Minimal model for signal-induced Ca2+ oscillations and for their frequency encoding through protein phosphorylation

In a variety of cells, hormonal or neurotransmitter signals elicit a train of intracellular Ca2+ spikes. The analysis of a minimal model based on Ca2(+)-induced Ca2+ release from intracellular stores shows how sustained oscillations of cytosolic Ca2+ may develop as a result of a rise in inositol 1,4,5-trisphosphate (InsP3) triggered by external stimulation. This rise elicits the release of a certain amount of Ca2+ from an InsP3-sensitive intracellular store. The subsequent rise in cytosolic Ca2+ in turn triggers the release of Ca2+ from a second store insensitive to InsP3. In contrast to the model proposed by Meyer and Stryer [Meyer, T. & Stryer, L. (1988) Proc. Natl. Acad. Sci. USA 85, 5051-5055], the present model, which contains only two variables, predicts the occurrence of periodic Ca2+ spikes in the absence of InsP3 oscillations. Such results indicate that repetitive Ca2+ spikes evoked by external stimuli do not necessarily require the concomitant, periodic variation of InsP3. The model is closely related to that proposed by Kuba and Takeshita [Kuba, K. & Takeshita, S. (1981) J. Theor. Biol. 93, 1009-1031] for Ca2+ oscillations in sympathetic neurones, based on Ca2(+)-induced Ca2+ release. We extend their results by showing the minimal conditions in which the latter process gives rise to periodic behavior and take into account the role of the rise in InsP3 caused by external stimulation. The analysis further shows how signal-induced Ca2+ oscillations might be effectively encoded in terms of their frequency through the phosphorylation of a cellular substrate by a protein kinase activated by cytosolic Ca2+.