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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
Article

Abstract

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.

Keywords

Mathematical Biology Generator Matrix Release Site Stochastic Excitability Transition State Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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