Analysis of multilocus zygotic associations

While nonrandom associations between zygotes at different loci (zygotic associations) frequently occur in Hardy-Weinberg disequilibrium populations, statistical analysis of such associations has received little attention. In this article, we describe the joint distributions of zygotes at multiple loci, which are completely characterized by heterozygosities at individual loci and various multilocus zygotic associations. These zygotic associations are defined in the same fashion as the usual multilocus linkage (gametic) disequilibria on the basis of gametic and allelic frequencies. The estimation and test procedures are described with details being given for three loci. The sampling properties of the estimates are examined through Monte Carlo simulation. The estimates of three-locus associations are not free of bias due to the presence of two-locus associations and vice versa. The power of detecting the zygotic associations is small unless different loci are strongly associated and/or sample sizes are large (>100). The analysis of zygotic associations not only offers an effective means of packaging numerous genic disequilibria required for a complete characterization of multilocus structure, but also provides opportunities for making inference about evolutionary and demographic processes through a comparative assessment of zygotic association vs. gametic disequilibrium for the same set of loci in nonequilibrium populations.     

Multilocus and Multitrait Measures of Differentiation for Gene Markers and Phenotypic Traits

Multilocus measures of differentiation taking into account gametic disequilibrium are developed. Even if coupling and repulsion heterozygotes cannot be separated at the multilocus level, a method is given to calculate a composite measure of differentiation (CF(st)) at the zygotic level, which accounts for allelic associations combining both gametic and nongametic effects. Mean and maximum differentiations may be relevant when multilocus measures are computed. Maximum differentiation is the highest eigenvalue of the F(st) matrix, whereas mean differentiation corresponds to the mean value of all eigenvalues of the F(st) matrix. Gametic disequilibrium has a stronger effect on maximum differentiation than on mean differentiation and takes into account the anisotropy that may exist between within- and between-population components of disequilibria. Multilocus mean and maximum differentiation are calculated for a set of 81 Quercus petraea (sessile oak) populations assessed with eight allozyme loci and two phenotypic traits (bud burst and height growth). The results indicate that maximum differentiation increases as more loci (traits) are considered whereas mean differentiation remains constant or decreases. Phenotypic traits exhibit higher population differentiation than allozymes. The applications and uses of mean and maximum differentiations are further discussed.     

Properties of the multiallelic trend test

Disease genes can be mapped on the basis of associations between genetic markers and disease status, with the case-control design having the advantage of not requiring individuals from different generations. When the marker loci have multiple alleles, there has been debate on whether the power of tests for association increases or decreases. We show here that the multiple-allele version of Armitage's trend test has increased power over the two-allele version under the requirement of equifrequent alleles, but not in general. The trend test has the advantage of remaining valid even when the sampled population is not in Hardy-Weinberg equilibrium. A departure from Hardy-Weinberg means that association tests depend on gametic and nongametic linkage disequilibrium between marker and disease loci, and we illustrate the magnitude of these effects with simulated data.     

Hitchhiking: A Comparison of Linkage and Partial Selfing

Genetic hitchhiking occurs when alleles at unselected loci are changed in frequency because of an association with alleles at a selected locus. This association may be mediated either by linkage or partial selfing (inbreeding) and can affect the gene frequency and gametic disequilibrium at the neutral loci. Hitchhiking from partial selfing (unlinked loci) occurs more quickly than linkage hitchhiking and generally has a greater effect. In addition, partial-selfing hitchhiking can cause increases or changes in sign in gametic disequilibrium between neutral loci. The effects of the two types of hitchhiking with different levels of dominance, zygotic frequencies and number of selected loci are also examined. The general conditions for linkage and partial-selfing hitchhiking are outlined and the implications of hitchhiking are discussed for marker or electrophoretic loci.     

Sampling Theory for Cytonuclear Disequilibria

We examine the statistical properties of cytonuclear disequilibria within a system including one diploid nuclear locus and one haploid cytoplasmic locus, each with two alleles. The results provide practical guidelines for the design and interpretation of cytonuclear surveys seeking to utilize the novel evolutionary information recorded in the observed pattern of cytonuclear associations. Important applications include population studies of nuclear allozymes in conjunction with genes from mitochondria, chloroplasts, or cytoplasmically inherited microorganisms. Our attention focuses on the allelic and genotypic disequilibria, which respectively measure the nonrandom associations between the cytotypes and the nuclear alleles and genotypes. We first derive the maximum likelihood estimators and their approximate large sample variances for each disequilibrium measure. These are each in turn used to set up an asymptotic test of the null hypothesis of no disequilibrium. We then calculate the minimum sample sizes required to detect the disequilibria under specified alternate hypotheses. The work also incorporates the deviation from Hardy-Weinberg equilibrium at the nuclear locus, which can significantly affect the results. The practical utility of this new sampling theory is illustrated through applications to two nuclear-mitochondrial data sets.     

The multiple-locus symmetric fertility model

Selection due to variation in the fecundity among matings of genotypes with respect to many loci each with two alleles is studied. The fitness of a mating depends only on the genotypic distinction between homozygote and heterozygote at each locus in the two individuals, and differences among loci are allowed. This symmetric fertility model is therefore a generalization of the multiple-locus symmetric viability model. The phenomena seen in the two-locus symmetric fertility model generalize—e.g., the possibility of joint stability of equilibria with linkage equilibrium and with linkage disequilibrium, and the existence of different types of totally polymorphic equilibria with the gametic proportions in linkage equilibrium. The central equilibrium with genotypic frequencies in Hardy-Weinberg proportions and gametic frequencies in Robbins proportions exists for all symmetric fertility models. For some symmetric fertility regimes additional equilibria exist with gametic frequencies in linkage equilibrium and with genotypic frequencies in Hardy-Weinberg proportions at all except one locus. These equilibria may exist in the dioecious symmetric viability model, and then they will be locally stable. For free recombination the stable equilibria show linkage equilibrium, but several of these with different numbers of polymorphic loci may be stable simultaneously.     

Characterization of 12 polymorphic microsatellite markers for a facultatively eusocial sweat bee (Megalopta genalis)

We developed a library of twelve polymorphic di- and tri-nucleotide microsatellite markers for Megalopta genalis, a facultatively eusocial sweat bee. We tested each locus in a panel of 23 unrelated females and found 7-20 alleles per locus. Observed and expected heterozygosities ranged from 0.65 to 0.96 and from 0.69 to 0.95 respectively. None of the loci deviated from Hardy-Weinberg equilibrium proportions or was found to be in gametic disequilibrium.     

Estimation of Hardy-Weinberg and pairwise disequilibrium in the apolipoprotein AI-CIII-AIV gene cluster.

Departures from Hardy-Weinberg (HW) equilibria and pairwise disequilibria were estimated in a sample of unrelated healthy individuals typed for six RFLPs in the apo AI-CIII-AIV gene region. The sample was composed of males and females, selected for health, from two populations, those of exclusively French-Canadian (FC) and those of some non-French-Canadian (NFC) ancestry. An approach suggested by Weir and Cockerham, which includes estimates of nonrandom association (disequilibria) between three and four alleles at two loci as well as the traditional associations between two alleles, at two loci was used. The pattern of departures from HW equilibria suggested that the genetic structures of the FC and NFC are different. Departure from HW equilibrium at an RFLP locus could not be predicted from information about other loci in the same gene region. Nonrandom associations were also evident from the pairwise analyses. Two pairs of loci had significant diallelic disequilibria, while two other pairs had significant triallelic disequilibria. All of the RFLP pairs had at least one measure of disequilibrium at its maximum value determined by allele frequencies. Inferences about pairwise disequilibria depended on the statistical approach used. Sizes of the pairwise disequilibria were not correlated with the physical distance between loci. The impact of these disequilibria on RFLP-phenotype association studies is discussed.     

Variance Analysis of Immunoglobulin Alleles in Natural Populations of Rabbit (Oryctolagus Cuniculus): The Extensive Interallelic Divergence at the B Locus Could Be the Outcome of Overdominance-Type Selection

Population genetic data are presented which should contribute to evaluation of the hypothesis that the extraordinary evolutionary patterns observed at the b locus of the rabbit immunoglobulin light chain constant region can be the outcome of overdominance-type selection. The analysis of allele correlations in natural populations revealed an excess of heterozygotes of about 10% at the b locus while heterozygote excess was not observed at loci determining the immunoglobulin heavy chain. Data from the published literature, where homozygote advantage was suggested, were reevaluated and found in agreement with data here presented. Gene diversity was evenly distributed among populations and showed similarities with patterns reported for histocompatibility loci. Analysis of genotypic disequilibria revealed strong digenic associations between the leading alleles of heavy and light chain constant region loci in conjunction with trigenic disequilibria corresponding to a preferential association of b locus heterozygosity with the predominant allele of the heavy chain e locus. It is argued that this may indicate compensatory or nonadditive aspects of a putative heterozygosity enhancing mechanism, implying that effects at the light chain might be more pronounced in populations fixed for the heavy chain polymorphism.     

Evolution of Zygotic Linkage Disequilibrium in a Finite Local Population

One crucial feature of zygotic linkage disequilibrium (LD) analysis is its direct use of diploid genotyping data, irrespective of the type of mating system. Previous theories from an evolutionary perspective mainly focus on gametic LD, but the equivalent development for zygotic LD is not available. Here I study the evolution of zygotic LD and the covariances between gametic and zygotic LDs or between distinct zygotic LDs in a finite local population under constant immigration from a continent population. I derive the analytical theory under genetic hitchhiking effects or in a neutral process. Results indicate that zygotic LDs (diploid level) are more informative than gametic LD (haploid level) in indicating the effects of different evolutionary forces. Zygotic LDs may be greater than or comparable to gametic LD under the epistatic selection process, but smaller than gametic LD under the non epistatic selection process. The covariances between gametic and zygotic LDs are strongly affected by the mating system, linkage distance, and genetic drift effects, but weakly affected by seed and pollen flow and natural selection. The covariances between different zygotic LDs are generally robust to the effects of gene flow, selection, and linkage distance, but sensitive to the effects of genetic drift and mating system. Consistent patterns exist for the covariances between the zygotic LDs for the two-locus genotypes with one common genotype at one locus or without any common genotype at each locus. The results highlight that zygotic LDs can be applied to detecting natural population history.     

Genetic structure of five susceptibility gene regions for coronary artery disease: disequilibria within and among regions

We analyzed two-locus disequilibria for 16 polymorphic loci of seven susceptibility genes for coronary artery disease located in five chromosomal regions distributed across four chromosomes. Included were the genes coding for apolipoprotein B (ApoB, chromosome 2, four marker loci), lipoprotein lipase (LPL, chromosome 8, three marker loci), apolipoproteins AI, CIII, AIV (ApoAI–CIII–AIV, chromosome 11, three marker loci), apolipoprotein E (ApoE, chromosome 19, two marker loci), and the low density lipoprotein receptor (LDLR, chromosome 19, four marker loci). Our sample included 540 unrelated individuals from the Rochester, Minn. population. There were no statistically significant deviations of single-locus genotypes from Hardy-Weinberg equilibrium. The strongest associations within genes were for composite diallelic disequilibria; 17/19 were significant (13 at Pr <0.001, 1 at Pr <0.01, 3 at Pr <0.05). These observations suggest marker alleles within genes have a shared evolutionary history reflected by disequilibria that have not been dissipated by recombination. Disequilibrium was not generally concordant with the physical orderings of markers. Only two significant higher-order disequilibria were observed although 12 triallelic disequilibria were at maximum possible values. We observed 19 statistically significant disequilibria (Pr <0.05; 4 composite diallelic, 13 triallelic, and 2 quadriallelic) between 101 pairs of marker loci, where each locus in a pair was from a different unlinked region. These unexpected results are most likely explained by recent historical factors, including worldwide population expansion and amalgamation with continuous admixture, that influence the genetic structure (organization of alleles and non-alleles into genotypes) of a population. We conclude that disequilibria between loci from unlinked regions may be more extensive than is commonly assumed. Our findings also suggest that it is, on average, at least 15 times more likely to not detect significant disequilibrium among unlinked loci when it is really present than to make a false positive inference. Disequilibria between functional loci within or between regions will impact estimates of genetic variance associated with particular functional mutations within a susceptibility gene region. Received: 15 January 1998 / Accepted: 24 June 1998     

The New Rga Locus Encodes a Negative Regulator of Gibberellin Response in Arabidopsis Thaliana

We have identified a new locus involved in gibberellin (GA) signal transduction by screening for suppressors of the Arabidopsis thaliana GA biosynthetic mutant ga1-3. The locus is named RGA for repressor of ga1-3. Based on the recessive phenotype of the digenic rga/ga1-3 mutant, the wild-type gene product of RGA is probably a negative regulator of GA responses. Our screen for suppressors of ga1-3 identified 17 mutant alleles of RGA as well as 10 new mutant alleles at the previously identified SPY locus. The digenic (double homozygous) rga/ga1-3 mutants are able to partially repress several defects of ga1-3 including stem growth, leaf abaxial trichome initiation, flowering time, and apical dominance. The phenotype of the trigenic mutant (triple homozygous) rga/spy/ga1-3 shows that rga and spy have additive effects regulating flowering time, abaxial leaf trichome initiation and apical dominance. This trigenic mutant is similar to wild type with respect to each of these developmental events. Because rga/spy/ga1-3 is almost insensitive to GA for hypocotyl growth and its bolting stem is taller than the wild-type plant, the combined effects of the rga and spy mutations appear to allow GA-independent stem growth. Our studies indicate that RGA lies on a separate branch of the GA signal transduction pathway from SPY, which leads us to propose a modified model of the GA response pathway.     

Genetic Structure and the Search for Genotype-Phenotype Relationships: An Example from Disequilibrium in the Apo B Gene Region

We analyzed allelic associations (disequilibria) for four restriction fragment length polymorphisms (RFLPs) in the region of the 43-kb Apo B gene in a sample of 233 unrelated individuals from Montreal, Canada, sampled for health. This total sample (T) included 160 individuals of known French Canadian (FC) ancestry. We present a rigorous application of current methodology to these samples, including estimation of type II error probabilities and correlations between markers for estimates of disequilibria. We then consider the utility of these estimates of allelic disequilibria for the interpretation of genotype-phenotype relations. Significant deviations from Hardy-Weinberg equilibrium were not predicted by proximity to other markers in disequilibrium. We found significant quadri-allelic disequilibrium for two marker pairs despite absence of significant deviations from Hardy-Weinberg equilibrium for either marker or tri-allelic disequilibrium, respectively. Altogether these results underscore the complexity of the genotypic structure of the data. A combination of nonevolutionary factors, including sampling for health, small sample size and data exclusion due to methodological constraints of not successfully typing all members of the sample for every RFLP, is a likely explanation for this complexity. These types of factors are common to many RFLP studies. Patterns of composite di-allelic disequilibrium indicated that some RFLP allele pairs may have a longer shared evolutionary history than others and that disequilibrium is not predicted by distance between RFLPs. Type II error probabilities were generally much higher than those for type I errors. Correlations between marker pairs for disequilibria were generally not high. We show from a review of 14 published studies of association between the XbaI RFLP and variation in a total of 15 lipid traits that deviations from Hardy-Weinberg equilibrium can cause substantial differences in the estimation of variability associated with phenotypic differences among marker genotypes relative to Hardy-Weinberg conditions.     

The effect of positive assortative mating at one locus on a second linked locus

Summary Considerations proceed from a model of positive assortative mating based on genotype at one locus, with an arbitrary number of alleles, assuming no selection, mutation, or migration, hypothetically infinite population size, and discrete non-overlapping generations. From these conditions, inferences are made about the genotypic structure at a linked locus, as well as about the corresponding 2-locus gametic structure.The following main results are presented: in the course of the generations, the genotypic structure at the second locus and the 2-locus gametic structure always tend to a limit responsive to the initial conditions concerning the joint genotypic structure at the two loci and the degree of assortativity and linkage. A complete, analytical representation of the limits is given. In particular, if assortative mating is only partial and at the same time linkage is not complete, a population is not able to maintain a permanent deviation of the gametic structure from linkage equilibrium, and thus the genotypic structure at the second locus tends to Hardy-Weinberg proportions. On the other hand, if initial linkage disequilibrium is combined with partial assortative mating and complete linkage (or with complete assortative mating and unlinked loci) the population maintains this disequilibrium and thus the genotypic structure at the second locus need not tend to Hardy-Weinberg proportions. It turns out that the conditions not only of complete linkage, but also of unlinked loci together with complete assortativity, imply no change in gametic structure from the initial structure.In order to demonstrate the influence of several parameters on the speed of convergence to and the magnitude of the respective limits, several graphs are included.     

Effect of gene conversion on variances of digenic identity measures

The variances and covariances of digenic descent measures are studied for a two-locus model incorporating mutation, gene conversion, recombination, drift, and finite sampling. Gene conversion can occur between allelic pairs of genes or between non-allelic pairs on the same or different gametes within individuals. Most interest therefore centers on pairs of genes, and five digenic identity measures are required. The behavior over time of these measures is studied, with an emphasis on the effects of gene conversion. Because of the stochastic nature of the forces of drift, recombination, mutation, and conversion, the actual identity status of gene pairs can vary from expectation among replicate populations. To study this variation we compute the expected variances and covariances of the measures, and show that this requires the introduction of trigenic and quadrigenic measures. Allowing for conversion between genes on different gametes requires a large number of these higher-order measures.     

Isolation and characterization of 17 microsatellite loci for the house finch (Carpodacus mexicanus)

The house finch (Carpodacus mexicanus) has emerged recently as a model species in studies of sexual selection, reproductive physiology, population genetics, and epizootic disease ecology. Here we describe 17 highly polymorphic microsatellite loci for this species. In a sample of 36 individuals, we observed an average of 16 alleles per locus and heterozygosity ranged from 0.61 to 0.97. One locus showed significant deviation from Hardy-Weinberg proportions, but no significant gametic disequilibrium was observed among any of the loci. Amplification by polymerase chain reaction was optimized under similar parameters across loci, thereby facilitating multiplexing and rapid multilocus genotyping.     

Definition and Properties of Disequilibrium Statistics for Associations between Nuclear and Cytoplasmic Genotypes

We define and establish the interrelationships of four components of statistical association between a diploid nuclear gene and a uniparentally transmitted, haploid cytoplasmic gene: an allelic (gametic) disequilibrium (D), which measures associations between alleles at the two loci; and three genotypic disequilibria (D1, D2, D3), which measure associations between two cytotypes and the three respective nuclear backgrounds. We also consider an alternative set of measures, including D and the residual disequilibrium (d). The dynamics of these disequilibria are then examined under three conventional models of the mating system: (1) random mating; (2a) assortative mating without dominance (the "mixed-mating model"); and (2b) assortative mating with dominance ("O'Donald's model"). The trajectories of gametic disequilibria are similar to those for pairs of unlinked nuclear loci. The dynamics of genotypic disequilibria exhibit a variety of behaviors depending on the model and the initial conditions. Procedures for statistical estimation of cytonuclear disequilibria are developed and applied to several real and hypothetical data sets. Special attention is paid to the biological interpretations of various categories of allelic and genotypic disequilibria in hybrid zones. Genetic systems for which these statistics might be appropriate include nuclear genotype frequencies in conjunction with those for mitochondrial DNA, chloroplast DNA, or cytoplasmically inherited microorganisms.     

Linkage disequilibrium among multiple neutral alleles produced by mutation in finite population.

An analysis is undertaken for a finite random mating population of the linkage disequilibrium between two loci, at both of which all alleles are neutral, all mutant alleles differ from existing ones and several may be segregating at any time. Formulae are derived for the expected total squared disequilibrium, measured as the sum of squares of disequilibria between all pairs of alleles. The ratio of this quantity to the expected value of the product of the heterozygosities at the two loci is similar to that obtained previously by Ohta and Kimura for two nucleotide sites at each of which not more than two mutant types can segregate at any time.