Beneficial fitness effects are not exponential for two viruses

The distribution of fitness effects during adaptive walks using a simple genetic network

The tempo and mode of adaptation depends on the availability of beneficial alleles. Genetic interactions arising from gene networks can restrict this availability. However, the extent to which networks affect adaptation remains largely unknown. Current models of evolution consider additive genotype-phenotype relationships while often ignoring the contribution of gene interactions to phenotypic variance. In this study, we model a quantitative trait as the product of a simple gene regulatory network, the negative autoregulation motif. Using forward-time genetic simulations, we measure adaptive walks towards a phenotypic optimum in both additive and network models. A key expectation from adaptive walk theory is that the distribution of fitness effects of new beneficial mutations is exponential. We found that both models instead harbored distributions with fewer large-effect beneficial alleles than expected. The network model also had a complex and bimodal distribution of fitness effects among all mutations, with a considerable density at deleterious selection coefficients. This behavior is reminiscent of the cost of complexity, where correlations among traits constrain adaptation. Our results suggest that the interactions emerging from genetic networks can generate complex and multimodal distributions of fitness effects.

Changing fitness effects of mutations through long-term bacterial evolution

The distribution of fitness effects of new mutations shapes evolution, but it is challenging to observe how it changes as organisms adapt. Using Escherichia coli lineages spanning 50,000 generations of evolution, we quantify the fitness effects of insertion mutations in every gene. Macroscopically, the fraction of deleterious mutations changed little over time whereas the beneficial tail declined sharply, approaching an exponential distribution. Microscopically, changes in individual gene essentiality and deleterious effects often occurred in parallel; altered essentiality is only partly explained by structural variation. The identity and effect sizes of beneficial mutations changed rapidly over time, but many targets of selection remained predictable because of the importance of loss-of-function mutations. Taken together, these results reveal the dynamic-but statistically predictable-nature of mutational fitness effects.

The effect of weak clonal interference on average fitness trajectories in the presence of macroscopic epistasis

Adaptation dynamics on fitness landscapes is often studied theoretically in the strong-selection, weak-mutation regime. However, in a large population, multiple beneficial mutants can emerge before any of them fixes in the population. Competition between mutants is known as clonal interference, and while it is known to slow down the rate of adaptation (when compared to the strong-selection, weak-mutation model with the same parameters), how it affects the shape of long-term fitness trajectories in the presence of epistasis is an open question. Here, by considering how changes in fixation probabilities arising from weak clonal interference affect the dynamics of adaptation on fitness-parameterized landscapes, we find that the change in the shape of fitness trajectory arises only through changes in the supply of beneficial mutations (or equivalently, the beneficial mutation rate). Furthermore, a depletion of beneficial mutations as a population climbs up the fitness landscape can speed up the rescaled fitness trajectory (where adaptation speed is measured relative to its value at the start of the experiment), while an enhancement of the beneficial mutation rate does the opposite of slowing it down. Our findings suggest that by carrying out evolution experiments in both regimes (with and without clonal interference), one could potentially distinguish the different sources of macroscopic epistasis (fitness effect of mutations vs change in fraction of beneficial mutations).

Selection bias in mutation accumulation

Mutation accumulation (MA) experiments, in which de novo mutations are sampled and subsequently characterized, are an essential tool in understanding the processes underlying evolution. In microbial populations, MA protocols typically involve a period of population growth between severe bottlenecks, such that a single individual can form a visible colony. While it has long been appreciated that the action of positive selection during this growth phase cannot be eliminated, it is typically assumed to be negligible. Here, we quantify the effect of both positive and negative selection in MA studies, demonstrating that selective effects can substantially bias the distribution of fitness effects (DFE) and mutation rates estimated from typical MA protocols in microbes. We then present a simple correction for this bias which applies to both beneficial and deleterious mutations, and can be used to correct the observed DFE in multiple environments. We use simulated MA experiments to illustrate the extent to which the MA-inferred DFE differs from the underlying true DFE, and demonstrate that the proposed correction accurately reconstructs the true DFE over a wide range of scenarios; we also provide an example of these corrections applied to experimental data. These results highlight that positive selection during microbial MA experiments is in fact not negligible, but can be corrected to gain a more accurate understanding of fundamental evolutionary parameters. This article is protected by copyright. All rights reserved.

A branching process model of evolutionary rescue

Evolutionary rescue is the process whereby a declining population may start growing again, thus avoiding extinction, via an increase in the frequency of fitter genotypes. These genotypes may either already be present in the population in small numbers, or arise by mutation as the population declines. We present a simple two-type discrete-time branching process model and use it to obtain results such as the probability of rescue, the shape of the population growth curve of a rescued population, and the time until the first rescuing mutation occurs. Comparisons are made to existing results in the literature in cases where both the mutation rate and the selective advantage of the beneficial mutations are small.

Extreme value theory as a framework for understanding mutation frequency distribution in cancer genomes

Currently, the population dynamics of preclonal cancer cells before clonal expansion of tumors has not been sufficiently addressed thus far. By focusing on preclonal cancer cell population as a Darwinian evolutionary system, we formulated and analyzed the observed mutation frequency among tumors (MFaT) as a proxy for the hypothesized sequence read frequency and beneficial fitness effect of a cancer driver mutation. Analogous to intestinal crypts, we assumed that sample donor patients are separate culture tanks where proliferating cells follow certain population dynamics described by extreme value theory (EVT). To validate this, we analyzed three large-scale cancer genome datasets, each harboring > 10000 tumor samples and in total involving > 177898 observed mutation sites. We clarified the necessary premises for the application of EVT in the strong selection and weak mutation (SSWM) regime in relation to cancer genome sequences at scale. We also confirmed that the stochastic distribution of MFaT is likely of the Fréchet type, which challenges the well-known Gumbel hypothesis of beneficial fitness effects. Based on statistical data analysis, we demonstrated the potential of EVT as a population genetics framework to understand and explain the stochastic behavior of driver-mutation frequency in cancer genomes as well as its applicability in real cancer genome sequence data.

Effects of synonymous mutations beyond codon bias: The evidence for adaptive synonymous substitutions from microbial evolution experiments

Synonymous mutations are often assumed to be neutral with respect to fitness because they do not alter the encoded amino acid and so cannot be 'seen' by natural selection. Yet a growing body of evidence suggests that synonymous mutations can have fitness effects that drive adaptive evolution through their impacts on gene expression and protein folding. Here, we review what microbial experiments have taught us about the contribution of synonymous mutations to adaptation. A survey of site-directed mutagenesis experiments reveals the distributions of fitness effects for nonsynonymous and synonymous mutations are more similar, especially for beneficial mutations, than expected if all synonymous mutations were neutral, suggesting they should drive adaptive evolution more often than is typically observed. A review of experimental evolution studies where synonymous mutations have contributed to adaptation shows they can impact fitness through a range of mechanisms including the creation of illicit RNA polymerase binding sites impacting transcription and changes to mRNA folding stability that modulate translation. We suggest that clonal interference in evolving microbial populations may be the reason synonymous mutations play a smaller role in adaptive evolution than expected based on their observed fitness effects. We finish by discussing the impacts of falsely assuming synonymous mutations are neutral and discuss directions for future work exploring the role of synonymous mutations in adaptive evolution.

The distribution of fitness effects among synonymous mutations in a gene under directional selection

The fitness effects of synonymous mutations, nucleotide changes that do not alter the encoded amino acid, have often been assumed to be neutral, but a growing body of evidence suggests otherwise. We used site-directed mutagenesis coupled with direct measures of competitive fitness to estimate the distribution of fitness effects among synonymous mutations for a gene under directional selection and capable of adapting via synonymous nucleotide changes. Synonymous mutations had highly variable fitness effects, both deleterious and beneficial, resembling those of nonsynonymous mutations in the same gene. This variation in fitness was underlain by changes in transcription linked to the creation of internal promoter sites. A positive correlation between fitness and the presence of synonymous substitutions across a phylogeny of related Pseudomonads suggests these mutations may be common in nature. Taken together, our results provide the most compelling evidence to date that synonymous mutations with non-neutral fitness effects may in fact be commonplace.

Simulations reveal challenges to artificial community selection and possible strategies for success

Multispecies microbial communities often display "community functions" arising from interactions of member species. Interactions are often difficult to decipher, making it challenging to design communities with desired functions. Alternatively, similar to artificial selection for individuals in agriculture and industry, one could repeatedly choose communities with the highest community functions to reproduce by randomly partitioning each into multiple "Newborn" communities for the next cycle. However, previous efforts in selecting complex communities have generated mixed outcomes that are difficult to interpret. To understand how to effectively enact community selection, we simulated community selection to improve a community function that requires 2 species and imposes a fitness cost on one or both species. Our simulations predict that improvement could be easily stalled unless various aspects of selection are carefully considered. These aspects include promoting species coexistence, suppressing noncontributors, choosing additional communities besides the highest functioning ones to reproduce, and reducing stochastic fluctuations in the biomass of each member species in Newborn communities. These considerations can be addressed experimentally. When executed effectively, community selection is predicted to improve costly community function, and may even force species to evolve slow growth to achieve species coexistence. Our conclusions hold under various alternative model assumptions and are therefore applicable to a variety of communities.

The fitness landscape of the codon space across environments

Fitness landscapes map the relationship between genotypes and fitness. However, most fitness landscape studies ignore the genetic architecture imposed by the codon table and thereby neglect the potential role of synonymous mutations. To quantify the fitness effects of synonymous mutations and their potential impact on adaptation on a fitness landscape, we use a new software based on Bayesian Monte Carlo Markov Chain methods and re-estimate selection coefficients of all possible codon mutations across 9 amino acid positions in Saccharomyces cerevisiae Hsp90 across 6 environments. We quantify the distribution of fitness effects of synonymous mutations and show that it is dominated by many mutations of small or no effect and few mutations of larger effect. We then compare the shape of the codon fitness landscape across amino acid positions and environments, and quantify how the consideration of synonymous fitness effects changes the evolutionary dynamics on these fitness landscapes. Together these results highlight a possible role of synonymous mutations in adaptation and indicate the potential mis-inference when they are neglected in fitness landscape studies.

Mutation-Driven Parallel Evolution during Viral Adaptation

Convergent evolution has been demonstrated across all levels of biological organization, from parallel nucleotide substitutions to convergent evolution of complex phenotypes, but whether instances of convergence are the result of selection repeatedly finding the same optimal solution to a recurring problem or are the product of mutational biases remains unsettled. We generated 20 replicate lineages allowed to fix a single mutation from each of four bacteriophage genotypes under identical selective regimes to test for parallel changes within and across genotypes at the levels of mutational effect distributions and gene, protein, amino acid, and nucleotide changes. All four genotypes shared a distribution of beneficial mutational effects best approximated by a distribution with a finite upper bound. Parallel adaptation was high at the protein, gene, amino acid, and nucleotide levels, both within and among phage genotypes, with the most common first-step mutation in each background fixing on an average in 7 of 20 replicates and half of the substitutions in two of the four genotypes occurring at shared sites. Remarkably, the mutation of largest beneficial effect that fixed for each genotype was never the most common, as would be expected if parallelism were driven by selection. In fact, the mutation of smallest benefit for each genotype fixed in a total of 7 of 80 lineages, equally as often as the mutation of largest benefit, leading us to conclude that adaptation was largely mutation-driven, such that mutational biases led to frequent parallel fixation of mutations of suboptimal effect.

Additive Phenotypes Underlie Epistasis of Fitness Effects

Gene interactions, or epistasis, play a large role in determining evolutionary outcomes. The ruggedness of fitness landscapes, and thus the predictability of evolution and the accessibility of high-fitness genotypes, is determined largely by the pervasiveness of epistasis and the degree of correlation between similar genotypes. We created all possible pairings of three sets of five beneficial first-step mutations fixed during adaptive walks under three different regimes: selection on growth rate alone, on growth rate and thermal stability, and on growth rate and pH stability. All 30 double-mutants displayed negative, antagonistic epistasis with regard to growth rate and fitness, but positive epistasis and additivity were common for the stability phenotypes. This suggested that biophysically simple phenotypes, such as capsid stability, may on average behave more additively than complex phenotypes like viral growth rate. Growth rate epistasis was also smaller in magnitude when the individual effects of single mutations were smaller. Significant sign epistasis, such that the effect of a mutation that is beneficial in the wild-type background is deleterious in combination with a second mutation, emerged more frequently in intragenic mutational pairings than in intergenic pairs, and was evident in nearly half of the double-mutants, indicating that the fitness landscape is moderately uncorrelated and of intermediate ruggedness. Together, our results indicated that mutations may interact additively with regard to phenotype when considered at a basic, biophysical level, but that epistasis arises as a result of pleiotropic interactions between the individual components of complex phenotypes and diminishing returns arising from intermediate phenotypic optima.

Single-mutation fitness landscapes for an enzyme on multiple substrates reveal specificity is globally encoded

Our lack of total understanding of the intricacies of how enzymes behave has constrained our ability to robustly engineer substrate specificity. Furthermore, the mechanisms of natural evolution leading to improved or novel substrate specificities are not wholly defined. Here we generate near-comprehensive single-mutation fitness landscapes comprising >96.3% of all possible single nonsynonymous mutations for hydrolysis activity of an amidase expressed in E. coli with three different substrates. For all three selections, we find that the distribution of beneficial mutations can be described as exponential, supporting a current hypothesis for adaptive molecular evolution. Beneficial mutations in one selection have essentially no correlation with fitness for other selections and are dispersed throughout the protein sequence and structure. Our results further demonstrate the dependence of local fitness landscapes on substrate identity and provide an example of globally distributed sequence-specificity determinants for an enzyme.

Systematically understanding the sequence determinants to substrate specificity for enzymes has implications in areas from evolutionary biology to biocatalysis. Here, Whitehead and colleagues generate and analyse near-comprehensive single-mutation fitness landscapes for an amidase with three different substrates.

Dynamics of molecular evolution in RNA virus populations depend on sudden versus gradual environmental change

Understanding the dynamics of molecular adaptation is a fundamental goal of evolutionary biology. While adaptation to constant environments has been well characterized, the effects of environmental complexity remain seldom studied. One simple but understudied factor is the rate of environmental change. Here we used experimental evolution with RNA viruses to investigate whether evolutionary dynamics varied based on the rate of environmental turnover. We used whole-genome next-generation sequencing to characterize evolutionary dynamics in virus populations adapting to a sudden versus gradual shift onto a novel host cell type. In support of theoretical models, we found that when populations evolved in response to a sudden environmental change, mutations of large beneficial effect tended to fix early, followed by mutations of smaller beneficial effect; as predicted, this pattern broke down in response to a gradual environmental change. Early mutational steps were highly parallel across replicate populations in both treatments. The fixation of single mutations was less common than sweeps of associated "cohorts" of mutations, and this pattern intensified when the environment changed gradually. Additionally, clonal interference appeared stronger in response to a gradual change. Our results suggest that the rate of environmental change is an important determinant of evolutionary dynamics in asexual populations.

Dynamics and Fate of Beneficial Mutations Under Lineage Contamination by Linked Deleterious Mutations

Beneficial mutations drive adaptive evolution, yet their selective advantage does not ensure their fixation. Haldane's application of single-type branching process theory showed that genetic drift alone could cause the extinction of newly arising beneficial mutations with high probability. With linkage, deleterious mutations will affect the dynamics of beneficial mutations and might further increase their extinction probability. Here, we model the lineage dynamics of a newly arising beneficial mutation as a multitype branching process. Our approach accounts for the combined effects of drift and the stochastic accumulation of linked deleterious mutations, which we call lineage contamination We first study the lineage-contamination phenomenon in isolation, deriving dynamics and survival probabilities (the complement of extinction probabilities) of beneficial lineages. We find that survival probability is zero when [Formula: see text] where U is deleterious mutation rate and [Formula: see text] is the selective advantage of the beneficial mutation in question, and is otherwise depressed below classical predictions by a factor bounded from below by [Formula: see text] We then put the lineage contamination phenomenon into the context of an evolving population by incorporating the effects of background selection. We find that, under the combined effects of lineage contamination and background selection, ensemble survival probability is never zero but is depressed below classical predictions by a factor bounded from below by [Formula: see text] where [Formula: see text] is mean selective advantage of beneficial mutations, and [Formula: see text] This factor, and other bounds derived from it, are independent of the fitness effects of deleterious mutations. At high enough mutation rates, lineage contamination can depress fixation probabilities to values that approach zero. This fact suggests that high mutation rates can, perhaps paradoxically, (1) alleviate competition among beneficial mutations, or (2) potentially even shut down the adaptive process. We derive critical mutation rates above which these two events become likely.

How Good Are Statistical Models at Approximating Complex Fitness Landscapes?

Fitness landscapes determine the course of adaptation by constraining and shaping evolutionary trajectories. Knowledge of the structure of a fitness landscape can thus predict evolutionary outcomes. Empirical fitness landscapes, however, have so far only offered limited insight into real-world questions, as the high dimensionality of sequence spaces makes it impossible to exhaustively measure the fitness of all variants of biologically meaningful sequences. We must therefore revert to statistical descriptions of fitness landscapes that are based on a sparse sample of fitness measurements. It remains unclear, however, how much data are required for such statistical descriptions to be useful. Here, we assess the ability of regression models accounting for single and pairwise mutations to correctly approximate a complex quasi-empirical fitness landscape. We compare approximations based on various sampling regimes of an RNA landscape and find that the sampling regime strongly influences the quality of the regression. On the one hand it is generally impossible to generate sufficient samples to achieve a good approximation of the complete fitness landscape, and on the other hand systematic sampling schemes can only provide a good description of the immediate neighborhood of a sequence of interest. Nevertheless, we obtain a remarkably good and unbiased fit to the local landscape when using sequences from a population that has evolved under strong selection. Thus, current statistical methods can provide a good approximation to the landscape of naturally evolving populations.

Monotonicity of fitness landscapes and mutation rate control

A common view in evolutionary biology is that mutation rates are minimised. However, studies in combinatorial optimisation and search have shown a clear advantage of using variable mutation rates as a control parameter to optimise the performance of evolutionary algorithms. Much biological theory in this area is based on Ronald Fisher’s work, who used Euclidean geometry to study the relation between mutation size and expected fitness of the offspring in infinite phenotypic spaces. Here we reconsider this theory based on the alternative geometry of discrete and finite spaces of DNA sequences. First, we consider the geometric case of fitness being isomorphic to distance from an optimum, and show how problems of optimal mutation rate control can be solved exactly or approximately depending on additional constraints of the problem. Then we consider the general case of fitness communicating only partial information about the distance. We define weak monotonicity of fitness landscapes and prove that this property holds in all landscapes that are continuous and open at the optimum. This theoretical result motivates our hypothesis that optimal mutation rate functions in such landscapes will increase when fitness decreases in some neighbourhood of an optimum, resembling the control functions derived in the geometric case. We test this hypothesis experimentally by analysing approximately optimal mutation rate control functions in 115 complete landscapes of binding scores between DNA sequences and transcription factors. Our findings support the hypothesis and find that the increase of mutation rate is more rapid in landscapes that are less monotonic (more rugged). We discuss the relevance of these findings to living organisms.

Hierarchy and extremes in selections from pools of randomized proteins

Variation and selection are the core principles of Darwinian evolution, but quantitatively relating the diversity of a population to its capacity to respond to selection is challenging. Here, we examine this problem at a molecular level in the context of populations of partially randomized proteins selected for binding to well-defined targets. We built several minimal protein libraries, screened them in vitro by phage display, and analyzed their response to selection by high-throughput sequencing. A statistical analysis of the results reveals two main findings. First, libraries with the same sequence diversity but built around different "frameworks" typically have vastly different responses; second, the distribution of responses of the best binders in a library follows a simple scaling law. We show how an elementary probabilistic model based on extreme value theory rationalizes the latter finding. Our results have implications for designing synthetic protein libraries, estimating the density of functional biomolecules in sequence space, characterizing diversity in natural populations, and experimentally investigating evolvability (i.e., the potential for future evolution).

Exploiting the Adaptation Dynamics to Predict the Distribution of Beneficial Fitness Effects

Adaptation of asexual populations is driven by beneficial mutations and therefore the dynamics of this process, besides other factors, depends on the distribution of beneficial fitness effects. It is known that on uncorrelated fitness landscapes, this distribution can only be of three types: truncated, exponential and power law. We performed extensive stochastic simulations to study the adaptation dynamics on rugged fitness landscapes, and identified two quantities that can be used to distinguish the underlying distribution of beneficial fitness effects. The first quantity studied here is the fitness difference between successive mutations that spread in the population, which is found to decrease in the case of truncated distributions, remains nearly a constant for exponentially decaying distributions and increases when the fitness distribution decays as a power law. The second quantity of interest, namely, the rate of change of fitness with time also shows quantitatively different behaviour for different beneficial fitness distributions. The patterns displayed by the two aforementioned quantities are found to hold good for both low and high mutation rates. We discuss how these patterns can be exploited to determine the distribution of beneficial fitness effects in microbial experiments.

The implications of small stem cell niche sizes and the distribution of fitness effects of new mutations in aging and tumorigenesis

Somatic tissue evolves over a vertebrate's lifetime due to the accumulation of mutations in stem cell populations. Mutations may alter cellular fitness and contribute to tumorigenesis or aging. The distribution of mutational effects within somatic cells is not known. Given the unique regulatory regime of somatic cell division, we hypothesize that mutational effects in somatic tissue fall into a different framework than whole organisms; one in which there are more mutations of large effect. Through simulation analysis, we investigate the fit of tumor incidence curves generated using exponential and power‐law distributions of fitness effects (DFE) to known tumorigenesis incidence. Modeling considerations include the architecture of stem cell populations, that is, a large number of very small populations, and mutations that do and do not fix neutrally in the stem cell niche. We find that the typically quantified DFE in whole organisms is sufficient to explain tumorigenesis incidence. Further, deleterious mutations are predicted to accumulate via genetic drift, resulting in reduced tissue maintenance. Thus, despite there being a large number of stem cells throughout the intestine, its compartmental architecture leads to the accumulation of deleterious mutations and significant aging, making the intestinal stem cell niche a prime example of Muller's Ratchet.

Greedy adaptive walks on a correlated fitness landscape

We study adaptation of a haploid asexual population on a fitness landscape defined over binary genotype sequences of length L. We consider greedy adaptive walks in which the population moves to the fittest among all single mutant neighbors of the current genotype until a local fitness maximum is reached. The landscape is of the rough mount Fuji type, which means that the fitness value assigned to a sequence is the sum of a random and a deterministic component. The random components are independent and identically distributed random variables, and the deterministic component varies linearly with the distance to a reference sequence. The deterministic fitness gradient c is a parameter that interpolates between the limits of an uncorrelated random landscape (c=0) and an effectively additive landscape (c→∞). When the random fitness component is chosen from the Gumbel distribution, explicit expressions for the distribution of the number of steps taken by the greedy walk are obtained, and it is shown that the walk length varies non-monotonically with the strength of the fitness gradient when the starting point is sufficiently close to the reference sequence. Asymptotic results for general distributions of the random fitness component are obtained using extreme value theory, and it is found that the walk length attains a non-trivial limit for L→∞, different from its values for c=0 and c=∞, if c is scaled with L in an appropriate combination.

Can the experimental evolution programme help us elucidate the genetic basis of adaptation in nature?

There have been a variety of approaches taken to try to characterize and identify the genetic basis of adaptation in nature, spanning theoretical models, experimental evolution studies and direct tests of natural populations. Theoretical models can provide formalized and detailed hypotheses regarding evolutionary processes and patterns, from which experimental evolution studies can then provide important proofs of concepts and characterize what is biologically reasonable. Genetic and genomic data from natural populations then allow for the identification of the particular factors that have and continue to play an important role in shaping adaptive evolution in the natural world. Further to this, experimental evolution studies allow for tests of theories that may be difficult or impossible to test in natural populations for logistical and methodological reasons and can even generate new insights, suggesting further refinement of existing theories. However, as experimental evolution studies often take place in a very particular set of controlled conditions--that is simple environments, a small range of usually asexual species, relatively short timescales--the question remains as to how applicable these experimental results are to natural populations. In this review, we discuss important insights coming from experimental evolution, focusing on four key topics tied to the evolutionary genetics of adaptation, and within those topics, we discuss the extent to which the experimental work compliments and informs natural population studies. We finish by making suggestions for future work in particular a need for natural population genomic time series data, as well as the necessity for studies that combine both experimental evolution and natural population approaches.

The evolutionarily stable distribution of fitness effects

The distribution of fitness effects (DFE) of new mutations is a key parameter in determining the course of evolution. This fact has motivated extensive efforts to measure the DFE or to predict it from first principles. However, just as the DFE determines the course of evolution, the evolutionary process itself constrains the DFE. Here, we analyze a simple model of genome evolution in a constant environment in which natural selection drives the population toward a dynamic steady state where beneficial and deleterious substitutions balance. The distribution of fitness effects at this steady state is stable under further evolution and provides a natural null expectation for the DFE in a population that has evolved in a constant environment for a long time. We calculate how the shape of the evolutionarily stable DFE depends on the underlying population genetic parameters. We show that, in the absence of epistasis, the ratio of beneficial to deleterious mutations of a given fitness effect obeys a simple relationship independent of population genetic details. Finally, we analyze how the stable DFE changes in the presence of a simple form of diminishing-returns epistasis.

Inference for one-step beneficial mutations using next generation sequencing

Experimental evolution is an important research method that allows for the study of evolutionary processes occurring in microorganisms. Here we present a novel approach to experimental evolution that is based on application of next generation sequencing. Under this approach population level sequencing is applied to an evolving population in which multiple first-step beneficial mutations occur concurrently. As a result, frequencies of multiple beneficial mutations are observed in each replicate of an experiment. For this new type of data we develop methods of statistical inference. In particular, we propose a method for imputing selection coefficients of first-step beneficial mutations. The imputed selection coefficient are then used for testing the distribution of first-step beneficial mutations and for estimation of mean selection coefficient. In the case when selection coefficients are uniformly distributed, collected data may also be used to estimate the total number of available first-step beneficial mutations.

Quantitative evolutionary dynamics using high-resolution lineage tracking

Evolution of large asexual cell populations underlies ∼30% of deaths worldwide, including those caused by bacteria, fungi, parasites, and cancer. However, the dynamics underlying these evolutionary processes remain poorly understood because they involve many competing beneficial lineages, most of which never rise above extremely low frequencies in the population. To observe these normally hidden evolutionary dynamics, we constructed a sequencing-based ultra high-resolution lineage tracking system in Saccharomyces cerevisiae that allowed us to monitor the relative frequencies of ∼500,000 lineages simultaneously. In contrast to some expectations, we found that the spectrum of fitness effects of beneficial mutations is neither exponential nor monotonic. Early adaptation is a predictable consequence of this spectrum and is strikingly reproducible, but the initial small-effect mutations are soon outcompeted by rarer large-effect mutations that result in variability between replicates. These results suggest that early evolutionary dynamics may be deterministic for a period of time before stochastic effects become important.

Contrasting modes and tempos of venom expression evolution in two snake species

Selection is predicted to drive diversification within species and lead to local adaptation, but understanding the mechanistic details underlying this process and thus the genetic basis of adaptive evolution requires the mapping of genotype to phenotype. Venom is complex and involves many genes, but the specialization of the venom gland toward toxin production allows specific transcripts to be correlated with specific toxic proteins, establishing a direct link from genotype to phenotype. To determine the extent of expression variation and identify the processes driving patterns of phenotypic diversity, we constructed genotype-phenotype maps and compared range-wide toxin-protein expression variation for two species of snake with nearly identical ranges: the eastern diamondback rattlesnake (Crotalus adamanteus) and the eastern coral snake (Micrurus fulvius). We detected significant expression variation in C. adamanteus, identified the specific loci associated with population differentiation, and found that loci expressed at all levels contributed to this divergence. Contrary to expectations, we found no expression variation in M. fulvius, suggesting that M. fulvius populations are not locally adapted. Our results not only linked expression variation at specific loci to divergence in a polygenic, complex trait but also have extensive conservation and biomedical implications. C. adamanteus is currently a candidate for federal listing under the Endangered Species Act, and the loss of any major population would result in the irrevocable loss of a unique venom phenotype. The lack of variation in M. fulvius has significant biomedical application because our data will assist in the development of effective antivenom for this species.

Adaptation in tunably rugged fitness landscapes: the rough Mount Fuji model

Much of the current theory of adaptation is based on Gillespie's mutational landscape model (MLM), which assumes that the fitness values of genotypes linked by single mutational steps are independent random variables. On the other hand, a growing body of empirical evidence shows that real fitness landscapes, while possessing a considerable amount of ruggedness, are smoother than predicted by the MLM. In the present article we propose and analyze a simple fitness landscape model with tunable ruggedness based on the rough Mount Fuji (RMF) model originally introduced by Aita et al. in the context of protein evolution. We provide a comprehensive collection of results pertaining to the topographical structure of RMF landscapes, including explicit formulas for the expected number of local fitness maxima, the location of the global peak, and the fitness correlation function. The statistics of single and multiple adaptive steps on the RMF landscape are explored mainly through simulations, and the results are compared to the known behavior in the MLM model. Finally, we show that the RMF model can explain the large number of second-step mutations observed on a highly fit first-step background in a recent evolution experiment with a microvirid bacteriophage.

Effects of new mutations on fitness: insights from models and data

The rates and properties of new mutations affecting fitness have implications for a number of outstanding questions in evolutionary biology. Obtaining estimates of mutation rates and effects has historically been challenging, and little theory has been available for predicting the distribution of fitness effects (DFE); however, there have been recent advances on both fronts. Extreme-value theory predicts the DFE of beneficial mutations in well-adapted populations, while phenotypic fitness landscape models make predictions for the DFE of all mutations as a function of the initial level of adaptation and the strength of stabilizing selection on traits underlying fitness. Direct experimental evidence confirms predictions on the DFE of beneficial mutations and favors distributions that are roughly exponential but bounded on the right. A growing number of studies infer the DFE using genomic patterns of polymorphism and divergence, recovering a wide range of DFE. Future work should be aimed at identifying factors driving the observed variation in the DFE. We emphasize the need for further theory explicitly incorporating the effects of partial pleiotropy and heterogeneity in the environment on the expected DFE.

The fates of mutant lineages and the distribution of fitness effects of beneficial mutations in laboratory budding yeast populations

The outcomes of evolution are determined by which mutations occur and fix. In rapidly adapting microbial populations, this process is particularly hard to predict because lineages with different beneficial mutations often spread simultaneously and interfere with one another's fixation. Hence to predict the fate of any individual variant, we must know the rate at which new mutations create competing lineages of higher fitness. Here, we directly measured the effect of this interference on the fates of specific adaptive variants in laboratory Saccharomyces cerevisiae populations and used these measurements to infer the distribution of fitness effects of new beneficial mutations. To do so, we seeded marked lineages with different fitness advantages into replicate populations and tracked their subsequent frequencies for hundreds of generations. Our results illustrate the transition between strongly advantageous lineages that decisively sweep to fixation and more moderately advantageous lineages that are often outcompeted by new mutations arising during the course of the experiment. We developed an approximate likelihood framework to compare our data to simulations and found that the effects of these competing beneficial mutations were best approximated by an exponential distribution, rather than one with a single effect size. We then used this inferred distribution of fitness effects to predict the rate of adaptation in a set of independent control populations. Finally, we discuss how our experimental design can serve as a screen for rare, large-effect beneficial mutations.

Influenza virus drug resistance: a time-sampled population genetics perspective

The challenge of distinguishing genetic drift from selection remains a central focus of population genetics. Time-sampled data may provide a powerful tool for distinguishing these processes, and we here propose approximate Bayesian, maximum likelihood, and analytical methods for the inference of demography and selection from time course data. Utilizing these novel statistical and computational tools, we evaluate whole-genome datasets of an influenza A H1N1 strain in the presence and absence of oseltamivir (an inhibitor of neuraminidase) collected at thirteen time points. Results reveal a striking consistency amongst the three estimation procedures developed, showing strongly increased selection pressure in the presence of drug treatment. Importantly, these approaches re-identify the known oseltamivir resistance site, successfully validating the approaches used. Enticingly, a number of previously unknown variants have also been identified as being positively selected. Results are interpreted in the light of Fisher's Geometric Model, allowing for a quantification of the increased distance to optimum exerted by the presence of drug, and theoretical predictions regarding the distribution of beneficial fitness effects of contending mutations are empirically tested. Further, given the fit to expectations of the Geometric Model, results suggest the ability to predict certain aspects of viral evolution in response to changing host environments and novel selective pressures.

A bayesian MCMC approach to assess the complete distribution of fitness effects of new mutations: uncovering the potential for adaptive walks in challenging environments

The role of adaptation in the evolutionary process has been contentious for decades. At the heart of the century-old debate between neutralists and selectionists lies the distribution of fitness effects (DFE)—that is, the selective effect of all mutations. Attempts to describe the DFE have been varied, occupying theoreticians and experimentalists alike. New high-throughput techniques stand to make important contributions to empirical efforts to characterize the DFE, but the usefulness of such approaches depends on the availability of robust statistical methods for their interpretation. We here present and discuss a Bayesian MCMC approach to estimate fitness from deep sequencing data and use it to assess the DFE for the same 560 point mutations in a coding region of Hsp90 in Saccharomyces cerevisiae across six different environmental conditions. Using these estimates, we compare the differences in the DFEs resulting from mutations covering one-, two-, and three-nucleotide steps from the wild type—showing that multiple-step mutations harbor more potential for adaptation in challenging environments, but also tend to be more deleterious in the standard environment. All observations are discussed in the light of expectations arising from Fisher’s geometric model.

An ABC method for estimating the rate and distribution of effects of beneficial mutations

Determining the distribution of adaptive mutations available to natural selection is a difficult task. These are rare events and most of them are lost by chance. Some theoretical works propose that the distribution of newly arising beneficial mutations should be close to exponential. Empirical data are scarce and do not always support an exponential distribution. Analysis of the dynamics of adaptation in asexual populations of microorganisms has revealed that these can be summarized by two effective parameters, the effective mutation rate, U_e, and the effective selection coefficient of a beneficial mutation, S_e. Here, we show that these effective parameters will not always reflect the rate and mean effect of beneficial mutations, especially when the distribution of arising mutations has high variance, and the mutation rate is high. We propose a method to estimate the distribution of arising beneficial mutations, which is motivated by a common experimental setup. The method, which we call One Biallelic Marker Approximate Bayesian Computation, makes use of experimental data consisting of periodic measures of neutral marker frequencies and mean population fitness. Using simulations, we find that this method allows the discrimination of the shape of the distribution of arising mutations and that it provides reasonable estimates of their rates and mean effects in ranges of the parameter space that may be of biological relevance.

Environment determines epistatic patterns for a ssDNA virus

Despite the accumulation of substantial quantities of information about epistatic interactions among both deleterious and beneficial mutations in a wide array of experimental systems, neither consistent patterns nor causal explanations for these interactions have yet emerged. Furthermore, the effects of mutations depend on the environment in which they are characterized, implying that the environment may also influence epistatic interactions. Recent work with beneficial mutations for the single-stranded DNA bacteriophage ID11 demonstrated that interactions between pairs of mutations could be understood by means of a simple model that assumes that mutations have additive phenotypic effects and that epistasis arises through a nonlinear phenotype-fitness map with a single intermediate optimum. To determine whether such a model could also explain changes in epistatic patterns associated with changes in environment, we measured epistatic interactions for these same mutations under conditions for which we expected to find the wild-type ID11 at different distances from its phenotypic optimum by assaying fitnesses at three different temperatures: 33°, 37°, and 41°. Epistasis was present and negative under all conditions, but became more pronounced as temperature increased. We found that the additive-phenotypes model explained these patterns as changes in the parameters of the phenotype-fitness map, but that a model that additionally allows the phenotypes to vary across temperatures performed significantly better. Our results show that ostensibly complex patterns of fitness effects and epistasis across environments can be explained by assuming a simple structure for the genotype-phenotype relationship.

Rates of fitness decline and rebound suggest pervasive epistasis

Unraveling the factors that determine the rate of adaptation is a major question in evolutionary biology. One key parameter is the effect of a new mutation on fitness, which invariably depends on the environment and genetic background. The fate of a mutation also depends on population size, which determines the amount of drift it will experience. Here, we manipulate both population size and genotype composition and follow adaptation of 23 distinct Escherichia coli genotypes. These have previously accumulated mutations under intense genetic drift and encompass a substantial fitness variation. A simple rule is uncovered: the net fitness change is negatively correlated with the fitness of the genotype in which new mutations appear—a signature of epistasis. We find that Fisher's geometrical model can account for the observed patterns of fitness change and infer the parameters of this model that best fit the data, using Approximate Bayesian Computation. We estimate a genomic mutation rate of 0.01 per generation for fitness altering mutations, albeit with a large confidence interval, a mean fitness effect of mutations of −0.01, and an effective number of traits nine in mutS⁻E. coli. This framework can be extended to confront a broader range of models with data and test different classes of fitness landscape models.

Evolutionary rescue and adaptation to abrupt environmental change depends upon the history of stress

Whether evolution will be rapid enough to rescue declining populations will depend upon population size, the supply of genetic variation, the degree of maladaptation and the historical direction of selection. We examined whether the level of environmental stress experienced by a population prior to abrupt environmental change affects the probability of evolutionary rescue (ER). Hundreds of populations of two species of yeast, Saccharomyces cerevisiae and Saccharomyces paradoxus were exposed to a range of sublethal concentrations of salt for approximately a hundred generations before transfer to a concentration of salt lethal to the ancestor (150 g l(-1) NaCl). The fitness of surviving populations of both species was a quadratic function of yield: fitness was greatest for large populations that had been selected on low salt concentrations (less than 20 g l(-1) NaCl) and small populations that had adapted to high salt (more than 80 g l(-1) NaCl). However, differences occurred between species in the probability of ER. The frequency of ER was positively correlated with salt concentration for S. cerevisiae, but negatively correlated with salt concentration in S. paradoxus. These results not only demonstrate that past environmental conditions can determine the probability of ER after abrupt environmental change, but also suggest that there may even be differences between closely related species that are worth further exploration.

Virus replication as a phenotypic version of polynucleotide evolution

In this paper, we revisit and adapt to viral evolution an approach based on the theory of branching process advanced by Demetrius et al. (Bull. Math. Biol. 46:239-262, 1985), in their study of polynucleotide evolution. By taking into account beneficial effects, we obtain a non-trivial multivariate generalization of their single-type branching process model. Perturbative techniques allows us to obtain analytical asymptotic expressions for the main global parameters of the model, which lead to the following rigorous results: (i) a new criterion for "no sure extinction", (ii) a generalization and proof, for this particular class of models, of the lethal mutagenesis criterion proposed by Bull et al. (J. Virol. 18:2930-2939, 2007), (iii) a new proposal for the notion of relaxation time with a quantitative prescription for its evaluation, (iv) the quantitative description of the evolution of the expected values in four distinct "stages": extinction threshold, lethal mutagenesis, stationary "equilibrium", and transient. Finally, based on these quantitative results, we are able to draw some qualitative conclusions.

The consistency of beneficial fitness effects of mutations across diverse genetic backgrounds

Parallel and convergent evolution have been remarkably common observations in molecular adaptation but primarily in the context of the same genotype adapting to the same conditions. These phenomena therefore tell us about the stochasticity and limitations of adaptation. The limited data on convergence and parallelism in the adaptation of different genotypes conflict as to the importance of such events. If the effects of beneficial mutations are highly context dependent (i.e., if they are epistatic), different genotypes should adapt through different mutations. Epistasis for beneficial mutations has been investigated but mainly through measurement of interactions between individually beneficial mutations for the same genotype. We examine epistasis for beneficial mutations at a broader genetic scale by measuring the fitness effects of two mutations beneficial for the ssDNA bacteriophage ID11 in eight different, related genotypes showing 0.3-3.7% nucleotide divergence from ID11. We found no evidence for sign epistasis, but the mutations tended to have much smaller or no effects on fitness in the new genotypes. We found evidence for diminishing-returns epistasis; the effects were more beneficial for lower-fitness genotypes. The patterns of epistasis were not determined by phylogenetic relationships to the original genotype. To improve our understanding of the patterns of epistasis, we fit the data to a model in which each mutation had a constant, nonepistatic phenotypic effect across genotypes and the phenotype-fitness map had a single optimum. This model fit the data well, suggesting that epistasis for these mutations was due to nonlinearity in the phenotype-fitness mapping and that the likelihood of parallel evolution depends more on phenotype than on genotype.

Estimating the number of one-step beneficial mutations

Mutations that confer a selective advantage to an organism are the raw material upon which natural selection acts. The number of such mutations that are available is a central quantity of interest for understanding the tempo and trajectory of adaptive evolution. While this quantity is typically unknown, it can be estimated with varying levels of accuracy based on data obtained experimentally. We propose a method for estimating the number of beneficial mutations that accounts for the evolutionary forces that generate the data. Our model-based parametric approach is compared to an adjusted nonparametric abundance-based coverage estimator. We show that, in general, our estimator performs better. When the number of mutations is small, however, the performances of the two estimators are similar.

Quantifying the adaptive potential of an antibiotic resistance enzyme

For a quantitative understanding of the process of adaptation, we need to understand its "raw material," that is, the frequency and fitness effects of beneficial mutations. At present, most empirical evidence suggests an exponential distribution of fitness effects of beneficial mutations, as predicted for Gumbel-domain distributions by extreme value theory. Here, we study the distribution of mutation effects on cefotaxime (Ctx) resistance and fitness of 48 unique beneficial mutations in the bacterial enzyme TEM-1 β-lactamase, which were obtained by screening the products of random mutagenesis for increased Ctx resistance. Our contributions are threefold. First, based on the frequency of unique mutations among more than 300 sequenced isolates and correcting for mutation bias, we conservatively estimate that the total number of first-step mutations that increase Ctx resistance in this enzyme is 87 [95% CI 75-189], or 3.4% of all 2,583 possible base-pair substitutions. Of the 48 mutations, 10 are synonymous and the majority of the 38 non-synonymous mutations occur in the pocket surrounding the catalytic site. Second, we estimate the effects of the mutations on Ctx resistance by determining survival at various Ctx concentrations, and we derive their fitness effects by modeling reproduction and survival as a branching process. Third, we find that the distribution of both measures follows a Fréchet-type distribution characterized by a broad tail of a few exceptionally fit mutants. Such distributions have fundamental evolutionary implications, including an increased predictability of evolution, and may provide a partial explanation for recent observations of striking parallel evolution of antibiotic resistance.

Real time forecasting of near-future evolution

A metaphor for adaptation that informs much evolutionary thinking today is that of mountain climbing, where horizontal displacement represents change in genotype, and vertical displacement represents change in fitness. If it were known a priori what the 'fitness landscape' looked like, that is, how the myriad possible genotypes mapped onto fitness, then the possible paths up the fitness mountain could each be assigned a probability, thus providing a dynamical theory with long-term predictive power. Such detailed genotype-fitness data, however, are rarely available and are subject to change with each change in the organism or in the environment. Here, we take a very different approach that depends only on fitness or phenotype-fitness data obtained in real time and requires no a priori information about the fitness landscape. Our general statistical model of adaptive evolution builds on classical theory and gives reasonable predictions of fitness and phenotype evolution many generations into the future.

Distribution of fixed beneficial mutations and the rate of adaptation in asexual populations

When large asexual populations adapt, competition between simultaneously segregating mutations slows the rate of adaptation and restricts the set of mutations that eventually fix. This phenomenon of interference arises from competition between mutations of different strengths as well as competition between mutations that arise on different fitness backgrounds. Previous work has explored each of these effects in isolation, but the way they combine to influence the dynamics of adaptation remains largely unknown. Here, we describe a theoretical model to treat both aspects of interference in large populations. We calculate the rate of adaptation and the distribution of fixed mutational effects accumulated by the population. We focus particular attention on the case when the effects of beneficial mutations are exponentially distributed, as well as on a more general class of exponential-like distributions. In both cases, we show that the rate of adaptation and the influence of genetic background on the fixation of new mutants is equivalent to an effective model with a single selection coefficient and rescaled mutation rate, and we explicitly calculate these effective parameters. We find that the effective selection coefficient exactly coincides with the most common fixed mutational effect. This equivalence leads to an intuitive picture of the relative importance of different types of interference effects, which can shift dramatically as a function of the population size, mutation rate, and the underlying distribution of fitness effects.

Correlations of record events as a test for heavy-tailed distributions

A record is an entry in a time series that is larger or smaller than all previous entries. If the time series consists of independent, identically distributed random variables with a superimposed linear trend, record events are positively (negatively) correlated when the tail of the distribution is heavier (lighter) than exponential. Here we use these correlations to detect heavy-tailed behavior in small sets of independent random variables. The method consists of converting random subsets of the data into time series with a tunable linear drift and computing the resulting record correlations.

Multiple genetic pathways to similar fitness limits during viral adaptation to a new host

The gain in fitness during adaptation depends on the supply of beneficial mutations. Despite a good theoretical understanding of how evolution proceeds for a defined set of mutations, there is little understanding of constraints on net fitness-whether fitness will reach a limit despite ongoing selection and mutation, and if there is a limit, what determines it. Here, the dsDNA bacteriophage SP6, a virus of Salmonella, was adapted to Escherichia coli K-12. From an isolate capable of modest growth on E. coli, four lines were adapted for rapid growth by protocols differing in use of mutagen, propagation method, and duration, but using the same media, temperature, and a continual excess of the novel host. Nucleotide changes underlying those adaptations differed greatly in number and identity, but the four lines achieved similar absolute fitness at the end, an increase of more than 4000-fold phage descendants per hour. Thus, the fitness landscape allows multiple genetic paths to the same approximate fitness limit. The existence and causes of fitness limits have ramifications to genome engineering, vaccine design, and "lethal mutagenesis" treatments to cure viral infections.

Cost of antibiotic resistance and the geometry of adaptation

The distribution of effects of beneficial mutations is key to our understanding of biological adaptation. Yet, empirical estimates of this distribution are scarce, and its functional form is largely unknown. Theoretical models of adaptation predict that the functional form of this distribution should depend on the distance to the optimum. Here, we estimate the rate and distribution of adaptive mutations that compensate for the effect of a single deleterious mutation, which causes antibiotic resistance. Using a system with multiple molecular markers, we estimate the distribution of fitness effects of mutations at two distances from the adaptive peak in 60 populations of Escherichia coli. We find that beneficial mutations, which can contribute to compensatory evolution, occur at very high rates, of the order of 10⁻⁵ per genome per generation and can be detected within a few tens of generations. They cause an average fitness increase of 2.5% and 3.6%, depending on the cost of resistance, which is expected under Fisher's geometrical model of adaptation. Moreover, we provide the first description of the distribution of beneficial mutations, segregating during the process of compensatory evolution, to antibiotic resistances bearing different costs. Hence, these results have important implications to understanding the spread and maintenance of antibiotic resistance in bacteria.

Estimation of the rate and effect of new beneficial mutations in asexual populations

The rate and effect of available beneficial mutations are key parameters in determining how a population adapts to a new environment. However, these parameters are poorly known, in large part because of the difficulty of designing and interpreting experiments to examine the rare and intrinsically stochastic process of mutation occurrence. We present a new approach to estimate the rate and selective advantage of beneficial mutations that underlie the adaptation of asexual populations. We base our approach on the analysis of experiments that track the effect of newly arising beneficial mutations on the dynamics of a neutral marker in evolving bacterial populations and develop efficient estimators of mutation rate and selective advantage. Using extensive simulations, we evaluate the accuracy of our estimators and conclude that they are quite robust to the use of relatively low experimental replication. To validate the predictions of our model, we compare theoretical and experimentally determined estimates of the selective advantage of the first beneficial mutation to fix in a series of ten replicate populations. We find that our theoretical predictions are not significantly different from experimentally determined selection coefficients. Application of our method to suitably designed experiments will allow estimation of how population evolvability depends on demographic and initial fitness parameters.

The phenotype-fitness map in experimental evolution of phages

Evolutionary biologists commonly interpret adaptations of organisms by reference to a phenotype-fitness map, a model of how different states of a phenotype affect fitness. Notwithstanding the popularity of this approach, it remains difficult to directly test these mappings, both because the map often describes only a small subset of phenotypes contributing to total fitness and because direct measures of fitness are difficult to obtain and compare to the map. Both limitations can be overcome for bacterial viruses (phages) grown in the experimental condition of unlimited hosts. A complete accounting of fitness requires 3 easily measured phenotypes, and total fitness is also directly measurable for arbitrary genotypes. Yet despite the presumed transparency of this system, directly estimated fitnesses often differ from fitnesses calculated from the phenotype-fitness map. This study attempts to resolve these discrepancies, both by developing a more exact analytical phenotype-fitness map and by exploring the empirical foundations of direct fitness estimates. We derive an equation (the phenotype-fitness map) for exponential phage growth that allows an arbitrary distribution of lysis times and burst sizes. We also show that direct estimates of fitness are, in many cases, plausibly in error because the population has not attained stable age distribution and thus violates the model underlying the phenotype-fitness map. In conjunction with data provided here, the new understanding appears to resolve a discrepancy between the reported fitness of phage T7 and the substantially lower value calculated from its phenotype-fitness map.

Stickbreaking: a novel fitness landscape model that harbors epistasis and is consistent with commonly observed patterns of adaptive evolution

In relating genotypes to fitness, models of adaptation need to both be computationally tractable and qualitatively match observed data. One reason that tractability is not a trivial problem comes from a combinatoric problem whereby no matter in what order a set of mutations occurs, it must yield the same fitness. We refer to this as the bookkeeping problem. Because of their commutative property, the simple additive and multiplicative models naturally solve the bookkeeping problem. However, the fitness trajectories and epistatic patterns they predict are inconsistent with the patterns commonly observed in experimental evolution. This motivates us to propose a new and equally simple model that we call stickbreaking. Under the stickbreaking model, the intrinsic fitness effects of mutations scale by the distance of the current background to a hypothesized boundary. We use simulations and theoretical analyses to explore the basic properties of the stickbreaking model such as fitness trajectories, the distribution of fitness achieved, and epistasis. Stickbreaking is compared to the additive and multiplicative models. We conclude that the stickbreaking model is qualitatively consistent with several commonly observed patterns of adaptive evolution.

Adaptive walks and extreme value theory

We study biological evolution in a high-dimensional genotype space in the regime of rare mutations and strong selection. The population performs an uphill walk which terminates at local fitness maxima. Assigning fitness randomly to genotypes, we show that the mean walk length is logarithmic in the number of initially available beneficial mutations, with a prefactor determined by the tail of the fitness distribution. This result is derived analytically in a simplified setting where the mutational neighborhood is fixed during the adaptive process, and confirmed by numerical simulations.

Multiple adaptive substitutions during evolution in novel environments

We consider an asexual population under strong selection-weak mutation conditions evolving on rugged fitness landscapes with many local fitness peaks. Unlike the previous studies in which the initial fitness of the population is assumed to be high, here we start the adaptation process with a low fitness corresponding to a population in a stressful novel environment. For generic fitness distributions, using an analytic argument we find that the average number of steps to a local optimum varies logarithmically with the genotype sequence length and increases as the correlations among genotypic fitnesses increase. When the fitnesses are exponentially or uniformly distributed, using an evolution equation for the distribution of population fitness, we analytically calculate the fitness distribution of fixed beneficial mutations and the walk length distribution.