A stochastic automata network descriptor for Markov chain models of instantaneously coupled intracellular Ca2+ channels

Abstract

Although there is consensus that localized Ca(2+) elevations known as Ca(2+) puffs and sparks arise from the cooperative activity of intracellular Ca(2+) channels, the precise relationship between single-channel kinetics and the collective phenomena of stochastic Ca(2+) excitability is not well understood. Here we present a formalism by which mathematical models for Ca(2+)-regulated Ca(2+) release sites are derived from stochastic models of single-channel gating that include Ca(2+) activation, Ca(2+) inactivation, or both. Such models are stochastic automata networks (SANs) that involve a large number of functional transitions, that is, the transition probabilities of the infinitesimal generator matrix of one of the automata (i.e., an individual channel) may depend on the local [Ca(2+)] and thus the state of the other channels. Simulation and analysis of the SAN descriptors representing homogeneous clusters of intracellular Ca(2+) channels show that (1) release site density can modify both the steady-state open probability and stochastic excitability of Ca(2+) release sites, (2) Ca(2+) inactivation is not a requirement for Ca(2+) puffs or sparks, and (3) a single-channel model with a bell-shaped open probability curve does not lead to release site activity that is a biphasic function of release site density. These findings are obtained using iterative, memory-efficient methods (novel in this biophysical context and distinct from Monte Carlo simulation) that leverage the highly structured SAN descriptor to unambiguously calculate the steady-state probability of each release site configuration and puff statistics such as puff duration and inter-puff interval. The validity of a mean field approximation that neglects the spatial organization of Ca(2+) release sites is also discussed.