In any natural population, mutation is the primary source of genetic variation required for evolutionary novelty and adaptation. Nevertheless, most mutations, especially those with phenotypic effects, are harmful and are consequently removed by natural selection. For this reason, under natural selection, an organism will evolve to a lower mutation rate. Overall, the action of natural selection on mutation rate is related to population size and mutation effects. Although theoretical work has intensively investigated the relationship between natural selection and mutation rate, most of these studies have focused on individual competition within a population, rather than on competition among populations. The aim of the present study was to use computer simulations to investigate how natural selection adjusts mutation rate among asexually reproducing subpopulations with different mutation rates.The competition results for the different subpopulations showed that a population could evolve to an "optimum" mutation rate during long-term evolution, and that this rate was modulated by both population size and mutation effects. A larger population could evolve to a higher optimum mutation rate than could a smaller population. The optimum mutation rate depended on both the fraction and the effects of beneficial mutations, rather than on the effects of deleterious ones. The optimum mutation rate increased with either the fraction or the effects of beneficial mutations. When strongly favored mutations appeared, the optimum mutation rate was elevated to a much higher level. The competition time among the subpopulations also substantially shortened.Competition at the population level revealed that the evolution of the mutation rate in asexual populations was determined by both population size and mutation effects. The most striking finding was that beneficial mutations, rather than deleterious mutations, were the leading force that modulated the optimum mutation rate. The initial configuration of the population appeared to have no effect on these conclusions, confirming the robustness of the simulation method developed in the present study. These findings might further explain the lower mutation rates observed in most asexual organisms, as well as the higher mutation rates in some viruses. Understanding the genetic structure of populations requires knowledge of the mutation rate, an important parameter of evolution. One of the essential problems in population genetics is determining how natural selection acts on the mutation rate of an organism during long-term evolution. Although mutation provides the ultimate source of genetic variation, it typically leads to decreased fitness. Even when a population is in the process of adaptation, the majority of its mutations are still deleterious and will ultimately be eliminated by selection. This type of selection pressure was first observed by Sturtevant , who queU) in DNA-based microbes was about 0.0034 per generation despite a wide variation in genome size. This relatively constant observed value indicates that the genome mutation rate in microbes has evolved perfectly to fit the pace of environmental changes through natural selection. Several theoretical methods have the potential to explain Drake's observation from different perspectives [Since Sturtevant's pioneering work, the evolution of mutation rate has been researched by many evolutionary biologists and our understanding of this question has been improved in many respects. At present, several methods have been proposed for characterization of the evolution of mutation rate, including direct estimates from mutation accumulation experiments -5, indirpectives ,12,14,15pectives . He descIn any finite population, the process of evolution is well known to be influenced by population size and mutation effects . BeneficWhen both deleterious and beneficial mutations are present, it is necessary to explore whether an organism could evolve to an "optimum" mutation rate under these two opposing forces. The nature of the dominating factors that shape the optimum mutation rate also needs to be determined. In the present paper, we have developed a simulation method based on competition among subpopulations with different mutation rates to examine how selection may impact the evolution of genome mutation rate. Our results indicate that a larger population could tolerate a higher mutation rate than could a smaller one. The optimum mutation rate depends almost exclusively on the effects of beneficial mutations regardless of the extent of deleterious mutation effects. Possible reasons for these findings are discussed in comparison with previous studies.N) that comprises 10 subpopulations, each of which has N/10 individuals and a different mutation rate, with everything else equal. The rationale of the method is that these subpopulations compete for existence under natural selection and random drift. At the end of a simulation, only one subpopulation remains and the rest are extinct. The mutation rate of the remaining population becomes the "fixed" mutation rate in that competition. By simulating the process many times, we can define the most frequently fixed mutation rate as the "optimum" mutation rate.We consider a finite strictly asexual haploid population (with constant population size b pand d p(i.e. 1- bp), respectively. The effects (selection coefficients) of both beneficial and deleterious mutations are drawn from continuous probability distributions. We denote b sas the effects of beneficial mutations (in which case fitness w is increased by a factor 1+ bs), while d srepresents the effects of deleterious mutations (in which case fitness w is decreased by a factor 1- ds)[Each of the ten subpopulations is assigned with a distinct mutation rate per genome per generation (see parameters). Both deleterious and beneficial mutations occur in each subpopulation with fractions for beneficial and deleterious mutations represented by or 1- sd).b sfollows an exponential distribution:d sin analytical calculations; however, empirical studies support a gamma distribution with shape parameter smaller than one (other distributions are not necessarily excluded)[d sfollows a skewed gamma distribution We assume that s models -24. The excluded),26. In tst cases ,28.U)) is roughly uniformly distributed between -4 and -1. In addition, we adopt several ranges consisting of different mutation rates, which are shown in Table In our simulations, the sizes of fractions and effects of both beneficial and deleterious mutations are the most important quantitative parameters. Numerous experimental studies on microbes have shed some light on this area and some estimates of these parameters are summarized in Table m) appearing in an individual belonging to the i-th subpopulation is drawn from a Poisson distribution i Uis the genome mutation rate of the i-th subpopulation. The deleterious mutation rate is then given by iU×d pand the beneficial mutation rate is iU×bp. Given that a deleterious  mutation occurs, the fitness w of the individual is decreased (or increased) by 1- d s(or 1+ bs), where d s(or bs) is randomly drawn from a gamma  distribution. Here, we assume that no epistasis occurs; therefore, all mutations have independent effects on fitness and act multiplicatively. It is possible that an individual may carry multiple mutations within a single generation. In this case, the fitness of an individual in the n-th generation (nw) is a function of the mutation numbers the individual carries (m), their mutation effects (js), and the fitness of its parent in the (n-1)-th generation (n-1w). This function can be described asThroughout the study, we assume that generations are discrete and non-overlapping. In each generation, the number of new mutations . Some initial conditions of the population are also relaxed to test the robustness of the method (see Discussion).We trace the numbers of individuals of each subpopulation until the population size of one subpopulation reaches opt Uas the optimum mutation rate and G as the mean number of generations required for competition in one simulation. We also used u Rto represent which group of mutation rates was adopted in Table Our extensive simulations were designed to test whether natural selection could shape the optimum mutation rate, given the initial configuration of the population. Each curve represents one simulation result with 300 competitions among all the figures and each point represents the frequency of being fixed of the corresponding mutation rate. For convenience of description, we used the symbol N). Clearly, N had a significant effect on the frequencies of fixed mutation rates. With all other factors held constant, increasing in N caused the curve to shift to the right, which demonstrated that a larger population tolerated a higher mutation rate. Large populations could benefit from relatively higher mutation rates because beneficial mutations appeared more frequently and selection was more efficient in removing deleterious mutations in large populations than in small ones [4 to 107.Figure all ones . As Figubp), while holding the effects of both deleterious and beneficial mutations constant. The curve was shifted to the right with increasing bp, indicating that a population could evolve to a higher mutation rate when beneficial mutations appeared more frequently. As Figure b pvaried from 1% to 10%.To investigate the influence of the relative fraction of deleterious and beneficial mutations, and the combined effects of both, on the optimum mutation rate, we ran simulations where one factor was controlled and the other was varied. In Figure β) of the gamma distribution for describing deleterious mutations effects, while holding the shape parameter and the effects of beneficial mutations constant. The frequencies of the fixed mutation rates varied only slightly and the optimum mutation rate held constant when β was varied, although the mean effects of deleterious mutations changed substantially. Evaluation of how the shape parameter (α) of the gamma distribution influenced the results was also important. In Figure α, while holding the scale parameter and the effects of beneficial mutations constant. As α decreased, the deleterious mutations that had small effects increased while those with large effects decreased, shifting the curve slightly to the right. Varying the shape parameter therefore had more effect on the frequencies of fixed mutation rates than did varying the scale parameter. Nevertheless, the optimum mutation rate remained constant in this case, indicating that deleterious mutations were effectively eliminated by relentless selection and that their effects had little influence on the optimum mutation rate in long-term evolution. Only when the deleterious mutation effects became small enough to compensate for the beneficial mutation effects could the optimum mutation rate evolve to a higher level (data not shown).In Figure λ) of exponential distribution, while holding the effects of deleterious mutations constant. In contrast to the very small impact seen in response to varying the deleterious mutation effects, the beneficial mutation effects contributed significantly to the optimum mutation rate. The curve was substantially shifted to the right with increasing G = 210) shortened sharply compared to Finally, in Figure In this study, we developed a simulation method based on competition among subpopulations in order to explore the pattern of evolution of mutation rate in large asexual populations. Our simulation results showed that populations tended to form an optimum mutation rate based on their initial configuration. This optimum mutation rate depended on the influx of favorable mutations as well as on their corresponding effects. Below, we first discuss the influence of the initial configuration of the population. We then discuss why beneficial mutations are important in asexual populations. Finally, we compare the present results to previous studies about mutators.To assess whether different mutation ranges and different initial fitness influenced the optimum mutation rate, we performed new simulations, yielding the following results. In Figure We also assigned rugged initial fitness to replace the assumption that all individuals had unified initial fitness value of 1.0. In Figure Finally, we show the influence of organism's fertility limitation on the optimum mutation rate in Figure To summarize, the initial configuration of the population had little influence on the optimum mutation rate, demonstrating the robustness of the developed method based on competition among subpopulations.In this study, we have made three assumptions about mutation effects: first, the mean effects of deleterious mutations are much larger than those of beneficial ones; second, beneficial mutation effects are exponentially distributed; and finally, deleterious mutations effects follow a gamma distribution. However, our simulation results hinge mainly on the first two assumptions. The first assumption is likely to be biologically realistic in many cases, although surely not universally true. Indeed, theory analysis ,28 and ein silico system to show that beneficial mutations with small effects have always existed in the process of evolution. Although beneficial mutations are much rarer compared to deleterious mutations, they supply the driving force for adaptive evolution and contribute to survival of populations in tough environments [bp) and effects (bs) [In general, organisms are well adapted to their living environments, so only a few changes lead to fitness increases and these beneficial mutations have very small effects ,34,35,37ronments . As showronments -44. How cts (sb) ,43.bp) is relatively high, the clonal interference becomes important. However, in this case, there will also be more chances for multiple beneficial mutations to occur in the same genetic background. Whenever clonal interference is important, so are multiple mutations. As Desai and Fisher showed, evolution in asexual budding yeast populations was dominated by the accumulation of multiple mutations of moderate effect [If the fraction of beneficial mutations (e effect . Individbs) are small, these mutations need more generations to be fixed and remain at low frequency in the population for quite a long time [b sincreased. As Wilke pointed out, in the presence of clonal interference, adaptation speed in asexuals still continued to grow with the mean beneficial mutation effects [b smay further increase the adaptation rate of populations considerably. In this case, the population favored a much higher mutation rate. Our simulation results indicated that if strong beneficial mutations (Uopt = 0.05).On the other hand, if the effects of the majority of new arising beneficial mutations (ong time . This prong time . By contong time ,45,47. T effects , althougThis might provide an alternate explanation for why viruses are capable of evolving to a much higher mutation rate  under thThe action of selection on mutation rate can be classified as either direct or indirect: direct action is dependent on the effects of modifier alleles on fitness, while indirect action is dependent on the "linkage disequilibrium" between modifier alleles and alleles at other loci affecting fitness . Strong Nevertheless, it should be noticed that in all of the previous studies, high genome mutation rates were generally disfavored in asexual populations except when organisms were under extreme conditions. Gerrish et al. suggested that in the case of complete linkage, the mutation rate would continue to increase until it reached an intolerable level and then lead to organism extinction, rather than elevate without a ceiling . The intOther studies involving modifiers also suggested that even if a high mutation rate increased the rate of adaptation in the short term, due to deleterious mutations, selection would be likely to decrease the mutation rate in the long term evolution ,52-54. TBased on competition among subpopulations with different mutation rates, we investigated the evolution of mutation rates in finite asexual populations. The efficiency of natural selection on mutation rate was shown to depend on population size and mutation effects. Large populations tend to have high mutation rates. The optimum rate is also the result of a balance between two opposing forces: a decreasing rate caused by deleterious mutations and adaptation caused by beneficial mutations. However, the influx of favorable mutations is the critical factor and largely determines the optimum mutation rate in large asexual populations. Contrary to our intuition, the effects of deleterious mutations have little impact on this rate as long as there is no an abundance of deleterious mutations with tiny effects. We hope this simulation method and these findings provide useful inspiration for further modeling of the evolution of mutation rates in asexual populations.XJ carried out the simulation experiments, analyzed the data, and drafted the manuscript. BM and ZH participated in writing the computer programs. MZ and XW proposed important suggestions for conceiving the simulation process. ST conceived of the study and supported the research. All authors have read and approved the final version of the manuscript.


A large number of medically important viruses, including HIV, hepatitis C virus, and influenza, have RNA genomes. These viruses replicate with extremely high mutation rates and exhibit significant genetic diversity. This diversity allows a viral population to rapidly adapt to dynamic environments and evolve resistance to vaccines and antiviral drugs. For the last 30 years, quasispecies theory has provided a population-based framework for understanding RNA viral evolution. A quasispecies is a cloud of diverse variants that are genetically linked through mutation, interact cooperatively on a functional level, and collectively contribute to the characteristics of the population. Many predictions of quasispecies theory run counter to traditional views of microbial behavior and evolution and have profound implications for our understanding of viral disease. Here, we discuss basic principles of quasispecies theory and describe its relevance for our understanding of viral fitness, virulence, and antiviral therapeutic strategy. The rapid evolution of RNA viruses complicates the management of chronic infections and the control of emerging infectious agents The evolutionary dynamics of RNA viruses are complex and their high mutation rates, rapid replication kinetics, and large population sizes present a challenge to traditional population genetics Here we explain basic aspects of quasispecies theory, describe key experiments that define “quasispecies effects,” and highlight how these results may shape our view of viral pathogenesis, antiviral drug development, and vaccine design. We will stress three clinically relevant principles. First, the fitness of a particular virus sequence may be determined more by its freedom to mutate into related sequences than by its own replicative capacity. Second, many viruses operate near a threshold of “error catastrophe” and may be combated by increasing their replication error rates. Third, increasing the fidelity of genome replication may paradoxically attenuate viruses.−4 mutations per nucleotide copied, which is orders of magnitude greater than those of nearly all DNA-based viruses and organisms Most viruses encode enzymes responsible for replicating their DNA or RNA genomes. The intrinsic error rate, or fidelity, of the replicase determines the mutation rate for that virus and the range of genetic variation upon which natural selection can act. Viral RNA polymerases exhibit characteristically low fidelity with measured mutation rates of roughly 10The genetic organization of populations is often depicted using the concept of sequence space, a geometric representation of all possible sequences where physical distance reflects genetic similarity. For example, a given viral genome will undergo replication and generate hundreds of progeny that differ at roughly one position . SubsequViral populations evolve within a fitness landscape where the “ground level” is a representation of the range of genotypes in sequence space. The “altitude” at any given location is the fitness associated with that particular genotype. The environment and its selective pressures determine the contours of the corresponding landscape, and adaptation to an environment involves a mutational walk from one point in the fitness landscape to another . In quasA viral quasispecies, then, is a cloud of diverse variants that are genetically linked through mutation, interact cooperatively on a functional level, and collectively contribute to the characteristics of the population. The unit of selection is the population as a whole, and the nature of the functional interactions among genetically distinct variants is therefore of critical importance to pathogenesis in infected hosts. These effects and their biomedical implications are described below.Mutation and selection are the most fundamental processes in evolution. In Darwinian evolution, natural selection acts on existing genetic variation to optimize fitness. Conceptually, fitness refers to how well a given organism “fits” into its environment, often reflected in how well it survives and reproduces Measuring the fitness of individual variants within a population may misrepresent the fitness of a quasispecies. Early experiments with vesicular stomatitis virus (VSV) established that high fitness variants could be suppressed to low levels within a complex population A flat quasispecies with an expansive mutant repertoire can explore vast regions of sequence space without consequence and is poised to adapt to rapid environmental change. This framework may explain many observed phenomena with direct clinical relevance. Arboviruses must successfully adapt to insect and mammalian hosts and their associated fitness landscapes. A quasispecies that occupies a broad, flat region of sequence space could access neutral networks and local fitness peaks in either environment. Similarly, retrospective studies of primary HIV isolates suggest that HIV could be moving to a flatter and less fit region of sequence space Given their high mutation rates, it is not surprising that many discussions of RNA virus evolution focus on the relationship between genetic diversity and adaptability. While it is clear that RNA viruses have the capacity to quickly explore large regions of sequence space, genome size and selective constraints place significant limitations on the amount of diversity that is actually expressed Quasispecies theory accepts this trade-off, and proposes an upper limit to mutation rate, the “error threshold” . AccordiRecent work on host cell restriction of retroviral infection suggests that lethal mutagenesis is a natural form of antiviral defense These observations suggest that lethal mutagenesis could be an effective therapeutic strategy for RNA virus infections While these results are encouraging, much work needs to be done before lethal mutagenesis can be considered as a quasispecies-inspired therapeutic strategy. Some have argued for a distinction between lethal mutagenesis and error catastrophe, and have suggested, on theoretical grounds, that the above experiments do not show a true phase transition with loss of a master sequence. Indeed, careful studies of lethal mutagenesis by Lowenstein and colleagues have shown an imperfect correlation between mutational load and population extinction The potential for mutagen resistance is another important concern. In principle, a virus could evolve biochemical resistance to nucleosides either by excluding the drug from its active site or by lowering its intrinsic error rate (see below). Quasispecies theory suggests that viruses could also achieve resistance by moving to flatter regions of the fitness landscape, where the density of neutral mutations is higher. Combination therapy with mutagens and traditional antivirals may minimize the probability of mutagenic escape, and work in the FMDV system is encouraging Evolutionary theory predicts that high mutation rates are favored in a dynamic environment, and viral error rates may have been optimized by natural selection G64S) Recent work by two groups in the poliovirus system provides experimental support for this model. Drawing on experience with ribavirin and lethal mutagenesis, they hypothesized that a mutant with a low mutation rate would be less sensitive to lethal mutagenesis and resistant to ribavirin. Both groups sought to isolate ribavirin resistant mutants from a poliovirus quasispecies and recovered a variant with a single amino acid substitution in the viral polymerase , the site of disease This work shows how quasispecies theory and its predictions can lead to the rational development of live, attenuated vaccines. Because variants with low mutation rates would be less likely to revert to wild type in a vaccinee, attenuation should be a stable phenotype and increase the safety of candidate vaccines. Caution is warranted, however, since the error rates of the high fidelity variants are still orders of magnitude greater than those of DNA viruses. It is also unclear whether reducing mutation rate will similarly affect the adaptability and virulence of other RNA viruses. Further work in other systems is clearly needed before this work can be translated into new vaccines Quasispecies theory has had a profound influence on virology, and experiments with model RNA viruses have validated many of its predictions. Considerably less is known about how its population-based models apply to the evolutionary behavior of RNA viruses in infected hosts, and it will be challenging to translate them to the complex reality of viral infection in patients. The initial studies pose as many questions as they answer. What is the best measure of viral fitness in a dynamic population? How can we use a fitness landscape model to understand selection pressure in vivo, where many such landscapes coexist? Do the principles of mutational robustness and survival of the flattest determine the behavior of populations in the host ecosystem? How does population diversity influence pathogenesis? What are the mechanisms by which variants or subpopulations cooperate, and does this have implications for coinfection?While early forays into complex experimental systems have provided tantalizing results, much more work is needed in this area if quasispecies theory is to be translated from bench to bedside. Improved assays for characterizing viral populations are critical, and the advent of ultra high throughput, or “deep” sequencing, is particularly exciting. Several groups have used this technology to explore the complexity of mutant spectra in clinical samples from HIV- and HBV-infected patients, although the sequencing error rate presents certain limitations for studies of RNA as opposed to DNA viruses


The rate of mutation is central to evolution. Mutations are required for adaptation, yet most mutations with phenotypic effects are deleterious. As a consequence, the mutation rate that maximizes adaptation will be some intermediate value. Here, we used digital organisms to investigate the ability of natural selection to adjust and optimize mutation rates. We assessed the optimal mutation rate by empirically determining what mutation rate produced the highest rate of adaptation. Then, we allowed mutation rates to evolve, and we evaluated the proximity to the optimum. Although we chose conditions favorable for mutation rate optimization, the evolved rates were invariably far below the optimum across a wide range of experimental parameter settings. We hypothesized that the reason that mutation rates evolved to be suboptimal was the ruggedness of fitness landscapes. To test this hypothesis, we created a simplified landscape without any fitness valleys and found that, in such conditions, populations evolved near-optimal mutation rates. In contrast, when fitness valleys were added to this simple landscape, the ability of evolving populations to find the optimal mutation rate was lost. We conclude that rugged fitness landscapes can prevent the evolution of mutation rates that are optimal for long-term adaptation. This finding has important implications for applied evolutionary research in both biological and computational realms. Natural selection is shortsighted and therefore does not necessarily drive populations toward improved long-term performance. Some traits may evolve because they provide immediate gains, even though they are less successful in the long run than some alternatives. Here, we use digital organisms to analyze the ability of evolving populations to optimize their mutation rate, a fundamental evolutionary parameter. We show that when the mutation rate is constrained to be high, populations adapt considerably faster over the long term than when the mutation rate is allowed to evolve. By varying the fitness landscape, we show that natural selection tends to reduce the mutation rate on rugged landscapes (but not on smooth ones) so as to avoid the production of harmful mutations, even though this short-term benefit limits adaptation over the long term. The mechanisms of DNA replication and repair are themselves genetically encoded and variable Experiments have shown that genotypes with increased mutation rates can be favored by selection if they face novel or changing environments These limitations can be overcome using evolution with digital organisms owing to the speed and ease of data collection. Digital organisms are self-replicating computer programs that inhabit a virtual world where they reproduce, mutate, compete for resources, and evolve according to the same fundamental processes as biological organisms −5 to 10 mutations per genome per generation). The final fitness values confirmed that there was an optimal mutation rate at an intermediate value, with optU≈4.641 (−5) produced the highest fitness values, whereas for generations 50–230 a mutation rate of 2.2 gave the highest fitness values.We studied the evolution of mutation rates using the Avida digital evolution platform pt≈4.641 . An analoptU, we ran additional experiments in which mutation rates were allowed to change (see −3) or above (10) the optimum. Strikingly, mutation rates evolved to levels far below the long-term optU, regardless of the starting value  for the populations starting at high and low initial rates, respectively. Both of these values are significantly lower than the 14.45±0.64 obtained for populations evolved at optU . The fitness advantage for optU is also clear for nearly all intermediate time points . We conclude, therefore, that selection fails to optimize mutation rates for long-term adaptation in a broad range of experimental conditions.The finding that mutation rates evolved to be suboptimal was robust to diverse and substantial changes in the experimental conditions. First, we tested whether our results depended on the particular ancestral organism used. In the original experiments, the ancestor was a default, hand-coded organism. To assess whether this condition substantively influenced our results, we let a population founded by this organism adapt for 50,000 updates to an environment without any rewarded functions, using optU is that selection favored those genotypes that minimized the short-term fitness costs caused by deleterious mutations. This explanation is supported by the observations that, during the earliest generations of the evolution experiments, the lowest mutation rate yielded the highest fitness values. To test whether short-term selection would favor low mutation rates, we performed competition experiments between two kinds of organisms, designated A and B. These organisms were identical except for their mutation rate, which was set to optU for A and 0 for B; neither mutation rate was allowed to change during the competition. All competitions were conducted with the same environmental configurations as in the main experiments. In all of 50 runs, B drove A extinct in fewer than 40 generations. Competitions were also performed using U = 1.0 and U = 2.154 for B in order to address whether selection would also favor less extreme reductions in mutation rate. In both treatments, B drove A extinct in all 50 trials in fewer than 800 generations. These experiments confirm our hypothesis that natural selection was shortsighted and favored low mutation rates, even when such low rates precluded further adaptation.A possible explanation for why mutation rates evolved to be much lower than We conclude from the results presented thus far that the failure of the evolving populations to achieve or even maintain the mutation rates that maximize long-term adaptation reflect the conflict between the short-term cost of deleterious mutations and the long-term potential for adaptive evolution. We further hypothesize that the resolution of this tension may depend on the topology of the fitness landscape on which evolution occurs. In a rugged fitness landscape, where there are multiple peaks separated by maladaptive valleys −5) or above (1) the long-term optimum. Near-optimal values were efficiently selected in those landscapes without a fitness valley or with a narrow valley (optU (optU (P<0.001 in both cases), which indicates a small benefit of adjusting mutation rates during evolution optU (P<0.001 in both cases), confirming that the evolved mutation rates were suboptimal on the these rugged landscapes.A quantitative investigation of mutation rates spanning orders of magnitude revealed, once again, that intermediate mutation rates were optimal over the long-term . We thenw valley . Howeverey , which we call Π, affects the evolutionarily stable mutation rate. Our intuition was that lower values of Π would make contests between lineages with different mutation rates less frequent, but that the long-term results of many such contests would remain the same. To test this prediction we again used the explicit landscape with a valley size of three. Even when Π varied over four orders of magnitude, it did not affect the final mutation rate that was reached . Hence, We have shown that mutation rates evolve to near-optimal levels on extremely smooth fitness landscapes. However, if fitness landscapes are rugged, and the maladapted valleys between nearby fitness peaks are wide, then the scarcity of immediately accessible beneficial mutations tips the scale such that short-term selection favors mutation rates that are far below the optimum that would produce the fastest long-term adaptation. Moreover, this process is self-reinforcing because the lower the mutation rate, the less likely it becomes to produce a genotype on the other side of the fitness valley, thereby effectively widening the valley. The digital organisms in the standard Avida configuration used in our first set of experiments exhibit extensive and variable genetic interactions, making the fitness landscape rugged Our experiments were performed under conditions that were favorable for the optimization of mutation rates. First, the organisms reproduced asexually. Both theoretical The inability of evolving populations to optimize their mutation rates for long-term adaptation, even with such favorable conditions, indicates that mutation rates will be suboptimal under a wide range of circumstances, at least when fitness landscapes are rugged and populations are far from a global fitness peak. While novel environments can promote increases in the mutation rate if many beneficial mutations become accessible Recent theoretical work by Gerrish et al. Beyond their implications for understanding nature, our findings are also relevant for applied fields that use evolution to improve the performance of biological and computational systems, from molecular and microbial engineering to robotics and evolutionary computation In summary, natural selection is not universally effective at optimizing mutation rates for long-term adaptation; in fact, it is very poor in this respect for populations that evolve on complex fitness landscapes. Also, our results caution against making generalizations based on analyses of simple fitness landscapes, whether one is studying natural systems or using evolution for engineering. As we have shown, the mere inclusion of fitness valleys—which are presumably common to the vast majority of fitness landscapes—can yield radically different conclusions from those based on smooth fitness landscapes.A general description of the Avida software can be found elsewhere μ, the genomic mutation rate being U = 100×μ. All instructions were equally likely to result from any given mutation. The mutation rate was held constant in some experiments, while in others the rate could change by evolving over time. In treatments where the mutation rate could change, μ had a constant and high probability Π of changing by a small amount during any replication cycle. The magnitude of any resulting change was obtained by drawing log2(μoffspring/μparent) values from a Gaussian distribution . For the experiments in which mutation rates were more likely to increase than to decrease, we drew log2(μoffspring/μparent) from a Gaussian , where b controls the upward bias, and tested values such that mutation rates were up to ∼1.6 times more likely to increase than decrease .During replication, each genomic instruction could mutate to another with probability Organisms died when another organism's offspring replaced them or when they executed 2,000 instructions without producing an offspring of their own. All experiments using the standard configuration lasted 150,000 updates. Updates are an arbitrary unit of time in Avida; they represent the time during which each organism, on average, executes 30 instructions 10 transformed, and finally averaged over all replicate populations  in an experimental treatment.Each organism's phenotype depended on the complex rules that governed how its genomic program was executed, and its fitness depended on the interaction between the resulting phenotype and its environment To perform logic functions, organisms used inputs consisting of three randomly generated 32-bit strings, which they manipulated to produce an output. The manipulation of these numbers occurred as organisms moved them on and off stacks or between registers by executing instructions such as push, pop, add (combines the numbers in the two specified registers and places the result in a third), shift-r (bit shift right), and so on. A function was rewarded only if the input to output conversion conformed to one of the 77 canonical one-, two- or three-input logic operations. For example, the two-input EQU  function requires inputting two strings and outputting a third string that had a 1 for each of the 32 bits where both inputs had the same value and a 0 where they differed.Avida runs are inherently stochastic with respect to mutation and death. Therefore, we performed 50 replicate runs for each treatment. Those replicates had identical initial conditions except for a random number seed. That seed affects the outcome of all subsequent stochastic events.The standard and the explicit Avida configurations differed in the instruction set, the fitness calculation and the mode of replication. We modified Avida to mimic a two-allele, 10-locus bit-string model used in a previous study In the standard configuration, digital organisms had to copy their genomic instructions in order to replicate, and their fitness depended on their speed of replication as well as any rewards they obtained for performing computational functions. Under this alternative configuration, the organisms did not copy themselves, and only the number of A or B instructions mattered to their fitness. The rest of the setup, such as population size, was identical to the standard configuration.http://devolab.cse.msu.edu/software/avida. Default settings were used unless otherwise indicated.All experiments were performed with the Avida software, which can be downloaded for free at


Pseudomonas aeruginosa as a model system, we investigate the underlying distribution of fitness effects of beneficial mutations on which natural selection acts. Consistent with theory, the effects of beneficial mutations are exponentially distributed where the fitness of the wild type is moderate to high. However, when the fitness of the wild type is low, the data no longer follow an exponential distribution, because many beneficial mutations have large effects on fitness. There is no existing population genetic theory to explain this bias towards mutations of large effects, but it can be readily explained by the underlying biochemistry of rifampicin–RNA polymerase interactions. These results demonstrate the limitations of current population genetic theory for predicting adaptation to severe sources of stress, such as antibiotics, and they highlight the utility of integrating statistical and biophysical approaches to adaptation.Understanding how beneficial mutations affect fitness is crucial to our understanding of adaptation by natural selection. Here, using adaptation to the antibiotic rifampicin in the opportunistic pathogen  P. aeruginosa. When the fitness of the wild type is high, most beneficial mutations have small effects. This finding is consistent with existing population genetic models of adaptation based on statistical theory. When the fitness of the wild type is low, most beneficial mutations have large effects. This distribution cannot be explained by population genetic theory, but it can be readily understood by considering the biochemical basis of resistance. This study confirms an important prediction of population genetic theory, and it highlights the need to integrate statistical and biochemical approaches to adaptation in order to understand evolution in stressful environments, such as those provided by antibiotics.Adaptation by natural selection depends on the spread of novel beneficial mutations, and one of the most important challenges in our understanding of adaptation is to be able to predict how beneficial mutations impact fitness. Here, we investigate the underlying distribution of fitness effects of beneficial mutations that natural selection acts on during the evolution of antibiotic resistance in the opportunistic human pathogen  Adaptation by natural selection ultimately depends on the spread of novel beneficial mutations that increase fitness. Can we predict the fitness effects of beneficial mutations? GillespiePseudomonas aeruginosa as a model system. Experimental studies of the underlying distribution of fitness effects of beneficial mutationsPseudomonas aeruginosa with mutations in the β-subunit of RNA polymerase (rpoB) that are beneficial in the presence of the drug rifampicin rpoB of the mutants. To test the hypothesis that the fitness effects of beneficial mutations are exponentially distributed, we used log-likelihood tests that have been specifically developed to test this hypothesis using this experimental designIn this paper, we investigate how the distribution of fitness effects of beneficial mutations changes with the fitness of the wildtype population using the evolution of antibiotic resistance in the opportunistic pathogen Using this approach, we show that the distribution of fitness effects of beneficial mutations is variable: under conditions where the fitness of the wild-type is high, the fitness effects of beneficial mutations are exponentially distributed, as predicted by theory. However, when the fitness of the wildtype is low, the data may no longer fit an exponential distribution because many beneficial mutations have large effects on fitness. We show that this non-exponential distribution of fitness effects emerges as a direct consequence of the molecular interactions that are under selection in this system and we argue that existing theory on the fitness effects of beneficial mutations cannot be applied to understand adaptation to novel stressful environments, such as those provided by antibiotics.To investigate how the distribution of fitness effects of beneficial mutations changes with the fitness of the wild-type, we measured the fitness of beneficial mutations isolated at a high concentration of rifampicin  and the One limitation of this study is that our power to test the null hypothesis is weakest under conditions where the fitness of the wild-type is high, because only half of the mutants that we isolated increase fitness at low concentrations of rifampicin. Given that we had to sample 80 mutants in order to identify 15 beneficial mutations , it unlikely that increasing the sample size of our study would have substantially increased the power of our analysis. It is also important to note that this limitation is not unique to this study: beneficial mutations are rare events, and all other comparable experimental evolution studies are based on a similar sample size of beneficial mutationsr = .86–.9, P<.0001). The genetic variation in growth rate in the absence of antibiotics generated by spontaneous mutation is normally distributed , hence the fitness effects of beneficial mutations are exponentially distributed because only mutations in the right tail of the distribution . Residues that surround the binding pocket interact only indirectly with rifampicin, and it has been argued that resistance arises at these residues due to amino acid changes that alter the folding of the protein in the binding pocket. We identified only a small number of beneficial mutations (n = 4) in residues that are involved in indirect Rif-RNAP interactions and mutations at these residues give rise to intermediate levels of rifampicin resistance . This biophysical approach to understanding the effects of beneficial mutations suggests that the data may no longer fit an exponential distribution because of the high specificity of interactions between rifampicin and RNA polymerase: changes to the majority of amino acids that are involved in rifampicin-RNAP interactions results in large increases in resistance and, therefore, large increases in fitness at high concentrations of rifampicin. To test this hypothesis further, we assayed fitness in the presence of sorangicinP = .09).The large effect of beneficial mutations on resistance is consistent with the molecular interactions that occur between rifampicin and RNA polymerase. Structural studies have shown that rifampicin binds to a small, highly conserved pocket of the β-subunit of RNAP and only 12 amino acid residues are involved in direct interactions with rifampicinIt is important to note that these results do not necessarily imply that the distribution of fitness effects of mutations that are beneficial in the presence of sorangicin is universally exponential. Most mutations that increase sorangicin resistance do so by altering membrane permeability, instead of altering the structure of RNAPet al.The variability in the distribution of fitness effects of beneficial mutations in this study is consistent with population genetic theory. When the fitness of the wild-type is high, beneficial mutations can be viewed from a statistical perspective as representing draws from the extreme tail of the distribution of fitness effects of mutations, hence the fitness effects of beneficial mutations will be exponentially distributed. However, EVT does not specify how high fitness has to be in order for this theory to apply. In our experimental system, this distribution held over a wide range of parameter space: we failed to detect significant deviations from the exponential distribution when the fitness of the wild-type was reduced by 20–30%. When the fitness of the wild-type is low, statistical theory does not make any predictions regarding the form of the distribution of fitness effects of beneficial mutations and hence there is no reason to expect an exponential distribution. Comparable experimental studies in viral systems are in agreement with this idea: Sanjuan and colleagues a priori). Specifically, the distribution is biased towards mutations of large effect as a result of the high specificity of interactions between rifampicin and RNA polymerase that arises from the low conformational flexibility of rifampicin. Antibacterial and antiviral drugs are usually involved in highly specific interactions with their target proteins Despite our lack of certainty of the statistical explanation for the observed distribution when the fitness of the wild-type is low, we have a good molecular mechanistic explanation . This overnight culture was diluted down 10−6 into fresh M9KB and 120 uL aliquots of this diluted culture were used to setup 480 cultures on 5 96 well microplates. These cultures were incubated overnight at 37 C without shaking. To isolate beneficial mutations, 5 uL of each of the 480 overnight cultures was plated out on M9KB supplemented with 62.11 ug/mL of rifampicin, the minimal concentration required for complete inhibition of growth of the wildtype strain. To isolate beneficial mutations, we isolated a single colony from each of the first 80 cultures that gave samples containing exactly 1 colony on agar plate containing rifampicin.A single clone of rpoB gene in each of the 80 colonies that we isolated in our fluctuation test. Genomic DNA was isolated from each colony using a Wizard Genomic DNA extraction kit  as per the manufacturer's instructions. Our sequencing strategy was to first sequence a highly-conserved domain of rpoB that is known to be important for rifampicin resistance in all 80 clones. This region was amplified with primers rpoB_fwd (5′-GTTCTTCAGCGCCGAGCG-3′) and rpoB_rev (5′-GCGATGACGTGGTCGGC-3′) that amplify the region of the rpoB gene between nucleotides 1178 and 1864. Reaction mixtures consisted of BIOTAQ polymerase , 1 mM dNTPs, 16 nM (NH4)2SO4, 62.5 mM Tris-HCL (pH 8.8), .01% Tween 20, 2 mM MgCl2, and each primer at a concentration of .2pM. Amplification reactions were carried out as follows: 94 C for 5 minutes, followed by 35 cycles of 94 C for 30 seconds, 60 C for 30 seconds, and 72 C for 1 minute, followed by a final incubation at 72 C for 10 minutes. PCR products were purified using a MultiScreen PCR96 filter plate  as per the manufacturer's instructions. Purified PCR products were sequenced with both forward and reverse primers using BigDye 3.1 sequencing  followed by ethanol/EDTA precipitation of sequencing products. In all cases, this strategy identified either a single mutation in this region of rpoB or no mutations.To determine the mutations underlying adaptation, we sequenced the rpoB were subsequently sequenced for a second region that has previously been implicated in rifampicin resistance spanning nucleotides 1 to 1012 of the rpoB gene. This region was amplified and sequenced with primers rpoB_up (5′–ATGGCTTACTCATACACTGAG-3′) and rpo_B1 (5′-CTCGATGCG CACGACCTG-3′). The protocol was the same as described above, except that the annealing temperature used in the PCR reactions was 54 C instead of 60 C. We idenfied a single mutation in this region in all of the clones that did not contain a mutation in the highly conserved domain of rpoB. As a further control, we sequenced the entire rpoB gene in six randomly chosen clones. We failed to detect any second site mutations in rpoB using this approach.Clones that lacked a mutation in the highly conserved domain of r, of each beneficial mutation and wildtype PAO1 at 4 different concentrations of rifampicin . Pre-assay overnight cultures of each mutant were prepared by growth in M9KB. These cultures were then diluted 100 fold into fresh culture medium and we measured the growth rate of each culture using an automated microplate reader by taking hourly measurements of optical density at 600 nm (OD600) over a period of approximately 12 hours. All incubations were carried out at 37 C. Assays at low concentrations of rifampicin  were carried out with 12 fold replication and assays at high concentrations of rifampicin (64 mg/L) were carried out with 18 fold replication. A further assay was carried out using the same method to measure fitness in the presence of sorangicin (20 ug/mL). OD600 is proportional to the log of cell density, and the slope of OD600 against time (mOD/min) in exponential growth phase therefore provides an estimate of r, the growth rate of the bacterial clone, such that i,jr is the growth rate to the ith genotype at the jth concentration of rifampicin.To assay fitness, we estimated the growth rate, i,jw = i,jr−1,jr where i,jw is the fitness of the ith beneficial mutation in the jth environment, i,jr is the growth rate of the ith beneficial mutation in the jth environment, and 1,jr is the growth rate of the least fit beneficial mutation in the the jth environment .To test the hypothesis that the fitness effects of beneficial mutations are exponentially distributed, we used a likelihood ratio test developed by Beisel and colleagues 0:κ = 0, HA: κ≠0). P values were calculated by performing 10000 parametric bootstrap replicates using a software package (EVDA) developed for R (software available at http://www.webpages.uidaho.edu/˜joyce/Lab20Page/Computer-Programs.html). This test is potentially sensitive to measurement error, but the accuracy of our fitness measurements was high enough (Average CV = 19.8%) that measurement error should not inflate the probability of making a type I error The likelihood ratio test developed by Beisel and colleagues calculates −2logΛ, negative twice the difference in log-likelihood between two statistical models, one model in which the shape parameter of the GPD is set to 0 and the other in which the scale parameter of the GPD is free to vary : 0, 3.9, 7.8, 15.6, 31.3, 62.5, 125, 250, 500, and 1000. Assay cultures were incubated at 37 C and we measured the optical density of cultures after exactly 24 hours of incubation (+/−10 minutes) at 600 nM using an automated microplate reader. We assayed the resistance of 12 cultures of each of the rpoB mutants that we identified and 12 replicates of PAO1. Resistance was calculated as IC50, the concentration of rifampicin necessary to cause a 50% reduction in optical density, using the following regression model: y is optical density, measured in absorbance units, x is the concentration of rifampicin, measured in ug/mL, and H is parameter that estimates the rate of decay in optical density with increasing rifampicin concentration.To measure the resistance conferred by


Recent outbreaks of novel infectious diseases  have highlighted the threat of cross-species pathogen transmission. When first introduced to a population, a pathogen is often poorly adapted to its new host and must evolve in order to escape extinction. Theoretical arguments and empirical studies have suggested various factors to explain why some pathogens emerge and others do not, including host contact structure, pathogen adaptive pathways and mutation rates. Using a multi-type branching process, we model the spread of an introduced pathogen evolving through several strains. Extending previous models, we use a network-based approach to separate host contact patterns from pathogen transmissibility. We also allow for arbitrary adaptive pathways. These generalizations lead to novel predictions regarding the impact of hypothesized risk factors. Pathogen fitness depends on the host population in which it circulates, and the ‘riskiest’ contact distribution and adaptive pathway depend on initial transmissibility. Emergence probability is sensitive to mutation probabilities and number of adaptive steps required, with the possibility of large adaptive steps (e.g. simultaneous point mutations or recombination) having a dramatic effect. In most situations, increasing overall mutation probability increases the risk of emergence; however, notable exceptions arise when deleterious mutations are available. Such ‘sA number of risk factors have been proposed to explain why some pathogens emerge and others do not. These include the breadth of the pathogen's host range , susceptThe association between transmissibility and likelihood of emergence is, perhaps, not surprising. Any factor that increases the expected number of transmitted infections, such as higher initial transmissibility between members of the new host species, means that the pathogen can circulate for longer after the cross-species jump, and thus has greater potential for ultimate evolutionary adaptation . A high mutation rate might lead to a greater likelihood of the appropriate adaptive mutations occurring, implying greater evolutionary potential . Indeed,In this paper, we investigate the above factors through mathematical modelling. We ask two main questions: (i) what types of host contact structure lead to the greatest risk of evolutionary emergence? and (ii) how do patterns of mutation affect the risk of evolutionary emergence? We address these questions using a branching process model that tracks the number of infected hosts as a newly introduced pathogen spreads and evolves. There have been two main approaches to using such models in the epidemiological literature, a network-based approach and a phenomenological approach. The former explicitly considers the patterns of contact among individuals in the population and the probability of transmission through any given contact e.g. , but has2.In order to develop a unified modelling framework that encompasses both the network-based and phenomenological approaches, it is useful first to focus on single-strain models that ignore evolution.2.1.We model in discrete time, using a Galton–Watson process. To define the branching process, we require a distribution for the number of ‘offspring’ produced by each individual. In the present context, infected hosts are the individuals of interest, and an ‘offspring’ is viewed as a contact to whom the infection is transmitted. The network-based approach and the phenomenological approach arrive at this offspring distribution in different ways. The network-based approach builds up the offspring distribution from underlying assumptions about the processes of host–host contact and pathogen transmission. The phenomenological approach, on the other hand, simply specifies this offspring distribution directly, without explicit consideration of the underlying contact and transmission processes through which it might arise.n infective  are considered generation n + 1 infectives, and ‘later-generation’ infectives (i.e. generation ≥ 1) receive their infections from other individuals within the population.The network-based approach represents the host population by a graph: each vertex or node corresponds to an individual, and an edge between two nodes signifies that the two individuals are acquaintances, i.e. can contact and potentially transmit disease to one another. We assume that this network is static, and restrict attention to random graphs in the limit of infinite population size. In this situation, the network becomes a tree graph, with the number of ‘branches’ from each node drawn independently from the arbitrary degree, or contact, distribution p̃d} given by the PGFd〉 = g′(1) denotes the expected value of degree d.Following the derivation presented by independently with probability T, called the transmissibility, taken to be a constant. That is, if d is the number of still-susceptible contacts, then the number of infections transmitted is distributed as Binomial. For the initial infective, who has been infected by a source outside the population, all contacts (given by the degree) remain susceptible. Thus, the PGF γ(s) for the number of infections transmitted by a randomly chosen initial infective is (Γ(s) for the number of infections transmitted by a later-generation infective isR0, can then be defined as the mean number of infections transmitted by one typical (later-generation) infective:G′(1)) and pathogen transmission (represented by T). R0 is an appropriate measure of the fitness of a single strain of pathogen.An infective is assumed to transmit infection to each still-susceptible contact ctive is A laterX, produced by an infected individual. The approach typically uses a Poisson distribution of infections, implicitly assuming homogeneous mixing in the population, but sometimes incorporates individual heterogeneity by allowing the mean of the distribution itself to be a random variable. For example, ν, from a Gamma distribution with mean R0 and dispersion parameter β; then X has a negative binomial distribution with mean R0 and dispersion β. As special cases, β = 1 corresponds to ν ∼ Exp(R0) and X ∼ Geometric(R0), while β → ∞ corresponds to ν = R0  and X ∼ Poisson(R0). In any case, the key difference from the network-based approach is that Γ(s) is effectively specified directly. As before, one then defines the basic reproductive number as R0 = Γ′(1). Notice, however, that unlike the network-based approach, no distinction is made here between generation 0 infectives (i.e. those who receive the infection from an outside source) and later-generation infectives.  If we consider only later-generation infectives, the phenomenological approach can be obtained via the network-based approach, provided that the contact distribution in the latter is chosen to obtain the same end result for the offspring distribution (appendix A.3).The phenomenological approach (e.g. λ and dispersion β (appendix A.1), i.e. G(z) = (1 + (λ/β)(1 − z))β− (λT and dispersion β: Γ(s) = G(1 − T + Ts) = (1 + (λT/β) (1 − s))β−. Since R0 = λT, this gives exact correspondence with the phenomenological approach of For example, if the contact distribution is such that the number of still-susceptible contacts has a negative binomial distribution with mean extinction probabilityq of the process—that is, the probability that the disease outbreak eventually ends, infecting only a finite number of people—is the smallest non-negative solution of the equation Γ(s) = s. The epidemic is guaranteed to go extinct if R0 ≤ 1, but persists with non-zero probability if R0 > 1 (g(1 − T + Tq) ≡ γ(q)  ≡ γ(q) .2.2.et al.A phenomenological approach has also been used to explore how pathogen evolution affects disease emergence by accounting for multiple pathogen strains et al.. In the i  denotes which one of the m possible pathogen strains is infecting that host. We allow for arbitrary contact distribution and assume that the contact network is determined by the host, independently of the pathogen strain. Thus, as before, degree is described by the PGF g(z) and the excess degree by G(z) for every individual, regardless of type. We retain the assumption that transmissions occur independently to each contact, and further assume that transmissibility T is strain-specific, denoted Ti for strain i. Thus, the probability of making an infectious contact depends only on the current infective's strain type. However, a different strain may be transmitted as a result of pathogen mutation. Specifically, given that a type i infective makes an infectious contact, strain j is transmitted with probability μij  allowed in the evolution of the pathogen and their probabilities.We account for pathogen evolution using a multi-type branching process. An individual's type within individual hosts, but we account for this in a phenomenological way. The most accurate interpretation of our approach would be that a host currently infected with strain i transmits only strain i, but that each infective produced in the next generation has probability μij of converting to strain j over the course of infection, before any further transmission occurs. In this approach, μij is an ‘effective conversion rate’ from strain i to strain j within one host, summarizing the results of within-host dynamics (§4.3). This is a common type of simplification involving a separation of the time scales on which within- and between-host processes occur. We use the word ‘mutation’ loosely to mean any processes resulting in a change in the strain identified as a host's type. These processes actually occur i infective with d susceptible contacts now has distribution Multinomial, where Ui· is the ith row of U and the probability vector is given in the order: no transmission, transmit strain 1, … , transmit strain m. The corresponding PGF for the number of transmitted infections of each type, , given d susceptible contacts, isi isReturning to our multi-type process, the number of transmissions of each type made by a type i, denoted qi, is obtained as the smallest non-negative root of the equation qi = Γi, solved simultaneously for all i (appendix D). More compactly, we can write this as a vector fixed-point equation: i, chosen uniformly at random, the overall probability of extinction is then given by g(1 − Ti + Ti ∑ j=1mμijqj) ≡ γii is introduced to a randomly chosen individual, is to allow the initial infective to be type j with probability μij. We generally assume that strain 1 is, by definition, initially introduced to the host population.The probability of extinction starting from one later-generation infective of type i’, denoted Ri0, and defined as the expected total number of infectious contacts made by a typical (later-generation) type i infective. Since this number is distributed as Binomial, we havei were the only strain present, with no possibility of mutation, Ri0, would be the value of its basic reproductive number in the single-strain model.In what follows, it is sometimes useful to refer to the ‘basic reproductive number of strain Ri0, > 1. Probability of emergence is thus the complement of probability of extinction. We present numerical results for the probability of emergence beginning from one later-generation infective of type 1, 1 − q1, but these results may be extended to account for generation 0 and/or type conversion as described above. Note that the branching process may be either indecomposable or decomposable, depending on the mutational scheme. Thus, if the epidemic persists, the complement of strains that will be present must be treated on a case-by-case basis (appendix D).We define emergence as the situation in which the pathogen escapes extinction, which necessarily requires evolution to a strain having a priori restrictions on the choices of contact distribution, transmissibilities and mutation probabilities. However, in the numerical results to follow, we limit ourselves to specific examples.Our derivation so far is quite general, imposing no Contact distributions. We consider the following contact distributions, each with mean λ .Deterministic. Every individual has exactly λ contacts.— Mixed deterministic. There are n types of individuals, where the kth type occurs in proportion pk and has exactly λk contacts.— Poisson.— Mixed Poisson. There are n types of individuals, where the kth type occurs in proportion pk and has a Poisson(λk) distribution of contacts.— Negative binomial, with dispersion β.— Mutational schemes. We consider two broad types of mutational schemes: ‘linear’ and ‘hub-and-spoke’ . In all cases, we assume that strain m is well adapted to the host; that is, Tm is chosen such that Rm0, > 1. On the other hand, any other strain i is poorly adapted to the host  unless otherwise specified.m − 1 are intermediates between strains 1 and m. We consider the following possibilities.One-step irreversible. The pathogen must acquire m − 1 point mutations, one at a time and in a fixed order. Thus, mutation occurs only from strain i to i + 1. This scheme is presented by — Multi-step irreversible. Again the pathogen must acquire m − 1 point mutations in a fixed order, but now possibly simultaneously, implying that mutations can occur from strain i to any strain j > i, though with diminishing probabilities. Essentially, higher order terms are being included where they were neglected in the previous scheme. This scheme is presented by — Interchangeable and irreversible. This scheme, also presented by i in our model representing any ‘real’ strain having i − 1 of the required mutations. Our model thus implicitly assumes that any collection of i − 1 mutations yields the same transmissibility, Ti.— Point mutation and recombination. One-step point mutation occurs as in scheme 1, but we simplistically model an additional mechanism of more extensive genetic change .— One-step irreversible mutations, coupled with a Poisson(λ) distribution of contacts, corresponds to the model of Ri0, = λTi.Linear mutational schemes represent single directions through strain space, where strains 2, … , m − 1 no longer represent intermediates on the path to strain m. For simplicity, in our results we consider only paths of length 1 , with equal probability of proceeding along any path. However, this set-up could obviously be extended to incorporate pathways of various lengths and mutation probabilities. We consider two possible scenarios.One-step irreversible.— One-step reversible.— Finally, we must specify the transmissibility of each strain, m − 1 have identical transmissibility, and hence the same value of Ri0, < 1. Other biologically interesting possibilities include a fitness valley, where intermediate strains have lower fitness than strain 1, or an additive model, where fitness increases linearly with strain number . Though still numbered sequentially for convenience, strains 2, … , 3.3.1.R0, is widely used as a predictor of when a disease has epidemic potential, as well as a descriptor of disease spread T. Using the relationship G(z) = g′(z)/g′(1) gives R0 = (g″(1)/g′(1))T. Then substituting g″(1) = σ2 − g′(1) + (g′(1))2, where σ2 is the variance of the contact distribution, yieldsc = σ/g′(1) is the coefficient of variation of the contact distribution. For a given transmissibility and mean number of contacts, the contact distribution that maximizes variance thus maximizes R0. The above equation is essentially the same as the expression for R0 given by c, and is similar to the expression for R0 used in deterministic network models  and mutation probabilities given by matrix U = [μij], we have aij = G′(1)Tiμij = (g′(1)(1 + c2) − 1)Tiμij. We can thus express the mean matrix asM—and hence its dominant eigenvalue—on the mean and variance of the contact distribution, the transmissibility of each strain and the mutation scheme.In our multiple strain model, with contact distribution given by PGF 3.2.R0, and on the probability of emergence starting from one later-generation infective, 1 − q. Figure 2a shows how R0 increases with transmissibility T: in all cases, this increase is linear, but at a different rate for each contact distribution depending on the variance in contacts. Figure 2b plots the probability of emergence versus T for each contact distribution. This probability becomes non-zero at the point where R0 passes the critical value of one, which occurs at a different value of T for each distribution (as illustrated in figure 2a). Thus, at low values of T, the disease can persist in some host contact structures but not others, despite the fact that they all have the same mean number of contacts. As T increases, however, the probability of emergence increases at different rates for different contact structures, such that the ordering is not preserved. This observation is highlighted in the inset, showing emergence probability over an extended range of T. In contrast, the ordering is consistent in figure 2c, which plots the probability of emergence versus R0, comparable to Brauer's presentation Tm. Hence, criticality is independent of the precise details of a uni-directional mutation scheme. By contrast, a scheme that allows both forward and reverse mutation can affect criticality. R0  changes with the probability of reverse mutation (ν) in the linear one-step reversible scheme; similar considerations apply to the hub-and-spoke one-step reversible scheme as well. A high rate of reverse mutation from a well-adapted to a poorly adapted strain can tip the branching process from supercritical to subcritical. Population R0 increases almost linearly with Tm, but is relatively insensitive to number of strains (m), earlier strain transmissibility (T1) and forward mutation probability (μ). This is because population R0 is very close to amm = G′(1)Tm(1 − ν), but slightly influenced by contributions from small populations of other (poorly adapted) strains that persist asymptotically if the process escapes extinction.When there are multiple strains of pathogen, the pathways of mutation among them may also affect the threshold parameter. In the case of irreversible mutation, we can number strains such that 3.4.q1, we compute the extinction probability si < 1, −12 or 10−16 in the figures we present.We can compare not only the qualitative result of criticality, but also the quantitative probability of emergence across various supercritical branching processes. To obtain this probability, 1 − Tm is far from the critical threshold, and reverse mutation, even at a high rate equal to that of forward mutation, makes a negligible impact on the probability of emergence. This agrees with Sagitov and Serra's result that reverse mutation is negligible in a similar model with an arbitrary offspring distribution .ribution , and wit  varies. Although we have typically used the jackpot model, one might also consider other choices of transmissibilities. As one would expect, the higher the intermediate strain fitness, the higher the probability of emergence as observed by 3.5.et al.For any given mutation scheme, we can gain greater insight into the effect of mutation probabilities and number of strains by estimating the probability of emergence analytically. As other authors et al. have conet al. derived the following approximation for the probability of evolution to strain m (starting from strain 1) in the case of a Poisson-distributed number of infectious transmissions and one-step irreversible mutation  having Rm0, > 1; all other types i ≠ m have Ri0, < 1.— There is a single ‘escape’ type .— All mutation probabilities Under these conditions, the above approximation has error of O(u2). However, the neglected error term increases as Ri0, → 1, implying that the approximation will break down if strain fitness approaches the critical threshold. Ri0,, including the near-critical case, though only for a specific offspring distribution.More generally, y et al. , 2004. Sq1) scales as O(μm−1) + O(ρ), where μ is the probability of one-step forward ‘point mutation’ and ρ is the probability of jump-to-m ‘recombination’. m = 3 strains, illustrating the relative contributions of the different adaptive pathways. When ρ ≫ μ2, the probability of emergence is almost constant in μ and scales approximately linearly in ρ. When ρ ≪ μ2, the probability of emergence is almost constant in ρ and scales proportionally to μ2.As an example, consider our ‘mutation and recombination’ scheme. Either by applying an Antia-like argument to this specific scenario or by substituting mutation probabilities into Serra and Haccou's result, we can show that the probability of emergence starting from a type 1 individual  or where the assumptions underlying these approximations are broken or pushed to their limits (e.g. relatively large mutation probabilities), as we will encounter in the next subsection.3.6.The same biological mechanisms  contribute to all mutation rates, not only those for beneficial mutations. Given this constraint, it is not immediately clear whether pathogens with the highest mutation rates will have the largest or smallest probability of emergence. We explore this question for two mutational schemes in which deleterious steps are possible: one-step reversible mutation and hub-and-spoke irreversible mutation.μ, and from 2 to 1 with probability ν, which we scale proportionally to μ. Results demonstrate a non-monotonic relationship between mutation rate and probability of emergence , the poorly adapted strain 1 and the well-adapted strain 2. Mutation from 1 to 2 occurs with probability mergence . The proT1 for various mutation probabilities. When T1 is well below the critical threshold , emergence probability increases linearly with mutation probability, as we would predict from analytical approximations (§3.5). Although this figure plots results for m = 10 strains, a virtually identical increase in emergence probability versus μ at fixed small T1 was observed for m = 2, 3, 5 as well (results not shown). This indicates that regardless of how high the risk of deleterious steps, a pathogen that is guaranteed to go extinct unless it acquires a beneficial mutation (evolves to strain m) is always better off having a large mutation rate. Near the critical threshold, emergence probability shows a sharp jump; once strain 1 is sufficiently fit to escape extinction without further evolution, mutation probability makes only a small difference in results. Shortly after this point, a very high mutation probability becomes detrimental to the pathogen, even while strain 1 is not as well adapted as strain m. The risk of mutating to a poorly adapted strain now outweighs the potential benefit of mutating to a better adapted strain. Adding reverse mutation at a probability equal to that of forward mutation makes negligible difference to quantitative results (results not shown), as also observed with linear mutation schemes where Rm0, is not too close to one. We might suspect that increasing reverse mutation probability independently of forward mutation probability could provide a benefit to the pathogen if the transmissibility of deleterious strains is non-zero, such that mutation could ‘rescue’ the pathogen by a return to strain 1 (the hub). However, this does not appear to be the case within tested parameter ranges (results not shown): increasing reverse mutation probability decreases probability of emergence, though only marginally when Rm0, is well above one.With an irreversible hub-and-spoke mutation scheme, we assume that the initial strain is equally likely to mutate to any other, but only one mutational pathway is beneficial. As an extreme case, we set transmissibility of all deleterious strains to be zero. 4.We have linked phenomenological and contact network-based approaches to modelling disease spread as a branching process, clarifying how they may be viewed within a common framework. By explicitly considering the number of contacts along with a model of how transmission occurs to these contacts, a network-based model offers more detailed insights into factors contributing to pathogen fitness and probability of emergence. Such a model lends itself to comparison of the relative impact of various public health interventions, which may act by reducing either number of contacts or probability of transmission per contact .R0 = (g′(1)(1 + c2) − 1)T—where g′(1) is the mean of the contact distribution, c is its coefficient of variation and T is transmissibility—highlights the contributions of both ecological factors (contact structure) and epidemiological factors (transmissibility) to pathogen fitness. That is, a pathogen's fitness depends on the particular host population in which it circulates, an observation that has been made more generally for pathogen fitness measures (M = (g′(1)(1 + c2) − 1)diagU is the mutation matrix. These expressions make it clear that for fixed transmissibilities, mean number of contacts and mutation scheme, maximizing pathogen fitness is equivalent to maximizing the variance of the host population's contact distribution; this agrees with the general observation that increasing heterogeneity among hosts increases R0 (Expressing the basic reproductive number as measures . Similareases R0 . Furthereases R0 .R0). This comparison more easily lends itself to questions about the sort of host population contact structure that is associated with greatest risk of disease emergence. In contrast, holding R0 fixed in comparisons requires that we simultaneously alter pathogen transmissibility when altering the contact distribution, thereby confounding these two factors.We have argued that the comparison of emergence probability across contact distributions should be made at given pathogen transmissibility, rather than at given pathogen fitness  Through comparison of contact distributions at given transmissibility, we found that heterogeneity can in fact increase the risk of emergence over a significant range.Although previous authors have often found that increased population heterogeneity increases the probability of extinction  by which the pathogen may possibly evolve . ProvideWe also considered more realistic situations where deleterious as well as beneficial evolutionary steps are available to the pathogen, either by reverse mutation or by including multiple pathways in strain space. Here, we constrained all types of mutations to occur with proportional probabilities, based on some overall mutation rate of the pathogen. We found that, when the pathogen was initially poorly adapted, increasing overall mutation rate was usually beneficial to an emerging pathogen (i.e. riskier to the host). On the other hand, as the pathogen's initial fitness increases beyond the critical threshold where survival without adaptation is possible, the risk of deleterious mutations comes to outweigh the benefit of potential mutation to a more fit strain, and a high mutation rate becomes a liability . Thus, o4.3.We have modelled the spread of an emerging pathogen following its introduction to a new host species. This model does not incorporate the dynamics of the introduction itself (or possibly multiple introductions), such as interspecific interactions  and the strain conversion process have been treated only phenomenologically in our present between-host model. We (and other authors) have implicitly assumed that any conversion of strains within an individual is instantaneous, with no possibility for co-infection. We have also ignored variation in the precise number of pathogen copies in the body. Realistically, once a mutation arises in a host there will be some dynamical process leading to fixation or loss over time, with coexistence of strains at least temporarily. Both R0 and conversion rates) supports a move to modelling at the within-host level.Furthermore, both the level of transmissibility (An important avenue for future work will thus be to develop an explicit model of within-host processes, and then to link the within- and between-host scales. In recent years, a number of authors have developed such ‘nested models’ (reviewed in


Evolution involves both deterministic and random processes, both of which are known to contribute to directional evolutionary change. A number of studies have shown that when fitness is treated as a random variable, meaning that each individual has a distribution of possible fitness values, then both the mean and variance of individual fitness distributions contribute to directional evolution. Unfortunately the most general mathematical description of evolution that we have, the Price equation, is derived under the assumption that both fitness and offspring phenotype are fixed values that are known exactly. The Price equation is thus poorly equipped to study an important class of evolutionary processes.I present a general equation for directional evolutionary change that incorporates both deterministic and stochastic processes and applies to any evolving system. This is essentially a stochastic version of the Price equation, but it is derived independently and contains terms with no analog in Price's formulation. This equation shows that the effects of selection are actually amplified by random variation in fitness. It also generalizes the known tendency of populations to be pulled towards phenotypes with minimum variance in fitness, and shows that this is matched by a tendency to be pulled towards phenotypes with maximum positive asymmetry in fitness. This equation also contains a term, having no analog in the Price equation, that captures cases in which the fitness of parents has a direct effect on the phenotype of their offspring.Directional evolution is influenced by the entire distribution of individual fitness, not just the mean and variance. Though all moments of individuals' fitness distributions contribute to evolutionary change, the ways that they do so follow some general rules. These rules are invisible to the Price equation because it describes evolution retrospectively. An equally general prospective evolution equation compliments the Price equation and shows that the influence of stochastic processes on directional evolution is more diverse than has generally been recognized. Evolution involves both deterministic processes, such as selection, and random processes such as drift. When deterministic and stochastic processes are combined in the same model it is common to use the "diffusion approximation" – essentially assuming that populations are large (so that evolution can be approximated as a continuous process), that population size is relatively stable, and that selection is weak -4. The di.e. other moments) should do so as well. However, most of the models that have been studied have used methods  but also by the variances in possible fitness values associated with each strategy -10. If tcalculus ) which mThe most general (in the sense of making the fewest simplifying assumptions) mathematical description of evolution that we currently have, the Price equation , does noDespite this apparent limitation, the Price equation has been used extensively to study social evolution -16, the Below, I present a general equation for directional evolutionary change that treats fitness and offspring phenotype as random variables, rather than numbers, but imposes no restrictions on the distributions associated with these random variables. This is essentially a stochastic version of the Price equation, though it is derived independently and contains a term not found in Price's formulation. This theory accommodates all processes that influence directional evolution, both deterministic and stochastic. Using this result, I show that deterministic and stochastic processes interact in complex ways. One result is that stochastic variation in fitness amplifies the effects of selection in small or fluctuating populations. Furthermore, the role of fitness variation within an individual is more complex than has generally been recognized. The well known tendency for populations to be pulled towards phenotypes with minimum variance in fitness turns out to be one instance of a more general rule that, all else held equal, populations are pulled towards phenotypes with minimum symmetrical variation in fitness, as measured by all of the even moments of an individual's fitness distribution. This process can actually cause the variance in fitness to increase (so long as higher even moments decrease). There is also a tendency for populations to be pulled towards phenotypes with maximum positive asymmetry in fitness (as measured by the odd moments). Finally, this equation contains a term, capturing the direct effects of reproduction on offspring phenotype, that has no analog in the Price equation.w) measures the number of descendants that the individual has at some future time, potentially including the individual itself  is the nth central moment of x (this is just the delta method). We can now write For a random variable, w and w and The Taylor series expansion of Equation 10 does converge st mixed central moment of wk and 2 = 0, so If the actual number of descendants of different individuals are independent – meaning that the number of descendants of individual ϕ, Substituting Equation 14 into cov. In such a case, we can write individual fitness as wi = si, where i, and si is the deviation from this expected fitness due to pure demographic stochasticity. In this case,Consider a case in which individuals with certain phenotypes are consistently influenced the same way by environmental variation across generations  = 2, var(w1) = 1.5, μ3(w0) = 2, μ3(w1) = 1.5, μ4(w0) = 6, μ4(w1) = 4.5.Equation 20 is derived using the fact that f0 and f1, the general rule is:Next, we need to specify the current frequencies of each phenotype and solve for the covariance terms. For the case of only two phenotypes, assigned values 0 and 1 and having frequencies n0 = n1 = N/2. For this case, we have ϕ, ϕ, var(w)) = -0.125, cov) = -0.125, cov) = -0.375, The dashed curves in the figure were derived by using the moments of the fitness distributions for each phenotype to approximate 1/H. We can thus iterate this process forward in time. If there are more than two phenotypes, then iteration is not possible unless we make further assumptions , that allow us to specify the entire distribution given only the mean.4(It is sometimes necessary to use moments higher than μN individuals, evenly divided between the two phenotypic values (so initially The monte-carlo simulations used asexual individuals with non overlapping generations. The value of ΔSHR did the work and wrote the paper.