Detecting gene-gene interactions using affected sib pair analysis with covariates

Interest has recently focussed on allowing for interactions between loci as a way to increase power to detect linkage. In this paper, a simplified logistic regression method was used to perform affected sib pair analyses allowing for the inclusion of data from other loci. A systematic search of two-locus disease models was carried out to determine the situations in which this was advantageous. If IBD information is available (e.g. from a genome scan), it is unlikely that allowing for interactions will give a large lod score in the absence of linkage evidence from sinlge-locus analysis. Furthermore, allowing for interactions rarely gave a significant increase in power to detect linkage over a single-locus analysis, except for heterogeneity models with low K(P). Conversely, the availability of disease-associated genotypes may greatly increase the power both to detect linkage to a second locus and interaction between the loci. These results indicate that when only IBD information is available, two-locus analysis of genome scan data should be restricted to regions giving peaks under single-locus analysis. If disease-associated genotypes are available, it may be worth re-analysing the whole genome.     

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 likelihood approach for quantitative-trait-locus mapping with selected pedigrees

Selective sampling is a cost-effective design for mapping quantitative trait loci (QTLs). A unified framework, which naturally combines two complementary sources of linkage information in the data, is proposed for the mapping of QTLs using selected pedigrees. Score statistics for detecting linkage are introduced for single-locus models (univariate or bivariate phenotypes) and two-locus epistasis models. A computer implementation of the methods for single-locus univariate-phenotype models is provided for nuclear families with arbitrary number of sibs and is freely available.     

Log-linear models for gene mapping with affected sib pair data

In genome-wide screens of genetic marker loci, non-mendelian inheritance of a marker is taken to indicate its vicinity to a disease locus. Heritable complex traits are thought to be under the influence of multiple possibly interacting susceptibility loci yet the most frequently used methods of linkage and association analysis focus on one susceptibility locus at a time. Here we introduce log-linear models for the joint analysis of multiple marker loci and interaction effects between them. Our approach focuses on affected sib pair data and identical by descent (IBD) allele sharing values observed on them. For each heterozygous parent, the IBD values at linked markers represent a sequence of dependent binary variables. We develop log-linear models for the joint distribution of these IBD values. An independence log-linear model is proposed to model the marginal means and the neighboring interaction model is advocated to account for associations between adjacent markers. Under the assumption of conditional independence, likelihood methods are applied to simulated data containing one or two susceptibility loci. It is shown that the neighboring interaction log-linear model is more efficient than the independence model, and incorporating interaction in the two-locus analysis provides increased power and accuracy for mapping of the trait loci.     

Two-stage two-locus models in genome-wide association

Studies in model organisms suggest that epistasis may play an important role in the etiology of complex diseases and traits in humans. With the era of large-scale genome-wide association studies fast approaching, it is important to quantify whether it will be possible to detect interacting loci using realistic sample sizes in humans and to what extent undetected epistasis will adversely affect power to detect association when single-locus approaches are employed. We therefore investigated the power to detect association for an extensive range of two-locus quantitative trait models that incorporated varying degrees of epistasis. We compared the power to detect association using a single-locus model that ignored interaction effects, a full two-locus model that allowed for interactions, and, most important, two two-stage strategies whereby a subset of loci initially identified using single-locus tests were analyzed using the full two-locus model. Despite the penalty introduced by multiple testing, fitting the full two-locus model performed better than single-locus tests for many of the situations considered, particularly when compared with attempts to detect both individual loci. Using a two-stage strategy reduced the computational burden associated with performing an exhaustive two-locus search across the genome but was not as powerful as the exhaustive search when loci interacted. Two-stage approaches also increased the risk of missing interacting loci that contributed little effect at the margins. Based on our extensive simulations, our results suggest that an exhaustive search involving all pairwise combinations of markers across the genome might provide a useful complement to single-locus scans in identifying interacting loci that contribute to moderate proportions of the phenotypic variance.     

Effect of genetic heterogeneity and assortative mating on linkage analysis: a simulation study.

Linkage studies of complex genetic traits raise questions about the effects of genetic heterogeneity and assortative mating on linkage analysis. To further understand these problems, I have simulated and analyzed family data for a complex genetic disease in which disease phenotype is determined by two unlinked disease loci. Two models were studied, a two-locus threshold model and a two-locus heterogeneity model. Information was generated for a marker locus linked to one of the disease-defining loci. Random-mating and assortative-mating samples were generated. Linkage analysis was then carried out by use of standard methods, under the assumptions of a single-locus disease trait and a random-mating population. Results were compared with those from analysis of a single-locus homogeneous trait in samples with the same levels of assortative mating as those considered for the two-locus traits. The results show that (1) introduction of assortative mating does not, in itself, markedly affect the estimate of the recombination fraction; (2) the power of the analysis, reflected in the LOD scores, is somewhat lower with assortative rather than random mating. Loss of power is greater with increasing levels of assortative mating; and (3) for a heterogeneous genetic disease, regardless of mating type, heterogeneity analysis permits more accurate estimate of the recombination fraction but may be of limited use in distinguishing which families belong to each homogeneous subset. These simulations also confirmed earlier observations that linkage to a disease "locus" can be detected even if the disease is incorrectly defined as a single-locus (homogeneous) trait, although the estimated recombination fraction will be significantly greater than the true recombination fraction between the linked disease-defining locus and the marker locus.     

A Symmetric Two-Locus Fertility Model

A model in which selection is mediated by differential fertilities among the genotypes at two diallelic loci is proposed. Fertility depends only on the number of heterozygous loci participating in the mating. Classes analogous to symmetric equilibria in symmetric viability models are determined explicitly and shown to exhibit stability behavior very different from the viability results. Linkage equilibrium is shown to occur in a relatively asymmetric fashion and to overlap in stability with linkage disequilibrium. In many cases single-locus or two-locus polymorphism is shown to be stable simultaneously with chromosome fixation even under very tight linkage. It is suggested that historical effects may be of great significance in the evolution of systems in which fertility is the primary agent of natural selection.     

The relationship between the sibling recurrence-risk ratio and genotype relative risk

The recurrence-risk ratio of disease in siblings, lambdaS, is a standard parameter used in genetic analysis to estimate the statistical power for detection of a disease locus. However, the relationship between the underlying risk conferred by a disease-susceptibility allele and lambdaS has not been well described. The former is generally quantified as a genotype relative risk, gamma, and measures the ratio of disease risks between those with and those without the susceptibility genotype(s). We demonstrate that lambdaS varies significantly more with respect to gamma and the disease-allele frequency for two-locus multiplicative models than for other two-locus and for single-locus models. For the single- and two-locus dominant-inheritance models that we studied, when a disease-susceptibility allele had a frequency >/=.2, lambdaS had an upper limit of <10. In general, lambdaS values >10 are possible only under recessive inheritance, dominant inheritance with relatively rare (<5%) disease-susceptibility alleles, or when two or more disease loci have alleles acting either epistatically or multiplicatively. We introduce the idea of a restricted sib recurrence-risk ratio (lambda*S) estimated by restriction of sibships to those ascertained through a proband who already has a putative high-risk allele. A lambda*S larger than the lambdaS value estimated from randomly selected probands can serve as an indirect way of testing whether the posited susceptibility allele increases disease risk. Our results demonstrate that a lambdaS of 2-3 may portend successful mapping for a variety of genetic models but that, for some two-locus models, a lambdaS as high as 10 does not guarantee underlying genes easily mapped by linkage.     

Two-locus linkage analysis of cutaneous malignant melanoma/dysplastic nevi.

Previous linkage analyses of 19 cutaneous malignant melanoma/dysplastic nevi (CMM/DN) kindreds showed significant evidence of linkage and heterogeneity to both chromosomes 1p and 9p. Five kindreds also showed evidence of linkage (Z>0.7) to both regions. To further examine these findings, we conducted two-trait-locus, two-marker-locus linkage analysis. We examined one homogeneity and one heterogeneity single-locus model (SL-Hom and SL-Het), and two-locus (2L) models: an epistatic model (Ep), in which CMM was treated as a genuine 2L disease, and a heterogeneity model (Het), in which CMM could result from disease alleles at either locus. Both loci were modeled as autosomal dominant. The LOD scores for CMM alone were highest using the SL-Het model (Z = 8.48, theta = .0). There was much stronger evidence of linkage to chromosome 9p than to 1p for CMM alone; the LOD scores were approximately two times greater on 9p than on 1p. The change in LOD scores from an evaluation of CMM alone to CMM/DN suggested that a chromosome 1p locus (or loci) contributed to both CMM and CMM/DN, whereas a 9p locus contributed more to CMM alone. For both 2L models, the LOD scores from 1p were greater for CMM/DN than for CMM alone (Ep: Z=4.63 vs. 3.83; Het: 4.94 vs. 3.80, respectively). In contrast, for 9p, the LOD scores were substantially lower with CMM/DN than with CMM alone (Ep: 4.64 vs. 7.06; Het: 5.38 vs. 7.99, respectively). After conditioning on linkage to the other locus, only the 9p locus consistently showed significant evidence for linkage to CMM alone. Thus, the application of 2L models may be useful to help unravel the complexities of familial melanoma.     

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.     

Efficiency robust tests for mapping quantitative trait loci using extremely discordant sib pairs

In 1972, Haseman and Elston proposed a pioneering regression method for mapping quantitative trait loci using randomly selected sib pairs. Recently, the statistical power of their method was shown to be increased when extremely discordant sib pairs are ascertained. While the precise genetic model may not be known, prior information that constrains IBD probabilities is often available. We investigate properties of tests that are robust against model uncertainty and show that the power gain from further constraining IBD probabilities is marginal. The additional linkage information contained in the trait values can be incorporated by combining the Haseman-Elston regression method and a robust allele sharing test.     

Analysis of genes for alcoholism using two-disease-locus models

Using model-based two-locus methods for mapping genes, we analyzed the family data from the Collaborative Study on the Genetics of Alcoholism. Microsatellite data from 143 families ascertained through having three or more individuals affected with alcohol dependence were used for this investigation. Four regions showing evidence for linkage were identified using single-locus models from previous investigations. We investigated the genetic linkage, pattern of disease inheritance, and pair-wise genetic epistasis of these loci using the TLINKAGE program for two-disease-locus analysis.     

Matrix representation of the Haseman-Elston method.

The Hasemann-Elston method of linkage detection is based on the probabilities of a sib pair having 0, 1, or 2 alleles identical by descent (IBD) at a marker and a trait locus. These probabilities form a 3x3 matrix. Here, the characteristic values and characteristic vectors of this matrix were used to clarify the structure of the equations and to simplify calculations. As examples, the regression coefficients were derived for three genetic systems: a trait and a marker, two epistatic traits and two markers, and one trait locus and two markers. The last model was studied under the assumption of no crossover interference, the expression for allele IBD sharing at a trait locus was derived as a function of allele IBD sharing at two marker loci, and the regression is shown to be non-linear.     

Identification of quantitative trait loci and epistatic interactions for plant height and heading date in rice

Appropriate heading date and plant height are prerequisites for attaining the desired yield level in rice breeding programs. In this study, we analyzed the genetic bases of heading date and plant height at both single- locus and two-locus levels, using a population of 240 F_2:3 families derived from a cross between two elite rice lines. Measurements for the traits were obtained over 2 years in replicated field trials. A linkage map was constructed with 151 polymorphic marker loci, based on which interval mapping was performed using Mapmaker/QTL. The analyses detected six QTLs for plant height and six QTLs for heading date; collectively the QTLs for heading date accounted for a much greater amount of phenotypic variation than did the QTLs for plant height. Two-way analyses of variance, with all possible two-locus combinations, detected large numbers (from 101 to 257) of significant digenic interactions in the 2 years for both traits involving markers distributed in the entire genome; 22 and 39 were simultaneously detected in both years for plant height and heading date, respectively. Each of the interactions individually accounted for only a very small portion of the phenotypic variation. The majority of the significant interactions involved marker loci that did not detect significant effects by single-locus analyses, and many of the QTLs detected by single-locus analyses were involved in epistatic interactions. The results clearly demonstrated the importance of epistatic interactions in the genetic bases of heading date and plant height. Received: 5 May 2001 / Accepted: 3 August 2001     

Simulation studies of segregation analysis: application to two-locus models

We tested the power of a segregation analysis method (first proposed by Elandt-Johnson) to distinguish between single-locus and two-locus models, with and without environmentally caused reduced penetrance. We also looked at the effect of ascertainment probability on the analysis and at the proband-conditioned ascertainment correction proposed by Cannings and Thompson. We found that: (1) the segregation analysis has sufficient power to distinguish between the fully-penetrant double-recessive (RR) model and the fully-penetrant single-locus dominant and recessive models; (2) the method can also distinguish fairly well between the dominant-recessive (DR) and RR models, even when one does not take into account the population prevalence; (3) the method has much less power to distinguish between the fully-penetrant RR model and the single-locus models with reduced penetrance; (4) when environmental penetrance is taken account of in the analysis, the power of the method to distinguish between the one- and two-locus models improved substantially; (5) the estimates of ascertainment probability, pi, were robust, regardless of the model under which the data were generated; and (6) the Cannings-Thompson approach to ascertainment correction worked well only when the pi used to generate the data was less than .1.     

A Two-Locus Forensic Match Probability for Subdivided Populations

A two-locus match probability is presented that incorporates the effects of within-subpopulation inbreeding (consanguinity) in addition to population subdivision. The usual practice of calculating multi-locus match probabilities as the product of single-locus probabilities assumes independence between loci. There are a number of population genetics phenomena that can violate this assumption: in addition to consanguinity, which increases homozygosity at all loci simultaneously, gametic disequilibrium will introduce dependence into DNA profiles. However, in forensics the latter problem is usually addressed in part by the careful choice of unlinked loci. Hence, as is conventional, we assume gametic equilibrium here, and focus instead on between-locus dependence due to consanguinity. The resulting match probability formulae are an extension of existing methods in the literature, and are shown to be more conservative than these methods in the case of double homozygote matches. For two-locus profiles involving one or more heterozygous genotypes, results are similar to, or smaller than, the existing approaches. This revised version was published online in August 2006 with corrections to the Cover Date.     

Two-locus models of disease: comparison of likelihood and nonparametric linkage methods.

The power to detect linkage for likelihood and nonparametric (Haseman-Elston, affected-sib-pair, and affected-pedigree-member) methods is compared for the case of a common, dichotomous trait resulting from the segregation of two loci. Pedigree data for several two-locus epistatic and heterogeneity models have been simulated, with one of the loci linked to a marker locus. Replicate samples of 20 three-generation pedigrees (16 individuals/pedigree) were simulated and then ascertained for having at least 6 affected individuals. The power of linkage detection calculated under the correct two-locus model is only slightly higher than that under a single locus model with reduced penetrance. As expected, the nonparametric linkage methods have somewhat lower power than does the lod-score method, the difference depending on the mode of transmission of the linked locus. Thus, for many pedigree linkage studies, the lod-score method will have the best power. However, this conclusion depends on how many times the lod score will be calculated for a given marker. The Haseman-Elston method would likely be preferable to calculating lod scores under a large number of genetic models (i.e., varying both the mode of transmission and the penetrances), since such an analysis requires an increase in the critical value of the lod criterion. The power of the affected-pedigree-member method is lower than the other methods, which can be shown to be largely due to the fact that marker genotypes for unaffected individuals are not used.     

Gene networks in hexaploid wheat: interacting quantitative trait loci for grain protein content

In hexaploid wheat, single-locus and two-locus quantitative trait loci (QTL) analyses for grain protein content (GPC) were conducted using two different mapping populations (PI and PII). Main effect QTLs (M-QTLs), epistatic QTLs (E-QTLs) and QTL x environment interactions (QE, QQE) were detected using two-locus analyses in both the populations. Only a few QTLs were common in both the analyses, and the QTLs and the interactions detected in the two populations differed, suggesting the superiority of two-locus analysis and the need for using several mapping populations for QTL analysis. A sizable proportion of genetic variation for GPC was due to interactions (28.59% and 54.03%), rather than to M-QTL effects (7.24% and 7.22%), which are the only genetic effects often detected in the majority of QTL studies. Even E-QTLs made a marginal contribution to genetic variation (2.68% and 6.04%), thus suggesting that the major part of genetic variation is due to changes in gene networks rather than the presence or absence of specific genes. This is in sharp contrast to the genetic dissection of pre-harvest sprouting tolerance conducted by us earlier, where interaction effects were not substantial, suggesting that the nature of genetic variation also depends on the nature of the trait.     

Strategies for conditional two-locus nonparametric linkage analysis

In this article we deal with two-locus nonparametric linkage (NPL) analysis, mainly in the context of conditional analysis. This means that one incorporates single-locus analysis information through conditioning when performing a two-locus analysis. Here we describe different strategies for using this approach. Cox et al. [Nat Genet 1999;21:213-215] implemented this as follows: (i) Calculate the one-locus NPL process over the included genome region(s). (ii) Weight the individual pedigree NPL scores using a weighting function depending on the NPL scores for the corresponding pedigrees at speci fi c conditioning loci. We generalize this by conditioning with respect to the inheritance vector rather than the NPL score and by separating between the case of known (prede fi ned) and unknown (estimated) conditioning loci. In the latter case we choose conditioning locus, or loci, according to prede fi ned criteria. The most general approach results in a random number of selected loci, depending on the results from the previous one-locus analysis. Major topics in this article include discussions on optimal score functions with respect to the noncentrality parameter (NCP), and how to calculate adequate p values and perform power calculations. We also discuss issues related to multiple tests which arise from the two-step procedure with several conditioning loci as well as from the genome-wide tests.     

Evolution of migration under kin selection and local adaptation

We present here a stochastic two-locus, two-habitat model for the evolution of migration with local adaptation and kin selection. One locus determines the migration rate while the other causes local adaptation. We show that the opposing forces of kin competition and local adaptation can lead to the existence of one or two convergence stable migration rates, notably depending on the recombination rate between the two loci. We show that linkage between migration and local adaptation loci has two antagonist effects: when linkage is tight, cost of local adaptation increases, leading to smaller equilibrium migration rates. However, when linkage is tighter, the population structure at the migration locus tends to be very high because of the indirect selection, and thus equilibrium migration rates increases. This result, qualitatively different from results obtained with other models of migration evolution, indicates that ignoring drift or the detail of the genetic architecture may lead to incorrect conclusions.