The statistical mechanics of a polygenic character under 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 five 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.

Mapping Mendelian traits in asexual progeny using changes in marker allele frequency.

Linkage between markers and genes that affect a phenotype of interest may be determined by examining differences in marker allele frequency in the extreme progeny of a cross between two inbred lines. This strategy is usually employed when pooling is used to reduce genotyping costs. When the cross progeny are asexual, the extreme progeny may be selected by multiple generations of asexual reproduction and selection. We analyse this method of measuring phenotype in asexual progeny and examine the changes in marker allele frequency due to selection over many generations. Stochasticity in marker frequency in the selected population arises due to the finite initial population size. We derive the distribution of marker frequency as a result of selection at a single major locus, and show that in order to avoid spurious changes in marker allele frequency in the selected population, the initial population size should be in the low to mid hundreds.

Cryptic variation between species and the basis of hybrid performance.

Crosses between closely related species give two contrasting results. One result is that species hybrids may be inferior to their parents, for example, being less fertile [1]. The other is that F1 hybrids may display superior performance (heterosis), for example with increased vigour [2]. Although various hypotheses have been proposed to account for these two aspects of hybridisation, their biological basis is still poorly understood [3]. To gain further insights into this issue, we analysed the role that variation in gene expression may play. We took a conserved trait, flower asymmetry in Antirrhinum, and determined the extent to which the underlying regulatory genes varied in expression among closely related species. We show that expression of both genes analysed, CYC and RAD, varies significantly between species because of cis-acting differences. By making a quantitative genotype-phenotype map, using a range of mutant alleles, we demonstrate that the species lie on a plateau in gene expression-morphology space, so that the variation has no detectable phenotypic effect. However, phenotypic differences can be revealed by shifting genotypes off the plateau through genetic crosses. Our results can be readily explained if genomes are free to evolve within an effectively neutral zone in gene expression space. The consequences of this drift will be negligible for individual loci, but when multiple loci across the genome are considered, we show that the variation may have significant effects on phenotype and fitness, causing a significant drift load. By considering these consequences for various gene-expression-fitness landscapes, we conclude that F1 hybrids might be expected to show increased performance with regard to conserved traits, such as basic physiology, but reduced performance with regard to others. Thus, our study provides a new way of explaining how various aspects of hybrid performance may arise through natural variation in gene activity.

Effects of spontaneous mutation accumulation on sex ratio traits in a parasitoid wasp.

Sex allocation theory has proved extremely successful at predicting when individuals should adjust the sex of their offspring in response to environmental conditions. However, we know rather little about the underlying genetics of sex ratio or how genetic architecture might constrain adaptive sex-ratio behavior. We examined how mutation influenced genetic variation in the sex ratios produced by the parasitoid wasp Nasonia vitripennis. In a mutation accumulation experiment, we determined the mutability of sex ratio, and compared this with the amount of genetic variation observed in natural populations. We found that the mutability (h(2)(m)) ranges from 0.001 to 0.002, similar to estimates for life-history traits in other organisms. These estimates suggest one mutation every 5-60 generations, which shift the sex ratio by approximately 0.01 (proportion males). In this and other studies, the genetic variation in N. vitripennis sex ratio ranged from 0.02 to 0.17 (broad-sense heritability, H(2)). If sex ratio is maintained by mutation-selection balance, a higher genetic variance would be expected given our mutational parameters. Instead, the observed genetic variance perhaps suggests additional selection against sex-ratio mutations with deleterious effects on other fitness traits as well as sex ratio (i.e., pleiotropy), as has been argued to be the case more generally.

The Hill-Robertson effect and the evolution of recombination.

In finite populations, genetic drift generates interference between selected loci, causing advantageous alleles to be found more often on different chromosomes than on the same chromosome, which reduces the rate of adaptation. This "Hill-Robertson effect" generates indirect selection to increase recombination rates. We present a new method to quantify the strength of this selection. Our model represents a new beneficial allele (A) entering a population as a single copy, while another beneficial allele (B) is sweeping at another locus. A third locus affects the recombination rate between selected loci. Using a branching process model, we calculate the probability distribution of the number of copies of A on the different genetic backgrounds, after it is established but while it is still rare. Then, we use a deterministic model to express the change in frequency of the recombination modifier, due to hitchhiking, as A goes to fixation. We show that this method can give good estimates of selection for recombination. Moreover, it shows that recombination is selected through two different effects: it increases the fixation probability of new alleles, and it accelerates selective sweeps. The relative importance of these two effects depends on the relative times of occurrence of the beneficial alleles.

Chromosomal speciation and molecular divergence--accelerated evolution in rearranged chromosomes.

Humans and their closest evolutionary relatives, the chimpanzees, differ in approximately 1.24% of their genomic DNA sequences. The fraction of these changes accumulated during the speciation processes that have separated the two lineages may be of special relevance in understanding the basis of their differences. We analyzed human and chimpanzee sequence data to search for the patterns of divergence and polymorphism predicted by a theoretical model of speciation. According to the model, positively selected changes should accumulate in chromosomes that present fixed structural differences, such as inversions, between the two species. Protein evolution was more than 2.2 times faster in chromosomes that had undergone structural rearrangements compared with colinear chromosomes. Also, nucleotide variability is slightly lower in rearranged chromosomes. These patterns of divergence and polymorphism may be, at least in part, the molecular footprint of speciation events in the human and chimpanzee lineages.

Accumulating postzygotic isolation genes in parapatry: a new twist on chromosomal speciation.

Chromosomal rearrangements can promote reproductive isolation by reducing recombination along a large section of the genome. We model the effects of the genetic barrier to gene flow caused by a chromosomal rearrangement on the rate of accumulation of postzygotic isolation genes in parapatry. We find that, if reproductive isolation is produced by the accumulation in parapatry of sets of alleles compatible within but incompatible across species, chromosomal rearrangements are far more likely to favor it than classical genetic barriers without chromosomal changes. New evidence of the role of chromosomal rearrangements in parapatric speciation suggests that postzygotic isolation is often due to the accumulation of such incompatibilities. The model makes testable qualitative predictions about the genetic signature of speciation.

The effect of deleterious alleles on adaptation in asexual populations.

We calculate the fixation probability of a beneficial allele that arises as the result of a unique mutation in an asexual population that is subject to recurrent deleterious mutation at rate U. Our analysis is an extension of previous works, which make a biologically restrictive assumption that selection against deleterious alleles is stronger than that on the beneficial allele of interest. We show that when selection against deleterious alleles is weak, beneficial alleles that confer a selective advantage that is small relative to U have greatly reduced probabilities of fixation. We discuss the consequences of this effect for the distribution of effects of alleles fixed during adaptation. We show that a selective sweep will increase the fixation probabilities of other beneficial mutations arising during some short interval afterward. We use the calculated fixation probabilities to estimate the expected rate of fitness improvement in an asexual population when beneficial alleles arise continually at some low rate proportional to U. We estimate the rate of mutation that is optimal in the sense that it maximizes this rate of fitness improvement. Again, this analysis relaxes the assumption made previously that selection against deleterious alleles is stronger than on beneficial alleles.

Extending the coalescent to multilocus systems: the case of balancing selection.

Natural populations are structured spatially into local populations and genetically into diverse 'genetic backgrounds' defined by different combinations of selected alleles. If selection maintains genetic backgrounds at constant frequency then neutral diversity is enhanced. By contrast, if background frequencies fluctuate then diversity is reduced. Provided that the population size of each background is large enough, these effects can be described by the structured coalescent process. Almost all the extant results based on the coalescent deal with a single selected locus. Yet we know that very large numbers of genes are under selection and that any substantial effects are likely to be due to the cumulative effects of many loci. Here, we set up a general framework for the extension of the coalescent to multilocus scenarios and we use it to study the simplest model, where strong balancing selection acting on a set of n loci maintains 2n backgrounds at constant frequencies and at linkage equilibrium. Analytical results show that the expected linked neutral diversity increases exponentially with the number of selected loci and can become extremely large. However, simulation results reveal that the structured coalescent approach breaks down when the number of backgrounds approaches the population size, because of stochastic fluctuations in background frequencies. A new method is needed to extend the structured coalescent to cases with large numbers of backgrounds.

The effects of multilocus balancing selection on neutral variability.

We studied the effect of multilocus balancing selection on neutral nucleotide variability at linked sites by simulating a model where diallelic polymorphisms are maintained at an arbitrary number of selected loci by means of symmetric overdominance. Different combinations of alleles define different genetic backgrounds that subdivide the population and strongly affect variability. Several multilocus fitness regimes with different degrees of epistasis and gametic disequilibrium are allowed. Analytical results based on a multilocus extension of the structured coalescent predict that the expected linked neutral diversity increases exponentially with the number of selected loci and can become extremely large. Our simulation results show that although variability increases with the number of genetic backgrounds that are maintained in the population, it is reduced by random fluctuations in the frequencies of those backgrounds and does not reach high levels even in very large populations. We also show that previous results on balancing selection in single-locus systems do not extend to the multilocus scenario in a straightforward way. Different patterns of linkage disequilibrium and of the frequency spectrum of neutral mutations are expected under different degrees of epistasis. Interestingly, the power to detect balancing selection using deviations from a neutral distribution of allele frequencies seems to be diminished under the fitness regime that leads to the largest increase of variability over the neutral case. This and other results are discussed in the light of data from the Mhc.

Neutral evolution in spatially continuous populations.

We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum. The recursion allows for the interactions induced by density-dependent regulation of the population, which are inevitable in a continuous population. We give explicit series expansions for large neighbourhood size and for low mutation rates respectively and investigate the accuracy of the classical Malécot formula for these general models. When neighbourhood size is small, this formula does not give the identity even over large scales. However, for large neighbourhood size, it is an accurate approximation which summarises the local population structure in terms of three quantities: the effective dispersal rate, sigma(e); the effective population density, rho(e); and a local scale, kappa, at which local interactions become significant. The results are illustrated by simulations.

HYBRID DYSFUNCTION IN FIRE-BELLIED TOADS (BOMBINA).

Reproductive isolation between two taxa may be due to endogenous selection, which is generated by incompatibilities between the respective genomes, to exogenous selection, which is generated by differential adaptations to alternative environments, or to both. The continuing debate over the relative importance of either mode of selection has highlighted the need for unambiguous data on the fitness of hybrid genotypes. The hybrid zone between the fire-bellied toad (Bombina bombina) and the yellow-bellied toad (B. variegata) in central Europe involves adaptation to different environments, but evidence of hybrid dysfunction is equivocal. In this study, we followed the development under laboratory conditions of naturally laid eggs collected from a transect across the Bombina hybrid zone in Croatia. Fitness was significantly reduced in hybrid populations: Egg batches from the center of the hybrid zone showed significantly higher embryonic and larval mortality and higher frequencies of morphological abnormalities relative to either parental type. Overall mortality from day of egg collection to three weeks after hatching reached 20% in central hybrid populations, compared to 2% in pure populations. There was no significant difference in fitness between two parental types. Within hybrid populations, there was considerable variation in fitness, with some genotypes showing no evidence of reduced viability. We discuss the implications of these findings for our understanding of barriers to gene flow between species.