Fisher's Geometric Model Research Articles

The Nearly Neutral and Selection Theories of Molecular Evolution Under the Fisher Geometrical Framework: Substitution Rate, Population Size, and Complexity

The general theories of molecular evolution depend on relatively arbitrary assumptions about the relative distribution and rate of advantageous, deleterious, neutral, and nearly neutral mutations. The Fisher geometrical model (FGM) has been used to make distributions of mutations biologically interpretable. We explored an FGM-based molecular model to represent molecular evolutionary processes typically studied by nearly neutral and selection models, but in which distributions and relative rates of mutations with different selection coefficients are a consequence of biologically interpretable parameters, such as the average size of the phenotypic effect of mutations and the number of traits (complexity) of organisms. A variant of the FGM-based model that we called the static regime (SR) represents evolution as a nearly neutral process in which substitution rates are determined by a dynamic substitution process in which the population's phenotype remains around a suboptimum equilibrium fitness produced by a balance between slightly deleterious and slightly advantageous compensatory substitutions. As in previous nearly neutral models, the SR predicts a negative relationship between molecular evolutionary rate and population size; however, SR does not have the unrealistic properties of previous nearly neutral models such as the narrow window of selection strengths in which they work. In addition, the SR suggests that compensatory mutations cannot explain the high rate of fixations driven by positive selection currently found in DNA sequences, contrary to what has been previously suggested. We also developed a generalization of SR in which the optimum phenotype can change stochastically due to environmental or physiological shifts, which we called the variable regime (VR). VR models evolution as an interplay between adaptive processes and nearly neutral steady-state processes. When strong environmental fluctuations are incorporated, the process becomes a selection model in which evolutionary rate does not depend on population size, but is critically dependent on the complexity of organisms and mutation size. For SR as well as VR we found that key parameters of molecular evolution are linked by biological factors, and we showed that they cannot be fixed independently by arbitrary criteria, as has usually been assumed in previous molecular evolutionary models.

FISHER'S GEOMETRICAL MODEL OF FITNESS LANDSCAPE AND VARIANCE IN FITNESS WITHIN A CHANGING ENVIRONMENT

The fitness of an individual can be simply defined as the number of its offspring in the next generation. However, it is not well understood how selection on the phenotype determines fitness. In accordance with Fisher's fundamental theorem, fitness should have no or very little genetic variance, whereas empirical data suggest that is not the case. To bridge these knowledge gaps, we follow Fisher's geometrical model and assume that fitness is determined by multivariate stabilizing selection toward an optimum that may vary among generations. We assume random mating, free recombination, additive genes, and uncorrelated stabilizing selection and mutational effects on traits. In a constant environment, we find that genetic variance in fitness under mutation-selection balance is a U-shaped function of the number of traits (i.e., of the so-called "organismal complexity"). Because the variance can be high if the organism is of either low or high complexity, this suggests that complexity has little direct costs. Under a temporally varying optimum, genetic variance increases relative to a constant optimum and increasingly so when the mutation rate is small. Therefore, mutation and changing environment together can maintain high genetic variance. These results therefore lend support to Fisher's geometric model of a fitness landscape.

Universal pleiotropy is not a valid null hypothesis: reply to Hill and Zhang

Relating to our recent Review (The pleiotropic structure of the genotype–phenotype map: the evolvability of complex organisms. Nature Reviews Genetics 12, 204–213 (2011))1, Hill and Zhang argue in their Correspondence (Assessing pleiotropy and its evolutionary consequences: pleiotropy is not necessarily limited, nor need it hinder the evolution of complexity. Nature Reviews Genetics 21 Feb 2012 (doi:10.1038/nrg2949-c1))2 that the experimental data summarized and discussed in our article are unable to reject the hypothesis of universal pleiotropy (HUP), which asserts that every mutation (or gene) affects every trait. This influential idea originated from Fisher's geometric model of evolution3. Hill and Zhang2 argue that any empirical detection of a gene effect relies on some statistical significance threshold and is thus likely to underestimate the true degree of pleiotropy. We agree with the substance of this argument and have discussed its implications in our paper1. However, we disagree with the conclusion that the HUP is a viable model of genetic architecture. We think this disagreement has deeper methodological roots, which we discuss below.

Cost of Antibiotic Resistance and the Geometry of Adaptation

The distribution of effects of beneficial mutations is key to our understanding of biological adaptation. Yet, empirical estimates of this distribution are scarce, and its functional form is largely unknown. Theoretical models of adaptation predict that the functional form of this distribution should depend on the distance to the optimum. Here, we estimate the rate and distribution of adaptive mutations that compensate for the effect of a single deleterious mutation, which causes antibiotic resistance. Using a system with multiple molecular markers, we estimate the distribution of fitness effects of mutations at two distances from the adaptive peak in 60 populations of Escherichia coli. We find that beneficial mutations, which can contribute to compensatory evolution, occur at very high rates, of the order of 10−5 per genome per generation and can be detected within a few tens of generations. They cause an average fitness increase of 2.5% and 3.6%, depending on the cost of resistance, which is expected under Fisher's geometrical model of adaptation. Moreover, we provide the first description of the distribution of beneficial mutations, segregating during the process of compensatory evolution, to antibiotic resistances bearing different costs. Hence, these results have important implications to understanding the spread and maintenance of antibiotic resistance in bacteria.

Heterozygote advantage as a natural consequence of adaptation in diploids

Molecular adaptation is typically assumed to proceed by sequential fixation of beneficial mutations. In diploids, this picture presupposes that for most adaptive mutations, the homozygotes have a higher fitness than the heterozygotes. Here, we show that contrary to this expectation, a substantial proportion of adaptive mutations should display heterozygote advantage. This feature of adaptation in diploids emerges naturally from the primary importance of the fitness of heterozygotes for the invasion of new adaptive mutations. We formalize this result in the framework of Fisher's influential geometric model of adaptation. We find that in diploids, adaptation should often proceed through a succession of short-lived balanced states that maintain substantially higher levels of phenotypic and fitness variation in the population compared with classic adaptive walks. In fast-changing environments, this variation produces a diversity advantage that allows diploids to remain better adapted compared with haploids despite the disadvantage associated with the presence of unfit homozygotes. The short-lived balanced states arising during adaptive walks should be mostly invisible to current scans for long-term balancing selection. Instead, they should leave signatures of incomplete selective sweeps, which do appear to be common in many species. Our results also raise the possibility that balancing selection, as a natural consequence of frequent adaptation, might play a more prominent role among the forces maintaining genetic variation than is commonly recognized.

COMPLEXITY, PLEIOTROPY, AND THE FITNESS EFFECT OF MUTATIONS

One of the assumptions underlying many theoretical predictions in evolutionary biology concerns the distribution of the fitness effect of mutations. Approximations to this distribution have been derived using various theoretical approaches, of which Fisher's geometrical model is among the most popular ones. Two key concepts in this model are complexity and pleiotropy. Recent studies have proposed different methods for estimating how complexity varies across species, but their results have been contradictory. Here, we show that contradictory results are to be expected when the assumption of universal pleiotropy is violated. We develop a model in which the two key parameters are the total number of traits and the mean number of traits affected by a single mutation. We derive approximations for the distribution of the fitness effect of mutations when populations are either well-adapted or away from the optimum. We also consider drift load in a well-adapted population and show that it is independent of the distribution of the fitness effect of mutations. We show that mutation accumulation experiments can only measure the effect of the mean number of traits affected by mutations, whereas drift load only provides information about the total number of traits. We discuss the plausibility of the model.

The emergence of complexity and restricted pleiotropy in adapting networks

BackgroundThe emergence of organismal complexity has been a difficult subject for researchers because it is not readily amenable to investigation by experimental approaches. Complexity has a myriad of untested definitions and our understanding of its evolution comes primarily from static snapshots gleaned from organisms ranked on an intuitive scale. Fisher's geometric model of adaptation, which defines complexity as the number of phenotypes an organism exposes to natural selection, provides a theoretical framework to study complexity. Yet investigations of this model reveal phenotypic complexity as costly and therefore unlikely to emerge.ResultsWe have developed a computational approach to study the emergence of complexity by subjecting neural networks to adaptive evolution in environments exacting different levels of demands. We monitored complexity by a variety of metrics. Top down metrics derived from Fisher's geometric model correlated better with the environmental demands than bottom up ones such as network size. Phenotypic complexity was found to increase towards an environment-dependent level through the emergence of restricted pleiotropy. Such pleiotropy, which confined the action of mutations to only a subset of traits, better tuned phenotypes in challenging environments. However, restricted pleiotropy also came at a cost in the form of a higher genetic load, as it required the maintenance by natural selection of more independent traits. Consequently, networks of different sizes converged in complexity when facing similar environment.ConclusionsPhenotypic complexity evolved as a function of the demands of the selective pressures, rather than the physical properties of the network architecture, such as functional size. Our results show that complexity may be more predictable, and understandable, if analyzed from the perspective of the integrated task the organism performs, rather than the physical architecture used to accomplish such tasks. Thus, top down metrics emphasizing selection may be better for describing biological complexity than bottom up ones representing size and other physical attributes.

FISHER'S MODEL AND THE GENOMICS OF ADAPTATION: RESTRICTED PLEIOTROPY, HETEROGENOUS MUTATION, AND PARALLEL EVOLUTION

Genetic theories of adaptation generally overlook the genes in which beneficial substitutions occur, and the likely variation in their mutational effects. We investigate the consequences of heterogeneous mutational effects among loci on the genetics of adaptation. We use a generalization of Fisher's geometrical model, which assumes multivariate Gaussian stabilizing selection on multiple characters. In our model, mutation has a distinct variance-covariance matrix of phenotypic effects for each locus. Consequently, the distribution of selection coefficients s varies across loci. We assume each locus can only affect a limited number of independent linear combinations of phenotypic traits (restricted pleiotropy), which differ among loci, an effect we term "orientation heterogeneity." Restricted pleiotropy can sharply reduce the overall proportion of beneficial mutations. Orientation heterogeneity has little impact on the shape of the genomic distribution, but can substantially increase the probability of parallel evolution (the repeated fixation of beneficial mutations at the same gene in independent populations), which is highest with low pleiotropy. We also consider variation in the degree of pleiotropy and in the mean s across loci. The latter impacts the genomic distribution of s, but has a much milder effect on parallel evolution. We discuss these results in the light of evolution experiments.

A Theory of Age-Dependent Mutation and Senescence

Laboratory experiments show us that the deleterious character of accumulated novel age-specific mutations is reduced and made less variable with increased age. While theories of aging predict that the frequency of deleterious mutations at mutation-selection equilibrium will increase with the mutation's age of effect, they do not account for these age-related changes in the distribution of de novo mutational effects. Furthermore, no model predicts why this dependence of mutational effects upon age exists. Because the nature of mutational distributions plays a critical role in shaping patterns of senescence, we need to develop aging theory that explains and incorporates these effects. Here we propose a model that explains the age dependency of mutational effects by extending Fisher's geometrical model of adaptation to include a temporal dimension. Using a combination of simple analytical arguments and simulations, we show that our model predicts age-specific mutational distributions that are consistent with observations from mutation-accumulation experiments. Simulations show us that these age-specific mutational effects may generate patterns of senescence at mutation-selection equilibrium that are consistent with observed demographic patterns that are otherwise difficult to explain.

Mean curvature versus normality: A comparison of two approximations of Fisher's geometrical model

Fisher's geometrical model amounts to a description of mutation and selection for individuals characterised by a number of quantitative traits. In the present work the fitness landscape is not assumed to be spherically symmetric, hence different points, i.e. phenotypes, on a surface of constant fitness generally have different curvatures. We investigate two different approximations of Fisher's geometrical model that have appeared in the literature. One approximation uses the average curvature of the fitness surface at the parental phenotype. The other approach is based on a normal approximation of a distribution associated with new mutations. Analytical results and simulations are used to compare the accuracy of the two approximations.

Determining the Effective Dimensionality of the Genetic Variance–Covariance Matrix

Determining the dimensionality of G provides an important perspective on the genetic basis of a multivariate suite of traits. Since the introduction of Fisher's geometric model, the number of genetically independent traits underlying a set of functionally related phenotypic traits has been recognized as an important factor influencing the response to selection. Here, we show how the effective dimensionality of G can be established, using a method for the determination of the dimensionality of the effect space from a multivariate general linear model introduced by Amemiya (1985). We compare this approach with two other available methods, factor-analytic modeling and bootstrapping, using a half-sib experiment that estimated G for eight cuticular hydrocarbons of Drosophila serrata. In our example, eight pheromone traits were shown to be adequately represented by only two underlying genetic dimensions by Amemiya's approach and factor-analytic modeling of the covariance structure at the sire level. In contrast, bootstrapping identified four dimensions with significant genetic variance. A simulation study indicated that while the performance of Amemiya's method was more sensitive to power constraints, it performed as well or better than factor-analytic modeling in correctly identifying the original genetic dimensions at moderate to high levels of heritability. The bootstrap approach consistently overestimated the number of dimensions in all cases and performed less well than Amemiya's method at subspace recovery.

A GENERAL MULTIVARIATE EXTENSION OF FISHER'S GEOMETRICAL MODEL AND THE DISTRIBUTION OF MUTATION FITNESS EFFECTS ACROSS SPECIES

The evolution of complex organisms is a puzzle for evolutionary theory because beneficial mutations should be less frequent in complex organisms, an effect termed "cost of complexity." However, little is known about how the distribution of mutation fitness effects (f(s)) varies across genomes. The main theoretical framework to address this issue is Fisher's geometric model and related phenotypic landscape models. However, it suffers from several restrictive assumptions. In this paper, we intend to show how several of these limitations may be overcome. We then propose a model of f(s) that extends Fisher's model to account for arbitrary mutational and selective interactions among n traits. We show that these interactions result in f(s) that would be predicted by a much smaller number of independent traits. We test our predictions by comparing empirical f(s) across species of various gene numbers as a surrogate to complexity. This survey reveals, as predicted, that mutations tend to be more deleterious, less variable, and less skewed in higher organisms. However, only limited difference in the shape of f(s) is observed from Escherichia coli to nematodes or fruit flies, a pattern consistent with a model of random phenotypic interactions across many traits. Overall, these results suggest that there may be a cost to phenotypic complexity although much weaker than previously suggested by earlier theoretical works. More generally, the model seems to qualitatively capture and possibly explain the variation of f(s) from lower to higher organisms, which opens a large array of potential applications in evolutionary genetics.

Fisher's geometrical model of evolutionary adaptation—Beyond spherical geometry

Fisher's geometrical model of evolutionary adaptation has recently been used in a variety of contexts of interest to evolutionary biologists. The renewed interest in this model strongly motivates generalizations that make it a more realistic description of evolutionary adaptation. Previously, the distribution of mutant effects has, for analytical tractability, rather than biological realism, been taken as spherically symmetric. Here we substantially extend Fisher's model, by allowing a wider class of mutational distributions that incorporate mutational bias and more general deviations from spherical symmetry such as correlations between mutant effects. We also incorporate work on generalized fitness landscapes, thereby reducing the number of artificial assumptions underlying the model. The generalized model exhibits a substantially increased flexibility and a far richer underlying geometry. We find that the distribution characterizing selection coefficients of new mutations is expressed in terms of a number of geometrical invariants associated with mutation, selection and the parental phenotype.

A GENERAL MULTIVARIATE EXTENSION OF FISHER'S GEOMETRICAL MODEL AND THE DISTRIBUTION OF MUTATION FITNESS EFFECTS ACROSS SPECIES

The evolution of complex organisms is a puzzle for evolutionary theory because beneficial mutations should be less frequent in complex organisms, an effect termed “cost of complexity.” However, little is known about how the distribution of mutation fitness effects (f(s)) varies across genomes. The main theoretical framework to address this issue is Fisher's geometric model and related phenotypic landscape models. However, it suffers from several restrictive assumptions. In this paper, we intend to show how several of these limitations may be overcome. We then propose a model of f(s) that extends Fisher's model to account for arbitrary mutational and selective interactions among n traits. We show that these interactions result in f(s) that would be predicted by a much smaller number of independent traits. We test our predictions by comparing empirical f(s) across species of various gene numbers as a surrogate to complexity. This survey reveals, as predicted, that mutations tend to be more deleterious, less variable, and less skewed in higher organisms. However, only limited difference in the shape of f(s) is observed from Escherichia coli to nematodes or fruit flies, a pattern consistent with a model of random phenotypic interactions across many traits. Overall, these results suggest that there may be a cost to phenotypic complexity although much weaker than previously suggested by earlier theoretical works. More generally, the model seems to qualitatively capture and possibly explain the variation of f(s) from lower to higher organisms, which opens a large array of potential applications in evolutionary genetics.

Fisher’s Microscope and Haldane’s Ellipse

Fisher's geometrical model was introduced to study the phenotypic size of mutations contributing to adaptation. However, as pointed out by Haldane, the model involves a simplified picture of the action of natural selection, and this calls into question its generality. In particular, Fisher's model assumes that each trait contributes independently to fitness. Here, we show that Haldane's concerns may be incorporated into Fisher's model solely by allowing the intensity of selection to vary between traits. We further show that this generalization may be achieved by introducing a single, intuitively defined quantity that describes the phenotype prior to adaptation. Comparing the process of adaptation under the original and generalized models, we show that the generalization may bias results toward either larger or smaller mutations. The applicability of Fisher's model is then discussed.

The distribution of fitness effects among beneficial mutations in Fisher's geometric model of adaptation

Recent models of adaptation at the DNA sequence level assume that the fitness effects of new mutations show certain statistical properties. In particular, these models assume that the distribution of fitness effects among new mutations is in the domain of attraction of the so-called Gumbel-type extreme value distribution. This assumption has not, however, been justified on any biological or theoretical grounds. In this note, I study random mutation in one of the simplest models of mutation and adaptation—Fisher's geometric model. I show that random mutation in this model yields a distribution of mutational effects that belongs to the Gumbel type. I also show that the distribution of fitness effects among rare beneficial mutations in Fisher's model is asymptotically exponential. I confirm these analytic findings with exact computer simulations. These results provide some support for the use of Gumbel-type extreme value theory in studies of adaptation and point to a surprising connection between recent phenotypic- and sequence-based models of adaptation: in both, the distribution of fitness effects among rare beneficial mutations is approximately exponential.

Theories of adaptation: what they do and don?t say

Theoretical work on adaptation has lagged behind experimental. But two classes of adaptation model have been partly explored. One is phenotypic and the other DNA sequence based. I briefly consider an example of each--Fisher's geometric model and Gillespie's mutational landscape model, respectively--reviewing recent results. Despite their fundamental differences, these models give rise to several strikingly similar results. I consider possible reasons for this congruence. I also emphasize what predictions do and, as important, do not follow from these models.

The population genetics of adaptation: the adaptation of DNA sequences.

I describe several patterns characterizing the genetics of adaptation at the DNA level. Following Gillespie (1983, 1984, 1991), I consider a population presently fixed for the ith best allele at a locus and study the sequential substitution of favorable mutations that results in fixation of the fittest DNA sequence locally available. Given a wild type sequence that is less than optimal, I derive the fitness rank of the next allele typically fixed by natural selection as well as the mean and variance of the jump in fitness that results when natural selection drives a substitution. Looking over the whole series of substitutions required to reach the best allele, I show that the mean fitness jumps occurring throughout an adaptive walk are constrained to a twofold window of values, assuming only that adaptation begins from a reasonably fit allele. I also show that the first substitution and the substitution of largest effect account for a large share of the total fitness increase during adaptation. I further show that the distribution of selection coefficients fixed throughout such an adaptive walk is exponential (ignoring mutations of small effect), a finding reminiscent of that seen in Fisher's geometric model of adaptation. Last, I show that adaptation by natural selection behaves in several respects as the average of two idealized forms of adaptation, perfect and random.

THE POPULATION GENETICS OF ADAPTATION: THE ADAPTATION OF DNA SEQUENCES

I describe several patterns characterizing the genetics of adaptation at the DNA level. Following Gillespie (1983, 1984, 1991), I consider a population presently fixed for the ith best allele at a locus and study the sequential substitution of favorable mutations that results in fixation of the fittest DNA sequence locally available. Given a wild type sequence that is less than optimal, I derive the fitness rank of the next allele typically fixed by natural selection as well as the mean and variance of the jump in fitness that results when natural selection drives a substitution. Looking over the whole series of substitutions required to reach the best allele, I show that the mean fitness jumps occurring throughout an adaptive walk are constrained to a twofold window of values, assuming only that adaptation begins from a reasonably fit allele. I also show that the first substitution and the substitution of largest effect account for a large share of the total fitness increase during adaptation. I further show that the distribution of selection coefficients fixed throughout such an adaptive walk is exponential (ignoring mutations of small effect), a finding reminiscent of that seen in Fisher's geometric model of adaptation. Last, I show that adaptation by natural selection behaves in several respects as the average of two idealized forms of adaptation, perfect and random.

Compensating for our load of mutations: freezing the meltdown of small populations.

We have investigated the reduction of fitness caused by the fixation of new deleterious mutations in small populations within the framework of Fisher's geometrical model of adaptation. In Fisher's model, a population evolves in an n-dimensional character space with an adaptive optimum at the origin. The model allows us to investigate compensatory mutations, which restore fitness losses incurred by other mutations, in a context-dependent manner. We have conducted a moment analysis of the model, supplemented by the numerical results of computer simulations. The mean reduction of fitness (i.e., expected load) scaled to one is approximately n/(n+2Ne), where Ne is the effective population size. The reciprocal relationship between the load and Ne implies that the fixation of deleterious mutations is unlikely to cause extinction when there is a broad scope for compensatory mutations, except in very small populations. Furthermore, the dependence of load on n implies that pleiotropy plays a large role in determining the extinction risk of small populations. Differences and similarities between our results and those of a previous study on the effects of Ne and n are explored. That the predictions of this model are qualitatively different from studies ignoring compensatory mutations implies that we must be cautious in predicting the evolutionary fate of small populations and that additional data on the nature of mutations is of critical importance.