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Relying on quantitative measurements of Ca2+ activation and inhibition of the inositol 1,4,5-trisphosphate (1P3) receptor in the endoplasmic reticulum, we construct a simplified kinetic model to describe the properties of this channel. Selecting rate constants to fit key kinetic and equilibrium data, we find that the model reproduces a variety of in vivo and in vitro experiments. In combination with Ca2+-ATPase activity for Ca2+ uptake into the endoplasmic reticulum, the model leads to cytoplasmic oscillations in Ca2+ concentration at fixed IP3 concentration and only a single pool of releasable Ca2 , the endoplasmic reticulum. Incorporation of a positive-feedback mechanism of Ca2+ on IP3 production by phospholipase C enriches the properties of the oscillations and leads to oscillations in Ca2+ concentration accompanied by oscillations in I1P3 concentration. We discuss the possible significance of these results for the interpretation of experiments. We construct a simplified model of the IP3 receptor/ channel by assuming that three equivalent and independent subunits are involved in conduction. While it is probable that the subunits are in fact not independent of one another, improvements in this aspect of the model need to await more detailed kinetic measurements. We further assume that each subunit has one IP3 binding site and two Ca2+ binding sites, one for activation, the other for inhibition. Thus, each subunit may exist in eight states with transitions governed by second-order (ai) and first-order (b,) rate constants (Fig. 1) 9895The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

Based on realistic mechanisms of Ca2+ buffering that include both stationary and mobile buffers, we derive and investigate models of Ca2+ diffusion in the presence of rapid buffers. We obtain a single transport equation for Ca2+ that contains the effects caused by both stationary and mobile buffers. For stationary buffers alone, we obtain an expression for the effective diffusion constant of Ca2+ that depends on local Ca2+ concentrations. Mobile buffers, such as fura-2, BAPTA, or small endogenous proteins, give rise to a transport equation that is no longer strictly diffusive. Calculations are presented to show that these effects can modify greatly the manner and rate at which Ca2+ diffuses in cells, and we compare these results with recent measurements by Allbritton et al. (1992). As a prelude to work on Ca2+ waves, we use a simplified version of our model of the activation and inhibition of the IP3 receptor Ca2+ channel in the ER membrane to illustrate the way in which Ca2+ buffering can affect both the amplitude and existence of Ca2+ oscillations.

Pancreatic beta-cells in an intact Islet of Langerhans exhibit bursting electrical behavior. The Chay-Keizer model describes this using a calcium-activated potassium (K-Ca) channel, but cannot account for the irregular spiking of isolated beta-cells. Atwater I., L. Rosario, and E. Rojas, Cell Calcium. 4:451-461, proposed that the K-Ca channels, which are rarely open, are shared by several cells. This suggests that the chaotic behavior of isolated cells is stochastic. We have revised the Chay-Keizer model to incorporate voltage clamp data of Rorsman and Trube and extended it to include stochastic K-Ca channels. This model can describe the behavior of single cells, as well as that of clusters of cells tightly coupled by gap junctions. As the size of the clusters is increased, the electrical activity shows a transition from chaotic spiking to regular bursting. Although the model of coupling is over-simplified, the simulations lend support to the hypothesis that bursting is the result of channel sharing.

Following the experimental findings of Atwater et al. (In Biochemistry Biophysics of the Pancreatic-beta-Cell, George Thieme Verlag, New York, 100-107), we have formulated a mathematical model for ionic and electrical events that take place in pancreatic-beta-cells. Our formulation incorporates a Hodgkin-Huxley type gating mechanism for Ca2+ and K+ channels, in addition to Ca2+ gated K+-channels. Consistent with the experimental observations, our model generates spikes and bursts in beta-cell membrane potentials and gives the correct responses to additions of glucose, quinine, and tetraethylammonium ions. The response of the oscillations to ouabain and changing concentrations of external K+ can be incorporated into the present model, although a more complete treatment would require inclusion of the Na+/K+ pump.