The concept of a fitness landscape is a powerful metaphor that offers 10 insight into various aspects of evolutionary processes and guidance for the study of 11 evolution. Until recently, empirical evidence on the ruggedness of these landscapes 12 was lacking, but since it became feasible to construct all possible genotypes containing 13 combinations of a limited set of mutations, the number of studies has grown to a 14 point where a classification of landscapes becomes possible. The aim of this review 15 is to identify measures of epistasis that allow a meaningful comparison of fitness 16 landscapes and then apply them to the empirical landscapes to discern factors that 17 affect ruggedness. The various measures of epistasis that have been proposed in the 18 literature appear to be equivalent. Our comparison shows that the ruggedness of 19 the empirical landscape is affected by whether the included mutations are beneficial 20 or deleterious and by whether intra-or intergenic epistasis is involved. Finally, the 21 empirical landscapes are compared to landscapes generated with the Rough Mt. Fuji 22 model. Despite the simplicity of this model, it captures the features of the experimental 23 landscapes remarkably well.

To gauge the relative importance of contingency and determinism in evolution is a fundamental problem that continues to motivate much theoretical and empirical research. In recent evolution experiments with microbes, this question has been explored by monitoring the repeatability of adaptive changes in replicate populations. Here, we present the results of an extensive computational study of evolutionary predictability based on an experimentally measured eight-locus fitness landscape for the filamentous fungus Aspergillus niger. To quantify predictability, we define entropy measures on observed mutational trajectories and endpoints. In contrast to the common expectation of increasingly deterministic evolution in large populations, we find that these entropies display an initial decrease and a subsequent increase with population size N, governed, respectively, by the scales Nμ and Nμ 2 , corresponding to the supply rates of single and double mutations, where μ denotes the mutation rate. The amplitude of this pattern is determined by μ. We show that these observations are generic by comparing our findings for the experimental fitness landscape to simulations on simple model landscapes.clonal interference | epistasis | experimental evolution E volutionary adaptations arise from an intricate interplay of deterministic selective forces and random reproductive or mutational events, and the relative roles of these two types of influences on the outcome of evolution has been subject to longstanding controversy with significant philosophical implications (1, 2). Although the vision of "replaying the tape of life" on Earth or on some extrasolar planet remains confined to the realm of imagination (3, 4), evolution experiments with microbial populations have begun to address predictability of adaptation on a microevolutionary scale (5-9). In particular, strong signatures of parallel evolution have been observed in the context of the evolution of antibiotic resistance in pathogens, a finding that is of direct relevance to strategies of drug design and deployment (10-14). As lack of knowledge of crucial parameters (e.g., the frequency of beneficial mutations) in such experiments prevents forward predictions, predictability is used in a weaker, a posteriori sense implying repeatability of evolutionary trajectories in replicate populations. For this reason, the two terms will often be used interchangeably in the following (15).The repeatability of adaptive trajectories is expected to depend on the genetic constraints imposed by epistatic interactions as well as on parameters such as population size N, mutation rate μ, and the typical scale s of selection coefficients (16-18). To be specific, consider a population evolving in the regime of strong selection and weak mutation (SSWM), where mutations are so rare that normally not more than one mutant is present simultaneously and the population can be represented as a single entity that performs an adaptive walk in the space of genotypes (19)(20)(21). Such walks are constrained to ...

Understanding epistasis is central to biology. For instance, epistatic interactions determine the topography of the fitness landscape and affect the dynamics and determinism of adaptation. However, few empirical data are available, and comparing results is complicated by confounding variation in the system and the type of mutations used. Here, we take a systematic approach by quantifying epistasis in two sets of four beneficial mutations in the antibiotic resistance enzyme TEM-1 β-lactamase. Mutations in these sets have either large or small effects on cefotaxime resistance when present as single mutations. By quantifying the epistasis and ruggedness in both landscapes, we find two general patterns. First, resistance is maximal for combinations of two mutations in both fitness landscapes and declines when more mutations are added due to abundant sign epistasis and a pattern of diminishing returns with genotype resistance. Second, large-effect mutations interact more strongly than small-effect mutations, suggesting that the effect size of mutations may be an organizing principle in understanding patterns of epistasis. By fitting the data to simple phenotype resistance models, we show that this pattern may be explained by the nonlinear dependence of resistance on enzyme stability and an unknown phenotype when mutations have antagonistically pleiotropic effects. The comparison to a previously published set of mutations in the same gene with a joint benefit further shows that the enzyme's fitness landscape is locally rugged but does contain adaptive pathways that lead to high resistance.

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.

We study Lyapunov vectors (LVs) corresponding to the largest Lyapunov exponents in systems with spatiotemporal chaos. We focus on characteristic LVs and compare the results with backward LVs obtained via successive Gram-Schmidt orthonormalizations. Systems of a very different nature such as coupled-map lattices and the (continuous-time) Lorenz '96 model exhibit the same features in quantitative and qualitative terms. Additionally, we propose a minimal stochastic model that reproduces the results for chaotic systems. Our work supports the claims about universality of our earlier results [I. G. Szendro, Phys. Rev. E 76, 025202(R) (2007)] for a specific coupled-map lattice.

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.T HE genetic adaptation of an asexual population to a novel environment is governed by the number and fitness effects of available beneficial mutations, their epistatic interactions, and the rate at which they are supplied (Sniegowski and Gerrish 2010). Despite the inherent complexity of this process, recent theoretical work has identified several robust statistical patterns of adaptive evolution (Orr 2005a,b). Most of these predictions were derived in the framework of Gillespie's mutational landscape model (MLM), which is based on three key assumptions (Gillespie 1983(Gillespie , 1984(Gillespie , 1991Orr 2002). First, selection is strong enough to prevent the fixation of deleterious mutations and mutation is sufficiently weak such that mutations emerge and fix one at a time [the strong selection/weak mutation (SSWM) regime]. Second, wild-type fitness is high, which allows one to describe the statistics of beneficial mutations using extreme value theory (EVT). Third, the fitness values of new mutants are uncorrelated with the fitness of the ancestor from which they arise. This last assumption implies that the fitness landscape underlying the adaptive process is maximally rugged with many local maxima and minima (Kauffman and Levin 1987;Kauffman 1993;Jain and Krug 2007), a limiting situation that is often referred to as the house of cards (HoC) landscape (Kingman 1978). Thus, the MLM is concerned with a population evolving in a HoC landscape under SSWM dynamics, starting from an initial state of high fitness.The validity of the SSWM assumption depends primarily on the population size N. Denoting the mutation rate by u and the typical selection strength by s, the criterion for the SSWM regime reads Nu ( 1 ( Ns, which can always be satisfied by a suitable choice o...

Starting from fitness correlation functions, we calculate exact expressions for the amplitude spectra of fitness landscapes as defined by Stadler [1996. Landscapes and their correlation functions. J. Math. Chem. 20, 1] 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.

The spatiotemporal dynamics of Lyapunov vectors ͑LVs͒ in spatially extended chaotic systems is studied by means of coupled-map lattices. We determine intrinsic length scales and spatiotemporal correlations of LVs corresponding to the leading unstable directions by translating the problem to the language of scale-invariant growing surfaces. We find that the so-called characteristic LVs exhibit spatial localization, strong clustering around given spatiotemporal loci, and remarkable dynamic scaling properties of the corresponding surfaces. In contrast, the commonly used backward LVs ͑obtained through Gram-Schmidt orthogonalization͒ spread all over the system and do not exhibit dynamic scaling due to artifacts in the dynamical correlations by construction. Lyapunov exponents ͑LEs͒ measure the exponential separation ͑or convergence͒ of nearby trajectories and provide the most common tool to characterize spatiotemporal chaos ͑STC͒ ͓1-3͔. Not only exponential separation rates but also spatial correlations are crucial to deal with predictability questions in extended systems ͓4͔. Basically, ͑almost͒ any initial infinitesimal perturbation evolves in time asymptotically aligning along the most unstable direction, i.e., the main Lyapunov vector ͑LV͒. In extended systems, the main LV is found to be localized in space at any given time. However, due to spatial homogeneity, all directions in tangent space are actually equivalent and the localization center keeps moving all over the system. This is known as "dynamic localization" of the main LV ͓5,6͔.In the last decade, there has been some progress in the study of STC with tools borrowed from nonequilibrium statistical physics. In particular, the evolution of perturbations in spatially extended chaotic systems has been shown to be described by multiplicative Langevin-type equations ͓5-8͔. In many cases ͓6͔, the dynamics of perturbations can be expressed in terms of the prototypical stochastic surface growth equation of Kardar-Parisi-Zhang ͑KPZ͒, ‫ץ‬ t h = ‫ץ͑‬ x h͒ 2 + ‫ץ‬ xx h + , where is a white noise ͓9͔. In this Rapid Communication we study the spatiotemporal structure of STC encoded by the LVs. We show that a family of vectors-that we shall call characteristic LVs-carry important information about the real-space structure, localization properties, and space-time correlations. These properties are disclosed only after a logarithmic transformation, so that each LV is mapped into a rough surface. Our results greatly strengthen the link between STC and certain nonequilibrium surface growth models. We also find that the most widely used orthogonal LVs-that appear in the standard GramSchmidt procedure to compute the LEs ͓10,11͔-lack many of these features due to the construction procedure and, therefore, have much less physical significance.We illustrate our results by numerical simulations of coupled-map lattices ͑CMLs͒, which are a prototype of spatially extended dynamical systems exhibiting chaos. The main advantage of CMLs, as compared with e.g., partial differential equat...