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.     

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     

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.     

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.     

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.     

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.     

Estimation of Selection Parameters using Multi‐generation Cytonuclear Data

We have developed an approximate maximum likelihood framework for the problem of estimating the selection coefficients in a simple fertility selection model via random union of zygotes. We consider a sampling scheme where a random sample from each (discrete) generation of a population observed over several generations is collected and genotyped based on one nuclear locus and a cytonuclear locus, simultaneously. Simulation results show excellent small sample performance of the resulting approximate MLE. Asymptotic variance‐covariance matrix of our estimator is also obtained. We further show that these estimates can be used to obtain simple test statistics for testing various types of selection hypotheses including a test of neutrality.     

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.     

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.     

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.     

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.     

Cytonuclear genetic structure of a hybrid zone in lizards of the Sceloporus grammicus complex (Sauria, Phrynosomatidae)

Lizards of the Sceloporus grammicus complex are comprised of multiple chromosome races that form several zones of parapatric hybridization in central Mexico. We scored diagnostic mitochondrial DNA (mtDNA) haplotypes and autosomal chromosome markers in a sample of 342 lizards from one well-defined zone between 2n = 34 and 2n = 46 races. A two-part analysis was performed on this data set in an attempt to infer the predominant evolutionary forces shaping the cytonuclear structure of this zone. The complications posed by its spatial structure were addressed by analysing a hierarchical series of smaller subsamples chosen to approximate single mating units. Two critical conclusions were drawn from this first-stage analysis. First and foremost, the three chromosomes have largely concordant cytonuclear disequilibrium patterns within each subsample with adequate numbers of individuals for detecting nonrandom cytonuclear associations. This suggests that the cytonuclear structure of this zone is predominantly a result of deterministic genome-wide forces rather than genetic drift or deterministic forces specific to individual chromosomes or loci. Second, the fit of a series of migration models to the data shows that the cytonuclear structure of the subsamples is well accounted for by continued gene flow from the two parental races alone, with random mating with respect to cytonuclear genotype and no other evolutionary forces. These results motivate several further empirical and theoretical investigations to refine our understanding of the relative roles of migration and other potentially important forces such as natural selection and genetic drift, in this and other hybrid zones.     

ARE ENZYME LOCI SELECTIVELY NEUTRAL IN HAPLOID POPULATIONS OF NONVASCULAR PLANTS?

Enzyme data from 12 previously published studies of haploid bryophyte populations were subjected to tests of selective neutrality and population equilibrium. Altogether 280 samples were tested using the Ewens-Watterson test, the Ewens exact test, and Chakraborty's test. The Ewens-Watterson and the Ewens exact tests revealed neutrality in 84.6% of the cases, whereas the Chakraborty test revealed neutrality in 100% of the cases. Neutrality may thus explain the relatively high amounts of genetic variation found in these organisms. Possible hidden population sub-structuring and nonrandom field sampling of genotypes in many of the studies could make Chakraborty's test more reliable than the two others.     

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.     

On the neutrality of molecular genetic markers: pedigree analysis of genetic variation in fragmented populations

Many studies employ molecular markers to infer ecological and evolutionary processes, assuming that variation found at genetic loci offers a reliable representation of stochastic events in natural populations. Increasingly, evidence emerges that molecular markers might not always be selectively neutral. However, only a few studies have analysed how deviations from neutrality could affect estimates of genetic variation, using populations with known genealogy. We monitored changes in allozyme variation over eight generations in captive metapopulations of the butterfly Bicyclus anynana. Population demography was recorded by individually marking 35 000 butterflies and constructing pedigrees. We designed a computer program that simulated the inheritance of founder allozyme alleles in butterfly pedigrees. We thus tested whether the observed transmission of allozyme alleles could be explained by random genetic drift alone, or whether there was evidence for positive or negative selection. This analysis showed that in the smallest metapopulations the loss of allozyme variation exceeded the neutral rate. Possibly, linkage disequilibria between deleterious mutations and marker alleles resulted in background selection and a faster erosion of allozyme variation. In larger metapopulations, one locus (MDH) showed a significant heterozygote excess and smaller than expected loss in heterozygosity, observations consistent with (associative) overdominance. This study demonstrates that the neutrality of molecular markers cannot always be assumed, particularly in small populations with a high mutation load.     

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.     

Association of genetic traits to estimated haplotypes from SNP genotypes using EM algorithm and Markov Chain Monte Carlo techniques

In order to find association of genetic traits to some haplotypes from closely spaced multilocus phase-unknown genotypes we use a three-stage approach. Haplotype frequencies and the most likely haplotype pair for each individual are estimated from random samples of individual or small (nuclear) family genotypes via an EM algorithm. If the most likely haplotype pair configuration of the whole sample outweighs the less likely ones, we may consider the estimated haplotypes as alleles of a multi-allelic marker and carry out the conventional statistics, TDT or ANOVA for quantitative traits. If the most likely haplotype pair configuration and the less likely ones do not differ much in their weight, we sample the TDT or ANOVA statistic over all haplotype pair configurations using the Metropolis-Hastings algorithm. Applications of our method to simulated data as well as real data are given.     

Estimation of genetically effective breeding numbers using a rejection algorithm approach

Polygynous mating results in nonrandom sampling of the adult male gamete pool in each generation, thereby increasing the rate of genetic drift. In principle, genetic paternity analysis can be used to infer the effective number of breeding males (Nebm). However, this requires genetic data from an exhaustive sample of candidate males. Here we describe a new approach to estimate Nebm using a rejection algorithm in association with three statistics: Euclidean distance between the frequency distributions of maternally and paternally inherited alleles, average number of paternally inherited alleles and average gene diversity of paternally inherited alleles. We quantify the relationship between these statistics and Nebm using an individual-based simulation model in which the male mating system varied continuously between random mating and extreme polygyny. We evaluate this method using genetic data from a natural population of highly polygynous fruit bats (Cynopterous sphinx). Using data in the form of mother-offspring genotypes, we demonstrate that estimates of Nebm are very similar to independent estimates based on a direct paternity analysis that included data on candidate males. Our method also permits an evaluation of uncertainty in estimates of Nebm and thus facilitates inferences about the mating system from genetic data. Finally, we investigate the sensitivity of our method to sample size, model assumptions, adult population size and the mating system. These analyses demonstrate that the rejection algorithm provides accurate estimates of Nebm across a broad range of demographic scenarios, except when the true Nebm is high.     

A new method to uncover signatures of divergent and stabilizing selection in quantitative traits

While it is well understood that the pace of evolution depends on the interplay between natural selection, random genetic drift, mutation, and gene flow, it is not always easy to disentangle the relative roles of these factors with data from natural populations. One popular approach to infer whether the observed degree of population differentiation has been influenced by local adaptation is the comparison of neutral marker gene differentiation (as reflected in FST) and quantitative trait divergence (as reflected in QST). However, this method may lead to compromised statistical power, because FST and QST are summary statistics which neglect information on specific pairs of populations, and because current multivariate tests of neutrality involve an averaging procedure over the traits. Further, most FST-QST comparisons actually replace QST by its expectation over the evolutionary process and are thus theoretically flawed. To overcome these caveats, we derived the statistical distribution of population means generated by random genetic drift and used the probability density of this distribution to test whether the observed pattern could be generated by drift alone. We show that our method can differentiate between genetic drift and selection as a cause of population differentiation even in cases with FST=QST and demonstrate with simulated data that it disentangles drift from selection more accurately than conventional FST-QST tests especially when data sets are small.     

Genic VERSUS Chromosomal Variation in Natural Populations of DROSOPHILA SUBOBSCURA

Gametic frequencies in one mainland and one island population of D. subobscura were obtained by means of extracting wild chromosomes and subsequently analyzing them for inversions and allozymes. The high degree of cytological heterogeneity which characterizes these populations is not reflected in the genetic data. Two cases of non-random association were observed among eighteen pair-wise comparisons involving gene alleles and inversions to which the locus is linked. In both cases exchange of alleles at the locus is completely suppressed by the inversions. Four cases of linkage disequilibrium were detected among eighteen pairs of loci; two of them could best be explained as transient associations generated by random drift. The results suggest that disequilibria among enzyme loci are not widespread in natural populations—Populations with a lower degree of chromosomal variation are genetically as variable as populations with a higher degree of chromosomal variation. This observation does not support the hypothesis that selection in marginal homokaryotypic populations is for specialized homozygous genotypes.