Multilocus Behavior in Random Environments. II. Linkage Disequilibrium in an Additive Model

The effect of a stochastic environment on an additive, two-locus model of a diploid population is examined. The appropriate diffusion equation is derived and its asymptotic properties are approximated by an Orstein-Uhlenbeck process. The first and second order moments of this approximating process are given. The mean linkage disequilibrium will be nonzero if the alleles at the different loci are correlated. The sign of the mean disequilibrium is determined by the sign of the correlation.     

A semi-symmetric two-locus model

The two-locus symmetric viability model characterized by its invariance with respect to the exchange of alleles at each locus, is a well-studied model of classical two-locus theory. The symmetric model introduced by Lewontin and Kojima is among the few multi-locus models with epistatic interactions between loci for which a polymorphism with linkage equilibrium can be stable and this happens when recombination is sufficiently large. We show that an analogous property holds true for a different model, in which symmetry need exist at only one locus. The properties of this new semi-symmetric model are compared with those of the classical symmetric model. For tight linkage, two classes of polymorphisms are possible, depending on the magnitude of additive epistasis. The recombination rate above which linkage equilibrium becomes stable is derived analytically. As in the symmetric model, intervals of recombination in which no polymorphism is stable are possible, and stable polymorphisms can coexist with stable fixations.     

Linkage and association analysis of angiotensin I-converting enzyme (ACE)-gene polymorphisms with ACE concentration and blood pressure

Considerable effort has been expended to determine whether the gene for angiotensin I-converting enzyme (ACE) confers susceptibility to cardiovascular disease. In this study, we genotyped 13 polymorphisms in the ACE gene in 1,343 Nigerians from 332 families. To localize the genetic effect, we first performed linkage and association analysis of all the markers with ACE concentration. In multipoint variance-component analysis, this region was strongly linked to ACE concentration (maximum LOD score 7.5). Likewise, most of the polymorphisms in the ACE gene were significantly associated with ACE (P<.0013). The two most highly associated polymorphisms, ACE4 and ACE8, accounted for 6% and 19% of the variance in ACE, respectively. A two-locus additive model with an additive x additive interaction of these polymorphisms explained most of the ACE variation associated with this region. We next analyzed the relationship between these two polymorphisms (ACE4 and ACE8) and blood pressure (BP). Although no evidence of linkage was detected, significant association was found for both systolic and diastolic BP when a two-locus additive model developed for ACE concentration was used. Further analyses demonstrated that an epistasis model provided the best fit to the BP variation. In conclusion, we found that the two polymorphisms explaining the greatest variation in ACE concentration are significantly associated with BP, through interaction, in this African population sample. Our study also demonstrates that greater statistical power can be anticipated with association analysis versus linkage, when markers in strong linkage disequilibrium with a trait locus have been identified. Furthermore, allelic interaction may play an important role in the dissection of complex traits such as BP.     

Covariances of relatives stemming from a population undergoing mixed self and random mating

We consider covariances of all parent and first-generation relatives from outcrossing or self-fertilization in a parent population that is in equilibrium with respect to these processes. The results, which are for any number of alleles and loci with additive and dominance effects, are phrased in terms of six quadratic genetic components whose coefficients are given by descent measures for equilibrium populations. Because of the variation in the inbreeding coefficients for this system of mating, the expressions include joint contributions of loci to the variances and covariances of relatives. By inclusion of the full complement of relatives, all quadratic components can be estimated. The findings of Ghai (1982, Biometrics 38, 87-92) for compound functions of the covariances with two alleles at a single locus are analyzed in terms of the more general model.     

Analysis of Some Two-Locus Systems for Traits Exhibiting Continuous Variation

Some generalized two-locus, major-gene models for traits exhibiting continuous expression are investigated. Conditions of gametic equilibria are stated as functions of the parameters characterizing the within phenotypic-genotypic distributions. Selection is stabilizing in favor of an optimum phenotype. It is established that an increase in the phenotypic variance facilitates stability of a central polymorphic state with tighter linkage. Also, with increased phenotypic variance in an additive two-locus selection model, a Hardy-Weinberg type of equilibrium will be less central. The rate of convergence, in this case, is slowed down with an augmented environmental background.     

New properties of the two-locus partial selfing model with selection

Analytic solutions are obtained for the equilibria of a simple two-locus, heterotic selection model with mixed selfing and random outcrossing. Two general phenomena are possible, depending upon the viabilities and the degree of selfing: (1) Negative disequilibrium potential, under which only gametic disequilibrium is possible; and (2) positive disequilibrium potential, which can result in permanent gametic disequilibrium provided that linkage is sufficiently tight. Under random mating (s = 0), these two situations correspond to negative and positive additive epistasis, respectively. With partial self-fertilization, however, this is no longer true, and a more appropriate measure of gametc disequilibrium potential, Δ(s), is introduced. A numerically aided examination of the model results in the discovery of two new properties of partial selfing with selection: (1) With negative disequilibrium potential (Δ(s) < 0), the equilibrium mean fitness increases with increasing recombination. With positive disequilibrium potential (Δ(s) > 0), the opposite is true. (2) Gametic disequilibrium can increase or decrease as the degree of selfing is increased. Therefore, it is apparent that partial selfing and linkage are not analogous as regards the maintenance of disequilibrium.     

Evidence for two unlinked loci regulating total serum IgE levels.

Studies investigating the genetic control of total serum IgE levels are of major importance in understanding basic pathophysiologic mechanisms in atopy and asthma, since IgE levels predict onset and correlate with the clinical expression of these disorders. Previous analysis of data from 92 families, ascertained through a parent with asthma, showed evidence for recessive inheritance of high IgE levels with linkage to chromosome 5q. Since there was significant residual familial correlation in the one-locus segregation analysis, two-locus segregation and linkage analyses were performed. Segregation analyses provided evidence for a second major locus unlinked to the locus on 5q. Utilization of this two-locus model corroborates the previous evidence for linkage between this trait and markers on 5q31-q33. The LODs for the most informative marker D5S436 increased from 3.00 at 10% recombination to 4.67 at 9% recombination, when the two-locus model was used. Additional linkage studies are needed to map this second locus. These results demonstrate the importance of performing multilocus segregation and linkage analyses for quantitative traits that are related to the phenotype of a complex disorder. This approach has given further insight into the genetics of allergy and asthma by providing evidence for a two-locus model.     

The covariance between relatives conditional on genetic markers

The development of molecular genotyping techniques makes it possible to analyze quantitative traits on the basis of individual loci. With marker information, the classical theory of estimating the genetic covariance between relatives can be reformulated to improve the accuracy of estimation. In this study, an algorithm was derived for computing the conditional covariance between relatives given genetic markers. Procedures for calculating the conditional relationship coefficients for additive, dominance, additive by additive, additive by dominance, dominance by additive and dominance by dominance effects were developed. The relationship coefficients were computed based on conditional QTL allelic transmission probabilities, which were inferred from the marker allelic transmission probabilities. An example data set with pedigree and linked markers was used to demonstrate the methods developed. Although this study dealt with two QTLs coupled with linked markers, the same principle can be readily extended to the situation of multiple QTL. The treatment of missing marker information and unknown linkage phase between markers for calculating the covariance between relatives was discussed.     

How to model a complex trait. 2. Analysis with two disease loci

Complex traits are often governed by more than one trait locus. The first step towards an adequate model for such diseases is a linkage analysis with two trait loci. Such an analysis can be expected to have higher power to detect linkage than a standard single-trait-locus linkage analysis. However, it is crucial to accurately specify the parameters of the two-locus model. Here, we recapitulate the general two-locus model with and without genomic imprinting. We relate heterogeneity, multiplicative, and additive two-locus models to biological or pathophysiological mechanisms, and give the corresponding averaged ("best-fitting") single-trait-locus models for each of the two loci. Furthermore, we derive the two-locus penetrances from the averaged single-locus models, under the assumption of one of the three model classes mentioned above. Using these formulae, if the best-fitting single-locus models are available, investigators may perform a two-trait-locus linkage analysis under a realistic model. This procedure will maximize the power to detect linkage for traits which are governed by two or more loci, and lead to more accurate estimates of the disease-locus positions.     

A complete classification of epistatic two-locus models

Background

The study of epistasis is of great importance in statistical genetics in fields such as linkage and association analysis and QTL mapping. In an effort to classify the types of epistasis in the case of two biallelic loci Li and Reich listed and described all models in the simplest case of 0/1 penetrance values. However, they left open the problem of finding a classification of two-locus models with continuous penetrance values.

Results

We provide a complete classification of biallelic two-locus models. In addition to solving the classification problem for dichotomous trait disease models, our results apply to any instance where real numbers are assigned to genotypes, and provide a complete framework for studying epistasis in QTL data. Our approach is geometric and we show that there are 387 distinct types of two-locus models, which can be reduced to 69 when symmetry between loci and alleles is accounted for. The model types are defined by 86 circuits, which are linear combinations of genotype values, each of which measures a fundamental unit of interaction.

Conclusion

The circuits provide information on epistasis beyond that contained in the additive × additive, additive × dominance, and dominance × dominance interaction terms. We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.     

Domestication reshaped the genetic basis of inbreeding depression in a maize landrace compared to its wild relative,teosinte

Inbreeding depression is the reduction in fitness and vigor resulting from mating of close relatives observed in many plant and animal species. The extent to which the genetic load of mutations contributing to inbreeding depression is due to large-effect mutations versus variants with very small individual effects is unknown and may be affected by population history. We compared the effects of outcrossing and self-fertilization on 18 traits in a landrace population of maize, which underwent a population bottleneck during domestication, and a neighboring population of its wild relative teosinte. Inbreeding depression was greater in maize than teosinte for 15 of 18 traits, congruent with the greater segregating genetic load in the maize population that we predicted from sequence data. Parental breeding values were highly consistent between outcross and selfed offspring, indicating that additive effects determine most of the genetic value even in the presence of strong inbreeding depression. We developed a novel linkage scan to identify quantitative trait loci (QTL) representing large-effect rare variants carried by only a single parent, which were more important in teosinte than maize. Teosinte also carried more putative juvenile-acting lethal variants identified by segregation distortion. These results suggest a mixture of mostly polygenic, small-effect partially recessive effects in linkage disequilibrium underlying inbreeding depression, with an additional contribution from rare larger-effect variants that was more important in teosinte but depleted in maize following the domestication bottleneck. Purging associated with the maize domestication bottleneck may have selected against some large effect variants, but polygenic load is harder to purge and overall segregating mutational burden increased in maize compared to teosinte.     

Combined segregation and linkage analysis of Graves disease with a thyroid autoantibody diathesis.

Combined segregation and linkage analysis is a powerful technique for modeling linkage to diseases whose etiology is more complex than the effect of a well-described single genetic locus and for investigating the influence of single genes on various aspects of the disease phenotype. Graves disease is familial and is associated with human leukocyte antigen (HLA) allele DR3. Probands with Graves disease, as well as close relatives, have raised levels of thyroid autoantibodies. This phenotypic information additional to affection status may be considered by the computer program COMDS for combined segregation and linkage analysis, when normals are classified into diathesis classes of increasing thyroid autoantibody titer. The ordinal model considers the cumulative odds of lying in successive classes, and a single additional parameter is introduced for each gene modeled. Distributional assumptions are avoided by providing estimates of the population frequencies of each class. Evidence for linkage was increased by considering the thyroid autoantibody diathesis and by testing two-locus models. The analysis revealed evidence for linkage to HLA-DR when the strong coupling of the linked locus to allele DR3 was considered (lod score of 6.6). Linkage analysis of the residual variation revealed no evidence of linkage to Gm, but a suggestion of linkage to Km.     

Change in Genetic Variance under Selection in a Self-Fertilizing Population

In this study we show how the genetic variance of a quantitative trait changes in a self-fertilizing population under repeated cycles of truncation selection, with the analysis based on the infinitesimal model in which it is assumed that the trait is determined by an infinite number of unlinked loci without epistasis. The genetic variance is reduced not as a consequence of the genotypic frequency change but due to the build-up of linkage disequilibrium under truncation selection in this model. We assume that the order of the genotypic contribution from each locus is n(-1/2), where n is the number of loci involved, and investigate the change in linkage disequilibrium resulting from selection and self-fertilization using genotypic frequency dynamics in order to analyze the change in the genetic variance. Our analysis gives recurrence relations of genetic variance among the succeeding generations for the three cases of gene action, i.e., purely additive action, pure dominance without additive effect and the presence of both additive effect and dominance, respectively. Numerical examples are also given as a check on the recurrence formulas.     

Linkage analysis of a large African family segregating stuttering suggests polygenic inheritance

We describe a pedigree of 71 individuals from the Republic of Cameroon in which at least 33 individuals have a clinical diagnosis of persistent stuttering. The high concentration of stuttering individuals suggests that the pedigree either contains a single highly penetrant gene variant or that assortative mating led to multiple stuttering-associated variants being transmitted in different parts of the pedigree. No single locus displayed significant linkage to stuttering in initial genome-wide scans with microsatellite and SNP markers. By dividing the pedigree into five subpedigrees, we found evidence for linkage to previously reported loci on 3q and 15q, and to novel loci on 2p, 3p, 14q, and a different region of 15q. Using the two-locus mode of Superlink, we showed that combining the recessive locus on 2p and a single-locus additive representation of the 15q loci is sufficient to achieve a two-locus score over 6 on the entire pedigree. For this 2p + 15q analysis, we show LOD scores ranging from 4.69 to 6.57, and the scores are sensitive to which marker is chosen for 15q. Our findings provide strong evidence for linkage at several loci.     

Multilocus linkage tests based on affected relative pairs

For complex diseases, recent interest has focused on methods that take into account joint effects at interacting loci. Conditioning on effects of disease loci at known locations can lead to increased power to detect effects at other loci. Moreover, use of joint models allows investigation of the etiologic mechanisms that may be involved in the disease. Here we present a method for simultaneous analysis of the joint genetic effects at several loci that uses affected relative pairs. The method is a generalization of the two-locus LOD-score analysis for affected sib pairs proposed by Cordell et al. We derive expressions for the relative risk, lambdaR, to a relative of an affected individual, in terms of the additive and epistatic components of variance at an arbitrary number of disease loci, and we show how these can be used to fit a likelihood model to the identity-by-descent sharing among pairs of affected relatives in extended pedigrees. We implement the method by use of a stepwise strategy in which, given evidence of linkage to disease at m-1 locations on the genome, we calculate the conditional likelihood curve across the genome for an mth disease locus, using multipoint methods similar to those proposed by Kruglyak et al. We evaluate the properties of our method by use of simulated data and present an application to real data from families with insulin-dependent diabetes mellitus.     

Analysis on additive effects and additive-by-additive epistatic effects of QTLs for yield traits in a recombinant inbred line population of rice

A linkage map consisting of 158 DNA markers were constructed by using a recombinant inbred line (RIL) population derived from the indica-indica rice cross Zhenshan 97B 2 Milyang 46. Quantitative trait loci (QTLs) conditioning grain yield and five yield component traits were determined at the one-locus and two-locus levels, and genotype-by-environment (GE) interactions were analyzed. Thirty-one QTLs were detected to have significant additive effects for yield traits, of which 12 also exhibited significant epistatic effects. Sixteen significant additive-by-additive (AA) interactions were detected, of which nine occurred between QTLs with own additive effects (M_epQTLs), four occurred between QTLs showing epistatic effects only (epQTLs), and three occurred between M_epQTLs and epQTLs. Significant GE interactions were found for six QTLs with additive effects and one AA interaction. Generally, the contributions to the phenotypic variation were higher due to QTL main effects than to epistatic effects. The detection of additive effects and AA effects of a QTL interfered with each other, indicating that the detection of QTLs with main effects, as well as the magnitude and directions of the additive effects, might vary depending on their interactions with other loci.     

Direct power comparisons between simple LOD scores and NPL scores for linkage analysis in complex diseases.

Several methods have been proposed for linkage analysis of complex traits with unknown mode of inheritance. These methods include the LOD score maximized over disease models (MMLS) and the "nonparametric" linkage (NPL) statistic. In previous work, we evaluated the increase of type I error when maximizing over two or more genetic models, and we compared the power of MMLS to detect linkage, in a number of complex modes of inheritance, with analysis assuming the true model. In the present study, we compare MMLS and NPL directly. We simulated 100 data sets with 20 families each, using 26 generating models: (1) 4 intermediate models (penetrance of heterozygote between that of the two homozygotes); (2) 6 two-locus additive models; and (3) 16 two-locus heterogeneity models (admixture alpha = 1.0,.7,.5, and.3; alpha = 1.0 replicates simple Mendelian models). For LOD scores, we assumed dominant and recessive inheritance with 50% penetrance. We took the higher of the two maximum LOD scores and subtracted 0.3 to correct for multiple tests (MMLS-C). We compared expected maximum LOD scores and power, using MMLS-C and NPL as well as the true model. Since NPL uses only the affected family members, we also performed an affecteds-only analysis using MMLS-C. The MMLS-C was both uniformly more powerful than NPL for most cases we examined, except when linkage information was low, and close to the results for the true model under locus heterogeneity. We still found better power for the MMLS-C compared with NPL in affecteds-only analysis. The results show that use of two simple modes of inheritance at a fixed penetrance can have more power than NPL when the trait mode of inheritance is complex and when there is heterogeneity in the data set.     

A complete enumeration and classification of two-locus disease models

There are 512 two-locus, two-allele, two-phenotype, fully penetrant disease models. Using the permutation between two alleles, between two loci, and between being affected and unaffected, one model can be considered to be equivalent to another model under the corresponding permutation. These permutations greatly reduce the number of two-locus models in the analysis of complex diseases. This paper determines the number of nonredundant two-locus models (which can be 102, 100, 96, 51, 50, or 58, depending on which permutations are used, and depending on whether zero-locus and single-locus models are excluded). Whenever possible, these nonredundant two-locus models are classified by their property. Besides the familiar features of multiplicative models (logical AND), heterogeneity models (logical OR), and threshold models, new classifications are added or expanded: modifying-effect models, logical XOR models, interference and negative interference models (neither dominant nor recessive), conditionally dominant/recessive models, missing lethal genotype models, and highly symmetric models. The following aspects of two-locus models are studied: the marginal penetrance tables at both loci, the expected joint identity-by-descent (IBD) probabilities, and the correlation between marginal IBD probabilities at the two loci. These studies are useful for linkage analyses using single-locus models while the underlying disease model is two-locus, and for correlation analyses using the linkage signals at different locations obtained by a single-locus model.     

Linkage strategies for genetically complex traits. I. Multilocus models

In order to investigate linkage detection strategies for genetically complex traits, multilocus models of inheritance need to be specified. Here, two types of multilocus model are described: (1) a multiplicative model, representing epistasis (interaction) among loci, and (2) an additive model, which is shown to closely approximate genetic heterogeneity, which is characterized by no interlocus interaction. A ratio lambda R of risk for type R relatives that is compared with population prevalence is defined. For a single-locus model, lambda R - 1 decreases by a factor of two with each degree of relationship. The same holds true for an additive multilocus model. For a multiplicative (epistasis) model, lambda R - 1 decreases more rapidly than by a factor of two with degree of relationship. Examination of lambda R values for various classes of relatives can potentially suggest the presence of multiple loci and epistasis. For example, data for schizophrenia suggest multiple loci in interaction. It is shown in the second paper of this series that lambda R is the critical parameter in determining power to detect linkage by using affected relative pairs.     

The role of recombination and selection in the modifier theory of sex-ratio distortion

The equilibrium configurations for a two-locus multialle model of sex-linked meiotic drive are studied with regard to the recombination fraction:limit cycles can occur in the case of small recombination while stable equilibrium points associated with linkage equilibrium can exist for an intermediate range of recombination values depending on the equilibrium sex ratio, linkage disequilibrium at nearby equilibrium points taking turn with loser linkage. The evolutionary dynamics in two-locus sex-ratio distortion systems is enlightened: while equilibria with a sex ratio closer to 1/2 are more likely to be stable with respect to perturbations on the frequencies of sex-ratio distorters that are represented at equilibrium, such equilibria are also more vulnerable to the invasion of mutant distorters when there is some degree of linkage with the sex-determining locus. For X-linked multimodifier systems of sex-ratio distortion, differential fertilities and viabilities are incorporated and a maximum principle is suggested.