The mitotic oscillator

A. Goldbeter, A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase. Proc. Natl. Acad. Sci. USA 88, 9107-9111 (1991). 

A. Goldbeter, Modeling the mitotic oscillator driving the cell division cycle. Comments Theor. Biol. 3, 75-107 (1993). 

The molecular bases of Ca oscillations, as well as those of the mitotic oscillator, are addressed at the end of this book, where minimal models closely related to recent experimental observations are analysed for the two phenomena and shown to admit limit cycle oscillations. 

Modelling the mitotic oscillator driving the cell division cycle... 

These results have opened the way to the construction of more realistic models for the mitotic oscillator. The purpose of this chapter is briefly to present these models and to classify them according to the type of regulation responsible for oscillatory behaviour. The way sustained oscillations are generated is examined in detail in a minimal model based on the cascade of phosphorylation-dephosphorylation cycles that controls the onset of mitosis in embryonic cells. Extensions of the cascade model taking into account additional, recently uncovered phosphorylation-dephosphorylation cycles are considered. Ways of arresting the cell division cycle in that model and the control of the mitotic oscillator by growth factors are also discussed. 

The interaction of cdc2 kinase with cyclin is in fact central to the mechanism of the mitotic oscillator in all eukaryotic cells (Nurse, 1990). The situation encountered in the early stages of amphibian development represents the simplest form of mitotic trigger mechanism... 

Murray Kirschner, 1989a) in such rapidly dividing embryonic cells, where the period of the mitotic oscillator is of the order of 30 min, the accumulation of cyclin alone suffices to drive the cell cycle (Murray Kirschner, 1989b), which may be viewed as a cdc2 cycle (Miuray, 1989a). Oscillations in cdc2 kinase activity have been demonstrated in Xenopus egg extracts (Felix et al, 1989). 

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... 

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... 

A phosphorylation-dephosphorylation cascade model for the mitotic oscillator in embryonic cells... 

Fig. 10.4. Minimal cascade model for the mitotic oscillator. The cascade incorporates cyclin synthesis and degradation, activation of the phosphorylated form of cdc2 kinase through dephosphorylation into the form M by phosphatase cdc25 (El), which is itself activated by cyclin, inactivation of active cdc2 kinase M into M by the kinase weel (Ej), phosphorylation of inactive cyclin protease into the active form X by cdc2 kinase (E3), and inactivation of X into X by the phosphatase E4 (see text for details).

Fig. 10.7. Limit cycle behaviour of the cascade model for the mitotic oscillator. The curves are obtained by projecting the trajectory of the three-variable system governed by eqns (10.1) onto the cyclin-cdc2 kinase (C, M) plane. Two sets of initial conditions are considered, one inside and the other outside the limit cycle arrows indicate the direction of the time evolution. Parameter values are X, = 0.1 (i = 1,... 4), = 0.5 min" Vj = 0.167 min = 0.2 min ...

Fig. 10.11. Sustained oscillations in the extended cascade model of fig. 10.10 for the mitotic oscillator, in the absence of autocatalysis by cdc2 kinase. Shown are the fraction of active cdc25 phosphatase (P), the fraction of active cdc2 kinase (M), the cyclin concentration (C), and the fraction of active cyclin protease (2l). The curves are obtained by numerical integration of eqns (lO.la-c), and (10.10)-(10.12). Parameter values are (in min" ) T i = 4, V = 1.5, = 1,...

Any model for the biochemical oscillator controlling the onset of mitosis should account for the fact that, in some cells or at certain stages of development, mitosis can be prevented and cells cease to divide. In dynamic terms, such a transient or permanent suppression of mitotic activity can be viewed as a switch of the mitotic control system from an oscillatory into a nonoscillatory regime corresponding to a stable steady state. Models for the mitotic oscillator allow the discussion of possible mechanisms whereby the mitotic clock might be arrested. 

Fig. 10.15. Arrest of the mitotic oscillator through inhibiting the cdc25 phosphatase that activates cdc2 kinase. The evolution of the fraction of active cdc2 kinase (M) in the minimal cascade model of fig. 10.4 is shown, together with the cyclin concentration (C) under the conditions of fig. 10.6, with K = K2 = K =...

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