A chip off the old block: a model for the evolution of genomic imprinting via selection for parental similarity

A consequence of genomic imprinting is that offspring are more similar to one parent than to the other, depending on which parent's genes are inactivated in those offspring. We hypothesize that genomic imprinting may have evolved at some loci because of selection to be similar to the parent of one sex or the other. We construct and analyze an evolutionary-genetic model of a two-locus two-deme system, in which one locus codes for a character under local selection and the second locus is a potential cis-acting modifier of imprinting. A proportion of males only migrate between demes every generation, and prebreeding males are less fit, on average, than females. We examine the conditions in which an imprinting modifier allele can invade a population fixed for a nonimprinting modifier allele and vice versa. We find that the conditions under which the imprinting modifier invades are biologically restrictive (high migration rates and high values of recombination between the two loci) and thus this hypothesis is unlikely to explain the evolution of imprinting. Our modeling also shows that, as with several other hypotheses, polymorphism of imprinting status may evolve under certain circumstances, a feature not predicted by verbal accounts.     

The effect of genetic conflict on genomic imprinting and modification of expression at a sex-linked locus

We examine how genomic imprinting may have evolved at an X-linked locus, using six diallelic models of selection in which one allele is imprintable and the other is not. Selection pressures are generated by genetic conflict between mothers and their offspring. The various models describe cases of maternal and paternal inactivation, in which females may be monogamous or bigamous. When inactivation is maternal, we examine the situations in which only female offspring exhibit imprinting as well as when both sexes do. We compare our results to those previously obtained for an autosomal locus and to four models in which a dominant modifier of biallelic expression is subjected to the same selection pressures. We find that, in accord with verbal predictions, maternal inactivation of growth enhancers and paternal inactivation of growth inhibitors are more likely than imprinting in the respective opposite directions, although these latter outcomes are possible for certain parameter combinations. The expected outcomes are easier to evolve than the same outcomes for autosomal loci, contradicting the available evidence concerning the direction of imprinting on mammalian sex chromosomes. In most of our models stable polymorphism of imprinting status is possible, a behavior not predicted by verbal accounts.     

Dynamics of the classical genetic model for the evolution of dominance

Using a generalized method of Ljapunov functions, the dynamics of the classical genetic model for the evolution of dominance is studied. The model is treated as a two locus two allele system of a primary and a modifying locus with selection, mutation, and recombination. Its behavior may be described either by a system of four differential equations or by a system of four difference equations. In particular, it is proved that under very general conditiones on the five parameters involved, in both cases the well-known fixed point for the mutation-selection balance at the primary locus when the modifier is completely selected is globally asymptotically stable. If, however, the unmodified heterozygote is completely recessive or underdominant, the modifier is only selected if at the beginning of the evolution its frequency and that of the favorable primary allele is not extremely low. Otherwise, it may happen that the favorable primary allele becomes extinct.     

Segregation and the evolution of sex under overdominant selection

The segregation of alleles disrupts genetic associations at overdominant loci, causing a sexual population to experience a lower mean fitness compared to an asexual population. To investigate whether circumstances promoting increased sex exist within a population with heterozygote advantage, a model is constructed that monitors the frequency of alleles at a modifier locus that changes the relative allocation to sexual and asexual reproduction. The frequency of these modifier alleles changes over time as a correlated response to the dynamics at a fitness locus under overdominant selection. Increased sex can be favored in partially sexual populations that inbreed to some extent. This surprising finding results from the fact that inbred populations have an excess of homozygous individuals, for whom sex is always favorable. The conditions promoting increased levels of sex depend on the selection pressure against the homozygotes, the extent of sex and inbreeding in the population, and the dominance of the invading modifier allele.     

Evolution of assortative mating in a population expressing dominance

In this article, we study the influence of dominance on the evolution of assortative mating. We perform a population-genetic analysis of a two-locus two-allele model. We consider a quantitative trait that is under a mixture of frequency-independent stabilizing selection and density- and frequency-dependent selection caused by intraspecific competition for a continuum of resources. The trait is determined by a single (ecological) locus and expresses intermediate dominance. The second (modifier) locus determines the degree of assortative mating, which is expressed in females only. Assortative mating is based on similarities in the quantitative trait ('magic trait' model). Analytical conditions for the invasion of assortment modifiers are derived in the limit of weak selection and weak assortment. For the full model, extensive numerical iterations are performed to study the global dynamics. This allows us to gain a better understanding of the interaction of the different selective forces. Remarkably, depending on the size of modifier effects, dominance can have different effects on the evolution of assortment. We show that dominance hinders the evolution of assortment if modifier effects are small, but promotes it if modifier effects are large. These findings differ from those in previous work based on adaptive dynamics.     

ESS gene expression of X-linked imprinted genes subject to sexual selection

We define ESS (Evolutionary Stable Strategy) conditions for the evolution of genomic imprinting at an X-linked locus. The system analysed is designed for mammalian imprinting in which X-linked genes typically undergo random X-inactivation and lack Y-linked homologues. We consider two models that map cellular gene expression to fitness in females subject to random X-inactivation. In the first model, female fitness is simply a function of the average gene expression across all cells. In the second model, each cell contributes independently to fitness, and female fitness is assessed as the average of these contributions across all cells. In both models, imprinting readily evolves when sexual selection favours different levels of gene expression in the two sexes. Imprinting is beneficial as it improves adaptation in both sexes. There are limits to the improvement in adaptation when sexual selection is strong and favours greater gene expression in males (the heterogametic sex). We also consider the consequences of an active Y-linked homologue on the evolution of imprinting. Our analysis suggests that restrictive conditions apply for the evolution of polymorphic ESSs at an X-linked imprinted loci.     

Evolution of dominance under frequency-dependent intraspecific competition

A population-genetic analysis is performed of a two-locus two-allele model, in which the primary locus has a major effect on a quantitative trait that is under frequency-dependent disruptive selection caused by intraspecific competition for a continuum of resources. The modifier locus determines the degree of dominance at the trait level. We establish the conditions when a modifier allele can invade and when it becomes fixed if sufficiently frequent. In general, these are not equivalent because an unstable internal equilibrium may exist and the condition for successful invasion of the modifier is more restrictive than that for eventual fixation from already high frequency. However, successful invasion implies global fixation, i.e., fixation from any initial condition. Modifiers of large effect can become fixed, and also invade, in a wider parameter range than modifiers of small effect. We also study modifiers with a direct, frequency-independent deleterious fitness effect. We show that they can invade if they induce a sufficiently high level of dominance and if disruptive selection on the ecological trait is strong enough. For deleterious modifiers, successful invasion no longer implies global fixation because they can become stuck at an intermediate frequency due to a stable internal equilibrium. Although the conditions for invasion and for fixation if sufficiently frequent are independent of the linkage relation between the two loci, the rate of spread depends strongly on it. The present study provides further support to the view that evolution of dominance may be an efficient mechanism to remove unfit heterozygotes that are maintained by balancing selection. It also demonstrates that an invasion analysis of mutants of very small effect is insufficient to obtain a full understanding of the evolutionary dynamics under frequency-dependent selection.     

On the evolutionary stability of Mendelian segregation

We present a model of a primary locus subject to viability selection and an unlinked locus that causes sex-specific modification of the segregation ratio at the primary locus. If there is a balanced polymorphism at the primary locus, a population undergoing Mendelian segregation can be invaded by modifier alleles that cause sex-specific biases in the segregation ratio. Even though this effect is particularly strong if reciprocal heterozygotes at the primary locus have distinct viabilities, as might occur with genomic imprinting, it also applies if reciprocal heterozygotes have equal viabilities. The expected outcome of the evolution of sex-specific segregation distorters is all-and-none segregation schemes in which one allele at the primary locus undergoes complete drive in spermatogenesis and the other allele undergoes complete drive in oogenesis. All-and-none segregation results in a population in which all individuals are maximally fit heterozygotes. Unlinked modifiers that alter the segregation ratio are unable to invade such a population. These results raise questions about the reasons for the ubiquity of Mendelian segregation.     

The Evolution of the Y Chromosome with X-Y Recombination

A theoretical population genetic model is developed to explore the consequences of X-Y recombination in the evolution of sex chromosome polymorphism. The model incorporates one sex-determining locus and one locus subject to natural selection. Both loci have two alleles, and the rate of classical meiotic recombination between the loci is r. The alleles at the sex-determining locus specify whether the chromosome is X or Y, and the alleles at the selected locus are arbitrarily labeled A and a. Natural selection is modeled as a process of differential viabilities. The system can be expressed in terms of three recurrence equations, one for the frequency of A on the X-bearing gametes produced by females, one for each of the frequency of A on the X- and Y-bearing gametes produced by males. Several special cases are examined, including X chromosome dominance and symmetric selection. Unusual equilibria are found with the two sexes having very different allele frequencies at the selected locus. A significant finding is that the allowance of recombination results in a much greater opportunity for polymorphism of the Y chromosome. Tighter linkage results in a greater likelihood for equilibria with a large difference between the sex chromosomes in allele frequency.     

Failure of imprinting at Igf-2: two models of mutation-selection balance.

The failure of maternal imprinting at the insulin-like growth factor II (Igf-2) locus predisposes individuals to several clinical conditions, including Wilms tumor. Having two functional Igf-2 genes, therefore, is selectively disadvantageous, and the condition is probably maintained in human populations by recurrent mutation. We propose two models that predict the expected frequency of functionally diploid individuals in a large population, in terms of a mutation rate, mu, and the selection coefficient against functionally diploid individuals, s. In the first model a mutant Igf-2 allele that cannot be imprinted arises from the standard, imprintable allele at a rate mu. Our second model hypothesizes a second modifier locus at which a recessive allele arises at rate mu. Mothers who are homozygous for this recessive modifier allele fail to imprint their eggs. Both models predict the expected frequency of affecteds to be 2 mu/s(1 + mu), approximately twice that predicted by the standard one-locus model of a recessive allele in mutation-selection balance. This frequency suggests that < or = 25% of the cases of Wilms tumor are due to the failure to imprint the maternal Igf-2 gene.     

Sex‐specific recombination rates and allele frequencies affect the invasion of sexually antagonistic variation on autosomes

The introduction and persistence of novel, sexually antagonistic alleles can depend upon factors that differ between males and females. Understanding the conditions for invasion in a two‐locus model can elucidate these processes. For instance, selection can act differently upon the sexes, or sex linkage can facilitate the invasion of genetic variation with opposing fitness effects between the sexes. Two factors that deserve further attention are recombination rates and allele frequencies – both of which can vary substantially between the sexes. We find that sex‐specific recombination rates in a two‐locus diploid model can affect the invasion outcome of sexually antagonistic alleles and that the sex‐averaged recombination rate is not necessarily sufficient to predict invasion. We confirm that the range of permissible recombination rates is smaller in the sex benefitting from invasion and larger in the sex harmed by invasion. However, within the invasion space, male recombination rate can be greater than, equal to or less than female recombination rate in order for a male‐benefit, female‐detriment allele to invade (and similarly for a female‐benefit, male‐detriment allele). We further show that a novel, sexually antagonistic allele that is also associated with a lowered recombination rate can invade more easily when present in the double heterozygote genotype. Finally, we find that sexual dimorphism in resident allele frequencies can impact the invasion of new sexually antagonistic alleles at a second locus. Our results suggest that accounting for sex‐specific recombination rates and allele frequencies can determine the difference between invasion and non‐invasion of novel, sexually antagonistic alleles in a two‐locus model.     

EVOLUTION OF DOMINANCE UNDER FREQUENCY-DEPENDENT INTRASPECIFIC COMPETITION IN AN ASSORTATIVELY MATING POPULATION

We study the evolution of higher levels of dominance as a response to negative frequency-dependent selection. In contrast to previous studies, we focus on the effect of assortative mating on the evolution of dominance under frequency-dependent intraspecific competition. We analyze a two-locus two-allele model, in which the primary locus has a major effect on a quantitative trait that is under a mixture of frequency-independent stabilizing selection, density-dependent selection, and frequency-dependent selection caused by intraspecific competition for a continuum of resources. The second (modifier) locus determines the degree of dominance at the trait level. Additionally, the population mates assortatively with respect to similarities in the ecological trait. Our analysis shows that the parameter region in which dominance can be established decreases if small levels of assortment are introduced. In addition, the degree of dominance that can be established also decreases. In contrast, if assortment is intermediate, sexual selection for extreme types can be established, which leads to evolution of higher levels of dominance than under random mating. For modifiers with large effects, intermediate levels of assortative mating are most favorable for the evolution of dominance. For large modifiers, the speed of fixation can even be higher for intermediate levels of assortative mating than for random mating.     

Population models of genomic imprinting. I. Differential viability in the sexes and the analogy with genetic dominance

Many single-locus, two-allele selection models of genomic imprinting have been shown to reduce formally to one-locus Mendelian models with a modified parameter for genetic dominance. One exception is the model where selection at the imprinted locus affects the sexes differently. We present two models of maternal inactivation with differential viability in the sexes, one with complete inactivation, and the other with a partial penetrance for inactivation. We show that, provided dominance relations at the imprintable locus are the same in both sexes, a globally stable polymorphism exists for a range of viabilities that is independent of the penetrance of imprinting. The conditions for a polymorphism are the same as in previous models with differential viability in the sexes but without imprinting and in a model of the paternal X-inactivation system in marsupials. The model with incomplete inactivation is used to illustrate the analogy between imprinting and dominance by comparing equilibrium bifurcation plots for fixed values of dominance and penetrance. We also derive a single expression for the dominance parameter that leaves the frequency and stability of equilibria unchanged for all levels of inactivation. Although an imprinting model with sex differences does not formally reduce to a nonimprinting scheme, close theoretical parallels clearly exist.     

Additional evidence for the genomic imprinting model of sex determination in the haplodiploid wasp Nasonia vitripennis: isolation of biparental diploid males after X-ray mutagenesis

The primary sex-determining signal in the haplodiploid wasp Nasonia vitripennis is not known. In haplodiploid reproduction, unfertilized eggs typically develop into uniparental haploid males and fertilized eggs into biparental diploid females. Although this reproductive strategy is common to all Hymenoptera, sex-determination is not strictly specified by the number of genome copies inherited. Furthermore, primary sex-determining signals differ among haplodiploid species. In the honeybee, for example, the primary signal is the genotype at a single, polymorphic locus: diploid animals that are homozygous develop into males while heterozygotes develop into females. Sex determination in Nasonia cannot be explained by this mechanism. Various lines of evidence show that the inheritance of a paternal genome is required for female sexual development and suggest a genomic imprinting mechanism involving an imprinted gene, expressed only from a paternal copy, that triggers female sexual development. In this model, haploid or diploid uniparental embryos develop into males due to a maternal imprint that silences this locus. The genomic imprinting model predicts that a loss-of-function mutation in the paternal copy of the imprinted gene would result in male sexual development in a biparental diploid embryo. In support of this model, we have identified rare biparental diploid males in the F1 progeny of X-ray mutagenized haploid males. Although uniparental diploid male progeny of virgin triploid females have been previously described, this is the first report of biparental diploid males in Nasonia. Our work provides a new, independent line of evidence for the genomic imprinting model of Nasonia sex determination.     

Mutation-Selection Balance at a Modifier-of-Imprinting Locus

We propose a pair of population genetic models for a modifier-of-imprinting locus for which different genotypes imprint different proportions of an imprintable target locus in their gametes. The two models examine the situations in which imprinting is advantageous or disadvantageous, and we discuss three cases for which the modifier is respectively partially dominant, dominant, or recessive. The models predict the stable equilibrium frequencies of the mutant modifier and functionally diploid individuals in a large population in terms of up to four parameters: the mutation rate at the modifier locus, V; the selection coefficient against the disadvantageous phenotype, s; the proportion of unimprinted eggs produced by homozygotes for the mutant modifier, θ, and, in the partially dominant models, the dominance parameter, k. The equilibrium frequency of the mutant phenotypes is shown to be approximately twice that of standard Mendelian models: 2V/s or 4V/s when the modifier is recessive or dominant, respectively. Mathematical equivalences between these and nonimprinting models are noted.     

Differential selection between the sexes and selection for sex

Anisogamy is known to generate an important cost for sexual reproduction (the famous "twofold cost of sex"). However, male-female differences may have other consequences on the evolution of sex, due to the fact that selective pressures may differ among the sexes. On the one hand, intralocus sexual conflict should favor asexual females, which can fix female-beneficial, male-detrimental alleles. On the other hand, it has been suggested repeatedly that sexual selection among males may help to purge the mutation load, providing an advantage to sexual females. However, no analytical model has computed the strength of selection acting on a modifier gene affecting the frequency of sexual reproduction when selection differs between the sexes. In this article, we analyze a two-locus model using two approaches: a quasi-linkage-equilibrium (QLE) analysis and a local stability analysis, whose predictions are verified using a multilocus simulation. We find that costly sex can be maintained when selection is stronger in males than in females, but acts in the same direction in both. Complete asexuality, however, evolves under any other form of selection. Finally, we discuss how experimental measurements of fitness variances and covariances between sexes could be used to determine the overall direction and strength on selection for sex arising from differences in selection between males and females.     

Sex-Specific Aggression and Antipredator Behaviour in Young Brown Trout

Sex differences in adult behaviour are often interpreted as consequences of sexual selection and/or different reproductive roles in males and females. Sex-specific juvenile behaviour, however, has received less attention. Adult brown trout males are more aggressive than females during spawning and juvenile aggression may be genetically correlated with adult aggression in fish. We therefore tested the prediction that immature brown trout males are more aggressive and bolder than immature females. Because previous work has suggested that precocious maturation increases dominance in salmonids, we included precocious males in the study to test the prediction that early sexual maturation increase male aggression and boldness. Aggression and dominance relations were estimated in dyadic contests, whereas boldness was measured as a response to simulated predation risk using a model heron. Independent of maturity state, males initiated more than twice as many agonistic interactions as females in intersexual contests. However, males were not significantly more likely to win these contests than females. The response to a first predator attack did not differ between sex categories, but males reacted less to a second predator attack than females. Sexual maturity did not affect the antipredator response in males. Since there is no evidence from field studies that stream-living immature male and female salmonids differ in growth rate, it appears unlikely that the sex differences demonstrated are behavioural consequences of sex-specific investment in growth. It seems more likely that sex-specific behaviour arises as a correlated response to sexually selected gene actions promoting differential behaviour in adult males and females during reproduction. Alternatively, sex differences may develop gradually during juvenile life, because a gradual developmental program should be less costly than a sudden behavioural change at the onset of sexual maturity.     

Modifiers of mutation rate: a general reduction principle

A deterministic two-locus population genetic model with random mating is studied. The first locus, with two alleles, is subject to mutation and arbitrary viability selection. The second locus, with an arbitrary number of alleles, controls the mutation at the first locus. A class of viability-analogous Hardy-Weinberg equilibria is analyzed in which the selected gene and the modifier locus are in linkage equilibrium. It is shown that at these equilibria a reduction principle for the success of new mutation-modifying alleles is valid. A new allele at the modifier locus succeeds if its marginal average mutation rate is less than the mean mutation rate of the resident modifier allele evaluated at the equilibrium. Internal stability properties of these equilibria are also described.     

Mutation-Selection Balance under Genomic Imprinting at an Autosomal Locus

I model the effect of genomic imprinting on the equilibrium allele frequencies at an autosomal diallelic locus subject to viability selection and mutation. The population size is assumed to be very large; male and female mutation rates may be unequal. Different models examine cases of the inactivation of one gene (with both complete and partial penetrance) and of differential expression of genes according to the parent of origin. In the simplest cases the frequency of the deleterious allele is approximately twice that of a dominant nonimprinting mutant, but considerably less than that of a recessive nonimprinting mutant. Under imprinting, selection and unequal mutation rates interact: other things being equal, male-biased mutation leads to lower mutant frequencies under maternal imprinting and higher frequencies under paternal imprinting. I also model cases where just one allele is imprintable (and the other not). These models allow us to predict the frequency of a failure to imprint in a normally imprinting system, as well as the frequency of imprinting at a standard nonimprinting locus.     

Sex ratio variation in female-biased populations of Notostracans

Females from female-biased populations of the notostracan Triops newberryi produce viable eggs when reared in isolation. Clutches produced under such conditions exhibit sex-ratio polymorphism. One category of females (monogenics) produces only female offspring; the second category (amphigenics) produces clutches with a sex ratio of 3 females: 1 male. The relative proportions of the two categories of females varies significantly between populations and is correlated with population sex ratio. This correlation, and the pattern of offspring distribution in amphigenics, suggests that sex is determined by an autosomal Mendelian gene locus for which the male-determining allele is recessive. Limited pedigree analysis of lineages under selfing support the genetic model.