Figure 5.

Minimal oscillator with two steps of phosphorylation and dephosphorylation of M substrates (namely, APC). (A) Schematic diagram. Cdk/cyclin phosphorylates M and M_P, and the doubly phosphorylated form of M (M_PP) promotes the degradation of Cdk/cyclin. The different phosphoforms of M compete for phosphorylation by Cdk and dephosphorylation by PPase. (B and C) Time evolution of Cdk/cyclin, securin, and M_PP in the presence or absence, respectively, of securin. (D and E) For the cases with and without securin, the limit cycle (closed curve) is projected onto the APC_PP-versus-Cdk/cyclin phase plane. The S-shaped curve is the APC_PP nullcline. For intermediate values of Cdk/cyclin, a region of bistability is present in the phosphorylation state of APC, whether or not securin is present. Arrows indicate the direction of motion along the limit cycle. (F and G) Bifurcation diagrams for M_PP versus V_scdk in the presence or absence, respectively, of securin. Solid curves indicate stable steady states or maxima and minima of the sustained oscillations, and dashed curves indicate unstable states. With securin present, the limit-cycle region is bounded by a SNIC bifurcation at V_scdk ≈ 0.005 and a Hopf bifurcation at V_scdk ≈ 0.15. In the absence of securin, sustained oscillations are still possible, and in this case, the limit-cycle region is bounded by two Hopf bifurcations. Parameter values for the simulations are given in Section 4 of Supplement 1 in the Supporting Material.