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

Effects of selection stringency on the outcomes of directed evolution

Directed evolution makes mutant lineages compete in climbing complicated sequence-function landscapes. Given this underlying complexity it is unclear how selection stringency, a ubiquitous parameter of directed evolution, impacts the outcome. Here we approach this question in terms of the fitnesses of the candidate variants at each round and the heterogeneity of their distributions of fitness effects. We show that even if the fittest mutant is most likely to yield the fittest mutants in the next round of selection, diversification can improve outcomes by sampling a larger variety of fitness effects. We find that heterogeneity in fitness effects between variants, larger population sizes, and evolution over a greater number of rounds all encourage diversification.

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

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

Synergistic epistasis among cancer drivers can rescue early tumors from the accumulation of deleterious passengers

Epistasis among driver mutations is pervasive and explains relevant features of cancer, such as differential therapy response and convergence towards well-characterized molecular subtypes. Furthermore, a growing body of evidence suggests that tumor development could be hampered by the accumulation of slightly deleterious passenger mutations. In this work, we combined empirical epistasis networks, computer simulations, and mathematical models to explore how synergistic interactions among driver mutations affect cancer progression under the burden of slightly deleterious passengers. We found that epistasis plays a crucial role in tumor development by promoting the transformation of precancerous clones into rapidly growing tumors through a process that is analogous to evolutionary rescue. The triggering of epistasis-driven rescue is strongly dependent on the intensity of epistasis and could be a key rate-limiting step in many tumors, contributing to their unpredictability. As a result, central genes in cancer epistasis networks appear as key intervention targets for cancer therapy.

Machine learning-assisted discovery of growth decision elements by relating bacterial population dynamics to environmental diversity

Microorganisms growing in their habitat constitute a complex system. How the individual constituents of the environment contribute to microbial growth remains largely unknown. The present study focused on the contribution of environmental constituents to population dynamics via a high-throughput assay and data-driven analysis of a wild-type Escherichia coli strain. A large dataset constituting a total of 12,828 bacterial growth curves with 966 medium combinations, which were composed of 44 pure chemical compounds, was acquired. Machine learning analysis of the big data relating the growth parameters to the medium combinations revealed that the decision-making components for bacterial growth were distinct among various growth phases, e.g., glucose, sulfate, and serine for maximum growth, growth rate, and growth delay, respectively. Further analyses and simulations indicated that branched-chain amino acids functioned as global coordinators for population dynamics, as well as a survival strategy of risk diversification to prevent the bacterial population from undergoing extinction.

Hybridization alters the shape of the genotypic fitness landscape, increasing access to novel fitness peaks during adaptive radiation

Estimating the complex relationship between fitness and genotype or phenotype (i.e. the adaptive landscape) is one of the central goals of evolutionary biology. However, adaptive walks connecting genotypes to organismal fitness, speciation, and novel ecological niches are still poorly understood and processes for surmounting fitness valleys remain controversial. One outstanding system for addressing these connections is a recent adaptive radiation of ecologically and morphologically novel pupfishes (a generalist, molluscivore, and scale-eater) endemic to San Salvador Island, Bahamas. We leveraged whole-genome sequencing of 139 hybrids from two independent field fitness experiments to identify the genomic basis of fitness, estimate genotypic fitness networks, and measure the accessibility of adaptive walks on the fitness landscape. We identified 132 single nucleotide polymorphisms (SNPs) that were significantly associated with fitness in field enclosures. Six out of the 13 regions most strongly associated with fitness contained differentially expressed genes and fixed SNPs between trophic specialists; one gene (mettl21e) was also misexpressed in lab-reared hybrids, suggesting a potential intrinsic genetic incompatibility. We then constructed genotypic fitness networks from adaptive alleles and show that scale-eating specialists are the most isolated of the three species on these networks. Intriguingly, introgressed and de novo variants reduced fitness landscape ruggedness as compared to standing variation, increasing the accessibility of genotypic fitness paths from generalist to specialists. Our results suggest that adaptive introgression and de novo mutations alter the shape of the fitness landscape, providing key connections in adaptive walks circumventing fitness valleys and triggering the evolution of novelty during adaptive radiation.

A Family of Fitness Landscapes Modeled through Gene Regulatory Networks

Fitness landscapes are a powerful metaphor for understanding the evolution of biological systems. These landscapes describe how genotypes are connected to each other through mutation and related through fitness. Empirical studies of fitness landscapes have increasingly revealed conserved topographical features across diverse taxa, e.g., the accessibility of genotypes and "ruggedness". As a result, theoretical studies are needed to investigate how evolution proceeds on fitness landscapes with such conserved features. Here, we develop and study a model of evolution on fitness landscapes using the lens of Gene Regulatory Networks (GRNs), where the regulatory products are computed from multiple genes and collectively treated as phenotypes. With the assumption that regulation is a binary process, we prove the existence of empirically observed, topographical features such as accessibility and connectivity. We further show that these results hold across arbitrary fitness functions and that a trade-off between accessibility and ruggedness need not exist. Then, using graph theory and a coarse-graining approach, we deduce a mesoscopic structure underlying GRN fitness landscapes where the information necessary to predict a population's evolutionary trajectory is retained with minimal complexity. Using this coarse-graining, we develop a bottom-up algorithm to construct such mesoscopic backbones, which does not require computing the genotype network and is therefore far more efficient than brute-force approaches. Altogether, this work provides mathematical results of high-dimensional fitness landscapes and a path toward connecting theory to empirical studies.

Conditional prediction of consecutive tumor evolution using cancer progression models: What genotype comes next?

Accurate prediction of tumor progression is key for adaptive therapy and precision medicine. Cancer progression models (CPMs) can be used to infer dependencies in mutation accumulation from cross-sectional data and provide predictions of tumor progression paths. However, their performance when predicting complete evolutionary trajectories is limited by violations of assumptions and the size of available data sets. Instead of predicting full tumor progression paths, here we focus on short-term predictions, more relevant for diagnostic and therapeutic purposes. We examine whether five distinct CPMs can be used to answer the question "Given that a genotype with n mutations has been observed, what genotype with n + 1 mutations is next in the path of tumor progression?" or, shortly, "What genotype comes next?". Using simulated data we find that under specific combinations of genotype and fitness landscape characteristics CPMs can provide predictions of short-term evolution that closely match the true probabilities, and that some genotype characteristics can be much more relevant than global features. Application of these methods to 25 cancer data sets shows that their use is hampered by a lack of information needed to make principled decisions about method choice. Fruitful use of these methods for short-term predictions requires adapting method's use to local genotype characteristics and obtaining reliable indicators of performance; it will also be necessary to clarify the interpretation of the method's results when key assumptions do not hold.

Unbiased inference of the fitness landscape ruggedness from imprecise fitness estimates

Fitness landscapes map genotypes to their corresponding fitness under given environments and allow explaining and predicting evolutionary trajectories. Of particular interest is the landscape ruggedness or the unevenness of the landscape, because it impacts many aspects of evolution such as the likelihood that a population is trapped in a local fitness peak. Although the ruggedness has been inferred from a number of empirically mapped fitness landscapes, it is unclear to what extent this inference is affected by fitness estimation error, which is inevitable in the experimental determination of fitness landscapes. Here, we address this question by simulating fitness landscapes under various theoretical models, with or without fitness estimation error. We find that all eight examined measures of landscape ruggedness are overestimated due to imprecise fitness quantification, but different measures are affected to different degrees. We devise a method to use replicate fitness measures to correct this bias and show that our method performs well under realistic conditions. We conclude that previously reported fitness landscape ruggedness is likely upward biased owing to the negligence of fitness estimation error and advise that future fitness landscape mapping should include at least three biological replicates to permit an unbiased inference of the ruggedness.

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.

The Single-Stranded RNA Bacteriophage Qβ Adapts Rapidly to High Temperatures: An Evolution Experiment

Single-stranded (ss)RNA viruses are thought to evolve rapidly due to an inherently high mutation rate. However, it remains unclear how ssRNA viruses adapt to novel environments and/or how many and what types of substitutions are needed to facilitate this evolution. In this study, we followed the adaptation of the ssRNA bacteriophage Qβ using thermally adapted Escherichia coli as a host, which can efficiently grow at temperatures between 37.2 and 45.3 °C. This made it possible to evaluate Qβ adaptation to the highest known temperature that supports growth, 45.3 °C. We found that Qβ was capable of replication at this temperature; within 114 days (~1260 generations), we detected more than 34 novel point mutations in the genome of the thermally adapted Qβ population, representing 0.8% of the total Qβ genome. In addition, we returned the 45.3 °C-adapted Qβ populations to 37.2 °C and passaged them for 8 days (~124 generations). We found that the reverse-adapted Qβ population showed little to no decrease in fitness. These results indicate that Qβ can evolve in response to increasing temperatures in a short period of time with the accumulation of point mutations.

Predictable properties of fitness landscapes induced by adaptational tradeoffs

Fitness effects of mutations depend on environmental parameters. For example, mutations that increase fitness of bacteria at high antibiotic concentration often decrease fitness in the absence of antibiotic, exemplifying a tradeoff between adaptation to environmental extremes. We develop a mathematical model for fitness landscapes generated by such tradeoffs, based on experiments that determine the antibiotic dose-response curves of Escherichia coli strains, and previous observations on antibiotic resistance mutations. Our model generates a succession of landscapes with predictable properties as antibiotic concentration is varied. The landscape is nearly smooth at low and high concentrations, but the tradeoff induces a high ruggedness at intermediate antibiotic concentrations. Despite this high ruggedness, however, all the fitness maxima in the landscapes are evolutionarily accessible from the wild type. This implies that selection for antibiotic resistance in multiple mutational steps is relatively facile despite the complexity of the underlying landscape.

An uncertainty law for microbial evolution

Medical practice would benefit from a thorough understanding of constraints and uncertainty in microbial evolution. Higher order epistasis refers to unexpected effects of multiple mutations even if both single mutations and pairwise effects have been accounted for. Recent studies show that higher order epistasis is abundant in nature, for bacteria as well as higher organisms. However, the importance of higher order effects has been debated. It has been suggested that such effects cannot be interpreted, and should not be considered. Here, we show conclusively that higher order epistasis changes the adaptive prospects for a population. The conclusion is based on an exhaustive search of 193,270,310 hyper-cube graphs and applications of graph theory. Our results are more precise, yet more universal, than related research since they depend on mathematical theory, rather than sampling or simulations. Moreover, the uncertainty we establish for microbial evolution, due to higher order epistasis is not sensitive for detailed model assumptions, such as the baseline being additive or log-additive fitness.

Adaptive walks on high-dimensional fitness landscapes and seascapes with distance-dependent statistics

The dynamics of evolution is intimately shaped by epistasis - interactions between genetic elements which cause the fitness-effect of combinations of mutations to be non-additive. Analyzing evolutionary dynamics that involves large numbers of epistatic mutations is intrinsically difficult. A crucial feature is that the fitness landscape in the vicinity of the current genome depends on the evolutionary history. A key step is thus developing models that enable study of the effects of past evolution on future evolution. In this work, we introduce a broad class of high-dimensional random fitness landscapes for which the correlations between fitnesses of genomes are a general function of genetic distance. Their Gaussian character allows for tractable computational as well as analytic understanding. We study the properties of these landscapes focusing on the simplest evolutionary process: random adaptive (uphill) walks. Conventional measures of "ruggedness" are shown to not much affect such adaptive walks. Instead, the long-distance statistics of epistasis cause all properties to be highly conditional on past evolution, determining the statistics of the local landscape (the distribution of fitness-effects of available mutations and combinations of these), as well as the global geometry of evolutionary trajectories. In order to further explore the effects of conditioning on past evolution, we model the effects of slowly changing environments. At long times, such fitness "seascapes" cause a statistical steady state with highly intermittent evolutionary dynamics: populations undergo bursts of rapid adaptation, interspersed with periods in which adaptive mutations are rare and the population waits for more new directions to be opened up by changes in the environment. Finally, we discuss prospects for studying more complex evolutionary dynamics and on broader classes of high-dimensional landscapes and seascapes.

Recombination and mutational robustness in neutral fitness landscapes

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.

Every which way? On predicting tumor evolution using cancer progression models

Successful prediction of the likely paths of tumor progression is valuable for diagnostic, prognostic, and treatment purposes. Cancer progression models (CPMs) use cross-sectional samples to identify restrictions in the order of accumulation of driver mutations and thus CPMs encode the paths of tumor progression. Here we analyze the performance of four CPMs to examine whether they can be used to predict the true distribution of paths of tumor progression and to estimate evolutionary unpredictability. Employing simulations we show that if fitness landscapes are single peaked (have a single fitness maximum) there is good agreement between true and predicted distributions of paths of tumor progression when sample sizes are large, but performance is poor with the currently common much smaller sample sizes. Under multi-peaked fitness landscapes (i.e., those with multiple fitness maxima), performance is poor and improves only slightly with sample size. In all cases, detection regime (when tumors are sampled) is a key determinant of performance. Estimates of evolutionary unpredictability from the best performing CPM, among the four examined, tend to overestimate the true unpredictability and the bias is affected by detection regime; CPMs could be useful for estimating upper bounds to the true evolutionary unpredictability. Analysis of twenty-two cancer data sets shows low evolutionary unpredictability for several of the data sets. But most of the predictions of paths of tumor progression are very unreliable, and unreliability increases with the number of features analyzed. Our results indicate that CPMs could be valuable tools for predicting cancer progression but that, currently, obtaining useful predictions of paths of tumor progression from CPMs is dubious, and emphasize the need for methodological work that can account for the probably multi-peaked fitness landscapes in cancer.

Mapping a Systematic Ribozyme Fitness Landscape Reveals a Frustrated Evolutionary Network for Self-Aminoacylating RNA

Molecular evolution can be conceptualized as a walk over a “fitness landscape”, or the function of fitness (e.g., catalytic activity) over the space of all possible sequences. Understanding evolution requires knowing the structure of the fitness landscape and identifying the viable evolutionary pathways through the landscape. However, the fitness landscape for any catalytic biomolecule is largely unknown. The evolution of catalytic RNA is of special interest because RNA is believed to have been foundational to early life. In particular, an essential activity leading to the genetic code would be the reaction of ribozymes with activated amino acids, such as 5(4H)-oxazolones, to form aminoacyl-RNA. Here we combine in vitro selection with a massively parallel kinetic assay to map a fitness landscape for self-aminoacylating RNA, with nearly complete coverage of sequence space in a central 21-nucleotide region. The method (SCAPE: sequencing to measure catalytic activity paired with in vitro evolution) shows that the landscape contains three major ribozyme families (landscape peaks). An analysis of evolutionary pathways shows that, while local optimization within a ribozyme family would be possible, optimization of activity over the entire landscape would be frustrated by large valleys of low activity. The sequence motifs associated with each peak represent different solutions to the problem of catalysis, so the inability to traverse the landscape globally corresponds to an inability to restructure the ribozyme without losing activity. The frustrated nature of the evolutionary network suggests that chance emergence of a ribozyme motif would be more important than optimization by natural selection.

Molecular Fitness Landscapes from High-Coverage Sequence Profiling

The function of fitness (or molecular activity) in the space of all possible sequences is known as the fitness landscape. Evolution is a random walk on the fitness landscape, with a bias toward climbing hills. Mapping the topography of real fitness landscapes is fundamental to understanding evolution, but previous efforts were hampered by the difficulty of obtaining large, quantitative data sets. The accessibility of high-throughput sequencing (HTS) has transformed this study, enabling large-scale enumeration of fitness for many mutants and even complete sequence spaces in some cases. We review the progress of high-throughput studies in mapping molecular fitness landscapes, both in vitro and in vivo, as well as opportunities for future research. Such studies are rapidly growing in number. HTS is expected to have a profound effect on the understanding of real molecular fitness landscapes.

Stability of equilibria in quantitative genetic models based on modified-gradient systems

Motivated by questions in biology, we investigate the stability of equilibria of the dynamical system [Formula: see text] which arise as critical points of f, under the assumption that [Formula: see text] is positive semi-definite. It is shown that the condition [Formula: see text], where [Formula: see text] is the smallest eigenvalue of [Formula: see text], plays a key role in guaranteeing uniform asymptotic stability and in providing information on the basis of attraction of those equilibria.

Cancer progression models and fitness landscapes: a many-to-many relationship

Motivation

The identification of constraints, due to gene interactions, in the order of accumulation of mutations during cancer progression can allow us to single out therapeutic targets. Cancer progression models (CPMs) use genotype frequency data from cross-sectional samples to identify these constraints, and return Directed Acyclic Graphs (DAGs) of restrictions where arrows indicate dependencies or constraints. On the other hand, fitness landscapes, which map genotypes to fitness, contain all possible paths of tumor progression. Thus, we expect a correspondence between DAGs from CPMs and the fitness landscapes where evolution happened. But many fitness landscapes-e.g. those with reciprocal sign epistasis-cannot be represented by CPMs.

Results

Using simulated data under 500 fitness landscapes, I show that CPMs' performance (prediction of genotypes that can exist) degrades with reciprocal sign epistasis. There is large variability in the DAGs inferred from each landscape, which is also affected by mutation rate, detection regime and fitness landscape features, in ways that depend on CPM method. Using three cancer datasets, I show that these problems strongly affect the analysis of empirical data: fitness landscapes that are widely different from each other produce data similar to the empirically observed ones and lead to DAGs that infer very different restrictions. Because reciprocal sign epistasis can be common in cancer, these results question the use and interpretation of CPMs.

Availability and implementation

Code available from Supplementary Material.

Contact

ramon.diaz@iib.uam.es.

Supplementary information

Supplementary data are available at Bioinformatics online.

Selection for Protein Stability Enriches for Epistatic Interactions

A now classical argument for the marginal thermodynamic stability of proteins explains the distribution of observed protein stabilities as a consequence of an entropic pull in protein sequence space. In particular, most sequences that are sufficiently stable to fold will have stabilities near the folding threshold. Here, we extend this argument to consider its predictions for epistatic interactions for the effects of mutations on the free energy of folding. Although there is abundant evidence to indicate that the effects of mutations on the free energy of folding are nearly additive and conserved over evolutionary time, we show that these observations are compatible with the hypothesis that a non-additive contribution to the folding free energy is essential for observed proteins to maintain their native structure. In particular, through both simulations and analytical results, we show that even very small departures from additivity are sufficient to drive this effect.

Mutation-Driven Parallel Evolution during Viral Adaptation

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

Evolutionary constraints in fitness landscapes

In the last years, several genotypic fitness landscapes-combinations of a small number of mutations-have been experimentally resolved. To learn about the general properties of "real" fitness landscapes, it is key to characterize these experimental landscapes via simple measures of their structure, related to evolutionary features. Some of the most relevant measures are based on the selectively acessible paths and their properties. In this paper, we present some measures of evolutionary constraints based on (i) the similarity between accessible paths and (ii) the abundance and characteristics of "chains" of obligatory mutations, that are paths going through genotypes with a single fitter neighbor. These measures have a clear evolutionary interpretation. Furthermore, we show that chains are only weakly correlated to classical measures of epistasis. In fact, some of these measures of constraint are non-monotonic in the amount of epistatic interactions, but have instead a maximum for intermediate values. Finally, we show how these measures shed light on evolutionary constraints and predictability in experimentally resolved landscapes.

Key issues review: evolution on rugged adaptive landscapes

Adaptive landscapes represent a mapping between genotype and fitness. Rugged adaptive landscapes contain two or more adaptive peaks: allele combinations with higher fitness than any of their neighbors in the genetic space. How do populations evolve on such rugged landscapes? Evolutionary biologists have struggled with this question since it was first introduced in the 1930s by Sewall Wright. Discoveries in the fields of genetics and biochemistry inspired various mathematical models of adaptive landscapes. The development of landscape models led to numerous theoretical studies analyzing evolution on rugged landscapes under different biological conditions. The large body of theoretical work suggests that adaptive landscapes are major determinants of the progress and outcome of evolutionary processes. Recent technological advances in molecular biology and microbiology allow experimenters to measure adaptive values of large sets of allele combinations and construct empirical adaptive landscapes for the first time. Such empirical landscapes have already been generated in bacteria, yeast, viruses, and fungi, and are contributing to new insights about evolution on adaptive landscapes. In this Key Issues Review we will: (i) introduce the concept of adaptive landscapes; (ii) review the major theoretical studies of evolution on rugged landscapes; (iii) review some of the recently obtained empirical adaptive landscapes; (iv) discuss recent mathematical and statistical analyses motivated by empirical adaptive landscapes, as well as provide the reader with instructions and source code to implement simulations of evolution on adaptive landscapes; and (v) discuss possible future directions for this exciting field.

Additive Phenotypes Underlie Epistasis of Fitness Effects

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

With a little help from my friends: cooperation can accelerate the rate of adaptive valley crossing

Background

Natural selection favors changes that lead to genotypes possessing high fitness. A conflict arises when several mutations are required for adaptation, but each mutation is separately deleterious. The process of a population evolving from a genotype encoding for a local fitness maximum to a higher fitness genotype is termed an adaptive peak shift.

Results

Here we suggest cooperative behavior as a factor that can facilitate adaptive peak shifts. We model cooperation in a public goods scenario, wherein each individual contributes resources that are later equally redistributed among all cooperating individuals. We use mathematical modeling and stochastic simulations to study the effect of cooperation on peak shifts in both panmictic and structured populations. Our results show that cooperation can substantially affect the rate of complex adaptation. Furthermore, we show that cooperation increases the population diversity throughout the peak shift process, thus increasing the robustness of the population to sudden environmental changes.

Conclusions

We provide a new explanation to adaptive valley crossing in natural populations and suggest that the long term evolution of a species depends on its social behavior.

Electronic supplementary material

The online version of this article (doi:10.1186/s12862-017-0983-2) contains supplementary material, which is available to authorized users.

Genotypic Complexity of Fisher's Geometric Model

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.

On the (un)predictability of a large intragenic fitness landscape

The study of fitness landscapes, which aims at mapping genotypes to fitness, is receiving ever-increasing attention. Novel experimental approaches combined with next-generation sequencing (NGS) methods enable accurate and extensive studies of the fitness effects of mutations, allowing us to test theoretical predictions and improve our understanding of the shape of the true underlying fitness landscape and its implications for the predictability and repeatability of evolution. Here, we present a uniquely large multiallelic fitness landscape comprising 640 engineered mutants that represent all possible combinations of 13 amino acid-changing mutations at 6 sites in the heat-shock protein Hsp90 in Saccharomyces cerevisiae under elevated salinity. Despite a prevalent pattern of negative epistasis in the landscape, we find that the global fitness peak is reached via four positively epistatic mutations. Combining traditional and extending recently proposed theoretical and statistical approaches, we quantify features of the global multiallelic fitness landscape. Using subsets of the data, we demonstrate that extrapolation beyond a known part of the landscape is difficult owing to both local ruggedness and amino acid-specific epistatic hotspots and that inference is additionally confounded by the nonrandom choice of mutations for experimental fitness landscapes.

Inference of fitness values and putative appearance time points for evolvable self-replicating molecules from time series of occurrence frequencies in an evolution reactor

We have established a translation-coupled RNA replication system within a cell-like compartment, and conducted an experimental evolution of the RNA molecules in the system. Then, we obtained a time series of occurrence frequencies of 91 individual genotypes through random sampling and next-generation sequencing. The time series showed a complex clonal interference and a polymorphic population called the "quasispecies". By fitting a deterministic kinetic model of evolvable simple self-replicators to the time series, we estimated the fitness value and "putative appearance time point" for each of the 91 major genotypes identified, where the putative appearance time point is defined as a certain time point at which a certain mutant genotype is supposed to appear in the deterministic kinetic model. As a result, the kinetic model was well fitted and additionally we confirmed that the estimated fitness values for 11 genotypes were considerably close to the experimentally measured ones (Ichihashi et al., 2015). In this sequel paper, with the theoretical basis of the deterministic kinetic model, we present the details of inference of the fitness values and putative appearance time points for the 91 genotypes. It may be possible to apply this methodology to other self-replicating molecules, viruses and bacteria.

Greedy adaptive walks on a correlated fitness landscape

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

Scaling properties of evolutionary paths in a biophysical model of protein adaptation

The enormous size and complexity of genotypic sequence space frequently requires consideration of coarse-grained sequences in empirical models. We develop scaling relations to quantify the effect of this coarse-graining on properties of fitness landscapes and evolutionary paths. We first consider evolution on a simple Mount Fuji fitness landscape, focusing on how the length and predictability of evolutionary paths scale with the coarse-grained sequence length and alphabet. We obtain simple scaling relations for both the weak- and strong-selection limits, with a non-trivial crossover regime at intermediate selection strengths. We apply these results to evolution on a biophysical fitness landscape that describes how proteins evolve new binding interactions while maintaining their folding stability. We combine the scaling relations with numerical calculations for coarse-grained protein sequences to obtain quantitative properties of the model for realistic binding interfaces and a full amino acid alphabet.

Phase transition in random adaptive walks on correlated fitness landscapes

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∼lnL), and a phase at large c, where walks are long (D∼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.

Multidimensional epistasis and the transitory advantage of sex

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.

Length of adaptive walk on uncorrelated and correlated fitness landscapes

We consider the adaptation dynamics of an asexual population that walks uphill on a rugged fitness landscape which is endowed with a large number of local fitness peaks. We work in a parameter regime where only those mutants that are a single mutation away are accessible, as a result of which the population eventually gets trapped at a local fitness maximum and the adaptive walk terminates. We study how the number of adaptive steps taken by the population before reaching a local fitness peak depends on the initial fitness of the population, the extreme value distribution of the beneficial mutations, and correlations among the fitnesses. Assuming that the relative fitness difference between successive steps is small, we analytically calculate the average walk length for both uncorrelated and correlated fitnesses in all extreme value domains for a given initial fitness. We present numerical results for the model where the fitness differences can be large and find that the walk length behavior differs from that in the former model in the Fréchet domain of extreme value theory. We also discuss the relevance of our results to microbial experiments.