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.     

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.     

Network models of TEM β-lactamase mutations coevolving under antibiotic selection show modular structure and anticipate evolutionary trajectories

Understanding how novel functions evolve (genetic adaptation) is a critical goal of evolutionary biology. Among asexual organisms, genetic adaptation involves multiple mutations that frequently interact in a non-linear fashion (epistasis). Non-linear interactions pose a formidable challenge for the computational prediction of mutation effects. Here we use the recent evolution of β-lactamase under antibiotic selection as a model for genetic adaptation. We build a network of coevolving residues (possible functional interactions), in which nodes are mutant residue positions and links represent two positions found mutated together in the same sequence. Most often these pairs occur in the setting of more complex mutants. Focusing on extended-spectrum resistant sequences, we use network-theoretical tools to identify triple mutant trajectories of likely special significance for adaptation. We extrapolate evolutionary paths (n = 3) that increase resistance and that are longer than the units used to build the network (n = 2). These paths consist of a limited number of residue positions and are enriched for known triple mutant combinations that increase cefotaxime resistance. We find that the pairs of residues used to build the network frequently decrease resistance compared to their corresponding singlets. This is a surprising result, given that their coevolution suggests a selective advantage. Thus, β-lactamase adaptation is highly epistatic. Our method can identify triplets that increase resistance despite the underlying rugged fitness landscape and has the unique ability to make predictions by placing each mutant residue position in its functional context. Our approach requires only sequence information, sufficient genetic diversity, and discrete selective pressures. Thus, it can be used to analyze recent evolutionary events, where coevolution analysis methods that use phylogeny or statistical coupling are not possible. Improving our ability to assess evolutionary trajectories will help predict the evolution of clinically relevant genes and aid in protein design.     

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.     

Natural transformation increases the rate of adaptation in the human pathogen Helicobacter pylori

Gene exchange between individuals can lead to profound evolutionary effects at both the genomic and population levels. These effects have sparked widespread interest in examining the specific adaptive benefits of recombination. Although this work has primarily focused on the benefits of sex in eukaryotes, it is assumed that similar benefits of genetic exchange apply across eukaryotes and prokaryotes. Here we report a direct test of this assumption using the naturally transformable human gastric pathogen Helicobacter pylori as a model organism. We show that genetic exchange accelerates adaptation to a novel laboratory environment within bacterial populations and that a general adaptive advantage exists for naturally transformable strains when transfer occurs among conspecific backgrounds. This finding demonstrates that there are generalized benefits to adaptation in both eukaryotes and prokaryotes even though the underlying processes are mechanistically different.     

No accident: genetic codes freeze in error-correcting patterns of the standard genetic code

The standard genetic code poses a challenge in understanding the evolution of information processing at a fundamental level of biological organization. Genetic codes are generally coadapted with, or 'frozen' by, the protein-coding genes that they translate, and so cannot easily change by natural selection. Yet the standard code has a significantly non-random pattern that corrects common errors in the transmission of information in protein-coding genes. Because of the freezing effect and for other reasons, this pattern has been proposed not to be due to selection but rather to be incidental to other evolutionary forces or even entirely accidental. We present results from a deterministic population genetic model of code-message coevolution. We explicitly represent the freezing effect of genes on genetic codes and the perturbative effect of changes in genetic codes on genes. We incorporate characteristic patterns of mutation and translational error, namely, transition bias and positional asymmetry, respectively. Repeated selection over small successive changes produces genetic codes that are substantially, but not optimally, error correcting. In particular, our model reproduces the error-correcting patterns of the standard genetic code. Aspects of our model and results may be applicable to the general problem of adaptation to error in other natural information-processing systems.     

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.     

The evolution of dominance: Haldane v Fisher revisited.

Fisher's model for the evolution of dominance indicates that the accumulation of dominance modifiers will be accelerated by (1) an increased frequency of the mutant heterozygote, (2) increased selection for the phenotype of the normal homozygote. The model has been criticised by Haldane on the grounds that point (1) is not fulfilled, that is dominance appears to be more common in populations with a low frequency of mutant heterozygotes (populations of inbreeders). In support of Fisher's model it is argued that intense selection for the wild type phenotype is more common in inbreeders than outbreeders. This situation should promote the accumulation of dominance modifiers (point (2) above).     

Evaluating the contributions of purifying selection and progeny-skew in dictating within-host Mycobacterium tuberculosis evolution

The within-host evolutionary dynamics of tuberculosis (TB) remain unclear, and underlying biological characteristics render standard population genetic approaches based upon the Wright-Fisher model largely inappropriate. In addition, the compact genome combined with an absence of recombination is expected to result in strong purifying selection effects. Thus, it is imperative to establish a biologically relevant evolutionary framework incorporating these factors in order to enable an accurate study of this important human pathogen. Further, such a model is critical for inferring fundamental evolutionary parameters related to patient treatment, including mutation rates and the severity of infection bottlenecks. We here implement such a model and infer the underlying evolutionary parameters governing within-patient evolutionary dynamics. Results demonstrate that the progeny skew associated with the clonal nature of TB severely reduces genetic diversity and that the neglect of this parameter in previous studies has led to significant mis-inference of mutation rates. As such, our results suggest an underlying de novo mutation rate that is considerably faster than previously inferred, and a progeny distribution differing significantly from Wright-Fisher assumptions. This inference represents a more appropriate evolutionary null model, against which the periodic effects of positive selection, associated with drug-resistance for example, may be better assessed.     

The evolution of evolvability in genetic linkage patterns

A number of factors have been proposed that may affect the capacity for an evolutionary system to generate adaptation. One that has received little recent attention among biologists is linkage patterns, or the ordering of genes on chromosomes. In this study, a simple model of genetic interactions, implemented in an evolutionary simulation, demonstrates that clustering of epistatically interacting genes increases the rate of adaptation. Moreover, long-term evolution with inversion can reorganize linkage patterns from random gene ordering into this more modular organization, thereby facilitating adaptation. These results are consistent with a large body of biological observations and some mathematical theory. Although linkage patterns are neutral with respect to individual fitness in this model, they are subject to lineage level selection for evolvability. At least two candidate mechanisms may contribute to improved evolvability under epistatic clustering: clustering may reduce interference between selection on different traits, and it may allow the simultaneous optimization of different recombination rates for gene pairs with additive and epistatic fitness effects.     

The probability of fixation of a single mutant in an exchangeable selection model

The Cannings exchangeable model for a finite population in discrete time is extended to incorporate selection. The probability of fixation of a mutant type is studied under the assumption of weak selection. An exact formula for the derivative of this probability with respect to the intensity of selection is deduced, and developed in the case of a single mutant. This formula is expressed in terms of mean coalescence times under neutrality assuming that the coefficient of selection for the mutant type has a derivative with respect to the intensity of selection that takes a polynomial form with respect to the frequency of the mutant type. An approximation is obtained in the case where this derivative is a continuous function of the mutant frequency and the population size is large. This approximation is consistent with a diffusion approximation under moment conditions on the number of descendants of a single individual in one time step. Applications to evolutionary game theory in finite populations are presented.      

Evolution of fitnesses and allele frequencies in a population with spatially heterogeneous selection pressures

The level of gene flow considerably influences the outcome of evolutionary processes in structured populations with spatial heterogeneity in selection pressures; low levels of gene flow may allow local adaptation whereas high levels of gene flow may oppose this process thus preventing the stable maintenance of polymorphism. Indeed, proportions of fitness space that successfully maintain polymorphism are substantially larger in spatially heterogenous populations with lower to moderate levels of gene flow when compared to single-deme models. Nevertheless, the effect of spatial heterogeneity on the evolutionary construction of polymorphism is less clear. We have investigated the levels of polymorphism resulting from a simple two-deme construction model, which incorporates recurrent mutation as well as selection. We further compared fitness properties, stability of equilibria, and frequency distribution patterns emerging from the construction approach and compared these to the static fitness-space approach. The construction model either promotes or constrains the level of polymorphisms, depending on the levels of gene flow. Comparison of the fitness properties resulting from both approaches shows that they maintain variation in different parts of fitness space. The part of fitness space resulting from construction is more stable than that implied by the ahistoric fitness-space approach. Finally, the equilibrium allele-frequency distribution patterns vary substantially with different levels of gene flow, underlining the importance of correctly sampling spatial structure if these patterns are to be used to estimate population-genetic processes.     

Considering ecological dynamics in resource selection functions

1.  Describing distribution and abundance is requisite to exploring interactions between organisms and their environment. Recently, the resource selection function (RSF) has emerged to replace many of the statistical procedures used to quantify resource selection by animals.
2.  A RSF is defined by characteristics measured on resource units such that its value for a unit is proportional to the probability of that unit being used by an organism. It is solved using a variety of techniques, particularly the binomial generalized linear model.
3.  Observing dynamics in a RSF – obtaining substantially different functions at different times or places for the same species – alerts us to the varying ecological processes that underlie resource selection.
4.  We believe that there is a need for us to reacquaint ourselves with ecological theory when interpreting RSF models. We outline a suite of factors likely to govern ecologically based variation in a RSF. In particular, we draw attention to competition and density-dependent habitat selection, the role of predation, longitudinal changes in resource availability and functional responses in resource use.
5.  How best to incorporate governing factors in a RSF is currently in a state of development; however, we see promise in the inclusion of random as well as fixed effects in resource selection models, and matched case–control logistic regression.
6.  Investigating the basis of ecological dynamics in a RSF will allow us to develop more robust models when applied to forecasting the spatial distribution of animals. It may also further our understanding of the relative importance of ecological interactions on the distribution and abundance of species.     

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.     

The place of development in mathematical evolutionary theory

Development plays a critical role in structuring the joint offspring-parent phenotype distribution. It thus must be part of any truly general evolutionary theory. Historically, the offspring-parent distribution has often been treated in such a way as to bury the contribution of development, by distilling from it a single term, either heritability or additive genetic variance, and then working only with this term. I discuss two reasons why this approach is no longer satisfactory. First, the regression of expected offspring phenotype on parent phenotype can easily be nonlinear, and this nonlinearity can have a pronounced impact on the response to selection. Second, even when the offspring-parent regression is linear, it is nearly always a function of the environment, and the precise way that heritability covaries with the environment can have a substantial effect on adaptive evolution. Understanding these complexities of the offspring-parent distribution will require understanding of the developmental processes underlying the traits of interest. I briefly discuss how we can incorporate such complexity into formal evolutionary theory, and why it is likely to be important even for traits that are not traditionally the focus of evo-devo research. Finally, I briefly discuss a topic that is widely seen as being squarely in the domain of evo-devo: novelty. I argue that the same conceptual and mathematical framework that allows us to incorporate developmental complexity into simple models of trait evolution also yields insight into the evolution of novel traits.     

Powerful detection of polygenic selection and evidence of environmental adaptation in US beef cattle

Selection on complex traits can rapidly drive evolution, especially in stressful environments. This polygenic selection does not leave intense sweep signatures on the genome, rather many loci experience small allele frequency shifts, resulting in large cumulative phenotypic changes. Directional selection and local adaptation are changing populations; but, identifying loci underlying polygenic or environmental selection has been difficult. We use genomic data on tens of thousands of cattle from three populations, distributed over time and landscapes, in linear mixed models with novel dependent variables to map signatures of selection on complex traits and local adaptation. We identify 207 genomic loci associated with an animal’s birth date, representing ongoing selection for monogenic and polygenic traits. Additionally, hundreds of additional loci are associated with continuous and discrete environments, providing evidence for historical local adaptation. These candidate loci highlight the nervous system’s possible role in local adaptation. While advanced technologies have increased the rate of directional selection in cattle, it has likely been at the expense of historically generated local adaptation, which is especially problematic in changing climates. When applied to large, diverse cattle datasets, these selection mapping methods provide an insight into how selection on complex traits continually shapes the genome. Further, understanding the genomic loci implicated in adaptation may help us breed more adapted and efficient cattle, and begin to understand the basis for mammalian adaptation, especially in changing climates. These selection mapping approaches help clarify selective forces and loci in evolutionary, model, and agricultural contexts.