The statistical mechanics of a polygenic characterunder stabilizing selection, mutation and drift

By exploiting an analogy between population genetics and statistical mechanics, we study the evolution of a polygenic trait under stabilizing selection, mutation, and genetic drift. This requires us to track only four macroscopic variables, instead of the distribution of all the allele frequencies that influence the trait. These macroscopic variables are the expectations of: the trait mean and its square, the genetic variance, and of a measure of heterozygosity, and are derived from a generating function that is in turn derived by maximizing an entropy measure. These four macroscopics are enough to accurately describe the dynamics of the trait mean and of its genetic variance (and in principle of any other quantity). Unlike previous approaches that were based on an infinite series of moments or cumulants, which had to be truncated arbitrarily, our calculations provide a well-defined approximation procedure. We apply the framework to abrupt and gradual changes in the optimum, as well as to changes in the strength of stabilizing selection. Our approximations are surprisingly accurate, even for systems with as few as 5 loci. We find that when the effects of drift are included, the expected genetic variance is hardly altered by directional selection, even though it fluctuates in any particular instance. We also find hysteresis, showing that even after averaging over the microscopic variables, the macroscopic trajectories retain a memory of the underlying genetic states.Comment: 35 pages, 8 figure

Emergence of clones in sexual populations

In sexual population, recombination reshuffles genetic variation and produces novel combinations of existing alleles, while selection amplifies the fittest genotypes in the population. If recombination is more rapid than selection, populations consist of a diverse mixture of many genotypes, as is observed in many populations. In the opposite regime, which is realized for example in the facultatively sexual populations that outcross in only a fraction of reproductive cycles, selection can amplify individual genotypes into large clones. Such clones emerge when the fitness advantage of some of the genotypes is large enough that they grow to a significant fraction of the population despite being broken down by recombination. The occurrence of this "clonal condensation" depends, in addition to the outcrossing rate, on the heritability of fitness. Clonal condensation leads to a strong genetic heterogeneity of the population which is not adequately described by traditional population genetics measures, such as Linkage Disequilibrium. Here we point out the similarity between clonal condensation and the freezing transition in the Random Energy Model of spin glasses. Guided by this analogy we explicitly calculate the probability, Y, that two individuals are genetically identical as a function of the key parameters of the model. While Y is the analog of the spin-glass order parameter, it is also closely related to rate of coalescence in population genetics: Two individuals that are part of the same clone have a recent common ancestor.Comment: revised versio

Asymptotic power law of moments in a random multiplicative process with weak additive noise

It is well known that a random multiplicative process with weak additive noise generates a power-law probability distribution. It has recently been recognized that this process exhibits another type of power law: the moment of the stochastic variable scales as a function of the additive noise strength. We clarify the mechanism for this power-law behavior of moments by treating a simple Langevin-type model both approximately and exactly, and argue this mechanism is universal. We also discuss the relevance of our findings to noisy on-off intermittency and to singular spatio-temporal chaos recently observed in systems of non-locally coupled elements.Comment: 11 pages, 9 figures, submitted to Phys. Rev.

A simple mathematical model of gradual Darwinian evolution: Emergence of a Gaussian trait distribution in adaptation along a fitness gradient

We consider a simple mathematical model of gradual Darwinian evolution in continuous time and continuous trait space, due to intraspecific competition for common resource in an asexually reproducing population in constant environment, while far from evolutionary stable equilibrium. The model admits exact analytical solution. In particular, Gaussian distribution of the trait emerges from generic initial conditions.Comment: 21 pages, 2 figures, as accepted to J Math Biol 2013/03/1

The genetics of mate preferences in hybrids between two young and sympatric Lake Victoria cichlid species

The genetic architecture of mate preferences is likely to affect significant evolutionary processes, including speciation and hybridization. Here, we investigate laboratory hybrids between a pair of sympatric Lake Victoria cichlid fish species that appear to have recently evolved from a hybrid population between similar predecessor species. The species demonstrate strong assortative mating in the laboratory, associated with divergent male breeding coloration (red dorsum versus blue). We show in a common garden experiment, using DNA-based paternity testing, that the strong female mate preferences among males of the two species are fully recovered in a large fraction of their F2 hybrid generation. Individual hybrid females often demonstrated consistent preferences in multiple mate choice trials (more than or equal to five) across a year or more. This result suggests that female mate preference is influenced by relatively few major genes or genomic regions. These preferences were not changed by experience of a successful spawning event with a male of the non-preferred species in a no-choice single-male trial. We found no evidence for imprinting in the F2 hybrids, although the F1 hybrid females may have been imprinted on their mothers. We discuss this nearly Mendelian inheritance of consistent innate mate preferences in the context of speciation theory

Monte carlo simulations of parapatric speciation

Parapatric speciation is studied using an individual--based model with sexual reproduction. We combine the theory of mutation accumulation for biological ageing with an environmental selection pressure that varies according to the individuals geographical positions and phenotypic traits. Fluctuations and genetic diversity of large populations are crucial ingredients to model the features of evolutionary branching and are intrinsic properties of the model. Its implementation on a spatial lattice gives interesting insights into the population dynamics of speciation on a geographical landscape and the disruptive selection that leads to the divergence of phenotypes. Our results suggest that assortative mating is not an obligatory ingredient to obtain speciation in large populations at low gene flow.Comment: submitted to Phys.Rev.

How Gaussian competition leads to lumpy or uniform species distributions

A central model in theoretical ecology considers the competition of a range of species for a broad spectrum of resources. Recent studies have shown that essentially two different outcomes are possible. Either the species surviving competition are more or less uniformly distributed over the resource spectrum, or their distribution is 'lumped' (or 'clumped'), consisting of clusters of species with similar resource use that are separated by gaps in resource space. Which of these outcomes will occur crucially depends on the competition kernel, which reflects the shape of the resource utilization pattern of the competing species. Most models considered in the literature assume a Gaussian competition kernel. This is unfortunate, since predictions based on such a Gaussian assumption are not robust. In fact, Gaussian kernels are a border case scenario, and slight deviations from this function can lead to either uniform or lumped species distributions. Here we illustrate the non-robustness of the Gaussian assumption by simulating different implementations of the standard competition model with constant carrying capacity. In this scenario, lumped species distributions can come about by secondary ecological or evolutionary mechanisms or by details of the numerical implementation of the model. We analyze the origin of this sensitivity and discuss it in the context of recent applications of the model.Comment: 11 pages, 3 figures, revised versio

Using evolutionary demography to link life history theory, quantitative genetics and population ecology

1. There is a growing number of empirical reports of environmental change simultaneously influencing population dynamics, life history and quantitative characters. We do not have a well-developed understanding of links between the dynamics of these quantities

Predicting evolution and visualizing high-dimensional fitness landscapes

The tempo and mode of an adaptive process is strongly determined by the structure of the fitness landscape that underlies it. In order to be able to predict evolutionary outcomes (even on the short term), we must know more about the nature of realistic fitness landscapes than we do today. For example, in order to know whether evolution is predominantly taking paths that move upwards in fitness and along neutral ridges, or else entails a significant number of valley crossings, we need to be able to visualize these landscapes: we must determine whether there are peaks in the landscape, where these peaks are located with respect to one another, and whether evolutionary paths can connect them. This is a difficult task because genetic fitness landscapes (as opposed to those based on traits) are high-dimensional, and tools for visualizing such landscapes are lacking. In this contribution, we focus on the predictability of evolution on rugged genetic fitness landscapes, and determine that peaks in such landscapes are highly clustered: high peaks are predominantly close to other high peaks. As a consequence, the valleys separating such peaks are shallow and narrow, such that evolutionary trajectories towards the highest peak in the landscape can be achieved via a series of valley crossingsComment: 12 pages, 7 figures. To appear in "Recent Advances in the Theory and Application of Fitness Landscapes" (A. Engelbrecht and H. Richter, eds.). Springer Series in Emergence, Complexity, and Computation, 201