In dividing cells, entry into mitosis is caused by maturation promoting factor (MPF), which is formed autocatalytically by activation of a complex of p34cdc2 and cyclin B. This biochemical system may oscillate, causing repeated mitosis. It is shown mathematically that the oscillatory tendency would be enhanced by a cofactor which binds to MPF and inhibits its autocatalytic action. A candidate for such a cofactor is the suc1 gene product p13, which binds to p34cdc2/cyclin B complex and inhibits MPF-induced MPF activation. At a steady rate of cyclin biosynthesis, with small amounts converted to MPF, p13suc1 would have to be titrated by MPF before autocatalysis could begin. This would have three possibly important effects: (1) it would determine the 'threshold' cyclin accumulation (and hence the corresponding time-delay) for MPF activation; (2) it would cause the accumulation of a backlog of MPF precursor (tyrosine-phosphorylated p34cdc2/cyclin B) sufficient to produce a substantial MPF pulse when MPF autocatalysis begins; (3) it would give the autocatalysis a high reaction order, which tends to destabilize the steady state, promote autonomous oscillations, and enhance the triggering property (excitability) of the system. The MPF pulse generated by this system may be essential for the proper triggering of the events of M phase, including the cyclin degradation which inactivates MPF at the end of M phase. This model offers explanations for several puzzling effects of p13suc1, including the fact that p13suc1, though an inhibitor of MPF activation, is nevertheless necessary for mitosis.