Fig. 1. Inferring regulation types using regulation-detection functions and scores.

a Because X positively regulates Y, as X increases, $\stackrel{°}{Y}$ increases. Thus, whenever X^d(t, t^*) = X(t) − X(t^*) > 0, ${\stackrel{°}{Y}}^d\left(t,t^{*}\right)=\stackrel{°}{Y}\left(t\right)-\stackrel{°}{Y}\left(t^{*}\right)>0$. b Therefore, when X^d(t, t^*) > 0, the regulation-detection function $I_{X^{+}}^Y\left(t,t^{*}\right):=X^d\left(t,t^{*}\right)\cdot {\stackrel{°}{Y}}^d\left(t,t^{*}\right)$ is always positive. Here, I is in the range [−1, 1] since all the time series are normalized. c If X negatively regulates Y, $I_{X^{-}}^Y:=\left(-X^d\right)\cdot {\stackrel{°}{Y}}^d$ is always positive when X^d(t, t^*) < 0. d–i When X₁ and X₂ positively regulate Y, as X₁ and X₂ increase ($X_1^d>0$, $X_2^d>0$), $\stackrel{°}{Y}$ increases (${\stackrel{°}{Y}}^d>0$) (d). Thus, when $X_1^d\left(t,t^{*}\right)>0$ and $X_2^d\left(t,t^{*}\right)>0$, $I_{X_1^{+}{X}_2^{+}}^Y:=X_1^d\cdot X_2^d\cdot {\stackrel{°}{Y}}^d$ is positive (e). When X₁ and X₂ positively and negatively regulate Y, respectively (g), $I_{X_1^{+}{X}_2^{-}}^Y:=X_1^d\cdot \left(-X_2^d\right)\cdot {\stackrel{°}{Y}}^d$ is always positive when $X_1^d\left(t,t^{*}\right)>0$ and $X_2^d\left(t,t^{*}\right)<0$ (i). Such positivity disappears for the regulation-detection functions, which do not match with the actual regulation type (f, h). j–l When X₁ positively regulates Y and X₂ does not regulate Y (j), both $I_{X_1^{+}{X}_2^{+}}^Y:=X_1^d\cdot X_2^d\cdot {\stackrel{°}{Y}}^d$ (k) and $I_{X_1^{+}{X}_2^{-}}^Y:=X_1^d\cdot \left(-X_2^d\right)\cdot {\stackrel{°}{Y}}^d$ (l) are positive because the regulation type of X₂ does not matter. Here, we use $X_1\left(t\right)=\cos \left(2\pi t\right)$ and $X_2\left(t\right)=\sin \left(2\pi t\right)$ as the input signal and Y(0) = 0 for simulation on [0, 1]. Source data are provided as a Source Data file.