Mathematical analysis of binary activation of a cell cycle kinase which down-regulates its own inhibitor.

In mammalian cells, the heterodimeric kinase cyclin E/CDK2 (EK2) mediates cell cycle progress from G1 phase into S phase. The protein p27Kip1 (p27) binds to and inhibits EK2; but EK2 can phosphorylate p27, and that leads to the deactivation of p27, presumably liberating more EK2 and forming a positive-feedback loop. It has been proposed that this positive-feedback loop gives rise to binary (all-or-none) release of EK2 from its inactive complex with p27. Binary release suggests a bistable biochemical system in which a stable steady state with low EK2 activity is extinguished in a saddle-node bifurcation, causing the system to shift abruptly to a stable steady state with high EK2 activity. Two mathematical models are discussed, one in which free EK2 deactivates p27 in the EK2-p27 inhibitory complex as well as free p27, and one in which the rate of EK2-catalyzed deactivation of free p27 has saturable kinetics with respect to free p27. In general, if inhibitory binding is approximately in equilibrium, bistability requires that there be a potential unstable steady state where the reaction order of p27 deactivation is greater with respect to EK2 than with respect to p27.