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

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

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.     

Effect of strong directional selection on weakly selected mutations at linked sites: implication for synonymous codon usage

The fixation of weakly selected mutations can be greatly influenced by strong directional selection at linked loci. Here, I investigate a two-locus model in which weakly selected, reversible mutations occur at one locus and recurrent strong directional selection occurs at the other locus. This model is analogous to selection on codon usage at synonymous sites linked to nonsynonymous sites under strong directional selection. Two approximations obtained here describe the expected frequency of the weakly selected preferred alleles at equilibrium. These approximations, as well as simulation results, show that the level of codon bias declines with an increasing rate of substitution at the strongly selected locus, as expected from the well-understood theory that selection at one locus reduces the efficacy of selection at linked loci. These solutions are used to examine whether the negative correlation between codon bias and nonsynonymous substitution rates recently observed in Drosophila can be explained by this hitchhiking effect. It is shown that this observation can be reasonably well accounted for if a large fraction of the nonsynonymous substitutions on genes in the data set are driven by strong directional selection.     

DYNAMICS OF SEXUAL SELECTION IN DIPLOID POPULATIONS

We describe results for a diploid, two-locus model for the evolution of a female mating preference directed at an attractive male trait that is subject to viability and/or fertility selection. Using computer simulation, we studied a large, random sample of parameter values, assuming additivity of alleles at the preference locus and partial dominance at the trait locus. Simulation results were classifiable into nine types of parameter sets, each differing in equilibria, evolutionary trajectories, and rates of evolution. For many parameters, evolutionary trajectories converged on curves within the allelic frequency plane and subsequently evolved along the curves toward fixation. Neutrally stable curves of equilibria did not occur in Fisherian models that assume only viability and sexual selection unless there is complete dominance at the trait locus. The Fisherian models also exhibited oscillation of allelic frequencies and unique polymorphic equilibria. “Sexy son” models in which attractive males had reduced fertility were much less likely to lead to increase in traits and preferences than were the Fisherian models. However, if less fertile males had increased viability, trait polymorphisms and fixation of rare “sexy” alleles occurred. In general, the behavior of the diploid model was much more complex than that of analogous haploid or polygenic models.     

Genetic variation maintained in multilocus models of additive quantitative traits under stabilizing selection.

Stabilizing selection for an intermediate optimum is generally considered to deplete genetic variation in quantitative traits. However, conflicting results from various types of models have been obtained. While classical analyses assuming a large number of independent additive loci with individually small effects indicated that no genetic variation is preserved under stabilizing selection, several analyses of two-locus models showed the contrary. We perform a complete analysis of a generalization of Wright's two-locus quadratic-optimum model and investigate numerically the ability of quadratic stabilizing selection to maintain genetic variation in additive quantitative traits controlled by up to five loci. A statistical approach is employed by choosing randomly 4000 parameter sets (allelic effects, recombination rates, and strength of selection) for a given number of loci. For each parameter set we iterate the recursion equations that describe the dynamics of gamete frequencies starting from 20 randomly chosen initial conditions until an equilibrium is reached, record the quantities of interest, and calculate their corresponding mean values. As the number of loci increases from two to five, the fraction of the genome expected to be polymorphic declines surprisingly rapidly, and the loci that are polymorphic increasingly are those with small effects on the trait. As a result, the genetic variance expected to be maintained under stabilizing selection decreases very rapidly with increased number of loci. The equilibrium structure expected under stabilizing selection on an additive trait differs markedly from that expected under selection with no constraints on genotypic fitness values. The expected genetic variance, the expected polymorphic fraction of the genome, as well as other quantities of interest, are only weakly dependent on the selection intensity and the level of recombination.     

Polymorphism in the two-locus Levene model with nonepistatic directional selection

For the Levene model with soft selection in two demes, the maintenance of polymorphism at two diallelic loci is studied. Selection is nonepistatic and dominance is intermediate. Thus, there is directional selection in every deme and at every locus. We assume that selection is in opposite directions in the two demes because otherwise no polymorphism is possible. If at one locus there is no dominance, then a complete analysis of the dynamical and equilibrium properties is performed. In particular, a simple necessary and sufficient condition for the existence of an internal equilibrium and sufficient conditions for global asymptotic stability are obtained. These results are extended to deme-independent degree of dominance at one locus. A perturbation analysis establishes structural stability within the full parameter space. In the absence of genotype-environment interaction, which requires deme-independent dominance at both loci, nongeneric equilibrium behavior occurs, and the introduction of arbitrarily small genotype-environment interaction changes the equilibrium structure and may destroy stable polymorphism. The volume of the parameter space for which a (stable) two-locus polymorphism is maintained is computed numerically. It is investigated how this volume depends on the strength of selection and on the dominance relations. If the favorable allele is (partially) dominant in its deme, more than 20% of all parameter combinations lead to a globally asymptotically stable, fully polymorphic equilibrium.     

CYCLICAL ENVIRONMENTAL CHANGES AS A FACTOR MAINTAINING GENETIC POLYMORPHISM. 2. DIPLOID SELECTION FOR AN ADDITIVE TRAIT

The subject of this paper is polymorphism maintenance due to stabilizing selection with a moving optimum. It was shown that in case of two-locus additive control of the selected trait, global polymorphism is possible only when the geometric mean fitnesses of double homozygotes averaged over the period are lower than that of the single heterozygotes and of the double heterozygote (with a multiplier [1 – r]^p, which depends on recombination rate r and period length p). But local stability of polymorphism cannot be excluded even if geometric mean fitnesses of all double homozygotes are higher than that of all heterozygotes. We proved, that for logarithmically convex fitness functions, cyclical changes of the optimum cannot help in polymorphism maintenance in case of additive control of the selected trait by two equal loci. However, within the same class of fitness functions, nonequal gene action and/or dominance effect for one or both loci may lead to local polymorphism stability with large enough polymorphism attracting domain. The higher the intensity of selection and closer the linkage between selected loci the larger is this domain. Note that even simple cyclical selection could result in two forms of polymorphic limiting behavior: (a) usually expected forced cycle with a period equal to that of environmental changes; and (b) “supercycles,” nondumping auto-oscillations with a period comprising of hundreds of forced oscillation periods.     

The evolution of recombination in permanent translocation heterozygotes

The evolution of a selectively neutral locus that controls the degree to which alleles at a single selected locus are linked with a particular set of chromosomes in a permanent translocation heterozygote is studied. With complete selfing and fitness overdominance a new allele at the modifying locus will increase in frequency if it increases the linkage of all alleles at the selected locus to a particular set of chromosomes. With random mating a new allele at the modifying locus will increase when rare if it increases the linkage of alleles at the selected locus to a particular set of chromosomes. In addition, a parameter analogous to the coefficient of linkage disequilibrium in usual two-locus models with random mating must be nonzero if a new allele at the modifying locus is to increase in frequency at a geometric rate when rare. With mixed selfing and random mating a new allele at the modifying locus will apparently increase when rare only if it increases the linkage of alleles at the selected locus to a particular set of chromosomes.     

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     

Definition and Estimation of Higher-Order Gene Fixation Indices

Fixation indices summarize the associations between genes that arise from the joint effects of inbreeding and selection. In this paper, fixation indices are derived for pairs, triplets and quadruplets of genes at a single multiallelic locus. The fixation indices are obtained by dividing cumulants by constants; the cumulants describe the statistical distribution of alleles and the constants are functions of gene frequency. The use of cumulants instead of moments is necessary only for four-gene indices, when the fourth cumulant is used. A second type of four-gene index is also required, and this index is based upon the covariation of second-order cumulants. At multiallelic loci, a large number of indices is possible. If alleles are selectively neutral, the number of indices is reduced and the relationship between gene identity and gene cumulants is shown.--Two-gene indices can always be estimated from genotypic frequency data at a single polymorphic locus. Three-gene indices are also estimable except when allele frequency equals one-half. Four-gene indices are not estimable unless selection is assumed to have an equal effect upon each allele (such as under selective neutrality) and the locus contains at least three alleles of unequal frequency. For diallelic or selected loci, an alternative four-gene fixation index is proposed. This index incorporates both types of four-gene associations but cannot be related to gene identity.     

The Maintenance of Genetic Variability in Two-Locus Models of Stabilizing Selection

The maintenance of genetic variability at two diallelic loci under stabilizing selection is investigated. Generations are discrete and nonoverlapping; mating is random; mutation and random genetic drift are absent; selection operates only through viability differences. The determination of the genotypic values is purely additive. The fitness function has its optimum at the value of the double heterozygote and decreases monotonically and symmetrically from its optimum, but is otherwise arbitrary. The resulting fitness scheme is identical to the symmetric viability model. Linkage disequilibrium is neglected, but the results are otherwise exact. Explicit formulas are found for all the equilibria, and explicit conditions are derived fro their existence and stability. A complete classification of the six possible global convergence patterns is presented. In addition to the symmetric equilibrium (with gene frequency 1/2 at both loci), a pair of unsymmetric equilibria may exist; the latter are usually, but not always, unstable. If the ratio of the effect of the major locus to that of the minor one exceeds a critical value, both loci will be stably polymorphic. If selection is weak at the minor locus, the more rapidly the fitness function decreases near the optimum, the lower is this critical value; for rapidly decreasing fitness functions, the critical value is close to one. If the fitness function is smooth at the optimum, then a stable polymorphism exists at both loci only if selection is strong at the major locus.     

Fixation probability in a two-locus model by the ancestral recombination-selection graph

We use the ancestral influence graph (AIG) for a two-locus, two-allele selection model in the limit of a large population size to obtain an analytic approximation for the probability of ultimate fixation of a single mutant allele A. We assume that this new mutant is introduced at a given locus into a finite population in which a previous mutant allele B is already segregating with a wild type at another linked locus. We deduce that the fixation probability increases as the recombination rate increases if allele A is either in positive epistatic interaction with B and allele B is beneficial or in no epistatic interaction with B and then allele A itself is beneficial. This holds at least as long as the recombination fraction and the selection intensity are small enough and the population size is large enough. In particular this confirms the Hill-Robertson effect, which predicts that recombination renders more likely the ultimate fixation of beneficial mutants at different loci in a population in the presence of random genetic drift even in the absence of epistasis. More importantly, we show that this is true from weak negative epistasis to positive epistasis, at least under weak selection. In the case of deleterious mutants, the fixation probability decreases as the recombination rate increases. This supports Muller's ratchet mechanism to explain the accumulation of deleterious mutants in a population lacking recombination.     

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.     

Use of restriction fragment length polymorphisms for genetic counseling: Population genetic considerations

Two-locus population genetic models are analyzed to evaluate the utility of restriction fragment length polymorphisms for purposes of genetic counseling. It is shown that the linkage disequilibrium between a neutral marker and a tightly linked overdominant mutant will increase rapidly as the mutant moves to its polymorphic equilibrium. The linkage disequilibrium decays for deleterious recessive mutants. Two measures involving the linkage disequilibrium are investigated to determine how much information the transmission of the neutral marker provides about the transmission of the selected gene. In certain kinds of matings, where the parental two-locus genotypes and linkage phases are known, it is possible to determine whether or not a progeny is homozygous for the selected gene on the basis of the fetal genotype at the marker locus. A quantity of primary interest is the fraction of matings between individuals heterozygous for the selected gene in which exact diagnosis can be made in this way. The expected proportion of such matings, taken over all two-locus matings involving heterozygotes at the selected locus, is calculated as a function of the gene frequencies at the two loci and the linkage disequilibrium between them. This expected value is maximized when the linkage disequilibrium is at its maximum in absolute value. Fewer than half of all matings are informative if the linkage disequilibrium is small in magnitude or if the gene frequencies at the two loci are quite different. Consideration is also given to various conditional measures of association that may be useful when the parental two-locus genotypes are unknown. The results suggest that the utility of tightly linked neutral marker genes in predicting the transmission of a selected gene is generally less when selection acts against a recessive gene than for overdominant selection.     

Additive genetic variation under intraspecific competition and stabilizing selection: a two-locus study

A diallelic two-locus model is investigated in which the loci determine the genotypic value of a quantitative trait additively. Fitness has two components: stabilizing selection on the trait and a frequency-dependent component, as induced, for instance, if the ability to utilize different food resources depends on this trait. Since intraspecific competition induces disruptive selection, this model leads to a conflict of selective forces. We study how the underlying genetics (recombination rate and allelic effects) interacts with the selective forces, and explore the resulting equilibrium structure. For the special case of equal effects, global stability results are proved. Unless the locus effects are sufficiently different, the genetic variance maintained at equilibrium displays a threshold-like dependence on the strength of competition. For loci with equal effects, the equilibrium fitnesses of genotypic values exhibit disruptive selection if and only if competition is strong enough to maintain a stable two-locus polymorphism. For unequal effects, disruptive selection can be observed for weaker competition and in the absence of a stable polymorphism.     

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.     

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.     

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.     

Social selection in human populations. I. Modification of the fitness of offspring by an affected parent.

The concept of social selection for deleterious genes has been introduced by considering two alleles at one locus. A social selection model is constructed by assuming that the fitness of an individual is determined by his or her own as well as the parental phenotypes. It is shown that the equilibrium gene frequency depends on the loss of fitness of an individual due to the trait (gamma), due to affected parents (beta), and the probability that the heterozygote develops the trait (h). With mutational changes from the wild-type allele to the deleterious gene at a rate of alpha per generation, the equilibrium frequency of deleterious genes is approximately alpha/hs for 0 less than h less than or equal to 1 and square root alpha/s for h = 0, where s = gamma + beta(1 -- gamma)/2. Implications of the social selection model have been discussed for several diseases in man.