Fig. 4

Epistasis domain structure. The biochemical model and epistasis analysis (Fig. 3) suggests a radial epistasis domain structure centred at the optimum, with e.g. downstream sign epistasis (red) in a single domain towards the top and bottom, upstream sign epistasis (green) towards the right and left and reciprocal sign epistasis (yellow) in between (see a). Here, we test whether the empirical data are consistent with the domain structure. a Angle normalisation scheme, which can be seen as straightening the dashed lines. More specifically, for any point P, the angular coordinates \(\alpha _{\mathrm{A}}\), \(\alpha _{\mathrm{B}}\), \(\alpha _{\mathrm{C}}\) and \(\alpha _{\mathrm{D}}\) on the black circle passing through P are the intersections with the parameterised optima (red and green dashed lines). These angles are normalised to respectively \(\theta _{\mathrm{A}} = 0\), \(\theta _{\mathrm{B}} = {\mathrm{\pi }}/2\), \(\theta _{\mathrm{C}} = {\mathrm{\pi }}\) and \(\theta _{\mathrm{D}} = 3{\mathrm{\pi }}/2\). The normalised angle \(\theta _{\mathrm{P}}\) for point P is interpolated linearly between these angles. b Normalised angular histograms of epistasis counts for the empirical data (Fig. 1), aggregated overall input ranges, for all possible initial and final mutants and for all phenotypic parameters. Faded bars: non-significant counts, p < 0.05 (one-tailed binomial test, Methods section). Upstream and downstream sign epistasis counts (resp. green and red) are grouped in peaks, which is consistent with the radial organisation of domains (a, left). Note that empirical data does not cover the full phenotype space, and hence some angles (e.g. bottom right) are less represented in the data. Some deviations are observed, such as the yellow peak towards the bottom right