Pathways to extinction: beyond the error threshold

Since the introduction of the quasispecies and the error catastrophe concepts for molecular evolution by Eigen and their subsequent application to viral populations, increased mutagenesis has become a common strategy to cause the extinction of viral infectivity. Nevertheless, the high complexity of virus populations has shown that viral extinction can occur through several other pathways apart from crossing an error threshold. Increases in the mutation rate enhance the appearance of defective forms and promote the selection of mechanisms that are able to counteract the accelerated appearance of mutations. Current models of viral evolution take into account more realistic scenarios that consider compensatory and lethal mutations, a highly redundant genotype-to-phenotype map, rough fitness landscapes relating phenotype and fitness, and where phenotype is described as a set of interdependent traits. Further, viral populations cannot be understood without specifying the characteristics of the environment where they evolve and adapt. Altogether, it turns out that the pathways through which viral quasispecies go extinct are multiple and diverse.     

Theory of lethal mutagenesis for viruses

Mutation is the basis of adaptation. Yet, most mutations are detrimental, and elevating mutation rates will impair a population's fitness in the short term. The latter realization has led to the concept of lethal mutagenesis for curing viral infections, and work with drugs such as ribavirin has supported this perspective. As yet, there is no formal theory of lethal mutagenesis, although reference is commonly made to Eigen's error catastrophe theory. Here, we propose a theory of lethal mutagenesis. With an obvious parallel to the epidemiological threshold for eradication of a disease, a sufficient condition for lethal mutagenesis is that each viral genotype produces, on average, less than one progeny virus that goes on to infect a new cell. The extinction threshold involves an evolutionary component based on the mutation rate, but it also includes an ecological component, so the threshold cannot be calculated from the mutation rate alone. The genetic evolution of a large population undergoing mutagenesis is independent of whether the population is declining or stable, so there is no runaway accumulation of mutations or genetic signature for lethal mutagenesis that distinguishes it from a level of mutagenesis under which the population is maintained. To detect lethal mutagenesis, accurate measurements of the genome-wide mutation rate and the number of progeny per infected cell that go on to infect new cells are needed. We discuss three methods for estimating the former. Estimating the latter is more challenging, but broad limits to this estimate may be feasible.     

Effect of lethality on the extinction and on the error threshold of quasispecies

In this paper the effect of lethality on error threshold and extinction has been studied in a population of error-prone self-replicating molecules. For given lethality and a simple fitness landscape, three dynamic regimes can be obtained: quasispecies, error catastrophe, and extinction. Using a simple model in which molecules are classified as master, lethal and non-lethal mutants, it is possible to obtain the mutation rates of the transitions between the three regimes analytically. The numerical resolution of the extended model, in which molecules are classified depending on their Hamming distance to the master sequence, confirms the results obtained in the simple model and shows how an error catastrophe regime changes when lethality is taken in account.     

Stochastic Simulations Suggest that HIV-1 Survives Close to Its Error Threshold

The use of mutagenic drugs to drive HIV-1 past its error threshold presents a novel intervention strategy, as suggested by the quasispecies theory, that may be less susceptible to failure via viral mutation-induced emergence of drug resistance than current strategies. The error threshold of HIV-1, , however, is not known. Application of the quasispecies theory to determine poses significant challenges: Whereas the quasispecies theory considers the asexual reproduction of an infinitely large population of haploid individuals, HIV-1 is diploid, undergoes recombination, and is estimated to have a small effective population size in vivo. We performed population genetics-based stochastic simulations of the within-host evolution of HIV-1 and estimated the structure of the HIV-1 quasispecies and . We found that with small mutation rates, the quasispecies was dominated by genomes with few mutations. Upon increasing the mutation rate, a sharp error catastrophe occurred where the quasispecies became delocalized in sequence space. Using parameter values that quantitatively captured data of viral diversification in HIV-1 patients, we estimated to be substitutions/site/replication, ∼2–6 fold higher than the natural mutation rate of HIV-1, suggesting that HIV-1 survives close to its error threshold and may be readily susceptible to mutagenic drugs. The latter estimate was weakly dependent on the within-host effective population size of HIV-1. With large population sizes and in the absence of recombination, our simulations converged to the quasispecies theory, bridging the gap between quasispecies theory and population genetics-based approaches to describing HIV-1 evolution. Further, increased with the recombination rate, rendering HIV-1 less susceptible to error catastrophe, thus elucidating an added benefit of recombination to HIV-1. Our estimate of may serve as a quantitative guideline for the use of mutagenic drugs against HIV-1.     

Genomic Imprinting Leads to Less Selectively Maintained Polymorphism on X Chromosomes

Population-genetic models are developed to investigate the consequences of viability selection at a diallelic X-linked locus subject to genomic imprinting. Under complete paternal-X inactivation, a stable polymorphism is possible under the same conditions as for paternal-autosome inactivation with differential selection on males and females. A necessary but not sufficient condition is that there is sexual conflict, with selection acting in opposite directions in males and females. In contrast, models of complete maternal-X inactivation never admit a stable polymorphism and alleles will either be fixed or lost from the population. Models of complete paternal-X inactivation are more complex than corresponding models of maternal-X inactivation, as inactivation of paternally derived X chromosomes in females screens these chromosomes from selection for a generation. We also demonstrate that polymorphism is possible for incomplete X inactivation, but that the parameter conditions are more restrictive than for complete paternal-X inactivation. Finally, we investigate the effects of recurrent mutation in our models and show that deleterious alleles in mutation–selection balance at imprinted X-linked loci are at frequencies rather similar to those with corresponding selection pressures and mutation rates at unimprinted loci. Overall, our results add to the reasons for expecting less selectively maintained allelic variation on X chromosomes.     

Lethal mutagenesis of bacteria

Lethal mutagenesis, the killing of a microbial pathogen with a chemical mutagen, is a potential broad-spectrum antiviral treatment. It operates by raising the genomic mutation rate to the point that the deleterious load causes the population to decline. Its use has been limited to RNA viruses because of their high intrinsic mutation rates. Microbes with DNA genomes, which include many viruses and bacteria, have not been considered for this type of treatment because their low intrinsic mutation rates seem difficult to elevate enough to cause extinction. Surprisingly, models of lethal mutagenesis indicate that bacteria may be candidates for lethal mutagenesis. In contrast to viruses, bacteria reproduce by binary fission, and this property ensures their extinction if subjected to a mutation rate >0.69 deleterious mutations per generation. The extinction threshold is further lowered when bacteria die from environmental causes, such as washout or host clearance. In practice, mutagenesis can require many generations before extinction is achieved, allowing the bacterial population to grow to large absolute numbers before the load of deleterious mutations causes the decline. Therefore, if effective treatment requires rapid population decline, mutation rates 0.69 may be necessary to achieve treatment success. Implications for the treatment of bacteria with mutagens, for the evolution of mutator strains in bacterial populations, and also for the evolution of mutation rate in cancer are discussed.     

The classical hitchhiking model with continuous mutational pressure and purifying selection

Detecting selective sweeps driven by strong positive selection and localizing the targets of selection in the genome play a major role in modern population genetics and genomics. Most of these analyses are based on the classical model of genetic hitchhiking proposed by Maynard Smith and Haigh (1974, Genetical Research, 23, 23). Here, we consider extensions of the classical two‐locus model. Introducing mutation at the strongly selected site, we analyze the conditions under which soft sweeps may arise. We identify a new parameter (the ratio of the beneficial mutation rate to the selection coefficient) that characterizes the occurrence of multiple‐origin soft sweeps. Furthermore, we quantify the hitchhiking effect when the polymorphism at the linked locus is not neutral but maintained in a mutation‐selection balance. In this case, we find a smaller relative reduction of heterozygosity at the linked site than for a neutral polymorphism. In our analysis, we use a semi‐deterministic approach; i.e., we analyze the frequency process of the beneficial allele in an infinitely large population when its frequency is above a certain threshold; however, for very small frequencies in the initial phase after the onset of selection we rely on diffusion theory.     

Mutation induced extinction in finite populations: lethal mutagenesis and lethal isolation

Reproduction is inherently risky, in part because genomic replication can introduce new mutations that are usually deleterious toward fitness. This risk is especially severe for organisms whose genomes replicate "semi-conservatively," e.g. viruses and bacteria, where no master copy of the genome is preserved. Lethal mutagenesis refers to extinction of populations due to an unbearably high mutation rate (U), and is important both theoretically and clinically, where drugs can extinguish pathogens by increasing their mutation rate. Previous theoretical models of lethal mutagenesis assume infinite population size (N). However, in addition to high U, small N can accelerate extinction by strengthening genetic drift and relaxing selection. Here, we examine how the time until extinction depends jointly on N and U. We first analytically compute the mean time until extinction (τ) in a simplistic model where all mutations are either lethal or neutral. The solution motivates the definition of two distinct regimes: a survival phase and an extinction phase, which differ dramatically in both how τ scales with N and in the coefficient of variation in time until extinction. Next, we perform stochastic population-genetics simulations on a realistic fitness landscape that both (i) features an epistatic distribution of fitness effects that agrees with experimental data on viruses and (ii) is based on the biophysics of protein folding. More specifically, we assume that mutations inflict fitness penalties proportional to the extent that they unfold proteins. We find that decreasing N can cause phase transition-like behavior from survival to extinction, which motivates the concept of "lethal isolation." Furthermore, we find that lethal mutagenesis and lethal isolation interact synergistically, which may have clinical implications for treating infections. Broadly, we conclude that stably folded proteins are only possible in ecological settings that support sufficiently large populations.     

MULLER'S RATCHET AND MUTATIONAL MELTDOWNS

We extend our earlier work on the role of deleterious mutations in the extinction of obligately asexual populations. First, we develop analytical models for mutation accumulation that obviate the need for time-consuming computer simulations in certain ranges of the parameter space. When the number of mutations entering the population each generation is fairly high, the number of mutations per individual and the mean time to extinction can be predicted using classical approaches in quantitative genetics. However, when the mutation rate is very low, a fixation-probability approach is quite effective. Second, we show that an intermediate selection coefficient (s) minimizes the time to extinction. The critical value of s can be quite low, and we discuss the evolutionary implications of this, showing that increased sensitivity to mutation and loss of capacity for DNA repair can be selectively advantageous in asexual organisms. Finally, we consider the consequences of the mutational meltdown for the extinction of mitochondrial lineages in sexual species.     

Does Mutational Robustness Inhibit Extinction by Lethal Mutagenesis in Viral Populations?

Lethal mutagenesis is a promising new antiviral therapy that kills a virus by raising its mutation rate. One potential shortcoming of lethal mutagenesis is that viruses may resist the treatment by evolving genomes with increased robustness to mutations. Here, we investigate to what extent mutational robustness can inhibit extinction by lethal mutagenesis in viruses, using both simple toy models and more biophysically realistic models based on RNA secondary-structure folding. We show that although the evolution of greater robustness may be promoted by increasing the mutation rate of a viral population, such evolution is unlikely to greatly increase the mutation rate required for certain extinction. Using an analytic multi-type branching process model, we investigate whether the evolution of robustness can be relevant on the time scales on which extinction takes place. We find that the evolution of robustness matters only when initial viral population sizes are small and deleterious mutation rates are only slightly above the level at which extinction can occur. The stochastic calculations are in good agreement with simulations of self-replicating RNA sequences that have to fold into a specific secondary structure to reproduce. We conclude that the evolution of mutational robustness is in most cases unlikely to prevent the extinction of viruses by lethal mutagenesis.     

Evolution at a High Imposed Mutation Rate: Adaptation Obscures the Load in Phage T7

Evolution at high mutation rates is expected to reduce population fitness deterministically by the accumulation of deleterious mutations. A high enough rate should even cause extinction (lethal mutagenesis), a principle motivating the clinical use of mutagenic drugs to treat viral infections. The impact of a high mutation rate on long-term viral fitness was tested here. A large population of the DNA bacteriophage T7 was grown with a mutagen, producing a genomic rate of 4 nonlethal mutations per generation, two to three orders of magnitude above the baseline rate. Fitness—viral growth rate in the mutagenic environment—was predicted to decline substantially; after 200 generations, fitness had increased, rejecting the model. A high mutation load was nonetheless evident from (i) many low- to moderate-frequency mutations in the population (averaging 245 per genome) and (ii) an 80% drop in average burst size. Twenty-eight mutations reached high frequency and were thus presumably adaptive, clustered mostly in DNA metabolism genes, chiefly DNA polymerase. Yet blocking DNA polymerase evolution failed to yield a fitness decrease after 100 generations. Although mutagenic drugs have caused viral extinction in vitro under some conditions, this study is the first to match theory and fitness evolution at a high mutation rate. Failure of the theory challenges the quantitative basis of lethal mutagenesis and highlights the potential for adaptive evolution at high mutation rates.THE evolutionary consequences of a high mutation rate are mysterious. It is widely considered that mutations are essential for adaptation, but that the rate maximizing adaptation is far below what can be tolerated (e.g., Trobner and Piechocki 1984; Sniegowski 1997, 2001). In this “twilight zone” of higher-than-optimal mutation rates, the population experiences unique challenges. In one process, the “error catastrophe,” the best genotype is driven out of the population deterministically because the onslaught of viable, mutant genotypes simply overwhelms it (Eigen et al. 1988). With Muller''s ratchet, a phenomenon of finite asexual populations, high mutation rates and genetic drift combine to cause loss of the wild-type genome, and the absence of recombination blocks its recreation (Muller 1964); fitness gradually decays as mutations continue their stochastic accumulation. Yet another high mutation rate process is the straightforward, deterministic decline in population fitness as deleterious mutations accumulate (Kimura and Maruyama 1966), leading to extinction if fecundity is too low to compensate (Maynard Smith 1978; Bull et al. 2007).The problem with our understanding of evolution at a high mutation rate is that it is piecemeal. We do not yet know how to combine these different processes nor do we know their relative importance. For example, the fitness loss at a high mutation rate can be offset both by adaptation and by the error catastrophe, but for realistic models, there is no formal basis for predicting the magnitude of adaptation or even for recognizing an error catastrophe (Bull et al. 2005, 2007). Empirical studies are needed. Several studies of viruses have explored extinction through elevated mutation rate (lethal mutagenesis) (Domingo et al. 2001; Anderson et al. 2004; also see discussion), but they have not been tied to any quantitative model. The practical value of such work is that mutagenic drugs are sometimes used to treat viral infections, yet we do not know how the elevated mutation rate is affecting the virus.Here we develop an empirical system to enforce viral evolution at a high mutation rate and test theory developed for lethal mutagenesis. A mutagen is applied to the culture in which the DNA bacteriophage T7 is grown, the mutation input per generation is measured on a genomewide scale, and the system is used to observe both molecular and fitness evolution. Comparison of data and theory provides new insights into the process that underlies lethal mutagenesis. However, existing theory must also be modified to address some empirical properties of the system.

Theory of fitness evolution at high mutation rate:

The objective is to develop a theory for data that are readily obtained. The most basic theory requires one population property (the deleterious mutation rate) to predict another population property (mean fitness), but other properties are not predicted. In experimental systems, mean fitness is easily measured, and the deleterious mutation rate can be estimated within bounds. A fully comprehensive model of evolution at a high mutation rate, one predicting full distributions of genotypes, could be developed if mutation rates and fitness effects were known for each individual mutation and for combinations of mutations, including recombination frequencies. However, the full spectrum of mutations and their fitness effects is too vast to allow those measurements in any biological system, so the only applicable theory describes just mean fitness.If the fitness (e.g., viability) of the mutation-free genotype is assigned the value 1, the mean fitness of an infinite, asexual population at equilibrium is e^−U, where U is the genomic deleterious mutation rate (discrete generations) (Kimura and Maruyama 1966). By itself, this result does not indicate whether a population will survive or not, but one simple modification extends the model to address lethal mutagenesis: fecundity. For an asexual population to survive, a minimal condition is that each parent must produce at least one surviving offspring. In the case of a virus, if each infection produces b viable progeny (in the absence of mutation), the inequality be^−U < 1 ensures eventual extinction. When this inequality is met, the number of progeny in each generation starts out smaller than the number in the parent generation, so the population size declines (Bull et al. 2007).This decline in fitness is not due to stochastic effects in small populations; extinction in this model formally requires a finite population, but the effect of deleterious mutations is treated deterministically. Finite population size can contribute to extinction at mutation rates below the threshold (e.g., from Muller''s ratchet), but we limit ourselves to nearly infinite population sizes.A useful property of the model is that the fitness effects of deleterious mutations and their individual rates need not be known, only the overall rate. Yet this elegance of the Kimura–Maruyama result starts to fade when considering empirical reality. The model considers only deleterious mutations, including lethals; neutral mutations are allowed but ignored, and beneficial mutations are not even allowed. Maximum fitness is assigned to the starting, mutation-free genotype, so any mutation that elevates fitness is excluded. Compensatory mutations that ameliorate the effect of deleterious mutations, and thus are beneficial only within mutated genomes, are also not allowed.To consider a simple model with beneficial mutations, if the initial genotype does not have maximum possible fitness, but a fitness of W relative to the starting genotype is attainable by beneficial mutations (W > 1), then a modified equilibrium is simply We^−U relative to a starting fitness of 1.0. In a virus whose initial fitness is b progeny, adaptive evolution could be accommodated in the model by increasing fecundity to B. The extent to which B exceeds b represents the extent to which the initial (wild-type) virus is poorly adapted to the mutagenic environment, which is unknown. Furthermore, this threshold relaxation omits compensatory mutations that ameliorate specific deleterious mutations and neglects any interference of deleterious mutations on the ascent of beneficial ones.Two further empirical limitations of the Kimura–Maruyama model are evident. Following the onset of an increased mutation rate, the fitness equilibrium may require few or many generations to be approached closely and potentially could require more generations than would be experienced by any real population (Crow and Kimura 1970; Bull and Wilke 2008). The rate of approach depends on the details of the mutation rate and fitness effects, whereas the equilibrium mean fitness does not. We thus attempt to carry out experiments long enough to assume that fitness has neared equilibrium. Second, the Kimura–Maruyama model was developed explicitly for asexuals; the same equilibrium applies with free recombination and no epistasis, but not necessarily when either of these conditions is violated (Maynard Smith 1978; Kondrashov 1982, 1984; Keightley and Otto 2006).In the Kimura–Maruyama model (Kimura and Maruyama 1966), fitness is measured per discrete generation as relative number of surviving offspring. In our viral study, fitness is measured as a growth rate, essentially the log of fitness in the Kimura–Maruyama model. This discrepancy can be resolved by deriving new results for growth rate, again assuming asexuality. Neglecting viral loss from death and other causes, a model of viral growth rate (r) is given by(1)where C is cell (host) density, k is the adsorption rate of virus to cells, b is burst size (average number of progeny per infected cell), and L is lysis time in minutes (Bull 2006). Cell density is assumed to be constant, and cells always outnumber virus (a condition that can be enforced experimentally). r is an exponential or geometric growth rate: at equilibrium, the number of virus at time t, N_t, as a function of initial density, N₀, is given by N_t = e^rtN₀. This model is tailored to the conditions used here, and a model for treatment of a mammalian infection would need to contend with spatial structure and the possibility that the viral population had reached a dynamic equilibrium in which exponential growth no longer applied (see also Steinmeyer and Wilke 2009).With a deleterious, genomic mutation rate U per generation, the deterministic growth rate of the mutation-free class is simply(2)By assumption, all mutation classes in the population are derived ultimately from the mutation-free class and, because all mutations in U are deleterious (neutral mutations are allowed but not counted), all mutants have slower growth rates than the mutation-free genotype. Back mutations and other forms of beneficial mutations are not allowed. It follows that the growth rate of the entire population at mutation–selection equilibrium is given by (2). This result is convenient because the average population growth rate can be understood from the growth rate of the mutation-free class. It is important to emphasize that the solution to (2) [and (1)] is an equilibrium that may require thousands of generations to be reached. Thus, if the solution is negative (r < 0), implying that the population will ultimately decline, the population may go extinct before attaining approximate equilibrium.Equation 2 does not lend itself to an explicit solution, but it is easily solved numerically. Although the parameters in (2) are meant to apply across all mutation rates, the reality for any chemical mutagen or drug is that higher doses of mutagen will not only increase U but also directly reduce viral fitness, such as by reducing burst size. To address this issue, parameters should be estimated in the mutagenic environment. In turn, estimating parameters in the mutagenic environment creates the complication that lethal mutations kill progeny and reduce the apparent burst size (when burst size is determined by plaque counts). To overcome this latter problem, we partition the total deleterious mutation rate into the sum of the lethal rate (U_X) and the nonlethal rate (U_d), U = U_X + U_d, and rewrite Equation 2 as(3)where , the viable burst size. Now, the direct effect of mutagen on burst size is inseparable from the effects of lethal mutations.

Population variation:

An important but subtle implication of the theory is that, when the mutation rate is high, the population will be genetically heterogeneous for deleterious mutations maintained at low to moderate frequencies (Haldane 1927; Crow and Kimura 1970; Eigen et al. 1988). Although every genome may contain many deleterious mutations, different genomes have different sets of deleterious mutations. Only a small proportion of the population may be of the best genotype, in which case, most individuals sampled will have lower fitness than that characterizing the population''s growth (Rouzine et al. 2003, 2008). This heterogeneity has the effect of complicating one means of estimating population fitness. When fitness involves component life history parameters such as burst size and lysis time, a fitness calculation based on separate estimates of life history components appears to underestimate actual population fitness. We have observed this effect in unpublished simulations and suspect that it is a parallel to the principle that the average of a ratio is not the ratio of averages. The T7 system that we use here has the advantage that the intrinsic mutation rate of the virus is low. Thus the starting phage and isolates are genetically uniform and are not subject to this problem. Estimation of fitness directly (as population growth rate rather than from separate fitness components) avoids this problem as well.     

Mutation-selection balance: ancestry,load, and maximum principle

We analyze the equilibrium behavior of deterministic haploid mutation-selection models. To this end, both the forward and the time-reversed evolution processes are considered. The stationary state of the latter is called the ancestral distribution, which turns out as a key for the study of mutation-selection balance. We find that the ancestral genotype frequencies determine the sensitivity of the equilibrium mean fitness to changes in the corresponding fitness values and discuss implications for the evolution of mutational robustness. We further show that the difference between the ancestral and the population mean fitness, termed mutational loss, provides a measure for the sensitivity of the equilibrium mean fitness to changes in the mutation rate. The interrelation of the loss and the mutation load is discussed. For a class of models in which the number of mutations in an individual is taken as the trait value, and fitness is a function of the trait, we use the ancestor formulation to derive a simple maximum principle, from which the mean and variance of fitness and the trait may be derived; the results are exact for a number of limiting cases, and otherwise yield approximations which are accurate for a wide range of parameters. These results are applied to threshold phenomena caused by the interplay of selection and mutation (known as error thresholds). They lead to a clarification of concepts, as well as criteria for the existence of error thresholds.     

Optimality of Mutation and Selection in Germinal Centers

The population dynamics theory of B cells in a typical germinal center could play an important role in revealing how affinity maturation is achieved. However, the existing models encountered some conflicts with experiments. To resolve these conflicts, we present a coarse-grained model to calculate the B cell population development in affinity maturation, which allows a comprehensive analysis of its parameter space to look for optimal values of mutation rate, selection strength, and initial antibody-antigen binding level that maximize the affinity improvement. With these optimized parameters, the model is compatible with the experimental observations such as the ∼100-fold affinity improvements, the number of mutations, the hypermutation rate, and the “all or none” phenomenon. Moreover, we study the reasons behind the optimal parameters. The optimal mutation rate, in agreement with the hypermutation rate in vivo, results from a tradeoff between accumulating enough beneficial mutations and avoiding too many deleterious or lethal mutations. The optimal selection strength evolves as a balance between the need for affinity improvement and the requirement to pass the population bottleneck. These findings point to the conclusion that germinal centers have been optimized by evolution to generate strong affinity antibodies effectively and rapidly. In addition, we study the enhancement of affinity improvement due to B cell migration between germinal centers. These results could enhance our understanding of the functions of germinal centers.     

Critical Mutation Rate Has an Exponential Dependence on Population Size in Haploid and Diploid Populations

Understanding the effect of population size on the key parameters of evolution is particularly important for populations nearing extinction. There are evolutionary pressures to evolve sequences that are both fit and robust. At high mutation rates, individuals with greater mutational robustness can outcompete those with higher fitness. This is survival-of-the-flattest, and has been observed in digital organisms, theoretically, in simulated RNA evolution, and in RNA viruses. We introduce an algorithmic method capable of determining the relationship between population size, the critical mutation rate at which individuals with greater robustness to mutation are favoured over individuals with greater fitness, and the error threshold. Verification for this method is provided against analytical models for the error threshold. We show that the critical mutation rate for increasing haploid population sizes can be approximated by an exponential function, with much lower mutation rates tolerated by small populations. This is in contrast to previous studies which identified that critical mutation rate was independent of population size. The algorithm is extended to diploid populations in a system modelled on the biological process of meiosis. The results confirm that the relationship remains exponential, but show that both the critical mutation rate and error threshold are lower for diploids, rather than higher as might have been expected. Analyzing the transition from critical mutation rate to error threshold provides an improved definition of critical mutation rate. Natural populations with their numbers in decline can be expected to lose genetic material in line with the exponential model, accelerating and potentially irreversibly advancing their decline, and this could potentially affect extinction, recovery and population management strategy. The effect of population size is particularly strong in small populations with 100 individuals or less; the exponential model has significant potential in aiding population management to prevent local (and global) extinction events.     

A Bayesian Mutation–Selection Framework for Detecting Site-Specific Adaptive Evolution in Protein-Coding Genes

In recent years, codon substitution models based on the mutation–selection principle have been extended for the purpose of detecting signatures of adaptive evolution in protein-coding genes. However, the approaches used to date have either focused on detecting global signals of adaptive regimes—across the entire gene—or on contexts where experimentally derived, site-specific amino acid fitness profiles are available. Here, we present a Bayesian site-heterogeneous mutation–selection framework for site-specific detection of adaptive substitution regimes given a protein-coding DNA alignment. We offer implementations, briefly present simulation results, and apply the approach on a few real data sets. Our analyses suggest that the new approach shows greater sensitivity than traditional methods. However, more study is required to assess the impact of potential model violations on the method, and gain a greater empirical sense its behavior on a broader range of real data sets. We propose an outline of such a research program.     

Age- and sex-distribution of the mutation load

We investigate the age and sex distribution of genetic fitness under mutation–selection balance by means of simple one-locus two-allele models. We find that the extent of age and sex variation in the mutation load is very dependent on the average effect of new mutations. If the average heterozygote selective effect of new mutations is large, then age and sex differences may constitute a significant fraction of the total load, and be significant as compared to standing genetic variation. Whether the mutation load will increase or decrease with age depends on the age- and sex-specific effects of the new mutations, and on the rate of accumulation of mutations in the germ line as individuals age. We argue that the load will most likely increase with age in animals with continuous germ-cell division throughout life, and that this will occur even when mutations have unconditionally deleterious effects. We show that a male-biased mutation rate is likely to result in both a male-biased mutation load and a load that increases with male age. This revised version was published online in July 2006 with corrections to the Cover Date.     

Lethal mutagenesis and evolutionary epidemiology

The lethal mutagenesis hypothesis states that within-host populations of pathogens can be driven to extinction when the load of deleterious mutations is artificially increased with a mutagen, and becomes too high for the population to be maintained. Although chemical mutagens have been shown to lead to important reductions in viral titres for a wide variety of RNA viruses, the theoretical underpinnings of this process are still not clearly established. A few recent models sought to describe lethal mutagenesis but they often relied on restrictive assumptions. We extend this earlier work in two novel directions. First, we derive the dynamics of the genetic load in a multivariate Gaussian fitness landscape akin to classical quantitative genetics models. This fitness landscape yields a continuous distribution of mutation effects on fitness, ranging from deleterious to beneficial (i.e. compensatory) mutations. We also include an additional class of lethal mutations. Second, we couple this evolutionary model with an epidemiological model accounting for the within-host dynamics of the pathogen. We derive the epidemiological and evolutionary equilibrium of the system. At this equilibrium, the density of the pathogen is expected to decrease linearly with the genomic mutation rate U. We also provide a simple expression for the critical mutation rate leading to extinction. Stochastic simulations show that these predictions are accurate for a broad range of parameter values. As they depend on a small set of measurable epidemiological and evolutionary parameters, we used available information on several viruses to make quantitative and testable predictions on critical mutation rates. In the light of this model, we discuss the feasibility of lethal mutagenesis as an efficient therapeutic strategy.     

An Improved Codon Modeling Approach for Accurate Estimation of the Mutation Bias

Phylogenetic codon models are routinely used to characterize selective regimes in coding sequences. Their parametric design, however, is still a matter of debate, in particular concerning the question of how to account for differing nucleotide frequencies and substitution rates. This problem relates to the fact that nucleotide composition in protein-coding sequences is the result of the interactions between mutation and selection. In particular, because of the structure of the genetic code, the nucleotide composition differs between the three coding positions, with the third position showing a more extreme composition. Yet, phylogenetic codon models do not correctly capture this phenomenon and instead predict that the nucleotide composition should be the same for all three positions. Alternatively, some models allow for different nucleotide rates at the three positions, an approach conflating the effects of mutation and selection on nucleotide composition. In practice, it results in inaccurate estimation of the strength of selection. Conceptually, the problem comes from the fact that phylogenetic codon models do not correctly capture the fixation bias acting against the mutational pressure at the mutation–selection equilibrium. To address this problem and to more accurately identify mutation rates and selection strength, we present an improved codon modeling approach where the fixation rate is not seen as a scalar, but as a tensor. This approach gives an accurate representation of how mutation and selection oppose each other at equilibrium and yields a reliable estimate of the mutational process, while disentangling the mean fixation probabilities prevailing in different mutational directions.     

Persistence of Common Alleles in Two Related Populations or Species

Mathematical studies are conducted on three problems that arise in molecular population genetics. (1) The time required for a particular allele to become extinct in a population under the effects of mutation, selection, and random genetic drift is studied. In the absence of selection, the mean extinction time of an allele with an initial frequency close to 1 is of the order of the reciprocal of the mutation rate when 4Nv << 1, where N is the effective population size and v is the mutation rate per generation. Advantageous mutations reduce the extinction time considerably, whereas deleterious mutations increase it tremendously even if the effect on fitness is very slight. (2) Mathematical formulae are derived for the distribution and the moments of extinction time of a particular allele from one or both of two related populations or species under the assumption of no selection. When 4Nv << 1, the mean extinction time is about half that for a single population, if the two populations are descended from a common original stock. (3) The expected number as well as the proportion of common neutral alleles shared by two related species at the tth generation after their separation are studied. It is shown that if 4Nv is small, the two species are expected to share a high proportion of common alleles even 4N generations after separation. In addition to the above mathematical studies, the implications of our results for the common alleles at protein loci in related Drosophila species and for the degeneration of unused characters in cave animals are discussed.