BIOLOGICAL EVOLUTION THROUGH MUTATION, SELECTION, AND DRIFT: AN INTRODUCTORY REVIEW

In this review some simple models of asexual populations evolving on smooth landscapes are studied. The basic model is based on a cellular automaton, which is analyzed here in the spatial mean-field limit. Firstly, the evolution on a fixed fitness landscape is considered. The correspondence between the time evolution of the population and equilibrium properties of a statistical mechanics system is investigated, finding the limits for which this mapping holds. The mutational meltdown, Eigen's error threshold and Muller's ratchet phenomena are studied in the framework of a simplified model. Finally, the shape of a quasi-species and the condition of coexistence of multiple species in a static fitness landscape are analyzed. In the second part, these results are applied to the study of the coexistence of quasi-species in the presence of competition, obtaining the conditions for a robust speciation effect in asexual populations.

BackgroundPrimary bacterial endosymbionts of insects (p-endosymbionts) are thought to be undergoing the process of Muller's ratchet where they accrue slightly deleterious mutations due to genetic drift in small populations with negligible recombination rates. If this process were to go unchecked over time, theory predicts mutational meltdown and eventual extinction. Although genome degradation is common among p-endosymbionts, we do not observe widespread p-endosymbiont extinction, suggesting that Muller's ratchet may be slowed or even stopped over time. For example, selection may act to slow the effects of Muller's ratchet by removing slightly deleterious mutations before they go to fixation thereby causing a decrease in nucleotide substitutions rates in older p-endosymbiont lineages.Methodology/Principal FindingsTo determine whether selection is slowing the effects of Muller's ratchet, we determined the age of the Candidatus Riesia/sucking louse assemblage and analyzed the nucleotide substitution rates of several p-endosymbiont lineages that differ in the length of time that they have been associated with their insect hosts. We find that Riesia is the youngest p-endosymbiont known to date, and has been associated with its louse hosts for only 13–25 My. Further, it is the fastest evolving p-endosymbiont with substitution rates of 19–34% per 50 My. When comparing Riesia to other insect p-endosymbionts, we find that nucleotide substitution rates decrease dramatically as the age of endosymbiosis increases.Conclusions/SignificanceA decrease in nucleotide substitution rates over time suggests that selection may be limiting the effects of Muller's ratchet by removing individuals with the highest mutational loads and decreasing the rate at which new mutations become fixed. This countering effect of selection could slow the overall rate of endosymbiont extinction.

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.

BackgroundThe 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.MethodsWe 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.ResultsWe 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.

Background: The accumulation of deleterious mutations of a population directly contributes to the fate as to how long the population would exist. Muller's ratchet provides a quantitative framework to study the effect of accumulation. Adaptive landscape as a powerful concept in system biology provides a handle to describe complex and rare biological events. In this article we study the evolutionary process of a population exposed to Muller's ratchet from the new viewpoint of adaptive landscape which allows us estimate the single click of the ratchet starting with an intuitive understanding. Methods: We describe how Wright-Fisher process maps to Muller's ratchet. We analytically construct adaptive landscape from general diffusion equation. It shows that the construction is dynamical and the adaptive landscape is independent of the existence and normalization of the stationary distribution. We generalize the application of diffusion model from adaptive landscape viewpoint. Results: We develop a novel method to describe the dynamical behavior of the population exposed to Muller's ratchet, and analytically derive the decaying time of the fittest class of populations as a mean first passage time. Most importantly, we describe the absorption phenomenon by adaptive landscape, where the stationary distribution is non-normalizable. These results suggest the method may be used to understand the mechanism of populations evolution and describe the biological processes quantitatively.

Error thresholds of replication in finite populations mutation frequencies and the onset of muller's ratchet

Two independently derived theories predict upper limits to the mutation rate beyond which evolution cannot be controlled by natural selection. One is the theory of Muller's ratchet, explaining the low phylogenetic age of parthenogenetic clones, the other one is the theory of error thresholds, predicting the maximal information content of selfreplicating molecules in prebiotic evolution. Both theories are based on similiar mathematical models but reach qualitatively different conclusions. Muller's ratchet only works in finite populations, while error thresholds are a deterministic phenomenon. In this paper it is shown that this discrepancy is due to different assumptions about the fitness values the selfreplicative units are allowed to assume. If no lower limit for the fitness values is assumed then the deterministic equilibrium frequency of the currently best genotype is strictly positive, no matter how strong mutation is, and random drift is required to cause its extinction (Muller's ratchet). On the other hand, positive lower limits for the fitness values lead to zero equilibrium frequencies in the deterministic description provided the mutation rate is high enough and no back mutations occur.

Why sex exists remains an unsolved problem in biology. If mutations are on the average deleterious, a high mutation rate can account for the evolution of sex. One form of this mutational hypothesis is Muller's ratchet. If the mutation rate is high, mutation-free individuals become rare and they can be lost by genetic drift in small populations. In asexual populations, as Muller noted, the loss is irreversible and the load of deleterious mutations increases in a ratchet-like manner with the successive loss of the least-mutated individuals. Sex can be advantageous because it increases the fitness of sexual populations by re-creating mutation-free individuals from mutated individuals and stops (or slows) Muller's ratchet. Although Muller's ratchet is an appealing hypothesis, it has been investigated and documented experimentally in only one group of organisms--ciliated protozoa. I initiated a study to examine the role of Muller's ratchet on the evolution of sex in RNA viruses and report here a significant decrease in fitness due to Muller's ratchet in 20 lineages of the RNA bacteriophage phi 6. These results show that deleterious mutations are generated at a sufficiently high rate to advance Muller's ratchet in an RNA virus and that beneficial, backward and compensatory mutations cannot stop the ratchet in the observed range of fitness decrease.

The emergence of life depended on the ability of the first biopolymer populations to thrive and approach larger population sizes and longer sequences. The evolution of these populations likely occurred under circumstances under which Muller's Ratchet in synergism with random drift could have caused large genetic deterioration of the biopolymers. This deterioration can drive the populations to extinction unless there is a mechanism to counteract it. The effect of the mutation rate and the effective population size on the time to extinction was tested on clonal populations of B16-19 ligase ribozymes, evolved with the continuous evolution in vitro system. Populations of 100, 300, 600 and/or 3000 molecules were evolved with and without the addition of Mn(II). The times to extinction for populations evolved without Mn(II) were found to be directly related to the effective size of the population. The small populations approached extinction at an average of 24.3 cycles; while the large populations did so at an average of 44.5 cycles. Genotypic characterization of the populations showed the presence of deleterious mutations in the small populations, which are the likely cause of their genetic deterioration and extinction via mutational meltdown. These deleterious mutations were not observed in the large populations; in contrast an advantageous mutant was present. Populations of 100 and 3000 molecules were evolved with Mn(II). None of the populations showed signs of genetic deterioration nor did they become extinct. Genotypic characterization of the 100-molecule population indicated the presence of a cloud of mutants closely genetically-related, forming a "quasispecies" structure.. The close connectedness of the mutants facilitates the recovery of one from another in the event of being removed from the population by random genetic drift. Thus, quasispecies shift the target of selection from individual to group. The total fitness of the molecules was measured by identifying the fitness component of the system that effect the ligase replication cycles: ligation, reverse transcription and transcription. It was found that the strength of the three components of fitness varied and that each one has a differential effect in the total absolute fitness of the ligases.

Wright meets AD: not all landscapes are adaptive

SummaryWe study multi-locus models for the accumulation of disadvantagenous mutant alleles in diploid populations. The theory used is closely related to the quasi-species theory of molecular evolution. The stationary mutant distribution may either be localized close to a peak in the fitness landscape or delocalized throughout sequence space. In some cases there is a sharp transition between these two cases known as an error threshold. We study a multiplicative fitness landscape where the fitness of an individual withjhomozygous mutant loci andkheterozygous loci iswjk= (1 −s)j(1 −hs)k. For a sexual population in this landscape there are two types of solution separated by an error threshold. For a parthenogenetic population there may be three types of solution and two error thresholds for some values ofh. For a population reproducing by selfing the solution is independent ofh, since the frequency of heterozygous individuals is negligible. The mean fitnesses of the populations depend on the reproductive method even for the multiplicative landscape. The sexual may have a higher or lower fitness than the parthenogen, depending on the values ofhandu/s. Selfing leads to a higher mean fitness than either sexual reproduction or parthenogenesis. We also study a fitness landscape with epistatic interactions withwjk= exp(−s(2j+k)α). The sexual population has a higher fitness than the parthenogen when α > 1. This confirms previous theories that sexual reproduction is advantageous in cases of synergistic epistasis. The mean fitness of a selfing population was found to be higher than both the sexual and the parthenogen over the range of parameter values studied. We discuss these results in relation to the theory of the evolution of sex. The fitness of the stationary distribution in cases where unfavourable mutations accumulation is one factor which could explain the observed prevalence of sexual reproduction in natural populations, although other factors may be more important in many cases.

The evolution of sexual reproduction has long been a major problem in biology. According to one theory, sex opposes the fitness-destroying process of Muller's ratchet, which occurs by the stochastic loss of high-fitness genotypes in small populations. Sex opposes the ratchet by allowing genotypes with different deleterious mutations to produce mutation-free offspring. We used the Avida digitalevolution software to investigate sex in relation to Muller's ratchet. Populations of digital organisms mutated, competed, and evolved in a complex environment. Populations were either asexual or sexual; in the latter case, parental genomes recombined to produce offspring. We also varied genomic mutation rates and population sizes, which at extreme values often caused mutational meltdowns and population extinctions. Our results demonstrate that sex is advantageous for population survival under some conditions. However, differences in extinction probabilities were usually small, occurred over a narrow range of mutation rates and population sizes, and the advantage of sex for population survival required many generations. Also, the mean fitness of surviving asexual populations was often greater than in sexual populations. This last result indicates the need for work that compares the statistical distribution of mutational effects and epistatic interactions in asexual and sexual populations.

Evolution of sex and the molecular clock in RNA viruses

Population genetic models have shown that if genetic drift is strong and the rate of deleterious mutations is high, Muller's ratchet provides an advantage to sex. A previous study tested for the possibility that Muller's ratchet could work in RNA viruses, which are known to have very high mutation rates. Muller's ratchet was found to operate when lineages of the RNA bacteriophage φ6 were subjected to intensified genetic drift. The study did not determine, however, whether sex is advantageous to these viruses. We have examined whether sex can reverse the effects of Muller's ratchet by crossing nine φ6 lineages that were subjected to the ratchet in Chao's study. To determine whether there was a net advantage to sex, we analyzed the effect of crossing three lineages to all other lineages. Crossing increased significantly the fitness of two lineages, but it did not significantly affect the fitness of the third lineage. We argue that the minimal advantage of sex to these nine lineages is small, but positive. These results provide a possible scenario for the evolution of sex in an RNA phage like φ6.

Patterns of Stem Cell Divisions Contribute to Plant Longevity