PLOIDY AND EVOLUTION BY SEXUAL SELECTION: A COMPARISON OF HAPLOID AND DIPLOID FEMALE CHOICE MODELS NEAR FIXATION EQUILIBRIA

We compare the stability properties of haploid and diploid models of Fisherian sexual selection (with male contribution limited to sperm) by examining both models at equilibria for which a male trait is fixed or absent. Haploid and diploid two locus diallelic models share the property that the stability of such fixation equilibria is determined by the relationship between the harmonic mean of relative preference values for the common male trait, weighted by the frequency of the preferences, and the relative viability associated with the common male trait. When diploid females with heterozygotic-based preferences express preference strengths intermediate between homozygote-based preferences, then boundary equilibria of haploid and diploid models share many stability properties. However, even with intermediate heterozygote preferences, haploid and diploid models do differ: (1) for a particular frequency of the preference allele, both fixation boundaries can be stable for the diploid model, and (2) with over- or underdominance at the preference locus (a possibility precluded in the haploid model), a fixation boundary in the diploid model may show two switches in its stability state for increasing frequencies of one of the preference alleles. These differences are due not just to the impossibility of dominance in haploid models, but also to the larger number of diploid genotypes.     

EVOLUTION BY FISHERIAN SEXUAL SELECTION IN DIPLOIDS

Most models of Fisherian sexual selection assume haploidy. However, analytical models that focus on dynamics near fixation boundaries and simulations show that the resulting behavior depends on ploidy. Here we model sexual selection in a diploid to characterize behaviour away from fixation boundaries. The model assumes two di-allelic loci, a male-limited trait locus subject to viability selection, and a preference locus that determines a female's tendency to mate with males based on their genotype at the trait locus. Using a quasi-linkage equilibrium (QLE) approach, we find a general equation for the curves of quasi-neutral equilibria, and the conditions under which they are attracting or repelling. Unlike in the haploid model, the system can move away from the internal curve of equilibria in the diploid model. We show that this is the case when the combined forces of natural and sexual selection induce underdominance at the trait locus.     

On the logic of Fisherian sexual selection*

In Fisher's model of sexual selection, a female preference for a male trait spreads together with the trait because their genetic bases become correlated. This can be interpreted as a “greenbeard” system: a preference gene, by inducing a female to mate with a trait-bearing male, favors itself because the male is disproportionately likely also to carry the preference gene. Here, we use this logic to argue that Fisherian sexual selection in diploids proceeds via two channels: (i) trait-bearing males are disproportionately the product of matings between preference-bearing mothers and trait-bearing fathers, and thus trait and preference genes are correlated “in trans”; (ii) trait and preference genes come into gametic phase disequilibrium, and thus are correlated “in cis.” Gametic phase disequilibrium is generated by three distinct mechanisms that we identify. The trans channel does not operate when sexual selection is restricted to the haploid phase, and therefore represents a fundamental difference between haploid and diploid models of sexual selection. We show that the cis and trans channels contribute equally to the spread of the preference when recombination between the preference and trait loci is free, but that the trans channel is substantially more important when linkage is tight.     

Runaway sexual selection with paternal transmission of the male trait and gene-culture determination of the female preference

Sexual selection is modeled with a male viability-reducing trait and a female mating preference for that trait both of which are culturally transmitted. Both the male trait and the female preference are transmitted only between same-sex individuals, so that non-random association between the trait and the preference, which would give rise to a Fisherian runaway process, cannot arise. Inclusion of an autosomal gene that confers a female predisposition to acquire a certain preference is shown to allow the coevolution of the male trait and the female preference by a Fisherian process. This holds true even when the female preference has a slight viability cost, provided the male cultural transmission is not perfect. It is also suggested that a Fisherian process can be more easily initiated in these models than in the conventional genetic models. Furthermore, a Fisherian process may cause cultural transmission of female preference to evolve. Additionally, polymorphism can be maintained at the predisposition locus if heterozygous females have a stronger predisposition to acquire the preference than homozygotes. Our models may be applicable to the case when the male trait is a Y-linked genetic or environmentally determined trait.     

SEXUAL SELECTION BY THE HANDICAP MECHANISM

The handicap mechanism of sexual selection by female choice has been strongly criticized because it does not cause sexual selection to reinforce viability selection and it cannot account for the origin of mating preferences. However, several models indicate that the handicap mechanism can have important effects when operating in conjunction with Fisher's mechanism in polygynous populations. These models have been criticized because they require that fitness remains heritable indefinitely. I develop a simple haploid model of the handicap mechanism based on nonheritable variation in paternal investment, thus eliminating the problem of heritable fitness. This model produces the same evolutonary dynamics as both simple and quantitative genetic models of the handicap mechanism based on heritable fitness. If the parameters are such that Fisherian runaway selection does not occur in the null model (i.e., the polymorphic equilibria, which lie along the “Fisher line,” are stable), then the handicap mechanism turns the Fisher line into an evolutionary trajectory upon which all other trajectories converge. This occurs because Fisher's mechanism generates no net selection on female preference when the population is on the Fisher line, so that any additional source of selection (direct or indirect) on female choice causes the population to evolve deterministically along the Fisher line. This change in the evolutionary dynamics has the important consequence of eliminating the potential for rapid population divergence for mating systems via genetic drift along the Fisher line.     

General analysis of frequency-dependent sexual selection at a multi-allelic locus

A general model is analyzed in which arbitrarily frequency-dependent selection acts on one sex of a diploid population with several alleles at one locus, as a result of viability or mating-success differences. The existence of boundary and polymorphic equilibria is examined, and conditions for local stability, internal and external, are obtained. The status of Hardy-Weinberg approximations in studying stability and approach to equilibria is also considered. The general principles are then applied to two specific models: one where genotypes fall into two phenotypic classes; and one with a hierarchy of dominance where viability and sexual selection are opposed. In the latter case it is found that, of all the equilibria present, there is one and only one which could possibly be stable: the existence of a unique globally stable equilibrium might then be inferred.     

Complex dynamics occur in a single-locus,multiallelic model of general frequency-dependent selection

We examine the characteristics of non-equilibrium dynamics produced by a simple well-known model of frequency-dependent selection at a single diploid locus. An examination of the parameter space of this “pairwise-interaction model” (PIM) revealed non-equilibrium dynamics for polymorphisms of 3, 4 and 5 alleles; both allele-frequency cycling and aperiodic trajectories were detected. We measured the number, cycle length and domains of attraction of the various attractors produced by the model. The domains of attraction tended to be smaller, and the cycles longer, for systems with larger number of alleles. Fitnesses that parametrized negative frequency-dependent selection were more likely to allow cycling, and these cycles also had larger domains of attraction. Aperiodic trajectories were detected only in cases with 4 or 5 alleles. The genetic cycles produced by the model do not have periods as short as those predicted in ecological models with cycling (such as predator–prey population cycles, etc.). Consequently, in a real-world system, PIM allele-frequency cycling is likely to be indistinguishable from stable equilibria when observed over short time scales.     

Fisherian and “good genes” benefits of mate choice: how (not) to distinguish between them

“Good genes” models of mate choice are commonly tested by examining whether attractive males sire offspring with improved survival. If offspring do not survive better (or indeed survive less well), but instead inherit the attractiveness of their father, results are typically interpreted to support the Fisherian process, which allows the evolution of preferences for arbitrary traits. Here, I show that the above view is mistaken. Because of life‐history trade‐offs, an attractive male may perform less well in other components of fitness. A female obtains a “good genes” benefit whenever males show heritable variation in quality, even if high‐quality males invest so much in sexual advertisement that attractiveness has no positive correlation with any other life‐history trait than male mating success itself. Therefore, a negative correlation between attractiveness and viability does not falsify good genes, if mating with a high‐quality male results on average in superior offspring performance (mating success of sons included). The heritable “good genes” benefit can be sustained even if sexually antagonistic genes cause female offspring sired by high‐quality males to survive and reproduce less well. Neglecting the component of male mating success from measurements of fitness returns from sons and daughters will bias the advantage of mating with a high‐quality male downwards. This result may partly account for the rather weak “good genes” effects found in a recent meta‐analysis.     

ON EVOLUTION UNDER SEXUAL AND VIABILITY SELECTION: A TWO-LOCUS DIPLOID MODEL

A two-locus diploid model of sexual selection is presented in which the two loci govern, respectively, a trait limited in expression in one sex (generally male) and the mating preferences of the other sex (generally female). The viability of a male depends on its genotype at the trait locus. In contrast, all females are equally viable and all individuals are equally fertile with respect to the two loci. Near fixation at both loci, evolution at the mating locus is neutral and hence a new mating preference allele will increase only through random genetic drift or through a correlated response to the increase of a new advantageous trait allele. If, however, a polymorphism is already maintained at the trait locus through overdominance in fitness then the increase of a rare preference allele depends only on the recombination rate between the loci and not on the new preference scheme.     

The evolution of mate choice in a fluctuating environment

This paper analyzes the evolutionary dynamics of a locus controlling the degree of female mating preference in a temporally fluctuating environment. Preference for mating with males with respect to their genotypes at a locus that is subject to temporally varying natural selection pressure is considered first. With weak selection and free recombination between the choice locus and the selected locus, preference for mating with heterozygotes appears to be favored. With strong selection, preference for homozygous mates may be favored. In each case, choice alleles may increase from very low initial frequencies to near fixation, in contrast to previous models of mate choice in varying environments. Linkages between the two loci has complex effects on the strength and direction of selection for mate choice. Preference for mating with males with the currently fitter genotypes at the locus under natural selection is also modelled. Provided that the environmental period is not too short, a rare allele conferring such preference may be favored and spread to fixation. Strong natural selection, tight linkage and a short environmental period may produce polymorphism for the level of mate choice.     

Patterns of multiallelic polymorphism maintained by migration and selection

Evolution at a multiallelic locus under the joint action of migration and viability selection is investigated. Generations are discrete and nonoverlapping. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. The forward migration matrix is arbitrary, but time independent and ergodic (i.e., irreducible and aperiodic). Several examples of globally attracting multiallelic equilibria are presented. Migration can cause global fixation even if, without migration, there is a globally attracting multiallelic equilibrium in every colony. Migration can also cause the global fixation of an allele that, without migration, is eliminated in every colony. Without dominance, generically, the number of alleles present at equilibrium cannot exceed the number of colonies. Some general properties and examples of the Levene model are studied in detail. If in each colony there is either no dominance or, without migration, a globally attracting internal equilibrium, then there exists a globally attracting equilibrium with migration. Therefore, if an internal equilibrium exists, it is the global attractor.     

DIPLOID MALES IN A PRIMITIVELY EUSOCIAL BEE,LASIOGLOSSUM (DIALICTUS) ZEPHYRUM (HYMENOPTERA: HALICTIDAE)

Hymenoptera are characterized by a haplo-diploid mechanism of sex determination. Females are diploid and males are haploid. However, in many species diploid males may occur if individuals are homozygous at a sex determining locus. Diploid males were found in three out of four populations (nest aggregations) of the primitively eusocial, halictine bee Lasioglossum zephyrum for which samples of males were examined electrophoretically. The frequency of diploid males was greater in a small, geographically isolated population (the “Robinson” nest aggregation) than in a large population that had nearby neighboring populations (the “Salmon Creek A” nest aggregation). In addition, the proportion of polymorphic loci was lower in the Robinson nest aggregation suggesting that a bottleneck event or loss of alleles due to small population size occurred in the Robinson population that involved a loss in the number of alleles at the sex determining locus.     

Some circumstances assuring monomorphism in subdivided populations

It is well known that in a subdivided population subject to soft selection with two alleles at one locus, instability of both fixation states (a “protected polymorphism”) entails at least one stable polymorphic equilibrium. Although stable polymorphic and monomorphic equilibria can coexist in general, a stable fixation state (monomorphic equilibrium) precludes the existence of any polymorphic equilibrium under the circumstances of haploid or submultiplicative diploid viabilities. This provides that a stable monomorphism is robust against random fluctuations in allele frequencies. It also increases the known circumstances where there is a unique globally attracting stable equilibrium, i.e., where allele frequencies are determined by the selection-migration structure independent of the history of the system.     

Analysis of some nonrandom mating models

In this paper a few asymmetric models are presented taking account of the effects of assortative mating on an autosomal trait controlled by a single locus possibly with multiple alleles. The models are developed by specifying the intensities for preference mating for various phenotypes. The analysis is confined to the case in which preference is exercised by the individuals of one sex only. It is assumed that males possess unlimited fertility.The dynamics of the population and its equilibrium distribution are discussed. The gene frequency usually changes with time and equilibrium distribution in most cases depends only on the assortment parameters. Expressions are obtained giving the additive and dominance components of variance, and covariances for relatives for populations in equilibrium for some of the models.     

Selective sweeps in multilocus models of quantitative traits

We study the trajectory of an allele that affects a polygenic trait selected toward a phenotypic optimum. Furthermore, conditioning on this trajectory we analyze the effect of the selected mutation on linked neutral variation. We examine the well-characterized two-locus two-allele model but we also provide results for diallelic models with up to eight loci. First, when the optimum phenotype is that of the double heterozygote in a two-locus model, and there is no dominance or epistasis of effects on the trait, the trajectories of selected mutations rarely reach fixation; instead, a polymorphic equilibrium at both loci is approached. Whether a polymorphic equilibrium is reached (rather than fixation at both loci) depends on the intensity of selection and the relative distances to the optimum of the homozygotes at each locus. Furthermore, if both loci have similar effects on the trait, fixation of an allele at a given locus is less likely when it starts at low frequency and the other locus is polymorphic (with alleles at intermediate frequencies). Weaker selection increases the probability of fixation of the studied allele, as the polymorphic equilibrium is less stable in this case. When we do not require the double heterozygote to be at the optimum we find that the polymorphic equilibrium is more difficult to reach, and fixation becomes more likely. Second, increasing the number of loci decreases the probability of fixation, because adaptation to the optimum is possible by various combinations of alleles. Summaries of the genealogy (height, total length, and imbalance) and of sequence polymorphism (number of polymorphisms, frequency spectrum, and haplotype structure) next to a selected locus depend on the frequency that the selected mutation approaches at equilibrium. We conclude that multilocus response to selection may in some cases prevent selective sweeps from being completed, as described in previous studies, but that conditions causing this to happen strongly depend on the genetic architecture of the trait, and that fixation of selected mutations is likely in many instances.     

A classical setting for associations between markers and loci affecting quantitative traits

We examine the relationships between a genetic marker and a locus affecting a quantitative trait by decomposing the genetic effects of the marker locus into additive and dominance effects under a classical genetic model. We discuss the structure of the associations between the marker and the trait locus, paying attention to non-random union of gametes, multiple alleles at the marker and trait loci, and non-additivity of allelic effects at the trait locus. We consider that this greater-than-usual level of generality leads to additional insights, in a way reminiscent of Cockerham's decomposition of genetic variance into five terms: three terms in addition to the usual additive and dominance terms. Using our framework, we examine several common tests of association between a marker and a trait.     

Population genetic models of male and mutual mate choice

Examples of male mate choice are becoming increasingly common, even in polygynous species. We create a series of population genetic models to examine the evolutionary equilibria and dynamics resulting from male mate choice during polygyny, alone and in the context of mutual mate choice by both sexes. We find that unless males with a preference are able to increase their overall courtship output, male preference will be lost. This loss can be counteracted if males choose females not based on arbitrary traits, but based on a trait that indicates high fertility or viability. We also conclude that if male and female preferences and traits are all controlled by different loci, the male and female mate choice systems are decoupled; the presence of a male preference then has no influence on the equilibria or dynamics of female mate choice. If male and female traits are coupled by pleiotropy, it becomes possible for a male preference to be maintained, regardless of whether preferences between the sexes are pleiotropic or controlled by separate loci.     

The population genetics of X-autosome synthetic lethals and steriles

Epistatic interactions are widespread, and many of these interactions involve combinations of alleles at different loci that are deleterious when present in the same individual. The average genetic environment of sex-linked genes differs from that of autosomal genes, suggesting that the population genetics of interacting X-linked and autosomal alleles may be complex. Using both analytical theory and computer simulations, we analyzed the evolutionary trajectories and mutation-selection balance conditions for X-autosome synthetic lethals and steriles. Allele frequencies follow a set of fundamental trajectories, and incompatible alleles are able to segregate at much higher frequencies than single-locus expectations. Equilibria exist, and they can involve fixation of either autosomal or X-linked alleles. The exact equilibrium depends on whether synthetic alleles are dominant or recessive and whether fitness effects are seen in males, females, or both sexes. When single-locus fitness effects and synthetic incompatibilities are both present, population dynamics depend on the dominance of alleles and historical contingency (i.e., whether X-linked or autosomal mutations occur first). Recessive synthetic lethality can result in high-frequency X-linked alleles, and dominant synthetic lethality can result in high-frequency autosomal alleles. Many X-autosome incompatibilities in natural populations may be cryptic, appearing to be single-locus effects because one locus is fixed. We also discuss the implications of these findings with respect to standing genetic variation and the origins of Haldane's rule.     

Consequences of genetic linkage for the maintenance of sexually antagonistic polymorphism in hermaphrodites

When selection differs between males and females, pleiotropic effects among genes expressed by both sexes can result in sexually antagonistic selection (SA), where beneficial alleles for one sex are deleterious for the other. For hermaphrodites, alleles with opposing fitness effects through each sex function represent analogous genetic constraints on fitness. Recent theory based on single‐locus models predicts that the maintenance of SA genetic variation should be greatly reduced in partially selfing populations. However, selfing also reduces the effective rate of recombination, which should facilitate selection on linked allelic combinations and expand opportunities for balancing selection in a multilocus context. Here, I develop a two‐locus model of SA selection for simultaneous hermaphrodites, and explore the joint influence of linkage, self‐fertilization, and dominance on the maintainance of SA polymorphism. I find that the effective reduction in recombination caused by selfing significantly expands the parameter space where SA polymorphism can be maintained relative to single‐locus models. In particular, linkage facilitates the invasion of male‐beneficial alleles, partially compensating for the “female‐bias” in the net direction of selection created by selfing. I discuss the implications of accounting for linkage among SA loci for the maintenance of SA genetic variation and mixed mating systems in hermaphrodites.