Models based on positive feedback by cdc2 kinase

103 Models based on positive feedback by cdc2 kinase 

In their early theoretical studies of the mitotic oscillator, Kauffman et al (Kauffman, 1975 Kauffman Wille, 1975 Tyson Kauffman, 1975) resorted to the abstract, Brusselator model (Lefever Nicolis, 1971) for their simulations of mixing experiments in which Physarum plasmodia taken at different phases of the cell cycle were fused. Like most models proposed for limit cycle behaviour, the Brusselator relies on an autocatalytic step for producing the instability leading to oscillations an advantage of this simple model is that the temporal evolution is governed by two polynomial, nonlinear kinetic equations (Lefever Nicolis, 1971). 

The first models for the mitotic oscillator specifically based on the interaction between cyclin and cdc2 kinase also relied on positive feedback. The experimental basis for autocatalytic regulation of cdc2 kinase stems primarily from observations showing that catalytic amounts of active MPF promote the transition from inactive to active MPF, which consists of a complex between cyclin and the active form of cdc2 kinase spontaneous activation of this transition, however, does not normally 

In the model proposed by Hyver Le Guyader (1990), two systems of equations are considered. In the first version, inactive p34 (i.e. cdc2 kinase) transforms into active p34, either spontaneously or in an auto-catalytic manner active p34 then combines with cyclin to yield active MPF. This situation is described by four differential equations, of a polynomial nature, in which the highest nonlinearities are of the quadratic type. In a second version of this model, governed by three kinetic equations of a similar form, the authors consider the effect of an activation of MPF by MPF itself as well as cyclin, and show that oscillations develop when the degradation of cyclin is brought about by the formation of a complex between cyclin and MPF. That study was the first to show the occurrence of sustained oscillations in a model based on the interactions between cyclin and cdc2 kinase. The type of kinetics considered for these interactions remained, however, remote from the actual kinetics of phosphorylation-dephosphorylation cycles. 

Building on similar ideas but starting from a more detailed reaction scheme, Tyson (1991) proposed a model for the mitotic oscillator based on the formation of a complex between cycUn and cdc2 kinase, followed by the activation of this complex. Essential to the oscillatory mechanism is the assumption that the active complex, i.e. MPF, promotes its own activation in a nonlinear memner. The kinetic equations, of a polynomial form, reduce under some simplifying assumptions to the equations of the two-variable Brusselator model. Inactivation of MPF is not 

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