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.     

On the evolution of genetic incompatibility systems. IV. Modification of response to an existing antigen polymorphism under partial selfing

A 2-locus model of the evolution of self-incompatibility in a population practicing partial selfing is presented. An allele is introduced at a modifier locus which influences the strength of the rejection reaction expressed by the style in response to antigens recognized in pollen. Two causes of inbreeding depression are investigated. First, offspring viability depends solely on the source (self or non-self) of the fertilizing pollen. Second, offspring viability declines with the expression of recessive deleterious alleles, segregating at a third (disease) locus, which exhibit an imperfect association with antigen alleles. Evolutionary changes occurring at the disease locus are not considered in this study. The condition under which a modifier allele that intensifies the incompatibility reaction increases when rare depends upon the number of antigens, the frequency of recessive deleterious alleles at the disease locus, and the level of association between the antigen locus and the disease locus. It is the improvement of viability among offspring derived by outcrossing, rather than the prevention of self-fertilization, that may represent the primary evolutionary function of genetic incompatibility systems.     

A quantitative model of the relationship between phenotypic variance and heterozygosity at marker loci under partial selfing.

Negative relationships between allozyme heterozygosity and morphological variance have often been observed and interpreted as evidence for increased developmental stability in heterozygotes. However, inbreeding can also generate such relationships by decreasing heterozygosity at neutral loci and redistributing genetic variance at the same time. I here provide a quantitative genetic model of this process by analogy with heterozygosity-fitness relationships. Inbreeding generates negative heterozygosity-variance relationships irrespective of the genetic architecture of the trait. This holds for fitness traits as well as neutral traits, the effect being stronger for fitness traits under directional dominance or overdominance. The order of magnitude of heterozygosity-variance regressions is compatible with empirical data even with very low inbreeding. Although developmental stability effects cannot be excluded, inbreeding is a parsimonious explanation that should be seriously considered to explain correlations between heterozygosity and both mean and variance of phenotypes in natural populations.     

Dominance and the maintenance of polymorphism in multiallelic migration-selection models with two demes

The maintenance of genetic variation in a spatially heterogeneous environment has been one of the main research themes in theoretical population genetics. Despite considerable progress in understanding the consequences of spatially structured environments on genetic variation, many problems remain unsolved. One of them concerns the relationship between the number of demes, the degree of dominance, and the maximum number of alleles that can be maintained by selection in a subdivided population. In this work, we study the potential of maintaining genetic variation in a two-deme model with deme-independent degree of intermediate dominance, which includes absence of G×E interaction as a special case. We present a thorough numerical analysis of a two-deme three-allele model, which allows us to identify dominance and selection patterns that harbor the potential for stable triallelic equilibria. The information gained by this approach is then used to construct an example in which existence and asymptotic stability of a fully polymorphic equilibrium can be proved analytically. Noteworthy, in this example the parameter range in which three alleles can coexist is maximized for intermediate migration rates. Our results can be interpreted in a specialist-generalist context and (among others) show when two specialists can coexist with a generalist in two demes if the degree of dominance is deme independent and intermediate. The dominance relation between the generalist allele and the specialist alleles play a decisive role. We also discuss linear selection on a quantitative trait and show that G×E interaction is not necessary for the maintenance of more than two alleles in two demes.     

Stability analysis of the partial selfing selection model

We undertake a detailed study of the one-locus two-allele partial selfing selection model. We show that a polymorphic equilibrium can exist only in the cases of overdominance and underdominance and only for a certain range of selfing rates. Furthermore, when it exists, we show that the polymorphic equilibrium is unique. The local stability of the polymorphic equilibrium is investigated and exact analytical conditions are presented. We also carry out an analysis of local stability of the fixation states and then conclude that only overdominance can maintain polymorphism in the population. When the linear local analysis is inconclusive, a quadratic analysis is performed. For some sets of selective values, we demonstrate global convergence. Finally, we compare and discuss results under the partial selfing model and the random mating model.     

Selection in complex genetic systems. VI. Equilibrium properties of two locus selection models with partial selfing

The results of a combined analytical and numerical study of two locus selection models with partial selfing indicate that several commonly held opinions about the effects of partial self-fertilization do not hold in general. For example, the heterozygosity of a population may actually increase as the selfing rate is increased. Similarly, selection strong enough to guarantee a two locus polymorphism with complete selfing does not necessarily guarantee a two locus polymorphism with intermediate amounts of self-fertilization. The results presented here and a brief review of previously existing results indicate that the predictions of population genetic models based on the assumption of random mating will not be greatly altered by a small amount of self-fertilization, unless the loci involved are tightly linked. On the other hand, the results presented indicate that a very small amount of outcrossing may lead to marked differences from the expectation based on complete self-fertilization.     

Predictions of single-nucleotide polymorphism differentiation between two populations in terms of mutual information

Mutual information (I) provides a robust measure of genetic differentiation for the purposes of estimating dispersal between populations. At present, however, there is little predictive theory for I. The growing importance in population biology of analyses of single-nucleotide and other single-feature polymorphisms (SFPs) is a potent reason for developing an analytic theory for I with respect to a single locus. This study represents a first step towards such a theory. We present theoretical predictions of I between two populations with respect to a single haploid biallelic locus. Dynamical and steady-state forecasts of I are derived from a Wright-Fisher model with symmetrical mutation between alleles and symmetrical dispersal between populations. Analytical predictions of a simple Taylor approximation to I are in good agreement with numerical simulations of I and with data on I from SFP analyses of dispersal experiments on Drosophila fly populations. The theory presented here also provides a basis for the future inclusion of selection effects and extension to multiallelic loci.     

Effect of partial selfing and polygenic selection on establishment in a new habitat

This article analyzes how partial selfing in a large source population influences its ability to colonize a new habitat via the introduction of a few founder individuals. Founders experience inbreeding depression due to partially recessive deleterious alleles as well as maladaptation to the new environment due to selection on a large number of additive loci. I first introduce a simplified version of the inbreeding history model to characterize mutation‐selection balance in a large, partially selfing source population under selection involving multiple nonidentical loci. I then use individual‐based simulations to study the eco‐evolutionary dynamics of founders establishing in the new habitat under a model of hard selection. The study explores how selfing rate shapes establishment probabilities of founders via effects on both inbreeding depression and adaptability to the new environment, and also distinguishes the effects of selfing on the initial fitness of founders from its effects on the long‐term adaptive response of the populations they found. A high rate of (but not complete) selfing is found to aid establishment over a wide range of parameters, even in the absence of mate limitation. The sensitivity of the results to assumptions about the nature of polygenic selection is discussed.     

Genetic polymorphism and close linkage of two plasma protein loci in dogs

By using a simple method of two-dimensional horizontal electrophoresis, phenotypes of an unidentified plasma protein (PA4) were determined in 967 dogs belonging to 43 different breeds. Two codominant, autosomal alleles (F and S) of PA4 were reported. While many of the breeds of middle and north-eastern Asia (akita inu, Alaskan malamute, chow chow, samoyed, Siberian husky and Tibetan terrier) showed a substantial frequency (0.1 to 0.6) of the S allele, a majority of the European breeds had only the F allele. Evidence was provided that the PA4 locus is closely linked to the plasma pretransferrin 1 locus (PRT1). No recombinant was observed in 45 informative offspring studied. In nearly all breeds, the PA4 S allele was almost always in coupling phase with the PRT1 F allele.     

On the maintenance of sex chromosome polymorphism by sex-antagonistic selection

Complex sex determination systems are a priori unstable and require specific selective forces for their maintenance. Analytical derivations suggest that sex antagonistic selection may play such a role, but this assumes absence of recombination between the sex-determining and sex-antagonistic genes. Using individual-based simulations and focusing on the sex chromosome and coloration polymorphisms of platy fishes as a case study, we show that the conditions for polymorphism maintenance induce female biases in primary sex ratios, so that sex ratio selection makes the system collapse toward male or female heterogamety as soon as recombinant genotypes appear. However, a polymorphism can still be maintained under scenarios comprising strong sexual selection against dull males, mild natural selection against bright females, and low recombination rates. Though such conditions are plausibly met in natural populations of fishes harboring such polymorphisms, quantitative empirical evaluations are required to properly test whether sex antagonistic selection is a causal agent or whether other selective processes are required (such as local mate competition favoring female-biased sex ratios).     

Different mechanisms drive the maintenance of polymorphism at loci subject to strong versus weak fluctuating selection

The long‐running debate about the role of selection in maintaining genetic variation has been given new impetus by the discovery of hundreds of seasonally oscillating polymorphisms in wild Drosophila, possibly stabilized by an alternating summer‐winter selection regime. Historically, there has been skepticism about the potential of temporal variation to balance polymorphism, because selection must be strong to have a meaningful stabilizing effect—unless dominance also varies over time (“reversal of dominance”). Here, we develop a simplified model of seasonally variable selection that simultaneously incorporates four different stabilizing mechanisms, including two genetic mechanisms (“cumulative overdominance” and reversal of dominance), as well as ecological “storage” (“protection from selection” and boom‐bust demography). We use our model to compare the stabilizing effects of these mechanisms. Although reversal of dominance has by far the greatest stabilizing effect, we argue that the three other mechanisms could also stabilize polymorphism under plausible conditions, particularly when all three are present. With many loci subject to diminishing returns epistasis, reversal of dominance stabilizes many alleles of small effect. This makes the combination of the other three mechanisms, which are incapable of stabilizing small effect alleles, a better candidate for stabilizing the detectable frequency oscillations of large effect alleles.     

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.     

Estimation of parameters of deleterious mutations in partial selfing or partial outcrossing populations and in nonequilibrium populations

The Deng-Lynch method was developed to estimate the rate and effects of deleterious genomic mutations (DGM) in natural populations under the assumption that populations are either completely outcrossing or completely selfing and that populations are at mutation-selection (M-S) balance. However, in many plant and animal populations, selfing or outcrossing is often incomplete in that a proportion of populations undergo inbreeding while the rest are outcrossing. In addition, the degrees of deviation of populations from M-S balance are often not known. Through computer simulations, we investigated the robustness and the applicability of the Deng-Lynch method under different degrees of partial selfing or partial outcrossing and for nonequilibrium populations approaching M-S balance at different stages. The investigation was implemented under constant, variable, and epistatic mutation effects. We found that, generally, the estimation by the Deng-Lynch method is fairly robust if the selfing rate (S) is <0.10 in outcrossing populations and if S > 0.8 in selfing populations. The estimation may be unbiased under partial selfing with variable and epistatic mutation effects in predominantly outcrossing populations. The estimation is fairly robust in nonequilibrium populations at different stages approaching M-S balance. The dynamics of populations approaching M-S balance under various parameters are also studied. Under mutation and selection, populations approach balance at a rapid pace. Generally, it takes 400-2000 generations to reach M-S balance even when starting from homogeneous individuals free of DGM. Our investigation here provides a basis for characterizing DGM in partial selfing or outcrossing populations and for nonequilibrium populations.     

Detection of highly polymorphic microsatellite loci in a species with little allozyme polymorphism

Microsatellite loci are regions of DNA containing tandem repeats of a short sequence motif; they occur abundantly in all eukaryotic genomes and have been shown to be a rich source of highly polymorphic genetic markers in humans and other mammals. These loci are particularly suitable for population studies because they can be relatively easily scored using a combination of polymerase chain reaction (PCR) amplification of each locus followed by electrophoresis to separate alleles. This paper details a method for finding these loci in any species. This method demonstrates that trinucleotide microsatellite loci are abundant and highly polymorphic in the social wasp Polistes annularis , whereas allozyme electrophoresis reveals very little polymorphism. The first six loci examined were all polymorphic with a mean observed heterozygosity of 0.62; in comparison average heterozygosity of 33 allozymes was 0.035. We suggest that this method can be used to detect variation where other methods have failed, making it an ideal tool for population and conservation geneticists who must deal with populations lacking other types of genetic variability.     

On the mutual influence of individuals in a society

On the basis of previously proposed mathematical models of social behavior, the present note investigates the possibility of the control of behavior remaining permanently in the hands of one class, if this class possesses sufficient means for influencing mass behavior. The conclusion is reached that, with the assumptions made, if the behavior imposed by the controlling class leads to sufficiently strong dissatisfaction, the control will pass to another class, no matter how strong the controlling power of the first.