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.     

Simon-Ando decomposability and fitness landscapes

In this paper, we investigate fitness landscapes (under point mutation and recombination) from the standpoint of whether the induced evolutionary dynamics have a “fast-slow” time scale associated with the differences in relaxation time between local quasi-equilibria and the global equilibrium. This dynamical hevavior has been formally described in the econometrics literature in terms of the spectral properties of the appropriate operator matrices by Simon and Ando (Econometrica 29 (1961) 111), and we use the relations they derive to ask which fitness functions and mutation/recombination operators satisfy these properties. It turns out that quite a wide range of landscapes satisfy the condition (at least trivially) under point mutation given a sufficiently low mutation rate, while the property appears to be difficult to satisfy under genetic recombination. In spite of the fact that Simon-Ando decomposability can be realized over fairly wide range of parameters, it imposes a number of restriction on which landscape partitionings are possible. For these reasons, the Simon-Ando formalism does not appear to be applicable to other forms of decomposition and aggregation of variables that are important in evolutionary systems.     

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.     

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.     

Fourier and Taylor series on fitness landscapes

Holland's hyperplane transform of a fitness landscape, a random, real valued function of the verticies of a regular finite graph, is shown to be a special case of the Fourier transform of a function of a finite group. It follows that essentially all of the powerful Fourier theory, which assumes a simple form for commutative groups, can be used to characterize such landscapes. In particular, an analogue of the KarhunenLoève expansion can be used to prove that the Fourier coefficients of landscapes on commutative groups are uncorrelated and to infer their variance from the autocorrelation function of a random walk on the landscape. There is also a close relationship between the Fourier coefficients and Taylor coefficients, which provide information about the landscape's local properties. Special attention is paid to a particularly simple, but ubiquitous class of landscapes, so-called AR(1) landscapes.     

Random field models for fitness landscapes

 In many cases fitness landscapes are obtained as particular instances of random fields by randomly assigning a large number of parameters. Models of this type are often characterized reasonably well by their covariance matrices. We characterize isotropic random fields on finite graphs in terms of their Fourier series expansions and investigate the relation between the covariance matrix of the random field model and the correlation structure of the individual landscapes constructed from this random field. Correlation measures are a good characteristic of “rugged landscapes” models as they are closely related to quantities like the number of local optima or the length of adaptive walks. Our formalism suggests to approximate landscape with known autocorrelation function by a random field model that has the same correlation structure. Received: 10 November 1995 / Revised version: 19 February 1996     

Predictability of evolutionary trajectories in fitness landscapes

Experimental studies on enzyme evolution show that only a small fraction of all possible mutation trajectories are accessible to evolution. However, these experiments deal with individual enzymes and explore a tiny part of the fitness landscape. We report an exhaustive analysis of fitness landscapes constructed with an off-lattice model of protein folding where fitness is equated with robustness to misfolding. This model mimics the essential features of the interactions between amino acids, is consistent with the key paradigms of protein folding and reproduces the universal distribution of evolutionary rates among orthologous proteins. We introduce mean path divergence as a quantitative measure of the degree to which the starting and ending points determine the path of evolution in fitness landscapes. Global measures of landscape roughness are good predictors of path divergence in all studied landscapes: the mean path divergence is greater in smooth landscapes than in rough ones. The model-derived and experimental landscapes are significantly smoother than random landscapes and resemble additive landscapes perturbed with moderate amounts of noise; thus, these landscapes are substantially robust to mutation. The model landscapes show a deficit of suboptimal peaks even compared with noisy additive landscapes with similar overall roughness. We suggest that smoothness and the substantial deficit of peaks in the fitness landscapes of protein evolution are fundamental consequences of the physics of protein folding.     

Multidimensional epistasis and fitness landscapes in enzyme evolution

The conventional analysis of enzyme evolution is to regard one single salient feature as a measure of fitness, expressed in a milieu exposing the possible selective advantage at a given time and location. Given that a single protein may serve more than one function, fitness should be assessed in several dimensions. In the present study we have explored individual mutational steps leading to a triple-point-mutated human GST (glutathione transferase) A2-2 displaying enhanced activity with azathioprine. A total of eight alternative substrates were used to monitor the diverse evolutionary trajectories. The epistatic effects of the mutations on catalytic activity were variable in sign and magnitude and depended on the substrate used, showing that epistasis is a multidimensional quality. Evidently, the multidimensional fitness landscape can lead to alternative trajectories resulting in enzymes optimized for features other than the selectable markers relevant at the origin of the evolutionary process. In this manner the evolutionary response is robust and can adapt to changing environmental conditions.     

Intraindividual variability in behavior shapes fitness landscapes

Although intraindividual variability (IIV) in behavior is fundamental to ecological dynamics, the factors that contribute to the expression of IIV are poorly understood. Using an individual‐based model, this study examined the effects of stochasticity on the evolution of IIV represented by the residual variability of behavior. The model describes a population of prey with nonoverlapping generations, in which prey take refuge upon encountering a predator. The strategy of a prey is characterized by the mean and IIV (i.e., standard deviation) of hiding duration. Prey with no IIV will spend the same duration hiding in a refuge at each predator encounter, while prey with IIV will have variable hiding durations among encounters. For the sources of stochasticity, within‐generation stochasticity (represented by random predator encounters) and between‐generation stochasticity (represented by random resource availability) were considered. Analysis of the model indicates that individuals with high levels of IIV are maintained in a population in the presence of between‐generation stochasticity even though the optimal strategy in each generation is a strategy with no IIV, regardless of the presence or absence of within‐generation stochasticity. This contradictory pattern emerges because the mean behavioral trait and IIV do not independently influence fitness (e.g., the sign of the selection gradient with respect to IIV depends on the mean trait). Consequently, even when evolution eventually leads toward a strategy with no IIV (i.e., the optimal strategy), greater IIV may be transiently selected. Between‐generation stochasticity consistently imposes such transient selection and maintain high levels of IIV in a population.     

Estimates of the rate and distribution of fitness effects of spontaneous mutation in Saccharomyces cerevisiae

The per-genome, per-generation rate of spontaneous mutation affecting fitness (U) and the mean fitness cost per mutation (s) are important parameters in evolutionary genetics, but have been estimated for few species. We estimated U and sh (the heterozygous effect of mutations) for two diploid yeast strains differing only in the DNA mismatch-repair deficiency used to elevate the mutation rate in one (mutator) strain. Mutations were allowed to accumulate in 50 replicate lines of each strain, during 36 transfers of randomly chosen single colonies (approximately 600 generations). Among wild-type lines, fitnesses were bimodal, with one mode showing no change in mean fitness. The other mode showed a mean 29.6% fitness decline and the petite phenotype, usually caused by partial deletion of the mitochondrial genome. Excluding petites, maximum-likelihood estimates adjusted for the effect of selection were U = 9.5 x 10(-5) and sh = 0.217 for the wild type. Among the mutator lines, the best fit was obtained with 0.005 < or = U < or = 0.94 and 0.049 > or = sh > or = 0.0003. Like other recently tested model organisms, wild-type yeast have low mutation rates, with high mean fitness costs per mutation. Inactivation of mismatch repair increases the frequency of slightly deleterious mutations by approximately two orders of magnitude.     

Combinatorial vector fields and the valley structure of fitness landscapes

Adaptive (downhill) walks are a computationally convenient way of analyzing the geometric structure of fitness landscapes. Their inherently stochastic nature has limited their mathematical analysis, however. Here we develop a framework that interprets adaptive walks as deterministic trajectories in combinatorial vector fields and in return associate these combinatorial vector fields with weights that measure their steepness across the landscape. We show that the combinatorial vector fields and their weights have a product structure that is governed by the neutrality of the landscape. This product structure makes practical computations feasible. The framework presented here also provides an alternative, and mathematically more convenient, way of defining notions of valleys, saddle points, and barriers in landscape. As an application, we propose a refined approximation for transition rates between macrostates that are associated with the valleys of the landscape.     

Quantifying slow evolutionary dynamics in RNA fitness landscapes

We re-examine the evolutionary dynamics of RNA secondary structures under directional selection towards an optimum RNA structure. We find that the punctuated equilibria lead to a very slow approach to the optimum, following on average an inverse power of the evolutionary time. In addition, our study of the trajectories shows that the out-of-equilibrium effects due to the evolutionary process are very weak. In particular, the distribution of genotypes is close to that arising during equilibrium stabilizing selection. As a consequence, the evolutionary dynamics leave almost no measurable out-of-equilibrium trace, only the transition genotypes (close to the border between different periods of stasis) have atypical mutational properties.     

Cluster partitions and fitness landscapes of the Drosophila fly microbiome

Journal of Mathematical Biology - The concept of genetic epistasis defines an interaction between two genetic loci as the degree of non-additivity in their phenotypes. A fitness landscape describes...     

Error-threshold exists in fitness landscapes with lethal mutants

Background  

One of the important insights of quasi-species theory is an error-threshold. The error-threshold is the error rate of replication above which the sudden onset of the population delocalization from the fittest genotype occurs despite Darwinian selection; i.e., the break down of evolutionary optimization. However, a recent article by Wilke in this journal, after reviewing the previous studies on the error-threshold, concluded that the error-threshold does not exist if lethal mutants are taken into account in a fitness landscape. Since lethal mutants obviously exist in reality, this has a significant implication about biological evolution. However, the study of Wagner and Krall on which Wilke's conclusion was based considered mutation-selection dynamics in one-dimensional genotype space with the assumption that a genotype can mutate only to an adjoining genotype in the genotype space. In this article, we study whether the above conclusion holds in high-dimensional genotype space without the assumption of the adjacency of mutations, where the consequences of mutation-selection dynamics can be qualitatively different.     

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.     

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.     

Deterministic and stochastic regimes of asexual evolution on rugged fitness landscapes

We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length L. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is deterministic or stochastic depends on the effective mutational distance d(eff) up to which the population can spread in genotype space. For d(eff) = L, the deterministic quasi-species theory operates while for d(eff) < 1, the evolution is completely stochastic. Between these two limiting cases, the dynamics are described by a local quasi-species theory below a crossover time T(x) while above T(x) the population gets trapped at a local fitness peak and manages to find a better peak via either stochastic tunneling or double mutations. In the stochastic regime d(eff) < 1, we identify two subregimes associated with clonal interference and uphill adaptive walks, respectively. We argue that our findings are relevant to the interpretation of evolution experiments with microbial populations.     

Cambrian explosion and Permian quiescence: Implications of rugged fitness landscapes

Summary The transition from the late Precambrian to Cambrian coincided with a massive increase in diversity of multicellular organisms and the rapid establishment of a large number of phyla. In contrast, the Permian extinction 200 million years ago was followed by as rapid an increase at the family level, but no new phyla or classes emerged. This asymmetry has suggested alternative theories based on greater ecological opportunity in the Cambrian, or special developmental canalization locking in development by the Permian. I suggest instead that the asymmetry reflects generic features of adaptive evolution on rugged fitness landscapes.     

Correlated and uncorrelated fitness landscapes and how to tell the difference

The properties of multi-peaked fitness landscapes have attracted attention in a wide variety of fields, including evolutionary biology. However, relaively little attention has been paid to the properties of the landscapes themselves. Herein, we suggest a framework for the mathematical treatment of such landscapes, including an explicit mathematical model. A central role in this discussion is played by the autocorrelation of fitnesses obtained from a random walk on the landscape. Our ideas about average autocorrelations allow us to formulate a condition (satisfied by a wide class of landscapes we call AR(1) landscapes) under which the average autocorrelation approximates a decaying exponential. We then show how our mathematical model can be used to estimate both the globally optimal fitnesses of AR(1) landscapes and their local structure. We illustrate some aspects of our method with computer experiments based on a single family of landscapes (Kauffman's N-k model), that is shown to be a generic AR(1) landscape. We close by discussing how these ideas might be useful in the tuning of combinatorial optimization algorithms, and in modelling in the experimental sciences.