A complete classification of epistatic two-locus models

Abstract

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 x additive, additive x dominance, and dominance x dominance interaction terms. We discuss the connection between our classification and standard epistatic models and demonstrate its utility by analyzing a previously published dataset.

Figure 1

Example of epistasis. Example of epistasis in QTL data. The data is on chicken growth [24]. (a) The phenotypic means of the two-locus genotypes, (b) a wiggle plot of the data, where each line corresponds to a row in the table, (c) bar plot of the data, (d) the two-locus shape.

Figure 2

Symmetry classes. Shapes. The 69 symmetry classes of the shapes of two-locus models.

Figure 3

Shapes of two-locus models. Two-locus models. The model shapes for multiplicative and heterogeneous two-locus models.

Figure 4

Subdivisions. 0/1 models. The subdivisions for the 50 Li and Reich 0/1 penetrance models.

Figure 5

Shapes of two-locus models. Epistastic models. The tables list the genotype values associated with four epistatic models and below each table is the shape induced by a model with purely add × add, add × dom, dom × add or dom × dom interaction.

Figure 6

Example. Example of observed epistasis. A visual representation of the 30 trait/QTL pairs. The phenotype average for each genotype is given by the heights of the bars, the corresponding shape is also given, and the trait (A-E) and QTL pair (1–21) listed. Under each panel we list the cluster it falls into (1–4) and the group given by [24] (A, B, H, L, U).

Figure 7

Power to detect association. Power to detect association. The plots show the maximum value of the likelihood ratio test statistic observed for randomly generated data from each of the 387 shapes.