Fig. 2. Framework for inferring regulatory networks.

a With ODE describing the network (left), various time series are simulated with different initial conditions (middle). Then, from each time series, the regulation-detection score $S_{X^{\sigma }}^Y$ is calculated for every 1D regulation type σ (Step 1). The criteria $S_{{\mathbf{X}}^{\sigma }}^Y=1$ infers A ⊣ B. Next, $S_{{\mathbf{X}}^{\sigma }}^Y$ is calculated for every 2D regulation type σ (Step 2). Among the three types of regulations with $S_{{\mathbf{X}}^{\sigma }}^Y=1$, only one passes the Δ test (Step 3). By merging the inferred 2D regulation with the 1D regulation from Step 1, the regulatory network is successfully inferred. Here, data are presented as box plots (n = 100), in which the box bounds the IQR divided by the median, and whiskers extend to a maximum of 1.5× IQR beyond the box. b–f This framework successfully infers the network structures of the Kim–Forger model (b), Frzilator (c), the 4-state Goodwin oscillator (d), the Goldbeter model for the Drosophila circadian clock (e), and the cAMP oscillator of Dictyostelium (f). For each model, 100 time-series data are simulated from randomly selected initial conditions, which lie in the range of the original limit cycle. Source data are provided as a Source Data file.