Fig. 2

Sign epistasis resulting from geometric fitness models. Diagrams illustrating the principles to determine sign epistasis from phenotype–fitness relations. Here, we consider Gaussian fitness functions that do not specifically describe cascades nor molecular interactions. a Non-rotated Gaussian fitness function indicated by colour gradient and elliptical iso-fitness lines. Importantly, we consider mutations that affect X but not Y, and mutations that affect Y but not X, which is typically the case if X and Y have a distinct genetic basis. Mutations in X and Y are therefore described by orthogonal vectors (arrows). We consider mutations that lead from a suboptimal phenotype XY to the optimal X′Y′(arrows). For any starting phenotype XY in this landscape, all four mutations to X′Y' produce increase in fitness, resulting in magnitude epistasis (grey) everywhere (bottom). b Rotated Gaussian fitness function (angle = π/12). Starting from the red dot phenotype, one mutation decreases fitness, which implies sign epistasis (see arrows). To understand in mathematical terms, consider the dashed red line (X^opt), which indicates where the tangent of the iso-fitness lines is horizontal (white horizontal line). When moving along this tangent (which changes X but not Y), the fitness is optimal at X^opt. Thus, starting at or around X^opt, when changing X but not Y, fitness can only decrease. When the line X^opt is non-vertical as here, the same mutation can increase fitness in another Y background (Y’), because the optimum then shifts, and hence the phenotype (XY’) no longer lies at the optimum. The same analysis holds for the green dashed line (Y^opt). c Rotated Gaussian fitness function (angle = π/4). When domains for sign epistasis in X- and Y-effect (red and green) overlap, one obtains reciprocal sign epistasis (yellow dot). To link these features, such as the degree of rotation or skewness to the molecular level, one can consider a mechanistic fitness function (Fig. 3)