The effects of linkage and gene flow on local adaptation: a two-locus continent-island model

Population subdivision and migration are generally considered to be important causes of linkage disequilibrium (LD). We explore the combined effects of recombination and gene flow on the amount of LD, the maintenance of polymorphism, and the degree of local adaptation in a subdivided population by analyzing a diploid, deterministic continent–island model with genic selection on two linked loci (i.e., no dominance or epistasis). For this simple model, we characterize explicitly all possible equilibrium configurations. Simple and intuitive approximations for many quantities of interest are obtained in limiting cases, such as weak migration, weak selection, weak or strong recombination. For instance, we derive explicit expressions for the measures and r² (the squared correlation in allelic state) of LD. They depend in qualitatively different ways on the migration rate. Remarkably high values of r² are maintained between weakly linked loci, especially if gene flow is low. We determine how the maximum amount of gene flow that admits preservation of the locally adapted haplotype, hence of polymorphism at both loci, depends on recombination rate and selection coefficients. We also investigate the evolution of differentiation by examining the invasion of beneficial mutants of small effect that are linked to an already present, locally adapted allele. Mutants of much smaller effect can invade successfully than predicted by naive single-locus theory provided they are at least weakly linked. Finally, the influence of linkage on the degree of local adaptation, the migration load, and the effective migration rate at a neutral locus is explored. We discuss possible consequences for the evolution of genetic architecture, in particular, for the emergence of clusters of tightly linked, slightly beneficial mutations and the evolution of recombination and chromosome inversions.     

The Effect of Linkage on Establishment and Survival of Locally Beneficial Mutations

We study invasion and survival of weakly beneficial mutations arising in linkage to an established migration–selection polymorphism. Our focus is on a continent–island model of migration, with selection at two biallelic loci for adaptation to the island environment. Combining branching and diffusion processes, we provide the theoretical basis for understanding the evolution of islands of divergence, the genetic architecture of locally adaptive traits, and the importance of so-called “divergence hitchhiking” relative to other mechanisms, such as “genomic hitchhiking”, chromosomal inversions, or translocations. We derive approximations to the invasion probability and the extinction time of a de novo mutation. Interestingly, the invasion probability is maximized at a nonzero recombination rate if the focal mutation is sufficiently beneficial. If a proportion of migrants carries a beneficial background allele, the mutation is less likely to become established. Linked selection may increase the survival time by several orders of magnitude. By altering the timescale of stochastic loss, it can therefore affect the dynamics at the focal site to an extent that is of evolutionary importance, especially in small populations. We derive an effective migration rate experienced by the weakly beneficial mutation, which accounts for the reduction in gene flow imposed by linked selection. Using the concept of the effective migration rate, we also quantify the long-term effects on neutral variation embedded in a genome with arbitrarily many sites under selection. Patterns of neutral diversity change qualitatively and quantitatively as the position of the neutral locus is moved along the chromosome. This will be useful for population-genomic inference. Our results strengthen the emerging view that physically linked selection is biologically relevant if linkage is tight or if selection at the background locus is strong.     

Evolution of migration under kin selection and local adaptation

We present here a stochastic two-locus, two-habitat model for the evolution of migration with local adaptation and kin selection. One locus determines the migration rate while the other causes local adaptation. We show that the opposing forces of kin competition and local adaptation can lead to the existence of one or two convergence stable migration rates, notably depending on the recombination rate between the two loci. We show that linkage between migration and local adaptation loci has two antagonist effects: when linkage is tight, cost of local adaptation increases, leading to smaller equilibrium migration rates. However, when linkage is tighter, the population structure at the migration locus tends to be very high because of the indirect selection, and thus equilibrium migration rates increases. This result, qualitatively different from results obtained with other models of migration evolution, indicates that ignoring drift or the detail of the genetic architecture may lead to incorrect conclusions.     

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.     

Multilocus selection in subdivided populations II. Maintenance of polymorphism under weak or strong migration

The potential of maintaining multilocus polymorphism by migration-selection balance is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, selection may vary across demes, and loci are multiallelic. It is shown that if migration and recombination are strong relative to selection, then with weak or no epistasis and intermediate dominance at every locus and in every deme, arbitrarily many alleles can be maintained at arbitrarily many loci at a stable equilibrium. If migration is weak relative to selection and recombination, then with weak or no epistasis and intermediate dominance at every locus and in every deme, as many alleles as there are demes can be maintained at arbitrarily many loci at equilibrium. In both cases open sets of such parameter combinations are constructed, thus the results are robust with respect to small, but arbitrary, perturbations in the parameters. For weak migration, the number of demes is, in fact, a generic upper bound to the number of alleles that can be maintained at any locus. Thus, several scenarios are identified under which multilocus polymorphism can be maintained by migration-selection balance when this is impossible in a panmictic population.      

The effect of recurrent mutation on the linkage disequilibrium under a selective sweep

A selective sweep describes the reduction of diversity due to strong positive selection. If the mutation rate to a selectively beneficial allele is sufficiently high, Pennings and Hermisson (Mol Biol Evol 23(5):1076–1084, 2006a) have shown, that it becomes likely, that a selective sweep is caused by several individuals. Such an event is called a soft sweep and the complementary event of a single origin of the beneficial allele, the classical case, a hard sweep. We give analytical expressions for the linkage disequilibrium (LD) between two neutral loci linked to the selected locus, depending on the recurrent mutation to the beneficial allele, measured by D and ${\widehat{\sigma_D^2}}$ , a quantity introduced by Ohta and Kimura (Genetics 63(1):229–238, 1969), and conclude that the LD-pattern of a soft sweep differs substantially from that of a hard sweep due to haplotype structure. The analytical results are compared with simulations.     

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.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

Evidence for adaptation from standing genetic variation on an antimicrobial peptide gene in the mussel Mytilus edulis

Genome scans of population differentiation identify candidate loci for adaptation but provide little information on how selection has influenced the genetic structure of these loci. Following a genome scan, we investigated the nature of the selection responsible for the outlying differentiation observed between populations of the marine mussel Mytilus edulis at a leucine/arginine polymorphism (L31R) in the antimicrobial peptide MGD2. We analysed DNA sequence polymorphisms, allele frequencies and population differentiation of polymorphisms closely linked to L31R, and pairwise and third‐order linkage disequilibria. An outlying level of population differentiation was observed at L31R only, while no departure from panmixia was observed at linked loci surrounding L31R, as in most of the genome. Selection therefore seems to affect L31R directly. Three hypotheses can explain the lack of differentiation in the chromosomal region close to L31R: (i) hitchhiking has occurred but migration and recombination subsequently erased the signal, (ii) selection was weak enough and recombination strong enough to limit the hitchhiking effect to a very small chromosomal region or (iii) selection acted on a pre‐existing polymorphism (i.e. standing variation) at linkage equilibrium with its background. Linkage equilibrium was observed between L31R and linked polymorphisms in every population analysed, as expected under the three hypotheses. However, linkage disequilibrium was observed in some populations between pairs of loci located upstream and downstream to L31R, generating a complex pattern of third‐order linkage disequilibria which is best explained by the hypothesis of selection on a pre‐existing polymorphism. We hypothesise that selection could be either balanced, maintaining alleles at different frequencies depending on the pathogen community encountered locally by mussels, or intermittent, resulting in sporadic fluctuations in allele frequency.     

The effect of positive assortative mating at one locus on a second linked locus

Summary A model for positive assortative mating based on genotype for one locus is employed to investigate the effect of this mating system on the genotypic structure of a second linked locus as well as on the joint genotypic structure of these two loci. It is shown that the second locus does not attain a precise positive assortative mating structure, but yet it shares a property that is characteristic of positive assortative mating, namely an increase in the frequency of homozygotes over that typically found in panmictic structures. Given any arbitrary genotypic structure for the parental population, the resulting offspring generation possesses a structure at the second locus that does not depend on the recombination frequency, while the joint structure of course does. In case assortative mating as well as linkage are not complete, there exists a unique joint equilibrium state for the two loci, which is characterized by complete stochastic independence between the two loci as well as by Hardy-Weinberg proportions at the second locus. For the second locus alone, Hardy-Weinberg equilibrium is realized if and only if gametic linkage equilibrium and an additionally specified condition are realized.     

Isolation and characterisation of microsatellite loci in the wasp Mischocyttarus cassununga

Mischocyttarus cassununga, a primitively eusocial Brazilian wasp, commonly found in urban environments, is an interesting model for studies on the evolution of social behaviour in hymenopteran insects. In this study, we constructed a microsatellite-enriched genomic library and presented primers for 18 microsatellite loci. For the analysis, 20 unrelated females were screened and all loci obtained were polymorphic. PCR amplification revealed from 3 (Mcas5b) to 17 (Mcas53b) alleles per locus ( $ \overline{x} = 8.3 $ ). We detected the levels of observed (H _o) and expected (H _e) heterozygosities ranging from 0.150 to 0.950 and 0.261 to 0.920, respectively. The polymorphic information content ranged from 0.238 to 0.915, averaging 0.680. All loci were in Hardy–Weinberg equilibrium and linkage disequilibrium was not detected after sequential Bonferroni correction (P?>?0.05). These molecular markers will allow further studies on sociogenetic structure, extensive population genetic analysis and diversity of M. cassununga and other Mischocyttarini species.     

Linkage strategies for genetically complex traits. II. The power of affected relative pairs.

The power to detect disease-susceptibility loci through linkage analysis using pairs of affected relatives and affected-unaffected pairs is examined. Allelic identity by descent (ibd) for a completely polymorphic marker for sibling, uncle-nephew, grandparent-grandchild, half-sib, and first-cousin pairs is considered. Affected-unaffected pairs generally represent a poor strategy. For single-locus models, ibd depends on lambda R, the risk ratio for type R relatives compared with population prevalence, and the recombination fraction theta. The ibd for grandparent-grandchild pairs is least affected by recombination, followed by sibs, half-sib, uncle-nephew, and first-cousin pairs. For diseases with large lambda values and for small theta values, distant relatives offer greater power. For larger theta values, grandparent-grandchild pairs are best; for small lambda values, sibs are best. Additive and multiplicative multilocus models are considered. For the multiplicative model, the same formulas as in the single-locus model apply, except that lambda iR (for the ith contributing locus) is substituted for lambda R. For the additive model, the deviation from null expectation for ibd is divided among all contributing loci. Compared with the multiplicative model, for an additive model there is usually greater advantage in distant relationships. Multipoint analysis using linked marker loci for affected relative pairs is described. Simultaneous use of multiple markers diminishes the effect of recombination and allows for localization of the disease-susceptibility locus.     

Occasional recombination of a selfish X‐chromosome may permit its persistence at high frequencies in the wild

The sex‐ratio X‐chromosome (SR) is a selfish chromosome that promotes its own transmission to the next generation by destroying Y‐bearing sperm in the testes of carrier males. In some natural populations of the fly Drosophila neotestacea, up to 30% of the X‐chromosomes are SR chromosomes. To investigate the molecular evolutionary history and consequences of SR, we sequenced SR and standard (ST) males at 11 X‐linked loci that span the ST X‐chromosome and at seven arbitrarily chosen autosomal loci from a sample of D. neotestacea males from throughout the species range. We found that the evolutionary relationship between ST and SR varies among individual markers, but genetic differentiation between SR and ST is chromosome‐wide and likely due to large chromosomal inversions that suppress recombination. However, SR does not consist of a single multilocus haplotype: we find evidence for gene flow between ST and SR at every locus assayed. Furthermore, we do not find long‐distance linkage disequilibrium within SR chromosomes, suggesting that recombination occurs in females homozygous for SR. Finally, polymorphism on SR is reduced compared to that on ST, and loci displaying signatures of selection on ST do not show similar patterns on SR. Thus, even if selection is less effective on SR, our results suggest that gene flow with ST and recombination between SR chromosomes may prevent the accumulation of deleterious mutations and allow its long‐term persistence at relatively high frequencies.     

Quasi equilibrium,variance effective size and fixation index for populations with substructure

In this paper, we develop a method for computing the variance effective size \(N_{eV}\) , the fixation index \(F_{ST}\) and the coefficient of gene differentiation \(G_{ST}\) of a structured population under equilibrium conditions. The subpopulation sizes are constant in time, with migration and reproduction schemes that can be chosen with great flexibility. Our quasi equilibrium approach is conditional on non-fixation of alleles. This is of relevance when migration rates are of a larger order of magnitude than the mutation rates, so that new mutations can be ignored before equilibrium balance between genetic drift and migration is obtained. The vector valued time series of subpopulation allele frequencies is divided into two parts; one corresponding to genetic drift of the whole population and one corresponding to differences in allele frequencies among subpopulations. We give conditions under which the first two moments of the latter, after a simple standardization, are well approximated by quantities that can be explicitly calculated. This enables us to compute approximations of the quasi equilibrium values of \(N_{eV}\) , \(F_{ST}\) and \(G_{ST}\) . Our findings are illustrated for several reproduction and migration scenarios, including the island model, stepping stone models and a model where one subpopulation acts as a demographic reservoir. We also make detailed comparisons with a backward approach based on coalescence probabilities.     

Pleiotropy or linkage? Their relative contributions to the genetic correlation of quantitative traits and detection by multitrait GWA studies

Genetic correlations between traits may cause correlated responses to selection. Previous models described the conditions under which genetic correlations are expected to be maintained. Selection, mutation, and migration are all proposed to affect genetic correlations, regardless of whether the underlying genetic architecture consists of pleiotropic or tightly linked loci affecting the traits. Here, we investigate the conditions under which pleiotropy and linkage have different effects on the genetic correlations between traits by explicitly modeling multiple genetic architectures to look at the effects of selection strength, degree of correlational selection, mutation rate, mutational variance, recombination rate, and migration rate. We show that at mutation-selection(-migration) balance, mutation rates differentially affect the equilibrium levels of genetic correlation when architectures are composed of pairs of physically linked loci compared to architectures of pleiotropic loci. Even when there is perfect linkage (no recombination within pairs of linked loci), a lower genetic correlation is maintained than with pleiotropy, with a lower mutation rate leading to a larger decrease. These results imply that the detection of causal loci in multitrait association studies will be affected by the type of underlying architectures, whereby pleiotropic variants are more likely to be underlying multiple detected associations. We also confirm that tighter linkage between nonpleiotropic causal loci maintains higher genetic correlations at the traits and leads to a greater proportion of false positives in association analyses.     

The Effect of a Selected Locus on Linked Neutral Loci

The effects produced on linked neutral loci as a selected locus evolves towards its equilibrium value are considered. Significant effects on the neutral loci arise if the recombination fraction between the neutral and selected loci is smaller than the order of magnitude of the selective differences at the selected locus. The effect on gene frequencies at the neutral loci, that is, the hitchhiking effect, is determined, as well as the linkage disequilibrium generated by this hitchhiking effect. One of the more important findings is that significant disequilibrium can be generated between two neutral loci by the evolution of a linked selected locus. Consideration is given to the problem of determining how the effect of selection operating in natural populations can be detected, the question of the establishment of inversions in populations, and also to the nonequilibrium properties of populations.     

Central equilibria in multilocus systems. I. Generalized nonepistatic selection regimes

The generalized nonepistatic selection regime encompasses combinations of multiplicative and neutral viability effects distributed across a set of loci. These subsume, in particular, mixtures of the classical modes of multiplicative and additive fitness evaluations for multilocus traits. Exact analytic conditions for existence and stability of a multilocus Hardy-Weinberg (H-W) polymorphic equilibrium configuration are ascertained. It is established that the central H-W polymorphism is stable only if the component loci are "over-dominant" and sufficient recombination is in force. The H-W central equilibrium is never stable for tight linkage whenever some multiplicative selection effects are contributed by at least two of the loci involved. In the case of additive selection expression and individual overdominant loci, the H-W polymorphism is stable independently of the level of recombination. In the context of "natural" recombination schemes, "more recombination" enhances the stability of the H-W polymorphic equilibrium.     

Genome Changes After Gene Duplication: Haploidy vs. Diploidy

Since genome size and the number of duplicate genes observed in genomes increase from haploid to diploid organisms, diploidy might provide more evolutionary probabilities through gene duplication. It is still unclear how diploidy promotes genomic evolution in detail. In this study, we explored the evolution of segmental gene duplication in haploid and diploid populations by analytical and simulation approaches. Results show that (1) under the double null recessive (DNR) selective model, given the same recombination rate, the evolutionary trajectories and consequences are very similar between the same-size gene-pool haploid vs. diploid populations; (2) recombination enlarges the probability of preservation of duplicate genes in either haploid or diploid large populations, and haplo-insufficiency reinforces this effect; and (3) the loss of duplicate genes at the ancestor locus is limited under recombination while under complete linkage the loss of duplicate genes is always random at the ancestor and newly duplicated loci. Therefore, we propose a model to explain the advantage of diploidy: diploidy might facilitate the increase of recombination rate, especially under sexual reproduction; more duplicate genes are preserved under more recombination by originalization (by which duplicate genes are preserved intact at a special quasi-mutation-selection balance under the DNR or haplo-insufficient selective model), so genome sizes and the number of duplicate genes in diploid organisms become larger. Additionally, it is suggested that small genomic rearrangements due to the random loss of duplicate genes might be limited under recombination.USUALLY genome size becomes larger from haploid to diploid organisms (Lynch and Conery 2003), and so does the number of duplicate genes observed in genomes (Zhang 2003). It is extensively hypothesized that diploidy might facilitate the preservation and accumulation of duplicate genes, but it is still unclear how diploidy supports the evolution of duplicate genes in detail. The superiority of diploidy is classically attributed to preventing expression of deleterious mutations (Crow and Kimura 1965), but it is also argued that the sheltering of deleterious mutations cannot adequately explain the advantages of diploidy (Perrot et al. 1991).Recombination is a common phenomenon in all three kingdoms of life, Bacteria, Eukarya, and Archaea. It has been reported that recombination influences the loss of duplicate genes (Zhang and Kishino 2004; Xue et al. 2010). In diploid organisms, if recombination between the ancestor locus and the newly duplicated locus is free, the rate of recombination is maximally 0.5, which is commonly observed especially when the two loci are located on different chromosomes. Although recombination should not be regarded as an exception in haploid organisms (Fraser et al. 2007), recombination events usually occur more frequently in diploid populations than they do in haploid populations. In other words, diploidy might facilitate the occurrence of recombination. The difference of recombination behaviors between haploid and diploid organisms is an obvious and important feature during genomic evolution.In our recent studies of genomic duplication, we proposed a new possible way of preserving and accumulating duplicate genes in genomes—originalization (Xue and Fu 2009a). As is well known, for a locus in an infinite diploid population, the frequencies of wild-type and degenerative alleles will move to an equilibrium under purifying selection and mutation, which is known as the mutation–selection balance. After genomic duplication, under two simple selective models, double null recessive (DNR, under which valid individuals require at least one active wild-type allele on the ancestor and newly duplicated loci) and haplo-insufficient (HI or partial dominant, under which valid individuals require at least two active wild-type alleles on both loci) models, a special equilibrium of allele frequencies at the ancestor and newly duplicated loci will be reached under recombination, in which the frequency of wild-type allele is kept high at both loci. Under the HI selective model this balance becomes so stable and flexible that the fixation of a degenerative allele at one of these two loci (or the balance being broken) becomes very difficult even in a modest population (Xue and Fu 2009a,b). However, if the two loci are tightly linked (recombination rate r = 0), this balance of allele frequencies does not appear. As r increases, the balance becomes more stable and the frequency of the wild-type allele at two loci becomes higher. High frequency of the wild-type allele at both loci means that duplicate genes are preserved intact in genomes, so this phenomenon was named originalization.Although many duplicate genes originated from genomic duplications in some species, such as yeast, maize, and fish (Li et al. 2005), those from segmental duplications are also very popular (Zhang et al. 2000; Leister 2004). In haploid populations, most duplication events are small segmental duplications. Therefore, to understand genomic evolution comprehensively, it is necessary to explore the evolution of segmental genomic duplication.Lynch et al. (2001) and Tanaka et al. (2009) have studied the evolution of segmental gene duplication in diploid populations theoretically. However, in this study, we further compared the evolution of segmental gene duplication in haploid vs. diploid populations by numerical and simulation approaches under the DNR and HI selective models. We observed that haploid and diploid populations with the same-size gene pool are very similar under the DNR model and the same recombination rate. Recombination enlarges the probability of preservation of duplicate genes in either haploid or diploid populations via originalization, and haplo-insufficiency reinforces this effect. The loss of duplicate genes at the ancestor locus might be limited under recombination, while under complete linkage, the loss of duplicate genes is random at the ancestor and newly duplicated loci. According to these results, we propose a model with which to explain the revolutionary genomic transition from haploidy to diploidy.     

The role of recombination and selection in the modifier theory of sex-ratio distortion

The equilibrium configurations for a two-locus multialle model of sex-linked meiotic drive are studied with regard to the recombination fraction:limit cycles can occur in the case of small recombination while stable equilibrium points associated with linkage equilibrium can exist for an intermediate range of recombination values depending on the equilibrium sex ratio, linkage disequilibrium at nearby equilibrium points taking turn with loser linkage. The evolutionary dynamics in two-locus sex-ratio distortion systems is enlightened: while equilibria with a sex ratio closer to 1/2 are more likely to be stable with respect to perturbations on the frequencies of sex-ratio distorters that are represented at equilibrium, such equilibria are also more vulnerable to the invasion of mutant distorters when there is some degree of linkage with the sex-determining locus. For X-linked multimodifier systems of sex-ratio distortion, differential fertilities and viabilities are incorporated and a maximum principle is suggested.