Calcium Oscillations and Membrane Transport: The Importance of Two Time Scales

Abstract

It has been observed experimentally that, in many cell types, calcium fluxes across the plasma membrane affect inositol trisphosphate (IP$_3$)-induced calcium oscillations. This is somewhat puzzling; since IP$_3$-induced calcium oscillations involve the cycling of calcium to and from the endoplasmic reticulum it is not well understood how they can be so strongly affected by membrane fluxes. We use a mathematical model to answer this question, a model that relies on the introduction of a slow variable, the Ca$^{2+}$ load of the cell. We carry out a bifurcation analysis of the model in two time scales, treating the slow variable as a bifurcation parameter, thus simulating a closed-cell model, in which there is no Ca$^{2+}$ transport to and from the external medium. The presence of a homoclinic bifurcation in the closed-cell model accounts for the existence of a critical value of the Ca$^{2+}$ load which controls the existence of cytosolic Ca$^{2+}$ oscillations. Our model predictions are confirmed by experimental results. Since similar behavior is observed in two other models of IP$_3$-induced Ca$^{2+}$ oscillations, it is possible that this bifurcation structure is a generic feature of Ca$^{2+}$ oscillation models.