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.     

Migration load in males and females

Many theories have been proposed for interpreting the relative sexual mortalities in natural populations, but the effects of maladaptive genes from migrants on sexual mortality have not been examined. In this study, I systematically explore the properties of migration load in dioecious plants where gender dimorphism is determined by sex chromosomes (the XX-XY or similar system). Two stages of gene flow (pollen and seed) and two stages of selection (gametophyte and sporophyte) are jointly examined in the one- and two-locus cases. The general results show that several factors, including the maladaptive allele frequencies in migrants, migration rate, selection, and linkage disequilibria, can individually or jointly alter the relative sexual mortalities at either the gametophyte or the sporophyte stage, or both. When there are the same values between sexes in the parameters relevant to the migrant gene frequencies and two-stage selection coefficients under the additive viability model (without Y-chromosome and linkage disequilibrium effects), the average mortality is higher in ovules than in pollen since the males do not inherit the maladaptive genes from migrating pollen. The average mortality per individual in the homogametic sex is almost twice as large as that in heterogametic sex at the sporophyte stage. Under the additive viability model, the relative sexual mortalities at the gametophyte stage are negatively related to the relative sexual mortalities at the sporophyte stage.     

Seed and pollen flow in expanding a species' range

The distinct processes of gene flow via seeds and pollen in hermaphrodite plants provide a biological basis for interpreting their different roles in expanding a species' range. A species' range is primarily expanded through the colonization process by seed dispersal and followed by the joint effects of both seed and pollen flow. Here we examined the effects of seed and pollen flow on shaping a species' distribution in one-dimensional space. Our results demonstrate that pollen flow can enhance range expansion when immigrating genes are adaptive to recipient populations, but can shrink a species' range when immigrating genes are maladaptive. The incompletely purging of maladaptive genes from immigrating pollen grains at the gametophyte stage can reinforce the biological barrier to range expansion. The linkage disequilibria attained by immigrating seeds and pollen grains indirectly amplify the effects of the reaction component and further limit a species' range. The cumulative effect from multiple loci each with a small effect can be substantial on altering a species' range when these genes are maladaptive. These theoretical predictions can help understand the role of pollen flow that is incapable of colonizing new habitats in range expansion.     

THE EFFECT OF POLLEN VERSUS SEED FLOW ON THE MAINTENANCE OF NUCLEAR‐CYTOPLASMIC GYNODIOECY

Gynodioecy, where females co‐occur with hermaphrodites, is a relatively common sexual system in plants that is often the result of a genetic conflict between maternally inherited male sterility genes in the mitochondrial genome and the biparentally inherited male fertility restorer genes in the nucleus. Previous models have shown that nuclear‐cytoplasmic gynodioecy can be maintained under certain conditions by negative frequency‐dependent selection, but the effect of other evolutionary processes such as genetic drift and population subdivision is only partially understood. Here, we investigate the joint effects of frequency‐dependent selection, drift, and migration through either pollen or seeds on the maintenance of nuclear‐cytoplasmic gynodioecy in a subdivided population. We find that the combination of drift and selection causes the loss of gynodioecy under scenarios that would maintain it under the influence of selection alone, and that both seed and, more surprisingly, pollen flow can maintain the polymorphism. In particular, although pollen flow could not avoid the loss of cytoplasmic polymorphism within demes, it allowed the maintenance of nuclear‐cytoplasmic polymorphism at the metapopulation level.     

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 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.     

Cline coupling and uncoupling in a stickleback hybrid zone

Strong ecological selection on a genetic locus can maintain allele frequency differences between populations in different environments, even in the face of hybridization. When alleles at divergent loci come into tight linkage disequilibrium, selection acts on them as a unit and can significantly reduce gene flow. For populations interbreeding across a hybrid zone, linkage disequilibria between loci can force clines to share the same slopes and centers. However, strong ecological selection on a locus can also pull its cline away from the others, reducing linkage disequilibrium and weakening the barrier to gene flow. We looked for this “cline uncoupling” effect in a hybrid zone between stream resident and anadromous sticklebacks at two genes known to be under divergent natural selection (Eda and ATP1a1) and five morphological traits that repeatedly evolve in freshwater stickleback. These clines were all steep and located together at the top of the estuary, such that we found no evidence for cline uncoupling. However, we did not observe the stepped shape normally associated with steep concordant clines. It thus remains possible that these clines cluster together because their individual selection regimes are identical, but this would be very surprising given their diverse roles in osmoregulation, body armor, and swimming performance.     

A comparison of nuclear and mitochondrial cline shapes in a hybrid zone in the Sceloporus grammicus complex (Squamata; Phrynosomatidae)

The F5 and FM2 chromosome races of the Sceloporus grammicus complex form a hybrid zone in the Mexican state of Hidalgo. Previous studies of this zone have assessed genetic structure by averaging estimates of shape and width across three diagnostic chromosome markers. This approach is likely to mask subtle differences in cline shape among loci (e.g. selected vs. neutral), and obscure any displacement of cline centres (if present). Here we use maximum likelihood methods to construct the best fitting individual clines for three chromosomal markers, and also add two new markers; the mitochondrial DNA (mtDNA) locus, and the nuclear ribosomal DNA (rDNA) repeat. For each locus, hybrid zone models were fitted by cline shape and width, and the position and number of segments describing the centre of the zone. Pairwise comparisons between all clines revealed concordance between chromosomes 2 and 6, but significant discordance in cline structure among all other paired combinations. The concordance of chromosomes 2 and 6 suggests that these clines are maintained by genome-wide forces. The discordance of the chromosome 1 cline suggests an influence of asymmetric introgression, while the mtDNA cline is probably influenced by selection and drift. The rDNA locus reveals a pattern best explained by either extreme asymmetric introgression or gene conversion. The structure of zone indicates that genome-wide processes and locus specific selective forces as well as drift, are operating to different degrees on different loci. The locus-by-locus approach used here permits a finer discrimination among possible mechanisms responsible for the maintenance of the individual clines.     

Inference from Clines Stabilized by Frequency-Dependent Selection

Frequency-dependent selection against rare forms can maintain clines. For weak selection, s, in simple linear models of frequency-dependence, single locus clines are stabilized with a maximum slope of between square root of s/square root of 8 sigma and square root of s/square root of 12 delta, where sigma is the dispersal distance. These clines are similar to those maintained by heterozygote disadvantage. Using computer simulations, the weak-selection analytical results are extended to higher selection pressures with up to three unlinked genes. Graphs are used to display the effect of selection, migration, dominance, and number of loci on cline widths, speeds of cline movements, two-way gametic correlations ("linkage disequilibria"), and heterozygote deficits. The effects of changing the order of reproduction, migration, and selection, are also briefly explored. Epistasis can also maintain tension zones. We show that epistatic selection is similar in its effects to frequency-dependent selection, except that the disequilibria produced in the zone will be higher for a given level of selection. If selection consists of a mixture of frequency-dependence and epistasis, as is likely in nature, the error made in estimating selection is usually less than twofold. From the graphs, selection and migration can be estimated using knowledge of the dominance and number of genes, of gene frequencies and of gametic correlations from a hybrid zone.     

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.     

Estimates of Selection and Gene Flow from Measures of Cline Width and Linkage Disequilibrium in Heliconius Hybrid Zones

Hybrid zones can yield estimates of natural selection and gene flow. The width of a cline in gene frequency is approximately proportional to gene flow (sigma) divided by the square root of per-locus selection (square root of s). Gene flow also causes gametic correlations (linkage disequilibria) between genes that differ across hybrid zones. Correlations are stronger when the hybrid zone is narrow, and rise to a maximum roughly equal to s. Thus cline width and gametic correlations combine to give estimates of gene flow and selection. These indirect measures of sigma and s are especially useful because they can be made from collections, and require no field experiments. The method was applied to hybrid zones between color pattern races in a pair of Peruvian Heliconius butterfly species. The species are Müllerian mimics of one another, and both show the same changes in warning color pattern across their respective hybrid zones. The expectations of cline width and gametic correlation were generated using simulations of clines stabilized by strong frequency-dependent selection. In the hybrid zone in Heliconius erato, clines at three major color pattern loci were between 8.5 and 10.2 km wide, and the pairwise gametic correlations peaked at R approximately 0.35. These measures suggest that s approximately 0.23 per locus, and that sigma approximately 2.6 km. In erato, the shapes of the clines agreed with that expected on the basis of dominance. Heliconius melpomene has a nearly coincident hybrid zone. In this species, cline widths at four major color pattern loci varied between 11.7 and 13.4 km. Pairwise gametic correlations peaked near R approximately 1.00 for tightly linked genes, and at R approximately 0.40 for unlinked genes, giving s approximately 0.25 per locus and sigma approximately 3.7 km. In melpomene, cline shapes did not perfectly fit theoretical shapes based on dominance; this deviation might be explained by long-distance migration and/or strong epistasis. Compared with erato, sample sizes in melpomene are lower and the genetics of its color patterns are less well understood. In spite of these problems, selection and gene flow are clearly of the same order of magnitude in the two species. The relatively high per locus selection coefficients agree with "major gene" theories for the evolution of Müllerian mimicry, but the genetic architecture of the color patterns does not. These results show that the genetics and evolution of mimicry are still only sketchily understood.     

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.     

Evidence of restoration cost in the annual gynodioecious Phacelia dubia

A negative pleiotropic effect on fitness of nuclear sex‐determining genes (cost of restoration) could explain nuclear–cytoplasmic gynodioecy but rarely has been demonstrated empirically. In a gynodioecious Phacelia dubia population, maternal lineages produce only hermaphroditic progenies irrespective of the pollen parent (N) or can segregate females (S). Natural progenies of N maternal plants had lower seed viability than that of S. Full‐sib progenies of unrelated hermaphrodites from all possible matings between N and S lineages had similar pollen filling but differed in sporophyte performance, mainly at seed germination stage. A discrete multivariate analysis reveals that the performance of N_♀ × S_♂ progeny at early stages of development was significantly lower than that of the other three types of mating in agreement with the silent‐cost‐of‐restoration hypothesis, affecting the sporophyte. The restoration cost and male sterility appear to be dominant and consequence of nuclear–cytoplasmic incompatibilities that may maintain nuclear–cytoplasmic polymorphism by frequency‐dependent selection.     

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.     

The spatial genetic structure of cytoplasmic (cpDNA) and nuclear (allozyme) markers within and among populations of the gynodioecious Thymus vulgaris (Labiatae) in southern France

Recent advances in molecular biology have allowed the development of techniques to contrast spatial differentiation in nuclear and cytoplasmic genes and thus provide important data on relative levels of gene flow by pollen and seed in higher plants. In this paper, we compare the spatial structure of nuclear (allozymes) and cytoplasmic (cpDNA) genes among populations of the gynodioecious Thymus vulgaris in southern France. Based on a combination of three restriction enzymes (CfoI, EcoRV, and PstI), eight chlorotypes (combination of three restriction enzyme patterns revealed by Southern hybridization of Beta vulgaris cpDNA) were identified in the 13 studied populations. One chlorotype was particularly abundant and was detected in nearly all populations. Only one chlorotype was specific to a single population. Up to four different chlorotypes were observed in some populations. An FST of 0.238 (P < 0.002) for cpDNA haplotypes indicates spatial structure of cytoplasmic genes among the studied populations. Similar patterns were found within a single young population (CAB) structured in patches and surrounded by a continuous cover of T. vulgaris where the FST is 0.546 (P < 0.002). No significant correlation between sex and chlorotype nor between cpDNA diversity and female frequency was detected. Allozyme markers showed markedly less spatial structure (FST = 0.021 among populations and 0.019 in the CAB population, P < 0.001). This difference between cpDNA and nuclear allozyme markers suggests that pollen dispersal is more important than seed dispersal both among and within populations.     

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

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

NATURAL SELECTION ON QUANTITATIVE TRAITS IN THE BOMBINA HYBRID ZONE

Observations on the means, variances, and covariances of quantitative traits across hybrid zones can give information similar to that from Mendelian markers. In addition, they can identify particular traits through which the cline is maintained. We describe a survey of six traits across the hybrid zone between Bombina bombina and Bombina variegata (Amphibia: Discoglossidae) near Pe??enica in Croatia. We obtained laboratory measuments of the belly pattern, skin thickness, mating call, skeletal form, egg size, and the developmental time of tadpoles. Although offspring from hybrid populations showed no evidence of reduced viability, a third of the F₁ families failed completely, irrespective of the direction of the cross. All traits differed significantly between the taxa. Clines in belly pattern, skin thickness, mating call, and skeletal form were closely concordant with clines in four diagnostic enzyme loci. However, the cline in developmental time was displaced towards bombina, and the cline in egg size was displaced towards variegata. This discordance could be because the traits are not inherited additively or because they are subject to different selection pressures. We favor the latter explanation in the case of developmental time. We show that moderate selection acting directly on a trait suffices to shift its position; rather stronger selection is needed to change its width appreciably. Within hybrid populations, there are significant associations among quantitative traits, and between traits and enzymes. Phenotypic variances also increase in hybrid populations. These observations can be explained by linkage disequilibria among the underlying loci. However, the average magnitude of the covariance between traits is about half that expected from the linkage disequilibria between enzyme loci. The discrepancy is not readily explained by nonadditive gene action. This puzzle is now unresolved and calls for further investigation.     

Barriers to the spread of neutral alleles in the cytonuclear system

Neutral alleles can eventually pass a hybrid zone and their initial clines generated by a pure diffusion process dissipate with time, irrespective of the presence or absence of physical barriers. However, the transient neutral clines at the nuclear or organelle sites can be reinforced by the cytonuclear disequilibrium generated by diploid seed and haploid pollen dispersal. In this study, the spread of a neutral allele in an ecological zone of hermaphrodite plants is examined under three cytonuclear systems for genomes with contrasting modes of inheritance (paternal, maternal, and biparental inheritance). The results show that the transient neutral clines can exhibit the spatial pattern similar to the selective clines from separate genomes although discordance between them exists. The spread of a neutral allele is not only related to the vectors of seed and pollen dispersal but also to the mode of its inheritance. Pollen dispersal facilitates the direct effects of the selective organelle sites with paternal inheritance on the spread of a neutral nuclear allele. It also enhances its indirect effects on the spread of a neutral organelle allele with maternal inheritance via modifying the cytonuclear disequilibrium. A positive relationship exists between the barriers to the spread of selective nuclear (or organelle) and neutral organelle (or nuclear) alleles. An asymmetric barrier to the spread of the neutral alleles exists on the two sides of the physical barrier, given the presence of symmetric barrier to the spread of the selective alleles. These theoretical predictions highlight the effects of cytonuclear disequilibrium on the spread of a neutral allele and draw attention to our empirical cline analysis with neutral markers.     

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.     

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.