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.     

Establishment of allelic two-locus polymorphisms

An exact analysis of necessary and sufficient conditions for the establishment and protectedness of biallelic two-locus polymorphisms is developed for the classical model with constant, sexually symmetric fitnesses and random association of the successful gametes. To demonstrate application of the results to common model types, the model of symmetric viabilities depending on the degree of heterozygosity only is chosen as a paradigm. It is pointed out that a unique locally stable internal equilibrium may exist even though all marginal equilibria (including the fixation states) are locally attractive. This example is quoted as an indication of the priority that analyses of protectedness deserve over analyses of local stability or instability of internal equilibria. Further applications of broader appeal concern the role that recombination plays in protecting polymorphisms. Probably the most interesting finding is that with increasing recombination frequency the chances for protectedness of a polymorphism generally decline. Yet, if a certain hierarchic ordering of the fitnesses with respect to the degree of heterozygosity is realized, the polymorphism is protected for arbitrary amounts of recombination. If recombination is rare, heterozygote advantage is not a universal precondition for persistence of polymorphisms. This phenomenon is utilized to derive conditions under which deleterious recessive mutants can be maintained in a population.     

Conditions for protected polymorphism in subdivided plant populations: 1. Uniform pollen dispersal

Pollen and seed migration patterns are not the same in most plant populations, and the differences affect conditions for protection of alleles. We analyzed conditions for protectedness when pollen is freely exchanged along all demes, while seeds are deposited within the female parents' deme. Protectedness was analyzed at the boundary of fixation and necessary conditions were derived.If no selection among female genotypes exists, then simple average heterozygote superiority in the males can guarantee protection. However, regardless of the form of selection in females, simply doubling the male heterozygote superiority can still guarantee protectedness. Conditions for guaranteeing protectedness with female selection were also derived but are more complicated.The effect of inter-demic variability on protectedness of a biallelic polymorphim is studied for a particular method of reducing the variances of the selection values. It is shown that decreasing the variance of the female selection values also decreases protectedness. This is not necessarily true for the male selection values.     

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.     

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.     

Selection in plant populations of effectively infinite size II. Protectedness of a biallelic polymorphism

The biologically important problem of protectedness of genetic polymorphisms in monoecious plant populations exhibiting genotypically determined variation in rates of self-fertilization and sexually asymmetrical fertilities has hitherto escaped exact, analytical treatment for the reason that appropriate mathematical techniques relying on allelic frequencies do not seem to exist. For the particular case of one locus and two alleles it was possible to develop such a technique which provides conditions of high precision for protectedness of an allele. A comparison of the results with those already known from models that appear to be specializations of the present model showed that some of the earlier conclusions can be generalized, while others have to be handled with great care or should even be rejected. Above all, this concerns the role of self-fertilization, which is frequently considered to counteract the establishment of genetic polymorphisms. However, it turned out that increasing the heterozygote selfing rate also increases protectedness for both alleles in all situations. Moreover, even if the amount of self-fertilization is the same for all genotypes, asymmetry in the production of ovules and pollen, which is more the rule than an exception, may imply protectedness only for comparatively large selfing rates. The probably most outstanding finding is that, depending on the ovule and pollen fertilities, protectedness may be realized only within small ranges of selfing rates, and these ranges may vary from arbitrarily low to arbitrarily high rates. On the other hand, if the ovule fertilities show strong overdominance for the heterozygote—more precisely, if the heterozygote produces more than twice as many ovules as either of the homozygotes—both alleles are protected irrespective of the pollen fertilities and rates of self-fertilization; this generalizes earlier results obtained for more specific models.     

The nucleo-mitochondrial conflict in cytoplasmic male sterilities revisited

Cytoplasmic male sterility (CMS) in plants is a classical example of genomic conflict, opposing maternally-inherited cytoplasmic genes (mitochondrial genes in most cases), which induce male sterility, and nuclear genes, which restore male fertility. In natural populations, this type of sex control leads to gynodioecy, that is, the co-occurrence of female and hermaphroditic individuals within a population. According to theoretical models, two conditions may maintain male sterility in a natural population: (1) female advantage (female plants are reproductively more successful than hermaphrodites on account of their global seed production); (2) the counter-selection of nuclear fertility restorers when the corresponding male-sterility-inducing cytoplasm is lacking. In this review, we re-examine the model of nuclear-mitochondrial conflict in the light of recent experimental results from naturally occurring CMS, alloplasmic CMS (appearing after interspecific crosses resulting from the association of nuclear and cytoplasmic genomes from different species), and CMS plants obtained in the laboratory and carrying mitochondrial mutations. We raise new hypotheses and discuss experimental models that would take physiological interactions between cytoplasmic and nuclear genomes into account.     

Effective population sizes for cytoplasmic and nuclear genes in a gynodioecious species. The role of the sex determination system

Equations are derived for the effective sizes of gynodioecious populations with respect to both nuclear and cytoplasmic genes (N(ec) and N(en), respectively). Compared to hermaphroditism, gynodioecy generally reduces effective population sizes for both kinds of loci to an extent depending on the frequency of females, the sex determination system, and the selfing rate of hermaphrodites. This reduction is due to fitness differences between the sexes and is highly influenced by the mode of inheritance of this fitness. In absence of selfing, nuclear gynodioecy results in a reduction of N(ec) that depends strongly on the dominance of male sterility alleles, while N(en) remains equal to the census number (N). With cytonuclear gynodioecy, both cytoplasmic and nuclear effective sizes are reduced, and at the extreme, dioecy results in the minimum N(ec) values and either minimum or maximum N(en) values (for low or high frequency of females, respectively). When selfing occurs, gynodioecy either increases or decreases N(en) as compared to hermaphroditism with the same selfing rate of hermaphrodites. Unexpectedly, N(ec) also varies with the selfing rate. Thus the genetic sex-determination system appears as a major factor for the nuclear and cytoplasmic genetic diversities of gynodioecious species.     

Mathematical Study of the Evolution of Gynodioecy with Cytoplasmic Inheritance under the Effect of a Nuclear Restorer Gene

A study is described of the influence of the introduction of a dominant nuclear restorer gene into a cytoplasmic gynodioecious plant population. This study includes the consideration of separate effects on the relative female fertility of nuclear, cytoplasmic and sex (phenotypic) factors. Under these assumptions, the introduction of a dominant nuclear restorer gene into a cytoplasmic gynodioecious population can lead to several different situations: persistence of cytoplasmic gynodioecy, appearance of a nuclear-cytoplasmic gynodioecy, appearance of a nuclear gynodioecy or complete restoration of male fertility. The development of stable nuclear-cytoplasmic gynodioecy in a mathematical model is new and is possible because of the consideration of the separate relative female fertilities. The possibility of a transformation of cytoplasmic gynodioecy into a nuclear one has never been obtained before. It could constitute a route for the appearance of this latter kind of gynodioecy in plant populations. Finally, the possibilities of evolution of gynodioecy from one kind to the other, and towards dioecy, are discussed, as are some theoretical schemes that seem to correspond to observed actual situations.     

DO RECENT FINDINGS IN PLANT MITOCHONDRIAL MOLECULAR AND POPULATION GENETICS HAVE IMPLICATIONS FOR THE STUDY OF GYNODIOECY AND CYTONUCLEAR CONFLICT?

The coexistence of females and hermaphrodites in plant populations, or gynodioecy, is a puzzle recognized by Darwin. Correns identified cytoplasmic inheritance of one component of sex expression, now known as cytoplasmic male sterility (CMS). Lewis established cytonuclear inheritance of gynodioecy as an example of genetic conflict. Although biologists have since developed an understanding of the mechanisms allowing the joint maintenance of CMS and nuclear male fertility restorer genes, puzzles remain concerning the inheritance of sex expression and mechanisms governing the origination of CMS. Much of the theory of gynodioecy rests on the assumption of maternal inheritance of the mitochondrial genome. Here we review recent studies of the genetics of plant mitochondria, and their implications for the evolution and transmission of CMS. New studies of intragenomic recombination provide a plausible origin for the chimeric ORFs that characterize CMS. Moreover, evidence suggests that nonmaternal inheritance of mitochondria may be more common than once believed. These findings may have consequences for the maintenance of cytonuclear polymorphism, mitochondrial recombination, generation of gynomonoecious phenotypes, and interpretation of experimental crosses. Finally we point out that CMS can alter the nature of the cytonuclear conflict that may have originally selected for uniparental inheritance.     

Polymorphisms for purely cytoplasmically inherited traits in bisexual plants

It is shown that cytoplasm polymorphisms transmitted only by the ovules can be maintained without gene-cytoplasmic interactions. The necessary prerequisites are asymmetry of the plasmotypes in production of ovules and pollen (sexual asymmetry), incomplete and frequency-dependent fertilization efficiency and differential selfing rates. These factors can generate the negative frequency dependence of cytoplasmic fitnesses required for a stable polymorphism. The model considered allows also for facultative fixation of either of two plasmotypes and, thus, may produce all of the dynamical characteristics known for nuclear selection with two alleles at one locus.

Strong sexual asymmetry, which probably occurs frequently in bisexual plants, may facilitate stable cytoplasmic polymorphisms. However, these polymorphisms may also endanger survival of the whole population in the absence of nuclear interactions. Gene-cytoplasmic interactions avoid this risk and, at the same time, utilize the advantages of sexual asymmetry in maintaining genetic polymorphisms.

     

The maintenance of gynodioecy and androdioecy in angiosperms

Algebraic models of gynodioecy show that the effects on the equilibrium sex ratio of the relative survival and seed production of the sexes and of inbreeding of male-fertile plants are identical for all genic modes of inheritance, provided that different genotypes among male-fertile plants (or among females) do not differ in average fitness. The effects of three modes of inbreeding on equilibrium sex ratios are examined. If there is competition between self- and cross-fertilization of male-fertile individuals, a stable sexual dimorphism can be maintained by an outbreeding advantage of females if both the proportion of cross-fertilized seeds among those borne on male-fertile individuals,t, and the inbreeding depression (fitness inbred/outbred seeds),i, are less than one half. A lower frequency of females is obtained for the same values oft andi if self-fertilization precedes cross-fertilization. If self-fertilization follows cross-fertilization, gynodioecy cannot be maintained by an outbreeding advantage of females. When the sex phenotypes of gynodioecious populations are determined by cytoplasmic inheritance, females need only a slight advantage over males in survival, ovule production or outbreeding to persist at equilibrium. When determined by nuclear genes, androdioecy can be maintained by greater fecundity or a higher survival rate of males than of female-fertile plants, but not by an outbreeding advantage. Androdioecy cannot be maintained with cytoplasmic inheritance of sex. The models suggest explanations for the more frequent occurrence of gynodioecy than of andrdioecy, the high frequency of gynodioecy in Hawaii and New Zealand, and the origin of gynodioecy from hermaphrodite but not from monoecious ancestors.     

Sex inheritance in gynodioecious species: a polygenic view

Gynodioecy is defined as the coexistence of two different sexual morphs in a population: females and hermaphrodites. This breeding system is found among many different families of angiosperms and is usually under nucleo-cytoplasmic inheritance, with maternally inherited genes causing male sterility and nuclear factors restoring male fertility. Numerous theoretical models have investigated the conditions for the stable coexistence of females and hermaphrodites. To date, all models rest on the assumption that restoration of a given male sterile genotype is controlled by a single Mendelian factor. Here, we review data bearing on the genetic determinism of sex inheritance in three gynodiecious plant species. We suggest that restoration of male fertility is probably best viewed as a quantitative trait controlled by many loci. We develop a threshold model that accommodates an underlying polygenic trait, which is resolved at the phenotypic level in discrete sexual morphs. We use this model to reanalyse data in Thymus vulgaris, Silene vulgaris and Plantago coronopus. A simple Mendelian inheritance of sex determinism is unlikely in all three species. We discuss how our model can shed additional light on the genetics of restoration and point towards future efforts in the modelling of gynodioecy.     

Modelling the maintenance of male-fertile cytoplasm in a gynodioecious population

Gynodioecy is the co-occurrence of females and hermaphrodites in populations. It is usually due to the combined action of cytoplasmic male sterility (CMS) genes and nuclear genes that restore male fertility. According to previous theoretical studies, it is very difficult to explain the maintenance of gynodioecy with CMS and male-fertile cytotypes, although it has been observed in some species. However, only very specific situations have been investigated so far. We present a model to investigate the conditions that promote the maintenance of this breeding system in the case of an outcrossed species when CMS and male-fertile (non-CMS) cytotypes are present in an infinite panmictic population. We show that the type of cost of restoration strongly affects the conditions for stable maintenance of gynodioecy. Stable nuclear-cytoplasmic gynodioecy requires a female advantage, which is a classical condition for gynodioecy, but also a cost of CMS for female fitness, which had been rarely investigated. A cost of restoration is also needed, which could affect either pollen or seeds. Finally, we found that gynodioecy was attainable for a large set of parameter values, including low differences in fitness among genotypes and phenotypes. Our theoretical predictions are compared with previous theoretical work and with results of empirical studies on various gynodioecious species.     

Multiple CMS-restorer gene polymorphism in gynodioecious Plantago coronopus

The mode of inheritance of the male sterility trait is crucial for understanding the evolutionary dynamics of the sexual system gynodioecy, which is the co-occurrence of female and hermaphrodite plants in natural populations. Both cytoplasmic (CMS) and nuclear (restorer) genes are known to be involved. Theoretical models usually assume a limited number of CMS genes with each a single restorer gene, while reality is more complex. In this study, it is shown that in the gynodioecious species Plantago coronopus two new CMS-restorer polymorphisms exist in addition to the two that were already known, which means four CMS-restorer systems at the species level. Furthermore, three CMS types were shown to co-occur within a single population. All new CMS types showed a multilocus system for male fertility restoration, in which both recessive and dominant restorer alleles occur. Our finding of more than two co-occurring CMS-restorer systems each with multiple restorer genes raises the question how this complex of male sterility systems is maintained in natural populations.     

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.     

HOW SELFING,INBREEDING DEPRESSION,AND POLLEN LIMITATION IMPACT NUCLEAR‐CYTOPLASMIC GYNODIOECY: A MODEL

Gynodioecy, the co‐occurrence of females and hermaphrodites, is often due to conflicting interactions between cytoplasmic male sterility genes and nuclear restorers. Although gynodioecy often occurs in self‐compatible species, the effect of self‐pollination, inbreeding depression, and pollen limitation acting differently on females and hermaphrodites remains poorly known in the case of nuclear‐cytoplasmic gynodioecy (NCG). In this study, we model NCG in an infinite population and we study the effect of selfing rate, inbreeding depression, and pollen limitation on the maintenance of gynodioecy and on sex ratios at equilibrium. We found that selfing and inbreeding depression have a strong impact, which depends on whether restorer cost acts on male or female fitness. When cost affects male fitness, the strength of cost has no effect, whereas selfing and inbreeding depression only impact gynodioecy by modifying the value of the female advantage. When cost affects female fitness, selfing facilitates NCG and reduces the role of strength of the cost, even when no inbreeding depression occurs, whereas inbreeding depression globally restricts the maintenance of the polymorphism. Finally, we found that pollen limitation could additionally strongly modify the dynamic of gynodioecy. We discuss our findings in the light of empirical data available in gynodioecious species.     

Gynodioecy inSaxifraga granulata L. (Saxifragaceae)

The occurrence of gynodioecy in two populations in northern England of the normally hermaphroditeSaxifraga granulata is reported. Female plants have aborted stamens, and smaller petals than hermaphrodites. At Staindrop, County Durham, an estimated 23% of the flowering stems were female; at Macclesfield, Cheshire, 4% were female. The inheritance of male sterility is not simple, and probably involves at least one cytoplasmic and two nuclear genes. The secondary sexual characteristics, hermaphrodite-predominant sex ratios, and complex inheritance of male sterility, are typical of gynodioecious populations.     

The evolution of gynodioecy on a lattice

Gynodioecy is a breeding system in plants where populations consist of hermaphrodites and females. The females result from a genetic mutation which impairs pollen production in hermaphrodite plants. Most previous models for the evolution of gynodioecy do not take into account any spatial detail, which might be expected to play an important role in populations with short range interactions caused by poor or no locomotion.In this article we present a generalised mean-field analysis (which ignores any spatial detail), together with stochastic spatial simulations, to investigate the spatial effect on the evolution of gynodioecy. We show that, in a population of hermaphrodites where male sterility is caused by a dominant allele in a nuclear gene, mean-field calculations greatly underestimate the reproductive advantage females require to become viable under spatial constraints. This suggests that gynodioecy is less likely to evolve in plants with more localised pollination and seed setting. This may have implications for the evolution of dioecy, a breeding system in plants where the population consists of males and females, as gynodioecy is thought to be a route to dioecy. Our results also demonstrate that a lower frequency of females should be expected for gynodioecious populations when interactions are local. This is relevant when comparing the results of breeding experiments with observations of female frequency in the wild.     

Paternal leakage sustains the cytoplasmic polymorphism underlying gynodioecy but remains invasible by nuclear restorers

Cytoplasmic male sterility (CMS) in plants often results in gynodioecious populations, composed of hermaphrodites and male-sterile females. All models of gynodioecy assume maternal inheritance of the cytoplasmic alleles and postulate a variety of negatively frequency-dependent mechanisms to maintain the cytoplasmic polymorphisms observed in many natural populations. However, in some plant species, mitochondria are transmitted at least occasionally by pollen, a process called paternal leakage. We show that even a small amount of paternal leakage is sufficient to sustain a permanent, stable cytoplasmic polymorphism. Because only hermaphrodites provide pollen in gynodioecious species, the effects of paternal leakage are biased and occur more often from the non-CMS male-fertile haplotype to the CMS male-sterile haplotype. We also show that a nuclear restorer disrupts the polymorphic cytoplasmic equilibrium, leading to fixation of both the CMS allele and the restorer. Although a dominant nuclear restorer fixes, it fixes much more slowly than in the standard CMS models. Although a stable cytonuclear polymorphism is possible with "matching alleles" nuclear restoration, oscillations to low frequencies present a risk of loss by drift. Paternal leakage enhances the stability of joint cytonuclear polymorphism by reducing the chance that a CMS allele is lost by drift.