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Biological evolution can be conceptualized as a search process in the space of gene sequences guided by the fitness landscape, a mapping that assigns a measure of reproductive value to each genotype. Here we discuss probabilistic models of fitness landscapes with a focus on their evolutionary accessibility, where a path in a fitness landscape is said to be accessible if the fitness values encountered along the path increase monotonically. For uncorrelated (random) landscapes with independent and identically distributed fitness values, the probability of existence of accessible paths between genotypes at a distance linear in the sequence length $L$ becomes nonzero at a nontrivial threshold value of the fitness difference between the initial and final genotype, which can be explicitly computed for large classes of genotype graphs. The behaviour in uncorrelated random landscapes is contrasted with landscape models that display additional, biologically motivated structural features. In particular, landscapes defined by a tradeoff between adaptation to environmental extremes have been found to display a combinatorially large number of accessible paths to all local fitness maxima. We show that this property is characteristic of a broad class of models that satisfy a certain global constraint, and provide further examples from this class., Comment: Accepted contribution to the STATPHYS 28 issue of JSTAT; 30 pages, 10 figures

The extent of parallel evolution at the genotypic level is quantitatively linked to the distribution of beneficial fitness effects (DBFE) of mutations. The standard view, based on light-tailed distributions (i.e. distributions with finite moments), is that the probability of parallel evolution in duplicate populations is inversely proportional to the number of available mutations, and moreover that the DBFE is sufficient to determine the probability when the number of available mutations is large. Here we show that when the DBFE is heavy-tailed, as found in several recent experiments, these expectations are defied. The probability of parallel evolution decays anomalously slowly in the number of mutations or even becomes independent of it, implying higher repeatability of evolution. At the same time, the probability of parallel evolution is non-self-averaging, that is, it does not converge to its mean value even when a large number of mutations are involved. This behavior arises because the evolutionary process is dominated by only a few mutations of high weight. Consequently, the probability varies widely across systems with the same DBFE. Contrary to the standard view, the DBFE is no longer sufficient to determine the extent of parallel evolution, making it much less predictable. We illustrate these ideas theoretically and through analysis of empirical data on antibiotic resistance evolution.

A significant proportion of the infections driving the current {SARS-CoV-2} pandemic are transmitted asymptomatically. Here we introduce and study a simple epidemic model with separate compartments comprising asymptomatic and symptomatic infected individuals. The linear dynamics determining the outbreak condition of the model is equivalent to a renewal theory approach with exponential waiting time distributions. Exploiting a nontrivial conservation law of the full nonlinear dynamics, we derive analytic bounds on the peak number of infections in the absence and presence of mitigation through isolation and testing. The bounds are compared to numerical solutions of the differential equations., Comment: 15 pages, 4 figures

Biological evolution of a population is governed by the fitness landscape, which is a map from genotype to fitness. However, a fitness landscape depends on the organisms environment, and evolution in changing environments is still poorly understood. We study a particular model of antibiotic resistance evolution in bacteria where the antibiotic concentration is an environmental parameter and the fitness landscapes incorporate tradeoffs between adaptation to low and high antibiotic concentration. With evolutionary dynamics that follow fitness gradients, the evolution of the system under slowly changing antibiotic concentration resembles the athermal dynamics of disordered physical systems under external drives. Exploiting this resemblance, we show that our model can be described as a system with interacting hysteretic elements. As in the case of the driven disordered systems, adaptive evolution under antibiotic concentration cycling is found to exhibit hysteresis loops and memory formation. We derive a number of analytical results for quasistatic concentration changes. We also perform numerical simulations to study how these effects are modified under driving protocols in which the concentration is changed in discrete steps. Our approach provides a general framework for studying motifs of evolutionary dynamics in biological systems in a changing environment.

Darwinian evolution is driven by random mutations, genetic recombination (gene shuffling) and selection that favors genotypes with high fitness. For systems where each genotype can be represented as a bitstring of length $L$, an overview of possible evolutionary trajectories is provided by the $L$-cube graph with nodes labeled by genotypes and edges directed toward the genotype with higher fitness. Peaks (sinks in the graphs) are important since a population can get stranded at a suboptimal peak. The fitness landscape is defined by the fitness values of all genotypes in the system. Some notion of curvature is necessary for a more complete analysis of the landscapes, including the effect of recombination. The shape approach uses triangulations (shapes) induced by fitness landscapes. The main topic for this work is the interplay between peak patterns and shapes. Because of constraints on the shapes for $L=3$ imposed by peaks, there are in total 25 possible combinations of peak patterns and shapes. Similar constraints exist for higher $L$. Specifically, we show that the constraints induced by the staircase triangulation can be formulated as a condition of {\emph{universal positive epistasis}}, an order relation on the fitness effects of arbitrary sets of mutations that respects the inclusion relation between the corresponding genetic backgrounds. We apply the concept to a large protein fitness landscape for an immunoglobulin-binding protein expressed in Streptococcal bacteria., Comment: 25 pages, 6 figures

Fisher's geometric model describes biological fitness landscapes by combining a linear map from the discrete space of genotypes to an $n$-dimensional Euclidean phenotype space with a nonlinear, single-peaked phenotype-fitness map. Genotypes are represented by binary sequences of length $L$, and the phenotypic effects of mutations at different sites are represented by $L$ random vectors drawn from an isotropic Gaussian distribution. Recent work has shown that the interplay between the genotypic and phenotypic levels gives rise to a range of different landscape topographies that can be characterised by the number of local fitness maxima. Extending our previous study of the mean number of local maxima, here we focus on the distribution of the number of maxima when the limit $L \to \infty$ is taken at finite $n$. We identify the typical scale of the number of maxima for general $n$, and determine the full scaled probability density and two point correlation function of maxima for the one-dimensional case. We also elaborate on the close relation of the model to the anti-ferromagnetic Hopfield model with $n$ random continuous pattern vectors, and show that many of our results carry over to this setting. More generally, we expect that our analysis can help to elucidate the fluctuation structure of metastable states in various spin glass problems., Comment: 36 pages, 5 figures. Minor corrections

Understanding how genotypes map onto phenotypes, fitness, and eventually organisms is arguably the next major missing piece in a fully predictive theory of evolution. We refer to this generally as the problem of the genotype-phenotype map. Though we are still far from achieving a complete picture of these relationships, our current understanding of simpler questions, such as the structure induced in the space of genotypes by sequences mapped to molecular structures, has revealed important facts that deeply affect the dynamical description of evolutionary processes. Empirical evidence supporting the fundamental relevance of features such as phenotypic bias is mounting as well, while the synthesis of conceptual and experimental progress leads to questioning current assumptions on the nature of evolutionary dynamics-cancer progression models or synthetic biology approaches being notable examples. This work delves into a critical and constructive attitude in our current knowledge of how genotypes map onto molecular phenotypes and organismal functions, and discusses theoretical and empirical avenues to broaden and improve this comprehension. As a final goal, this community should aim at deriving an updated picture of evolutionary processes soundly relying on the structural properties of genotype spaces, as revealed by modern techniques of molecular and functional analysis., Comment: 111 pages, 11 figures uses elsarticle latex class

A fitness landscape is a mapping from a space of discrete genotypes to the real numbers. A path in a fitness landscape is a sequence of genotypes connected by single mutational steps. Such a path is said to be accessible if the fitness values of the genotypes encountered along the path increase monotonically. We study accessible paths on random fitness landscapes of the House-of-Cards type, on which fitness values are independent, identically and continuously distributed random variables. The genotype space is taken to be a Cartesian power graph $\mathcal{A}^L$, where $L$ is the number of genetic loci and the allele graph $\mathcal{A}$ encodes the possible allelic states and mutational transitions on one locus. The probability of existence of accessible paths between two genotypes at a distance linear in $L$ displays a transition from 0 to a positive value at a threshold $\beta_c$ for the fitness difference between the initial and final genotype. We derive a lower bound on $\beta_c$ for general $\mathcal{A}$ and show that this bound is tight for a large class of allele graphs. Our results generalize previous results for accessibility percolation on the biallelic hypercube, and compare favorably to published numerical results for multiallelic Hamming graphs., Comment: 43 pages, 9 figures, 2 tables

The fitness landscape encodes the mapping of genotypes to fitness and provides a succinct representation of possible trajectories followed by an evolving population. Evolutionary accessibility is quantified by the existence of fitness-monotonic paths connecting far away genotypes. Studies of accessibility percolation use probabilistic fitness landscape models to explore the emergence of such paths as a function of the initial fitness, the parameters of the landscape or the structure of the genotype graph. This chapter reviews these studies and discusses their implications for the predictability of evolutionary processes., Comment: 20 pages, 3 figures

Mutational robustness quantifies the effect of random mutations on fitness. When mutational robustness is high, most mutations do not change fitness or have only a minor effect on it. From the point of view of fitness landscapes, robust genotypes form neutral networks of almost equal fitness. Using deterministic population models it has been shown that selection favors genotypes inside such networks, which results in increased mutational robustness. Here we demonstrate that this effect is massively enhanced by recombination. Our results are based on a detailed analysis of mesa-shaped fitness landscapes, where we derive precise expressions for the dependence of the robustness on the landscape parameters for recombining and non-recombining populations. In addition, we carry out numerical simulations on different types of random holey landscapes as well as on an empirical fitness landscape. We show that the mutational robustness of a genotype generally correlates with its recombination weight, a new measure that quantifies the likelihood for the genotype to arise from recombination. We argue that the favorable effect of recombination on mutational robustness is a highly universal feature that may have played an important role in the emergence and maintenance of mechanisms of genetic exchange., Comment: 15 figures, Supplementary appendix, supplementary figures

Muller's ratchet describes the irreversible accumulation of deleterious mutations in asexual populations. In well-mixed populations the speed of fitness decline is exponentially small in the population size, and any positive rate of beneficial mutations is sufficient to reverse the ratchet in large populations. The behavior is fundamentally different in populations with spatial structure, because the speed of the ratchet remains nonzero in the infinite size limit when the deleterious mutation rate exceeds a critical value. Based on the relation between the spatial ratchet and directed percolation, we develop a scaling theory incorporating both deleterious and beneficial mutations. The theory is verified by extensive simulations in one and two dimensions., Comment: 7 pages, 6 figures

These lecture notes introduce key concepts of mathematical population genetics within the most elementary setting and describe a few recent applications to microbial evolution experiments. Pointers to the literature for further reading are provided, and some of the derivations are left as exercises for the reader., Comment: 18 pages, 7 figures

Kauffman's NK-model is a paradigmatic example of a class of stochastic models of genotypic fitness landscapes that aim to capture generic features of epistatic interactions in multilocus systems. Genotypes are represented as sequences of $L$ binary loci. The fitness assigned to a genotype is a sum of contributions, each of which is a random function defined on a subset of $k \le L$ loci. These subsets or neighborhoods determine the genetic interactions of the model. Whereas earlier work on the NK model suggested that most of its properties are robust with regard to the choice of neighborhoods, recent work has revealed an important and sometimes counter-intuitive influence of the interaction structure on the properties of NK fitness landscapes. Here we review these developments and present new results concerning the number of local fitness maxima and the statistics of selectively accessible (that is, fitness-monotonic) mutational pathways. In particular, we develop a unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of $L$, and identify two different universality classes of interaction structures that display different asymptotics of this quantity for large $k$. Moreover, we show that the probability that the fitness landscape can be traversed along an accessible path decreases exponentially in $L$ for a large class of interaction structures that we characterize as locally bounded. Finally, we discuss the impact of the NK interaction structures on the dynamics of evolution using adaptive walk models., Comment: 61 pages, 9 figures

Whether evolution can be predicted is a key question in evolutionary biology. Here we set out to better understand the repeatability of evolution. We explored experimentally the effect of mutation supply and the strength of selective pressure on the repeatability of selection from standing genetic variation. Different sizes of mutant libraries of an antibiotic resistance gene, TEM-1 $\beta$-lactamase in Escherichia coli, were subjected to different antibiotic concentrations. We determined whether populations went extinct or survived, and sequenced the TEM gene of the surviving populations. The distribution of mutations per allele in our mutant libraries- generated by error-prone PCR- followed a Poisson distribution. Extinction patterns could be explained by a simple stochastic model that assumed the sampling of beneficial mutations was key for survival. In most surviving populations, alleles containing at least one known large-effect beneficial mutation were present. These genotype data also support a model which only invokes sampling effects to describe the occurrence of alleles containing large-effect driver mutations. Hence, evolution is largely predictable given cursory knowledge of mutational fitness effects, the mutation rate and population size. There were no clear trends in the repeatability of selected mutants when we considered all mutations present. However, when only known large-effect mutations were considered, the outcome of selection is less repeatable for large libraries, in contrast to expectations. Furthermore, we show experimentally that alleles carrying multiple mutations selected from large libraries confer higher resistance levels relative to alleles with only a known large-effect mutation, suggesting that the scarcity of high-resistance alleles carrying multiple mutations may contribute to the decrease in repeatability at large library sizes., Comment: 31pages, 9 figures

Fisher's geometric model was originally introduced to argue that complex adaptations must occur in small steps because of pleiotropic constraints. When supplemented with the assumption of additivity of mutational effects on phenotypic traits, it provides a simple mechanism for the emergence of genotypic epistasis from the nonlinear mapping of phenotypes to fitness. Of particular interest is the occurrence of reciprocal sign epistasis, which is a necessary condition for multipeaked genotypic fitness landscapes. Here we compute the probability that a pair of randomly chosen mutations interacts sign epistatically, which is found to decrease with increasing phenotypic dimension $n$, and varies nonmonotonically with the distance from the phenotypic optimum. We then derive expressions for the mean number of fitness maxima in genotypic landscapes comprised of all combinations of $L$ random mutations. This number increases exponentially with $L$, and the corresponding growth rate is used as a measure of the complexity of the landscape. The dependence of the complexity on the model parameters is found to be surprisingly rich, and three distinct phases characterized by different landscape structures are identified. Our analysis shows that the phenotypic dimension, which is often referred to as phenotypic complexity, does not generally correlate with the complexity of fitness landscapes and that even organisms with a single phenotypic trait can have complex landscapes. Our results further inform the interpretation of experiments where the parameters of Fisher's model have been inferred from data, and help to elucidate which features of empirical fitness landscapes can be described by this model., Comment: 27 pages, 14 figures, 2 tables, minor changes (published version)

We study a modified record process where the $k$'th record in a series of independent and identically distributed random variables is defined recursively through the condition $Y_k > Y_{k-1} - \delta_{k-1}$ with a deterministic sequence $\delta_k > 0$ called the handicap. For constant $\delta_k \equiv \delta$ and exponentially distributed random variables it has been shown in previous work that the process displays a phase transition as a function of $\delta$ between a normal phase where the mean record value increases indefinitely and a stationary phase where the mean record value remains bounded and a finite fraction of all entries are records (Park \textit{et al} 2015 {\it Phys. Rev.} E \textbf{91} 042707). Here we explore the behavior for general probability distributions and decreasing and increasing sequences $\delta_k$, focusing in particular on the case when $\delta_k$ matches the typical spacing between subsequent records in the underlying simple record process without handicap. We find that a continuous phase transition occurs only in the exponential case, but a novel kind of first order transition emerges when $\delta_k$ is increasing. The problem is partly motivated by the dynamics of evolutionary adaptation in biological fitness landscapes, where $\delta_k$ corresponds to the change of the deterministic fitness component after $k$ mutational steps. The results for the record process are used to compute the mean number of steps that a population performs in such a landscape before being trapped at a local fitness maximum., Comment: minor changes. Published

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 ($c \to \infty$). 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 $L \to \infty$, different from its values for $c=0$ and $c = \infty$, if $c$ is scaled with $L$ in an appropriate combination., Comment: minor changes

Fitness landscapes are genotype to fitness mappings commonly used in evolutionary biology and computer science which are closely related to spin glass models. In this paper, we study the NK model for fitness landscapes where the interaction scheme between genes can be explicitly defined. The focus is on how this scheme influences the overall shape of the landscape. Our main tool for the analysis are adaptive walks, an idealized dynamics by which the population moves uphill in fitness and terminates at a local fitness maximum. We use three different types of walks and investigate how their length (the number of steps required to reach a local peak) and height (the fitness at the endpoint of the walk) depend on the dimensionality and structure of the landscape. We find that the distribution of local maxima over the landscape is particularly sensitive to the choice of interaction pattern. Most quantities that we measure are simply correlated to the rank of the scheme, which is equal to the number of nonzero coefficients in the expansion of the fitness landscape in terms of Walsh functions., Comment: 29 pages, 9 figures

We study biological evolution on a random fitness landscape where correlations are introduced through a linear fitness gradient of strength $c$. When selection is strong and mutations rare the dynamics is a directed uphill walk that terminates at a local fitness maximum. We analytically calculate the dependence of the walk length on the genome size $L$. When the distribution of the random fitness component has an exponential tail we find a phase transition of the walk length $D$ between a phase at small $c$ where walks are short $(D \sim \ln L)$ and a phase at large $c$ where walks are long $(D \sim L)$. For all other distributions only a single phase exists for any $c > 0$. The considered process is equivalent to a zero temperature Metropolis dynamics for the random energy model in an external magnetic field, thus also providing insight into the aging dynamics of spin glasses.

Identifying and quantifying the benefits of sex and recombination is a long standing problem in evolutionary theory. In particular, contradictory claims have been made about the existence of a benefit of recombination on high dimensional fitness landscapes in the presence of sign epistasis. Here we present a comparative numerical study of sexual and asexual evolutionary dynamics of haploids on tunably rugged model landscapes under strong selection, paying special attention to the temporal development of the evolutionary advantage of recombination and the link between population diversity and the rate of adaptation. We show that the adaptive advantage of recombination on static rugged landscapes is strictly transitory. At early times, an advantage of recombination arises through the possibility to combine individually occurring beneficial mutations, but this effect is reversed at longer times by the much more efficient trapping of recombining populations at local fitness peaks. These findings are explained by means of well established results for a setup with only two loci. In accordance with the Red Queen hypothesis the transitory advantage can be prolonged indefinitely in fluctuating environments, and it is maximal when the environment fluctuates on the same time scale on which trapping at local optima typically occurs., Comment: 34 pages, 9 figures and 8 supplementary figures; revised and final version

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 analyse a simple fitness landscape model with tunable ruggedness based on the Rough Mount Fuji (RMF) model originally introduced by Aita et al. [Biopolymers 54:64-79 (2000)] in the context of protein evolution. We provide a comprehensive collection of results pertaining to the topographical structure of RMF landscapes, including explicit formulae 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 backgound in a recent evolution experiment with a microvirid bacteriophage [Miller et al., Genetics 187:185-202 (2011)]., Comment: 43 pages, 12 figures; revised version with new results on the number of fitness maxima

A fitness landscape is a mapping from the space of genetic sequences, which is modeled here as a binary hypercube of dimension $L$, to the real numbers. We consider random models of fitness landscapes, where fitness values are assigned according to some probabilistic rule, and study the statistical properties of pathways to the global fitness maximum along which fitness increases monotonically. Such paths are important for evolution because they are the only ones that are accessible to an adapting population when mutations occur at a low rate. The focus of this work is on the block model introduced by A.S. Perelson and C.A. Macken [Proc. Natl. Acad. Sci. USA 92:9657 (1995)] where the genome is decomposed into disjoint sets of loci (`modules') that contribute independently to fitness, and fitness values within blocks are assigned at random. We show that the number of accessible paths can be written as a product of the path numbers within the blocks, which provides a detailed analytic description of the path statistics. The block model can be viewed as a special case of Kauffman's NK-model, and we compare the analytic results to simulations of the NK-model with different genetic architectures. We find that the mean number of accessible paths in the different versions of the model are quite similar, but the distribution of the path number is qualitatively different in the block model due to its multiplicative structure. A similar statement applies to the number of local fitness maxima in the NK-models, which has been studied extensively in previous works. The overall evolutionary accessibility of the landscape, as quantified by the probability to find at least one accessible path to the global maximum, is dramatically lowered by the modular structure., Comment: 26 pages, 12 figures; final version with some typos corrected

The adaptation of large asexual populations is hampered by the competition between independently arising beneficial mutations in different individuals, which is known as clonal interference. Fisher and Muller proposed that recombination provides an evolutionary advantage in large populations by alleviating this competition. Based on recent progress in quantifying the speed of adaptation in asexual populations undergoing clonal interference, we present a detailed analysis of the Fisher-Muller mechanism for a model genome consisting of two loci with an infinite number of beneficial alleles each and multiplicative fitness effects. We solve the infinite population dynamics exactly and show that, for a particular, natural mutation scheme, the speed of adaptation in sexuals is twice as large as in asexuals. Guided by the infinite population result and by previous work on asexual adaptation, we postulate an expression for the speed of adaptation in finite sexual populations that agrees with numerical simulations over a wide range of population sizes and recombination rates. The ratio of the sexual to asexual adaptation speed is a function of population size that increases in the clonal interference regime and approaches 2 for extremely large populations. The simulations also show that the imbalance between the numbers of accumulated mutations at the two loci is strongly suppressed even by a small amount of recombination. The generalization of the model to an arbitrary number $L$ of loci is briefly discussed. If each offspring samples the alleles at each locus from the gene pool of the whole population rather than from two parents, the ratio of the sexual to asexual adaptation speed is approximately equal to $L$ in large populations. A possible realization of this scenario is the reassortment of genetic material in RNA viruses with $L$ genomic segments., Comment: Title has been changed. Supporting Information (animation) can be found in the source file. 53 pages. 10 figures. To appear in Genetics

Competition between independently arising beneficial mutations is enhanced in spatial populations due to the linear rather than exponential growth of clones. Recent theoretical studies have pointed out that the resulting fitness dynamics is analogous to a surface growth process, where new layers nucleate and spread stochastically, leading to the build up of scale-invariant roughness. This scenario differs qualitatively from the standard view of adaptation in that the speed of adaptation becomes independent of population size while the fitness variance does not. Here we exploit recent progress in the understanding of surface growth processes to obtain precise predictions for the universal, non-Gaussian shape of the fitness distribution for one-dimensional habitats, which are verified by simulations. When the mutations are deleterious rather than beneficial the problem becomes a spatial version of Muller's ratchet. In contrast to the case of well-mixed populations, the rate of fitness decline remains finite even in the limit of an infinite habitat, provided the ratio $U_d/s^2$ between the deleterious mutation rate and the square of the (negative) selection coefficient is sufficiently large. Using again an analogy to surface growth models we show that the transition between the stationary and the moving state of the ratchet is governed by directed percolation.

Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. To each vertex of a graph a random number is assigned and a path through the graph is called accessible if all numbers along the path are in ascending order. For the case when the random variables are independent and identically distributed, we derive an asymptotically exact expression for the probability that there is at least one accessible path from the root to the leaves in an $n$-tree. This probability tends to 1 (0) if the branching number is increased with the height of the tree faster (slower) than linearly. When the random variables are biased such that the mean value increases linearly with the distance from the root, a percolation threshold emerges at a finite value of the bias., Comment: 6 pages, 4 figures

Starting from fitness correlation functions, we calculate exact expressions for the amplitude spectra of fitness landscapes as defined by P.F. Stadler [J. Math. Chem. 20, 1 (1996)] for common landscape models, including Kauffman's NK-model, rough Mount Fuji landscapes and general linear superpositions of such landscapes. We further show that correlations decaying exponentially with the Hamming distance yield exponentially decaying spectra similar to those reported recently for a model of molecular signal transduction. Finally, we compare our results for the model systems to the spectra of various experimentally measured fitness landscapes. We claim that our analytical results should be helpful when trying to interpret empirical data and guide the search for improved fitness landscape models., Comment: 13 pages, 5 figures; revised and final version

This is an extended abstract of lectures delivered at the 4th Warsaw Summer School on Statistical Physics in Kazimierz Dolny, June 25-July 2, 2011.

The adaptive evolution of a population under the influence of mutation and selection is strongly influenced by the structure of the underlying fitness landscape, which encodes the interactions between mutations at different genetic loci. Theoretical studies of such landscapes have been carried out for several decades, but only recently experimental fitness measurements encompassing all possible combinations of small sets of mutations have become available. The empirical studies have spawned new questions about the accessibility of optimal genotypes under natural selection. Depending on population dynamic parameters such as mutation rate and population size, evolutionary accessibility can be quantified through the statistics of accessible mutational pathways (along which fitness increases monotoncially), or through the study of the basin of attraction of the optimal genotype under greedy (steepest ascent) dynamics. Here we investigate these two measures of accessibility in the framework of Kauffman's $LK$-model, a paradigmatic family of random fitness landscapes with tunable ruggedness. The key parameter governing the strength of genetic interactions is the number $K$ of interaction partners of each of the $L$ sites in the genotype sequence. In general, accessibility increases with increasing genotype dimensionality $L$ and decreases with increasing number of interactions $K$. Remarkably, however, we find that some measures of accessibility behave non-monotonically as a function of $K$, indicating a special role of the most sparsely connected, non-trivial cases K=1 and 2. The relation between models for fitness landscapes and spin glasses is also addressed., Comment: 22 pages, 13 Figures

The concept of a fitness landscape is a powerful metaphor that offers insight into various aspects of evolutionary processes and guidance for the study of evolution. Until recently, empirical evidence on the ruggedness of these landscapes was lacking, but since it became feasible to construct all possible genotypes containing combinations of a limited set of mutations, the number of studies has grown to a point where a classification of landscapes becomes possible. The aim of this review is to identify measures of epistasis that allow a meaningful comparison of fitness landscapes and then apply them to the empirical landscapes to discern factors that affect ruggedness. The various measures of epistasis that have been proposed in the literature appear to be equivalent. Our comparison shows that the ruggedness of the empirical landscape is affected by whether the included mutations are beneficial or deleterious and by whether intra- or intergenic epistasis is involved. Finally, the empirical landscapes are compared to landscapes generated with the Rough Mt.\ Fuji model. Despite the simplicity of this model, it captures the features of the experimental landscapes remarkably well., Comment: 24 pages, 5 figures; to appear in Journal of Statistical Mechanics: Theory and Experiment

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., Comment: 4 pages, 2 figures; final version, to appear in Physical Review Letters

Functional effects of different mutations are known to combine to the total effect in highly nontrivial ways. For the trait under evolutionary selection (`fitness'), measured values over all possible combinations of a set of mutations yield a fitness landscape that determines which mutational states can be reached from a given initial genotype. Understanding the accessibility properties of fitness landscapes is conceptually important in answering questions about the predictability and repeatability of evolutionary adaptation. Here we theoretically investigate accessibility of the globally optimal state on a wide variety of model landscapes, including landscapes with tunable ruggedness as well as neutral `holey' landscapes. We define a mutational pathway to be accessible if it contains the minimal number of mutations required to reach the target genotype, and if fitness increases in each mutational step. Under this definition accessibility is high, in the sense that at least one accessible pathwayexists with a substantial probability that approaches unity as the dimensionality of the fitness landscape (set by the number of mutational loci) becomes large. At the same time the number of alternative accessible pathways grows without bound. We test the model predictions against an empirical 8-locus fitness landscape obtained for the filamentous fungus \textit{Aspergillus niger}. By analyzing subgraphs of the full landscape containing different subsets of mutations, we are able to probe the mutational distance scale in the empirical data. The predicted effect of high accessibility is supported by the empirical data and very robust, which we argue to reflect the generic topology of sequence spaces., Comment: 16 pages, 4 figures; supplementary material available on request

We study the evolution of a population in a two-locus genotype space, in which the negative effects of two single mutations are overcompensated in a high fitness double mutant. We discuss how the interplay of finite population size, $N$, and sexual recombination at rate $r$ affects the escape times $t_\mathrm{esc}$ to the double mutant. For small populations demographic noise generates massive fluctuations in $t_\mathrm{esc}$. The mean escape time varies non-monotonically with $r$, and grows exponentially as $\ln t_{\mathrm{esc}} \sim N(r - r^\ast)^{3/2}$ beyond a critical value $r^\ast$., Comment: 4 pages, 3 figures

Background: Recent experimental and theoretical studies have shown that small asexual populations evolving on complex fitness landscapes may achieve a higher fitness than large ones due to the increased heterogeneity of adaptive trajectories. Here we introduce a class of haploid three-locus fitness landscapes that allows to investigate this scenario in a precise and quantitative way. Results: Our main result derived analytically shows how the probability of choosing the path of largest initial fitness increase grows with the population size. This makes large populations more likely to get trapped at local fitness peaks and implies an advantage of small populations at intermediate time scales. The range of population sizes where this effect is operative coincides with the onset of clonal interference. Additional studies using ensembles of random fitness landscapes show that the results achieved for a particular choice of three-locus landscape parameters are robust and also persist as the number of loci increases. Conclusions: Our study indicates that an advantage for small populations is likely whenever the fitness landscape contains local maxima. The advantage appears at intermediate time scales, which are long enough for trapping at local fitness maxima to have occurred but too short for peak escape by the creation of multiple mutants., Comment: Version to appear in Evolution

The evolutionary effect of recombination depends crucially on the epistatic interactions between linked loci. A paradigmatic case where recombination is known to be strongly disadvantageous is a two-locus fitness landscape dis- playing reciprocal sign epistasis with two fitness peaks of unequal height. Focusing on the occurrence of bistability in the equilibrium solutions, we consider here the deterministic, haploid two-locus model with reversible mu- tations, selection and recombination. We find analytic formulae for the criti- cal recombination probability rc above which two stable stationary solutions appear which are localized on each of the two fitness peaks. We also derive the stationary genotype frequencies in various parameter regimes. When the recombination rate is close to rc and the fitness difference between the two peaks is small, we obtain a compact description in terms of a cubic polyno- mial which is analogous to the Landau theory of physical phase transitions., Comment: Sections 3.7 and 3.8 are added. 22 pages, 6 figures. To appear in J. Math. Biol

We consider an asexual biological population of constant size $N$ evolving in discrete time under the influence of selection and mutation. Beneficial mutations appear at rate $U$ and their selective effects $s$ are drawn from a distribution $g(s)$. After introducing the required models and concepts of mathematical population genetics, we review different approaches to computing the speed of logarithmic fitness increase as a function of $N$, $U$ and $g(s)$. We present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid. We then discuss approximate approaches to the finite population problem, distinguishing between the case of a single selection coefficient, $g(s) = \delta(s - s_b)$, and a continuous distribution of selection coefficients. Analytic estimates for the speed are compared to numerical simulations up to population sizes of order $10^{300}$., Comment: 33 pages, 10 figures

We investigate the fitness advantage associated with the robustness of a phenotype against deleterious mutations using deterministic mutation-selection models of quasispecies type equipped with a mesa shaped fitness landscape. We obtain analytic results for the robustness effect which become exact in the limit of infinite sequence length. Thereby, we are able to clarify a seeming contradiction between recent rigorous work and an earlier heuristic treatment based on a mapping to a Schr\"odinger equation. We exploit the quantum mechanical analogy to calculate a correction term for finite sequence lengths and verify our analytic results by numerical studies. In addition, we investigate the occurrence of an error threshold for a general class of epistatic landscape and show that diminishing epistasis is a necessary but not sufficient condition for error threshold behavior., Comment: 20 pages, 14 figures

The nature of epistasis has important consequences for the evolutionary significance of sex and recombination. Recent efforts to find negative epistasis as source of negative linkage disequilibrium and associated long-term sex advantage have yielded little support. Sign epistasis, where the sign of the fitness effects of alleles varies across genetic backgrounds, is responsible for ruggedness of the fitness landscape with implications for the evolution of sex that have been largely unexplored. Here, we describe fitness landscapes for two sets of strains of the asexual fungus \emph{Aspergillus niger} involving all combinations of five mutations. We find that $\sim 30$% of the single-mutation fitness effects are positive despite their negative effect in the wild-type strain, and that several local fitness maxima and minima are present. We then compare adaptation of sexual and asexual populations on these empirical fitness landscapes using simulations. The results show a general disadvantage of sex on these rugged landscapes, caused by the break down by recombination of genotypes escaping from local peaks. Sex facilitates escape from a local peak only for some parameter values on one landscape, indicating its dependence on the landscape's topography. We discuss possible reasons for the discrepancy between our results and the reports of faster adaptation of sexual populations.

We consider the evolution of an asexually reproducing population in an uncorrelated random fitness landscape in the limit of infinite genome size, which implies that each mutation generates a new fitness value drawn from a probability distribution $g(w)$. This is the finite population version of Kingman's house of cards model [J.F.C. Kingman, \textit{J. Appl. Probab.} \textbf{15}, 1 (1978)]. In contrast to Kingman's work, the focus here is on unbounded distributions $g(w)$ which lead to an indefinite growth of the population fitness. The model is solved analytically in the limit of infinite population size $N \to \infty$ and simulated numerically for finite $N$. When the genome-wide mutation probability $U$ is small, the long time behavior of the model reduces to a point process of fixation events, which is referred to as a \textit{diluted record process} (DRP). The DRP is similar to the standard record process except that a new record candidate (a number that exceeds all previous entries in the sequence) is accepted only with a certain probability that depends on the values of the current record and the candidate. We develop a systematic analytic approximation scheme for the DRP. At finite $U$ the fitness frequency distribution of the population decomposes into a stationary part due to mutations and a traveling wave component due to selection, which is shown to imply a reduction of the mean fitness by a factor of $1-U$ compared to the $U \to 0$ limit., Comment: Dedicated to Thomas Nattermann on the occasion of his 60th birthday. Submitted to JSTAT. Error in Section 3.2 was corrected

We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length $L$. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is deterministic or stochastic depends on the effective mutational distance $d_{\mathrm{eff}}$ upto which the population can spread in genotype space. For $d_{\mathrm{eff}}=L$, the deterministic quasispecies theory operates while for $d_{\mathrm{eff}} < 1$, the evolution is completely stochastic. Between these two limiting cases, the dynamics are described by a local quasispecies theory below a crossover time $T_{\times}$ while above $T_{\times}$, the population gets trapped at a local fitness peak and manages to find a better peak either via stochastic tunneling or double mutations. In the stochastic regime $d_\mathrm{eff} < 1$, we identify two subregimes associated with clonal interference and uphill adaptive walks, respectively. We argue that our findings are relevant to the interepretation of evolution experiments with microbial populations., Comment: Revised version, to appear in Genetics. Note on the role of selection in defining d_eff added; new figure 4 included

This is an introductory review of deterministic mutation-selection models for asexual populations (i.e., quasispecies theory) and related topics. First, the basic concepts of fitness, mutations, and sequence space are introduced. Different types of mutation-selection dynamics are defined and their relation to problems of statistical physics are outlined. Then the stationary population distribution in simple, single peak fitness landscapes is discussed at length, with particular emphasis on the error threshold phenomenon. Extensions of the theory covering e.g. epistatic interactions, diploid organisms, semiconservative replication and time-dependent fitness peaks are briefly described. A further section is devoted to randomly rugged fitness landscapes, which may display fitness correlations of various degree as well as extended neutral networks. The final two sections address evolutionary dynamics in both rugged and smooth fitness landscapes, and provide a brief overview of pertinent experiments., Comment: To appear in "Structural approaches to sequence evolution: Molecules, networks and populations", ed. by U. Bastolla, M. Porto, H.E. Roman and M. Vendruscolo

We consider the evolutionary trajectories traced out by an infinite population undergoing mutation-selection dynamics in static, uncorrelated random fitness landscapes. Starting from the population that consists of a single genotype, the most populated genotype \textit{jumps} from a local fitness maximum to another and eventually reaches the global maximum. We use a strong selection limit, which reduces the dynamics beyond the first time step to the competition between independent mutant subpopulations, to study the dynamics of this model and of a simpler one-dimensional model which ignores the geometry of the sequence space. We find that the fit genotypes that appear along a trajectory are a subset of suitably defined fitness \textit{records}, and exploit several results from the record theory for non-identically distributed random variables. The genotypes that contribute to the trajectory are those records that are not \textit{bypassed} by superior records arising further away from the initial population. Several conjectures concerning the statistics of bypassing are extracted from numerical simulations. In particular, for the one-dimensional model, we propose a simple relation between the bypassing probability and the dynamic exponent which describes the scaling of the typical evolution time with genome size. The latter can be determined exactly in terms of the extremal properties of the fitness distribution., Comment: Figures in color; minor revisions in text

We explore some aspects of the relationship between biological evolution processes and the mathematical theory of records. For Eigen's quasispecies model with an uncorrelated fitness landscape, we show that the evolutionary trajectories traced out by a population initially localized at a randomly chosen point in sequence space can be described in close analogy to record dynamics, with two complications. First, the increasing number of genotypes that become available with increasing distance from the starting point implies that fitness records are more frequent than for the standard case of independent, identically distributed random variables. Second, fitness records can be bypassed, which strongly reduces the number of genotypes that take part in an evolutionary trajectory. For exponential and Gaussian fitness distributions, this number scales with sequence length $N$ as $\sqrt{N}$, and it is of order unity for distributions with a power law tail. This is in strong contrast to the number of records, which is of order $N$ for any fitness distribution., Comment: Proceedings of 8th ICCMSP/Marrakech, to be published in Physica A