Difficulties in parentage analysis: the probability that an offspring and parent have the same heterozygous genotype.

Parentage studies often estimate the number of parents contributing to half-sib progeny arrays by counting the number of alleles attributed to unshared parents. This approach is compromised when an offspring has the same heterozygous genotype as the shared parent, for then the contribution of the unshared parent cannot be unambiguously deduced. To determine how often such cases occur, formulae for co-dominant markers with n alleles are derived here for Ph, the probability that a given heterozygous parent has an offspring with the same heterozygous genotype, and Pa, the probability that a randomly chosen offspring has the same heterozygous genotype as the shared parent. These formulae have been derived assuming Mendelian segregation with either (1) an arbitrary mating system, (2) random mating or (3) mixed mating. The maximum value of Pa under random mating is 0.25 and occurs with any two alleles each at a frequency of 0.5. The behaviour with partial selfing (where reproduction is by selfing with probability s, and random mating otherwise) is more complex. For n < or = 3 alleles, the maximum value of Pa occurs with any two alleles each at a frequency of 0.5 if s < 0.25, and with three equally frequent alleles otherwise. Numerically, the maximum value of Pa for n > or = 4 alleles occurs with n* < or = n alleles at equal frequencies, where the maximizing number of alleles n* is an increasing function of the selfing rate. Analytically, the maximum occurs with all n alleles present and equally frequent if s > or = 2/3. In addition, the potential applicability of these formulae for evolutionary studies is briefly discussed.

Accuracy and precision of methods to estimate the number of parents contributing to a half-sib progeny array.

Molecular technologies have made feasible large-scale studies of genetic parentage in nature by permitting the genotypic examination of hundreds or thousands of progeny. One common goal of such studies is to estimate the true number of unshared parents who contributed to a large half-sib progeny array. Here we introduce computer programs designed to count the number of gametotypes contributed by unshared parents to each such progeny array, as well as assess the accuracy and precision of various estimators for the true number of unshared parents via computer simulation. These simulations indicate that under most biological conditions (1) a traditional approach (the multilocus MINIMUM METHOD) that merely counts the number of distinct haplotypes in offspring and divides by 2L, where L is the number of loci assayed, often vastly underestimates the true number of unshared parents who contributed to a half-sib progeny array; (2) a recently developed HAPLOTYPES estimator is a considerable improvement over the MINIMUM METHOD when parental numbers are high; and (3) the accuracy and precision of the HAPLOTYPES estimator increase as marker polymorphism and sample size increase, or as reproductive skew and the number of parents contributing to the progeny array decrease. Generally, HAPLOTYPES-based estimates of parental numbers in large half-sib cohorts should improve the characterization of organismal reproductive strategies and mating systems from genetic data.

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.

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.

A formal assessment of gene flow and selection in the fire ant Solenopsis invicta.

Recent studies of the introduced fire ant Solenopsis invicta suggest that introduced polygyne (with multiple queens per nest) populations are strongly influenced by male-mediated gene flow from neighboring monogyne (single queen per nest) populations and selection acting on a single locus, general protein-9 (Gp-9). This investigation formally tests this hypothesis and determines if these processes can account for the genotypic structure of polygyne S. invicta. To increase the statistical power of this test, we considered the genotypes of polygyne queens and workers at both Gp-9 and the closely linked, selectively neutral locus Pgm-3. We then constructed and analyzed a novel mathematical model to delimit the effects of monogyne male gene flow and selection on the joint genotypes at the Pgm-3/Gp-9 superlocus. Using this framework, a hierarchical maximum-likelihood method was developed to estimate the best-fitting gene flow and selection parameters based on the fit of our model to data from both the current study and an earlier one of the same population. In each case, selection on polygyne queens and workers alone, with no monogyne male gene flow, provides the most parsimonious explanation for the observed genotype frequencies. The apparent discrepancy between this result and the empirical evidence for monogype male gene flow indicates that undocumented factors, such as other forms of selection in polygyne males or workers, are operating in introduced polygyne S. invicta.

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

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

The effects of 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 likelihood of homoploid hybrid speciation.

New species may be formed through hybridization and without an increase in ploidy. The challenge is for hybrid derivatives to escape the homogenizing effects of gene flow from parental species. The mechanisms hypothesized to underlie this process were modelled using a computer simulation. The model is of recombinational speciation, in which chromosomal rearrangements between parental species result in poor fertility of F1 hybrids, but through recombination, novel homozygous types are formed that have restored fertility. In simulations, stable populations bearing the recombinant karyotypes originated frequently and were maintained when the fertility of F1 hybrids was high. However, this high rate of origination was offset by low genetic isolation, and lower F1 hybrid fertility increased the evolutionary independence of derived populations. In addition, simulations showed that ecological and spatial isolation were required to achieve substantial reproductive isolation of incipient species. In the model, the opportunity for ecological isolation arose as a result of adaptation to extreme habitats not occupied by parental species, and any form of spatial isolation (e.g. founder events) contributed to genetic isolation. Our results confirmed the importance of the combination of factors that had been emphasized in verbal models and illustrate the trade-off between the frequency at which hybrid species arise and the genetic integrity of incipient species.

Symbiont survival and host-symbiont disequilibria under differential vertical transmission.

Interspecific genetic interactions in host-symbiont systems raise intriguing coevolutionary questions and may influence the effectiveness of public health and management policies. Here we present an analytical and numerical investigation of the effects of host genetic heterogeneity in the rate of vertical transmission of a symbiont. We consider the baseline case with a monomorphic symbiont and a single diallelic locus in its diploid host, where vertical transmission is the sole force. Our analysis introduces interspecific disequilibria to quantify nonrandom associations between host genotypes and alleles and symbiont presence/absence. The transient and equilibrium behavior is examined in simulations with randomly generated initial conditions and transmission parameters. Compared to the case where vertical transmission rates are uniform across host genotypes, differential transmission (i) increases average symbiont survival from 50% to almost 60%, (ii) dramatically reduces the minimum average transmission rate for symbiont survival from 0.5 to 0.008, and (iii) readily creates permanent host-symbiont disequilibria de novo, whereas uniform transmission can neither create nor maintain such associations. On average, heterozygotes are slightly more likely to carry and maintain the symbiont in the population and are more randomly associated with the symbiont. Results show that simple evolutionary forces can create substantial nonrandom associations between two species.

Effects of differential selection in the sexes on cytonuclear dynamics. Life stages with sex differences.

We extend our investigation of cytonuclear selection by determining when differential selection between the sexes will generate allele frequency changes or cytonuclear disequilibria in populations with constant viability selection and an adult census. We demonstrate analytically that there can be a cytonuclear hitchhiking effect upon a selectively neutral marker in either sex provided the other marker is selected in that sex and there is allelic disequilibrium between the loci in females. Cytonuclear disequilibria are generated de novo in both sexes when both loci affect fitness in females and there is a nonmultiplicative fitness interaction between them. Similar fitness interactions in males generate male disequilibria only. Through numerical analyses, we investigate the potential magnitude of such disequilibria, their qualitative dynamics, the expected frequency of detectable disequilibria under particular patterns or strengths of selection, and the possible disequilibrium sign patterns resulting from selection. These adult/viability results subsume those for populations with a gamete census and either constant fertility or viability selection. Although previous work suggests that the disequilibria generated by cytonuclear selection may be difficult to detect experimentally, this study shows that cytonuclear disequilibria at life stages with sex differences can be useful markers of the presence and strength of selection.

Detection of deleterious genotypes in multigenerational studies. I. Disruptions in individual Arabidopsis actin genes.

Plant actins are involved in numerous cytoskeletal processes effecting plant development, including cell division plane determination, cell elongation, and cell wall deposition. Arabidopsis thaliana has five ancient subclasses of actin with distinct patterns of spatial and temporal expression. To test their functional roles, we identified insertion mutants in three Arabidopsis actin genes, ACT2, ACT4, and ACT7, representing three subclasses. Adult plants homozygous for the act2-1, act4-1, and act7-1 mutant alleles appear to be robust, morphologically normal, and fully fertile. However, when grown as populations descended from a single heterozygous parent, all three mutant alleles were found at extremely low frequencies relative to the wild-type in the F2 generation. Thus, all three mutant alleles appear to be deleterious. The act2-1 mutant allele was found at normal frequencies in the F1, but at significantly lower frequencies than expected in the F2 and F3 generations. These data suggest that the homozygous act2-1/act2-1 mutant adult plants have a reduced fitness in the 2N sporophytic portion of the life cycle, consistent with the vegetative expression of ACT2. These data are interpreted in light of the extreme conservation of plant actin subclasses and genetic redundancy.

Detection of deleterious genotypes in multigenerational studies. II. Theoretical and experimental dynamics with selfing and selection.

A mathematical model was developed to help interpret genotype and allele frequency dynamics in selfing populations, with or without apomixis. Our analysis provided explicit time-dependent solutions for the frequencies at diallelic loci in diploid populations under any combination of fertility, viability, and gametic selection through meiotic drive. With no outcrossing, allelic variation is always maintained under gametic selection alone, but with any fertility or viability differences, variation will ordinarily be maintained if and only if the net fitness (fertility x viability) of heterozygotes exceeds that of both homozygotes by a substantial margin. Under pure selfing and Mendelian segregation, heterozygotes must have a twofold fitness advantage; the level of overdominance necessary to preserve genetic diversity declines with apomixis, and increases with segregation distortion if this occurs equally and independently in male and female gametes. A case study was made of the Arabidopsis act2-1 actin mutant over multiple generations initiated from a heterozygous plant. The observed genotypic frequency dynamics were consistent with those predicted by our model for a deleterious, incompletely recessive mutant in either fertility or viability. The theoretical framework developed here should be very useful in dissecting the form(s) and strength of selection on diploid genotypes in populations with negligible levels of outcrossing.

Genetic diversity at a single locus under viability selection and facultative apomixis: equilibrium structure and deviations from Hardy-Weinberg frequencies.

We extensively analyze the maintenance of genetic variation and deviations from Hardy-Weinberg frequencies at a diallelic locus under mixed mating with apomixis and constant viability selection. Analytical proofs show that: (1) at most one polymorphic equilibrium exists, (2) polymorphism requires overdominant or underdominant selection, and (3) a simple, modified overdominance condition is sufficient to maintain genetic variation. In numerical analyses, only overdominant polymorphic equilibria are stable, and these are stable whenever they exist, which happens for approximately 78% of random fitness and mating parameters. The potential for maintaining both alleles increases with increasing apomixis or outcrossing and decreasing selfing. Simulations also indicate that equilibrium levels of heterozygosity will often be statistically indistinguishable from Hardy-Weinberg frequencies and that adults, not seeds, should usually be censused to maximize detecting deviations. Furthermore, although both censuses more often have an excess rather than a deficit of heterozygotes, analytical sign analyses of the fixation indices prove that, overall, adults are more likely to have an excess and seeds a deficit at equilibrium.

Cytonuclear theory for haplodiploid species and X-linked genes. I. Hardy-Weinberg dynamics and continent-island, hybrid zone models.

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

The 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.

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 no nonmultiplicative fitness interaction between them. While computer simulations demonstrate that the majority of disequilibria produced by random selection are transient and small in magnitude, measurable permanent disequilibria can result from selective differences both within and between the two sexes. We derive analytic conditions for a protected cytonuclear polymorphism and use numerical simulations to quantitate the likelihood of obtaining permanent nuclear, cytoplasmic, and cytonuclear variation under various patterns of selection. The numerical analysis identifies special selection regimes more likely to generate disequilibria and maintain cytonuclear polymorphism and reveals a direct correlation to the strength of selection. As a byproduct, our models also provide the first decomposition of the different parental contributions to cytonuclear dynamics and the analytic conditions under which selection can cause cytoplasmic frequency changes or a cytonuclear hitchhiking effect.

Cytonuclear 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 of 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.

Constraints and normalized measures for cytonuclear disequilibria.

The full bounds are derived for cytonuclear disequilibria in two-locus systems with an arbitrary number of alleles at the cytoplasmic and nuclear markers. The associated marginal frequencies constrain the nonrandom associations between cytoplasmic alleles and nuclear genotypes in the same way that the allele frequencies constrain the linkage disequilibrium between two nuclear loci. Additional constraints are imposed on the nonrandom associations between cytoplasmic and nuclear alleles, however, by the marginal frequencies of nuclear genotypes carrying either two or no copies of the associated nuclear allele. These bounds are analysed and used to define normalized measures of cytonuclear disequilibria, whose practical utility is illustrated through applications to two sets of recent nuclear-mitochondrial data.

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.

Estimation of mitotic stability in conidial fungi: a theoretical framework.

Mitotic stability refers to the probability that genetic elements are transmitted to both daughters during mitosis. This is of practical importance in molecular genetics because autonomous cloning vectors should be transmitted at high frequency during mitosis. In filamentous coencytic fungi it is difficult to quantify mitotic stability because a fluctuation test is not feasible. We show how to get around this problem by formulating a general model of the transmission of nuclear genetic elements through the course of conidiogenesis. We derive formulas by two different methods for the expected proportion of conidiospores that retain the element as a function of its mitotic stability and the number of generations of spore production. An important by-product yields the exact probability distributions for the number of conidiospores retaining elements at each stage of conidiophore development. We outline, and illustrate through specific numerical examples, how to use these formulas to estimate mitotic stability. Although we use Aspergillus nidulans as our biological paradigm, the same general framework can be extended to other fungal species, and possibly to less closely related systems as well.