A unified model for functional and statistical epistasis and its application in quantitative trait Loci analysis

Abstract

Interaction between genes, or epistasis, is found to be common and it is a key concept for understanding adaptation and evolution of natural populations, response to selection in breeding programs, and determination of complex disease. Currently, two independent classes of models are used to study epistasis. Statistical models focus on maintaining desired statistical properties for detection and estimation of genetic effects and for the decomposition of genetic variance using average effects of allele substitutions in populations as parameters. Functional models focus on the evolutionary consequences of the attributes of the genotype-phenotype map using natural effects of allele substitutions as parameters. Here we provide a new, general and unified model framework: the natural and orthogonal interactions (NOIA) model. NOIA implements tools for transforming genetic effects measured in one population to the ones of other populations (e.g., between two experimental designs for QTL) and parameters of statistical and functional epistasis into each other (thus enabling us to obtain functional estimates of QTL), as demonstrated numerically. We develop graphical interpretations of functional and statistical models as regressions of the genotypic values on the gene content, which illustrates the difference between the models--the constraint on the slope of the functional regression--and when the models are equivalent. Furthermore, we use our theoretical foundations to conceptually clarify functional and statistical epistasis, discuss the advantages of NOIA over previous theory, and stress the importance of linking functional and statistical models.

Figure 1.—

Ideograph showing the main foundations of the NOIA model for the one-locus case. The starting point is a description of the genetic effects as allele substitutions on the reference genotype G₁₁ (solid circle). This description is extended to a general functional formulation (thick solid line) by means of the change-of-reference tool represented by the horizontal and the nearly horizontal arrows. The reference points for which criterion (7) holds are represented to the left of the vertical dashed line. For populations with those reference points as mean phenotype, the functional formulation is orthogonal, and it coincides with the statistical formulation (thick shaded line). Other reference points may be represented to the right of the vertical dashed line. For populations with those reference points as mean phenotype, the two formulations do not coincide, and the transformation tool, represented by the vertical arrow, can be used to transform the functional formulation into the statistical formulation and vice versa. The F₂ and the HW₃ populations, represented as labeled vertical bars on the functional and statistical formulations, are examples of these two situations (see text for details).

Figure 2.—

Graphical interpretation of the parameters of the NOIA model for an F₂ population in the one-locus case. The values of the parameters come from a regression of the genotypic values (G₁₁, G₁₂, G₂₂) on the gene content. These genotypic values are represented as solid circles, and their size is determined by their frequency in the population. We show a case of strong overdominance because it allows us to better visualize the parameters of interest. The functional regression (thick solid line) is constrained to have the same slope as the (dashed) line through G₁₁ and G₂₂. The statistical regression (thick shaded line) is a weighted linear regression on the gene content, and under condition (7) it has the same slope as the line through G₁₁ and G₂₂. In this case, therefore, the functional and statistical regressions coincide. The elements of the S = (s_ij) matrix are the natural and the orthogonal scales, in the functional and the statistical model formulations, respectively. Latin letters are the functional genetic effects, and Greek letters are the statistical genetic effects. The reference point, R = 1.25, is represented by a triangle. It is the intercept of the regressions and it occurs at the average gene content, which in this case is one. It is the starting point from which to measure the additive effects (α_I = a_i). The deviations of the regression, the dominance deviations (δ_ij = d_ij) would be zero if there were no dominance. We show their relationship to the parameters of the model.

Figure 3.—

Graphical interpretation of the parameters of the NOIA model for an HW₃ population, with frequencies p₁₁ = 0.09, p₁₂ = 0.42, p₂₂ = 0.49, in the one-locus case. (A) Statistical formulation. All the symbols have the same meaning as in Figure 2. The regression on the gene content (thick shaded line) does not have the same slope as the (dashed) line through G₁₁ and G₂₂, and therefore it does not coincide with the functional regression of the same population (shown below). This happens when the frequencies of the population do not fulfill condition (7), as shown in the ideograph (above). The reference point, R = 1.33, occurs at gene content 1.4. (B) Functional formulation. All the symbols have the same meaning as in Figures 2 and 3A. The regression on the gene content (thick solid line) is forced to be parallel to the (dashed) line through G₁₁ and G₂₂ and would have a different slope otherwise (see statistical regression of the same population above). However, the reference point is the same as in the statistical regression.