A perspective on epistasis: limits of models displaying no main effect

Abstract

The completion of a draft sequence of the human genome and the promise of rapid single-nucleotide-polymorphism-genotyping technologies have resulted in a call for the abandonment of linkage studies in favor of genome scans for association. However, there exists a large class of genetic models for which this approach will fail: purely epistatic models with no additive or dominance variation at any of the susceptibility loci. As a result, traditional association methods (such as case/control, measured genotype, and transmission/disequilibrium test [TDT]) will have no power if the loci are examined individually. In this article, we examine this class of models, delimiting the range of genetic determination and recurrence risks for two-, three-, and four-locus purely epistatic models. Our study reveals that these models, although giving rise to no additive or dominance variation, do give rise to increased allele sharing between affected sibs. Thus, a genome scan for linkage could detect genomic subregions harboring susceptibility loci. We also discuss some simple multilocus extensions of single-locus analysis methods, including a conditional form of the TDT.

Figure 1

Limits of two-locus, biallelic, purely epistatic (i.e., V_A=V_D=0 at each locus) models, with all alleles equally frequent. The bottom curve represents the maximum variance due to genotype (i.e., V_I), the middle curve represents the total variance as a function of disease prevalence (i.e., V_T=K(1-K)), and the top curve represents the maximum proportion of variance attributable to genotype (i.e., h²=V_I/V_T).

Figure 2

Limits of three-locus, biallelic, purely epistatic (i.e., V_A=V_D=0 at each locus) models, with all alleles equally frequent. The bottom curve represents the approximate maximum variance due to genotype (i.e., V_I), estimated by an iterative maximization algorithm from the SAS Institute (1995), the middle curve represents the total variance (i.e., V_T=K(1-K)) as a function of disease prevalence, and the top curve represents the values for h² that we have found for particular models; the dots on the top curve are the maximum proportion of variance attributable to genotype (i.e., h²=V_I/V_T), estimated by the iterative maximization method.

Figure 3

Limits of three-locus, biallelic, purely epistatic (i.e., V_A=V_D=0 at each locus) models. The bottom curve represents the estimated maximum variance due to genotype (i.e., V_I), the middle curve represents the total variance (i.e., V_T=K(1-K)) as a function of disease prevalence, and the top curve represents the estimated maximum proportion of variance attributable to genotype (i.e., h²=V_I/V_T).

Figure 4

Limits of four-locus, biallelic, purely epistatic (i.e., V_A=V_D=0 at each locus) models, with all alleles equally frequent. The bottom curve represents the estimated maximum variance due to genotype (i.e., V_I), the middle curve represents the total variance (i.e., V_T=K(1-K) as a function of disease prevalence, and the top curve represents the estimated maximum proportion of variance attributable to genotype (i.e., h²=V_I/V_T).

Figure 5

Limits of four-locus, biallelic, purely epistatic (i.e., V_A=V_D=0 at each locus) models. The bottom curve represents the estimated maximum variance due to genotype (i.e., V_I), the middle curve represents the total variance (i.e., V_T=K(1-K)) as a function of disease prevalence, and the top curve represents the estimated maximum proportion of variance attributable to genotype (i.e., h²=V_I/V_T).

Figure 6

Comparison of maximum heritabilities for three-locus, purely epistatic models with (top curve) and without (bottom curve) two-locus interactions. The maximum heritabilities for two-locus, purely epistatic models (middle curve) are included as a reference.