Figure 3. Incoherent Inputs Improve the Robustness of Biological Oscillators with Experimentally Estimated Parameters.

(A) Compared with a single input, incoherent or coherent inputs increase or decrease the nullcline range of a node, respectively. Left panel: heatmap of the nullcline ranges of node A that receives both an input of strength k₁ and a self-feedback of strength k₂. The value of k₁ (or k₂) can be positive or negative, representing activation or inhibition. The inset on the top shows the representative topologies for different combinations of k₁ and k₂ with positive or negative values. To eliminate any effect from parameters other than k₁ and k₂, the mean nullcline range is calculated from 100 simulations, with all parameters except for k₁ and k₂ randomly sampled within the parameter ranges listed in Table S1. The inset on the left shows three examples of nullcline for a node with the same negative input on one leg, but with an additional self-positive feedback, no additional input, or an additional self-negative feedback on the other leg (basal reaction rate = 0.1, self-regulation rate |k₂| = 1, input rate |k₁| = 10, EC₅₀ = 0.1, n = 2). Each nullcline is colored according to its nullcline range. Right panel: heat-map of the null-plane ranges of node A that receives both two inputs of strength |k₁| and of strength |k₂|. All notations are the same as in the left panel. Examples of nullplane are showing in Figures S4A–S4C.

(B and C) Topologies of the cell cycle (B) and p53 oscillator (C), where the nodes that receive incoherent inputs are labeled in green and the interactions of interest in yellow.

(D and E) Time courses of active Cdk1 levels (D) and total p53 levels (E), either with (labeled with *) or without (labeled with ○) the interactions labeled in yellow in (B and C). The results show that incoherent input is necessary for oscillation. The rest of parameter values are unchanged from the literature values (Batchelor et al., 2011; Tsai et al., 2014).

(F and G) Heatmaps of the nullcline ranges of Cdk1-cyclin B (F) and ATM (G), indicating that the strength of any of the incoherent inputs, such as Cdc25, Wip1–|ATM, or DSB, is positively correlated with the nullcline range, while the coherent input strength of Wee1 is negatively correlated with the nullcline range. The points labeled with * and ○ correspond to the same systems as in (D and E).

(H and I) Bifurcation analysis. The shaded regions denote the parameters compatible with sustained oscillations. The parameter ranges of the cyclin B synthesis rate constant k_synth (H) and DSB signal input strength (I), both as the essential clock inputs, become wider as the incoherent input strength k_Cdc25 (H) and k_Wip1–|ATM (I) increase. These results indicate that incoherent inputs increase the parameter choice for oscillation, and thus increase the robustness of the system. The points labeled with * and ○ correspond to the same systems as in (D and E).