Maximization principles for frequency-dependent selection I: the one-locus two-allele case

In this article we study the one-locus two-allele version of the pairwise-interaction model of frequency-dependent selection in discrete and continuous time. Our main aim is to provide necessary and sufficient conditions for the validity of maximization principles. We provide a systematic approach that covers all possible facets of the dynamical behavior of the model, and we illustrate our results by concrete examples. We show that the mean fitness of the population is nondecreasing if the interaction coefficients are symmetric and positive. Moreover, monotonic convergence to the set of equilibria always occurs, which is not true if we also consider negative interaction coefficients. For asymmetric interaction, we provide necessary conditions when the mean fitness is nondecreasing and sufficient conditions when it is not. Furthermore, in discrete time, we show that limit cycles cannot occur, unless some interaction coefficients are negative.     

Effects of genetic architecture on the evolution of assortative mating under frequency-dependent disruptive selection

We consider a model of sympatric speciation due to frequency-dependent competition, in which it was previously assumed that the evolving traits have a very simple genetic architecture. In the present study, we numerically analyze the consequences of relaxing this assumption. First, previous models assumed that assortative mating evolves in infinitesimal steps. Here, we show that the range of parameters for which speciation is possible increases when mutational steps are large. Second, it was assumed that the trait under frequency-dependent selection is determined by a single locus with two alleles and additive effects. As a consequence, the resultant intermediate phenotype is always heterozygous and can never breed true. To relax this assumption, here we add a second locus influencing the trait. We find three new possible evolutionary outcomes: evolution of three reproductively isolated species, a monomorphic equilibrium with only the intermediate phenotype, and a randomly mating population with a steep unimodal distribution of phenotypes. Both extensions of the original model thus increase the likelihood of competitive speciation.     

Stochastic fluctuations in frequency-dependent selection: a one-locus, two-allele and two-phenotype model

Stochastic fluctuations in a simple frequency-dependent selection model with one-locus, two-alleles and two-phenotypes are investigated. The steady-state statistics of allele frequencies for an interior stable phenotypic equilibrium are shown to be similar to the stochastic fluctuations in standard evolutionary game dynamics [Tao, Y., Cressman, R., 2007. Stochastic fluctuations through intrinsic noise in evolutionary game dynamics. Bull. Math. Biol. 69, 1377-1399]. On the other hand, for an interior stable phenotypic or genotypic equilibrium, our main results show that the deterministic model cannot be used to predict the expectation of phenotypic frequency. The variance of phenotypic frequency for an interior stable genotypic equilibrium is more sensitive to the expected population size than for an interior stable phenotypic equilibrium. Furthermore, the stochastic fluctuations of allele frequency and phenotypic frequency can be considered approximately independent of each other for these genotypic equilibria, but not for phenotypic.     

The influence of density on frequency-dependent food selection: a comparison of four experiments with wild birds

We compare the results of four experiments, conducted at different times and with different protocols, that explored the relationship between frequency-dependent selection and prey density in wild birds feeding on artificial populations of coloured baits. One (experiment 4) used pastry baits that differed only in the presence or absence of a red stripe, and this experiment provided no evidence for any kind of selective behaviour. The other three experiments used green and brown baits, and they all provided evidence for a trend towards increasing anti-apostatic selection with high densities (>100 baits m^–2). However, one of these (experiment 3) provided no evidence for frequency-dependent selection at low densities (0.5–20 baits m^–2), while the other two experiments concurred in suggesting a trend towards increasing apostatic selection with low densities (down to 2 baits m^–2). Together, these experiments both support and qualify the published findings of experiment 1 that frequency- dependent selection by wild birds on bait populations is modified by density. Experiment 4 indicates that frequency-dependent selection may break down entirely if bait types are too similar, while experiment 3 indicates that some details of this trend with density will depend either on the protocol used or on exogenous changes in the birds’ feeding behaviour. Received: 1 September 1999 / Accepted: 22 March 2000     

Frequency-dependent host selection by parasitic mites: a model and a case study

Previous studies on frequency-dependent food selection (changing food preferences in response to changes in relative food abundance) have focused on predators and parasitoids. These organisms utilize several victims during their lifetime. We introduce the case of parasites which, having accepted a host, do not change it. We propose two alternative models to explain the biased occurrence of parasites on different host types: (1) through the option of rejecting less-preferred hosts prior to accepting one of them; (2) by differential parasite survival on different host types. These models predict that host rejection, but not differential survival, can create frequency-dependent parasitism (FDP). Unlike previously described factors responsible for frequency dependence of food selection, which act through changing the foraging behaviour of individual predators or parasitoids, FDP involves no adjustment of parasite foraging strategy according to previous feeding experience. The mite Hemisarcoptes coccophagus is an obligate parasite of armoured scale insects (Homptera: Diaspididae). Our field data show that H. coccophagus is found more frequently on ovipositing than on young host females. Our model, combining the effects of host rejection and differential survival, is used to estimate the relative contribution of these factors to parasite biased occurrence on different hosts. The contribution of differential survival was dominant in H. coccophagus, and overode any effect of host rejection. Nevertheless, our prediction that FDP may be found in parasites is supported by literature data about a parasitic water mite.     

Diffusion approximations for one-locus multi-allele kin selection,mutation and random drift in group-structured populations: a unifying approach to selection models in population genetics

Diffusion approximations are ascertained from a two-time-scale argument in the case of a group-structured diploid population with scaled viability parameters depending on the individual genotype and the group type at a single multi-allelic locus under recurrent mutation, and applied to the case of random pairwise interactions within groups. The main step consists in proving global and uniform convergence of the distribution of the group types in an infinite population in the absence of selection and mutation, using a coalescent approach. An inclusive fitness formulation with coefficient of relatedness between a focal individual J affecting the reproductive success of an individual I, defined as the expected fraction of genes in I that are identical by descent to one or more genes in J in a neutral infinite population, given that J is allozygous or autozygous, yields the correct selection drift functions. These are analogous to the selection drift functions obtained with pure viability selection in a population with inbreeding. They give the changes of the allele frequencies in an infinite population without mutation that correspond to the replicator equation with fitness matrix expressed as a linear combination of a symmetric matrix for allozygous individuals and a rank-one matrix for autozygous individuals. In the case of no inbreeding, the mean inclusive fitness is a strict Lyapunov function with respect to this deterministic dynamics. Connections are made between dispersal with exact replacement (proportional dispersal), uniform dispersal, and local extinction and recolonization. The timing of dispersal (before or after selection, before or after mating) is shown to have an effect on group competition and the effective population size. In memory of Sam Karlin.     

Frequency-dependent selection by pollinators: mechanisms and consequences with regard to behaviour of bumblebees Bombus terrestris (L.) (Hymenoptera: Apidae)

Bombus terrestris , a typical pollinating insect species, was offered artificial flowers of two different corolla colours with the same sucrose solution reward in an array. Common colours were significantly preferred, and the strength of the frequency-dependent response increased as a result of learning. There were also frequency-independent biases towards blue flowers, probably because blue flowers appeared more conspicuous to bumblebees than yellow flowers, and the degree of preference for blue was greater when flowers had low nectar rewards. Flower-to-flower movements by individual bumblebees between flowers were non-random, were biased to movements within the same flower colour, and were also dependent on morph frequency. The mechanisms governing flower selection in bumblebees are discussed. Pollinators foraging similarly in a natural situation would induce positive frequency-dependent selection, assortative mating, and directional selection on different corolla colour morphs of the plant population being visited, resulting in stabilizing selection for a single flower colour.     

Differentiating the effects of origin and frequency in reciprocal transplant experiments used to test negative frequency-dependent selection hypotheses

Reciprocal transplant experiments have been used to estimate the probability that negative frequency-dependent selection by natural enemies has occurred in host populations by determining whether pest populations are less adapted to “foreign” (rare) hosts, which originate from a population with which the pests have not coevolved. However, these experiments usually confound the effects of frequency and origin: the rare genotypes are also genotypes that did not originate at a site. When clonal organisms are used, and the clones occur in more than one population, it is possible to separate the effects of origin and frequency. Here I present the results of an experiment in which Arabis clones of known frequency were reciprocally transplanted among sites. Contrary to expectations, clones at their site of origin had less disease, less herbivory, and higher fitness than foreign clones. However, variation within and among sites in herbivory and infection was large, suggesting that the number of sites and clones needed to thoroughly test the hypothesis of negative frequency-dependent selection in this system is very large: thus, these results are suggestive but not conclusive. Received: 20 October 1997 / Accepted: 8 February 1998     

Impact of density and disease on frequency-dependent selection and genetic polymorphism: experiments with stripe rust and wheat

Frequency-dependent disease impacts may contribute to the maintenance of genetic diversity and sexual reproduction in plant populations. In earlier work with experimental wheat (Triticum aestivum) populations at a single density, we found that stripe rust (caused by Puccinia striiformis) created frequency-dependent selection on its host but competitive interactions between host genotypes reduced the potential for disease to maintain genetic polymorphisms in this highly self-pollinated species; the weaker competitor actually exhibited positive disease-mediated frequency-dependent selection. Based on these results we predicted that at low density, where the overall level of competition is lower, disease would have a stronger impact relative to competition and thus be more likely to maintain genetic polymorphisms; at low densities the greatest effect of disease for negative frequency-dependent selection should be seen in the weak competitor. Here we report on results with wheat stripe rust in which we altered both the frequency and density of host genotypes in factorial combinations of two-way mixtures where each host genotype was attacked by its own specialized race of rust. Within each density disease levels increased with genotype frequencies, creating frequency-dependent disease attack at all densities. Similarly, disease created negative frequency-dependent selection on its host at all densities, as a genotype’s fitness was often greater at low than high frequency when disease was present. Disease levels increased with plant density in 1997 but decreased in 1998. While increasing plant density reduced absolute fitness, presumably as a result of increased competition, a genetic polymorphism was not more likely to be maintained at low than high density as we had predicted. Within each density, the impact of disease was insufficient to reverse the slope of the relationship between absolute fitness and planted frequency from positive to negative for the less competitive host genotype, thus preventing the maintenance of a genetic polymorphism.     

Evolutionary model selection with a genetic algorithm: a case study using stem RNA

The choice of a probabilistic model to describe sequence evolution can and should be justified. Underfitting the data through the use of overly simplistic models may miss out on interesting phenomena and lead to incorrect inferences. Overfitting the data with models that are too complex may ascribe biological meaning to statistical artifacts and result in falsely significant findings. We describe a likelihood-based approach for evolutionary model selection. The procedure employs a genetic algorithm (GA) to quickly explore a combinatorially large set of all possible time-reversible Markov models with a fixed number of substitution rates. When applied to stem RNA data subject to well-understood evolutionary forces, the models found by the GA 1) capture the expected overall rate patterns a priori; 2) fit the data better than the best available models based on a priori assumptions, suggesting subtle substitution patterns not previously recognized; 3) cannot be rejected in favor of the general reversible model, implying that the evolution of stem RNA sequences can be explained well with only a few substitution rate parameters; and 4) perform well on simulated data, both in terms of goodness of fit and the ability to estimate evolutionary rates. We also investigate the utility of several distance measures for comparing and contrasting inferred evolutionary models. Using widely available small computer clusters, our approach allows, for the first time, to evaluate the performance of existing RNA evolutionary models by comparing them with a large pool of candidate models and to validate common modeling assumptions. In addition, the new method provides the foundation for rigorous selection and comparison of substitution models for other types of sequence data.     

Fixation probability with multiple alleles and projected average allelic effect on selection

The first-order effect of selection on the probability of fixation of an allele, with respect to an intensity of selection s>0 in a diploid population of fixed finite size N, undergoing discrete, non-overlapping generations, is shown to be given by the sum of the average effects of that allele on the coefficient of selection in the current generation and all future generations, given the population state in the current generation. This projected average allelic effect is a weighted sum of average allelic effects in allozygous and autozygous offspring in the initial generation, with weights given in terms of expected coalescence times, under neutrality, for the lineages of two or three gametes chosen at random in the same generation. This is shown in the framework of multiple alleles at one locus, with genotypic values determining either viability or fertility differences, and with either multinomial or exchangeable reproduction schemes. In the limit of weak selection in a large population such that Ns tends to zero, the initial average allelic effects in allozygous offspring and autozygous offspring have the same weight on the fixation probability only in the domain of application of the Kingman coalescent. With frequency-dependent selection in a linear-game-theoretic context with two phenotypes determined by additive gene action, the first-order effect on the fixation probability is a combination of two effects of frequency-independent selection, one in a haploid population, the other in a diploid population. In the domain of application of the Kingman coalescent as the population size goes to infinity and Ns to zero, the first effect is three times more important than the second effect. This explains the one-third law of evolutionary dynamics in this domain, and shows how this law can be extended beyond this domain.     

Extremo-phage: In vitro selection of tolerance to a hostile environment

Extremophiles are microorganisms that thrive under extreme conditions such as temperatures above 65°C, pHs below 4 or above 10, salt concentrations above 0.5 m, or pressures of 600 atm. While studies of enzymes either isolated from extremophiles, or generated using site-specific mutagenesis, or adapted by in vivo or in vitro selection have established a precedent for the engineering and application of proteins at extreme conditions, generalization of the approaches to more complex multimolecular or multitask systems has remained elusive. Here we demonstrate that a significantly more complex system—a bacteriophage—can over a number of generations be adapted to tolerate a hostile and unnatural environment. An in vitro selection strategy was used to adapt phage to urea, a protein denaturing agent. As the concentration of urea employed in selections over 20 generations was gradually increased from 5 to 9 m, the surviving phages steadily improved their tolerance, finally achieving a greater than 350-fold stability enhancement over the original population.Correspondence to: J. Yin     

Phenotypic selection and response to selection in Lobularia maritima: importance of direct and correlational components of natural selection

By using selection differentials, gradients and structural equation modelling (SEM), I have quantified the phenotypic selection acting on Lobularia maritima (Cruciferae) flower size, display, colour and density, using data on lifetime female fitness. Furthermore, by analysing the resulting F₁ generation in field and greenhouse conditions, I estimated the actual intergenerational change in the value of these traits. Both pollinators preferred plants with many and large flowers. Strong directional selection for increased flower display was found in all years of the study, regardless of the technique used. Indirect selection due to a high significant correlation with flower display occurred on flower colour and size. SEM showed that pollinators played only a minor role in this observed phenotypic selection. The analysis of the phenotypes of F₁ plants showed that flower display actually increased across generations. In addition, white flowers were significantly more frequent in the offspring population than in the parental one, mostly due to the association between flower display and white coloured flowers. This suggests that both direct and indirect selection can play a role in the evolution of correlated traits in this crucifer.     

SSADH Variation in Primates: Intra- and Interspecific Data on a Gene with a Potential Role in Human Cognitive Functions

In the present study we focus on the nucleotide and the inferred amino acid variation occurring in humans and other primate species for mitochondrial NAD⁺-dependent succinic semialdehyde dehydrogenase, a gene recently supposed to contribute to cognitive performance in humans. We determined 2527 bp of coding, intronic, and flanking sequences from chimpanzee, bonobo, gorilla, orangutan, gibbon, and macaque. We also resequenced the entire coding sequence on 39 independent chromosomes from Italian families. Four variable coding sites were genotyped in additional populations from Europe, Africa, and Asia. A test for constancy of the nonsynonymous vs. synonymous rates of nucleotide changes revealed that primates are characterized by largely variable d_N/d_S ratios. On a background of strong conservation, probably controlled by selective constraints, the lineage leading to humans showed a ratio increased to 0.42. Human polymorphic levels fall in the range reported for other genes, with a pattern of frequency and haplotype structure strongly suggestive of nonneutrality. The comparison with the primate sequences allowed inferring the ancestral state at all variable positions, suggesting that the c.538(C) allele and the associated functional variant is indeed a derived state that is proceeding to fixation. The unexpected pattern of human polymorphism compared to interspecific findings outlines the possibility of a recent positive selection on some variants relevant to new cognitive capabilities unique to humans. [Reviewing Editor: Dr. Martin Kreitman] Paola Blasi and Francesca Palmerio contributed equally to the work.     

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.     

Interactions between the sexes: new perspectives on sexual selection and reproductive isolation

Understanding the processes underlying the origin of new species is a fundamental problem in evolutionary research. Whilst it has long been recognised that closely related taxa often differ markedly in reproductive characteristics, only relatively recently has sexual selection been evoked as a key promoter of speciation through its ability to generate reproductive isolation (RI). Sexual selection potentially can influence the probability that individuals from the same or different populations will reproduce successfully since it shapes precisely those traits involved in mating and reproduction. If reproductive characters diverge along different trajectories, then sexual selection can impact on the evolution of reproductive barriers operating both before and after mating. In this perspective, we consider some new advances in our understanding of the coevolution of male and female sexual signals and receptors and suggest how these developments may provide heretofore neglected insights into the mechanisms by which isolating barriers may emerge. Specifically, we explore how selfish genetic elements (SGEs) can mediate pre- and post-copulatory mate choice, thereby influencing gene flow and ultimately population divergence; we examine evidence from studies of intracellular sperm–egg interactions and propose that intracellular gametic incompatibilities may arise after sperm entry into the egg, and thus contribute to RI; we review findings from genomic studies demonstrating rapid, adaptive evolution of reproductive genes in both sexes and discuss whether such changes are causal in determining RI or simply associated with it; and finally, we consider genetic, developmental and functional mechanisms that might constrain reproductive trait diversification, thereby limiting the scope for reproductive barriers to arise via sexual selection. We hope to stimulate work that will further the understanding of the role sexual selection plays in generating RI and ultimately speciation.     

Towards a theory of the evolution of butterfly colour patterns under directional and disruptive selection

Two general models for the transspecific evolution of butterfly colour patterns are advanced: directional selection acting equally on both sexes, and disruptive selection involving periods of polymorphism. To consider possible outcomes of me latter process, a morphism notation based on an integrated classification for polymorphism and sexual dimorphism is developed. This notation is used to examine the properties of all morphism transformations possible from the minimal expressions of the nine morphism categories, as reached through defined minimum step changes. The significance of such pathway models is analysed in terms of general properties of butterfly polymorphism. The potential use of pathway models in evolutionary studies is briefly discussed, mainly with respect to phylogenetics, and ideas on the evolution of genetic dominance.     

Evolution in heterogeneous environments: Effects of migration on habitat specialization

Summary Richard Levins introduced fitness sets as a tool for investigating evolution within heterogeneous environments. Evolutionary game theory permits a synthesis and generalization of this approach by considering the evolutionary response of organisms to any scale of habitat heterogeneity. As scales of heterogeneity increase from fine to coarse, the evolutionary stable strategy (ESS) switches from a single generalist species to several species that become increasingly specialized on distinct habitats. Depending upon the organisms' ecology, the switch from one to two species may occur at high migration rates (relatively fine-grained environment), or may only occur at very low migration rates (coarse-grained environment). At the ESS, the evolutionary context of a species is the entire landscape, while its ecological context may be a single habitat.Evolution towards the ESS can be represented with adaptive landscapes. In the absence of frequency-dependence, shifting from a single strategy ESS to a two strategy ESS poses the problem of evolving across valleys in the adaptive surface to occupy new peaks (hence, Sewell Wright's shifting balance theory). Frequency-dependent processes facilitate evolution across valleys. If a system with a two strategy ESS is constrained to possess a single strategy, the population may actually evolve a strategy that minimizes fitness. Because the population now rests at the bottom of a valley, evolution by natural selection can drive populations to occupy both peaks.     

Triangulating the genetic basis of adaptation to multifarious selection

Understanding how natural populations adapt to their local environments is a major research theme for ecological genomics. This endeavour begins by sleuthing for shared genetic similarities among unrelated natural populations sharing adaptive traits to documented selective pressures. When the selective pressures have low dimensionality, and the genetic response is localized to a few genes of major effect, this detective work is relatively straightforward. However, in the real world, populations face a complex mixture of selective pressures and many adaptive responses are the result of changes in quantitative traits that have a polygenic genetic basis. This complex relationship between environment and adaptation presents a significant challenge. How can we begin to identify drivers of adaptation in natural settings? In this issue of Molecular Ecology, Orsini et al. (2012) take advantage of the biological attributes of the freshwater microcrustacean Daphnia ( Fig. 1 ) to disentangle multidimensional selection’s signature on the genome of populations that have repeatedly evolved adaptive responses to isolated selective pressures including predation, parasitism and anthropogenic changes in land use. Orsini et al. (2012) leverage a powerful combination of spatially structured populations in a geographic mosaic of environmental stressors, the historical archive of past genotypes preserved in lake‐bottom sediments and selection experiments to identify sets of candidate genomic regions associated with adaptation in response to these three environmental stressors. This study provides a template for future investigation in ecological genomics, combining multiple experimental approaches with the genomic investigation of a well‐studied ecological model species.

Figure 1 Open in figure viewer PowerPoint Adult Daphnia magna carrying a resting egg in the brood pouch. The water flea Daphnia is a renowned ecological model system and rapidly developing as an ecological and environmental genomics model species. Photo credit Joachim Mergeay.