APPLICABILITY OF THE HYPERGEOMETRIC PHENOTYPIC MODEL TO HAPLOID AND DIPLOID POPULATIONS

Sympatric ecological divergence with coevolution of niche preference

Reinforcement, the increase of assortative mating driven by selection against unfit hybrids, is conditional on pre-existing divergence. Yet, for ecological divergence to precede the evolution of assortment, strict symmetries between fitnesses in niches must hold, and/or there must be low gene flow between the nascent species. It has thus been argued that conditions favouring sympatric speciation are rarely met in nature. Indeed, we show that under disruptive selection, violating symmetries in niche sizes and increasing strength of the trade-off in selection between the niches quickly leads to loss of genetic variation, instead of evolution of specialists. The region of the parameter space where polymorphism is maintained further narrows with increasing number of loci encoding the diverging trait and the rate of recombination between them. Yet, evolvable assortment and pre-existing assortment both substantially broaden the parameter space within which polymorphism is maintained. Notably, pre-existing niche preference speeds up further increase of assortment, thus facilitating reinforcement in the later phases of speciation. We conclude that in order for sympatric ecological divergence to occur, niche preference must coevolve throughout the divergence process. Even if populations come into secondary contact, having diverged in isolation, niche preference substantially broadens the conditions for coexistence in sympatry and completion of the speciation process.

This article is part of the theme issue ‘Towards the completion of speciation: the evolution of reproductive isolation beyond the first barriers'.

The effect of intraspecific variation and heritability on community pattern and robustness

Intraspecific trait variation is widespread in nature, yet its effects on community dynamics are not well understood. Here we explore the consequences of intraspecific trait variation for coexistence in two- and multispecies competitive communities. For two species, the likelihood of coexistence is in general reduced by intraspecific variation, except when the species have almost equal trait means but different trait variances, such that one is a generalist and the other a specialist consumer. In multispecies communities, the only strong effect of non-heritable intraspecific variation is to reduce expected species richness. However, when intraspecific variation is heritable, allowing for the possibility of trait evolution, communities are much more resilient against environmental disturbance and exhibit far more predictable trait patterns. Our results are robust to varying model parameters and relaxing model assumptions.

Genetics, adaptation, and invasion in harsh environments

We analyze mathematical models to examine how the genetic basis of fitness affects the persistence of a population suddenly encountering a harsh environment where it would go extinct without evolution. The results are relevant for novel introductions and for an established population whose existence is threatened by a sudden change in the environment. The models span a range of genetic assumptions, including identical loci that contribute to absolute fitness, a two-locus quantitative genetic model with nonidentical loci, and a model with major and minor genes affecting a quantitative trait. We find as a general (though not universal) pattern that prospects for persistence narrow as more loci contribute to fitness, in effect because selection per locus is increasingly weakened with more loci, which can even overwhelm any initial enhancement of fitness that adding loci might provide. When loci contribute unequally to fitness, genes of small effect can significantly reduce extinction risk. Indeed, major and minor genes can interact synergistically to reduce the time needed to evolve growth. Such interactions can also increase vulnerability to extinction, depending not just on how genes interact but also on the initial genetic structure of the introduced, or newly invaded, population.

Multi-site adaptation in the presence of infrequent recombination

The adverse effect of co-inheritance linkage of a large number of sites on adaptation has been studied extensively for asexual populations. However, it is insufficiently understood for multi-site populations in the presence of recombination. In the present work, motivated by our studies of HIV evolution in infected patients, we consider a model of haploid populations with infrequent recombination. We assume that small quantities of beneficial alleles preexist at a large number of sites and neglect new mutation. Using a generalized form of the traveling wave method, we show that the effectiveness of recombination is impeded and the adaptation rate is decreased by inter-sequence correlations, arising due to the fact that some pairs of homologous sites have common ancestors existing after the onset of adaptation. As the recombination rate per individual becomes smaller, site pairs with common ancestors become more frequent, making recombination even less effective. In addition, an increasing number of sites become identical by descent across large samples of sequences, causing reversion of the direction of evolution and the loss of beneficial alleles at these sites. As a result, within a 10-fold range of the recombination rate, the average adaptation rate falls from 90% of the infinite-recombination value down to 10%. The entire transition from almost maximum to almost zero may occur at very small recombination rates. Interestingly, the strong effect of linkage on the adaptation rate is predicted in the absence of average linkage disequilibrium (Lewontin's measure).

The evolution of strong reproductive isolation

Felsenstein distinguished two ways by which selection can directly strengthen isolation. First, a modifier that strengthens prezygotic isolation can be favored everywhere. This fits with the traditional view of reinforcement as an adaptation to reduce deleterious hybridization by strengthening assortative mating. Second, selection can favor association between different incompatibilities, despite recombination. We generalize this "two allele" model to follow associations among any number of incompatibilities, which may include both assortment and hybrid inviability. Our key argument is that this process, of coupling between incompatibilities, may be quite different from the usual view of reinforcement: strong isolation can evolve through the coupling of any kind of incompatibility, whether prezygotic or postzygotic. Single locus incompatibilities become coupled because associations between them increase the variance in compatibility, which in turn increases mean fitness if there is positive epistasis. Multiple incompatibilities, each maintained by epistasis, can become coupled in the same way. In contrast, a single-locus incompatibility can become coupled with loci that reduce the viability of haploid hybrids because this reduces harmful recombination. We obtain simple approximations for the limits of tight linkage, and strong assortment, and show how assortment alleles can invade through associations with other components of reproductive isolation.

Connecting QTLS to the g-matrix of evolutionary quantitative genetics

Evolutionary quantitative genetics has recently advanced in two distinct streams. Many biologists address evolutionary questions by estimating phenotypic selection and genetic (co)variances (G matrices). Simultaneously, an increasing number of studies have applied quantitative trait locus (QTL) mapping methods to dissect variation. Both conceptual and practical difficulties have isolated these two foci of quantitative genetics. A conceptual integration follows from the recognition that QTL allele frequencies are the essential variables relating the G-matrix to marker-based mapping experiments. Breeding designs initiated from randomly selected parental genotypes can be used to estimate QTL-specific genetic (co)variances. These statistics appropriately distill allelic variation and provide an explicit population context for QTL mapping estimates. Within this framework, one can parse the G-matrix into a set of mutually exclusive genomic components and ask whether these parts are similar or dissimilar in their respective features, for example the magnitude of phenotypic effects and the extent and nature of pleiotropy. As these features are critical determinants of sustained response to selection, the integration of QTL mapping methods into G-matrix estimation can provide a concrete, genetically based experimental program to investigate the evolutionary potential of natural populations.

Evolutionary consequences of asymmetric dispersal rates

We study the consequences of asymmetric dispersal rates (e.g., due to wind or current) for adaptive evolution in a system of two habitat patches. Asymmetric dispersal rates can lead to overcrowding of the "downstream" habitat, resulting in a source-sink population structure in the absence of intrinsic quality differences between habitats or can even cause an intrinsically better habitat to function as a sink. Source-sink population structure due to asymmetric dispersal rates has similar consequences for adaptive evolution as a source-sink structure due to habitat quality differences: natural selection tends to be biased toward the source habitat. We demonstrate this for two models of adaptive evolution: invasion of a rare allele that improves fitness in one habitat but reduces it in the other and antagonistic selection on a quantitative trait determined by five additive loci. If a habitat can sustain a population without immigration, the conditions for adaptation to that habitat are most favorable if there is little or no immigration from the other habitat; the influence of emigration depends on the magnitude of the allelic effects involved and other parameters. If, however, the population is initially unable to persist in a given habitat without immigration, our model predicts that the population will be most likely to adapt to that habitat if the dispersal rates in both directions are high. Our results highlight the general message that the effect of gene flow upon local adaptation should depend profoundly on the demographic context of selection.

Diploid biological evolution models with general smooth fitness landscapes and recombination

Using a Hamilton-Jacobi equation approach, we obtain analytic equations for steady-state population distributions and mean fitness functions for Crow-Kimura and Eigen-type diploid biological evolution models with general smooth hypergeometric fitness landscapes. Our numerical solutions of diploid biological evolution models confirm the analytic equations obtained. We also study the parallel diploid model for the simple case of recombination and calculate the variance of distribution, which is consistent with numerical results.

Highly fit ancestors of a partly sexual haploid population

Earlier, using the semi-deterministic solitary wave approach, we have investigated accumulation of pre-existing beneficial alleles in genomes consisting of a large number of simultaneously evolving sites in the presence of selection and infrequent recombination with small rate r per genome. Our previous results for the dynamics of the fitness distribution of genomes are now interpreted in terms of the life cycle of recombinant clones. We show that, at sufficiently small r, the clones dominating fitness classes, at the moment of their birth, are nearly the best fit in a population. New progeny clones are mostly generated by parental genomes whose fitness falls within a narrow interval in the middle of the high-fitness tail of fitness distribution. We also derive the fitness distribution for the distant ancestors of sites of a randomly sampled genome and show that its form is controlled by a single composite model parameter proportional to r. The ancestral fitness distribution differs dramatically from the fitness distribution of the entire ancient population: it is much broader and localized in the high-fitness tail of the ancient population. We generalize these results to the case of moderately small r to conclude that, regardless of fitness of an individual, all its distant ancestors are exceptionally well fit.

Sympatric speciation under incompatibility selection

The existing theory of sympatric speciation assumes that a local population splits into two species under one-dimensional disruptive selection, which favors both of the opposite extreme values of a quantitative trait. Here we model sympatric speciation under selection that favors high values of either of the two independently inherited traits, each required to efficiently consume one of the two available resources, but acts, because of a tradeoff, against those possessing high values of both traits. Such two-dimensional incompatibility selection is similar to that involved in allopatric speciation. Using a hypergeometric phenotypic model, we show that incompatibility selection readily leads to sympatric speciation. In contrast to disruptive selection, two distinct modes of sympatric speciation exist under incompatibility selection: under strong tradeoffs both of the new species are specialists, each consuming its own resource, but under moderate tradeoffs speciation may be asymmetric and involve the origin of a specialist and a generalist species. Also, incompatibility selection may lead to irreversible specialization: under strong tradeoffs, the population speciates if it consists mostly of unspecialized individuals, but remains undivided if most of the individuals are specialized to consume one of the resources. Incompatibility selection appears to be more realistic than disruptive selection, implying that incompatibility between individually adaptive alleles or trait states drives both allopatric and sympatric speciation.

How do natural and sexual selection contribute to sympatric speciation?

I use explicit genetic models to investigate the importance of natural and sexual selection during sympatric speciation and to sort out how genetic architecture influences these processes. Assortative mating alone can lead to speciation, but rare phenotypes' disadvantage in finding mates and intermediate phenotypes' advantage due to stabilizing selection strongly impede speciation. Any increase in the number of loci also decreases the likelihood of speciation. Sympatric speciation is then harder to achieve than previously demonstrated by many theoretical studies which assume no mating disadvantage for rare phenotypes and consider a small number of loci. However, when a high level of assortative mating evolves, sexual selection might allow populations to split into dimorphic distributions with peaks corresponding to nearly extreme phenotypes. Competition then works against speciation by favouring intermediate phenotypes and preventing further divergence. The interplay between natural and sexual selection during speciation is then more complex than previously explained.

The Maintenance (or Not) of Polygenic Variation by Soft Selection in Heterogeneous Environments

On the basis of single-locus models, spatial heterogeneity of the environment coupled with strong population regulation within each habitat (soft selection) is considered an important mechanism maintaining genetic variation. We studied the capacity of soft selection to maintain polygenic variation for a trait determined by several additive loci, selected in opposite directions in two habitats connected by dispersal. We found three main types of stable equilibria. Extreme equilibria are characterized by extreme specialization to one habitat and loss of polymorphism. They are analogous to monomorphic equilibria in singe-locus models and are favored by similar factors: high dispersal, weak selection, and low marginal average fitness of intermediate genotypes. At the remaining two types of equilibria the population mean is intermediate but variance is very different. At fully polymorphic equilibria all loci are polymorphic, whereas at low-variance equilibria at most one locus remains polymorphic. For most parameters only one type of equilibrium is stable; the transition between the domains of fully polymorphic and low-variance equilibria is typically sharp. Low-variance equilibria are favored by high marginal average fitness of intermediate genotypes, in contrast to single-locus models, in which marginal overdominance is particularly favorable for maintenance of polymorphism. The capacity of soft selection to maintain polygenic variation is thus more limited than extrapolation from single-locus models would suggest, in particular if dispersal is high and selection weak. This is because in a polygenic model, variance can evolve independently of the mean, whereas in the single-locus two-allele case, selection for an intermediate mean automatically leads to maintenance of polymorphism.

Sex‐Biased Dispersal and Adaptation to Marginal Habitats

If gene flow occurs through both sexes but only females contribute to population growth, adaptation to marginal (sink) habitats should be differentially affected by male versus female dispersal. Here I address this problem with two models. First, I consider the fate of a rare allele that improves fitness in the marginal habitat but reduces fitness in the core (source) habitat. Then I study the evolution of a polygenic character mediating a trade-off in fitness between the habitats. Both approaches led to qualitatively similar predictions. The effect of a difference in the dispersal rate between the sexes depends on the degree to which immigration from the core habitat boosts the reproductive output from the marginal habitat. This boost is slight if the marginal habitat is able to sustain well a population without immigration. In that case, both female- and male-biased dispersal is more favorable for adaptation to marginal habitats than equal dispersal of both sexes (assuming that the dispersal rate averaged over the sexes is kept constant). In contrast, if the marginal habitat is an absolute sink unable to sustain a population without immigration, the conditions for adaptation to that habitat are least favorable under highly male-biased dispersal and most favorable under highly female-biased dispersal. Under some circumstances, high average (male+female) dispersal is more favorable than low dispersal. Thus, gene flow should not be seen solely as thwarting adaptation to marginal habitats. The results are interpreted in terms of how male and female dispersal affects the relative rate of gene flow from the source to the sink habitat and in the opposite direction. This study predicts that ecological niches of taxa with female-biased dispersal should tend to be broader and more evolutionarily flexible.

Sympatric speciation as a consequence of male pregnancy in seahorses

The phenomenon of male pregnancy in the family Syngnathidae (seahorses, pipefishes, and sea dragons) undeniably has sculpted the course of behavioral evolution in these fishes. Here we explore another potentially important but previously unrecognized consequence of male pregnancy: a predisposition for sympatric speciation. We present microsatellite data on genetic parentage that show that seahorses mate size-assortatively in nature. We then develop a quantitative genetic model based on these empirical findings to demonstrate that sympatric speciation indeed can occur under this mating regime in response to weak disruptive selection on body size. We also evaluate phylogenetic evidence bearing on sympatric speciation by asking whether tiny seahorse species are sister taxa to large sympatric relatives. Overall, our results indicate that sympatric speciation is a plausible mechanism for the diversification of seahorses, and that assortative mating (in this case as a result of male parental care) may warrant broader attention in the speciation process for some other taxonomic groups as well.

General models of multilocus evolution

In 1991, Barton and Turelli developed recursions to describe the evolution of multilocus systems under arbitrary forms of selection. This article generalizes their approach to allow for arbitrary modes of inheritance, including diploidy, polyploidy, sex linkage, cytoplasmic inheritance, and genomic imprinting. The framework is also extended to allow for other deterministic evolutionary forces, including migration and mutation. Exact recursions that fully describe the state of the population are presented; these are implemented in a computer algebra package (available on the Web at http://helios.bto.ed.ac.uk/evolgen). Despite the generality of our framework, it can describe evolutionary dynamics exactly by just two equations. These recursions can be further simplified using a "quasi-linkage equilibrium" (QLE) approximation. We illustrate the methods by finding the effect of natural selection, sexual selection, mutation, and migration on the genetic composition of a population.

Estimating multilocus linkage disequilibria

The state of a diploid population segregating for two alleles at each of n loci is described by 22n genotype frequencies, or equivalently, by allele frequencies and by multilocus moments or cumulants of various orders. These measures of linkage disequilibrium cannot usually be determined, both because one cannot tell whether a gene came from the maternal or paternal gamete, and because such a large number of parameters cannot be estimated even from large samples. Simplifying assumptions must therefore be made. This paper sets out methods for estimating multilocus genotype frequencies which are appropriate for unlinked neutral loci, and for populations that are ultimately derived by mixing of two source populations. In such a hybrid population, all multilocus associations depend primarily on the number of loci involved that derive from the maternal genome, and the number derived from the paternal genome. Allele frequencies may differ across loci, and the contribution of each locus to multilocus associations may be scaled by the difference in allele frequency between source populations for that locus (deltap </= 1). For example, the cumulant describing the association between genes i, j, k from the maternal genome, and genes i, l from the paternal genome is kappai,j,k,i*l*, = deltapi2 deltapj deltapk deltapl kappa3,2. The state of the population is described by n allele frequencies; n divergences, deltap; and by a symmetric matrix of cumulants, kappaJ,K (J=0,ellipsis, n, K=0, ellipsis, n). Expressions for these cumulants under short- and long-range migration are given. Two methods for estimating the cumulants are described: a simple method based on multivariate moments, and a maximum likelihood procedure, which uses the Metropolis algorithm. Both methods perform well when tested against simulations with two or four loci.

Interactions among quantitative traits in the course of sympatric speciation

Sympatric speciation, the origin of two or more species from a single local population, has almost certainly been involved in formation of several species flocks, and may be fairly common in nature. The most straightforward scenario for sympatric speciation requires disruptive selection favouring two substantially different phenotypes, and consists of the evolution of reproductive isolation between them followed by the elimination of all intermediate phenotypes. Here we use the hypergeometric phenotypic model to show that sympatric speciation is possible even when fitness and mate choice depend on different quantitative traits, so that speciation must involve formation of covariance between these traits. The increase in the number of variable loci affecting fitness facilitates sympatric speciation, whereas the increase in the number of variable loci affecting mate choice has the opposite effect. These predictions may enable more cases of sympatric speciation to be identified.