Bulletin of Mathematical Biology

, Volume 67, Issue 3, pp 393–432 | Cite as

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

  • Vien Nguyen
  • Roy Mathias
  • Gregory D. SmithEmail author


Although there is consensus that localized Ca2+ elevations known as Ca2+ puffs and sparks arise from the cooperative activity of intracellular Ca2+ channels, the precise relationship between single-channel kinetics and the collective phenomena of stochastic Ca2+ excitability is not well understood. Here we present a formalism by which mathematical models for Ca2+-regulated Ca2+ release sites are derived from stochastic models of single-channel gating that include Ca2+ activation, Ca2+ 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 [Ca2+] and thus the state of the other channels. Simulation and analysis of the SAN descriptors representing homogeneous clusters of intracellular Ca2+ channels show that (1) release site density can modify both the steady-state open probability and stochastic excitability of Ca2+ release sites, (2) Ca2+ inactivation is not a requirement for Ca2+ 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 Ca2+ release sites is also discussed.


Mathematical Biology Generator Matrix Release Site Stochastic Excitability Transition State Diagram 
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Copyright information

© Society for Mathematical Biology 2005

Authors and Affiliations

  1. 1.Department of Applied ScienceCollege of William and MaryWilliamsburgUSA
  2. 2.Department of MathematicsCollege of William and MaryWilliamsburgUSA

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