Thermodynamical interpretation of evolutionary dynamics on a fitness landscape in an evolution reactor,II

In our previous report [Aita, T., Morinaga, S., Hosimi, Y., 2004. Thermodynamical interpretation of evolutionary dynamics on a fitness landscape in an evolution reactor I. Bull. Math. Biol. 66, 1371–1403], an analogy between thermodynamics and adaptive walks on a Mt. Fuji-type fitness landscape in an artificial selection system was presented. Introducing the ‘free fitness’ as the sum of a fitness term and an entropy term and ‘evolutionary force’ as the gradient of free fitness on a fitness coordinate, we demonstrated that the adaptive walk (=evolution) is driven by the evolutionary force in the direction in which free fitness increases. In this report, we examine the effect of various modifications of the original model on the properties of the adaptive walk. The modifications were as follows: first, mutation distance d was distributed obeying binomial distribution; second, the selection process obeyed the natural selection protocol; third, ruggedness was introduced to the landscape according to the NK model; fourth, a noise was included in the fitness measurement. The effect of each modification was described in the same theoretical framework as the original model by introducing ‘effective’ quantities such as the effective mutation distance or the effective screening size.     

Thermodynamical interpretation of evolutionary dynamics on a fitness landscape in a evolution reactor,I

A theory for describing evolution as adaptive walks by a finite population with M walkers (M ≥ 1) on an anisotropic Mt. Fuji-type fitness landscape is presented, from a thermodynamical point of view. Introducing the ‘free fitness’ as the sum of a fitness term and an entropy term and ‘evolutionary force’ as the gradient of free fitness on a fitness coordinate, we demonstrate that the behavior of these theoretical walkers is almost consistent with the thermodynamical schemes. The major conclusions are as follows: (1) an adaptive walk (=evolution) is driven by an evolutionary force in the direction in which free fitness increases; (2) the expectation of the climbing rate obeys an equation analogous to the Einstein relation in Brownian motion; (3) the standard deviation of the climbing rate is a quantity analogous to the mean thermal energy of a particle, kT (×constant). In addition, on the interpretation that the walkers climb the landscape by absorbing ‘fitness information’ from the surroundings, we succeeded in quantifying the fitness information and formulating a macroscopic scheme from an informational point of view.     

The end of the adaptive landscape metaphor?

The concepts of adaptive/fitness landscapes and adaptive peaks are a central part of much of contemporary evolutionary biology; the concepts are introduced in introductory texts, developed in more detail in graduate-level treatments, and are used extensively in papers published in the major journals in the field. The appeal of visualizing the process of evolution in terms of the movement of populations on such landscapes is very strong; as one becomes familiar with the metaphor, one often develops the feeling that it is possible to gain deep insights into evolution by thinking about the movement of populations on landscapes consisting of adaptive valleys and peaks. But, since Wright first introduced the metaphor in 1932, the metaphor has been the subject of persistent confusion, from equivocation over just what the features of the landscape are meant to represent to how we ought to expect the landscapes to look. Recent advances—conceptual, empirical, and computational—have pointed towards the inadequacy and indeed incoherence of the landscapes as usually pictured. I argue that attempts to reform the metaphor are misguided; it is time to give up the pictorial metaphor of the landscape entirely and rely instead on the results of formal modeling, however difficult such results are to understand in ‘intuitive’ terms.

Reciprocal sign epistasis between frequently experimentally evolved adaptive mutations causes a rugged fitness landscape

The fitness landscape captures the relationship between genotype and evolutionary fitness and is a pervasive metaphor used to describe the possible evolutionary trajectories of adaptation. However, little is known about the actual shape of fitness landscapes, including whether valleys of low fitness create local fitness optima, acting as barriers to adaptive change. Here we provide evidence of a rugged molecular fitness landscape arising during an evolution experiment in an asexual population of Saccharomyces cerevisiae. We identify the mutations that arose during the evolution using whole-genome sequencing and use competitive fitness assays to describe the mutations individually responsible for adaptation. In addition, we find that a fitness valley between two adaptive mutations in the genes MTH1 and HXT6/HXT7 is caused by reciprocal sign epistasis, where the fitness cost of the double mutant prohibits the two mutations from being selected in the same genetic background. The constraint enforced by reciprocal sign epistasis causes the mutations to remain mutually exclusive during the experiment, even though adaptive mutations in these two genes occur several times in independent lineages during the experiment. Our results show that epistasis plays a key role during adaptation and that inter-genic interactions can act as barriers between adaptive solutions. These results also provide a new interpretation on the classic Dobzhansky-Muller model of reproductive isolation and display some surprising parallels with mutations in genes often associated with tumors.     

Simulating natural selection in landscape genetics

Linking landscape effects to key evolutionary processes through individual organism movement and natural selection is essential to provide a foundation for evolutionary landscape genetics. Of particular importance is determining how spatially-explicit, individual-based models differ from classic population genetics and evolutionary ecology models based on ideal panmictic populations in an allopatric setting in their predictions of population structure and frequency of fixation of adaptive alleles. We explore initial applications of a spatially-explicit, individual-based evolutionary landscape genetics program that incorporates all factors--mutation, gene flow, genetic drift and selection--that affect the frequency of an allele in a population. We incorporate natural selection by imposing differential survival rates defined by local relative fitness values on a landscape. Selection coefficients thus can vary not only for genotypes, but also in space as functions of local environmental variability. This simulator enables coupling of gene flow (governed by resistance surfaces), with natural selection (governed by selection surfaces). We validate the individual-based simulations under Wright-Fisher assumptions. We show that under isolation-by-distance processes, there are deviations in the rate of change and equilibrium values of allele frequency. The program provides a valuable tool (cdpop v1.0; http://cel.dbs.umt.edu/software/CDPOP/) for the study of evolutionary landscape genetics that allows explicit evaluation of the interactions between gene flow and selection in complex landscapes.     

ON THE FINDABILITY OF GENOTYPES

Can we define a measure that describes how easy or difficult it is for a population to evolve to a specific genotype? For populations evolving under weak mutation on a time‐invariant fitness landscape, I argue that one appropriate measure is the expected waiting time, starting from equilibrium, for a population to become fixed for a given genotype. Under this definition for the “findability” of genotypes, I show that for any pair of genotypes (1) a population at equilibrium is always more likely to fix at the more findable before the less findable genotype and (2) the expected time to evolve from the more findable to the less findable genotype is always greater that the expected time to evolve in the opposite direction. Although increasing the fitness of a genotype always increases its findability, in general there is no simple relationship between the rank ordering of genotypes by fitness and the rank ordering of genotypes by findability. I also present a method for quantifying the relative contributions of mutation, selection, substitution rate, and probability of reversion to a genotype's findability.     

适合度的相对性与路径依赖的自然选择

自然选择理论认为生物个体或者种群在进化的过程中, 其基因或者性状、行为策略的选择一定是能够提高其适合度或者达到某个可期的“目标”。然而, 随着某个突变基因或者性状特征、行为策略在种群中扩散, 其期望收益将随着其在种群中分布的密度变化或环境改变而发生改变, 这就是适合度景观的悖论, 即静态的、固定可期望的收益可能因此而不存在。基于动态而非静态适合度景观的概念, 我们提出路径依赖的自然选择概念。路径依赖的自然选择过程中, 一个突变的基因或表型在某种环境下随机产生, 但是该基因或表型在某些特定环境下会产生正反馈。尤其是在正反馈与随机漂变的共同作用下, 多条路径的演化就可能发生, 并且其路径的形成将同时受到其种群进化历史过程和空间特征分布等因素的强烈影响。而在不同路径下, 由于观测维度、角度和尺度的不同, 适合度意义将因此而存在不同。在此意义下, 自然选择更可能选择路径频率而不是适合度大小。基于上述概念, 我们借鉴现代物理学中复函数的方法, 来描述多重动力对物种形成或者生物特征、种群进化等路径依赖的演化过程, 以期为同域物种、隐存种形成以及生物多样性演化提供解释机制。     

Expected relative fitness and the adaptive topography of fluctuating selection

Wright's adaptive topography describes gene frequency evolution as a maximization of mean fitness in a constant environment. I extended this to a fluctuating environment by unifying theories of stochastic demography and fluctuating selection, assuming small or moderate fluctuations in demographic rates with a stationary distribution, and weak selection among the types. The demography of a large population, composed of haploid genotypes at a single locus or normally distributed phenotypes, can then be approximated as a diffusion process and transformed to produce the dynamics of population size, N, and gene frequency, p, or mean phenotype, . The expected evolution of p or is a product of genetic variability and the gradient of the long-run growth rate of the population, , with respect to p or . This shows that the expected evolution maximizes , the mean Malthusian fitness in the average environment minus half the environmental variance in population growth rate. Thus, as a function of p or represents an adaptive topography that, despite environmental fluctuations, does not change with time. The haploid model is dominated by environmental stochasticity, so the expected maximization is not realized. Different constraints on quantitative genetic variability, and stabilizing selection in the average environment, allow evolution of the mean phenotype to undergo a stochastic maximization of . Although the expected evolution maximizes the long-run growth rate of the population, for a genotype or phenotype the long-run growth rate is not a valid measure of fitness in a fluctuating environment. The haploid and quantitative character models both reveal that the expected relative fitness of a type is its Malthusian fitness in the average environment minus the environmental covariance between its growth rate and that of the population.     

The adaptive landscape of science

In 1988, David Hull presented an evolutionary account of science. This was a direct analogy to evolutionary accounts of biological adaptation, and part of a generalized view of Darwinian selection accounts that he based upon the Universal Darwinism of Richard Dawkins. Criticisms of this view were made by, among others, Kim Sterelny, which led to it gaining only limited acceptance. Some of these criticisms are, I will argue, no longer valid in the light of developments in the formal modeling of evolution, in particular that of Sergey Gavrilets’ work on adaptive landscapes. If we can usefully recast the Hullian view of science as being driven by selection in terms of Gavrilets’ and Kaufmann’s view of there being “giant components” of high-fitness networks through any realistic adaptive landscape, we may now find it useful to ask what the adaptive pressures on science are, and to extend the metaphor into a full analogy. This is in effect to reconcile the Fisherianism of the Dawkins–Hull approach to selection and replicators, with a Wrightean drift account of social constructionist views of science, preserving, it is to be hoped, the valuable aspects of both.

Evolutionary genetics of maternal effects

Maternal genetic effects (MGEs), where genes expressed by mothers affect the phenotype of their offspring, are important sources of phenotypic diversity in a myriad of organisms. We use a single‐locus model to examine how MGEs contribute patterns of heritable and nonheritable variation and influence evolutionary dynamics in randomly mating and inbreeding populations. We elucidate the influence of MGEs by examining the offspring genotype‐phenotype relationship, which determines how MGEs affect evolutionary dynamics in response to selection on offspring phenotypes. This approach reveals important results that are not apparent from classic quantitative genetic treatments of MGEs. We show that additive and dominance MGEs make different contributions to evolutionary dynamics and patterns of variation, which are differentially affected by inbreeding. Dominance MGEs make the offspring genotype‐phenotype relationship frequency dependent, resulting in the appearance of negative frequency‐dependent selection, while additive MGEs contribute a component of parent‐of‐origin dependent variation. Inbreeding amplifies the contribution of MGEs to the additive genetic variance and, therefore enhances their evolutionary response. Considering evolutionary dynamics of allele frequency change on an adaptive landscape, we show that this landscape differs from the mean fitness surface, and therefore, under some condition, fitness peaks can exist but not be “available” to the evolving population.     

Maynard Smith,optimization, and evolution

Maynard Smith’s defenses of adaptationism and of the value of optimization theory in evolutionary biology are both criticized. His defense does not adequately respond to the criticism of adaptationism by Gould and Lewontin. It is also argued here that natural selection cannot be interpreted as an optimization process if the objective function to be optimized is either (i) interpretable as a fitness, or (ii) correlated with the mean population fitness. This result holds even if fitnesses are frequency-independent; the problem is further exacerbated in the frequency-dependent context modeled by evolutionary game theory. However, Eshel and Feldman’s new results on “long-term” evolution may provide some hope for the continuing relevance of the game-theoretic framework. These arguments also demonstrate the irrelevance of attempts by Intelligent Design creationists to use computational limits on optimization algorithms as evidence against evolutionary theory. It is pointed out that adaptation, natural selection, and optimization are not equivalent processes in the context of biological evolution. It is a pleasure to dedicate this paper to the memory of John Maynard Smith. Thanks are due to James Justus and Samir Okasha for comments on an earlier draft.     

Adaptive walks on behavioural landscapes and the evolution of optimal behaviour by natural selection

Summary One of the main challenges to the adaptationist programme in general and to the use of optimality models in behavioural and evolutionary ecology in particular is that natural selection need not optimise fitness. This challenge is addressed by considering the evolution of optimal patch choice by natural selection. The behavioural model is based on a state variable approach in which a strategy consists of a sequence denoting the patch to be visited as a function of the organism's state and time. The optimal strategy maximises expected terminal reproduction. The fitnesses of alternative strategies are computed by iteration of the associated equations for fitness; this characterises the adaptive behavioural landscape. There may be enormous numbers of strategies that have near optimal fitnesses. A population model is used to connect frequencies of behavioural types from one generation to the next. Theories on adaptive walks on fitness landscapes are considered in the context of behaviour. The main result is that within the context of optimality arguments at selective equilibrium, sub-optimal behaviours can persist. General implications for research in behavioural ecology, including tests of behavioural theories, are discussed.     

Adaptive walks by the fittest among finite random mutants on a Mt. Fuji-type fitness landscape II. Effect of small non-additivity

We examined properties of adaptive walks by the fittest on “rough Mt. Fuji-type” fitness landscapes, which are modeled by superposing small uncorrelated random component on an additive fitness landscape. A single adaptive walk is carried out by repetition of the evolution cycle composed of (1) mutagenesis process that produces random d-fold point mutants of population size N and (2) selection process that picks out the fittest mutant among them. To comprehend trajectories of the walkers, the fitness landscape is mapped into a (x, y, z)-space, where x, y and z represent, respectively, normalized Hamming distance from the peak on the additive fitness landscape, scaled additive fitness and scaled non-additive fitness. Thus a single adaptive walk is expressed as the dynamics of a particle in this space. We drew the “hill-climbing” vector field, where each vector represents the most probable step for a walker in a single step. Almost all of the walkers are expected to move along streams of vectors existing on a particular surface that overlies the (x, y)-plane, toward the neighborhood of a characteristic point at which a mutation-selection-random drift balance is reached. We could theoretically predict this reachable point in the case of random sampling search strategy. Received: 1 March 2000 / Published online: 3 August 2000     

Thermodynamical interpretation of an adaptive walk on a Mt. Fuji-type fitness landscape: Einstein relation-like formula holds in a stochastic evolution

We have theoretically studied the statistical properties of adaptive walks (or hill-climbing) on a Mt. Fuji-type fitness landscape in the multi-dimensional sequence space through mathematical analysis and computer simulation. The adaptive walk is characterized by the "mutation distance" d as the step-width of the walker and the "population size" N as the number of randomly generated d-fold point mutants to be screened. In addition to the fitness W, we introduced the following quantities analogous to thermodynamical concepts: "free fitness" G(W) is identical with W+T x S(W), where T is the "evolutionary temperature" T infinity square root of d/lnN and S(W) is the entropy as a function of W, and the "evolutionary force" X is identical with d(G(W)/T)/dW, that is caused by the mutation and selection pressure. It is known that a single adaptive walker rapidly climbs on the fitness landscape up to the stationary state where a "mutation-selection-random drift balance" is kept. In our interpretation, the walker tends to the maximal free fitness state, driven by the evolutionary force X. Our major findings are as follows: First, near the stationary point W*, the "climbing rate" J as the expected fitness change per generation is described by J approximately L x X with L approximately V/2, where V is the variance of fitness distribution on a local landscape. This simple relationship is analogous to the well-known Einstein relation in Brownian motion. Second, the "biological information gain" (DeltaG/T) through adaptive walk can be described by combining the Shannon's information gain (DeltaS) and the "fitness information gain" (DeltaW/T).     

Geographic variation in the adaptive landscape for seed size at dispersal in the myrmecochorous <Emphasis Type="Italic">Helleborus foetidus</Emphasis>

Literature on seed dispersal mutualisms suggests that plant populations should hardly adapt to their current dispersers. We address the predictions that selection pressures exerted by ants on dispersal-related diaspore traits of the ant-dispersed Helleborus foetidus are highly variable in space, and that geographic (inter-population) variation in these traits is unrelated to selection by current dispersers. To test these predictions we use the concept of the quantitative adaptive landscape for seed size at dispersal. Such landscape depicts the relationship between the population’s mean trait value (mean seed size in the present study) and the population’s mean fitness (mean dispersal probability in the present study). Adaptive landscapes make it possible to assess whether the mean population’s phenotype agrees with one favored by selection. We first analyse, in 12 populations of H. foetidus from southern Spain, the extent of divergence among populations in seed and elaiosome size, and the abundance, composition, and behavior of the ant communities. Seeds from a fixed set of five of these populations were offered to ants in all the study sites to fit the adaptive landscape for seed size. In addition, seeds from the local population were also offered in each site. Our results show that seed size has undergone a larger divergence among populations than elaiosome size. Despite geographic variation in ant assemblages, the adaptive landscapes for seed size at dispersal were remarkably similar among sites: ants create disruptive selection on seed size in 10 out of 12 study sites. As predicted, the basic features of these adaptive landscapes (curvature and location of the minimum) varied geographically in accordance with variation in the size of seed dispersers. Also as predicted, in most populations, the observed mean seed size does not agree with that expected from the adaptive landscapes at dispersal. However, the relevance of dispersers for seed size evolution should not be neglected since the agreement between observed and optimum seed size was stronger where dispersers were more abundant. Thus, against the general view, our results evidence that, in H. foetidus, the observed geographic variation in dispersal-related plant traits is partly linked to selection exerted by current dispersers. Geographic variation in ant assemblages determines both the existence of a selection mosaic and the degree of adjustment of populations to the patterns of selection in the mosaic.     

When is sympatric speciation truly adaptive? An analysis of the joint evolution of resource utilization and assortative mating

The plausibility of sympatric speciation has long been debated among evolutionary ecologists. The process necessarily involves two key elements: the stable coexistence of at least two ecologically distinct types and the emergence of reproductive isolation. Recent theoretical studies within the theoretical framework of adaptive dynamics have shown how both these processes can be driven by natural selection. In the standard scenario, a population first evolves to an evolutionary branching point, next, disruptive selection promotes ecological diversification within the population, and, finally, the fitness disadvantage of intermediate types induces a selection pressure for assortative mating behaviour, which leads to reproductive isolation and full speciation. However, the full speciation process has been mostly studied through computer simulations and only analysed in part. Here I present a complete analysis of the whole speciation process by allowing for the simultaneous evolution of the branching ecological trait as well as a continuous trait controlling mating behaviour. I show how the joint evolution can be understood in terms of a gradient landscape, where the plausibility of different evolutionary paths can be evaluated graphically. I find sympatric speciation unlikely for scenarios with a continuous, unimodal, distribution of resources. Rather, ecological settings where the fitness inferiority of intermediate types is preserved during the ecological branching are more likely to provide opportunity for adaptive, sympatric speciation. Such scenarios include speciation due to predator avoidance or specialization on discrete resources. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Adaptive evolutionary walks require neutral intermediates in RNA fitness landscapes

In RNA fitness landscapes with interconnected networks of neutral mutations, neutral precursor mutations can play an important role in facilitating the accessibility of epistatic adaptive mutant combinations. I use an exhaustively surveyed fitness landscape model based on short sequence RNA genotypes (and their secondary structure phenotypes) to calculate the minimum rate at which mutants initially appearing as neutral are incorporated into an adaptive evolutionary walk. I show first, that incorporating neutral mutations significantly increases the number of point mutations in a given evolutionary walk when compared to estimates from previous adaptive walk models. Second, that incorporating neutral mutants into such a walk significantly increases the final fitness encountered on that walk - indeed evolutionary walks including neutral steps often reach the global optimum in this model. Third, and perhaps most importantly, evolutionary paths of this kind are often extremely winding in their nature and have the potential to undergo multiple mutations at a given sequence position within a single walk; the potential of these winding paths to mislead phylogenetic reconstruction is briefly considered.     

A minimum on the mean number of steps taken in adaptive walks

I consider the adaptation of a DNA sequence when mutant fitnesses are drawn randomly from a probability distribution. I focus on "gradient" adaptation in which the population jumps to the best mutant sequence available at each substitution. Given a random starting point, I derive the distribution of the number of substitutions that occur during adaptive walks to a locally optimal sequence. I show that the mean walk length is a constant:L = e-1, where e approximately 2.72. I argue that this result represents a limit on what is possible under any form of adaptation. No adaptive algorithm on any fitness landscape can arrive at a local optimum in fewer than a mean of L = e-1 steps when starting from a random sequence. Put differently, evolution must try out at least e wild-type sequences during an average bout of adaptation.