The mitotic oscillator: Temporal self‐organization in a phosphorylation‐dephosphorylation enzymatic cascade

Abstract

The conditions for temporal self‐organization in the form of sustained oscillations are determined in a minimal cascade model previously proposed (A. Goldbeter, Proc. Natl. Acad. Sci. USA 88, 9107–9111 (1991)) for the mitotic oscillator driving the embryonic cell division cycle. The model is based on a phosphorylation‐dephosphorylation cascade involving cyclin and cdc2 kinase. In the first cycle of the cascade, cdc2 kinase is activated through dephosphorylation triggered by the accumulation of cyclin, while in a second cycle the activation of a cyclin protease is brought about through phosphorylation by cdc2 kinase. The fact that cyclin promotes the activation of cdc2 kinase while the latter enzyme triggers cyclin degradation introduces a negative feedback loop which is at the core of the periodic operation of the cascade. We analyze the mechanism of oscillatory behavior by constructing stability diagrams as a function of some of the main parameters of the model. Investigated in turn are the roles of negative feedback and of phosphorylation‐dephosphorylation thresholds. Such thresholds arise from the phenomenon of zero‐order ultrasensitivity associated with the kinetics of covalent modification cycles. An extension of the minimal model allows one to address the effect of additional phosphorylation‐dephosphorylation cycles and the possible role of autocatalysis by cdc2 kinase in the generation of periodic behavior. The analysis shows that the existence of thresholds as well as an increase in the number of cycles in the cascade favor the occurrence of sustained oscillations. The results further indicate that negative and positive feedback may both contribute to the repetitive activation of cdc2 kinase.

A Model of Cell Cycle Behavior Dominated by Kinetics of a Pathway Stimulated by Growth Factors

“…Several models have been proposed in the past to describe phase-specific events of the cell cycle (Hyver and Le Guyader, 1990;Goldbeter, 1991;Norel and Agur, 1991;Thron, 1991;Tyson, 1991;Obeyesekere et al, 1994;Romond et al, 1994;Thron, 1994;Hatzimanikatis et al, 1995;Novak and Tyson, 1995;Obeyesekere et al, 1995;Tyson et al, 1995;Thron, 1997;Kohn, 1998). The previous work of Obeyesekere et al (1997) proposed a mathematical model which described the action of D-type cyclins, cdk4, cyclin E, cdk2, E2F, and RB on progression through the G1-phase of the mammalian cell cycle.…”

Thresholds and Oscillations in Enzymatic Cascades

“…24,25,35,40 Such positive feedback, as well as additional phosphorylation-dephosphorylation cycles on the path leading to activation of the cyclin protease, tend to increase the domain of oscillatory behavior and thereby make the requirement for steep thresholds less stringent. 35, 36 The present results suggest ways to arrest the oscillatory behavior of the cascade. Thus, oscillations can be suppressed if the Michaelis constant of one or several of the converter enzymes is sufficiently increased so that the threshold in the corresponding cycle is shifted, becomes less steep, or disappears.…”

Rescue of the Quasi‐Steady‐State Approximation in a Model for Oscillations in an Enzymatic Cascade

“…Time t, concentration C, and all parameters except the Michaelis constants have units as in [16,17], where simulations are compared to experiments. We keep these equations in this form because they appear in all previous studies [16,17,30,31].…”

“…Linear theory. The Hopf bifurcation boundaries were investigated only numerically [30,31]. In this appendix, we determine analytically the steady-state solution of (1)- (3) and its Hopf bifurcation point for the particular case when The steady-state solution (C, M, X) = (C s , M s , X s ) satisfies the following three conditions:…”