Properties of selected mutations and genotypic landscapes under Fisher's geometric model

The fitness landscape—the mapping between genotypes and fitness—determines properties of the process of adaptation. Several small genotypic fitness landscapes have recently been built by selecting a handful of beneficial mutations and measuring fitness of all combinations of these mutations. Here, we generate several testable predictions for the properties of these small genotypic landscapes under Fisher's geometric model of adaptation. When the ancestral strain is far from the fitness optimum, we analytically compute the fitness effect of selected mutations and their epistatic interactions. Epistasis may be negative or positive on average depending on the distance of the ancestral genotype to the optimum and whether mutations were independently selected, or coselected in an adaptive walk. Simulations show that genotypic landscapes built from Fisher's model are very close to an additive landscape when the ancestral strain is far from the optimum. However, when it is close to the optimum, a large diversity of landscape with substantial roughness and sign epistasis emerged. Strikingly, small genotypic landscapes built from several replicate adaptive walks on the same underlying landscape were highly variable, suggesting that several realizations of small genotypic landscapes are needed to gain information about the underlying architecture of the fitness landscape.     

Exploring phenotype space through neutral evolution

RNA secondary-structure folding algorithms predict the existence of connected networks of RNA sequences with identical secondary structures. Fitness landscapes that are based on the mapping between RNA sequence and RNA secondary structure hence have many neutral paths. A neutral walk on these fitness landscapes gives access to a virtually unlimited number of secondary structures that are a single point mutation from the neutral path. This shows that neutral evolution explores phenotype space and can play a role in adaptation. Received: 23 December 1995 / Accepted: 17 March 1996     

Detecting epistasis from an ensemble of adapting populations

The role that epistasis plays during adaptation remains an outstanding problem, which has received considerable attention in recent years. Most of the recent empirical studies are based on ensembles of replicate populations that adapt in a fixed, laboratory controlled condition. Researchers often seek to infer the presence and form of epistasis in the fitness landscape from the time evolution of various statistics averaged across the ensemble of populations. Here, we provide a rigorous analysis of what quantities, drawn from time series of such ensembles, can be used to infer epistasis for populations evolving under weak mutation on finite‐site fitness landscapes. First, we analyze the mean fitness trajectory—that is, the time course of the ensemble average fitness. We show that for any epistatic fitness landscape and starting genotype, there always exists a non‐epistatic fitness landscape that produces the exact same mean fitness trajectory. Thus, the presence of epistasis is not identifiable from the mean fitness trajectory. By contrast, we show that two other ensemble statistics—the time evolution of the fitness variance across populations, and the time evolution of the mean number of substitutions—can detect certain forms of epistasis in the underlying fitness landscape.     

Natural selection fails to optimize mutation rates for long-term adaptation on rugged fitness landscapes

The rate of mutation is central to evolution. Mutations are required for adaptation, yet most mutations with phenotypic effects are deleterious. As a consequence, the mutation rate that maximizes adaptation will be some intermediate value. Here, we used digital organisms to investigate the ability of natural selection to adjust and optimize mutation rates. We assessed the optimal mutation rate by empirically determining what mutation rate produced the highest rate of adaptation. Then, we allowed mutation rates to evolve, and we evaluated the proximity to the optimum. Although we chose conditions favorable for mutation rate optimization, the evolved rates were invariably far below the optimum across a wide range of experimental parameter settings. We hypothesized that the reason that mutation rates evolved to be suboptimal was the ruggedness of fitness landscapes. To test this hypothesis, we created a simplified landscape without any fitness valleys and found that, in such conditions, populations evolved near-optimal mutation rates. In contrast, when fitness valleys were added to this simple landscape, the ability of evolving populations to find the optimal mutation rate was lost. We conclude that rugged fitness landscapes can prevent the evolution of mutation rates that are optimal for long-term adaptation. This finding has important implications for applied evolutionary research in both biological and computational realms.     

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.     

Fluctuation domains in adaptive evolution

We derive an expression for the variation between parallel trajectories in phenotypic evolution, extending the well known result that predicts the mean evolutionary path in adaptive dynamics or quantitative genetics. We show how this expression gives rise to the notion of fluctuation domains-parts of the fitness landscape where the rate of evolution is very predictable (due to fluctuation dissipation) and parts where it is highly variable (due to fluctuation enhancement). These fluctuation domains are determined by the curvature of the fitness landscape. Regions of the fitness landscape with positive curvature, such as adaptive valleys or branching points, experience enhancement. Regions with negative curvature, such as adaptive peaks, experience dissipation. We explore these dynamics in the ecological scenarios of implicit and explicit competition for a limiting resource.     

Visualizing fitness landscapes

Fitness landscapes are a classical concept for thinking about the relationship between genotype and fitness. However, because the space of genotypes is typically high-dimensional, the structure of fitness landscapes can be difficult to understand and the heuristic approach of thinking about fitness landscapes as low-dimensional, continuous surfaces may be misleading. Here, I present a rigorous method for creating low-dimensional representations of fitness landscapes. The basic idea is to plot the genotypes in a manner that reflects the ease or difficulty of evolving from one genotype to another. Such a layout can be constructed using the eigenvectors of the transition matrix describing the evolution of a population on the fitness landscape when mutation is weak. In addition, the eigendecomposition of this transition matrix provides a new, high-level view of evolution on a fitness landscape. I demonstrate these techniques by visualizing the fitness landscape for selection for the amino acid serine and by visualizing a neutral network derived from the RNA secondary structure genotype-phenotype map.     

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.     

The foldability landscape of model proteins

Molecular evolution may be considered as a walk in a multidimensional fitness landscape, where the fitness at each point is associated with features such as the function, stability, and survivability of these molecules. We present a simple model for the evolution of protein sequences on a landscape with a precisely defined fitness function. We use simple lattice models to represent protein structures, with the ability of a protein sequence to fold into the structure with lowest energy, quantified as the foldability, representing the fitness of the sequence. The foldability of the sequence is characterized based on the spin glass model of protein folding. We consider evolution as a walk in this foldability landscape and study the nature of the landscape and the resulting dynamics. Selective pressure is explicitly included in this model in the form of a minimum foldability requirement. We find that different native structures are not evenly distributed in interaction space, with similar structures and structures with similar optimal foldabilities clustered together. Evolving proteins marginally fulfill the selective criteria of foldability. As the selective pressure is increased, evolutionary trajectories become increasingly confined to “neutral networks,” where the sequence and the interactions can be significantly changed while a constant structure is maintained. © 1997 John Wiley & Sons, Inc. Biopoly 42: 427–438, 1997     

Evolutionary walks through a land plant morphospace

A model for mimicking land plant evolution is here expanded and re-evaluated. The model consists of (1) a morphospace containing on the order of 10⁹ phenotypic variants, (2) 15 different fitness landscapes, each defined on the basis of performing one or more of four tasks (i.e. maximizing light interception, mechanical stability and reproduction, and minimizing total surface area), and (3) an algorithm driving a search through fitness landscapes for more fit variants. The model is used to predict the effects of the number of simultaneously performed tasks ('complexity'), abrupt changes in environmental conditions (mimicked by random replacement of one fitness landscape with another), and developmental barriers (mimicked by barring searches from entering specific subdomains in the morphospace) on number and accessibility of variants occupying fitness maxima. The model predicts that (1) the number and accessibility of fitness peaks will increase (while the difference between the relative fitness of peaks and valleys will decrease) in proportion to functional complexity, (2) abrupt shifts in landscapes will increase rather than decrease phenotypic diversity, and (3) obstructed searches have an equal or higher probability of reaching fitness peaks than unfettered searches. These results follow axiomatically from the way hypothetical variants are spatially ordered in the morphospace, the manner in which relative fitness is defined, and the protocol used to locate fitness peaks. A critique of the model's predictions and desiderata for future research are provided.     

Surveying a local fitness landscape of a protein with epistatic sites for the study of directed evolution

We present a method for analysis of a fitness landscape of a biopolymer with significantly epistatic sites. The analysis is based on a quasi-additive fitness model. The fitness model is constructed with additive terms conducted by "site-fitness" and epistatic terms conducted by "pair-fitness," where the site-fitness is a fitness contribution from an independent residue and the pair-fitness is a fitness contribution from a pair of epistatic residues. As a case study, we analyzed the sequence-fitness data for 45 clones of thermostable prolyl endopeptidase mutants. They were generated by a mutation scrambling method, which can accumulate advantageous mutations. The fitness contributions from 14 single-point mutations including E67Q and Q656R were identified by the analysis. As a result, we found that the fitness model with a significant epistatic term by a pair of the 67th site and 656th site was in good agreement with the experimental data and that the explored landscape in the binary 14-dimensional sequence space is still a mountainous landscape with twin peaks. The validity was supported by the analysis of mutant fitness distributions derived from another mutation scrambling experiment and by (3D) structural data.     

From Amino Acid Landscape to Protein Landscape: Analysis of Genetic Codes in Terms of Fitness Landscape

Assigning the values of a certain physicochemical property for individual amino acids to the corresponding codons, we can make an amino acid property "landscape" on a four valued three dimensional sequence space from a genetic code table. Eleven property landscapes made from the standard genetic code (SGC) were analyzed. The evaluation of correlation for each landscape is done by theta value, which represents the ratio of the mean slope (as an additive term) to the degree of roughness (as a nonadditive term). The theta-values for hydropathy indices, polarity, specific heat, and beta-sheet propensity were considerably large with respect to SGC. This implies that the additivity of the contribution from each letter holds for these properties. To clarify the meaning of the so-called mutational robustness of SGC, we next examined correlations between the amino acid property and the actual "site fitnesses" of a protein. The site fitnesses were derived from a set of binding preference scores of amino acid residues at every site in MHC class I molecule binding peptides (Udaka et al. in press). We found that the SGC's theta value for an amino acid property is correlated with the significance of the property in the protein function. Adaptive walk simulation on fitness (= affinity) landscapes in a base sequence space for these model peptides confirmed better evolvability due to the introduction of SGC.     

Male competition fitness landscapes predict both forward and reverse speciation

Speciation is facilitated when selection generates a rugged fitness landscape such that populations occupy different peaks separated by valleys. Competition for food resources is a strong ecological force that can generate such divergent selection. However, it is unclear whether intrasexual competition over resources that provide mating opportunities can generate rugged fitness landscapes that foster speciation. Here we use highly variable male F2 hybrids of benthic and limnetic threespine sticklebacks, Gasterosteus aculeatus Linnaeus, 1758, to quantify the male competition fitness landscape. We find that disruptive sexual selection generates two fitness peaks corresponding closely to the male phenotypes of the two parental species, favouring divergence. Most surprisingly, an additional region of high fitness favours novel hybrid phenotypes that correspond to those observed in a recent case of reverse speciation after anthropogenic disturbance. Our results reveal that sexual selection through male competition plays an integral role in both forward and reverse speciation.     

Replication and mutation on neutral networks

Folding of RNA sequences into secondary structures is viewed as a map that assigns a uniquely defined base pairing pattern to every sequence. The mapping is non-invertible since many sequences fold into the same minimum free energy (secondary) structure or shape. The pre-images of this map, called neutral networks, are uniquely associated with the shapes and vice versa. Random graph theory is used to construct networks in sequence space which are suitable models for neutral networks. The theory of molecular quasispecies has been applied to replication and mutation on single-peak fitness landscapes. This concept is extended by considering evolution on degenerate multi-peak landscapes which originate from neutral networks by assuming that one particular shape is fitter than all the others. On such a single-shape landscape the superior fitness value is assigned to all sequences belonging to the master shape. All other shapes are lumped together and their fitness values are averaged in a way that is reminiscent of mean field theory. Replication and mutation on neutral networks are modeled by phenomenological rate equations as well as by a stochastic birth-and-death model. In analogy to the error threshold in sequence space the phenotypic error threshold separates two scenarios: (i) a stationary (fittest) master shape surrounded by closely related shapes and (ii) populations drifting through shape space by a diffusion-like process. The error classes of the quasispecies model are replaced by distance classes between the master shape and the other structures. Analytical results are derived for single-shape landscapes, in particular, simple expressions are obtained for the mean fraction of master shapes in a population and for phenotypic error thresholds. The analytical results are complemented by data obtained from computer simulation of the underlying stochastic processes. The predictions of the phenomenological approach on the single-shape landscape are very well reproduced by replication and mutation kinetics of tRNA(phe). Simulation of the stochastic process at a resolution of individual distance classes yields data which are in excellent agreement with the results derived from the birth-and-death model.     

Environmental change exposes beneficial epistatic interactions in a catalytic RNA

Natural selection drives populations of individuals towards local peaks in a fitness landscape. These peaks are created by the interactions between individual mutations. Fitness landscapes may change as an environment changes. In a previous contribution, we discovered a variant of the Azoarcus group I ribozyme that represents a local peak in the RNA fitness landscape. The genotype at this peak is distinguished from the wild-type by four point mutations. We here report ribozyme fitness data derived from constructing all possible combinations of these point mutations. We find that these mutations interact epistatically. Importantly, we show that these epistatic interactions change qualitatively in the three different environments that we studied. We find examples where the relative fitness of a ribozyme can change from neutral or negative in one environment, to positive in another. We also show that the fitness effect of a specific GC-AU base pair switch is dependent on both the environment and the genetic context. Moreover, the mutations that we study improve activity at the cost of decreased structural stability. Environmental change is ubiquitous in nature. Our results suggest that such change can facilitate adaptive evolution by exposing new peaks of a fitness landscape. They highlight a prominent role for genotype-environment interactions in doing so.     

Evolutionary advantage of small populations on complex fitness landscapes

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 allow the investigation of this scenario in a precise and quantitative way. Our main result derived analytically shows how the probability of choosing the path of the 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. 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.     

路径依赖下的物种形成机制

生物科学几乎所有研究都需要物种概念作为基础, 生物多样性研究亦需要可操作的物种概念, 但现有物种概念存在不同程度的人为因素或难操作性, 对物种划分造成不利影响。本文引入“进化路径”这一概念, 说明适合度景观时刻变化着, 物种在每个进化时间点上依据瞬时适合度选择下一时刻的进化状态, 且总是沿着动态适合度景观中适合度增加的方向进化。基于演化博弈的方法, 以随机过程为例模拟物种的进化过程。进而提出路径依赖下的物种形成机制, 并在此基础上给出可操作的物种定义, 即: 针对基因、性状、生态过程等任一状态下两个群体内个体的多个变量做统计分析, 若群体之间同时在两个或多个维度状态下呈现出的不连续性d大于群体内变量呈现出的差异性σ_k, 则拥有相应变量的个体属于不同物种。     

The rank ordering of genotypic fitness values predicts genetic constraint on natural selection on landscapes lacking sign epistasis

Sewall Wright's genotypic fitness landscape makes explicit one mechanism by which epistasis for fitness can constrain evolution by natural selection. Wright distinguished between landscapes possessing multiple fitness peaks and those with only a single peak and emphasized that the former class imposes substantially greater constraint on natural selection. Here I present novel formalism that more finely partitions the universe of possible fitness landscapes on the basis of the rank ordering of their genotypic fitness values. In this report I focus on fitness landscapes lacking sign epistasis (i.e., landscapes that lack mutations the sign of whose fitness effect varies epistatically), which constitute a subset of Wright's single peaked landscapes. More than one fitness rank ordering lacking sign epistasis exists for L > 2 (where L is the number of interacting loci), and I find that a highly statistically significant effect exists between landscape membership in fitness rank-ordering partition and two different proxies for genetic constraint, even within this subset of landscapes. This statistical association is robust to population size, permitting general inferences about some of the characteristics of fitness rank orderings responsible for genetic constraint on natural selection.     

Predicting the Evolution of Sex on Complex Fitness Landscapes

Most population genetic theories on the evolution of sex or recombination are based on fairly restrictive assumptions about the nature of the underlying fitness landscapes. Here we use computer simulations to study the evolution of sex on fitness landscapes with different degrees of complexity and epistasis. We evaluate predictors of the evolution of sex, which are derived from the conditions established in the population genetic literature for the evolution of sex on simpler fitness landscapes. These predictors are based on quantities such as the variance of Hamming distance, mean fitness, additive genetic variance, and epistasis. We show that for complex fitness landscapes all the predictors generally perform poorly. Interestingly, while the simplest predictor, ΔVar_HD, also suffers from a lack of accuracy, it turns out to be the most robust across different types of fitness landscapes. ΔVar_HD is based on the change in Hamming distance variance induced by recombination and thus does not require individual fitness measurements. The presence of loci that are not under selection can, however, severely diminish predictor accuracy. Our study thus highlights the difficulty of establishing reliable criteria for the evolution of sex on complex fitness landscapes and illustrates the challenge for both theoretical and experimental research on the origin and maintenance of sexual reproduction.     

In silico modelling of directed evolution: Implications for experimental design and stepwise evolution

We model the process of directed evolution (DE) in silico using genetic algorithms. Making use of the NK fitness landscape model, we analyse the effects of mutation rate, crossover and selection pressure on the performance of DE. A range of values of K, the epistatic interaction of the landscape, are considered, and high- and low-throughput modes of evolution are compared. Our findings suggest that for runs of or around ten generations’ duration—as is typical in DE—there is little difference between the way in which DE needs to be configured in the high- and low-throughput regimes, nor across different degrees of landscape epistasis. In all cases, a high selection pressure (but not an extreme one) combined with a moderately high mutation rate works best, while crossover provides some benefit but only on the less rugged landscapes. These genetic algorithms were also compared with a “model-based approach” from the literature, which uses sequential fixing of the problem parameters based on fitting a linear model. Overall, we find that purely evolutionary techniques fare better than do model-based approaches across all but the smoothest landscapes.