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.     

The effect of variable environments on polymorphism at loci with several alleles I. A symmetric model

Much of the extant polymorphism has been attributed to spatial and temporal variation among selection regimes. Analysis of models entailing two alleles at a single locus has demonstrated that polymorphism may result from variation among selection regimes which prescribe monomorphism if constant. This relationship is studied in the context of several alleles at a locus.One result which is not valid with only two alleles is that variation among selection regimes which specify polymorphic equilibria may lead to a stable monomorphic equilibrium. The analyses of temporal variation and total panmixia spatial variation among environments show that temporal variation allows the simultaneous stability of equilibrium configurations which cannot be simultaneously stable under total panmixia spatial variation (hard or soft selection). The principle that polymorphism is more readily maintained with spatial than temporal variation is invalidated.Supported in part by Purdue Research Foundation and National Science Foundation (USA) grant MCS-8002227     

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.     

The effect of variable environments on polymorphism at loci with several alleles II. Submultiplicative viabilities

Circumstances assuring a unique stable equilibrium are investigated for a subdivided population with several alleles segregating at a single locus. For a broad class of selection regimes entailing heterozygote viabilities greater than the geometric mean of the corresponding homozygote viabilities, a stable fixation state precludes any other stable equilibria if either total-panmixia or temporal variation is operating. This extends known results for two alleles at a single locus and partially delimits when some of the bizarre behaviour engendered by multiple alleles may occur.Supported in part by National Science Foundation (USA) grant MCS-8002227     

A symmetric two locus model with viability and fertility selection

A two locus deterministic population genetic model is analysed. One locus is under viability selection, the other under fertility selection with both forms of selection completely symmetric. It is shown that linkage equilibrium may occur at two different equilibrium points. For a two-locus polymorphism to be stable, it is necessary that the viability locus be overdominant but not necessary that the fertility locus, considered separately, be able to support a stable polymorphism. The overlaps in stability are not as complex as under two locus symmetric fertilities, but considerably more complex than with symmetric viabilities. Extensions of the analysis for the central linkage equilibrium point with multiple viability and fertility loci are indicated.Research supported in part by NIH grants GM 28106 and GM 10452     

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.     

The generalized multiplicative model for viability selection at multiple loci

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

A Symmetric Two-Locus Fertility Model

A model in which selection is mediated by differential fertilities among the genotypes at two diallelic loci is proposed. Fertility depends only on the number of heterozygous loci participating in the mating. Classes analogous to symmetric equilibria in symmetric viability models are determined explicitly and shown to exhibit stability behavior very different from the viability results. Linkage equilibrium is shown to occur in a relatively asymmetric fashion and to overlap in stability with linkage disequilibrium. In many cases single-locus or two-locus polymorphism is shown to be stable simultaneously with chromosome fixation even under very tight linkage. It is suggested that historical effects may be of great significance in the evolution of systems in which fertility is the primary agent of natural selection.     

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.     

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.     

Linkage analysis of "necessary" disease loci versus "susceptibility" loci.

The association of some diseases with specific alleles of certain genetic markers has been difficult to explain. Several explanations have been proposed for the phenomenon of association, e.g. the existence of multiple, interacting genes (epistasis) or a disease locus in linkage disequilibrium with the marker locus. One might suppose that when marker data from families with associated diseases are analyzed for linkage, the existence of the association would assure that linkage will be found, and found at a tight recombination fraction. In fact, however, linkage analyses of some diseases associated with HLA, as well as diseases associated with alleles at other loci located throughout the genome, show significant evidence against linkage, and others show loose linkage, to the puzzlement of many researchers. In part, the puzzlement arises because linkage analysis is ideal for looking for loci that are necessary, even if not sufficient, for disease expression but may be much less useful for finding loci that are neither necessary nor sufficient for disease expression (so-called susceptibility loci). This work explores what happens when one looks for linkage to susceptibility loci. A susceptibility locus in this case means that the allele increases risk but is neither necessary nor sufficient for disease expression. It might be either an allele at the marker locus itself that is increasing susceptibility or an allele at a locus in linkage disequilibrium with the marker. This work uses computer simulation to examine how linkage analyses behave when confronted with data from such a model.(ABSTRACT TRUNCATED AT 250 WORDS)     

Population genetics of modifiers of meiotic drive III. Equilibrium analysis of a general model for the genetic control of segregation distortion

Prout, Bungaard and Bryant (1973, Theor. Popul. Biol. 4, 446–465) presented the first formal treatment of a model of meiotic drive involving a modifier locus which controls the intensity of drive. They studied the equilibrium behavior in the simplest model where it is assumed that drive is maximal when not suppressed. In that case there is one polymorphic equilibrium at which there is linkage disequilibrium. The equilibrium solutions in the general model of meiotic drive proposed by Prout, et al. are given in this paper together with a stability analysis. It is shown that up to three polymorphic equilibria may exist, two of which are in linkage disequilibrium and one in linkage equilibrium. These equilibria exhibit behavior qualitatively opposite to what is widely accepted as the usual for two locus systems and which is not seem in the simple case originally treated. The polymorphic equilibria with linkage disequilibrium may be stable for loose linkage and not for tight while that with linkage equilibrium is stable in an interval of relatively tight linkage values.     

THE EVOLUTION OF STRONG REPRODUCTIVE ISOLATION

Felsenstein distinguished two ways by which selection can directly strengthen isolation. First, a modifier that strengthens prezygotic isolation can be favored everywhere. This fits with the traditional view of reinforcement as an adaptation to reduce deleterious hybridization by strengthening assortative mating. Second, selection can favor association between different incompatibilities, despite recombination. We generalize this "two allele" model to follow associations among any number of incompatibilities, which may include both assortment and hybrid inviability. Our key argument is that this process, of coupling between incompatibilities, may be quite different from the usual view of reinforcement: strong isolation can evolve through the coupling of any kind of incompatibility, whether prezygotic or postzygotic. Single locus incompatibilities become coupled because associations between them increase the variance in compatibility, which in turn increases mean fitness if there is positive epistasis. Multiple incompatibilities, each maintained by epistasis, can become coupled in the same way. In contrast, a single-locus incompatibility can become coupled with loci that reduce the viability of haploid hybrids because this reduces harmful recombination. We obtain simple approximations for the limits of tight linkage, and strong assortment, and show how assortment alleles can invade through associations with other components of reproductive isolation.     

Epistasis in the multiple locus symmetric viability model

The n-locus two-allele symmetric viability model is considered in terms of the parameters measuring the additive epistasis in fitness. The dynamics is analysed using a simple linear transformation of the gametic frequencies, and then the recurrence equations depend on the epistatic parameters and Geiringer's recombination distribution only. The model exhibits an equilibrium, the central equilibrium, where the 2 ⁿ gametes are equally frequent. The transformation simplifies the stability analysis of the central point, and provides the stability conditions in terms of the existence conditions of other equilibria. For total negative epistasis (all epistatic parameters are negative) the central point is stable for all recombination distributions. For free recombination either a central point (segregating one, two, ... or n loci) or the n-locus fixation states are stable. For no recombination and some epistatic parameters positive the central point is unstable and several boundary equilibria may be locally stable. The sign structure of the additive epistasis is therefore an important determinant of the dynamics of the n-locus symmetric viability model. The non-symmetric multiple locus models previously analysed are dynamically related, and they all have an epistatic sign structure that resembles that of the multiplicative viability model. A non-symmetric model with total negative epistasis which share dynamical properties with the similar symmetric model is suggested.Supported in part by NIH grant GM 28016, and by grant 81-5458 from the Danish Natural Science Research Council     

A two-locus mutation—Selection model and some of its evolutionary implications

The deterministic properties of a two-locus model with mutation and selection have been investigated. The mutation process is unidirectional, and the model is so constructed that the genetic variation at one locus is selectively neutral in the absence of a mutant allele at the other locus. All genotypes with three or four mutant alleles are deleterious, while the double heterozygotes may have the same fitness as the standard genotype. If one of the mutant alleles becomes fixed in the population, then the other locus will show a regular one-locus mutation-selection balance. Such a boundary equilibrium may be unstable or stable in the full two-locus setting. In the symmetric case, which is analyzed in details, the population will either go to one of the two boundary equilibria, or to a fully polymorphic equilibrium at which both the mutant alleles are rare. The origin of reproductive separation between two populations via the fixation of complementary deleterious mutants at different loci, and the fixation of nonfunctional alleles at duplicated loci, are two biological processes which both can be studied with the present model. In the last part of the paper we show how the results from the deterministic analysis can be used to predict how different factors will influence the rates of evolution in these systems.     

A semi-symmetric two-locus model

The two-locus symmetric viability model characterized by its invariance with respect to the exchange of alleles at each locus, is a well-studied model of classical two-locus theory. The symmetric model introduced by Lewontin and Kojima is among the few multi-locus models with epistatic interactions between loci for which a polymorphism with linkage equilibrium can be stable and this happens when recombination is sufficiently large. We show that an analogous property holds true for a different model, in which symmetry need exist at only one locus. The properties of this new semi-symmetric model are compared with those of the classical symmetric model. For tight linkage, two classes of polymorphisms are possible, depending on the magnitude of additive epistasis. The recombination rate above which linkage equilibrium becomes stable is derived analytically. As in the symmetric model, intervals of recombination in which no polymorphism is stable are possible, and stable polymorphisms can coexist with stable fixations.     

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.     

Selection at the Esterase-2 Locus of Drosophila buzzatii? Perturbation-Reperturbation Experiments

Apparent selection affecting starch gel electrophoretic alleles at the Esterase-2 locus of Drosophila buzzatii has been detected in laboratory and natural populations. Perturbation-reperturbation of allele frequencies in replicated laboratory populations attempts to test direct selective effects at the locus versus effects of linked loci. Sequential gel electrophoresis has identified more alleles within starch classes, and three of these alleles (within the a, b and c starch alleles) were used in cage population experiments. Allele a/1.00/1.00/1.00 was set up in 10 replicate populations with allele c/1.00/1.00/1.00, and in an independent 10 replicate populations with allele b/0.99/1.01/1.00. For each set, three reperturbations were done. Replicate populations generally showed similar patterns of allele frequency change and clear directionality: effects of selection, not drift. However, four populations deviated from their replicates, indicating dissipation of linkage disequilibrium. Estimates of pre-adult viability in the F₂ of pair-wise crosses among 12 sequential gel electrophoretic alleles showed very variable modes of inheritance and relative viability fitnesses. Together with the diversity of patterns of allele frequency change in the cage populations, these results suggest a gene complex, with selection acting on an interacting set of loci which may include Esterase-2.     

Molecular evolution of two linked genes, Est-6 and Sod, in Drosophila melanogaster.

We have obtained 15 sequences of Est-6 from a natural population of Drosophila melanogaster to test whether linkage disequilibrium exists between Est-6 and the closely linked Sod, and whether natural selection may be involved. An early experiment with allozymes had shown linkage disequilibrium between these two loci, while none was detected between other gene pairs. The Sod sequences for the same 15 haplotypes were obtained previously. The two genes exhibit similar levels of nucleotide polymorphism, but the patterns are different. In Est-6, there are nine amino acid replacement polymorphisms, one of which accounts for the S-F allozyme polymorphism. In Sod, there is only one replacement polymorphism, which corresponds to the S-F allozyme polymorphism. The transversion/transition ratio is more than five times larger in Sod than in Est-6. At the nucleotide level, the S and F alleles of Est-6 make up two allele families that are quite different from each other, while there is relatively little variation within each of them. There are also two families of alleles in Sod, one consisting of a subset of F alleles, and the other consisting of another subset of F alleles, designed F(A), plus all the S alleles. The Sod F(A) and S alleles are completely or nearly identical in nucleotide sequence, except for the replacement mutation that accounts for the allozyme difference. The two allele families have independent evolutionary histories in the two genes. There are traces of statistically significant linkage disequilibrium between the two genes that, we suggest, may have arisen as a consequence of selection favoring one particular sequence at each locus.     

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.