Abstract
Localized Ca$^2+$ elevations known as Ca$^2+$ 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 Ca$^2+$ release sites on the surface of
the endoplasmic reticulum or sarcoplasmic reticulum. When Markov
chain models of these intracellular Ca$^2+$-regulated Ca$^2+$
channels are coupled via a mathematical representation of Ca$^2+$
microdomain, simulated Ca$^2+$ release sites may exhibit the
phenomenon of "stochastic Ca$^2+$ excitability" where the inositol
1,4,5-trisphosphate receptors (IP(3)Rs) 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 Ca$^2+$ 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 Ca$^2+$
channels, Bull. Math. Biol. 67 (2005) 393-432. In this case, termination
of the localized Ca$^2+$ 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 Ca$^2+$ release site composed of N two-state channels that
are activated by Ca$^2+$. Assuming that the local Ca$^2+$
experienced by a channel depends only on the number of open channels
at the Ca$^2+$ 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 Ca$^2+$
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 Ca$^2+$, channel kinetics, and the cooperactivity
of Ca$^2+$ binding. Because we explicitly model the Ca$^2+$
regulation of the intracellular channels, our results differ markedly
from (and in fact generalize) preliminary analyses that assume the
intracellular Ca$^2+$ channels are uncoupled and consequently
independent.
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