General Epistatic Models of the Risk of Complex Diseases

The range of possible gene interactions in a multilocus model of a complex inherited disease is studied by exploring genotype-specific risks subject to the constraint that the allele frequencies and marginal risks are known. We quantify the effect of gene interactions by defining the interaction ratio, , where K_R is the recurrence risk to relatives with relationship R for the true model and is the recurrence risk to relatives for a multiplicative model with the same marginal risks. We use a Markov chain Monte Carlo (MCMC) procedure to sample from the space of possible models. We find that the average of C_R increases with the number of loci for both low frequency (p = 0.03) and higher frequency (p = 0.25) causative alleles. Furthermore, the probability that C_R > 1 is nearly 1. Similar results are obtained when more weight is given to risk models that are closer to the comparable multiplicative model. These results imply that, in general, gene interactions will result in greater heritability of a complex inherited disease than is expected on the basis of a multiplicative model of interactions and hence may provide a partial explanation for the problem of missing heritability of complex diseases.ALTHOUGH many genome-wide association studies (GWAS) have been performed and have found hundreds of SNPs associated with higher risk of complex inherited diseases, those SNPs so far account for only a small fraction of the inherited risk of those diseases (Altshuler et al. 2008). Several not mutually exclusive explanations have been proposed for the “missing heritability,” i.e., the heritability that is not yet accounted for by SNPs found in GWAS (Manolio et al. 2009): (i) common alleles of small effect that have not been found because GWAS done so far have been underpowered, (ii) low-frequency alleles of moderate effect that are difficult to find using HapMap SNPs, (iii) rare copy-number variants that are not in strong linkage disequilibrium (LD) with HapMap SNPs, (iv) inherited epigenetic factors that are not in strong LD with HapMap SNPs, and (v) interactions among causative alleles that conceal their true contribution to heritability. In this article we investigate the last possibility and determine the extent to which interactions may account for missing heritability.Our analysis is in the same spirit as that of Culverhouse et al. (2002). We assume that the risk of being affected by a complex disease is determined by an individual''s genotype at two or more loci and that the frequencies of causative alleles and the average risks for each one-locus genotype (the marginal risks) are known. Culverhouse et al. (2002) assumed the marginal risks were the same for all genotypes and all loci. In that case, causative alleles have odds ratios of 1; they contribute to risk only through their interactions. Culverhouse et al. found the risk function that maximized the heritability and showed that the maximum possible heritability attributable to interactions increased with the number of loci. They concluded that it is quite possible that interactions among loci that have no main effect could contribute substantially to the heritability of a complex disease and indeed could account for “virtually all the variation in affection status for diseases with any prevalence” (Culverhouse et al. 2002, p. 468).We generalize the analysis of Culverhouse et al. in three ways. First, we allow causative alleles to have odds ratios >1. Second, we explore the entire space of models instead of focusing only on the risk model that maximizes heritability. Third, we examine how the importance of gene interactions depends on the “distance” between a risk model and a comparable multiplicative model. We show that gene interactions can substantially increase the heritability of risk as measured by recurrence risk, K_R, and that the effect increases with the number of loci carrying causative alleles. Furthermore, we show that these results are true even if more weight is given to models that are closer to a comparable multiplicative model.Geometrically, the space of feasible genotype-specific risks subject to the aforementioned constraints (i.e., that the allele frequencies and marginal risks are known) corresponds to a high-dimensional convex polytope, and the computational problem of interest involves integrating a quadratic function over the polytope. The dimension of the polytope grows exponentially with the number of loci, and, therefore, analytic computation is intractable for more than two loci. Hence, we devise a Monte Carlo approach to tackle the problem. Note that, because of high dimensionality, rejection algorithms are not appropriate for this kind of problem. We instead employ a Markov chain Monte Carlo (MCMC) algorithm based on a random walk that always stays inside the polytope. We present empirical results for up to five loci and obtain a closed-form formula for the minimum of K_R over the polytope; the latter result applies to an arbitrary number of loci. Interestingly, the minimum of K_R decreases as the number L of loci increases, but the average of K_R over the polytope increases with L.