Modeling epistasis of quantitative trait loci using Cockerham's model

We use the orthogonal contrast scales proposed by Cockerham to construct a genetic model, called Cockerham's model, for studying epistasis between genes. The properties of Cockerham's model in modeling and mapping epistatic genes under linkage equilibrium and disequilibrium are investigated and discussed. Because of its orthogonal property, Cockerham's model has several advantages in partitioning genetic variance into components, interpreting and estimating gene effects, and application to quantitative trait loci (QTL) mapping when compared to other models, and thus it can facilitate the study of epistasis between genes and be readily used in QTL mapping. The issues of QTL mapping with epistasis are also addressed. Real and simulated examples are used to illustrate Cockerham's model, compare different models, and map for epistatic QTL. Finally, we extend Cockerham's model to multiple loci and discuss its applications to QTL mapping.     

Modeling quantitative trait Loci and interpretation of models

A quantitative genetic model relates the genotypic value of an individual to the alleles at the loci that contribute to the variation in a population in terms of additive, dominance, and epistatic effects. This partition of genetic effects is related to the partition of genetic variance. A number of models have been proposed to describe this relationship: some are based on the orthogonal partition of genetic variance in an equilibrium population. We compare a few representative models and discuss their utility and potential problems for analyzing quantitative trait loci (QTL) in a segregating population. An orthogonal model implies that estimates of the genetic effects are consistent in a full or reduced model in an equilibrium population and are directly related to the partition of the genetic variance in the population. Linkage disequilibrium does not affect the estimation of genetic effects in a full model, but would in a reduced model. Certainly linkage disequilibrium would complicate the detection of QTL and epistasis. Using different models does not influence the detection of QTL and epistasis. However, it does influence the estimation and interpretation of genetic effects.     

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.     

Genetic expectations of quantitative trait loci main and interaction effects obtained with the triple testcross design and their relevance for the analysis of heterosis

Interpretation of experimental results from quantitative trait loci (QTL) mapping studies on the predominant type of gene action can be severely affected by the choice of statistical model, experimental design, and provision of epistasis. In this study, we derive quantitative genetic expectations of (i) QTL effects obtained from one-dimensional genome scans with the triple testcross (TTC) design and (ii) pairwise interactions between marker loci using two-way analyses of variance (ANOVA) under the F(2)- and the F(infinity)-metric model. The theoretical results show that genetic expectations of QTL effects estimated with the TTC design are complex, comprising both main and epistatic effects, and that genetic expectations of two-way marker interactions are not straightforward extensions of effects estimated in one-dimensional scans. We also demonstrate that the TTC design can partially overcome the limitations of the design III in separating QTL main effects and their epistatic interactions in the analysis of heterosis and that dominance x additive epistatic interactions of individual QTL with the genetic background can be estimated with a one-dimensional genome scan. Furthermore, we present genetic expectations of variance components for the analysis of TTC progeny tested in a split-plot design, assuming digenic epistasis and arbitrary linkage.     

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.     

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.     

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.     

A haplotype-based algorithm for multilocus linkage disequilibrium mapping of quantitative trait loci with epistasis

For tightly linked loci, cosegregation may lead to nonrandom associations between alleles in a population. Because of its evolutionary relationship with linkage, this phenomenon is called linkage disequilibrium. Today, linkage disequilibrium-based mapping has become a major focus of recent genome research into mapping complex traits. In this article, we present a new statistical method for mapping quantitative trait loci (QTL) of additive, dominant, and epistatic effects in equilibrium natural populations. Our method is based on haplotype analysis of multilocus linkage disequilibrium and exhibits two significant advantages over current disequilibrium mapping methods. First, we have derived closed-form solutions for estimating the marker-QTL haplotype frequencies within the maximum-likelihood framework implemented by the EM algorithm. The allele frequencies of putative QTL and their linkage disequilibria with the markers are estimated by solving a system of regular equations. This procedure has significantly improved the computational efficiency and the precision of parameter estimation. Second, our method can detect marker-QTL disequilibria of different orders and QTL epistatic interactions of various kinds on the basis of a multilocus analysis. This can not only enhance the precision of parameter estimation, but also make it possible to perform whole-genome association studies. We carried out extensive simulation studies to examine the robustness and statistical performance of our method. The application of the new method was validated using a case study from humans, in which we successfully detected significant QTL affecting human body heights. Finally, we discuss the implications of our method for genome projects and its extension to a broader circumstance. The computer program for the method proposed in this article is available at the webpage http://www.ifasstat.ufl.edu/genome/~LD.     

Identifying quantitative trait locus by genetic background interactions in association studies

Association studies are designed to identify main effects of alleles across a potentially wide range of genetic backgrounds. To control for spurious associations, effects of the genetic background itself are often incorporated into the linear model, either in the form of subpopulation effects in the case of structure or in the form of genetic relationship matrices in the case of complex pedigrees. In this context epistatic interactions between loci can be captured as an interaction effect between the associated locus and the genetic background. In this study I developed genetic and statistical models to tie the locus by genetic background interaction idea back to more standard concepts of epistasis when genetic background is modeled using an additive relationship matrix. I also simulated epistatic interactions in four-generation randomly mating pedigrees and evaluated the ability of the statistical models to identify when a biallelic associated locus was epistatic to other loci. Under additive-by-additive epistasis, when interaction effects of the associated locus were quite large (explaining 20% of the phenotypic variance), epistasis was detected in 79% of pedigrees containing 320 individuals. The epistatic model also predicted the genotypic value of progeny better than a standard additive model in 78% of simulations. When interaction effects were smaller (although still fairly large, explaining 5% of the phenotypic variance), epistasis was detected in only 9% of pedigrees containing 320 individuals and the epistatic and additive models were equally effective at predicting the genotypic values of progeny. Epistasis was detected with the same power whether the overall epistatic effect was the result of a single pairwise interaction or the sum of nine pairwise interactions, each generating one ninth of the epistatic variance. The power to detect epistasis was highest (94%) at low QTL minor allele frequency, fell to a minimum (60%) at minor allele frequency of about 0.2, and then plateaued at about 80% as alleles reached intermediate frequencies. The power to detect epistasis declined when the linkage disequilibrium between the DNA marker and the functional polymorphism was not complete.     

The Evolution of One- and Two-Locus Systems

Assuming age-independent fertilities and mortalities and random mating, continuous-time models for a monoecious population are investigated for weak selection. A single locus with multiple alleles and two alleles at each of two loci are considered. A slow-selection analysis of diallelic and multiallelic two-locus models with discrete nonoverlapping generations is also presented. The selective differences may be functions of genotypic frequencies, but their rate of change due to their explicit dependence on time (if any) must be at most of the second order in s, (i.e., O( s²)), where s is the intensity of natural selection. Then, after several generations have elapsed, in the continuous time models the time-derivative of the deviations from Hardy-Weinberg proportions is of O(s²), and in the two-locus models the rate of change of the linkage disequilibrium is of O(s²). It follows that, if the rate of change of the genotypic fitnesses is smaller than second order in s (i.e., o(s²)), then to O(s²) the rate of change of the mean fitness of the population is equal to the genic variance. For a fixed value of s, however, no matter how small, the genic variance may occasionally be smaller in absolute value than the (possibly negative) lower order terms in the change in fitness, and hence the mean fitness may decrease. This happens if the allelic frequencies are changing extremely slowly, and hence occurs often very close to equilibrium. Some new expressions are derived for the change in mean fitness. It is shown that, with an error of O( s), the genotypic frequencies evolve as if the population were in Hardy-Weinberg proportions and linkage equilibrium. Thus, at least for the deterministic behavior of one and two loci, deviations from random combination appear to have very little evolutionary significance.     

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

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

Zygotic associations and multilocus statistics in a nonequilibrium diploid population

The usual approach to characterizing and estimating multilocus associations in a diploid population assumes that the population is in Hardy-Weinberg equilibrium. The purpose of this study is to develop a set of summary statistics that can be used to characterize and estimate the multilocus associations in a nonequilibrium population. The concept of "zygotic associations" is first expanded to facilitate the development. The summary statistics are calculated using the distribution of a random variable, the number of heterozygous loci (K) found in diploid individuals in the population. In particular, the variance of K consists of single-locus and multilocus components with the latter being the sum of zygotic associations between pairs of loci. Simulation results show that the multilocus associations in the variance of K are detectable in a sample of moderate size (> or =30) when the sum of all pairwise zygotic associations is greater than zero and when gene frequency is intermediate. The method presented here is a generalization of the well-known development for the Hardy-Weinberg equilibrium population and thus may be of more general use in elucidating the multilocus organizations in nonequilibrium and equilibrium populations.     

Defining the assumptions underlying modeling of epistatic QTL using variance component methods

Variance component models are commonly used to detect quantitative trait loci (QTL) in general pedigrees. The variance-covariance structure of the random QTL effect is given by the identity by descent (IBD) between genotypes. Epistatic effects have previously been modeled, both for unlinked and linked loci, as a random effect with a variance-covariance structure given by the Hadamard product between the IBD matrices of the direct QTL effects. In the original papers, the model was given but not derived. Here, we identify the underlying assumptions of this previously proposed model. It assumes that either an unlinked QTL or a fully informative marker (i.e., all marker alleles are unique in the base generation) is located between the loci. We discuss the need of developing a general algorithm to estimate the variance-covariance structure of the random epistatic effect for linked loci.     

Linkage and the Maintenance of Heritable Variation by Mutation-Selection Balance

The role of linkage in influencing heritable variation maintained through a balance between mutation and stabilizing selection is investigated for two different models. In both cases one trait is considered and the interactions within and between loci are assumed to be additive. Contrary to most earlier investigations of this problem no a priori assumptions on the distribution of genotypic values are imposed. For a deterministic two-locus two-allele model with recombination and mutation, related to the symmetric viability model, a complete nonlinear analysis is performed. It is shown that, depending on the recombination rate, multiple stable equilibria may coexist. The equilibrium genetic and genic variances are calculated. For a polygenic trait in a finite population with a possible continuum of allelic effects a simulation study is performed. In both models the equilibrium genetic and genic variances are roughly equal to the house-of-cards prediction or its finite population counterpart as long as the recombination rate is not extremely low. However, negative linkage disequilibrium builds up. If the loci are very closely linked the equilibrium additive genetic variance is slightly lower than the house-of-cards prediction, but the genic variance is much higher. Depending on whether the parameters are in favor of the house-of-cards or the Gaussian approximation, different behavior of the genetic system occurs with respect to linkage.     

Simultaneous fine mapping of closely linked epistatic quantitative trait loci using combined linkage disequilibrium and linkage with a general pedigree

Causal mutations and their intra- and inter-locus interactions play a critical role in complex trait variation. It is often not easy to detect epistatic quantitative trait loci (QTL) due to complicated population structure requirements for detecting epistatic effects in linkage analysis studies and due to main effects often being hidden by interaction effects. Mapping their positions is even harder when they are closely linked. The data structure requirement may be overcome when information on linkage disequilibrium is used. We present an approach using a mixed linear model nested in an empirical Bayesian approach, which simultaneously takes into account additive, dominance and epistatic effects due to multiple QTL. The covariance structure used in the mixed linear model is based on combined linkage disequilibrium and linkage information. In a simulation study where there are complex epistatic interactions between QTL, it is possible to simultaneously map interacting QTL into a small region using the proposed approach. The estimated variance components are accurate and less biased with the proposed approach compared with traditional models.     

A general model for detecting genetic determinants underlying longitudinal traits with unequally spaced measurements and nonstationary covariance structure

A mixture model for determining quantitative trait loci (QTL) affecting growth trajectories has been proposed in the literature. In this article, we extend this model to a more general situation in which longitudinal traits for each subject are measured at unequally spaced time intervals, different subjects have different measurement patterns, and the residual correlation within subjects is nonstationary. We derive an EM-simplex hybrid algorithm to estimate the allele frequencies, Hardy-Weinberg disequilibrium, and linkage disequilibrium between QTL in the original population and parameters contained in the growth equation and in the covariance structure. A worked example of head circumference growth in 145 children is used to validate our extended model. A simulation study is performed to examine the statistical properties of the parameter estimation obtained from this example. Finally, we discuss the implications and extensions of our model for detecting QTL that affect growth trajectories.     

Deviations of Genotypic Structures from Hardy-Weinberg Proportions under Random Mating and Differential Selection between the Sexes

Population genetic models, such as differential viability selection between the sexes and differential multiplicative fecundity contributions of the sexes, are considered for a single multiallelic locus. These selection models usually produce deviations of the zygotic genotype frequencies from Hardy-Weinberg proportions. The deviations are investigated (with special emphasis put on equilibrium states) to quantify the effect of selective asymmetry in the two sexes. For many selection regimes, the present results demonstrate a strong affinity of zygotic genotype frequencies for Hardy-Weinberg proportions after two generations, at the latest. It is shown that the deviations of genotypic equilibria from the corresponding Hardy-Weinberg proportions can be expressed and estimated by means of selection components of only that sex with the lower selection intensity. This corresponds to the well-known fact that viability selection acting in only one sex yields Hardy-Weinberg equilibria.     

Contribution of genetic effects to genetic variance components with epistasis and linkage disequilibrium

Background

Cockerham genetic models are commonly used in quantitative trait loci (QTL) analysis with a special feature of partitioning genotypic variances into various genetic variance components, while the F_∞ genetic models are widely used in genetic association studies. Over years, there have been some confusion about the relationship between these two type of models. A link between the additive, dominance and epistatic effects in an F_∞ model and the additive, dominance and epistatic variance components in a Cockerham model has not been well established, especially when there are multiple QTL in presence of epistasis and linkage disequilibrium (LD).

Results

In this paper, we further explore the differences and links between the F_∞ and Cockerham models. First, we show that the Cockerham type models are allelic based models with a special modification to correct a confounding problem. Several important moment functions, which are useful for partition of variance components in Cockerham models, are also derived. Next, we discuss properties of the F_∞ models in partition of genotypic variances. Its difference from that of the Cockerham models is addressed. Finally, for a two-locus biallelic QTL model with epistasis and LD between the loci, we present detailed formulas for calculation of the genetic variance components in terms of the additive, dominant and epistatic effects in an F_∞ model. A new way of linking the Cockerham and F_∞ model parameters through their coding variables of genotypes is also proposed, which is especially useful when reduced F_∞ models are applied.

Conclusion

The Cockerham type models are allele-based models with a focus on partition of genotypic variances into various genetic variance components, which are contributed by allelic effects and their interactions. By contrast, the F_∞ regression models are genotype-based models focusing on modeling and testing of within-locus genotypic effects and locus-by-locus genotypic interactions. When there is no need to distinguish the paternal and maternal allelic effects, these two types of models are transferable. Transformation between an F_∞ model's parameters and its corresponding Cockerham model's parameters can be established through a relationship between their coding variables of genotypes. Genetic variance components in terms of the additive, dominance and epistatic genetic effects in an F_∞ model can then be calculated by translating formulas derived for the Cockerham models.

     

Isolation and characterization of 11 tetranucleotide microsatellite loci in the Egyptian mongoose (Herpestes ichneumon)

We report the isolation of 11 polymorphic tetranucleotide microsatellite loci in the Egyptian mongoose (Herpestes ichneumon). In a sample of 27 individuals, we observed between 4 and 7 alleles per locus and their observed and expected heterozygosities ranged from 0.37 to 0.85 and from 0.44 to 0.79, respectively. All genotypic frequencies conformed to Hardy-Weinberg equilibrium expectations and there were no instances of linkage disequilibrium detected between pairs of loci.