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.     

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.     

A Statistical Test of a Neutral Model Using the Dynamics of Cytonuclear Disequilibria

In this paper we use cytonuclear disequilibria to test the neutrality of mtDNA markers. The data considered here involve sample frequencies of cytonuclear genotypes subject to both statistical sampling variation as well as genetic sampling variation. First, we obtain the dynamics of the sample cytonuclear disequilibria assuming random drift alone as the source of genetic sampling variation. Next, we develop a test statistic using cytonuclear disequilibria via the theory of generalized least squares to test the random drift model. The null distribution of the test statistic is shown to be approximately chi-squared using an asymptotic argument as well as computer simulation. Power of the test statistic is investigated under an alternative model with drift and selection. The method is illustrated using data from cage experiments utilizing different cytonuclear genotypes of Drosophila melanogaster. A program for implementing the neutrality test is available upon request.     

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.     

Definition and Properties of Disequilibria within Nuclear-Mitochondrial-Chloroplast and Other Nuclear-Dicytoplasmic Systems

We define and determine the interrelationships among five sets of disequilibrium parameters that measure two- and three-locus nonrandom associations in nuclear-dicytoplasmic systems. These assume a diploid nuclear locus and two haploid cytoplasmic loci, with special reference to nuclear-mitochondrial-chloroplast systems. Three sets of two-locus disequilibria measure the association between haplotypes at the two cytoplasmic loci (DMC) and associations between each cytoplasmic locus and nuclear alleles or genotypes (DM, D1M, D2M, D3M; DC, D1C, D2C, D3C). In addition, we present two classes of higher-order disequilibria that measure nonrandom allelic or genotypic associations involving all three loci. The first class quantifies associations between the nuclear locus and the two cytoplasmic loci taken jointly (DA/MC, DAA/MC, DAa/MC, Daa/MC, etc.), whereas the second measures only those associations remaining after all two-locus associations have been taken into account (DA/M/C, DAA/M/C, DAa/M/C, Daa/M/C). Based on combinations of these five sets of measures, we suggest a variety of parameterizations of three-locus, nuclear-dicytoplasmic systems. The dynamics of these disequilibria are then investigated under models of random and mixed mating, either with both cytoplasmic genomes inherited through the same parent or through opposite parents. Except for associations between the cytoplasmic haplotypes, which are constant when the two cytoplasmic genomes are inherited through the same parent, all disequilibria ultimately decay to zero. These randomizations do not necessarily occur monotonically, however, and in some cases are preceded by an initial increase in magnitude or sign change. For both inheritance patterns, the asymptotic decay rates are steadily retarded by increasing levels of self-fertilization. This behavior contrasts with that in the extreme case of complete selfing, for which only the heterozygote disequilibria always decay to zero. For all models considered, the dynamics of the two-locus cytonuclear subsystems are solely a function of the mating system, whereas the dynamical behavior and sign patterns of the cytoplasmic and three-locus disequilibria also depend strongly on the mode of cytoplasmic inheritance.     

Empirical evaluation of cytonuclear models incorporating genetic drift and tests for neutrality of mtDNA variants: data from experimental Gambusia hybrid zones

Statistical tests of genetic drift and of the neutrality of mtDNA are presented using empirical time‐series data on multi‐generational changes in cytonuclear disequilibria within replicated experimental hybrid populations of two species of live‐bearing Poeciliid fishes (Gambusia holbrooki and G.affinis) which were monitored over a period of two years (three generations). Cytonuclear disequilibria D and D (which measure departures from random associations of cytoplasmic and nuclear genotypes) over the three generations of the experiment were non‐zero for all replicate populations. For each of five nuclear loci, the observed measures of D and D were highly concordant between replicates during each generation. Significant departures from expectations were observed after one and two generations. A statistical measure of goodness of fit of observed changes in cytonuclear disequilibria (and implicitly of the neutrality of the mtDNA markers) was calculated for each nuclear locus. When the results for the replicates were combined into an overall test of neutrality, the fit to the random union of zygotes (RUZ) model was rejected for four of the five nuclear loci (P < 0.05). A simple genetic drift model does not explain the temporal changes in composite cytonuclear genotypic frequencies. Frequencies of parental G. holbrooki mitochondrial alleles and nuclear genotypes exceeded expected values during most time periods, implying some selective advantage of offspring produced by G. holbrooki females. Expansion of cytonuclear models to explicitly address questions of genetic drift and neutrality have general relevance to studies of natural populations. This revised version was published online in July 2006 with corrections to the Cover Date.     

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 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.     

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.     

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.     

Modeling extent and distribution of zygotic disequilibrium: implications for a multigenerational canine pedigree

Unlike gametic linkage disequilibrium defined for a random-mating population, zygotic disequilibrium describes the nonrandom association between different loci in a nonequilibrium population that deviates from Hardy-Weinberg equilibrium. Zygotic disequilibrium specifies five different types of disequilibria simultaneously that are (1) Hardy-Weinberg disequilibria at each locus, (2) gametic disequilibrium (including two alleles in the same gamete, each from a different locus), (3) nongametic disequilibrium (including two alleles in different gametes, each from a different locus), (4) trigenic disequilibrium (including a zygote at one locus and an allele at the other), and (5) quadrigenic disequilibrium (including two zygotes each from a different locus). However, because of the uncertainty on the phase of the double heterozygote, gametic and nongametic disequilibria need to be combined into a composite digenic disequilibrium and further define a composite quadrigenic disequilibrium together with the quadrigenic disequilibrium. To investigate the extent and distribution of zygotic disequilibrium across the canine genome, a total of 148 dogs were genotyped at 247 microsatellite markers located on 39 pairs of chromosomes for an outbred multigenerational pedigree, initiated with a limited number of unrelated founders. A major portion of zygotic disequilibrium was contributed by the composite digenic and quadrigenic disequilibrium whose values and numbers of significant marker pairs are both greater than those of trigenic disequilibrium. All types of disequilibrium are extensive in the canine genome, although their values tend to decrease with extended map distances, but with a greater slope for trigenic disequilibrium than for the other types of disequilibrium. Considerable variation in the pattern of disequilibrium reduction was observed among different chromosomes. The results from this study provide scientific guidance about the determination of the number of markers used for whole-genome association studies.     

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.     

Disequilibrium among several linked neutral genes in finite population I. Mean changes in disequilibrium

Formulae are developed for computing changes in expected values in a finite population of linkage disequilibrium among neutral genes from more than two loci, although the exact analysis is taken up to only six loci. An essentially haploid model is used. As with two loci, the three-locus disequilibrium declines exponentially at all generations, but for m > 3 loci a matrix has to be constructed to give joint changes in the m-locus disequilibrium and products of disequilibria with fewer loci, for example of two m/2-locus disequilibria. The asymptotic rates of change in multilocus disequilibria depend on the arrangement of genes on the chromosome as well as its total length, but the initial rate of breakdown of disequilibrium from a line cross base is less dependent on the arrangement. With equally spaced loci the asymptotic rate of breakdown of m locus disequilibrium is roughly proportional to m. Although mutation and interference are excluded from the main analysis, it is shown how they can be incorporated.     

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.     

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 estimator of the optimal measure of allelic association: mean, variance and probability distribution when the sample size tends to infinity

The allelic association or linkage disequilibrium between two loci is a parameter of fundamental interest in modern population genetics for evolutionary inference and association mapping studies. Among the many measures available, the optimal measure of allelic association rho presents a strong evolutionary theory basis and is modeled on the physical distance along the chromosome with the Malécot equation for isolation by distance. Moreover, rho is equal to the absolute value of D', the standardized measure of gametic disequilibrium. We studied here the statistical properties of the rho sample estimator. We derived its asymptotic probability distribution and showed that it is neither asymptotically normal nor unbiased when rho=0 or when allelic frequencies are equal at both loci, in contrast to previous claims. This asymptotic study leads to propose a new test for absence of linkage disequilibrium. We compared it to Pearson's Chi2 test for independence in a contingency table and showed by simulations that the range in power of these two tests depends on the sign of D'. The new test outperformed slightly the Chi2 test, when D', polarized with respect to major alleles, is negative. Finally, we derived the asymptotic bias and information of the rho estimator that are due to the experimental sampling and showed by simulation that its bias is large in small samples. The consequences of these findings on applications using the rho measure are then discussed in particular for constructing LD unit maps, and call for a revised statistical treatment.     

Disequilibrium among several linked neutral genes in finite population 1. mean changes in disequilibrium

Formulae are developed for computing changes in expected values in a finite population of linkage disequilibrium among neutral genes from more than two loci, although the exact analysis is taken up to only six loci. An essentially haploid model is used. As with two loci, the three-locus disequilibrium declines exponentially at all generations, but for m > 3 loci a matrix has to be constructed to give joint changes in the m-locus disequilibrium and products of disequilibria with fewer loci, for example of two $\text{m}\text{2}\text{-}\text{locus}$ disequilibria. The asymptotic rates of change in multilocus disequilibria depend on the arrangement of genes on the chromosome as well as its total length, but the initial rate of breakdown of disequilibrium from a line cross base is less dependent on the arrangement. With equally spaced loci the asymptotic rate of breakdown of m locus disequilibrium is roughly proportional to m. Although mutation and interference are excluded from the main analysis, it is shown how they can be incorporated.     

The deviation from linkage equilibrium with multiple loci varying in a stepping-stone cline

The balance between the creation of associations between alleles at different loci by immigration and the convergence to linkage equilibrium due to the recombination process is studied in a theoretical model. The geographical structure of the model is a stepping-stone chain of populations linking two genetically constant source populations. The model assumes an arbitrary number of autosomal loci and considers genetic variation (two alleles at each locus) that is not subject to natural selection. The gene frequencies at each locus will then show a linear cline through the stepping-stone chain of populations. The deviation from linkage equilibrium through the stepping-stone cline is characterized by an equation for linear measures that provide the linkage disequilibrium measures for a given set of loci in terms of the gene frequencies and the linkage disequilibria in the source populations and in terms of the linkage disequilibrium measures through the cline for lower numbers of loci. Numerical examples of this iterative solution are given, and it is shown that the build-up of the higher order Bennett-disequilibria through the cline is considerably more pronounced than the build-up of two-locus disequilibria.     

Association studies under general disease models

There is great expectation that the levels of association found between genetic markers and disease status will play a role in the location of disease genes. This expectation follows from regarding association as being proportional to linkage disequilibrium and therefore inversely related to recombination value. For disease genes with more than two alleles, the association measure is instead a weighted average of linkage disequilibria, with the weights depending on allele frequencies and genotype susceptibilities at the disease loci. There is no longer a simple relationship, even in expectation, with recombination. We adopt a general framework to examine association mapping methods which helps to clarify the nature of case-control and transmission/disequilibrium-type tests and reveals the relationship between measures of association and coefficients of linkage disequilibrium. In particular, we can show the consequences of additive and nonadditive effects at the trait locus on the behavior of these tests. These concepts have a natural extension to marker haplotypes. The association of two-locus marker haplotypes with disease phenotype depends on a weighted average of three-locus disequilibria (two markers with each disease locus). It is likely that these two-marker analyses will provide additional information in association mapping studies.     

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.