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.     

CYTONUCLEAR THEORY FOR HAPLODIPLOID SPECIES AND X-LINKED GENES. II. STEPPING-STONE MODELS OF GENE FLOW AND APPLICATION TO A FIRE ANT HYBRID ZONE

We develop cytonuclear, hybrid zone models for haplodiploid species or X-linked genes in diploid species using a stepping-stone framework of migration, in which migration rates vary with both direction and sex. The equilibrium clines for the allele frequencies, cytonuclear disequilibria, and frequencies of pure parental types are examined for species with diagnostic markers, under four important migration schemes: uniform migration of both sexes in both directions, greater migration of both sexes from one direction, greater migration of females, and greater migration of males. Of the three cytonuclear variables examined, the allele frequency clines are the most informative in differentiating among the various migration patterns. The cytonuclear disequilibria and the frequency of the pure parental types tend to be useful only in revealing directional asymmetries in migration. The extent of hybrid zone subdivision has quantitative but not qualitative effects on the distribution of cytonuclear variables, in that the allele frequency clines become more gradual, the cytonuclear disequilibria decrease in magnitude, and the frequencies of pure parentals decline with increasing subpopulation number. Also, the only major difference between the X-linked and haplodiploid frameworks is that a higher frequency of pure parentals is found when considering haplodiploids, in which male production does not require mating. The final important theoretical result is that censusing after migration yields greater disequilibria and parental frequencies than censusing after mating. We analyzed cytonuclear data from two transects from a naturally occurring hybrid zone between two haplodiploid fire ant species, Solenopsis invicta and S. richteri, using our stepping-stone framework. The frequency of S. invicta mtDNA exceeds the frequency of the S. invicta nuclear markers through much of this hybrid zone, indicating that sex differences in migration or selection may be occurring. Maximum-likelihood estimates for the migration rates are very high, due to an unexpectedly large number of pure parental types in the hybrid zone, and differ substantially between the two transects. Overall, our model does not provide a good fit, in part because the S. invicta–S. richteri hybrid zone has not yet reached equilibrium.     

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 cytoplasmic male sterility on cytonuclear disequilibria in hybrid zones

Male sterility is studied in hybrid zones by different measures of cytonuclear disequilibria, D, D₁, D₂, and D₃. Of particular interest are the dynamics of disequilibria as the system evolves to equilibrium. Our first model, the hybrid swarm model, yields equilibrium results identical to those observed in a model with random mating. In our second model of a hybrid zone, predictions of the sign pattern of disequilibrium values can be made based on migration values. A characteristic sign pattern may help to distinguish cytoplasmic male sterility (CMS) from other mechanisms of selection. Our simple CMS model with migration is successfully fit to cytonuclear data on a hybrid population of cottonwoods.     

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.     

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.     

Cytonuclear models of epistatic mating with backcrossing and selection

We develop hybrid zone models that explore the combined effects of mating system and either backcrossing or viability selection on the disequilibria between nuclear and cytoplasmic genes. In the epistatic mating plus backcrossing model, we find patterns of permanent cytonuclear disequilibria like those found when epistatic mating is the only factor, as well as a novel combination of significant cytonuclear disequilibria sign patterns. The second group of models evaluates the potential of epistatic mating and postzygotic viability selection to maintain cytonuclear disequilibria. Simulations are used to evaluate nine patterns of selection, each of which represent differing forms of selection against hybrids, and show that while all disequilibria usually decay to zero, under certain circumstances a number of different patterns of significant cytonuclear disequilibria are possible at equilibrium. The results from these models are compared to the observed cytonuclear disequilibria previously found in a hybrid population of Hyla treefrogs.     

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.     

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

We explicitly solve and analyze a series of deterministic continent-island models to delimit the effects of pollen and seed migration on cytonuclear frequencies and disequilibria in random-mating, mixed-mating and self-fertilized populations. Given the critical assumption of maternal cytoplasmic inheritance, five major findings are (i) nonzero cytonuclear disequilibria will be maintained in the island population if and only if at least some migration occurs each generation through seeds with nonrandom cytonuclear associations; (ii) immigrant seeds with no cytonuclear disequilibria can strongly affect the genetic structure of the island population by generating significant and long-lasting transient associations; (iii) with all else being equal, substantially greater admixture disequilibria are generally found with higher rates of seed migration into, or higher levels of self-fertilization within, the island population (with the possible exception of the heterozygote disequilibrium); (iv) pollen migration can either enhance or reduce the cytonuclear disequilibria caused by seed migration, or that due to mixed-mating in the absence of seed migration, but the effect is usually small and appears primarily to make a noticeable difference in predominantly outcrossing populations; and (v) pollen migration alone cannot generate even transient disequilibria de novo in populations with completely random associations. This same basic behavior is exhibited as long as there is some random outcrossing in the island population. Self-fertilized populations represent a special case, however, in that they are necessarily closed to pollen migration, and nonzero disequilibria can be maintained even in the absence of seed migration. All of these general results hold whether the population is censused as adults or as seeds, but the ability to detect nonrandom cytonuclear associations can depend strongly on the life stage censused in populations with a significant level of random outcrossing. We suggest how these models might be used for the estimation of seed and pollen migration.     

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.     

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.     

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.     

Cytonuclear disequilibria in a spatially structured hybrid zone

A hybrid zone model was developed that uses a stepping-stone model of population structure along with both nuclear and cytoplasmic genotypes to evaluate the effect that migration has on random mating organisms when no selection is present. Numerical simulations indicate that a number of different allele frequency and cytonuclear disequilibrium patterns can be found across the hybrid zone when it has reached equilibrium, and these results are discussed in the context of relative migration rates of the two sexes and the two source populations. The importance of numbers of subpopulations sampled, census time, and the persistence of pure species individuals into the hybrid zone are also discussed. Although this model does not consider selection or assortative mating in the hybrid zone, it should provide a useful baseline for evaluating joint cytonuclear genetic data from a structurally complex hybrid zone.     

The spatial structure of sexual and cytonuclear polymorphism in the gynodioecious Beta vulgaris ssp. maritima: I/ at a local scale

We have analyzed the spatial distribution of the sex phenotypes and of mitochondrial, chloroplast, and nuclear markers within two gynodioecious populations of Beta vulgaris ssp. maritima. Within both populations, sexual phenotype variation is controlled mainly by the cytoplasmic genotype, although in one study population a joint polymorphism of cytonuclear factors is clearly involved. In spite of contrasts in the ecology (mainly due to different habitats), a clear common feature in both populations is the highly patchy distribution of cytoplasmic haplotypes, contrasting with the wide distribution of nuclear diversity. This high contrast between cytoplasmic vs. nuclear spatial structure may have important consequences for the maintenance of gynodioecy. It provides opportunities for differential selection since nuclear restorer alleles are expected to be selected for in the presence of their specific cytoplasmic male sterile (CMS) type, but to be neutral (or selected against if there is a cost of restoration) in the absence of their CMS type. Selective processes in such a cytonuclear landscape may explain the polymorphism we observed at restorer loci for two CMS types.     

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     

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.     

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.     

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.     

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.