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.     

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.     

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.     

The extinction time under mutational meltdown driven by high mutation rates

Mutational meltdown describes an eco‐evolutionary process in which the accumulation of deleterious mutations causes a fitness decline that eventually leads to the extinction of a population. Possible applications of this concept include medical treatment of RNA virus infections based on mutagenic drugs that increase the mutation rate of the pathogen. To determine the usefulness and expected success of such an antiviral treatment, estimates of the expected time to mutational meltdown are necessary. Here, we compute the extinction time of a population under high mutation rates, using both analytical approaches and stochastic simulations. Extinction is the result of three consecutive processes: (a) initial accumulation of deleterious mutations due to the increased mutation pressure; (b) consecutive loss of the fittest haplotype due to Muller''s ratchet; (c) rapid population decline toward extinction. We find accurate analytical results for the mean extinction time, which show that the deleterious mutation rate has the strongest effect on the extinction time. We confirm that intermediate‐sized deleterious selection coefficients minimize the extinction time. Finally, our simulations show that the variation in extinction time, given a set of parameters, is surprisingly small.     

From Molecular Genetics to Phylodynamics: Evolutionary Relevance of Mutation Rates Across Viruses

Although evolution is a multifactorial process, theory posits that the speed of molecular evolution should be directly determined by the rate at which spontaneous mutations appear. To what extent these two biochemical and population-scale processes are related in nature, however, is largely unknown. Viruses are an ideal system for addressing this question because their evolution is fast enough to be observed in real time, and experimentally-determined mutation rates are abundant. This article provides statistically supported evidence that the mutation rate determines molecular evolution across all types of viruses. Properties of the viral genome such as its size and chemical composition are identified as major determinants of these rates. Furthermore, a quantitative analysis reveals that, as expected, evolution rates increase linearly with mutation rates for slowly mutating viruses. However, this relationship plateaus for fast mutating viruses. A model is proposed in which deleterious mutations impose an evolutionary speed limit and set an extinction threshold in nature. The model is consistent with data from replication kinetics, selection strength and chemical mutagenesis studies.     

Antagonistic coevolution with parasites increases the cost of host deleterious mutations

The fitness consequences of deleterious mutations are sometimes greater when individuals are parasitized, hence parasites may result in the more rapid purging of deleterious mutations from host populations. The significance of host deleterious mutations when hosts and parasites antagonistically coevolve (reciprocal evolution of host resistance and parasite infectivity) has not previously been experimentally investigated. We addressed this by coevolving the bacterium Pseudomonas fluorescens and a parasitic bacteriophage in laboratory microcosms, using bacteria with high and low mutation loads. Directional coevolution between bacterial resistance and phage infectivity occurred in all populations. Bacterial population fitness, as measured by competition experiments with ancestral genotypes in the absence of phage, declined with time spent coevolving. However, this decline was significantly more rapid in bacteria with high mutation loads, suggesting the cost of bacterial resistance to phage was greater in the presence of deleterious mutations (synergistic epistasis). As such, resistance to phage was more costly to evolve in the presence of a high mutation load. Consistent with these data, bacteria with high mutation loads underwent less rapid directional coevolution with their phage populations, and showed lower levels of resistance to their coevolving phage populations. These data suggest that coevolution with parasites increases the rate at which deleterious mutations are purged from host populations.     

Impact of mutation rate on the adaptation of gut bacteria

To study the role of mutator bacteria in the evolution of bacterial populations, we followed the impact of the mutation rate of Escherichia coli strains in the colonisation of the gut of axenic mice and the evolution of the mutation rate of bacterial populations living in the gut. We show that mutator bacteria have an advantage during the colonization. This adaptive advantage comes from their ability to generate adaptive mutations faster than wild type strains, mutations that allow their maintenance in the ecosystem. However, while mutator bacteria are becoming specialised to the environment they are living in, they accumulate mutations that may be deleterious or lethal in secondary environments. By following the evolution of the mutation rate of bacterial populations living in the gut of mice receiving antibiotics, we show that this therapy selects not only for antibiotic resistant mutants but also for mutator alleles that enhance mutation rates and are responsible for the appearance of the resistance. The costs of a high mutation rate, due to the accumulation of mutations, is seen in environments where changes are recurrent. In an ever-changing situation where every change is new, mutator bacteria might help the evolution of bacterial populations.     

Optimal bacteriophage mutation rates for phage therapy

The mutability of bacteriophages offers a particular advantage in the treatment of bacterial infections not afforded by other antimicrobial therapies. When phage-resistant bacteria emerge, mutation may generate phage capable of exploiting and thus limiting population expansion among these emergent types. However, while mutation potentially generates beneficial variants, it also contributes to a genetic load of deleterious mutations. Here, we model the influence of varying phage mutation rate on the efficacy of phage therapy. All else being equal, phage types with historical mutation rates of approximately 0.1 deleterious mutations per genome per generation offer a reasonable balance between beneficial mutational diversity and deleterious mutational load. We determine that increasing phage inoculum density can undesirably increase the peak density of a mutant bacterial class by limiting the in situ production of mutant phage variants. For phage populations with minimal genetic load, engineering mutation rate increases beyond the mutation-selection balance optimum may provide even greater protection against emergent bacterial types, but only with very weak selective coefficients for de novo deleterious mutations (below approximately 0.01). Increases to the mutation rate beyond the optimal value at mutation-selection balance may therefore prove generally undesirable.     

Quasispecies made simple

Quasispecies are clouds of genotypes that appear in a population at mutation–selection balance. This concept has recently attracted the attention of virologists, because many RNA viruses appear to generate high levels of genetic variation that may enhance the evolution of drug resistance and immune escape. The literature on these important evolutionary processes is, however, quite challenging. Here we use simple models to link mutation–selection balance theory to the most novel property of quasispecies: the error threshold—a mutation rate below which populations equilibrate in a traditional mutation–selection balance and above which the population experiences an error catastrophe, that is, the loss of the favored genotype through frequent deleterious mutations. These models show that a single fitness landscape may contain multiple, hierarchically organized error thresholds and that an error threshold is affected by the extent of back mutation and redundancy in the genotype-to-phenotype map. Importantly, an error threshold is distinct from an extinction threshold, which is the complete loss of the population through lethal mutations. Based on this framework, we argue that the lethal mutagenesis of a viral infection by mutation-inducing drugs is not a true error catastophe, but is an extinction catastrophe.     

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.     

The Effects of a Bottleneck on Inbreeding Depression and the Genetic Load

We study the effects of a population bottleneck on the inbreeding depression and genetic load caused by deleterious mutations in an outcrossing population. The calculations assume that loci have multiplicative fitness effects and that linkage disequilibrium is negligible. Inbreeding depression decreases immediately after a sudden reduction of population size, but the drop is at most only several percentage points, even for severe bottlenecks. Highly recessive mutations experience a purging process that causes inbreeding depression to decline for a number of additional generations. On the basis of available parameter estimates, the absolute fall in inbreeding depression may often be only a few percentage points for bottlenecks of 10 or more individuals. With a very high lethal mutation rate and a very slow population growth, however, the decline may be on the order of 25%. We examine when purging might favor a switch from outbreeding to selfing and find it occurs only under very limited conditions unless population growth is very slow. In contrast to inbreeding depression, a bottleneck causes an immediate increase in the genetic load. Purging causes the load to decline and then overshoot its equilibrium value. The changes are typically modest: the absolute increase in the total genetic load will be at most a few percentage points for bottlenecks of size 10 or more unless the lethal mutation rate is very high and the population growth rate very slow.     

Population persistence under high mutation rate: From evolutionary rescue to lethal mutagenesis

Populations may genetically adapt to severe stress that would otherwise cause their extirpation. Recent theoretical work, combining stochastic demography with Fisher's geometric model of adaptation, has shown how evolutionary rescue becomes unlikely beyond some critical intensity of stress. Increasing mutation rates may however allow adaptation to more intense stress, raising concerns about the effectiveness of treatments against pathogens. This previous work assumes that populations are rescued by the rise of a single resistance mutation. However, even in asexual organisms, rescue can also stem from the accumulation of multiple mutations in a single genome. Here, we extend previous work to study the rescue process in an asexual population where the mutation rate is sufficiently high so that such events may be common. We predict both the ultimate extinction probability of the population and the distribution of extinction times. We compare the accuracy of different approximations covering a large range of mutation rates. Moderate increase in mutation rates favors evolutionary rescue. However, larger increase leads to extinction by the accumulation of a large mutation load, a process called lethal mutagenesis. We discuss how these results could help design “evolution‐proof” antipathogen treatments that even highly mutable strains could not overcome.     

Evolution of sex in RNA viruses

The distribution of deleterious mutations in a population of organisms is determined by the opposing effects of two forces, mutation pressure and selection. If mutation rates are high, the resulting mutation-selection balance can generate a substantial mutational load in the population. Sex can be advantageous to organisms experiencing high mutation rates because it can either buffer the mutation-selection balance from genetic drift, thus preventing any increases in the mutational load (Muller, 1964: Mut. Res. 1, 2), or decrease the mutational load by increasing the efficiency of selection (Crow, 1970: Biomathematics 1, 128). Muller's hypothesis assumes that deleterious mutations act independently, whereas Crow's hypothesis assumes that deleterious mutations interact synergistically, i.e., the acquisition of a deleterious mutation is proportionately more harmful to a genome with many mutations than it is to a genome with a few mutations. RNA viruses provide a test for these two hypotheses because they have extremely high mutation rates and appear to have evolved specific adaptations to reproduce sexually. Population genetic models for RNA viruses show that Muller's and Crow's hypotheses are also possible explanations for why sex is advantageous to these viruses. A re-analysis of published data on RNA viruses that are cultured by undiluted passage suggests that deleterious mutations in such viruses interact synergistically and that sex evolved there as a mechanism to reduce the mutational load.     

Nature of Deleterious Mutation Load in Drosophila

Much population genetics and evolution theory depends on knowledge of genomic mutation rates and distributions of mutation effects for fitness, but most information comes from a few mutation accumulation experiments in Drosophila in which replicated chromosomes are sheltered from natural selection by a balancer chromosome. I show here that data from these experiments imply the existence of a large class of minor viability mutations with approximately equivalent effects. However, analysis of the distribution of viabilities of chromosomes exposed to EMS mutagenesis reveals a qualitatively different distribution of effects lacking such a minor effects class. A possible explanation for this difference is that transposable element insertions, a common class of spontaneous mutation event in Drosophila, frequently generate minor viability effects. This explanation would imply that current estimates of deleterious mutation rates are not generally applicable in evolutionary models, as transposition rates vary widely. Alternatively, much of the apparent decline in viability under spontaneous mutation accumulation could have been nonmutational, perhaps due to selective improvement of balancer chromosomes. This explanation accords well with the data and implies a spontaneous mutation rate for viability two orders of magnitude lower than previously assumed, with most mutation load attributable to major effects.     

Evolution of mutation rates in bacteria

Evolutionary success of bacteria relies on the constant fine-tuning of their mutation rates, which optimizes their adaptability to constantly changing environmental conditions. When adaptation is limited by the mutation supply rate, under some conditions, natural selection favours increased mutation rates by acting on allelic variation of the genetic systems that control fidelity of DNA replication and repair. Mutator alleles are carried to high frequency through hitchhiking with the adaptive mutations they generate. However, when fitness gain no longer counterbalances the fitness loss due to continuous generation of deleterious mutations, natural selection favours reduction of mutation rates. Selection and counter-selection of high mutation rates depends on many factors: the number of mutations required for adaptation, the strength of mutator alleles, bacterial population size, competition with other strains, migration, and spatial and temporal environmental heterogeneity. Such modulations of mutation rates may also play a role in the evolution of antibiotic resistance.     

The Interaction between Selection,Demography and Selfing and How It Affects Population Viability

Population extinction due to the accumulation of deleterious mutations has only been considered to occur at small population sizes, large sexual populations being expected to efficiently purge these mutations. However, little is known about how the mutation load generated by segregating mutations affects population size and, eventually, population extinction. We propose a simple analytical model that takes into account both the demographic and genetic evolution of populations, linking population size, density dependence, the mutation load, and self-fertilisation. Analytical predictions were found to be relatively good predictors of population size and probability of population viability when verified using an explicit individual based stochastic model. We show that initially large populations do not always reach mutation-selection balance and can go extinct due to the accumulation of segregating deleterious mutations. Population survival depends not only on the relative fitness and demographic stochasticity, but also on the interaction between the two. When deleterious mutations are recessive, self-fertilisation affects viability non-monotonically and genomic cold-spots could favour the viability of outcrossing populations.     

Beneficial effects of population bottlenecks in an RNA virus evolving at increased error rate

RNA viruses replicate their genomes with a very high error rate and constitute highly heterogeneous mutant distributions similar to the molecular quasispecies introduced to explain the evolution of prebiotic replicators. The genetic information included in a quasispecies can only be faithfully transmitted below a critical error rate. When the error threshold is crossed, the population structure disorganizes, and it is substituted by a randomly distributed mutant spectrum. For viral quasispecies, the increase in error rate is associated with a decrease in specific infectivity that can lead to the extinction of the population. In contrast, a strong resistance to extinction has been observed in populations subjected to bottleneck events despite the increased accumulation of mutations. In the present study, we show that the mutagenic nucleoside analogue 5-azacytidine (AZC) is a potent mutagen for bacteriophage Qβ. We have evaluated the effect of the increase in the replication error rate in populations of the bacteriophage Qβ evolving either in liquid medium or during development of clonal populations in semisolid agar. Populations evolving in liquid medium in the presence of AZC were extinguished, while during plaque development in the presence of AZC, the virus experienced a significant increase in the replicative ability. Individual viruses isolated from preextinction populations could withstand high error rates during a number of plaque-to-plaque transfers. The response to mutagenesis is interpreted in the light of features of plaque development versus infections by free-moving virus particles and the distance to a mutation-selection equilibrium. The results suggest that clonal bacteriophage populations away from equilibrium derive replicative benefits from increased mutation rates. This is relevant to the application of lethal mutagenesis in vivo, in the case of viruses that encounter changing environments and are transmitted from cell to cell under conditions of limited diffusion that mimic the events taking place during plaque development.     

The Fitness Effects of Random Mutations in Single-Stranded DNA and RNA Bacteriophages

Mutational fitness effects can be measured with relatively high accuracy in viruses due to their small genome size, which facilitates full-length sequencing and genetic manipulation. Previous work has shown that animal and plant RNA viruses are very sensitive to mutation. Here, we characterize mutational fitness effects in single-stranded (ss) DNA and ssRNA bacterial viruses. First, we performed a mutation-accumulation experiment in which we subjected three ssDNA (ΦX174, G4, F1) and three ssRNA phages (Qβ, MS2, and SP) to plaque-to-plaque transfers and chemical mutagenesis. Genome sequencing and growth assays indicated that the average fitness effect of the accumulated mutations was similar in the two groups. Second, we used site-directed mutagenesis to obtain 45 clones of ΦX174 and 42 clones of Qβ carrying random single-nucleotide substitutions and assayed them for fitness. In ΦX174, 20% of such mutations were lethal, whereas viable ones reduced fitness by 13% on average. In Qβ, these figures were 29% and 10%, respectively. It seems therefore that high mutational sensitivity is a general property of viruses with small genomes, including those infecting animals, plants, and bacteria. Mutational fitness effects are important for understanding processes of fitness decline, but also of neutral evolution and adaptation. As such, these findings can contribute to explain the evolution of ssDNA and ssRNA viruses.     

Frequent beneficial mutations during single-colony serial transfer of Streptococcus pneumoniae

The appearance of new mutations within a population provides the raw material for evolution. The consistent decline in fitness observed in classical mutation accumulation studies has provided support for the long-held view that deleterious mutations are more common than beneficial mutations. Here we present results of a study using a mutation accumulation design with the bacterium Streptococcus pneumoniae in which the fitness of the derived populations increased. This rise in fitness was associated specifically with adaptation to survival during brief stationary phase periods between single-colony population bottlenecks. To understand better the population dynamics behind this unanticipated adaptation, we developed a maximum likelihood model describing the processes of mutation and stationary-phase selection in the context of frequent population bottlenecks. Using this model, we estimate that the rate of beneficial mutations may be as high as 4.8×10(-4) events per genome for each time interval corresponding to the pneumococcal generation time. This rate is several orders of magnitude higher than earlier estimates of beneficial mutation rates in bacteria but supports recent results obtained through the propagation of small populations of Escherichia coli. Our findings indicate that beneficial mutations may be relatively frequent in bacteria and suggest that in S. pneumoniae, which develops natural competence for transformation, a steady supply of such mutations may be available for sampling by recombination.     

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.