Phenogenetic drift and the evolution of genotype-phenotype relationships

Maintenance of a stable two-locus polymorphism is analyzed statistically by fitting a logistic regression with a quadratic function of genotypic fitnesses to the probability for a fitness set to maintain a polymorphism. The regression is fitted using a data set containing information on stable equilibria maintained by 32,00 randomly generated fitness sets with three recombination values (0. 005, 0.05, 0.5). Fitted logistic regressions discriminate with 88 to 90% accuracy between fitness sets maintaining and not maintaining a stable internal equilibrium, which implies the existence of a fitness structure (balance of fitnesses) maintaining a two-locus polymorphism. Aspects of the balance of fitnesses revealed by logistic regressions are discussed. It is demonstrated that logistic regression also discriminates between types of a stable polymorphism: globally stable polymorphism, several simultaneously stable polymorphisms, and stable equilibria in addition to a polymorphic one, which implies that different balances of fitnesses are responsible for the maintenance of different types of polymorphism.     

Natural Selection with Nuclear and Cytoplasmic Transmission. I. a Deterministic Model

A deterministic model allowing variation at a nuclear genetic locus in a population segregating two cytoplasmic types is formulated. Additive, multiplicative and symmetric viability matrices are analyzed for existence and stability of equilibria. The protectedness of polymorphisms in both nuclear genes and cytoplasmic types is also investigated in the general model. In no case is a complete polymorphism protected with this deterministic model. Results are discussed in light of the extensive variation in mtDNA that has recently been reported.     

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.     

The significance of over- and underdominance for the maintenance of genetic polymorphisms. I. Underdominance and stability

It is pointed out that the standard selection models in population genetics all require some form of heterozygote advantage in fitness in order to guarantee the maintenance or stability of genetic polymorphisms. Even more recent results demonstrating the existence of stable two-locus polymorphisms with marginal underdominance at both loci are based on certain epistatically acting heterosis assumptions. This raises the question as to whether heterozygote advantage in fitness is indeed a generally valid principle of maintaining polymorphisms. To avoid ambiguity in definition of heterozygote advantage (overdominance) as it appears in multiallele or multilocus systems, a one-locus-two-allele model is considered. This model allows for sexually asymmetric selection and random mating. It is shown that the model produces globally stable polymorphisms exhibiting underdominance in fitness for a considerable and biologically reasonable range of selection values. Having thus properly refuted the general validity of the common overdominance principle, a modified version is suggested which covers the classical viability selection model and its extension to arbitrary, sexually asymmetric viability and fertility selection. This modified overdominance principle is based on the notion of fractional fitnesses and relates protectedness of biallelic polymorphisms to the extent to which each genotype reproduces its own type. The fact that the model treated displays frequency dependent fitnesses which may change in ranking while approaching equilibrium is discussed in relation to problems of the evolution of overdominance and underdominance.     

Selection with gene-cytoplasm interactions. I. Maintenance of cytoplasm polymorphisms

General conditions for the protectedness of gene-cytoplasm polymorphisms are considered for a biallelic model with two cytoplasm types and under the assumption that nuclear polymorphisms cannot be maintained in the presence of only one cytoplasm type. Analytical results involving male fertilities, female fertilities, viabilities and selfing rates are obtained, and numerical results show spiral and cyclic behavior of population trajectories. It is shown that a maternally inherited cytoplasmic polymorphism cannot be maintained in the absence of a nuclear polymorphism, and that a gene-cytoplasm polymorphism can only be maintained if the population shows sexual asymmetry, i.e. , if the ratio of male to female fertility varies among genotypes. Thus, the classical viability selection model does not allow gene-cytoplasm polymorphisms.     

Should Individual Fitness Increase with Heterozygosity?

Natural selection influences not only gamete frequencies in populations but also the multilocus fitness structures associated with segregating gametes. In particular, only certain patterns of multilocus fitnesses are consistent with the maintenance of stable multilocus polymorphisms. This paper offers support for the proposition that, at stable, viability-maintained, multilocus polymorphisms, the fitness of a genotype tends to increase with the number of heterozygous loci it contains. Average fitness always increases with heterozygosity at stable product equilibria (i.e., those without linkage disequilibrium) maintained by either additive or multiplicative fitness schemes. Simulations suggest that it "generally" increases for arbitrary fitness schemes. The empirical literature correlating allozyme heterozygosity with fitness-correlated traits is discussed in the light of these and other theoretical results.     

On a genetic model of intraspecific competition and stabilizing selection

A genetic model is investigated in which two recombining loci determine the genotypic value of a quantitative trait additively. Two opposing evolutionary forces are assumed to act: stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the underlying genetics, in particular recombination rate and relative magnitude of allelic effects, interact with the conflicting selective forces and derive the resulting, surprisingly complex equilibrium patterns. We also investigate the conditions under which disruptive selection on the phenotypes can be observed and examine how much genetic variation can be maintained in such a model. We discovered a number of unexpected phenomena. For instance, we found that with little recombination the degree of stably maintained polymorphism and the equilibrium genetic variance can decrease as the strength of competition increases relative to the strength of stabilizing selection. In addition, we found that mean fitness at the stable equilibria is usually much lower than the maximum possible mean fitness and often even lower than the fitness at other, unstable equilibria. Thus, the evolutionary dynamics in this system are almost always nonadaptive.     

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.     

Selection with Gene-Cytoplasm Interactions. II. Maintenance of Gynodioecy

Gynodioecy is apparently frequently inherited through gene-cytoplasm interactions. General conditions for the protectedness of gene-cytoplasm polymorphisms for a biallelic model with two cytoplasm types were obtained previously, and these are applied to seven special cases of gene-cytoplasm interactions controlling gynodioecy and involving dominance. It is assumed that nuclear polymorphisms cannot be maintained in one cytoplasm type only. It is held that pure cytoplasmic inheritance of gynodioecy without nuclear interactions is unlikely, and it is shown that gynodioecy with gene-cytoplasm interactions is easier to establish than purely nuclear gynodioecy, for monogenic biallelic dominant or recessive inheritance. For three special cases, a resource-allocation model with simple assumptions always leads to conditions for protectedness of gynodioecy.     

Experimental and Theoretical Analysis of the "Sex-Ratio" Polymorphism in DROSOPHILA PSEUDOOBSCURA

The Sex-ratio chromosome (SR) is a widespread, multiply inverted rearrangement of the X chromosome present in several species of Drosophila. Male carriers transmit mostly X-bearing sperm. In the absence of strong counteracting selection, SR is expected to increase rapidly to fixation, causing extinction. The present study incorporates a selection-components analysis of SR in laboratory populations, using the closely linked Esterase-5 locus as a marker. Estimated fitnesses show directional viability selection against SR in both males and females, heterosis for fertility and no significant effects on virility, the male adult component of fitness. Estimated fitnesses satisfy conditions for protected polymorphism and accurately predict gene-frequency trajectories in experimental populations. A model of SR gene-frequency evolution is developed, which incorporates sex-linkage, meiotic drive, viability, fertility and virility selecton. We show that conditions for protected polymorphisms are not unduly restrictive and that differential fitness among males is not sufficient for protected polymorphism, irrespective of the degree of meiotic drive.     

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.     

Examples of the effect of genetic variation on competing species

An ordinary differential equation model for two competing populations with genetic variation in one population is presented. The degree of frequency dependence needed to produce various configurations of stable equilibria is discussed. For example, if the fitnesses are frequency independent then there may exist stable polymorphism although the genetically varying population becomes extinct in each fixation plane. Stable polymorphism where the genetically invariant population becomes extinct in each fixation plane requires frequency dependence in the fitness of the genetically invariant population.     

Temporally varying selection on multiple alleles: A diffusion analysis

A generalization of Gillespie's SAS-CFF model for natural selection acting on multiple alleles in a randomly fluctuating environment is presented that relaxes symmetry assumptions concerning the variances and covariances of allelic effects. The stationary density for a multidimensional diffusion approximation of the model is obtained and provides approximate necessary and sufficient conditions for the existence of stable polymorphisms. These conditions have exactly the same form as those derived by Kimura and Mandel for polymorphism under multiple allele selection in a constant environment, except that the time-invariant fitnesses are replaced by the approximate geometric mean fitnesses of the genotypes over time. An example illustrates that this simple relationship between random environment and constant environment conditions for polymorphism does not hold for more general selection schemes. The implications of these results for the maintenance of multiple alleles by balancing selection are discussed.     

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     

The significance of over- and underdominance for the maintenance of genetic polymorphisms. II. Overdominance and instability with random mating

Part I of the present series demonstrates that globally stable polymorphic equilibria may show underdominance in Darwinian fitness. Hence, overdominance in fitness can no longer be conceived of as a necessary condition for the stability of a polymorphism. In the present paper, the question is posed as to whether overdominance is at least sufficient for this stability. A population of randomly mating individuals is considered, where selection operates uniquely through differential fecundities of particular mating types and may generate either a heterozygote excess or deficit relative to Hardy-Weinberg proportions. It turns out that both unstable central overdominance and stable central underdominance are possible and that their occurrence is strongly related to an excess or a deficiency of heterozygotes in the vicinity of the regions of instability or stability. As one consequence, the above suggested sufficiency of heterozygote superiority is not valid, even in random mating populations. Based on the results of both papers of this series, which demonstrate the inadequacy of over- and underdominance as indicators of stability or instability, a modified overdominance principle is discussed. This principle states that a biallelic polymorphism is maintained if the heterozygote is superior in its degree of "heterogamous self-replication" to the degrees of "autogamous self-replication" of the corresponding homozygotes. It is derived with the help of fractional fitnesses, and it is pointed out that certain ratios of these may be more useful for finding evolutionary constants which govern the maintenance of genetic polymorphisms than are ratios of total fitnesses.     

The Two-Locus Model with Sex Differences in Recombination

The criteria for stability of the equilibrium with D=0 are obtained for the two locus model with multiplicative or symmetric fitnesses when the recombination values in males and females are different. It is shown that if r is defined to be equal to the average of the recombination values in males and females, then the criteria are exactly the same as in the standard two locus model.-The equilibrium values with D not equal0 are obtained for the symmetric fitness model. At this equilibrium, the absolute value of D is always greater (for the same average recombination value) if the recombination values in males and females differ than if they are equal.     

Density dependent selection 1: A stable feasible equilibrium may not be attainable

The joint evolution of gene frequency p, and population size N is studied. It is shown that when the genotypic fitnesses are logistic functions of the population size, sets of initial states exist which lead to bizarre behavior. Even though equilibria may be locally stable, these sets of initial conditions eventually produce negative fitnesses. Alternative models are discussed as are some general properties of the mean fitness.     

Equilibrium properties of a multi-locus, haploid-selection, symmetric-viability model

Under haploid selection, a multi-locus, diallelic, two-niche Levene (1953) model is studied. Viability coefficients with symmetrically opposing directional selection in each niche are assumed, and with a further simplification that the most and least favored haplotype in each niche shares no alleles in common, and that the selection coefficients monotonically increase or decrease with the number of alleles shared. This model always admits a fully polymorphic symmetric equilibrium, which may or may not be stable.We show that a stable symmetric equilibrium can become unstable via either a supercritical or subcritical pitchfork bifurcation. In the supercritical bifurcation, the symmetric equilibrium bifurcates to a pair of stable fully polymorphic asymmetric equilibria; in the subcritical bifurcation, the symmetric equilibrium bifurcates to a pair of unstable fully polymorphic asymmetric equilibria, which then connect to either another pair of stable fully polymorphic asymmetric equilibria through saddle-node bifurcations, or to a pair of monomorphic equilibria through transcritical bifurcations. As many as three fully polymorphic stable equilibria can coexist, and jump bifurcations can occur between these equilibria when model parameters are varied.In our Levene model, increasing recombination can act to either increase or decrease the genetic diversity of a population. By generating more hybrid offspring from the mating of purebreds, recombination can act to increase genetic diversity provided the symmetric equilibrium remains stable. But by destabilizing the symmetric equilibrium, recombination can ultimately act to decrease genetic diversity.     

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.     

Frequency-dependent selection and the maintenance of genetic variation: exploring the parameter space of the multiallelic pairwise interaction model

When individuals' fitnesses depend on the genetic composition of the population in which they are found, selection is then frequency dependent. Frequency-dependent selection (FDS) is often invoked as a heuristic explanation for the maintenance of large numbers of alleles at a locus. The pairwise interaction model is a general model of FDS via intraspecific competition at the genotypic level. Here we use a parameter-space approach to investigate the full potential for the maintenance of multiallelic equilibria under the pairwise interaction model. We find that FDS maintains full polymorphism more often than classic constant-selection models and produces more skewed equilibrium allele frequencies. Fitness sets with some degree of rare advantage maintained full polymorphism most often, but a wide variety of nonobvious fitness patterns were also found to have positive potential for polymorphism. An example is put forth suggesting possible explanations for multiallelic polymorphisms maintained despite positive FDS on individual alleles.