Fig. 5.

Neural network predictions enable comprehensive exploration of pattern formation dynamics. a Ensemble of deep neural networks enables screening through a vast parametric space. The parametric space consists of 13 parameters that were varied uniformly in the provided ranges (Supplementary Table 1). For each instance, we randomly generated all the varying parameters and used the neural network to predict the peak and distributional values for each parameter combination. We collected 10⁸ instances and discarded predictions with disagreement between ensemble predictions larger than 0.1. We then projected all the instances on all the possible 2 parameter plane. The majority of the instances generated patterns with no ring (gray), and they were distributed all over the projected parametric planes. Due to the huge number of instances, the parametric distribution of no ring (grey), one-ring (green), two-rings (blue) patterns on the projected 2D planes partially overlap. From the distribution of neural network predicted three-ring patterns (orange) over all the possible 2D parameter planes, the critical constraints to generate three-ring patterns are revealed: large domain radius (D), large synthesis rate of T7RNAP (α_T), small synthesis rate of T7 lysozyme (α_L), small half activation constant of T7RNAP (K_T), small half activation distance for gene expression ($K_{\phi }$). The analysis also suggested correlations between K_T and α_C (cell growth rate on agar), K_T and α_T, D and α_C. b–d Neural network predictions facilitate the evaluation the objective function of interest (generation of three-ring patterns). Based on the analysis above, we sought to further identify the correlation between K_T and α_C, K_T and α_T, D and α_C. For each of the screening, we varied two parameters of interest and fixed the rest. We collected 10⁷ instances and discarded predictions with disagreement between ensemble predictions larger than 0.1. We found generation of three-ring patterns requires a negative correlation between D and α_C and a negative correlation between K_T and α_C. We also found a positive linear correlation between K_T and α_T. α_A = 0.5, α = 0.5, β = 0.5, $K_{\varnothing }=0.3$, $n=0.5$, ${\alpha }_L=0.3$, $K_C=0.5$,$K_P=0.5$, $d_A=0.5$, b ${\alpha }_C=0.5$, $D=1.0$. c ${\alpha }_T=0.8$, $D=1.0$. d ${\alpha }_T=0.8$, $K_T=0.3$