Evolution and intraspecific exploitative competition I. One-locus theory for small additive gene effects

Using the model of exploitative competition of R. H. MacArthur and R. Levins (1967, Amer. Natur. 101, 377–385), evolution at a gene locus which influences the niche position is considered. The locus has multiple alleles, and the contributions of the alleles to the genotypic value are additive. The resource spectrum and the utilization functions of the genotypes are assumed to be Gaussian. Evolution will make the mode of the niche converge to the resource optimum, as long as the allele contributions are small compared to the distance between the mode of the niche and the resource optimum. When this distance is of the same order of magnitude as the allele contributions, then the globally stable equilibrium will maintain at most two alleles in the population, unless the allele contributions are large. Classical overdominance is not needed to maintain polymorphism. This result predicts high linkage disequilibrium in similar multilocus models. It is concluded that intraspecific competition can be a powerful force in maintaining two-allele polymorphisms, and that it can maintain high linkage disequilibrium among closely linked loci.     

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.     

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.     

Heterosis as an explanation for large amounts of genic polymorphism

By using both numerical and analytical approaches, we have shown that heterosis alone is not a mechanism for maintaining many alleles segregating at a locus. Even when all heterozygous are more fit than all homozygotes, the proportion of fitness arrays that will lead to a stable, feasible equilibrium of more than 6 or 7 alleles is vanishingly small. More alleles can be maintained if, in addition to heterosis, it is assumed that there is very little variation in fitness from heterozygote to heterozygote, with the ratio of mean heterosis to standard deviation of fitness among heterozygotes in the neighborhood of 10. When such conditions hold, the allelic frequency distribution and equilibrium will be very uniform, with all alleles very close to equal frequency (see PDF). It is much more likely that stable equilibria for multiple alleles will be best explained by multiple niche selection.     

The effect of variable environments on polymorphism at loci with several alleles II. Submultiplicative viabilities

Circumstances assuring a unique stable equilibrium are investigated for a subdivided population with several alleles segregating at a single locus. For a broad class of selection regimes entailing heterozygote viabilities greater than the geometric mean of the corresponding homozygote viabilities, a stable fixation state precludes any other stable equilibria if either total-panmixia or temporal variation is operating. This extends known results for two alleles at a single locus and partially delimits when some of the bizarre behaviour engendered by multiple alleles may occur.Supported in part by National Science Foundation (USA) grant MCS-8002227     

General analysis of frequency-dependent sexual selection at a multi-allelic locus

A general model is analyzed in which arbitrarily frequency-dependent selection acts on one sex of a diploid population with several alleles at one locus, as a result of viability or mating-success differences. The existence of boundary and polymorphic equilibria is examined, and conditions for local stability, internal and external, are obtained. The status of Hardy-Weinberg approximations in studying stability and approach to equilibria is also considered. The general principles are then applied to two specific models: one where genotypes fall into two phenotypic classes; and one with a hierarchy of dominance where viability and sexual selection are opposed. In the latter case it is found that, of all the equilibria present, there is one and only one which could possibly be stable: the existence of a unique globally stable equilibrium might then be inferred.     

Population structure and genetic analysis of different utility types of mango (Mangifera indica L.) germplasm of Andhra Pradesh state of India using microsatellite markers

Genetic analysis of 90 mango genotypes including juicy, table, dual and pickle types from different parts of Andhra Pradesh of India was carried out employing 143 mango-specific microsatellite markers. Of the 143, 34 were new mango-specific microsatellite loci isolated in the course of the present investigation by constructing an (CA) _n and (TG) _n -enriched genomic library. Characterization of the 90 genotypes resulted in the detection of 301 alleles from 106 polymorphic loci with an average of 2.87 alleles per locus and polymorphism information content of 0.67. UPGMA cluster analysis grouped all the genotypes into two major groups with a genetic similarity range of 47–88 %. Grouping of the genotypes based on the utility type was observed only at sub-cluster level. Study of population structure by a model-based STRUCTURE analysis revealed the germplasm to exist in four gene pools. Overall F _st of 0.11 indicated genetic differentiation between the populations to be low. Analysis of molecular variance revealed that major proportion of the variation was within the individuals (62.25 %). The molecular marker-based study of genetic diversity suggests that the germplasm studied representing the kind of variability would be a valuable genetic resource for future breeding and association mapping in search for new and novel alleles.     

Density- and Frequency-Dependent Selection at the Mdh-2 Locus in DROSOPHILA PSEUDOOBSCURA

We have studied differences in the number of Drosophila pseudoobscura produced in a culture when the flies differ with respect to two alleles (F and S) at the Mdh-2 locus, which codes for a malate dehydrogenase enzyme. The studies were done at low and at high density in two- and three-genotype combinations (S/S, F/F and S/F), with one-genotype cultures as controls.——Density affects the fitness of the Mdh-2 genotypes. Different genotypes are differently affected, and the genotype of the competitors also makes a difference on the fitness of a given genotype. When three genotypes are present in a culture, particularly at high density, intergenotypic competition is less intense than intragenotypic competition at several frequency combinations. That is, there is "overcompensation": the three genotypes together exploit the environmental resources better than one genotype alone.—The fitness of the genotypes is frequency dependent in both two-genotype and three-genotype combinations. An inverse relationship between frequency and fitness is observed at high density. This may lead to a stable polymorphism, because the fitness of a genotype increases as its frequency decreases.—Forty independent strains, sampled from a natural population, were used in the experiments. This ensures that more than 95% of the variation present in the genome in the natural population is also present is the experimental cultures. It also ensures that the genetic background of the Mdh-2 alleles is randomized in the same way as it is in nature. However, the possibility remains that Mdh-2 alleles in nature are nonrandomly associated with alleles at closely linked loci. If linkage disequilibrium is present in the experiments because it exists in nature, then the observed effects (such as frequency-dependent selection) would affect the Mdh-2 locus in nature as well.     

Density dependent selection incorporating intraspecific competition 1. A haploid model

A haploid model is introduced and analyzed in which intraspecific competition is incorporated within a density dependent framework. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when fixed for that type. Each genotypic fitness at a single multi-allelic locus is a function of a distinctive effective population size formed by adding the numbers of each genotype present, weighted by an intraspecific competition coefficient. As a result, the fitnesses depend upon the relative frequencies of the various genotypes as well as the total population size. Intergenotypic interactions can have a profound effect upon the outcome of the population. In particular, when the density effect of one individual upon another depends upon their respective genotypes, a unique stable interior equilibrium is possible in which all alleles are present. This stands in contrast to the purely density dependent haploid system in which the only possible stable state corresponds to fixation for the type with the highest carrying capacity. In the present model selective advantage is determined by a balance between carrying capacity and sensitivity to density pressures from other genotypes. Fixation for the genotype with the highest carrying capacity, for instance, will not be stable if it exerts a sufficiently weak competitive effect upon the other genotypes. In the diallelic case, maintenance of both alleles at a stable equilibrium requires that the net intragenotypic competition between individuals of like genotype be stronger than that between unlike types. As for purely density regulated systems, there may be no stable equilibria and/or regular and chaotic cycling may occur. The results may also be interpreted in terms of a discrete time model of interspecific competition with each haplotype representing a different species.     

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.     

Genetic Loads in Heterogeneous Environments

A model of population structure in heterogeneous environments is described with attention focused on genetic variation at a single locus. The existence of equilibria at which there is no genetic load is examined.--The absolute fitness of any genotype is regarded as a function of location in the niche space and the population density at that location. It is assumed that each organism chooses to live in that habitat in which it is most fit ("optimal habitat selection").--Equilibria at which there is no segregation load ("loadless equilibria") may exist. Necessary and sufficient conditions for the existence of such equilibria are very weak. If there is a sufficient amount of dominance or area in which the alleles are selectively neutral, then there exist equilibria without segregational loads. In the N2p phase plane defined by population size, N, and gene frequency, p, these equilibria generally consist of a line segment which is parallel to the p axis. These equilibria are frequently stable.     

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.     

The Effects of Overdominance on Linkage in a Multilocus System

Computer simulations were performed with overdominant multiple alleles among tightly linked multiple loci under a multiplicative fitness model. The quantity X²/N(n — 1) was introduced as a new measure of linkage disequilibrium which, unlike previously available measures, can be applied to multiple allele models, where N is the sample size, and n is the number of alleles at the locus possessing fewest alleles. Simulations showed that (1) With multiple (three or four) alleles, the approach to stable disequilibrium is slower and the amount of disequilibrium established is weaker than in a two allele system. (2) The number of complementary chromosomes is a function of number of alleles and of population size. (3) As population size increases, the rate of the approach to stable disequilibrium is slower. (4) There is an optimum selection coefficient which minimizes the transient fixation probability of alleles when linkage is present. (5) The absence of linkage disequilibrium is in most cases not a practical method of testing the hypothesis of balancing selection of genetic polymorphisms because it depends strongly on population size in determining linkage disequilibria.     

Nicotine Biosynthetic Enzyme Activities in Nicotiana tabacum L. Genotypes with Different Alkaloid Levels

Young plants of five Nicotiana tabacum L. genotypes were examined for activity of nicotine biosynthetic enzymes. Genotypes near isogenic except at two loci each with two alleles controlling nicotine level were used in a comparison of the four homozygous allelic combinations producing high, high intermediate, low intermediate, and low nicotine levels in a “Burley 21” background. Putrescine N-methyltransferase (EC 2.1.1.53) and quinolinic acid phosphoribosyltransferase (EC 2.4.2.19) activities in root tissue of these four genotypes were proportional to leaf nicotine level, whereas N-methylputrescine oxidase activity in root tissue differed in proportion and ranking. Quinolinic acid phosphoribosyltransferase activities in leaf tissue were lower than in roots, but no differences were found among the four genotypes. The homozygous recessive alleles at either locus affect levels of all three enzyme activities examined in roots. Each locus seems to be involved in regulation of nicotine metabolism, but whether directly as a regulatory locus or indirectly through the metabolic product of a structural locus is not known.     

Hardy-Weinberg equilibria in random mating populations

The structure of multiloci random mating populations is examined. Sufficient conditions for the existence of stable local Hardy-Weinberg equilibria for n loci and an arbitrary number of alleles per locus, are then derived for specified situations under the assumption of multiplicative gene action between loci. It is shown that a stable Hardy-Weinberg equilibrium can not be a local maximum of the mean fitness function with multiplicative gene action between loci. The stability of Hardy-Weinberg type border points and the condition for the increase of newly introduced genes are topics on which some n-loci results are also obtained for an arbitrary number of alleles per locus in systems that allow Hardy-Weinberg equilibria.     

The number of equilibria in the diallelic Levene model with multiple demes

The Levene model is the simplest mathematical model to describe the evolution of gene frequencies in spatially subdivided populations. It provides insight into how locally varying selection promotes a population’s genetic diversity. Despite its simplicity, interesting problems have remained unsolved even in the diallelic case.In this paper we answer an open problem by establishing that for two alleles at one locus and J demes, up to 2J−1 polymorphic equilibria may coexist. We first present a proof for the case of stable monomorphisms and then show that the result also holds for protected alleles. These findings allow us to prove that any odd number (up to 2J−1) of equilibria is possible, before we extend the proof to even numbers. We conclude with some numerical results and show that for J>2, the proportion of parameter space affording this maximum is extremely small.     

Adaptation at Specific Loci. I. Natural Selection on Phosphoglucose Isomerase of Colias Butterflies: Biochemical and Population Aspects

Electrophoretic variants of phosphoglucose isomerase (PGI) in Colias butterflies have been studied from field and laboratory viewpoints. The transmission pattern is that of a dimeric enzyme controlled by one structural gene locus. Populations usually harbor four to six allelic mobility classes. These mobility classes are shared among species complexes, though their frequencies differ widely. Preliminary Ferguson plot analysis of the variants has been carried out. Purified preparations of Colias PGI alleles are more effective in standardizing Ferguson plots than heterologous proteins, such as ferritin. Variation of Ferguson plot parameters is not an infallible guide to electrophoretically "cryptic alleles," as one putative case proved to be due to nonallele-specific effects. S, M, and F mobility classes in two Colias semispecies show the same retardation coefficients in Ferguson plots. Adults early in the flight periods of their nonoverlapping generations show genotype frequencies in Hardy-Weinberg equilibrium, but heterozygote excess develops as the insects age. Simple directional selection and large-scale population mixing are unlikely to be causes of this, although several other selection modes remain possible. Identical-by-descent lines of the four frequent-to-common alleles in C. eurytheme have been set up in culture, and enzyme has been purified from these for study of functional properties. Major differenecs in heat stability and in various kinetic parameters are found among the ten possible genotypes. In some cases, heterosis for kinetic parameters is seen; in other cases, opposing trends in kinetic function and heat stability create potential for net heterosis in function. Possible interpretations of these results in an adaptive metabolic context are discussed, and directions for further work are stated.     

Selection in plant populations of effectively infinite size. V. Biallelic models of trioecy

A one-locus two-allele model of trioecy (presence of hermaphrodites, males and females in one population) is considered, in order to study the conditions for the persistence of this system. All possible assignments of the three sex types to the three genotypes are considered. This leads to three different modes of inheritance of trioecy, namely (a) females heterozygous, (b) males heterozygous and (c) hermaphrodites heterozygous, where in each mode each of the remaining two sex types is homozygous for one of the alleles. For mode (c) trioecy is always persistent, and the dependence of the sex ratio (for the three sex types) on the ovule and pollen fertilities and on the hermaphrodite selfing rate is specified. For the other two modes, (a) and (b), trioecy is not protected, i.e., it may not persist for any fertilities, viabilities or selfing rates. Thus, in this situation it is important to study the conditions under which the "marginal" systems of sexuality of trioecy, i.e., hermaphroditism, dioecy and gynodioecy in mode (a), and hermaphroditism, dioecy and androdioecy in mode (b), may become established. The results show that each marginal system may evolve from each other via trioecy. The evolution of dioecy is easier in mode (a) than in (b), so that female heterogamety would be expected to occur more often than male heterogamety in the present model. Under some conditions the breeding system obtained in equilibrium populations may depend on the initial genotype frequencies.—The necessity of considering modes of inheritance for sexual polymorphisms is demonstrated by comparing our results with those obtained from an evolutionary stable strategy (ESS) analysis of a purely phenotypic model.     

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.     

When will a sexual population evolve to an ESS?

A diploid, Mendelian population is considered in which m alleles at a single autosomal locus uniquely determine the phenotype of each individual. In the population, a game-theoretical conflict is supposed. If the genetic system is able to uniquely realize the phenotypic evolutionarily stable strategy (ESS) state then the sexual population will evolve to this ESS.