The Genealogy of Samples in Models with Selection

We introduce the genealogy of a random sample of genes taken from a large haploid population that evolves according to random reproduction with selection and mutation. Without selection, the genealogy is described by Kingman''s well-known coalescent process. In the selective case, the genealogy of the sample is embedded in a graph with a coalescing and branching structure. We describe this graph, called the ancestral selection graph, and point out differences and similarities with Kingman''s coalescent. We present simulations for a two-allele model with symmetric mutation in which one of the alleles has a selective advantage over the other. We find that when the allele frequencies in the population are already in equilibrium, then the genealogy does not differ much from the neutral case. This is supported by rigorous results. Furthermore, we describe the ancestral selection graph for other selective models with finitely many selection classes, such as the K-allele models, infinitely-many-alleles models, DNA sequence models, and infinitely-many-sites models, and briefly discuss the diploid case.     

Ancestral Processes with Selection

In this paper, we show how to construct the genealogy of a sample of genes for a large class of models with selection and mutation. Each gene corresponds to a single locus at which there is no recombination. The genealogy of the sample is embedded in a graph which we call theancestral selection graph. This graph contains all the information about the ancestry; it is the analogue of Kingman's coalescent process which arises in the case with no selection. The ancestral selection graph can be easily simulated and we outline an algorithm for simulating samples. The main goal is to analyze the ancestral selection graph and to compare it to Kingman's coalescent process. In the case of no mutation, we find that the distribution of the time to the most recent common ancestor does not depend on the selection coefficient and hence is the same as in the neutral case. When the mutation rate is positive, we give a procedure for computing the probability that two individuals in a sample are identical by descent and the Laplace transform of the time to the most recent common ancestor of a sample of two individuals; we evaluate the first two terms of their respective power series in terms of the selection coefficient. The probability of identity by descent depends on both the selection coefficient and the mutation rate and is different from the analogous expression in the neutral case. The Laplace transform does not have a linear correction term in the selection coefficient. We also provide a recursion formula that can be used to approximate the probability of a given sample by simulating backwards along the sample paths of the ancestral selection graph, a technique developed by Griffiths and Tavaré (1994).     

Effect of balancing selection on spatial genetic structure within populations: theoretical investigations on the self-incompatibility locus and empirical studies in Arabidopsis halleri

The effect of selection on patterns of genetic structure within and between populations may be studied by contrasting observed patterns at the genes targeted by selection with those of unlinked neutral marker loci. Local directional selection on target genes will produce stronger population genetic structure than at neutral loci, whereas the reverse is expected for balancing selection. However, theoretical predictions on the intensity of this signal under precise models of balancing selection are still lacking. Using negative frequency-dependent selection acting on self-incompatibility systems in plants as a model of balancing selection, we investigated the effect of such selection on patterns of spatial genetic structure within a continuous population. Using numerical simulations, we tested the effect of the type of self-incompatibility system, the number of alleles at the self-incompatibility locus and the dominance interactions among them, the extent of gene dispersal, and the immigration rate on spatial genetic structure at the selected locus and at unlinked neutral loci. We confirm that frequency-dependent selection is expected to reduce the extent of spatial genetic structure as compared to neutral loci, particularly in situations with low number of alleles at the self-incompatibility locus, high frequency of codominant interactions among alleles, restricted gene dispersal and restricted immigration from outside populations. Hence the signature of selection on spatial genetic structure is expected to vary across species and populations, and we show that empirical data from the literature as well as data reported here on three natural populations of the herb Arabidopsis halleri confirm these theoretical results.     

A probabilistic view on the deterministic mutation–selection equation: dynamics,equilibria, and ancestry via individual lines of descent

We reconsider the deterministic haploid mutation–selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (random) structures inherent in this deterministic model. In a first step, we obtain a representation of the deterministic equation’s solution (and, in particular, of its equilibria) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error threshold phenomenon from a genealogical point of view. Next, we characterise the ancestral type distribution by means of the pruned lookdown ancestral selection graph and study its properties at equilibrium. We also provide an alternative characterisation in terms of a piecewise-deterministic Markov process. Throughout, emphasis is on the underlying dualities as well as on explicit results.

     

Inferences on the number and frequency of S-pollen gene (SFB) specificities in the polyploid Prunus spinosa

In flowering plants, self-incompatibility is a genetic mechanism that prevents self-fertilization. In gametophytic self-incompatibility (GSI), pollen specificity is encoded by the haploid genotype of the pollen tube. In GSI, specificities are maintained by frequency-dependent selection, and for diploid species, at equilibrium, equal specificity frequencies (isoplethy) are expected. This prediction has been tested in diploid, but never in polyploid self-incompatible species. For the latter, there is no theoretical expectation regarding isoplethy. Here, we report the first empirical study on specificity frequencies in a natural population of a polyploid self-incompatible species, Prunus spinosa. A total of 32 SFB (the pollen S gene) putative specificities are observed in a large sample from a natural population. Although P. spinosa is polyploid, the number of specificities found is similar to that reported for other diploid Rosaceae species. Unequal specificity frequencies are observed.     

The structured ancestral selection graph and the many-demes limit

We show that the unstructured ancestral selection graph applies to part of the history of a sample from a population structured by restricted migration among subpopulations, or demes. The result holds in the limit as the number of demes tends to infinity with proportionately weak selection, and we have also made the assumptions of island-type migration and that demes are equivalent in size. After an instantaneous sample-size adjustment, this structured ancestral selection graph converges to an unstructured ancestral selection graph with a mutation parameter that depends inversely on the migration rate. In contrast, the selection parameter for the population is independent of the migration rate and is identical to the selection parameter in an unstructured population. We show analytically that estimators of the migration rate, based on pairwise sequence differences, derived under the assumption of neutrality should perform equally well in the presence of weak selection. We also modify an algorithm for simulating genealogies conditional on the frequencies of two selected alleles in a sample. This permits efficient simulation of stronger selection than was previously possible. Using this new algorithm, we simulate gene genealogies under the many-demes ancestral selection graph and identify some situations in which migration has a strong effect on the time to the most recent common ancestor of the sample. We find that a similar effect also increases the sensitivity of the genealogy to selection.     

Migration rates, frequency-dependent selection and the self-incompatibility locus in Leavenworthia (brassicaceae)

Loci subject to negative frequency-dependent selection are expected to exhibit higher effective migration rates compared to reference loci. Although the number of gene copies transferred between populations by migration is the same for all genes, those subject to negative frequency-dependent selection are more likely to be retained in the immigrant population because rare alleles are selectively favored. So far, evidence for this prediction has been indirect, based on summary statistics rather than on migration rate estimates. Here, we introduce an approximate Bayesian procedure to jointly estimate migration rates at two predefined sets of loci between two populations. We applied the procedure to compare migration rate estimates at the self-incompatibility locus (S-locus) with that at 10 reference loci in two plant species, Leavenworthia alabamica and L. crassa (Brassicaceae). The maximum likelihood estimate for the proportion of migrants (m) was four times higher at the S-locus than at reference loci, but the difference was not statistically significant. Lack of significance might be due to insufficient data, but perhaps also to the recent divergence of the two species (311 ka), because we also show using simulations that the effective migration rate at the S-locus is expected to increase with increasing divergence time. These findings aid in understanding the evolutionary dynamics of negative frequency-dependent selection and they suggest that divergence time should be accounted for when employing migration rates to help detect negative frequency-dependent selection.     

DOES FREQUENCY‐DEPENDENT SELECTION WITH COMPLEX DOMINANCE INTERACTIONS ACCURATELY PREDICT ALLELIC FREQUENCIES AT THE SELF‐INCOMPATIBILITY LOCUS IN ARABIDOPSIS HALLERI?

Frequency-dependent selection is a major force determining the evolutionary dynamics of alleles at the self-incompatibility locus (S-locus) in flowering plants. We introduce a general method using numerical simulations to test several alternative models of frequency-dependent selection on S-locus data from sporophytic systems, taking into account both genetic drift and observed patterns of dominance interactions among S-locus haplotypes (S-haplotypes). Using a molecular typing method, we estimated S-haplotype frequencies in a sample of 322 adult plants and of 245 offspring obtained from seeds sampled on 22 maternal plants, collected in a single population of Arabidopsis halleri (Brassicaceae). We found eight different S-haplotypes and characterized their dominance interactions by controlled pollinations. We then compared the likelihood of different models of frequency-dependent selection: we found that the observed haplotype frequencies and observed frequency changes in one generation best fitted a model with (1) the observed dominance interactions and (2) no pollen limitation. Overall, our population genetic models of frequency-dependent selection, including patterns of dominance interactions among S-haplotypes and genetic drift, can reliably predict polymorphism at the S-locus. We discuss how these approaches allow detecting additional processes influencing the evolutionary dynamics of the S-locus, such as purifying selection on linked loci.     

ADAPTIVE DYNAMICS IN ALLELE SPACE: EVOLUTION OF GENETIC POLYMORPHISM BY SMALL MUTATIONS IN A HETEROGENEOUS ENVIRONMENT

We demonstrate how a genetic polymorphism of distinctly different alleles can develop during long-term frequency-dependent evolution in an initially monomorphic diploid population, if mutations have only small phenotypic effect. As a specific example, we use a version of Levene's (1953) soft selection model, where stabilizing selection acts on a continuous trait within each of two habitats. If the optimal phenotypes within the habitats are sufficiently different, then two distinctly different alleles evolve gradually from a single ancestral allele. In a wide range of parameter values, the two locally optimal phenotypes will be realized by one of the homozygotes and the heterozygote, rather than by the two homozygotes. Unlike in the haploid analogue of the model, there can be multiple polymorphic evolutionary attractors with different probabilities of convergence. Our results differ from the population genetic models of short-term evolution in two aspects: (1) a polymorphism that is population genetically stable may be invaded by a new mutant allele and, as a consequence, the population may fall back to monomorphism, (2) long-term evolution by allele substitutions may lead from a population where polymorphism is not possible into one where polymorphism is possible.     

Evolutionary games on cycles

Traditional evolutionary game theory explores frequency-dependent selection in well-mixed populations without spatial or stochastic effects. But recently there has been much interest in studying the evolutionary game dynamics in spatial settings, on lattices and other graphs. Here, we present an analytic approach for the stochastic evolutionary game dynamics on the simplest possible graph, the cycle. For three different update rules, called 'birth-death' (BD), 'death-birth' (DB) and 'imitation' (IM), we derive exact conditions for natural selection to favour one strategy over another. As specific examples, we consider a coordination game and Prisoner's Dilemma. In the latter case, selection can favour cooperators over defectors for DB and IM updating. We also study the case where the replacement graph of evolutionary updating remains a cycle, but the interaction graph for playing the game is a complete graph. In this setting, all three update rules lead to identical conditions in the limit of weak selection, where we find the '1/3-law' of well-mixed populations.     

Competitive divergence in non-random mating populations

A haploid model of frequency-dependent selection and assortative mating is introduced and analyzed for the case of a single multiallelic autosomal locus. Frequency-dependent selection is due to intraspecific competition mediated by a quantitative character under stabilizing or directional selection. Assortment is induced by the same trait. We analyze the equilibrium structure and the local stability properties of all possible equilibria. In the limit of weak selection we obtain global stability properties by finding a Lyapunov function. We provide necessary and sufficient conditions for the maintenance of polymorphism in terms of the strength of stabilizing selection, intraspecific competition and assortment. Our results also include criteria for the ability of extreme types to invade the population. Furthermore, we study the occurrence of disruptive selection and provide necessary and sufficient conditions for intraspecific divergence to occur.     

Repeated adaptive introgression at a gene under multiallelic balancing selection

Recently diverged species typically have incomplete reproductive barriers, allowing introgression of genetic material from one species into the genomic background of the other. The role of natural selection in preventing or promoting introgression remains contentious. Because of genomic co-adaptation, some chromosomal fragments are expected to be selected against in the new background and resist introgression. In contrast, natural selection should favor introgression for alleles at genes evolving under multi-allelic balancing selection, such as the MHC in vertebrates, disease resistance, or self-incompatibility genes in plants. Here, we test the prediction that negative, frequency-dependent selection on alleles at the multi-allelic gene controlling pistil self-incompatibility specificity in two closely related species, Arabidopsis halleri and A. lyrata, caused introgression at this locus at a higher rate than the genomic background. Polymorphism at this gene is largely shared, and we have identified 18 pairs of S-alleles that are only slightly divergent between the two species. For these pairs of S-alleles, divergence at four-fold degenerate sites (K = 0.0193) is about four times lower than the genomic background (K = 0.0743). We demonstrate that this difference cannot be explained by differences in effective population size between the two types of loci. Rather, our data are most consistent with a five-fold increase of introgression rates for S-alleles as compared to the genomic background, making this study the first documented example of adaptive introgression facilitated by balancing selection. We suggest that this process plays an important role in the maintenance of high allelic diversity and divergence at the S-locus in flowering plant families. Because genes under balancing selection are expected to be among the last to stop introgressing, their comparison in closely related species provides a lower-bound estimate of the time since the species stopped forming fertile hybrids, thereby complementing the average portrait of divergence between species provided by genomic data.     

Perturbation expansions of multilocus fixation probabilities for frequency-dependent selection with applications to the Hill-Robertson effect and to the joint evolution of helping and punishment

Natural populations are of finite size and organisms carry multilocus genotypes. There are, nevertheless, few results on multilocus models when both random genetic drift and natural selection affect the evolutionary dynamics. In this paper we describe a formalism to calculate systematic perturbation expansions of moments of allelic states around neutrality in populations of constant size. This allows us to evaluate multilocus fixation probabilities (long-term limits of the moments) under arbitrary strength of selection and gene action. We show that such fixation probabilities can be expressed in terms of selection coefficients weighted by mean first passages times of ancestral gene lineages within a single ancestor. These passage times extend the coalescence times that weight selection coefficients in one-locus perturbation formulas for fixation probabilities. We then apply these results to investigate the Hill-Robertson effect and the coevolution of helping and punishment. Finally, we discuss limitations and strengths of the perturbation approach. In particular, it provides accurate approximations for fixation probabilities for weak selection regimes only (Ns?1), but it provides generally good prediction for the direction of selection under frequency-dependent selection.     

A general model to explore complex dominance patterns in plant sporophytic self-incompatibility systems

We developed a general model of sporophytic self-incompatibility under negative frequency-dependent selection allowing complex patterns of dominance among alleles. We used this model deterministically to investigate the effects on equilibrium allelic frequencies of the number of dominance classes, the number of alleles per dominance class, the asymmetry in dominance expression between pollen and pistil, and whether selection acts on male fitness only or both on male and on female fitnesses. We show that the so-called "recessive effect" occurs under a wide variety of situations. We found emerging properties of finite population models with several alleles per dominance class such as that higher numbers of alleles are maintained in more dominant classes and that the number of dominance classes can evolve. We also investigated the occurrence of homozygous genotypes and found that substantial proportions of those can occur for the most recessive alleles. We used the model for two species with complex dominance patterns to test whether allelic frequencies in natural populations are in agreement with the distribution predicted by our model. We suggest that the model can be used to test explicitly for additional, allele-specific, selective forces.     

A tale of two continents: Baker's rule and the maintenance of self-incompatibility in Lycium (Solanaceae)

Over 50 years ago, Baker (1955, 1967) suggested that self-compatible species were more likely than self-incompatible species to establish new populations on oceanic islands. His logic was straightforward and rested on the assumption that colonization was infrequent; thus, mate limitation favored the establishment of self-fertilizing individuals. In support of Baker's rule, many authors have documented high frequencies of self-compatibility on islands, and recent work has solidified the generality of Baker's ideas. The genus Lycium (Solanaceae) has ca. 80 species distributed worldwide, and phylogenetic studies suggest that Lycium originated in South America and dispersed to the Old World a single time. Previous analyses of the S-RNase gene, which controls the stylar component of self-incompatibility, have shown that gametophytically controlled self-incompatibility is ancestral within the genus, making Lycium a good model for investigating Baker's assertions concerning reproductive assurance following oceanic dispersal. Lycium is also useful for investigations of reproductive evolution, given that species vary both in sexual expression and the presence of self-incompatibility. A model for the evolution of gender dimorphism suggests that polyploidy breaks down self-incompatibility, leading to the evolution of gender dimorphism, which arises as an alternative outcrossing mechanism. There is a perfect association of dimorphic gender expression, polyploidy, and self-compatibility (vs. cosexuality, diploidy, and self-incompatibility) among North American Lycium. Although the association between ploidy level and gender expression also holds for African Lycium, to date no studies of mating systems have been initiated in Old World species. Here, using controlled pollinations, we document strong self-incompatibility in two cosexual, diploid species of African Lycium. Further, we sequence the S-RNase gene in 15 individuals from five cosexual, diploid species of African Lycium and recover 24 putative alleles. Genealogical analyses indicate reduced trans-generic diversity of S-RNases in the Old World compared to the New World. We suggest that genetic diversity at this locus was reduced as a result of a founder event, but, despite the bottleneck, self-incompatibility was maintained in the Old World. Maximum-likelihood analyses of codon substitution patterns indicate that positive Darwinian selection has been relatively strong in the Old World, suggesting the rediversification of S-RNases following a bottleneck. The present data thus provide a dramatic exception to Baker's rule, in addition to supporting a key assumption of the Miller and Venable (2000) model, namely that self-incompatibility is associated with diploidy and cosexuality.     

Evolutionary genetics of self-incompatibility in the Solanaceae

The self-incompatibility (S) gene in flowering plants has long been appreciated as an example of extreme allelic polymorphism maintained by frequency-dependent selection. Recent studies of population samples of S-allele sequences obtained by RT-PCR from five species of Solanaceae now reveal a picture of conspicuous inter-specific variation in both S-allele number and age. Explanations for this variation are examined with reference to current theory. We propose that changes in species' effective population size, particularly those associated with the evolution of different life histories, best account for interspecific differences in both the number and average age of S alleles.     

Simulation of selected genealogies

Algorithms for generating genealogies with selection conditional on the sample configuration of n genes in one-locus, two-allele haploid and diploid models are presented. Enhanced integro-recursions using the ancestral selection graph, introduced by S. M. Krone and C. Neuhauser (1997, Theor. Popul. Biol. 51, 210-237), which is the non-neutral analogue of the coalescent, enables accessible simulation of the embedded genealogy. A Monte Carlo simulation scheme based on that of R. C. Griffiths and S. Tavaré (1996, Math. Comput. Modelling 23, 141-158), is adopted to consider the estimation of ancestral times under selection. Simulations show that selection alters the expected depth of the conditional ancestral trees, depending on a mutation-selection balance. As a consequence, branch lengths are shown to be an ineffective criterion for detecting the presence of selection. Several examples are given which quantify the effects of selection on the conditional expected time to the most recent common ancestor.     

A model for the evolutionary diversification of religions

We address the problem of cultural diversification by studying selection on cultural ideas that colonize human hosts and using diversification of religions as a conceptual example. In analogy to studying the evolution of pathogens or symbionts colonizing animal hosts, we use models for host-pathogen dynamics known from theoretical epidemiology. In these models, religious content colonizes individual humans. Rates of transmission of ideas between humans, i.e., transmission of cultural content, and rates of loss of ideas (loss of belief) are determined by the phenotype of the cultural content, and by interactions between hosts carrying different ideas. In particular, based on the notion that cultural non-conformism can be negative frequency-dependent (for example, religion can lead to oppression of lower classes and emergence of non-conformism and dissent once a religious belief has reached dominance), we assume that the rate of loss of belief increases as the number of humans colonized by a particular religious phenotype increases. This generates frequency-dependent selection on cultural content, and we use evolutionary theory to show that this frequency dependence can lead to the emergence of coexisting clusters of different cultural types. The different clusters correspond to different cultural traditions, and hence our model describes the emergence of distinct descendant cultures from a single ancestral culture in the absence of any geographical isolation.     

Populations adapt to fluctuating selection using derived and ancestral allelic diversity

Populations can adapt to changing environments by using allelic diversity, yet whether diversity is recently derived or ancestral is often debated. Although evolution could productively use both types of diversity in a changing environment, their relative frequency has not been quantified. We address this question experimentally using budding yeast strains that harbor a tandem repeat containing URA3 gene, which we expose to cyclical selection and counterselection. We characterize and quantify the dynamics of frameshift events in the URA3 gene in eight populations over 12 cycles of selection and find that ancestral alleles account for 10–20% of all adaptive events. Using a general model of fluctuating selection, we determine how these results depend on mutation rates, population sizes, and fluctuation timescales. We quantify the contribution of derived alleles to the adaptation process using the de novo mutation rate along the population's ancestral lineage, a novel measure that is applicable in a wide range of settings. We find that the adaptive dynamics undergoes a sharp transition from selection on ancestral alleles to selection on derived alleles as fluctuation timescales increase. Our results demonstrate that fluctuations can select between different modes of adaptation over evolutionary timescales.     

Genealogies and weak purifying selection

The assumption that selection alters the genealogical tree of a sample of alleles from a population relative to the neutral expectation underlies several "tests of neutrality." Two recent papers have studied the effect of purifying selection; their suggestive but incomplete results indicate that, in the single site case, the shape of a gene genealogy for a locus may differ only from the neutral expectation. We verify this finding for weak selection using the "ancestral selection graph." We consider a wider range of models, including both a four-allele single-site model and an infinite-sites model. Our results confirm the previous claim for the symmetric-mutation single site model. We emphasize, however, that a neutral-seeming genealogy is consistent with detectable effects of selection on the distribution of allele frequences within the sample. With selection operating, the information about a sample cannot be reduced to the genealogy. As a result, a distinction needs to be made between the selected sites themselves, for which the genealogy offers insufficient information, and linked neutral variation. This distinction seems to have been overlooked in previous papers, yet it has significant implications for the interpretation of data on DNA sequence variation. In particular, it predicts that under purifying selection, the frequency spectrum of neutral mutations will not reflect the skew toward rare polymorphisms at replacement sites even if there is no recombination between them. We caution, however, that the effect of weak selection on the genealogy is specific to the model; a (more realistic) model of multiple linked sites could lead to a more distorted genealogy than is observed for a single site.