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.     

A competitive selection model

A multi-locus model for complete positive assortative mating is discussed. For a two-locus model, if the gene frequencies for the two loci are different, as they are likely to be, it is shown that in equilibrium the population is not composed of only two homozygous types, as is usually thought. The limiting distribution will have three homozygous genotypes depending upon the initial gene frequencies. If there are m-loci such that gene frequencies at all loci are different, there will be (m+1) such homozygous genotypes present in the equilibrium population, one in each phenotypic group.     

Complex patterns of mating revealed in a Eucalyptus regnans seed orchard using allozyme markers and the neighbourhood model

The neighbourhood model apportions offspring of individual mother plants to self-fertilization, outcrossing to males within a circumscribed area around the mother plant (the neighbourhood), and outcrossing to males outside the neighbourhood. Formerly the model was applied only to haploid pollen gametes in the offspring of conifers, but is extended so that it can be used with genotypic data from diploid offspring of both angiosperms and gymnosperms. In addition, it is shown that the mating parameters can be estimated without independent estimates of allele frequencies in the pollen pools outside the neighbourhood; thus the model might be applied effectively to natural populations exposed to unknown external pollen sources. Parameters of the neighbourhood mating model were estimated for a 10-year-old seed orchard population of the insect-pollinated tree, Eucalyptus regnans, in southeast Australia, which contained a mixture of two geographical provenances (Victoria and Tasmania). The mating patterns revealed were complex. Crosses between trees of the same provenance occurred three times more often than crosses between trees of different provenances. Levels of self-fertilization and patterns of mating within neighbourhoods were influenced by provenance origin, crop fecundity and orchard position (central vs. edge) of mother trees. Gene dispersal, however, was extensive, with approximately 50% of effective pollen gametes coming from males more than 40 m away from mother trees (average distance between neighbouring trees was 7.4 m). Thus, insect pollinators are efficient promoters of cross-fertilization in this orchard, with the result that the effective number of males mating with each female is large.     

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.     

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.     

The genetic mating structure of subdivided populations I. Open-mating model

The genetic mating structure of a subdivided population can describe how parental genotypes gave rise to zygotes. When parents of the same genotype are considered together as one class (“open-mating”), three independent parameters of inbreeding and mating structure are needed to describe this structure at a diallelic locus. One is Wright's fixation index F. The other two are mating structure parameters, derived herein and termed the “effective selfing” rate E and the “inbreeding assortative selfing” rate D. E is the genetically equivalent proportion of self-fertilization at a single locus, and is given by standardized second and third central moments of gene frequencies of mates. E is a summary measure of inbreeding that includes effects due to self-fertilization and mating to relatives, as well as correlations between mates induced by Wahlund effects and/or selective diversification among neighborhoods. The second parameter D measures the tendency of inbred or more homozygous individuals to effectively self more (or less) than outbred or more heterozygous individuals. D is related to the maintenance of variation of inbreeding among individuals and/or to the prevalence of spatial variation of selection. D is independent of E, but together with E controls the generational change of inbreeding, ΔF. Extensions of the model to unequal allele frequencies in male vs female mates, and to multi-allelic loci, are also examined.     

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.     

Host mating system and the prevalence of disease in a plant population

A modified susceptible-infected-recovered (SIR) host-pathogen model is used to determine the influence of plant mating system on the outcome of a host-pathogen interaction. Unlike previous models describing how interactions between mating system and pathogen infection affect individual fitness, this model considers the potential consequences of varying mating systems on the prevalence of resistance alleles and disease within the population. If a single allele for disease resistance is sufficient to confer complete resistance in an individual and if both homozygote and heterozygote resistant individuals have the same mean birth and death rates, then, for any parameter set, the selfing rate does not affect the proportions of resistant, susceptible or infected individuals at equilibrium. If homozygote and heterozygote individual birth rates differ, however, the mating system can make a difference in these proportions. In that case, depending on other parameters, increased selfing can either increase or decrease the rate of infection in the population. Results from this model also predict higher frequencies of resistance alleles in predominantly selfing compared to predominantly outcrossing populations for most model conditions. In populations that have higher selfing rates, the resistance alleles are concentrated in homozygotes, whereas in more outcrossing populations, there are more resistant heterozygotes.     

Selection with two alleles and complete dominance

For a plant selection model with frequency-independent viabilities, fertilities and selfing rates, it is shown that apart from global fixation, for certain parameter combinations a protected polymorphism and facultative fixation (either allele may become fixed according to initial frequencies) may both occur. Facultative fixation requires different selling rates for the dominant and recessive type. Protection of the polymorphism requires resource allocation for male and female function. In this connection the problem of purely genetically caused population extinction is discussed.
For general frequency dependence and regular segregation, the chances for establishment of a completely recessive gene are compared to those of a completely dominant gene. It is proven that the process of establishment of the recessive gene, despite a fitness advantage, may be considerably endangered by drift effects if random mating prevails. The recessive gene may reach the same effectivity in establishment as a dominant gene, only if the recessive homozygote mates exclusively with its own type during the period of establishment.     

Linkage disequilibrium, gene trees and selfing: an ancestral recombination graph with partial self-fertilization

It is shown that partial self-fertilization can be introduced into neutral population genetic models with recombination as a simple change in the scaling of the parameters. This means that statistical and computational methods that have been developed under the assumption of random mating can be used without modification, provided the appropriate parameter changes are made. An important prediction is that all forms of linkage disequilibrium will be more extensive in selfing species. The implications of this are discussed.     

Associative overdominance: Evaluating the effects of inbreeding and linkage disequilibrium

Expressions are obtained for the expected phenotypic values of homozygous and heterozygous genotypes for a neutral marker locus linked to a locus segregating for a recessive deleterious gene. The phenotypic values are functions of the allele frequencies at the marker locus, the inbreeding coefficient and the degree of association of the deleterious gene with the marker alleles. The analysis is extended to more than two alleles at the marker locus. Either linkage disequilibrium or inbreeding alone can produce an apparent superiority of heterozygotes for the marker locus (unless specified otherwise, the terms ‘homozygote’ and ‘heterozygote’ will refer to the marker locus). The effect of linkage disequilibrium on the difference between the heterozygote and homozygote values can be positive (associative overdominance) or negative (associative underdominance), depending on the frequencies of the marker alleles and the degree of their association with the deleterious gene. Inbreeding has always a positive effect. In general, the expected value of a homozygote is a positive function of its allele frequency. When the various homozygous genotypes are combined into one class and the various heterozygous genotypes into another, the phenotypic difference of the two classes is a function of the evenness of the allelic frequency distribution. Inbreeding is a more likely explanation of associative overdominance if the frequency of the deleterious gene is low, but its effect on the character high. Conversely, linkage disequilibrium is more likely if the frequency is high and the effect low. The degrees of association between marker alleles and the deleterious gene can, in principle, be estimated from the observed phenotypic scores and used to calculate expected multi-locus genotype scores. This could provide the basis for statistical tests of the associative overdominance hypothesis as an explanation of observed correlations between multi-locus heterozygosity and phenotypic traits.     

Predominance of outcrossing in Lymnaea stagnalis despite low apparent fitness costs of self-fertilization

We have quantified the natural mating system in eight populations of the simultaneously hermaphroditic aquatic snail Lymnaea stagnalis, and studied the ecological and genetic forces that may be directing mating system evolution in this species. We investigated whether the natural mating system can be explained by the availability of mates, by the differential survival of self- and cross-fertilized snails in nature, and by the effects of mating system on parental fecundity and early survival. The natural mating system of L. stagnalis was found to be predominantly cross-fertilizing. Density of snails in the populations had no relationship with the mating system, suggesting that outcrossing rates are not limited by mate availability at the population densities observed. Contrary to expectations for outcrossing species, we detected no evidence for inbreeding depression in survival in nature with inferential population genetic methods. Further, experimental manipulations of mating system in the laboratory revealed that self-fertilization had no effect on parental fecundity, and only minor effects on offspring survival. Predominance of cross-fertilization despite low apparent fitness costs of self-fertilization is at odds with the paradigm that high self-fertilization depression is necessary for maintenance of cross-fertilization in self-compatible hermaphrodites.     

Continuous selective models

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

Evolutionary syndromes linking dispersal and mating system: the effect of autocorrelation in pollination conditions

Self-fertilization is classically thought to be associated with propagule dispersal because self-fertilization is a boon to colonizers entering environments devoid of pollinators or potential mates. Yet, it has been theoretically shown that random fluctuations in pollination conditions select for the opposite association of traits. In nature, however, various ecological factors may deviate from random variations, and thus create temporal correlation in pollination conditions. Here, we develop a model to assess the effects of pollination condition autocorrelation on the joint evolution of dispersal and self-fertilization. Basically, two syndromes are found: dispersing outcrossers and nondispersing (partial) selfers. Importantly, (1) selfers are never associated with dispersal, whereas complete outcrossers are, and (2) the disperser/outcrosser syndrome is favored (resp. disfavored) by negative (resp. positive) autocorrelation in pollination conditions. Our results suggest that observed dispersal/mating system syndromes may depend heavily on the regime of pollination condition fluctuations. We also point out potential negative evolutionary effects of anthropic management of the environment on outcrossing species.     

Estimation of mating system parameters in plant populations using marker loci with null alleles

Summary An Expectation-Maximization (EM)-algorithm procedure is presented that extends Cheliak et al. (1983) method of maximum-likelihood estimation of mating system parameters of mixed mating system models. The extension permits the estimation of the rate of self-fertilization (s) and allele frequencies (Pi) at loci in outcrossing pollen, at marker loci having recessive null alleles. The algorithm makes use of maternal and filial genotypic arrays obtained by the electrophoretic analysis of cohorts of progeny. The genotypes of maternal plants must be known. Explicit equations are given for cases when the genotype of the maternal gamete inherited by a seed can (gymnosperms) or cannot (angiosperms) be determined. The procedure can accommodate any number of codominant alleles, but only one recessive null allele at each locus. An example, using actual data from Pinus banksiana, is presented to illustrate the application of this EM algorithm to the estimation of mating system parameters using marker loci having both codominant and recessive alleles.Issued as AECL-8745     

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 plant populations of effectively infinite size VI. Overlapping generations

A selection model for iteroparous, monoecious, or hermaphroditic plant populations is considered which encompasses viabilities, pollen fertilities, ovule fertilities, and rates of self-fertilization which may arbitrarily depend on both age and genotype. The general conditions for establishment (which are also those for protectedness) of an allele are derived. The classical conjecture that the conditions of protectedness are the same for separated and overlapping generations if the intrinsic rates of increase are applied is discussed. For this purpose it is necessary to introduce two new intrinsic values: the intrinsic rate of self-fertilization and the intrinsic pollen-to-ovule ratio. The significance of the intrinsic values is demonstrated for complete self-fertilization, selection restricted to differential, partial self-fertilization, and sexual asymmetry (absence of proportionality between pollen and ovule production), including selection restricted to one sex. With the exception of asymmetric selection in both sexes, it turns out that the intrinsic values suffice to state the conditions for protectedness, but more information about the life histories is required to determine the exact speed of establishment. For asymmetric selection in both sexes, the concept of intrinsic value is inadequate for investigating the problem of establishment and thus the evolution of life histories. Since sexual asymmetry is rather the rule than the exception and selfing is common in plants, the consequences for finding optimal life histories are outlined.     

Sex differences in linkage between autosomal loci

In the case of conventional selection theory with multiplicative gene action between loci and no sex differences except in crossover frequencies, it is shown that the usual conditions for stability hold when the mean of the recombination frequencies for the two sexes is used. For additive gene action between loci, it is shown that, after one generation of random mating, the gene frequencies of male and female origin are the same. This equality implies the nondecreasing property of the mean fitness function. Some attention is also given to neutral loci.     

Aneuploidy and inbreeding depression in random mating and self-fertilizing autotetraploid populations

Summary Almost all autotetraploids produce aneuploid progeny because of irregularities at meiosis. Aneuploid plants produce high frequencies of aneuploids. If it were not for selection against aneuploid gametes and sporophytes the amount of aneuploidy would increase every generation. Most experimental and theoretical studies on population genetics and heterosis in autotetraploids have neglected aneuploidy as a factor. To take aneuploidy into account experimentally requires the cytological identification of all chromosomes and to consider it theoretically requires a huge amount of computations. Consequently, microcomputer programs have been devised to show the effects of random mating and self-fertilization in autotetraploid populations. According to the model aneuploidy rapidly increases in randomly mated and self-fertilized autotetraploid populations until they achieve an equilibrium where the amount of aneuploidy introduced into the population is balanced by the amount of aneuploidy removed from the population by selection. The model suggests that self-fertilized populations have greater frequencies of aneuploid gametes and zygotes than do randomly mated populations and therefore aneuploidy may be a significant cause of the great inbreeding depressions found in autotetraploids.Contribution from the Missouri Agricultural Experiment Station. Journal Series No. 9998