation but apply an approximation scheme which is especially well suited to describe extinction events from metastable states. This method also allows for a derivation of expressions for the quasi-stationary distribution of the fittest class. Additionally, we confirm numerically that the formulation with overlapping generations leads to the same results as the diffusion approximation and the corresponding Wright-Fisher model with non-overlapping generations.

Muller's ratchet is a paradigmatic model in population genetics which describes the fixation of a deleterious mutation in a population of finite size due to an unfortunate stochastic fluctuation. Obtaining quantitative predictions of the ratchet rate, i.e. the frequency with which such a mutation fixes, is believed to be important for understanding a broad range of effects ranging from the degeneration of the Y-chromosome to the evolution of sex as a means of avoiding the fixation of deleterious mutations. To obtain a better understanding of how Muller's ratchet operates, we have considered a model with overlapping generations, which allows for the application of methods specifically tailored for the analysis of rare stochastic fluctuations which drive the ratchet. We obtain concise and accurate results for the rate of Muller's ratchet. Additionally, we are able to predict the full distribution of the frequency of the fittest individuals, a quantity of central interest in understanding the ratchet rate and possibly experimentally much more accessible than the rate, in particular when the ratchet rate is very large.

Wright-Fisher dynamics on adaptive landscape

Adaptive landscape, proposed by Sewall Wright, has provided a conceptual framework to describe dynamical behaviours. However, it is still a challenge to explicitly construct such a landscape, and apply it to quantify interesting evolutionary processes. This is particularly true for neutral evolution. In this work, the authors study one-dimensional Wright Fisher process, and analytically obtain an adaptive landscape as a potential function. They provide the complete characterisation for dynamical behaviours of all possible mutation rates under the influence of mutation and random drift. This same analysis has been applied to situations with additive selection and random drift for all possible selection rates. The critical state dividing the basins of two stable states is directly obtained by the landscape. In addition, the landscape is able to handle situations with pure random drift, which would be non-normalisable for its stationary distribution. The nature of non-normalisation is from the singularity of adaptive landscape. In addition, they propose a new type of neutral evolution. It has the same probability for all possible states. The new type of neutral evolution describes the non-neutral alleles with 0%. They take the equal effect of mutation and random drift as an example.

Fluctuations in fitness distributions and the effects of weak linked selection on sequence evolution

Evolutionary dynamics and patterns of molecular evolution are strongly influenced by selection on linked regions of the genome, but our quantitative understanding of these effects remains incomplete. Recent work has focused on predicting the distribution of fitness within an evolving population, and this forms the basis for several methods that leverage the fitness distribution to predict the patterns of genetic diversity when selection is strong. However, in weakly selected populations random fluctuations due to genetic drift are more severe, and neither the distribution of fitness nor the sequence diversity within the population are well understood. Here, we briefly review the motivations behind the fitness-distribution picture, and summarize the general approaches that have been used to analyze this distribution in the strong-selection regime. We then extend these approaches to the case of weak selection, by outlining a perturbative treatment of selection at a large number of linked sites. This allows us to quantify the stochastic behavior of the fitness distribution and yields exact analytical predictions for the sequence diversity and substitution rate in the limit that selection is weak.

Absorbing phenomena and escaping time for Muller's ratchet in adaptive landscape

BACKGROUND

The accumulation of deleterious mutations of a population directly contributes to the fate as to how long the population would exist, a process often described as Muller's ratchet with the absorbing phenomenon. The key to understand this absorbing phenomenon is to characterize the decaying time of the fittest class of the population. Adaptive landscape introduced by Wright, a re-emerging powerful concept in systems biology, is used as a tool to describe biological processes. To our knowledge, the dynamical behaviors for Muller's ratchet over the full parameter regimes are not studied from the point of the adaptive landscape. And the characterization of the absorbing phenomenon is not yet quantitatively obtained without extraneous assumptions as well.

METHODS

We describe how Muller's ratchet can be mapped to the classical Wright-Fisher process in both discrete and continuous manners. Furthermore, we construct the adaptive landscape for the system analytically from the general diffusion equation. The constructed adaptive landscape is independent of the existence and normalization of the stationary distribution. We derive the formula of the single click time in finite and infinite potential barrier for all parameters regimes by mean first passage time.

RESULTS

We describe the dynamical behavior of the population exposed to Muller's ratchet in all parameters regimes by adaptive landscape. The adaptive landscape has rich structures such as finite and infinite potential, real and imaginary fixed points. We give the formula about the single click time with finite and infinite potential. And we find the single click time increases with selection rates and population size increasing, decreases with mutation rates increasing. These results provide a new understanding of infinite potential. We analytically demonstrate the adaptive and unadaptive states for the whole parameters regimes. Interesting issues about the parameters regions with the imaginary fixed points is demonstrated. Most importantly, we find that the absorbing phenomenon is characterized by the adaptive landscape and the single click time without any extraneous assumptions. These results suggest a graphical and quantitative framework to study the absorbing phenomenon.

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.

A unified treatment of the probability of fixation when population size and the strength of selection change over time

The fixation probability is determined when population size and selection change over time and differs from Kimura's result, with long-term implications for a population. It is found that changes in population size are not equivalent to the corresponding changes in selection and can result in less drift than anticipated.