As an important example of spatial and stochastic effects in cell signaling, we study the role of fluctuations during the nucleation and propagation of waves of cytosolic calcium. Briefly, calcium is an important second messenger in cellular communications. The liberation and uptake by cellular stores such as the endoplasmic reticulum (ER) and reactions with buffer proteins determine its intracellular dynamics. This system is a prime example for the importance of spatial and fluctuation effects by the following reasons. The release of calcium is mediated by ion channels, which are relatively scarce. Moreover, the channels frequently cluster. Thus the homogeneity assumption does not apply and, consequently, the release of calcium cannot be spatially averaged to yield ODE’s. Further, the number of channels in a cluster on the ER is so small (~20) that continuous Langevin equations cannot be used.

Instead our model for oscillations is based on the random transitions of a single channel in a cluster. The opening of a channel by binding of calcium increases the local calcium concentration by release through the channel. It thus causes further receptors to bind calcium and open. Since calcium also diffuses to adjacent clusters there is a small but finite probability for their channels to open. In this way the calcium signal spreads through the cell. Slow inhibition, diffusion into the interior of the cell and resequestration into the ER characterize the refractory phase of an oscillation cycle. A substantial part of the period of calcium oscillations is determined by the probability that one channel opens randomly and initiates the global release event.

(2) Hybrid code                                                                                                    

We use numerical simulation to study the model of calcium release sketched above. We incorporate the discreteness of calcium channels and the randomness in molecular processes by simulating a set of reaction-diffusion equations with localized source terms and a coupling to Markovian transitions of molecular states of membrane channels. We have set up a simulation package for a hybrid stochastic/deterministic model in two and three spatial dimensions. It utilizes finite elements for the deterministic part and an adapted Gillespie algorithm for the stochastic transitions. The dynamics of IP₃-controlled channels remains discrete and stochastic and is implemented in the numerical simulations by a variant of the Gillespie method. The strongly localized temporal behavior due to the on-off characteristics of channels as well as their spatial localization due to the weak diffusion is treated by an adaptive numerical method using finite elements.