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.     

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.     

Selection in Complex Genetic Systems I. the Symmetric Equilibria of the Three-Locus Symmetric Viability Model

The symmetric equilibria of the three-locus symmetric viability model are determined and their stability analyzed. For tight linkage there may be four stable equilibria, each characterized by having one pair of complementary chromosomes in high frequencies, with all others low. For looser linkage the only stable symmetric equilibrium is that with complete linkage equilibrium. For intermediate recombination values both types of equilibria may be stable. A new class of equilibria with all pairwise linkage disequilibria zero, but with third order linkage disequilibrium, has been discovered. It may be stable for tight linkage.     

Meiotic drive influences the outcome of sexually antagonistic selection at a linked locus

Most meiotic drivers, such as the t‐haplotype in Mus and the segregation distorter (SD) in Drosophila, act in a sex‐specific manner, gaining a transmission advantage through one sex although suffering only the fitness costs associated with the driver in the other. Their inheritance is thus more likely through one of the two sexes, a property they share with sexually antagonistic alleles. Previous theory has shown that pairs of linked loci segregating for sexually antagonistic alleles are more likely to remain polymorphic and that linkage disequilibrium accrues between them. I probe this similarity between drive and sexual antagonism and examine the evolution of chromosomes experiencing these selection pressures simultaneously. Reminiscent of previous theory, I find that: the opportunity for polymorphism increases for a sexually antagonistic locus that is physically linked to a driving locus; the opportunity for polymorphism at a driving locus also increases when linked to a sexually antagonistic locus; and stable linkage disequilibrium accompanies any polymorphic equilibrium. Additionally, I find that drive at a linked locus favours the fixation of sexually antagonistic alleles that benefit the sex in which drive occurs. Further, I show that under certain conditions reduced recombination between these two loci is selectively favoured. These theoretical results provide clear, testable predictions about the nature of sexually antagonistic variation on driving chromosomes and have implications for the evolution of genomic architecture.     

Selection in complex genetic systems IV. Multiple alleles and interactions between two loci

Summary A symmetric viability model for two loci with two alleles at one locus and m alleles at the other is suggested and analyzed. The analysis of the equilibria is complete if the two loci are absolutely linked, while if recombination is allowed the analysis is incomplete. The dynamics of the mode! resemble those of the two locus two allele model, namely that for loose linkage there will be no correlation between the loci and for tight linkage there may be strong correlation. The major caveats to this are: 1. The equilibria stable for tight linkage may belong to an array of different structures dependent on the selection and the number of alleles. 2. If both loci are overdominant in viability, the stable equilibria always contain all alleles segregating in the population; otherwise, the stable equilibria may only be two locus two allele high complementarity equilibria for tight linkage. 3. For intermediate linkage values and special selection values the boundary two locus two allele high complementarity equilibria may be stable simultaneously with the totally polymorphic central point at which there is no association between the loci.Dedicated to the memory of Ove Frydenberg.Research supported in part by a grant from the Danish Natural Science Research Council, a grant from National Science Foundation, U.S.A., and by USPHS grant NIH 10452-09-11.     

Population genetics of modifiers of meiotic drive. I. The solution of a special case and some general implications

This is a study of the formal population genetics of a two locus model where the alleles at one locus are subject to meiotic drive and zygotic selection and the only effect of the other locus is the modification of drive intensity. A complete analytic solution is obtained for a biologically reasonable special case. It is then argued, partially with the aid of computer analysis, that with moderate relaxation of assumptions of the special case, the conclusions derived from that case still hold. These conclusions are that if there is linkage a stable two locus polymorphism can result. There is permanent linkage disequilibrium with the loosing allele at the drive locus in coupling with the suppressor allele at the modifier locus, and the driven allele coupled with the modifier allele which enhances drive. It is suggested that this result explains how the SD system in Drosophila maintains its integrity in natural populations.     

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.     

Modifier theory of meiotic drive: Is Mendelian segregation stable?

The evolutionary fate of rare modifiers, based on the modifier theory of meiotic drive, is studied in this paper. It is shown that a polymorphism based on Mendelian segregation is never stable for any recombination frequencies between 0 and $\text{1}\text{2}$, and that, for tight linkage between the main locus and the modifier locus, the modifier locus tends towards heterozygosity.     

Dynamics of Genetic Variability in Two-Locus Models of Stabilizing Selection

We study a two locus model, with additive contributions to the phenotype, to explore the dynamics of different phenotypic characteristics under stabilizing selection and recombination. We demonstrate that the interaction of selection and recombination results in constraints on the mode of phenotypic evolution. Let V(g) be the genic variance of the trait and C(L) be the contribution of linkage disequilibrium to the genotypic variance. We demonstrate that, independent of the initial conditions, the dynamics of the system on the plane (V(g), C(L)) are typically characterized by a quick approach to a straight line with slow evolution along this line afterward. We analyze how the mode and the rate of phenotypic evolution depend on the strength of selection relative to recombination, on the form of fitness function, and the difference in allelic effect. We argue that if selection is not extremely weak relative to recombination, linkage disequilibrium generated by stabilizing selection influences the dynamics significantly. We demonstrate that under these conditions, which are plausible in nature and certainly the case in artificial stabilizing selection experiments, the model can have a polymorphic equilibrium with positive linkage disequilibrium that is stable simultaneously with monomorphic equilibria.     

The two-locus model of Gaussian stabilizing selection

We study the equilibrium structure of a well-known two-locus model in which two diallelic loci contribute additively to a quantitative trait that is under Gaussian stabilizing selection. The population is assumed to be infinitely large, randomly mating, and having discrete generations. The two loci may have arbitrary effects on the trait, the strength of selection and the recombination rate may also be arbitrary. We find that 16 different equilibrium patterns exist, having up to 11 equilibria; up to seven interior equilibria may coexist, and up to four interior equilibria, three in negative and one in positive linkage disequilibrium, may be simultaneously stable. Also, two monomorphic and two fully polymorphic equilibria may be simultaneously stable. Therefore, the result of evolution may be highly sensitive to perturbations in the initial conditions or in the underlying genetic parameters. For the special case of equal effects, global stability results are proved. In the general case, we rely in part on numerical computations. The results are compared with previous analyses of the special case of extremely strong selection, of an approximate model that assumes linkage equilibrium, and of the much simpler quadratic optimum model.     

Natural Selection and Y-Linked Polymorphism

Several population genetic models allowing natural selection to act on Y-linked polymorphism are examined. The first incorporates pleiotropic effects of a Y-linked locus, such that viability, segregation distortion, fecundity and sexual selection can all be determined by one locus. It is shown that no set of selection parameters can maintain a stable Y-linked polymorphism. Interaction with the X chromosome is allowed in the next model, with viabilities determined by both X- and Y-linked factors. This model allows four fixation equilibria, two equilibria with X polymorphism and a unique point with both X- and Y-linked polymorphism. Stability analysis shows that the complete polymorphism is never stable. When X- and Y-linked loci influence meiotic drive, it is possible to have all fixation equilibria simultaneously unstable, and yet there is no stable interior equilibrium. Only when viability and meiotic drive are jointly affected by both X- and Y-linked genes can a Y-linked polymorphism be maintained. Unusual dynamics, including stable limit cycles, are generated by this model. Numerical simulations show that only a very small portion of the parameter space admits Y polymorphism, a result that is relevant to the interpretation of levels of Y-DNA sequence variation in natural populations.     

The role of recombination in the modifier theory of autosomal segregation distortion

The effects of recombination on the equilibrium structures of two-locus systems of autosomal segregation distortion are studied. Exact conditions pertaining to the stability of polymorphic equilibria maintaining multiple distorters at the segregation-determination locus as well as their resistance to the invasion of mutant distorters are given. Evolutionary patterns of autosomal meiotic drive and the status of Mendelian segregation are reexamined.     

Evolution and intraspecific exploitative competition I. One-locus theory for small additive gene effects

Using the model of exploitative competition of R. H. MacArthur and R. Levins (1967, Amer. Natur. 101, 377–385), evolution at a gene locus which influences the niche position is considered. The locus has multiple alleles, and the contributions of the alleles to the genotypic value are additive. The resource spectrum and the utilization functions of the genotypes are assumed to be Gaussian. Evolution will make the mode of the niche converge to the resource optimum, as long as the allele contributions are small compared to the distance between the mode of the niche and the resource optimum. When this distance is of the same order of magnitude as the allele contributions, then the globally stable equilibrium will maintain at most two alleles in the population, unless the allele contributions are large. Classical overdominance is not needed to maintain polymorphism. This result predicts high linkage disequilibrium in similar multilocus models. It is concluded that intraspecific competition can be a powerful force in maintaining two-allele polymorphisms, and that it can maintain high linkage disequilibrium among closely linked loci.     

Selective sweeps in a 2-locus model for sex-ratio meiotic drive in Drosophila simulans

A way to identify loci subject to positive selection is to detect the signature of selective sweeps in given chromosomal regions. It is revealed by the departure of DNA polymorphism patterns from the neutral equilibrium predicted by coalescent theory. We surveyed DNA sequence variation in a region formerly identified as causing "sex-ratio" meiotic drive in Drosophila simulans. We found evidence that this system evolved by positive selection at 2 neighboring loci, which thus appear to be required simultaneously for meiotic drive to occur. The 2 regions are approximately 150-kb distant, corresponding to a genetic distance of 0.1 cM. The presumably large transmission advantage of chromosomes carrying meiotic drive alleles at both loci has not erased the individual signature of selection at each locus. This chromosome fragment combines a high level of linkage disequilibrium between the 2 critical regions with a high recombination rate. As a result, 2 characteristic traits of selective sweeps--the reduction of variation and the departure from selective neutrality in haplotype tests--show a bimodal pattern. Linkage disequilibrium level indicates that, in the natural population from Madagascar used in this study, the selective sweep may be as recent as 100 years.     

Pleiotropy and Multilocus Polymorphisms

It is demonstrated that systems of two pleiotropically related characters controlled by additive diallelic loci can maintain under Gaussian stabilizing selection a stable polymorphism in more than two loci. It is also shown that such systems may have multiple stable polymorphic equilibria. Stabilizing selection generates negative linkage disequilibrium, as a result of which the equilibrium phenotypic variances are quite low, even though the level of allelic polymorphisms can be very high. Consequently, large amounts of additive genetic variation can be hidden in populations at equilibrium under stabilizing selection on pleiotropically related characters.     

Linkage disequilibrium detected between dystrophia myotonica and APOC2 locus in the Finnish population

Summary Three polymorphic loci APOC2, CKMM and p134C were used to haplotype 15 Finnish dystrophia myotonica (DM) families representing about one third of all DM patients in this isolated population. Compound APOC2 and CKMM haplotypes reveal linkage disequilibrium: 90% of DM chromosomes co-occur with the haplotypes that occur in 31% of normal chromosomes only. The same disequilibrium is present when only polymorphisms occurring at the APOC2 locus are used. Surprisingly, no statistically significant linkage disequilibrium was discovered at the CKMM locus alone. Of the meiotic events, 84% were informative when both APO2 and CKMM loci were used. When studied selectively, 60% of meiotic events were informative at the APOC2 locus, whereas CKMM alone resulted in 65% meiotic informativeness. The distal marker p134C was found to have an unfortunately low information content in our population.     

More on recombination and selection in the modifier theory of sex-ratio distortion

G. Maffi and S.D. Jayakar suggested a model for the two-locus control of sex determination in the mosquito Aedes aegypti (1981, Theor. Pop. Biol. 19, 19-36). This model was extended to multiple alleles and analyzed in mathematical detail by S. Lessard (1987, Theor. Pop. Biol. 31, 339-358). The model supposes that males are "Mm" and females "mm" but the transmission from males is controlled by a second gene with alleles Ai. We show that in addition to the equilibrium in which mAi in females, MAi from males and mAi from males all have the same frequencies, a second class of polymorphic equilibria exists and can be stable. The former class was shown by Lessard to be stable for intermediate and/or loose linkage. The new class of equilibria may be stable for tight linkage under the conditions that preclude stability of the former. We also develop the theory of linkage modification from the neighborhood of the new equilibrium. Successful modifiers of recombination may either reduce or increase the recombination fraction with the outcome depending on the linkage of the modifier to the major genes.     

Some additional results about polymorphisms in models of an additive quantitative trait under stabilizing selection

 The existence of two stable, symmetric (allelic frequency 0.5 in each locus) polymorphic states is demonstrated for a two-locus model of an additive quantitative trait under strong Gaussian selection. Linkage disequilibrium at one of the states is negative whereas it is positive at the other state. For a three-locus model, it is shown that in order to maintain a stable polymorphism in all three loci, selection must be sufficiently but not exces- sively strong relative to recombination. Also, positive linkage disequilibrium can be maintained in a three-locus model under stabilizing selection that is not very strong. Received 15 July 1995     

Further data supporting linkage between cystic fibrosis and the met oncogene and haplotype analysis with met and pJ3.11

The linkage of cystic fibrosis (CF) and the polymorphic DNA markers pJ3.11, met, 7C22, DOCR1-917, COL1A2, and TCRB have jointly localized the mutation causing CF to chromosome 7q2.1-3.1. We report further linkage data with two polymorphic markers at the met oncogene locus, pmetH and pmetD, which supports the tight linkage found by White et al. between CF and met. One family shows evidence for meiotic recombination between CF and met. Analysis of haplotypes in CF pedigrees collected for linkage studies combined with data from single affected families requesting prenatal diagnosis (Farrall et al., Lancet i:1402-1404, 1986) shows CF and met to be in linkage equilibrium in our population while pJ3.11-CF haplotypes show a deviation from the equilibrium frequencies.     

Sexual antagonism and meiotic drive cause stable linkage disequilibrium and favour reduced recombination on the X chromosome

Sexual antagonism and meiotic drive are sex‐specific evolutionary forces with the potential to shape genomic architecture. Previous theory has found that pairing two sexually antagonistic loci or combining sexual antagonism with meiotic drive at linked autosomal loci augments genetic variation, produces stable linkage disequilibrium (LD) and favours reduced recombination. However, the influence of these two forces has not been examined on the X chromosome, which is thought to be enriched for sexual antagonism and meiotic drive. We investigate the evolution of the X chromosome under both sexual antagonism and meiotic drive with two models: in one, both loci experience sexual antagonism; in the other, we pair a meiotic drive locus with a sexually antagonistic locus. We find that LD arises between the two loci in both models, even when the two loci freely recombine in females and that driving haplotypes will be enriched for male‐beneficial alleles, further skewing sex ratios in these populations. We introduce a new measure of LD, , which accounts for population allele frequencies and is appropriate for instances where these are sex specific. Both models demonstrate that natural selection favours modifiers that reduce the recombination rate. These results inform observed patterns of congealment found on driving X chromosomes and have implications for patterns of natural variation and the evolution of recombination rates on the X chromosome.