Fig. 7: Two-component autoregulatory loops reproduce behaviors of autoregulation, but have additional behaviors describing the relative dynamics of the components.
From: Nonlinear delay differential equations and their application to modeling biological network motifs
a A two-component loop network motif, which is similar to autoregulation but with two explicitly modeled genes instead of one. b Parameter space of the two regulatory strength parameters showing phase diagram for a loop composed of two activators (cross-activating) or two repressors (cross-inhibitory). Shading shows results of simulations (with an interval of 0.1 for both γ and η axes); blue curve is the analytically derived phase (bifurcation) boundary from Eq. (32). c Parameter space varying both strength parameters. Because the Hopf bifurcations depend only on the total delay and transient oscillations most prominent for equal delays, we show only the cases γ1 = γ2. Blue curves show analytically derived Hopf bifurcations. Black dashed curves are the bifurcation boundaries from (b). Except for the activator/repressor case, all these curves lie above the bistability boundary given by the black curve in (b), meaning oscillations are always transient. d Representative simulations for specific initial conditions showing all possible qualitative behaviors for a two-component loop with two activators, two repressors, or one activator and one repressor. For two-activator bistability and oscillations, a second set of initial conditions is shown in dashed lines to demonstrate the bistability. Hill coefficients equal 2 for repressors, −2 for activators.
