The dynamics of stochastic attrition viewed as an absorption time on a terminating Markov chain

Localized Ca2+ elevations known as Ca2+ puffs and sparks are cellular signals that arise from the cooperative activity of clusters of inositol 1,4,5-trisphosphate receptors and ryanodine receptors clustered at Ca2+ release sites on the surface of the endoplasmic reticulum or sarcoplasmic reticulum. When Markov chain models of these intracellular Ca2+-regulated Ca2+ channels are coupled via a mathematical representation of Ca2+ microdomain, simulated Ca2+ release sites may exhibit the phenomenon of “stochastic Ca2+ excitability” where the inositol 1,4,5-trisphosphate receptors (IP3Rs) or ryanodine receptors (RyRs) open and close in a concerted fashion. Interestingly, under some conditions simulated puffs and sparks can be observed even when the single-channel model used does not include slow Ca2+ inactivation or, indeed, any long-lived closed/refractory state [V. Nguyen, R. Mathias, G. Smith, Stochastic automata network descriptor for Markov chain models of instantaneously-coupled intracellular Ca2+ channels, Bull. Math. Biol. 67 (2005) 393-432]. In this case, termination of the localized Ca2+ elevation occurs when all of the intracellular channels at a release site simultaneously close through a process referred to as stochastic attrition [M. Stern, Theory of excitation-contraction coupling in cardiac muscle, Biophys. J. 63 (1992) 497-517]. In this paper, we investigate the statistical properties of stochastic attrition viewed as an absorption time on a terminating Markov chain that represents a Ca2+ release site composed of N two-state channels that are activated by Ca2+. Assuming that the local [Ca2+] experienced by a channel depends only on the number of open channels at the Ca2+ release site (i.e., instantaneous mean-field coupling [ibid.], we derive the probability distribution function for the time until stochastic attrition occurs and present an analytical formula for the expectation of this random variable. We explore how the contribution of stochastic attrition to the termination of Ca2+ puffs and sparks depends on the number of channels at a release site, the source amplitude of the channels (i.e., the strength of the coupling), the background [Ca2+], channel kinetics, and the cooperactivity of Ca2+ binding. Because we explicitly model the Ca2+ regulation of the intracellular channels, our results differ markedly from (and in fact generalize) preliminary analyses that assume the intracellular Ca2+ channels are uncoupled and consequently independent.