Comparative Effects of Pollen and Seed Migration on the Cytonuclear Structure of Plant Populations. II. Paternal Cytoplasmic Inheritance

We continue our study of the effects of pollen and seed migration on the cytonuclear structure of mixed-mating plant populations by analyzing two deterministic continent-island models under the critical assumption of paternal cytoplasmic inheritance. The major results of this study that contrast with our previous conclusions based on maternal cytoplasmic inheritance are (i) pollen gene flow can significantly affect the cytonuclear structure of the island population, and in particular can help to generate cytonuclear disequilibria that greatly exceed the magnitude of those that would be produced by seed migration or mixed mating alone; (ii) with simultaneous pollen and seed migration, nonzero cytonuclear disequilibria will be maintained not only when there is disequilibrium in the immigrant pollen or seeds, but also through a variety of intermigrant admixture effects when the two pools of immigrants differ appropriately in their cytonuclear compositions; (iii) either immigrant pollen or immigrant seeds can generate disequilibria de novo in populations with initially random cytonuclear associations, but pollen migration alone generally produces lower levels of disequilibrium than does comparable seed migration, especially at high levels of self-fertilization when the overall fraction of immigrant pollen is low; (iv) the equilibrium state of the island population will be influenced by the rate of pollen gene flow whenever there is either allelic disequilibrium in the immigrant pollen or simultaneous seed migration coupled with different cytoplasmic or nuclear allele frequencies in immigrant pollen and seeds or nonzero allelic disequilibrium in either immigrant pool. The estimation of pollen migration should therefore be facilitated with paternal cytoplasmic inheritance relative to the case of maternal cytoplasmic inheritance. These basic conclusions hold whether the population is censused as seeds or as adults, but with simultaneous pollen and seed migration, the relationship between census time and the ability to detect nonrandom cytonuclear associations is complex. When migration is through pollen alone, however, the cytonuclear structure of the island population is independent of the life stage censused.     

The effects of pollen and seed migration on nuclear-dicytoplasmic systems. I. Nonrandom associations and equilibrium structure with both maternal and paternal cytoplasmic inheritance

We determine the nuclear-dicytoplasmic effects of unidirectional gene flow via pollen and seeds upon a mixed-mating plant population, focusing on nuclear-mitochondrial-chloroplast systems where mitochondria are inherited maternally and chloroplasts paternally, as in many conifers. After first delineating the general effects of admixture (via seeds or individuals) on the nonrandom associations in such systems, we derive the full dicytonuclear equilibrium structure, including when disequilibria may be indicators of gene flow. Substantial levels of permanent two- and three-locus disequilibria can be generated in adults by (i) nonzero disequilibria in the migrant pools or (ii) intermigrant admixture effects via different chloroplast frequencies in migrant pollen and seeds. Additionally, three-locus disequilibria can be generated by higher-order intermigrant effects such as different chloroplast frequencies in migrant pollen and seeds coupled with nuclear-mitochondrial disequilibria in migrant seeds, or different nuclear frequencies in migrant pollen and seeds coupled with mitochondrial-chloroplast disequilibria in migrant seeds. Further insight is provided by considering special cases with seed or pollen migration alone, complete random mating or selfing, or migrant pollen and seeds lacking disequilibria or intermigrant admixture effects. The results complete the theoretical foundation for a new method for estimating pollen and seed migration using joint cytonuclear or dicytonuclear data.     

The effects of admixture and population subdivision on cytonuclear disequilibria

We examine the generation of cytonuclear disequilibria by admixture and continued gene flow. General formulas analogous to the nuclear case are first derived showing that the allelic and genotypic disequilibria from admixture or population subdivision equal their expected value across the contributing (sub) populations plus the covariance across these sources between the cytoplasmic gene frequency and the relevant nuclear frequency. A detailed study is then presented of the cytonuclear dynamics, in a random-mating population under two different migration scenarios. In both cases closed-form solutions are given for all variables as a function of the initial conditions and relevant migration parameters. The dynamics of the gene frequencies and allelic disequilibria, which dominate each system, are the same as those involving two unlinked nuclear loci, while the dynamics of the genotypic disequilibria and cytonuclear frequencies have no nuclear counterpart. The continent-island formulation focuses on a population receiving continued immigration from a large source of constant composition. A major discovery is that cytonuclear disequilibria can transiently build up on the "island" to levels far exceeding those found at equilibrium. In contrast, the admixture formulation focuses on the dynamics within two populations undergoing continued intermigration. Although in this case all cytonuclear associations must ultimately decay to zero, long-term transient disequilibria can develop which are many times their initial admixture values. For both migration scenarios it is shown that the time of population censusing relative to migration and reproduction dramatically affects both the amount and pattern of the nonrandom associations produced. The empirical relevance of these models is discussed in light of nuclear-mitochondrial data from a hybrid zone between European and North American eels and from a zone of racial admixture in humans.     

The cytonuclear effects of facultative apomixis. I. Disequilibrium dynamics in diploid populations

We comprehensively analyze the cytonuclear effects of generalized mixed mating, including all combinations of selfing, outcrossing, and apomixis, the asexual production of seeds. After first deriving the time-dependent solutions for nonrandom associations (disequilibria) between a diallelic cytoplasmic marker and the alleles and genotypes at a diploid nuclear locus, we delimit all possible dynamical behaviors and the conditions under which each occurs. As in standard mixed mating systems, all disequilibria ultimately decay to zero except when outcrossing is absent, in which case permanent disequilibria result if the allelic association is initially nonzero. When at least some outcrossing is present, any initial allelic association decays at a constant geometric rate, whereas genotypic disequilibria may first increase in magnitude or change sign. Although selfing and apomixis tend to retard the decay of disequilibria (or approach to equilibrium) and often to the same extent, apomixis can have a stronger effect under some conditions. We also determine the dynamics of cytonuclear disequilibria in specific examples that may be of particular interest for empirical studies of hybrid zones. The results suggest several practical guidelines for experimental design and data analysis and show how the cytonuclear disequilibrium dynamics under mating system alone furnish a quantitative baseline for null hypotheses against which to test for the presence of other evolutionary forces.     

Cytonuclear Theory for Haplodiploid Species and X-Linked Genes. I. Hardy-Weinberg Dynamics and Continent-Island, Hybrid Zone Models

We develop models that describe the cytonuclear structure for either a cytoplasmic and nuclear marker in a haplodiploid species or a cytoplasmic and X-linked marker in a diploid species. Sex-specific disequilibrium statistics that summarize nonrandom cytonuclear associations in such systems are defined, and their basic Hardy-Weinberg dynamics and admixture formulae are delimited. We focus on the context of hybrid zones and develop continent-island models whereby individuals from two genetically differentiated source populations migrate into and mate within a single zone of admixture. We examine the effects of differential migration of the sexes, assortative mating by pure type females, and census time (relative to mating and migration), as well as special cases of random mating and migration subsumed under the general models. We show that pure type individuals and nonzero cytonuclear disequilibria can be maintained within a hybrid zone if there is continued migration from both source populations, and that females generally have a greater influence over these cytonuclear variables than males. The resulting theoretical framework can be used to estimate the rates of assortative mating and sex-specific gene flow in hybrid zones and other zones of admixture involving haplodiploid or sex-linked cytonuclear data.     

The cytonuclear effects of facultative apomixis. II. Definitions and dynamics of disequilibria in tetraploid populations

We develop a cytonuclear framework for tetraploid populations in which a diallelic nuclear marker exhibits tetrasomic inheritance. This system requires two separate parameterizations, with six cytonuclear disequilibria (nonrandom associations) in tetraploid individuals and four in their diploid gametes. Double reduction during meiosis adds further complexity by causing gametic output to vary with the distance of the nuclear locus from the centromere. We derive and analyze dynamical solutions for the disequilibria under generalized mixed mating, with any combination of apomixis, selfing, and outcrossing, with and without double reduction. As in comparable diploid systems, all disequilibria ultimately decay to zero, unless nuclear and cytoplasmic alleles are nonrandomly associated and outcrossing is absent, in which case permanent associations result. Selfing and apomixis retard the decay of disequilibria (or approach to equilibrium), and often to the same extent. In contrast, double reduction can accelerate the loss of tetraploid cytonuclear associations, but only negligibly in hybrid zones, and this loss is never faster than in diploids. Only in the absence of allelic associations or outcrossing is the asymptotic approach to equilibrium differentially affected by apomixis and selfing or slower under tetrasomic than disomic inheritance. To facilitate empirical applications, we also examine tetraploid hybrid zone dynamics and offer practical guidelines for experimental design and data analysis, showing how the consequences of the mating system alone provide a valuable baseline for drawing evolutionary inferences from the observed patterns of cytonuclear associations.     

The Effects of Assortative Mating and Migration on Cytonuclear Associations in Hybrid Zones

We examine the influence of nonrandom mating and immigration on the evolutionary dynamics of cytonuclear associations in hybrid zones. Recursion equations for allelic and genotypic cytonuclear disequilibria were generated under models of (1) migration alone, assuming hybrid zone matings are random with respect to cytonuclear genotype; and (2) migration in conjunction with refined epistatic mating, in which females of the pure parental species preferentially mate with conspecific males. Major results are as follows: (a) even the slightest migration removes the dependency of the final outcome on initial conditions, producing a unique equilibrium in which both pure parental genotypes are maintained in the hybrid zone; (b) in contrast to nuclear genes, the dynamics of cytoplasmic allele frequencies appear robust to changes in the assumed mating system, yet are particularly sensitive to gene flow; (c) continued immigration can generate permanent cytonuclear disequilibria, whether mating is random or assortative; and (d) the order of population censusing (before versus after reproduction by immigrants) can have a dramatic effect on the magnitude but not the pattern of cytonuclear disequilibria. Using the maximum likelihood method, the parameter space of migration rates and assortative mating rates was examined for best fit to observed cytonuclear disequilibria data in a hybrid population of Hyla tree frogs. An epistatic mating model with a total immigration rate of about 32% per generation produces equilibrium gene frequencies and cytonuclear disequilibria consistent with the empirical observations.     

Associations between cytoplasmic and nuclear Loci in hybridizing populations

We extend current multilocus models to describe the effects of migration, recombination, selection, and nonrandom mating on sets of genes in diploids with varied modes of inheritance, allowing us to consider the patterns of nuclear and cytonuclear associations (disequilibria) under various models of migration. We show the relationship between the multilocus notation recently presented by Kirkpatrick, Johnson, and Barton (developed from previous work by Barton and Turelli) and the cytonuclear parameterization of Asmussen, Arnold, and Avise and extend this notation to describe associations between cytoplasmic elements and multiple nuclear genes. Under models with sexual symmetry, both nuclear-nuclear and cytonuclear disequilibria are equivalent. They differ, however, in cases involving some type of sexual asymmetry, which is then reflected in the asymmetric inheritance of cytoplasmic markers. An example given is the case of different migration rates in males and females; simulations using 2, 3, 4, or 5 unlinked autosomal markers with a maternally inherited cytoplasmic marker illustrate how nuclear-nuclear and cytonuclear associations can be used to separately estimate female and male migration rates. The general framework developed here allows us to investigate conditions where associations between loci with different modes of inheritance are not equivalent and to use this nonequivalence to test for deviations from simple models of admixture.     

Seed and pollen flow and cline discordance among genes with different modes of inheritance

The relationships between seed and pollen flow and cline discordance/concordance between cytoplasmic and nuclear genes, with the incorporation of the effects of natural selection, are formulated for one locus with two alleles, under assumptions of random mating, no drift and no mutation. Results show that under certain conditions, the relative roles of seed and pollen flow in shaping cline discordance/ concordance are very similar to their roles in influencing population differentiation for selectively neutral markers with different modes of inheritance. Where the disequilibria between cytoplasmic and nuclear genes are of the order similar to selection coefficient, cline discordance/concordance can be predicted from the relative values of the ratio of pollen to seed flow and the ratio of selection coefficients. Where the disequilibria attained by seed and pollen flow are significant, the integrated cytonuclear data are recommended for cline analysis. In both cases, the relative rates of selection coefficients between cytoplasmic and nuclear genes can be roughly estimated according to their characteristic length.     

The effects of pollen and seed migration on nuclear-dicytoplasmic systems. II. A new method for estimating plant gene flow from joint nuclear-cytoplasmic data

A new maximum-likelihood method is developed for estimating unidirectional pollen and seed flow in mixed-mating plant populations from counts of joint nuclear-cytoplasmic genotypes. Data may include multiple unlinked nuclear markers with a single maternally or paternally inherited cytoplasmic marker, or with two cytoplasmic markers inherited through opposite parents, as in many conifer species. Migration rate estimates are based on fitting the equilibrium genotype frequencies under continent-island models of plant gene flow to the data. Detailed analysis of their equilibrium structures indicates when each of the three nuclear-cytoplasmic systems allows gene flow estimation and shows that, in general, it is easier to estimate seed than pollen migration. Three-locus nuclear-dicytoplasmic data only increase the conditions allowing seed migration estimates; however, the additional dicytonuclear disequilibria allow more accurate estimates of both forms of gene flow. Estimates and their confidence limits for simulated data sets confirm that two-locus data with paternal cytoplasmic inheritance provide better estimates than those with maternal inheritance, while three-locus dicytonuclear data with three modes of inheritance generally provide the most reliable estimates for both types of gene flow. Similar results are obtained for hybrid zones receiving pollen and seed flow from two source populations. An estimation program is available upon request.     

On migration load of seeds and pollen grains in a local population

We have extended Wright's model of migration load to hermaphrodite plants showing variation at a single locus with two alleles. The model incorporates independent migration of seeds and pollen grains, the selection at both the haploid gametophyte and the diploid sporophyte stages, and a mixed mating system. The analytical relations between migration load and migration rate of seeds and pollen grains are explicitly formulated. The results show that under certain conditions, seed flow can have a more effect on migration load than pollen flow. Pollen selection at the gametophyte stage cannot substantially affect the migration load at the sporophyte stage. Selection at the diploid sporophyte stage is critical in determining the migration load of pollen grains. The relative migration loads of pollen versus seeds can be approximately estimated in predominantly outcrossing populations by the ratio of pollen flow to twice the seed flow, when the selection coefficient (s(T)) is greater than, or approximately equal to, the migration rate (m).     

Dynamics of cytonuclear disequilibria in finite populations and comparison with a two-locus nuclear system.

We study the behavior of cytonuclear disequilibria in a finite monoecious population due to (1) random drift alone, (2) random drift and mutation, and (3) random drift and migration, using exact results on the RUZ (Random Union of Zygotes) model and diffusion approximations. We also show that the RUG (Random Union of Gametes) model is not suitable for a cytonuclear system. The study is also accompanied by a comparison with a two-locus nuclear system. We show that in a finite population of size N without mutation, the rate of decrease of the cytonuclear allelic disequilibrium is the same as that in the corresponding unlinked two-locus nuclear system. The principal rate of decrease of variance in allelic disequilibrium in a cytonuclear system is slightly faster than that in the corresponding nuclear system. However, the expected value of the variance in cytonuclear disequilibria is larger than that in a two-locus nuclear system for at least the first N generations. With mutation, the expected value of steady state variances of both systems are about the same; however, the normalized variance in linkage disequilibrium sigma 2d of the cytonuclear system is about twice as large as that for the corresponding nuclear system. For the migration process, two sets of steady state solutions are provided, one for the variables before migration and the other for the variables after migration. Diffusion approximations for both the principal rate of decay and steady state solutions in both systems are found to be satisfactory. A more accurate backward diffusion equation for a two-locus nuclear system is provided when the recombination fraction R is large.     

Effects of Differential Selection in the Sexes on Cytonuclear Polymorphism and Disequilibria

We develop a series of models that examine the effects of differential selection between the sexes on cytonuclear polymorphism and disequilibria. A detailed analysis is provided for populations under constant fertility or viability selection censused at life stages without frequency differences in the sexes. We show analytically that cytonuclear disequilibria can be generated de novo if the cytoplasmic and nuclear loci each affect female fitness and there is a nonmultiplicative fitness interaction between them. While computer simulations demonstrate that the majority of disequilibria produced by random selection are transient and small in magnitude, measurable permanent disequilibria can result from selective differences both within and between the two sexes. We derive analytic conditions for a protected cytonuclear polymorphism and use numerical simulations to quantitate the likelihood of obtaining permanent nuclear, cytoplasmic, and cytonuclear variation under various patterns of selection. The numerical analysis identifies special selection regimes more likely to generate disequilibria and maintain cytonuclear polymorphism and reveals a direct correlation to the strength of selection. As a byproduct, our models also provide the first decomposition of the different parental contributions to cytonuclear dynamics and the analytic conditions under which selection can cause cytoplasmic frequency changes or a cytonuclear hitchhiking effect.     

Sampling Theory for Cytonuclear Disequilibria

We examine the statistical properties of cytonuclear disequilibria within a system including one diploid nuclear locus and one haploid cytoplasmic locus, each with two alleles. The results provide practical guidelines for the design and interpretation of cytonuclear surveys seeking to utilize the novel evolutionary information recorded in the observed pattern of cytonuclear associations. Important applications include population studies of nuclear allozymes in conjunction with genes from mitochondria, chloroplasts, or cytoplasmically inherited microorganisms. Our attention focuses on the allelic and genotypic disequilibria, which respectively measure the nonrandom associations between the cytotypes and the nuclear alleles and genotypes. We first derive the maximum likelihood estimators and their approximate large sample variances for each disequilibrium measure. These are each in turn used to set up an asymptotic test of the null hypothesis of no disequilibrium. We then calculate the minimum sample sizes required to detect the disequilibria under specified alternate hypotheses. The work also incorporates the deviation from Hardy-Weinberg equilibrium at the nuclear locus, which can significantly affect the results. The practical utility of this new sampling theory is illustrated through applications to two nuclear-mitochondrial data sets.     

Definition and Properties of Disequilibrium Statistics for Associations between Nuclear and Cytoplasmic Genotypes

We define and establish the interrelationships of four components of statistical association between a diploid nuclear gene and a uniparentally transmitted, haploid cytoplasmic gene: an allelic (gametic) disequilibrium (D), which measures associations between alleles at the two loci; and three genotypic disequilibria (D1, D2, D3), which measure associations between two cytotypes and the three respective nuclear backgrounds. We also consider an alternative set of measures, including D and the residual disequilibrium (d). The dynamics of these disequilibria are then examined under three conventional models of the mating system: (1) random mating; (2a) assortative mating without dominance (the "mixed-mating model"); and (2b) assortative mating with dominance ("O'Donald's model"). The trajectories of gametic disequilibria are similar to those for pairs of unlinked nuclear loci. The dynamics of genotypic disequilibria exhibit a variety of behaviors depending on the model and the initial conditions. Procedures for statistical estimation of cytonuclear disequilibria are developed and applied to several real and hypothetical data sets. Special attention is paid to the biological interpretations of various categories of allelic and genotypic disequilibria in hybrid zones. Genetic systems for which these statistics might be appropriate include nuclear genotype frequencies in conjunction with those for mitochondrial DNA, chloroplast DNA, or cytoplasmically inherited microorganisms.     

The Exact Test for Cytonuclear Disequilibria

We extend the analysis of the statistical properties of cytonuclear disequilibria in two major ways. First, we develop the asymptotic sampling theory for the nonrandom associations between the alleles at a haploid cytoplasmic locus and the alleles and genotypes at a diploid nuclear locus, when there are an arbitrary number of alleles at each marker. This includes the derivation of the maximum likelihood estimators and their sampling variances for each disequilibrium measure, together with simple tests of the null hypothesis of no disequilibrium. In addition to these new asymptotic tests, we provide the first implementation of Fisher's exact test for the genotypic cytonuclear disequilibria and some approximations of the exact test. We also outline an exact test for allelic cytonuclear disequilibria in multiallelic systems. An exact test should be used for data sets when either the marginal frequencies are extreme or the sample size is small. The utility of this new sampling theory is illustrated through applications to recent nuclear-mtDNA and nuclear-cpDNA data sets. The results also apply to population surveys of nuclear loci in conjunction with markers in cytoplasmically inherited microorganisms.     

Dynamics and Equilibrium Behavior of Cytonuclear Disequilibria under Genetic Drift, Mutation, and Migration

A simple two-locus drift model for cytonuclear systems is developed, in which the stochastic dynamics of cytonuclear genotypic frequencies are specified. Random union of zygotes is assumed. Trajectories for the first two moments of both genotypic and allelic disequilibria are given under three scenarios: (i) random drift alone; (ii) random drift with mutation; and (iii) random drift with migration. Steady state solutions for the cytonuclear disequilibria are reported. The utility of this simple two-locus drift model in testing the neutrality of mitochondrial DNA markers in artificial hybrid zones is briefly illustrated     

用遗传标记监测不同环境和经营措施下的加勒比松森林群体遗传动态

对古巴加勒比松的6个群体（包括天然林、采伐林、母树林和种子园）进行了同功酶分析,根据5个酶系统8个位点的同功酶数据,对各群 交配系统以及群体遗传变异和结构进行了分析,天然林、种子园和母树林的多位点异交率和绝大多数单位点异交率都和完全异交无显著差异,过渡采伐的松树岛群体多位点异交率显著小于完全异交,而只有一半单位点异交率显著小于完全异交,而且该群体单位点平均异交率和多位点异交率均低于其它3个群体的估计值。采伐群体中同功酶变异和基因多样性与天然林群体JAG的相似,但低于其它群体,其近交系数较大,但小于天然林MAN和中国栽培群体的近交系数。中国引种栽培群体无论是同功酶变异还是基因多样性都显著高于古巴群体,与所有古巴群体的遗传距离都显著大于古巴群体之间的遗传距离。结果表明过度采伐导致群体自交程度增加,营建种子园可有效减少近交。自然分布区以外的引种栽培群体遗传变化量大,无论遗传变异和基因多样性都比参试其它群体大。     

Pollen pool heterogeneity in jack pine (Pinus banksiana Lamb.): a problem for estimating outcrossing rates?

Summary Pollen pool heterogeneity, which violates an assumption of the mixed-mating model, is a major potential problem in measuring plant mating systems. In this study, isozyme markers were used to examine pollen pool heterogeneity in two natural populations of jack pine, Pinus banksiana Lamb., in northwestern Ontario, Canada. Population multilocus estimates of outcrossing rate ranged from 0.83 to 0.95 and differed significantly between populations. Single-tree multilocus outcrossing rates were found to be homogeneous among trees in both populations. Computer simulation studies indicated that a consanguineous pollen pool (pollen gametes related to the mother tree) was capable of biasing population outcrossing estimates downward. Random pollen pool heterogeneity (uncorrelated with maternal genotypes) did not appear to affect population outcrossing estimates in the simulations. Heterogeneity G-tests and Spearman rank tests showed that pollen pool heterogeneity existed in the two natural populations examined; however, it did not have a major effect on population outcrossing estimates, since the consanguineous pollen pool detected was probably a relatively minor component of the outcross pollen pool in both populations. In addition, heterogeneity G-tests were found to be not sensitive in detecting pollen pool heterogeneity caused by consanguineous pollen pool.     

Cytonuclear disequilibrium and genetic drift in a natural population of ponderosa pine

We measured the cytonuclear disequilibrium between 11 nuclear allozyme loci and both mitochondrial and chloroplast DNA haplotypes in a natural population of ponderosa pine (Pinus ponderosa, Laws). Three allozyme loci showed significant associations with mtDNA variation, while two other loci showed significant association with cpDNA. However, the absolute number of individuals involved in any of the associations was small, such that in none of the nuclear-organellar combinations was the difference between observed and expected numbers >11 individuals. Patterns of association were not consistent across loci or organellar genomes, suggesting that they are not the result of mating patterns, which would act uniformly on all loci. This pattern of disequilibria is consistent with the action of genetic drift and with existing knowledge of the structure of this population and thus does not imply the action of other evolutionary processes. The overall magnitude (normalized disequilibrium) of associations was greater for maternally inherited mtDNA than for paternally inherited cpDNA, though this difference was neither large nor significant. Such significant disequilibria involving the paternally inherited organelle indicate that not only are there a limited number of seed parents, but the effective number of pollen parents is also limited.