It has been suggested that a ''master clock'' controls early embryonic divisions because these divisions are relatively fast, synchronous, and not dependent on cell growth. Although several components of this clock are known, the essential timing elements are not yet established. The clock is characterized by the activity of the maturation or M-phase-promoting factor (MPF), a cyclin-dependent kinase, that when active indicates that the cell is undergoing mitosis. The master clock, therefore, is set by the factors that determine the activity of MPF. The protein components that determine the activity of MPF seem common among eukaryotic cells and have been labeled a ''universal control mechanism'' (1) that regulates the onset of M phase. In the present study we call these protein components the MPF system. Studies in several systems also suggest that transient increases in the concentration of free cytosolic calcium ([Ca 2ϩ ] i ) contribute to cell cycle regulation. However, the extent to which these transients regulate early embryonic division is under discussion. One possibility is that these transients are essential, and that the mechanism that controls variations in [Ca 2ϩ ] i is a fundamental component of the master clock. The following observations (among others) suggest this scenario: (i) [Ca 2ϩ ] i transients* accompany early embryonic cell cycles (2-7); (ii) the injection of calcium buffers into intact Xenopus blastomeres delays or blocks division (8); and (iii) in the absence of division, [Ca 2ϩ ] i transients appear with the same frequency as that of division (3,5,6).A second scenario supposes that [Ca 2ϩ ] i transients are secondary. Here the master clock is timed chiefly through protein interactions. This scenario is supported by several experiments: (i) MPF activity in cycling extracts prepared from Xenopus laevis eggs is controlled by cyclin synthesis and destruction (9, 10); (ii) clam oocyte extracts continue cell cycle events in the presence of 5 mM EGTA (11), a strong calcium chelator; and (iii) in these extracts the addition of 1 mM CaCl 2 does not induce the destruction of cyclin. Observations in support of both scenarios are listed in Table 1 and reviewed below.Clearly the nature of calcium's role in early embryonic cycles is under investigation. Here we mathematically study the possibility that an endogenous [Ca 2ϩ ] i oscillator regulates or drives early embryonic cycles. To investigate this proposition, we couple [Ca 2ϩ ] i oscillator models to an established early embryonic cell cycle model based on the protein interactions that govern the activity of MPF. We hypothesize several dynamical states of the MPF system and construct a set of kinetic equations for each hypothesis. We then investigate how each system responds to [Ca 2ϩ ] i variations similar to those observed in Xenopus and sea urchin embryos by numerically solving the kinetic equations.The hypotheses differ in the degree to which [Ca 2ϩ ] i dynamics determine division rates and are listed in Table 2