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.     

Evolutionary accessibility of mutational pathways

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 pathway exists 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 bounds. We test the model predictions against an empirical 8-locus fitness landscape obtained for the filamentous fungus 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 is very robust, which we argue reflects the generic topology of sequence spaces. Together with the restrictive assumptions that lie in our definition of accessibility, this implies that the globally optimal configuration should be accessible to genome wide evolution, but the repeatability of evolutionary trajectories is limited owing to the presence of a large number of alternative mutational pathways.     

Unsupervised Inference of Protein Fitness Landscape from Deep Mutational Scan

The recent technological advances underlying the screening of large combinatorial libraries in high-throughput mutational scans deepen our understanding of adaptive protein evolution and boost its applications in protein design. Nevertheless, the large number of possible genotypes requires suitable computational methods for data analysis, the prediction of mutational effects, and the generation of optimized sequences. We describe a computational method that, trained on sequencing samples from multiple rounds of a screening experiment, provides a model of the genotype–fitness relationship. We tested the method on five large-scale mutational scans, yielding accurate predictions of the mutational effects on fitness. The inferred fitness landscape is robust to experimental and sampling noise and exhibits high generalization power in terms of broader sequence space exploration and higher fitness variant predictions. We investigate the role of epistasis and show that the inferred model provides structural information about the 3D contacts in the molecular fold.     

Evolution in alternating environments with tunable interlandscape correlations

Natural populations are often exposed to temporally varying environments. Evolutionary dynamics in varying environments have been extensively studied, although understanding the effects of varying selection pressures remains challenging. Here, we investigate how cycling between a pair of statistically related fitness landscapes affects the evolved fitness of an asexually reproducing population. We construct pairs of fitness landscapes that share global fitness features but are correlated with one another in a tunable way, resulting in landscape pairs with specific correlations. We find that switching between these landscape pairs, depending on the ruggedness of the landscape and the interlandscape correlation, can either increase or decrease steady‐state fitness relative to evolution in single environments. In addition, we show that switching between rugged landscapes often selects for increased fitness in both landscapes, even in situations where the landscapes themselves are anticorrelated. We demonstrate that positively correlated landscapes often possess a shared maximum in both landscapes that allows the population to step through sub‐optimal local fitness maxima that often trap single landscape evolution trajectories. Finally, we demonstrate that switching between anticorrelated paired landscapes leads to ergodic‐like dynamics where each genotype is populated with nonzero probability, dramatically lowering the steady‐state fitness in comparison to single landscape evolution.     

Coevolution to the edge of chaos: coupled fitness landscapes, poised states, and coevolutionary avalanches

We introduce a broadened framework to study aspects of coevolution based on the NK class of statistical models of rugged fitness landscapes. In these models the fitness contribution of each of N genes in a genotype depends epistatically on K other genes. Increasing epistatic interactions increases the rugged multipeaked character of the fitness landscape. Coevolution is thought of, at the lowest level, as a coupling of landscapes such that adaptive moves by one player deform the landscapes of its immediate partners. In these models we are able to tune the ruggedness of landscapes, how richly intercoupled any two landscapes are, and how many other players interact with each player. All these properties profoundly alter the character of the coevolutionary dynamics. In particular, these parameters govern how readily coevolving ecosystems achieve Nash equilibria, how stable to perturbations such equilibria are, and the sustained mean fitness of coevolving partners. In turn, this raises the possibility that an evolutionary metadynamics due to natural selection may sculpt landscapes and their couplings to achieve coevolutionary systems able to coadapt well. The results suggest that sustained fitness is optimized when landscape ruggedness relative to couplings between landscapes is tuned such that Nash equilibria just tenuously form across the ecosystem. In this poised state, coevolutionary avalanches appear to propagate on all length scales in a power law distribution. Such avalanches may be related to the distribution of small and large extinction events in the record.     

Mutational fitness landscape and drug resistance

Robust technology has been developed to systematically quantify fitness landscapes that provide valuable opportunities to improve our understanding of drug resistance and define new avenues to develop drugs with reduced resistance susceptibility. We outline the critical importance of drug resistance studies and the potential for fitness landscape approaches to contribute to this effort. We describe the major technical advancements in mutational scanning, which is the primary approach used to quantify protein fitness landscapes. There are many complex steps to consider in planning and executing mutational scanning projects including developing a selection scheme, generating mutant libraries, tracking the frequency of variants using next-generation sequencing, and processing and interpreting the data. Key experimental parameters impacting each of these steps are discussed to aid in planning fitness landscape studies. There is a strong need for improved understanding of drug resistance, and fitness landscapes provide a promising new approach.     

The population genetics of adaptation on correlated fitness landscapes: the block model

Several recent theoretical studies of the genetics of adaptation have focused on the mutational landscape model, which considers evolution on rugged fitness landscapes (i.e., ones having many local optima). Adaptation in this model is characterized by several simple results. Here I ask whether these results also hold on correlated fitness landscapes, which are smoother than those considered in the mutational landscape model. In particular, I study the genetics of adaptation in the block model, a tunably rugged model of fitness landscapes. Considering the scenario in which adaptation begins from a high fitness wild-type DNA sequence, I use extreme value theory and computer simulations to study both single adaptive steps and entire adaptive walks. I show that all previous results characterizing single steps in adaptation in the mutational landscape model hold at least approximately on correlated landscapes in the block model; many entire-walk results, however, do not.     

Comprehensive mutational scanning of a kinase in vivo reveals substrate-dependent fitness landscapes

Deep mutational scanning has emerged as a promising tool for mapping sequence–activity relationships in proteins, ribonucleic acid and deoxyribonucleic acid. In this approach, diverse variants of a sequence of interest are first ranked according to their activities in a relevant assay, and this ranking is then used to infer the shape of the fitness landscape around the wild-type sequence. Little is currently known, however, about the degree to which such fitness landscapes are dependent on the specific assay conditions from which they are inferred. To explore this issue, we performed comprehensive single-substitution mutational scanning of APH(3′)II, a Tn5 transposon-derived kinase that confers resistance to aminoglycoside antibiotics, in Escherichia coli under selection with each of six structurally diverse antibiotics at a range of inhibitory concentrations. We found that the resulting local fitness landscapes showed significant dependence on both antibiotic structure and concentration, and that this dependence can be exploited to guide protein engineering. Specifically, we found that differential analysis of fitness landscapes allowed us to generate synthetic APH(3′)II variants with orthogonal substrate specificities.     

Size scaling of mutation avalanches in a model for protein evolution

The tunably rugged NK-model is used to study avalanche-like events that occur when environmental change causes fitness optima to disappear. The probability of an event with Delta substitutions scales as exp(-c Delta) for smooth landscapes, and as exp(-c Delta(2)) for rugged landscapes. Increasing the ruggedness leads to two competing effects: (1) more possible routes by which single mutations can increase the fitness, which dominates at low ruggedness and acts to increase Delta; and (2) a higher density of fitness optima, which dominates at high ruggedness and acts to decrease Delta. Due to these competing effects, the largest average values of Delta occur at intermediate ruggedness. The effects of system size on the avalanche events are examined, and average values of Delta increase logarithmically with system size. The variance to mean ratios for the number of substitutions per unit time are consistent with experimental results for protein evolution.     

Adaptive landscapes in evolving populations of Pseudomonas fluorescens

The repeatability of adaptive evolution depends on the ruggedness of the underlying adaptive landscape. We contrasted the relative ruggedness of two adaptive landscapes by measuring the variance in fitness and metabolic phenotype within and among genetically distinct strains of Pseudomonas fluorescens in two environments differing only in the carbon source provided (glucose vs. xylose). Fitness increased in all lines, plateauing in one environment but not the other. The pattern of variance in fitness among replicate lines was unique to the selection environment; it increased over the course of the experiment in xylose but not in glucose. Metabolic phenotypes displayed two results: (1) populations adapted via changes that were distinctive to their selection environment, and (2) endpoint phenotypes were less variable in glucose than in xylose. These results indicate that although the response to selection is highly repeatable at the level of fitness, the underlying genetic routes taken were different for each environment and more variable in xylose. We suggest that this reflects a more rugged adaptive landscape in xylose compared to glucose. Our study demonstrates the utility of using replicate selection lines with different evolutionary starting points to try and quantify the relative ruggedness of adaptive landscapes.     

13 RNA fitness landscapes

The origin of life is believed to have progressed through an RNA World, in which RNA acted as both genetic material and functional molecules. Understanding early evolution requires systematic knowledge of the relationship between RNA sequence and functional activity. In particular, knowing the structure of the fitness landscape of RNA is critical in estimating the probability of the emergence of functional sequences and the role of historical accident during evolution. Much theoretical work has been devoted to fitness landscapes, but experimental maps have been relatively limited. We use in vitro selection on a pool of short RNA sequences that nearly saturates sequence space to reconstruct the form of a comprehensive fitness landscape. We also study mutations during non-enzymatic polymerization to understand how early RNA replicators would ‘move’ in sequence space.     

Cooperativity in RNA-protein interactions: global analysis of RNA binding specificity

The control and function of RNA are governed by the specificity of RNA binding proteins. Here, we describe a method for global unbiased analysis of RNA-protein interactions that uses in vitro selection, high-throughput sequencing, and sequence-specificity landscapes. The method yields affinities for a vast array of RNAs in a single experiment, including both low- and high-affinity sites. It is reproducible and accurate. Using this approach,we analyzed members of the PUF (Pumilio and FBF) family of eukaryotic mRNA regulators. Our data identify effects of a specific protein partner on PUF-RNA interactions, reveal subsets of target sites not previously detected, and demonstrate that designer PUF proteins can precisely alter specificity. The approach described here is, in principle, broadly applicable for analysis of any molecule that binds RNA, including proteins, nucleic acids, and small molecules.     

Fitness Landscape Analysis of a tRNA Gene Reveals that the Wild Type Allele is Sub-optimal,Yet Mutationally Robust

Fitness landscape mapping and the prediction of evolutionary trajectories on these landscapes are major tasks in evolutionary biology research. Evolutionary dynamics is tightly linked to the landscape topography, but this relation is not straightforward. Here, we analyze a fitness landscape of a yeast tRNA gene, previously measured under four different conditions. We find that the wild type allele is sub-optimal, and 8–10% of its variants are fitter. We rule out the possibilities that the wild type is fittest on average on these four conditions or located on a local fitness maximum. Notwithstanding, we cannot exclude the possibility that the wild type might be fittest in some of the many conditions in the complex ecology that yeast lives at. Instead, we find that the wild type is mutationally robust (“flat”), while more fit variants are typically mutationally fragile. Similar observations of mutational robustness or flatness have been so far made in very few cases, predominantly in viral genomes.     

Exploring the complexity of the HIV-1 fitness landscape

Although fitness landscapes are central to evolutionary theory, so far no biologically realistic examples for large-scale fitness landscapes have been described. Most currently available biological examples are restricted to very few loci or alleles and therefore do not capture the high dimensionality characteristic of real fitness landscapes. Here we analyze large-scale fitness landscapes that are based on predictive models for in vitro replicative fitness of HIV-1. We find that these landscapes are characterized by large correlation lengths, considerable neutrality, and high ruggedness and that these properties depend only weakly on whether fitness is measured in the absence or presence of different antiretrovirals. Accordingly, adaptive processes on these landscapes depend sensitively on the initial conditions. While the relative extent to which mutations affect fitness on their own (main effects) or in combination with other mutations (epistasis) is a strong determinant of these properties, the fitness landscape of HIV-1 is considerably less rugged, less neutral, and more correlated than expected from the distribution of main effects and epistatic interactions alone. Overall this study confirms theoretical conjectures about the complexity of biological fitness landscapes and the importance of the high dimensionality of the genetic space in which adaptation takes place.     

The Effect of Bacterial Recombination on Adaptation on Fitness Landscapes with Limited Peak Accessibility

There is ample empirical evidence revealing that fitness landscapes are often complex: the fitness effect of a newly arisen mutation can depend strongly on the allelic state at other loci. However, little is known about the effects of recombination on adaptation on such fitness landscapes. Here, we investigate how recombination influences the rate of adaptation on a special type of complex fitness landscapes. On these landscapes, the mutational trajectories from the least to the most fit genotype are interrupted by genotypes with low relative fitness. We study the dynamics of adapting populations on landscapes with different compositions and numbers of low fitness genotypes, with and without recombination. Our results of the deterministic model (assuming an infinite population size) show that recombination generally decelerates adaptation on these landscapes. However, in finite populations, this deceleration is outweighed by the accelerating Fisher-Muller effect under certain conditions. We conclude that recombination has complex effects on adaptation that are highly dependent on the particular fitness landscape, population size and recombination rate.     

Predicting changes in the distribution and abundance of species under environmental change

Environmental changes are expected to alter both the distribution and the abundance of organisms. A disproportionate amount of past work has focused on distribution only, either documenting historical range shifts or predicting future occurrence patterns. However, simultaneous predictions of abundance and distribution across landscapes would be far more useful. To critically assess which approaches represent advances towards the goal of joint predictions of abundance and distribution, we review recent work on changing distributions and on effects of environmental drivers on single populations. Several methods have been used to predict changing distributions. Some of these can be easily modified to also predict abundance, but others cannot. In parallel, demographers have developed a much better understanding of how changing abiotic and biotic drivers will influence growth rate and abundance in single populations. However, this demographic work has rarely taken a landscape perspective and has largely ignored the effects of intraspecific density. We advocate a synthetic approach in which population models accounting for both density dependence and effects of environmental drivers are used to make integrated predictions of equilibrium abundance and distribution across entire landscapes. Such predictions would constitute an important step forward in assessing the ecological consequences of environmental changes.     

The impact of population size on the evolution of asexual microbes on smooth versus rugged fitness landscapes

Background  

It is commonly thought that large asexual populations evolve more rapidly than smaller ones, due to their increased rate of beneficial mutations. Less clear is how population size influences the level of fitness an asexual population can attain. Here, we simulate the evolution of bacteria in repeated serial passage experiments to explore how features such as fitness landscape ruggedness, the size of the mutational target under selection, and the mutation supply rate, interact to affect the evolution of microbial populations of different sizes.     

The NK model of rugged fitness landscapes and its application to maturation of the immune response

Adaptive evolution is, to a large extent, a complex combinatorial optimization process. Such processes can be characterized as "uphill walks on rugged fitness landscapes". Concrete examples of fitness landscapes include the distribution of any specific functional property such as the capacity to catalyze a specific reaction, or bind a specific ligand, in "protein space". In particular, the property might be the affinity of all possible antibody molecules for a specific antigenic determinant. That affinity landscape presumably plays a critical role in maturation of the immune response. In this process, hypermutation and clonal selection act to select antibody V region mutant variants with successively higher affinity for the immunizing antigen. The actual statistical structure of affinity landscapes, although knowable, is currently unknown. Here, we analyze a class of mathematical models we call NK models. We show that these models capture significant features of the maturation of the immune response, which is currently thought to share features with general protein evolution. The NK models have the important property that, as the parameter K increases, the "ruggedness" of the NK landscape varies from a single peaked "Fujiyama" landscape to a multi-peaked "badlands" landscape. Walks to local optima on such landscapes become shorter as K increases. This fact allows us to choose a value of K that corresponds to the experimentally observed number of mutational "steps", 6-8, taken as an antibody sequence matures. If the mature antibody is taken to correspond to a local optimum in the model, tuning the model requires that K be about 40, implying that the functional contribution of each amino acid in the V region is affected by about 40 others. Given this value of K, the model then predicts several features of "antibody space" that are in qualitative agreement with experiment: (1) The fraction of fitter variants of an initial "roughed in" germ line antibody amplified by clonal selection is about 1-2%. (2) Mutations at some sites of the mature antibody hardly affect antibody function at all, but mutations at other sites dramatically decrease function. (3) The same "roughed in" antibody sequence can "walk" to many mature antibody sequences. (4) Many adaptive walks can end on the same local optimum. (5) Comparison of different mature sequences derived from the same initial V region shows evolutionary hot spots and parallel mutations. All these predictions are open to detailed testing by obtaining monoclonal antibodies early in the immune response and carrying out in vitro mutagenesis and adaptive hill climbing with respect to affinity for the immunizing antigen.     

Pattern generation in molecular evolution: Exploitation of the variation in RNA landscapes

Evolution of RNA secondary structure is studied using simulation techniques and statistical analysis of fitness landscapes. The transition from RNA sequence to RNA secondary structure leads to fitness landscapes that have local variations in their ruggedness. Evolution exploits these variations. In stable environments it moves the quasispecies toward relatively flat peaks, where not only the master sequence but also its mutants have a high fitness. In a rapidly changing environment, the situation is reversed; evolution moves the quasispecies to a region where the correlation between secondary structures of neighboring RNA sequences is relatively low. In selection for simple secondary structures the movement toward flat peaks leads to pattern generation in the RNA sequences. Patterns are generated at the level of polynucleotide frequencies and the distribution of purines and pyrimidines. The patterns increase the modularity of the sequence. They thereby prevent the formation of alternative secondary structures after mutations. The movement of the quasispecies toward relatively rugged parts of the landscape results in pattern generation at the level of the RNA secondary structure. The base-pairing frequency of the sequences increases. The patterns that are generated in the RNA sequences and the RNA secondary structures are not directly selected for and can be regarded as a side effect of the evolutionary dynamics of the system. Correspondence to: M.A. Huynen     

Evaluating the within-host fitness effects of mutations fixed during virus adaptation to different ecotypes of a new host

The existence of genetic variation for resistance in host populations is assumed to be essential to the spread of an emerging virus. Models predict that the rate of spread slows down with the increasing frequency and higher diversity of resistance alleles in the host population. We have been using the experimental pathosystem Arabidopsis thaliana—tobacco etch potyvirus (TEV) to explore the interplay between genetic variation in host''s susceptibility and virus diversity. We have recently shown that TEV populations evolving in A. thaliana ecotypes that differ in susceptibility to infection gained within-host fitness, virulence and infectivity in a manner compatible with a gene-for-gene model of host–parasite interactions: hard-to-infect ecotypes were infected by generalist viruses, whereas easy-to-infect ecotypes were infected by every virus. We characterized the genomes of the evolved viruses and found cases of host-driven convergent mutations. To gain further insights in the mechanistic basis of this gene-for-gene model, we have generated all viral mutations individually as well as in specific combinations and tested their within-host fitness effects across ecotypes. Most of these mutations were deleterious or neutral in their local ecotype and only a very reduced number had a host-specific beneficial effect. We conclude that most of the mutations fixed during the evolution experiment were so by drift or by selective sweeps along with the selected driver mutation. In addition, we evaluated the ruggedness of the underlying adaptive fitness landscape and found that mutational effects were mostly multiplicative, with few cases of significant epistasis.