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

Summary Considerations proceed from a model of positive assortative mating based on genotype at one locus, with an arbitrary number of alleles, assuming no selection, mutation, or migration, hypothetically infinite population size, and discrete non-overlapping generations. From these conditions, inferences are made about the genotypic structure at a linked locus, as well as about the corresponding 2-locus gametic structure.The following main results are presented: in the course of the generations, the genotypic structure at the second locus and the 2-locus gametic structure always tend to a limit responsive to the initial conditions concerning the joint genotypic structure at the two loci and the degree of assortativity and linkage. A complete, analytical representation of the limits is given. In particular, if assortative mating is only partial and at the same time linkage is not complete, a population is not able to maintain a permanent deviation of the gametic structure from linkage equilibrium, and thus the genotypic structure at the second locus tends to Hardy-Weinberg proportions. On the other hand, if initial linkage disequilibrium is combined with partial assortative mating and complete linkage (or with complete assortative mating and unlinked loci) the population maintains this disequilibrium and thus the genotypic structure at the second locus need not tend to Hardy-Weinberg proportions. It turns out that the conditions not only of complete linkage, but also of unlinked loci together with complete assortativity, imply no change in gametic structure from the initial structure.In order to demonstrate the influence of several parameters on the speed of convergence to and the magnitude of the respective limits, several graphs are included.     

Deviations of Genotypic Structures from Hardy-Weinberg Proportions under Random Mating and Differential Selection between the Sexes

Population genetic models, such as differential viability selection between the sexes and differential multiplicative fecundity contributions of the sexes, are considered for a single multiallelic locus. These selection models usually produce deviations of the zygotic genotype frequencies from Hardy-Weinberg proportions. The deviations are investigated (with special emphasis put on equilibrium states) to quantify the effect of selective asymmetry in the two sexes. For many selection regimes, the present results demonstrate a strong affinity of zygotic genotype frequencies for Hardy-Weinberg proportions after two generations, at the latest. It is shown that the deviations of genotypic equilibria from the corresponding Hardy-Weinberg proportions can be expressed and estimated by means of selection components of only that sex with the lower selection intensity. This corresponds to the well-known fact that viability selection acting in only one sex yields Hardy-Weinberg equilibria.     

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

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

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.     

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.     

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.     

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     

Isolation and characterization of 11 tetranucleotide microsatellite loci in the Egyptian mongoose (Herpestes ichneumon)

We report the isolation of 11 polymorphic tetranucleotide microsatellite loci in the Egyptian mongoose (Herpestes ichneumon). In a sample of 27 individuals, we observed between 4 and 7 alleles per locus and their observed and expected heterozygosities ranged from 0.37 to 0.85 and from 0.44 to 0.79, respectively. All genotypic frequencies conformed to Hardy-Weinberg equilibrium expectations and there were no instances of linkage disequilibrium detected between pairs of loci.     

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.     

Continuous selective models

Neglecting age-structure, but taking into account matings with differential fertility in Mendelian reproduction, continuous selective models are formulated for a single locus with an arbitrary number of alleles, with or without distinguishing the sexes, and for two alleles at each of two loci in a monoecious population. In each case, without restricting the mating system, differential equations are derived for the genotypic frequencies, and the validity of the customary Malthusian-parameter differential equations for the gametic frequencies is established. Particular attention is devoted to the conditions for Hardy-Weinberg proportions under random mating. For multiple alleles at a single locus in a monoecious population, exact solutions are obtained for the following three Hardy-Weinberg models: gametic selection, no dominance, and the same selective effect for all alleles but one. The last scheme includes, as special cases, a completely dominant or recessive distinguished allele, and arbitrary selection with only two alleles. Two single-locus assortative mating patterns are analyzed for a monoecious organism using the general formalism. One of these has an arbitrary number of alleles, all the genotypes being distinguishable, while the other involves two alleles, one of which is completely dominant to the other.     

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.     

Characterization of 12 polymorphic microsatellite markers for a facultatively eusocial sweat bee (Megalopta genalis)

We developed a library of twelve polymorphic di- and tri-nucleotide microsatellite markers for Megalopta genalis, a facultatively eusocial sweat bee. We tested each locus in a panel of 23 unrelated females and found 7-20 alleles per locus. Observed and expected heterozygosities ranged from 0.65 to 0.96 and from 0.69 to 0.95 respectively. None of the loci deviated from Hardy-Weinberg equilibrium proportions or was found to be in gametic disequilibrium.     

Isolation and characterization of 17 microsatellite loci for the house finch (Carpodacus mexicanus)

The house finch (Carpodacus mexicanus) has emerged recently as a model species in studies of sexual selection, reproductive physiology, population genetics, and epizootic disease ecology. Here we describe 17 highly polymorphic microsatellite loci for this species. In a sample of 36 individuals, we observed an average of 16 alleles per locus and heterozygosity ranged from 0.61 to 0.97. One locus showed significant deviation from Hardy-Weinberg proportions, but no significant gametic disequilibrium was observed among any of the loci. Amplification by polymerase chain reaction was optimized under similar parameters across loci, thereby facilitating multiplexing and rapid multilocus genotyping.     

Frequency-Dependent Selection: The High Potential for Permanent Genetic Variation in the Diallelic, Pairwise Interaction Model

A detailed analytic and numerical study is made of the potential for permanent genetic variation in frequency-dependent models based on pairwise interactions among genotypes at a single diallelic locus. The full equilibrium structure and qualitative gene-frequency dynamics are derived analytically for a symmetric model, in which pairwise fitnesses are chiefly determined by the genetic similarity of the individuals involved. This is supplemented by an extensive numerical investigation of the general model, the symmetric model, and nine other special cases. Together the results show that there is a high potential for permanent genetic diversity in the pairwise interaction model, and provide insight into the extent to which various forms of genotypic interactions enhance or reduce this potential. Technically, although two stable polymorphic equilibria are possible, the increased likelihood of maintaining both alleles, and the poor performance of protected polymorphism conditions as a measure of this likelihood, are primarily due to a greater variety and frequency of equilibrium patterns with one stable polymorphic equilibrium, in conjunction with a disproportionately large domain of attraction for stable internal equilibria.     

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.     

PERMANENT GENETIC RESOURCES: Isolation and characterization of 12 microsatellite markers in the middle-spotted woodpecker (Dendrocopos medius)

Twelve polymorphic microsatellite loci were developed for Dendrocopos medius. Polymorphism was assessed for 27 individuals from the southwesternmost population of this woodpecker species. The number of alleles per locus ranged from three to seven, with observed heterozygosity values from 0.444 to 0.852. Genotypic frequencies conformed to Hardy-Weinberg equilibrium, and no evidence for linkage disequilibrium was observed. Multilocus genotypes resulting from this set of markers will be useful to determine genetic diversity and differentiation within and among habitat patches inhabited by D. medius. Three of the loci were polymorphic for Picoides articus.     

Convergence of genotypic frequencies for differential selfing and positive assortative mating at a biallelic locus

For a biallelic model of differential self-fertilization and differential positive assortative mating based on genotype, it is shown that the genotypic frequencies converge for all sets of mating system parameters. Overdominance and underdominance with respect to the parameters are necessary but not sufficient conditions for global convergence to a polymorphic equilibrium and local attractiveness of both the fixation states, respectively. There are cases of overdominance and underdominance for which one fixation state is globally attractive. The relationship of the result to those known from the classical viability selection model are briefly discussed. For the multiallelic version, it is shown that after the first generation all of the homozygote frequencies are always in excess of the corresponding Hardy-Weinberg proportions if at least one homozygote rate of self-fertilization or assortment probability is positive.     

Continuous selective models with mutation and migration

The continuous selective model formulated previously for a single locus with multiple alleles in a monoecious population is extended to include mutation and migration. Somatic and germ line genotypic frequencies are distinguished, and the alternative hypotheses of constant mutation rates and age-independent mutation frequencies are analyzed in detail for arbitrary selection and mating schemes. With any mating pattern, if there is no selection, the equilibrium allelic frequencies are shown to be unaffected by the generalizations introduced in this paper. If, in addition, mating is at random, the equilibrium genotypic frequencies are proved to be in Hardy-Weinberg proportions. For both models, the nature of the approach to equilibrium is discussed. Migration is treated in the island model.     

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.