Genic Variation in Male Haploids under Deterministic Selection

Genic variation in male haploids and male diploids was compared assuming constant fitnesses (derived from computer-generated random numbers) and infinite population size. Several models were studied, differing by the fitness correlation between the sexes (r_s) and genotypes (r_g), and by the intensity of selection as measured by the coefficient of variation (CV) of the fitness distribution. Genic variation was quantified using the proportion of polymorphic loci, P, the gene diversity at polymorphic loci, H_p, and the gene diversity over all loci, H_a. The two genetic systems were compared via variation ratios: variation in male haploidy/variation in male diploidy.—P and H_a were markedly lower for male-haploids than for male diploids, the variation ratios declining with increasing r_s, r_g and CV, but the two genetic systems were similar for H_p. Except for male diploids with r_s = 1, the two sexes had different equilibrium gene frequencies but the sample sizes required to detect such differences reliably were larger than usually possible in surveys of natural populations.—Data from natural populations fit the above trends qualitatively, but the variation ratios are much lower than those from our analyses, except that for H_p, which is higher when Drosophila is excluded. Also, the frequency distribution of most common alleles from electrophoretic data has a deficiency of intermediate frequencies compared to that from the computer-generated sets of fitnesses, possibly reflecting either the influence of stochastic processes shifting frequencies away from equilibrium or the involvement of alleles under selection-mutation balance.——While electrophoretic data suggest that Drosophila has unusually high levels of genic variation, unusually low levels of genic variation in male haploids compared with male diploids are not strongly indicated. However, if further data confirm male haploids as having low levels of genic variation, likely explanations are that the bulk of electrophoretically detected variation involves fixed-fitness balancing selection, selection-mutation balance involving slightly deleterious recessive alleles, new favorable male haploid alleles moving more rapidly to fixation than under male diploidy and thus carrying linked loci to fixation faster, or some combination of these possible factors.     

Towards a genetic theory for the evolution of the sex ratio. III. Parental and sibling control of brood investment ratio under partial sib-mating

Models of sex ratio evolution under partial sib-mating are investigated in haplodiploids and diploids. In the cases of parental and sibling control of the brood investment ratio between the sexes in diploids, we find that the “unbeatable” investment ratio obtained by W. D. Hamilton (Science156, 477–488) for his local mate competition model corresponds in our inbreeding models to a weak form ESS (evolutionary stable strategy) fixation state and also to the population investment ratio at certain internal equilibria of our models. For haplodiploids, “strong form ESS” values exist under inbreeding in models involving father and sister control. Under brother and mother control, however, the ESS derived from local mate competition models is unstable in our inbreeding models to the introduction of any other investment ratio. We stress important qualitative differences between models involving local mate competition and inbreeding.     

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.     

The Maintenance of Genetic Variability in Two-Locus Models of Stabilizing Selection

The maintenance of genetic variability at two diallelic loci under stabilizing selection is investigated. Generations are discrete and nonoverlapping; mating is random; mutation and random genetic drift are absent; selection operates only through viability differences. The determination of the genotypic values is purely additive. The fitness function has its optimum at the value of the double heterozygote and decreases monotonically and symmetrically from its optimum, but is otherwise arbitrary. The resulting fitness scheme is identical to the symmetric viability model. Linkage disequilibrium is neglected, but the results are otherwise exact. Explicit formulas are found for all the equilibria, and explicit conditions are derived fro their existence and stability. A complete classification of the six possible global convergence patterns is presented. In addition to the symmetric equilibrium (with gene frequency 1/2 at both loci), a pair of unsymmetric equilibria may exist; the latter are usually, but not always, unstable. If the ratio of the effect of the major locus to that of the minor one exceeds a critical value, both loci will be stably polymorphic. If selection is weak at the minor locus, the more rapidly the fitness function decreases near the optimum, the lower is this critical value; for rapidly decreasing fitness functions, the critical value is close to one. If the fitness function is smooth at the optimum, then a stable polymorphism exists at both loci only if selection is strong at the major locus.     

Regions of Stable Equilibria for Models of Differential Selection in the Two Sexes under Random Mating

The equilibrium structure of models of differential selection in the sexes is investigated. It is shown that opposing additive selection leads to stable polymorphic equilibria for only a restricted set of selection intensities, and that for weak selection the selection intensities must be of approximately the same magnitude in the sexes. General models of opposing directional selection, with arbitrary dominance, are investigated by considering simultaneously the stability properties of the trivial equilibria and the curve along which multiple roots appear. Numerical calculations lead us to infer that the average degree of dominance determines the equilibrium characteristics of models of opposing selection. It appears that if the favored alleles are, on the average, recessive, there may be multiple polymorphic equilibria, whereas only a single polymorphic equilibrium can occur when the favored alleles are, on the average, dominant. The principle that the average degree of dominance controls equilibrium behavior is then extended to models allowing directional selection in one sex with overdominance in the other sex, by showing that polymorphism is maintained if and only if the average fitness in heterozygotes exceeds one.     

On a genetic model of intraspecific competition and stabilizing selection

A genetic model is investigated in which two recombining loci determine the genotypic value of a quantitative trait additively. Two opposing evolutionary forces are assumed to act: stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the underlying genetics, in particular recombination rate and relative magnitude of allelic effects, interact with the conflicting selective forces and derive the resulting, surprisingly complex equilibrium patterns. We also investigate the conditions under which disruptive selection on the phenotypes can be observed and examine how much genetic variation can be maintained in such a model. We discovered a number of unexpected phenomena. For instance, we found that with little recombination the degree of stably maintained polymorphism and the equilibrium genetic variance can decrease as the strength of competition increases relative to the strength of stabilizing selection. In addition, we found that mean fitness at the stable equilibria is usually much lower than the maximum possible mean fitness and often even lower than the fitness at other, unstable equilibria. Thus, the evolutionary dynamics in this system are almost always nonadaptive.     

Analysis of partial assortative mating and sexual selection models for a polygamous species involving a trait based on levels of heterozygosity

The consequences of preferential mating in the presence of partial assortative and sexual selection mechanisms are ascertained for a two-allele one-locus trait involving two phenotype classes C₁ = {all homozygotes} and C₂ = {heterozygotes}. Relevant biological cases may include Burley (1977, Proc. Nat. Acad. Sci. USA74, 3476–3479), Wilbur et al. (1978, Evolution32, 264–270), and Singh and Zouros (1978, Evolution32, 342–353). When the preference rate for the heterozygote exceeds that for homozygotes, it is established that the unique stable state is the central Hardy-Weinberg equilibrium. The rate of approach is faster with sexual selection than for the corresponding model of assortative mating. When the preference rates favor the homozygotes then in this symmetric model of sexual selection two asymmetric Hardy-Weinberg polymorphisms can evolve, and which succeeds depends on initial conditions. The models are also analyzed with natural selection acting on phenotypes superimposed on assortative mating. In this case we can have up to three coexisting stable states involving both fixation alternatives and a central polymorphism. The corresponding model with sexual selection maintains either the central equilibrium as in assortative mating or two asymmetric polymorphic equilibria.     

Polygyny in the nest-site limited acacia-ant Crematogaster mimosae

Polygyny is common in social insects despite inevitable decreases in nestmate relatedness and reductions to the inclusive fitness returns for cooperating non-reproductive individuals. We studied the prevalence and mode of polygyny in the African acacia-ant Crematogaster mimosae. These ants compete intensively with neighboring colonies of conspecifics and with three sympatric ant species for resources associated with the whistling-thorn acacias in which they all obligately nest. We used the genotypes of alate males at ten microsatellite loci to reconstruct queen genotypes and found that C. mimosae colonies are frequently secondarily polygynous, in that they include multiple closely related (and sometimes full-sib) queens, and (more rarely) unrelated queens. We also found that individual queens in both monogynous and polygynous colonies had mated with multiple males, making C. mimosae an interesting example of simultaneous polygyny and polyandry. The presence of polygyny in C. mimosae and the intense competition for nest-sites between C. mimosae and its conspecifics support the association between nest-site limitation and polygyny. Polygyny may allow for increased worker populations and a competitive advantage, as inter-colony conflicts are typically won by the colony with the larger number of workers.     

Mating system evolution and worker caste diversity in Pheidole ants

The efficiency of social groups is generally optimized by a division of labour, achieved through behavioural or morphological diversity of members. In social insects, colonies may increase the morphological diversity of workers by recruiting standing genetic variance for size and shape via multiply mated queens (polyandry) or multiple‐breeding queens (polygyny). However, greater worker diversity in multi‐lineage species may also have evolved due to mutual worker policing if there is worker reproduction. Such policing reduces the pressure on workers to maintain reproductive morphologies, allowing the evolution of greater developmental plasticity and the maintenance of more genetic variance for worker size and shape in populations. Pheidole ants vary greatly in the diversity of worker castes. Also, their workers lack ovaries and are thus invariably sterile regardless of the queen mating frequency and numbers of queens per colony. This allowed us to perform an across‐species study examining the genetic effects of recruiting more patrilines on the developmental diversity of workers in the absence of confounding effects from worker policing. Using highly variable microsatellite markers, we found that the effective mating frequency of the soldier‐polymorphic P. rhea (avg. me_N = 2.65) was significantly higher than that of the dimorphic P. spadonia (avg. me_N = 1.06), despite a significant paternity skew in P. rhea (avg. B = 0.10). Our findings support the idea that mating strategies of queens may co‐evolve with selection to increase the diversity of workers. We also detected patriline bias in the production of different worker sizes, which provides direct evidence for a genetic component to worker polymorphism.     

Maintenance of Genetic Variability under Strong Stabilizing Selection: A Two-Locus Model

We study a two locus model with additive contributions to the phenotype to explore the relationship between stabilizing selection and recombination. We show that if the double heterozygote has the optimum phenotype and the contributions of the loci to the trait are different, then any symmetric stabilizing selection fitness function can maintain genetic variability provided selection is sufficiently strong relative to linkage. We present results of a detailed analysis of the quadratic fitness function which show that selection need not be extremely strong relative to recombination for the polymorphic equilibria to be stable. At these polymorphic equilibria the mean value of the trait, in general, is not equal to the optimum phenotype, there exists a large level of negative linkage disequilibrium which ``hides' additive genetic variance, and different equilibria can be stable simultaneously. We analyze dependence of different characteristics of these equilibria on the location of optimum phenotype, on the difference in allelic effect, and on the strength of selection relative to recombination. Our overall result that stabilizing selection does not necessarily eliminate genetic variability is compatible with some experimental results where the lines subject to strong stabilizing selection did not have significant reductions in genetic variability.     

Sociogenetic organisation of two desert ants

Desert ants of the genus Cataglyphis evolved a remarkable diversity in their reproductive strategies. In Cataglyphis species where social organisation was described so far, colonies are headed by one or multiple queens, queens being singly or multiply mated, and workers and/or queens possess the ability to reproduce asexually via thelytokous parthenogenesis. Here, we investigate the social organisation of C. bombycina (group bombycinus) and C. theryi (group albicans) using highly polymorphic microsatellite markers. Our results show that both species are characterized by monogynous colonies and multiply mated queens, supporting the idea that polyandry is an ancestral trait of the genus. No evidence for parthenogenetic reproduction by queens was found. One distinctive feature of the species C. bombycina among the genus is the presence of a morphologically distinct soldier caste, with highly developed scythe blades jaws. In the only colony where a significant number of soldiers have been sampled, the distribution of patrilines is fundamentally different between the soldier and the worker caste. This result suggests a genetic contribution to worker caste determination in this species, and certainly awaits further investigation.     

The multiple-locus symmetric fertility model

Selection due to variation in the fecundity among matings of genotypes with respect to many loci each with two alleles is studied. The fitness of a mating depends only on the genotypic distinction between homozygote and heterozygote at each locus in the two individuals, and differences among loci are allowed. This symmetric fertility model is therefore a generalization of the multiple-locus symmetric viability model. The phenomena seen in the two-locus symmetric fertility model generalize—e.g., the possibility of joint stability of equilibria with linkage equilibrium and with linkage disequilibrium, and the existence of different types of totally polymorphic equilibria with the gametic proportions in linkage equilibrium. The central equilibrium with genotypic frequencies in Hardy-Weinberg proportions and gametic frequencies in Robbins proportions exists for all symmetric fertility models. For some symmetric fertility regimes additional equilibria exist with gametic frequencies in linkage equilibrium and with genotypic frequencies in Hardy-Weinberg proportions at all except one locus. These equilibria may exist in the dioecious symmetric viability model, and then they will be locally stable. For free recombination the stable equilibria show linkage equilibrium, but several of these with different numbers of polymorphic loci may be stable simultaneously.     

New properties of the two-locus partial selfing model with selection

Analytic solutions are obtained for the equilibria of a simple two-locus, heterotic selection model with mixed selfing and random outcrossing. Two general phenomena are possible, depending upon the viabilities and the degree of selfing: (1) Negative disequilibrium potential, under which only gametic disequilibrium is possible; and (2) positive disequilibrium potential, which can result in permanent gametic disequilibrium provided that linkage is sufficiently tight. Under random mating (s = 0), these two situations correspond to negative and positive additive epistasis, respectively. With partial self-fertilization, however, this is no longer true, and a more appropriate measure of gametc disequilibrium potential, Δ(s), is introduced. A numerically aided examination of the model results in the discovery of two new properties of partial selfing with selection: (1) With negative disequilibrium potential (Δ(s) < 0), the equilibrium mean fitness increases with increasing recombination. With positive disequilibrium potential (Δ(s) > 0), the opposite is true. (2) Gametic disequilibrium can increase or decrease as the degree of selfing is increased. Therefore, it is apparent that partial selfing and linkage are not analogous as regards the maintenance of disequilibrium.     

An application of kinship process to the gene frequencies: linkage disequilibrium due to random drift in mendelian genetics with reversible mutation and in molecular genetics.

Selection at the colony level in social Hymenoptera with colonies containing single, once-mated queens is examined under a simple two-allele model. The condition for balanced polymorphism is $\text{X>}\text{2V}{}^2\text{(V + 1)}$, where V is the fitness of colonies with all workers homozygous and X that of colonies with both heterozygous and homozygous workers, relative to the fitness of colonies with all workers heterozygous. For certain fitness combinations satisfying the above relationship and characterized by values of V and X much lower than one, iteration reveals the development of stable limit cycles of allele frequencies rather than convergence to an equilibrium point. Addition of a third allele, or overlap between generations, eliminates these cycles. Queen-level overdominance is sufficient but not necessary for balanced polymorphism when V < 1, is both sufficient and necessary when V = 1, and is necessary but not sufficient when V > 1. Colony-level selection is a potentially powerful force maintaining genetic variation in populations of social insects, but does not imply correspondence between queen and worker genotype frequencies.     

Sex-specific meiotic drive and selection at an imprinted locus

We present a one-locus model that breaks two symmetries of Mendelian genetics. Whereas symmetry of transmission is breached by allowing sex-specific segregation distortion, symmetry of expression is breached by allowing genomic imprinting. Simple conditions for the existence of at least one polymorphic stable equilibrium are provided. In general, population mean fitness is not maximized at polymorphic equilibria. However, mean fitness at a polymorphic equilibrium with segregation distortion may be higher than mean fitness at the corresponding equilibrium with Mendelian segregation if one (or both) of the heterozygote classes has higher fitness than both homozygote classes. In this case, mean fitness is maximized by complete, but opposite, drive in the two sexes. We undertook an extensive numerical analysis of the parameter space, finding, for the first time in this class of models, parameter sets yielding two stable polymorphic equilibria. Multiple equilibria exist both with and without genomic imprinting, although they occurred in a greater proportion of parameter sets with genomic imprinting.     

An Inclusive Fitness-Based Exploration of the Origin of Soldiers: The Roles of Sex Ratio, Inbreeding, and Soldier Reproduction

I present an inclusive fitness model for the origin of soldiers that incorporates soldier production of males and females, in haplodiploids and diploids. In this paper, the term soldier refers to an individual that is morphologically and/or behaviorally specialized for defense. However, the model presented here can usefully be extended to other helping behaviors that are episodic in nature and that impact the colony as a whole rather than helping behavior that is directed to specific individuals. In general, the results of the model show that proto-soldier reproduction via sib-mating increases the ease with which soldiering evolves. Male haplodiploids appear the most apt to evolve soldier behavior compared to female haplodiploids and diploids. Haplodiploid females are expected to produce a greater proportion of male dispersers relative to their production of female dispersers and thereby increase the predilection in females to evolve soldiering compared to diploid populations. Application of the model to gall-forming thrips in Australia reveals that species basal to the phylogenetic branch where soldiers are inferred to have evolved show a lower threshold for soldier evolution than more derived species. This phylogenetic pattern is consistent with soldier production of male and female offspring facilitating the origin of soldiering in the social gall-forming thrips.     

The generalized multiplicative model for viability selection at multiple loci

Selection due to differential viability is studied in an n-locus two-allele model using a set indexation that allows the simplicity of the one-locus two-allele model to be carried to multi-locus models. The existence condition is analyzed for polymorphic equilibria with linkage equilibrium: Robbins' equilibria. The local stability condition is given for the Robbins' equilibria on the boundaries in the generalized non-epistatic selection regimes of Karlin and Liberman (1979). These generalized non-epistatic regimes include the additive selection model, the multiplicative selection model and the multiplicative interaction model, and their symmetric versions cover all the symmetric viability models.Research supported by grant no. 11-7805 from the Danish Natural Science Research Council, by NIH grant GM 28016, by a fellowship from the Research Foundation of Aarhus University, and by a visiting fellowship from the University of New England, N.S.W.     

Queen developmental time and fitness consequences for queens of clonal social parasitic honeybees (A. m. capensis) and its host A. m. scutellata

Summary. The social parasitic honeybees of South Africa (Apis mellifera capensis) consist of a single clonal lineage, which has been selected for traits related to worker reproduction. Viable queens of this parasitic clonal lineage have never been observed. We tested if it is possible to rear queens from eggs of the social parasitic workers. In a competitive situation, using larvae of the parasitic clonal lineage and of the host, we tested the discriminatory ability of host colonies (A. m. scutellata) between parasitic and non-parasitic larvae. We found evidence for a reduced fitness of queens reared from the social parasite lineage, resulting from a longer developmental time. The results are discussed in the light of a fitness trade-off between queen and worker caste.Received 22 July 2004; revised 22 December 2004; accepted 5 January 2005.     

Natural selection and fitness entropy in a density-regulated population

The entropy H(p_o,p*) of a population with the initial allele frequency p_o given the equilibrium polymorphic frequency p* has been proposed as a measure of natural selection. In the present paper, we have extended this concept to include a particular aspect of density-dependent selection. We compared size trajectory of a population initially at genetic equilibrium, N(t), with the size trajectories of populations not initially at p*,N(t), but which do eventually converge to a common equilibrium allele frequency and equilibrium density, N*. The following experimentally testable hyopthesis was established. The total area defined by the difference between the trajectories of N(t) and N(t) as they converge to N* is directly proportional to the fitness entropy when population size is transformed using the density-dependent fitness value. Two properties of this relationship were noted. First, it is independent of the magnitude of natural selection and, secondly, it does not depend upon the initial population density as long as the equilibrium and nonequilibrium populations have the same initial numbers. This hypothesis was evaluated with experimental data on the flour beetle Tribolium castaneum.