Selection-migration regimes characterized by a globally stable equilibrium

The principle that a subdivided population subject to overdominance viability selection in each habitat will manifest a unique, globally attractng polymorphic equilibrium is posited. This follows as a corollary to the stronger principle that, if haploid selection or submultiplicative diploid selection (definition: the geometric mean of the homozygote viabilities is less than or equal to the heterozygote viability) is operating in each habitat,there is a unique, globally attracting stable equilibrium that may be monomorphic or polymorphic. These principles are proven for a broad spectrum of migration patterns. In all such migration selection systems, multiple fixation states cannot be simultaneously stable under submultiplicative viability regimes. Contrasting examples where submultiplicative viabilities are not in force are given.     

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 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     

Migration as a factor in natural selection

Migration is defined as the movement of genes from one locality or population to another. If certain genotypes tend to migrate more than others we have migrational selection. Evidence for migrational selection is discussed. A distinction is made between the dispersion of individuals from a point giving a leptokurtic distribution of migration distances, and large scale movements of individuals from one population to another.Models for large scale unidirectional migrations where genotypes vary in viability before and after migration are cosidered. These lead to different equilibrium points before and after migration.Models for reciprocal migration between two populations are discussed. It is shown that the gene frequencies of the two populations may diverge under the influence of varying migration rates and varying viabilities of the genotypes. Furthermore, more than one stable equilibrium point may be possible for a given set of viabilities. Equilibria are shown to be possible for viabilities not implying simple overdominance. In migratory birds, where there are two annual migrations between two localities, migrational selection may lead to differing gene frequencies in the two localities.The need for data on migration in different organisms is stressed. It is suggested that man is especially suitable for migration studies.     

The Genetic Structure of Natural Populations of DROSOPHILA MELANOGASTER. Xv. Nature of Developmental Homeostasis for Viability

Developmental homeostasis of relative viability was examined for homozygotes and heterozygotes using second chromosomes from two populations of Drosophila melanogaster. One was a chromosome population in which spontaneous mutations were allowed to accumulate since it was begun with a single near-normal second chromosome. The second was a natural population approximately at equilibrium. For the estimation of relative viability, the Cy method was employed (Wallace 1956), and environmental variance between simultaneously replicated cultures was used as the index of developmental homeostasis. A new method was used in the estimation of sampling variance for relative viability that was employed for the calculation of environmental variance (error variance between simultaneously replicated cultures - sampling variance). The following findings were obtained.: (1) The difference in environmental variance between homozygotes and heterozygotes could not be seen when a chromosome population with variation due to new mutations was tested. (2) When a chromosome group isolated from an approximate equilibrium population was examined, heterozygotes manifested a smaller environmental variance than the homozygotes if their relative viabilities were approximately the same. (3) There was a slight negative correlation between viability and environmental variance, although opposite results were found when the viabilities of individuals were high, especially when overdominance (coupling overdominance, Mukai 1969 a, b) was manifest. On the basis of these findings, it was concluded that developmental homeostasis was a product of natural selection, and its mechanism was discussed.     

The Genetic Variance for Viability and Its Components in a Local Population of DROSOPHILA MELANOGASTER

Two hundred and ninety second chromosomes extracted from a natural population of Drosophila melanogaster were analyzed to estimate the genetic variance of viability and its components by means of a partial diallel cross (Design II of Comstock and Robinson 1952). The additive and dominance variances are estimated to be 0.009 and 0.0012. Using the dominance variance and the inbreeding depression, the effective number of overdominant loci contributing to the variance in viability is estimated to be very small, a dozen or less. Either the actual number of loci is small, or the distribution of viabilities is strongly skewed with a large majority of very weakly selected loci. The additive variance in viability appears to be too large to be accounted for by recurrent harmful mutants or by overdominant loci at equilibrium with various genetic parameters estimated independently. The excess might be due to frequency-dependent selection, to negative correlations between viability and fertility, or possibly to the presence of a mutator. The selection for viability and fertility, or possibly to the presence of a mutator. The selection for viability at the average polymorphic locus must be very slight, of the order of 10(-3) or less.     

Two loci with two alleles: Linkage equilibrium and linkage disequilibrium can be simultaneously stable

A class of two-locus two-allele viability matrices is exhibited which has the property that, for a large range of recombination values, both linkage equilibrium and linkage disequilibrium are stable. The model is specified by five viabilities; the classical schemes previously analyzed involve at most four selection parameters.     

EQUILIBRIUM ANALYSIS OF SEXUAL SELECTION IN DROSOPHILA MELANOGASTER

Several models for sexual selection, both by male-male competition and female choice, predict that a character which covaries with mating success should be near an equilibrium where the intensity of sexual selection opposes viability selection. This prediction was used to design experiments for estimating the intensity of sexual and viability selection on wing length in a recently captured population of Drosophila melanogaster. Observations of matings by males color-marked for wing length indicated that the standardized sexual selection differential on wing length was 0.24 under a wide range of effective sex ratios. After estimating the heritability of wing length to be 0.62, the expected standardized response due to sexual selection was calculated as 0.15 (SE = 0.15). The response due to viability selection was then estimated by comparing wing lengths of progeny of flies that had been randomly mated, thereby preventing sexual selection, with progeny of flies that had been allowed to acquire mates in a mass-mating chamber. The results support an equilibrium model in that the standardized response due to viability selection (?0.31, SE = 0.08) was opposite in sign and similar in magnitude to the estimated response due to sexual selection. Observations of females orienting in front of males which differed in wing length indicated that the mating advantage accruing to long-winged males was not due to female choice. Instead, male-male competition in which the larger of two randomly chosen males succeeded in mating, explains the observed sexual selection. An experimental analysis of genotype-environment interaction revealed that larval density had a nonlinear effect on mean wing length within sibships. If a population is displaced from equilibrium, therefore, the evolutionary trajectory of mean wing length will depend both on the intensity of selection and the environment in which that selection is operating.     

On the evolution of genetic incompatibility systems. V. Origin of sporophytic self-incompatibility in response to overdominance in viability

Conditions for the origin of partial sporophytic self-incompatibility (SSI) are obtained from two quantitative models, which differ with respect to the determination of offspring viability. Offspring viability depends solely on the source (self or nonself) of the fertilizing pollen in the first model, which describes changes only at a primitive S-locus itself. Two loci evolve in the second model: overdominant viability selection maintains an arbitrary number of alleles at one locus, with SSI under the control of a separate locus. In both cases, the origin of SSI requires that the relative change in the numbers of offspring derived by the two reproductive modes compensate for the twofold cost of outcrossing. In the first model studied, the viability of inbred offspring fully determines the relative change in the numbers of inbred and outbred offspring produced. In the second model, the relative change in offspring numbers depends in addition on associations between the S-locus and the viability locus. Because these two-locus associations are comparable in magnitude to the differences between the viabilities of inbred and outbred offspring, SSI can arise under less restrictive conditions than expected from the one-locus model. Greater allelic multiplicity at the viability locus facilitates the origin of SSI by reducing the relative viability of inbred offspring. Tight linkage between the S-locus and the viability locus and high rates of receipt of self-pollen promote the generation and maintenance of associations between the S-locus and the viability locus. In populations in which more than two viability alleles are maintained, the active S-allele can invade even in the absence of linkage with the viability locus. The present study establishes that incompatibility systems can arise in response to identity disequilibrium between a modifier of incompatibility and a locus subject to overdominant viability selection; in particular, compensation for the twofold cost of outcrossing does not require preexisting gametic level disequilibria.     

Uniqueness of polymorphic equilibria under soft selection

Any two allele polymorphic equilibrium of a subdivided haploid population subject to soft selection is stable. This provides that for a two allele system in a subdivided haploid population there is a globally attracting equilibrium which is polymorphic if a polymorphic equilibrium exists, otherwise monomorphic. These results extend to diploid populations if within each habitat the heterozygote viability is greater than or equal to the geometric mean of the homozygote viabilities.     

Rates of decay of linkage disequilibrium under two-locus models of selection

The effect of selection and linkage on the decay of linkage disequilibrium, D, is investigated for a hierarchy of two-locus models. The method of analysis rests upon a qualitative classification of the dynamic of D under selection relative to the neutral dynamic. To eliminate the confounding effects of gene frequency change, the behavior of D is first studied with gene frequencies fixed at their invariant values. Second, the results are extended to certain special situations where gene frequencies are changing simultaneously.A wide variety of selection regimes can cause an acceleration of the rate of decay of D relative to the neutral rate. Specifically, the asymptotic rate of decay is always faster than the neutral rate in the neighborhood of a stable equilibrium point, when viabilities are additive or only one locus is selected. This is not necessarily the case for models in which there is nonzero additive epistasis. With multiplicative viabilities, decay is always accelerated near a stable boundary equilibrium, but decay is only faster near the stable central equilibrium (with = 0) if linkage is sufficiently loose. In the symmetric viability model, decay may even be retarded near a stable boundary equilibrium. Decay is only accelerated near a stable corner equilibrium when the double homozygote is more fit than the double heterozygotes. Decay near a stable edge equilibrium may be retarded if there is loose linkage. With symmetric viabilities there is usually an acceleration of the decay process for gene frequencies near 1/2 when the central equilibrium (with = 0) is stable. This is always the case when the sign of the epistasis is negative or zero.Conversely, the decay ofD is retarded in the neighborhood of a stable equilibrium in the multiplicative and symmetric viability models if any of the conditions above are violated. Near an unstable equilibrium of any of the models considered,D may either increase or decay at a rate slower than, equal to, or faster than the neutral rate. These analytic results are supplemented by numerical studies of the symmetric viability model.     

The Maintenance of Single-Locus Polymorphism. IV. Models with Mutation from Existing Alleles

The ability of viability selection to maintain allelic polymorphism is investigated using a constructionist approach. In extensions to the models we have previously proposed, a population is bombarded with a series of mutations whose fitnesses in conjunction with other alleles are functions of the corresponding fitnesses with a particular allele, the parent allele, already in the population. Allele frequencies are iterated simultaneously, thus allowing alleles to be driven to extinction by selection. Such models allow very high levels of polymorphism to evolve: up to 38 alleles in one case. Alleles that are lethal as homozygotes can evolve to surprisingly high frequencies. The joint evolution of allele frequencies and viabilities highlights the necessity to consider more than the current morphology of a population. Comparisons are made with the neutral theory of evolution and it is suggested that failure to reject neutrality using the Ewens-Watterson test cannot be regarded as evidence for the neutral theory.     

Some circumstances assuring monomorphism in subdivided populations

It is well known that in a subdivided population subject to soft selection with two alleles at one locus, instability of both fixation states (a “protected polymorphism”) entails at least one stable polymorphic equilibrium. Although stable polymorphic and monomorphic equilibria can coexist in general, a stable fixation state (monomorphic equilibrium) precludes the existence of any polymorphic equilibrium under the circumstances of haploid or submultiplicative diploid viabilities. This provides that a stable monomorphism is robust against random fluctuations in allele frequencies. It also increases the known circumstances where there is a unique globally attracting stable equilibrium, i.e., where allele frequencies are determined by the selection-migration structure independent of the history of the system.     

Opposing levels of selection can cause neutrality: mating patterns and maternal-fetal interactions

Abstract.— A biallelic viability model based on human data for maternal-fetal interactions reported by Hedrick (1997) gives the interesting result of neutral stability at all gene frequencies. I show that there are two levels of selection, within and among families, acting in opposing directions in this model and that the neutral stability occurs when the two levels of selection exactly balance one another, as they do in a randomly mating population. Deviations from random mating disrupt the balance and consequently destroy the neutral stability. However, with inbreeding avoidance, which characterizes the human histocompatibility loci, within-family selection is strengthened and among-family selection is weakened. This favors the invasion of new alleles and contributes to a high equilibrium level of genetic diversity at loci with maternal-fetal interactions affecting offspring viability in the pattern described by Hedrick. This pattern of selection is remarkably similar to that observed for the maternal effect selfish genes, Medea in flour beetles and scat in the mouse, and the Gp -9 gene in the fire ant.     

The Genetic Structure of Natural Populations of Drosophila melanogaster. Xx. Comparison of Genotype-Environment Interaction in Viability between a Northern and a Southern Population

In order to examine the operation of diversifying selection as the maintenance mechanism of excessive additive genetic variance for viability in southern populations in comparison with northern populations of Drosophila melanogaster, two sets of experiments were conducted using second chromosomes extracted from the Ogasawara population (a southern population in Japan) and from the Aomori population (a northern population in Japan). Chromosomal homozygote and heterozygote viabilities were estimated in eight kinds of artificially produced breeding environments. The main findings in the present investigation are as follows: (1) Significant genotype-environment interaction was observed using chromosomes extracted from the Ogasawara population. Indeed, the estimate of the genotype-environment interaction variance for heterozygotes was significantly larger than that of the genotypic variance. On the other hand, when chromosomes sampled from the Aomori population were examined, that interaction variance was significant only for homozygotes and its value was no more than one quarter of that for the chromosomes from the Ogasawara population. (2) The average genetic correlation between any two viabilities of the same lines estimated in the eight kinds of breeding environments for the chromosomes sampled from the Ogasawara population was smaller than that for the chromosomes from the Aomori population both in homozygotes and in heterozygotes, especially in the latter. (3) The stability of heterozygotes over homozygotes against fluctuations of environmental conditions was seen in the chromosomes from the Ogasawara population, but not from the Aomori population. (4) From the excessive genotype-environment interaction variance compared with the genotypic variance in heterozygotes, it was suggested for the chromosomes from the Ogasawara population that the reversal of viability order between homozygotes took place in some environments at the locus level. On the basis of these findings, it is strongly suggested that diversifying selection is operating in a southern population of D. melanogaster on some of the viability polygenes which are probably located outside the structural loci, and the excessive additive genetic variance of viability in southern populations is maintained by this type of selection.     

Hard selection in haploid species

The formulation of hard selection is reviwed in the context of haploid viabilities and the criteria for stability of the fixation states are given. In contrast to soft selection, both fixation states can be simultaneously stable. However, this is not possible if the migration matrix is positive definite. In the case of only two demes there is at most one polymorphic equilibrium as occurs under soft selection, but the internal equilibrium may be unstable in contrast to the soft selection case. The question of hard versus soft protection is posed in the context of haploid viabilities and the principle hard protection implies soft protection holds with a similar degree of generality as in the diploid case.     

Multilocus models of sympatric speciation: Bush versus Rice versus Felsenstein

Abstract. In populations of phytophagous insects that use the host plant as a rendezvous for mating, divergence in host preference could lead to sympatric speciation. Speciation requires the elimination of "generalist" genotypes, that is, those with intermediate host preference. This could occur because such genotypes have an inherent fitness disadvantage, or because preference alleles become associated with alleles that are oppositely selected on the two hosts. Although the former mechanism has been shown to be plausible, the latter mechanism has not been studied in detail. I consider a multilocus model (the "Bush model") in which one set of biallelic loci affects host preference, and a second set affects viability on the hosts once chosen. Alleles that increase viability on one host decrease viability on the other, and all loci are assumed to be unlinked. With moderately strong selection on the viability loci, preference alleles rapidly become associated with viability alleles, and the population splits into two reproductively isolated host specialist populations. The conditions for speciation to occur in this model, as measured by the strength of selection required, are somewhat more stringent than in a model in which preference and viability are controlled by the same loci (one-trait model). In contrast, the conditions are much less stringent than in a model in which speciation requires buildup of associations between viability loci and loci controlling a host-independent assortative mating trait (canonical two-trait model). Moreover, in the one-trait model, and to a lesser extent the Bush model, the strength of selection needed to initiate speciation is only slightly greater than that needed to complete it. This indicates that documenting instances of sympatric species that are reproductively isolated only by host or habitat preference would provide evidence for the plausibility of sympatric speciation in nature.     

On the evolutionary stability of Mendelian segregation

We present a model of a primary locus subject to viability selection and an unlinked locus that causes sex-specific modification of the segregation ratio at the primary locus. If there is a balanced polymorphism at the primary locus, a population undergoing Mendelian segregation can be invaded by modifier alleles that cause sex-specific biases in the segregation ratio. Even though this effect is particularly strong if reciprocal heterozygotes at the primary locus have distinct viabilities, as might occur with genomic imprinting, it also applies if reciprocal heterozygotes have equal viabilities. The expected outcome of the evolution of sex-specific segregation distorters is all-and-none segregation schemes in which one allele at the primary locus undergoes complete drive in spermatogenesis and the other allele undergoes complete drive in oogenesis. All-and-none segregation results in a population in which all individuals are maximally fit heterozygotes. Unlinked modifiers that alter the segregation ratio are unable to invade such a population. These results raise questions about the reasons for the ubiquity of Mendelian segregation.     

Inbreeding depression due to overdominance in partially self-fertilizing plant populations

The effect of the rate of partial self-fertilization and viability selection on the magnitude of inbreeding depression was investigated for the overdominance genetic model. The influence of these factors was determined for populations with equilibrium genotypic frequencies. Inbreeding depression was measured as the normalized disadvantage in mean viability of selfed progeny as compared to outcrossed progeny. When caused by symmetric homozygous disadvantage at a single locus it is shown always to be less than one-third. Moreover, for fixed rates of self-fertilization, its maximum value is found at intermediate levels of homozygous disadvantage. As the rate of self-fertilization increases, inbreeding depression increases and the homozygote viability that results in maximum depression tends toward one-half the heterozygote viability. Symmetric selection against homozygotes at multiple loci can lead to substantially higher values than selection at a single-locus. As the number of independent loci involved increases, inbreeding depression can reach high levels even though the selfing rate is low. Viability distributions for progenies produced from both random mating and self-fertilization were derived for the case of symmetric selection at independently assorting multiple loci. Distributions of viabilities in progenies resulting from mixtures of selfing and outcrossing were shown to be bimodal when inbreeding depression is high.