BIFURCATION-ANALYSIS OF A MODEL OF MITOTIC CONTROL IN FROG EGGS

Novak and Tyson have proposed a realistic mathematical model of the bi ochemical mechanism that regulates M-phase promoting factor (MPF), the major enzymatic activity controlling mitotic cycles in frog eggs, ear ly embryos, and cell-free egg extracts. We use bifurcation theory and numerical methods (AUTO) to characterize the codimension-one and -two bifurcation sets in this model. Our primary bifurcation parameter is t he rate constant for cyclin synthesis, which can be manipulated experi mentally by adding exogenously synthesized cyclin mRNA to extracts dep leted of all endogenous mRNA molecules. For the secondary bifurcation parameter we use the total amount of one of the principal regulatory e nzymes in the extract (APC, the enzyme complex that labels cyclin for degradation; Weel, the kinase that inhibits MPF; or Cdc25, the phospha tase that activates MPF). We find a rich array of physiologically dist inct behaviors exhibited by the model as these parameters are varied a round values that seem plausible for frog eggs and extracts. In additi on to unique, stable steady states (cell cycle arrest) and limit cycle oscillations (autonomous, periodic cell division), we find parameter combinations where the control system is bistable. For instance, an in terphase-arrested state may coexist with a metaphase-arrested state, o r two stable limit cycles of different amplitude and period may coexis t. We suggest that such strange behavior is nearly unavoidable in a co mplex regulatory system like the cell cycle. Perhaps cells exploit som e of these exotic bifurcations for control purposes that are as yet un recognized by physiologists. (C) 1998 Academic Press.