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.     

A numerical solution to the equilibria of the two-locus two-allele selection model.

Examination of the equilibria of the standard two-locus two-allele selection model leads to the construction of a polynomial with coefficients derived from selective values in the genotypic fitness matrix. This polynomial can be partially factored algebraically and numerical techniques are available to extract the roots of the remainder. Each root provides a possible value of the disequilibrium coefficient and the gametic frequencies at equilibrium, and these can be readily checked for stability.     

Evolution and intraspecific exploitative competition. II. A two-locus model for additive gene effects

A two-locus model corresponding to the model of Christiansen and Loeschcke (1980, theoret, Popul. Biol. 18, 297-313) is analysed. The two loci each have two alleles, and the loci influences a character which determines the utilization of resources in a one-dimensional continuum. The analysis of the model is supported by numerical iterations of the recurrence equations. The previous prediction of high linkage disequilibrium for small allele contributions to the character and close linkage between the loci is confirmed. For larger allele contributions results comparable to those for the symmetric viabilities model are obtained. The model degenerates when the allele contributions at the two loci are equal, i.e., in the most symmetric situation. The results are discussed as the outcome of a balance between optimizing selection and disruptive selection. For small allele contributions the results are virtually independent of which genotype is most favoured by the optimizing aspect of the selective forces.     

A single-locus model of speciation

The crucial phase of speciation is argued to be the evolution of mating cross-incompatibility (prezygotic incompatibility) between the genotypes distinguishing the prospective species populations. Based on this idea, a single-locus model of speciation is presented, which is shown to be biologically plausible and may help to settle the controversy as to the biological significance of single-locus modes of speciation. The model involves three alleles, two of which characterize in homozygous state the prospective species populations and in heterozygous state their hybrids. The third allele represents a mutant which is equivalent to one of the first two alleles with the exception that it inhibits mating with carriers of the third allele. This third allele is fixed in one population and immigrates into a second population which contains the mutant inhibiting matings with members of the former population. Migration in the reverse direction does not occur. Proceeding from a widely applicable concept of fitness and mating preference it is shown that postzygotic incompatibility (hybrid or heterozygote disadvantage) alone suffices to trigger evolutionary replacement of the extant mating relations in the population receiving immigrants by any arbitrary degree of prezygotic incompatibility. This corroborates Wallace's hypothesis and emphasizes the potential biological relevance of speciation by reinforcement (parapatric speciation) at single gene loci.     

Diffusion approximations of the two-locus Wright-Fisher model

Diffusion approximations are established for the multiallelic, two-locus Wright-Fisher model for mutation, selection, and random genetic drift in a finite, panmictic, monoecious, diploid population. All four combinations of weak or strong selection and tight or loose linkage are treated, though the proof in the case of strong selection and loose linkage is incomplete. Under certain conditions, explicit formulas are obtained for the stationary distributions of the two diffusions with loose linkage.Supported in part by NSF Grant DMS-8704369Supported in part by NSF Grant BSR-8512844     

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.     

Reinforcement of genetic coherence in a two-locus model

Background  

In order to maintain populations as units of reproduction and thus enable anagenetic evolution, genetic factors must exist which prevent continuing reproductive separation or enhance reproductive contact. This evolutionary principle is called genetic coherence and it marks the often ignored counterpart of cladistic evolution. Possibilities of the evolution of genetic coherence are studied with the help of a two-locus model with two alleles at each locus. The locus at which viability selection takes place is also the one that controls the fusion of gametes. The second locus acts on the first by modifying the control of the fusion probabilities. It thus acts as a mating modifier whereas the first locus plays the role of the object of selection and mating. Genetic coherence is enhanced by modifications which confer higher probabilities of fusion to heterotypic gametic combinations (resulting in heterozygous zygotes) at the object locus.     

New properties of the two-locus partial selfing model with selection

Analytic solutions are obtained for the equilibria of a simple two-locus, heterotic selection model with mixed selfing and random outcrossing. Two general phenomena are possible, depending upon the viabilities and the degree of selfing: (1) Negative disequilibrium potential, under which only gametic disequilibrium is possible; and (2) positive disequilibrium potential, which can result in permanent gametic disequilibrium provided that linkage is sufficiently tight. Under random mating (s = 0), these two situations correspond to negative and positive additive epistasis, respectively. With partial self-fertilization, however, this is no longer true, and a more appropriate measure of gametc disequilibrium potential, Δ(s), is introduced. A numerically aided examination of the model results in the discovery of two new properties of partial selfing with selection: (1) With negative disequilibrium potential (Δ(s) < 0), the equilibrium mean fitness increases with increasing recombination. With positive disequilibrium potential (Δ(s) > 0), the opposite is true. (2) Gametic disequilibrium can increase or decrease as the degree of selfing is increased. Therefore, it is apparent that partial selfing and linkage are not analogous as regards the maintenance of disequilibrium.     

A complete classification of epistatic two-locus models

Background

The study of epistasis is of great importance in statistical genetics in fields such as linkage and association analysis and QTL mapping. In an effort to classify the types of epistasis in the case of two biallelic loci Li and Reich listed and described all models in the simplest case of 0/1 penetrance values. However, they left open the problem of finding a classification of two-locus models with continuous penetrance values.

Results

We provide a complete classification of biallelic two-locus models. In addition to solving the classification problem for dichotomous trait disease models, our results apply to any instance where real numbers are assigned to genotypes, and provide a complete framework for studying epistasis in QTL data. Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for. The model types are defined by 86 circuits, which are linear combinations of genotype values, each of which measures a fundamental unit of interaction.

Conclusion

The circuits provide information on epistasis beyond that contained in the additive × additive, additive × dominance, and dominance × dominance interaction terms. We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.     

A two-locus molecular characterization of Paramecium calkinsi

Paramecium calkinsi (Ciliophora, Protozoa) is a euryhaline species which was first identified in freshwater habitats, but subsequently several strains were also collected from brackish water. It is characterized by clockwise spiral swimming movement and the general morphology of the "bursaria type." The present paper is the first molecular characterization of P. calkinsi strains recently collected in distant regions in Russia using ITS1-5.8S- ITS2-5'LSU rDNA (1100bp) and COI (620bp) mtDNA sequenced gene fragments. For comparison, our molecular analysis includes P. bursaria, exhibiting a similar "bursaria morphotype" as well as species representing the "aurelia type," i.e., P. caudatum, P. multimicronucleatum, P. jenningsi, and P. schewiakoffi, and some species of the P. aurelia species complex (P. primaurelia, P. tetraurelia, P. sexaurelia, and P. tredecaurelia). We also use data from GenBank concerning other species in the genus Paramecium and Tetrahymena (which used as an outgroup). The division of the genus Paramecium into four subgenera (proposed by Fokin et al. 2004) is clearly presented by the trees. There is a clear separation between P. calkinsi strains collected from different regions (races). Consequently, given the molecular distances between them, it seems that these races may represent different syngens within the species.     

Multilocus model of sympatric speciation. III. Computer simulations

In the previous papers some analytical results were obtained for the limit stages of sympatric speciation. The present paper aims at finding the scope of validity for these results. The Monte Carlo computer model of this process was created and studied. We deal with two aspects of the speciation process: the development of reproductive isolation between the forming species and the extinction of the intermediate individuals. It has been shown that the advantage of the allele of reproductive isolation increases with the growth of its frequency. The extinction of the intermediates goes differently with various numbers of loci involved in speciation. If reproductive isolation is due to differences in two or four loci, then the completion of extinction of the intermediates requires the strongest disruptive selection, so that the necessary conditions for speciation found previously are also proved to be sufficient. But with eight and probably with a larger number of loci, the selection required to promote speciation in a population that is far from it is considerably stronger than selection needed at the last stage of speciation. Consequently, under some intensities of disruptive selection the final state of a population depends initial state. The conditions under which the stationary state of a population is characterized by bimodal distribution of phenotypes are also found.     

Protected polymorphism in the two-locus haploid model with unpredictable fitnesses

 In an unpredictable environment, the distributions of alleles from which polymorphism can be maintained forever belong to a certain set, the C-viability kernel. Such a set is calculated in the two-locus haploid model, as well as the corresponding fitnesses at any time which make this maintenance possible. The dependence of the C-viability kernel on the set U of admissible fitnesses and on the recombination rate r is studied. Notably, the C-viability kernel varies rapidly in the neighborhood of equal fitness of AB and ab; it becomes empty when ab has a fitness below a certain function, which is delineated, of the recombination rate. The properties of the two-locus model under constraints, out of equilibrium and with unpredictable selection are thus presented. Received: 20 May 1999     

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.     

A quantitative genetic competition model for sympatric speciation

I use multilocus genetics to describe assortative mating in a competition model. The intensity of competition between individuals is influenced by a quantitative character whose value is determined additively by alleles from many loci. With assortative mating based on this character, frequency- and density-dependent competition can subdivide a population with an initially unimodal character distribution. The character distribution becomes bimodal, and the subpopulations corresponding to the two modes are reproductively separated because mating is assortative. This happens if the resource distribution is unimodal, i.e. even if selection due to phenotypic carrying capacities is not disruptive. The results suggest that sympatric speciation due to frequency-dependent selection can occur in quite general ecological scenarios if mating is assortative. I also discuss the evolution of assortative mating. Since it induces bimodal phenotype distributions, assortative mating leads to a better match of the resources if their distribution is also bimodal. Moreover, in a population with a bimodal phenotype distribution, the average strength of frequency-dependent competition is lower than in a unimodal population. Therefore, assortative mating permits higher equilibrium densities than random mating even if the resource distribution is unimodal. Thus, even though it may lead to a less efficient resource use, assortative mating is favoured over random mating because it reduces frequency-dependent effects of competition.