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

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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.

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  1. Department of Applied Science, College of William and Mary, McGlothlin-Street Hall, Rm 305, Williamsburg, VA, 23187, USA

    Vien Nguyen & Gregory D. Smith

  2. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187, USA

    Roy Mathias

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  1. Vien Nguyen
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Nguyen, V., Mathias, R. & Smith, G.D. A stochastic automata network descriptor for Markov chain models of instantaneously coupled intracellular Ca2+ channels. Bull. Math. Biol. 67, 393–432 (2005). https://doi.org/10.1016/j.bulm.2004.08.010

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