Fig. 4.

Optimal regulatory curves. (a) Optimal regulatory curves, $\overline{g}$(c), for C = 0.1 (left), C = 1 (middle), and C = 10 (right), and different values of M in the indirect translational regulation (ITR) model (colored curves): M = 10 (top), M = 100 (middle), M = 1000 (bottom). The direct transcriptional regulation (DTR) model is plotted in black for reference. Colors correspond to the vertical lines in Fig. 2. Here, F = 1 and y_max = 10. ΔI, the difference in capacity between ITR and DTR in bits, is shown in each panel. (b) and (c) Optimal regulation threshold, $K_{\text{eff}}^{*}$, and the optimal Hill coefficient, H*, that characterize the regulatory curves in (a), shown as a function of C. Since the overall regulation is repressive in either scheme, Hill coefficients are negative. In the DTR model, K_eff ≡ K; for ITR, K_eff = (K_c/K)My_max – K_c.