Markov chain models of coupled calcium channels: Kronecker representations and iterative solution methods.

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). Calcium release site models are stochastic automata networks that involve many functional transitions, that is, the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. We present a Kronecker-structured representation for calcium release site models and perform benchmark stationary distribution calculations using both exact and approximate iterative numerical solution techniques that leverage this structure. When it is possible to obtain an exact solution, response measures such as the number of channels in a particular state converge more quickly using the iterative numerical methods than occupation measures calculated via Monte Carlo simulation. In particular, multi-level methods provide excellent convergence with modest additional memory requirements for the Kronecker representation of calcium release site models. When an exact solution is not feasible, iterative approximate methods based on the power method may be used, with performance similar to Monte Carlo estimates. This suggests approximate methods with multi-level iterative engines as a promising avenue of future research for large-scale calcium release site models.

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

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.

Calcium release site ultrastructure and the dynamics of puffs and sparks.

When Markov chain models of intracellular Ca(2+)-regulated Ca(2+) channels are coupled via a mathematical representation of a Ca(2+) microdomain, simulated Ca(2+) release sites may exhibit the phenomenon of 'stochastic Ca(2+) excitability' reminiscent of Ca(2+) puffs and sparks. Interestingly, some single-channel models that include Ca(2+) inactivation are not particularly sensitive to channel density, so long as the requirement for inter-channel communication is satisfied, while other single-channel models that do not include Ca(2+) inactivation open and close synchronously only when the channel density is in a prescribed range. This observation led us to hypothesize that single-channel models with Ca(2+) inactivation would be less sensitive to the details of release site ultrastructure than models that lack a slow Ca(2+) inactivation process. To determine if this was the case, we simulated Ca(2+) release sites composed of instantaneously coupled Ca(2+)-regulated Ca(2+) channels whose random spatial locations were chosen from a uniform distribution on a disc of specified radius and compared the resulting release site dynamics to simulations with channels arranged on hexagonal lattices. Analysis of puff/spark statistics confirmed our hypothesis that puffs and sparks are less sensitive to the spatial organization of release sites when the single-channel model includes a slow inactivation process. We also investigated the validity of several different mean-field reductions that do not explicitly account for the details of release site ultrastructure. The most successful approximation maintains a distinction between each channel's substantial influence on its own stochastic gating and the collective contribution of elevated [Ca(2+)] from neighbouring channels.

Markov chain models of coupled intracellular calcium channels: Kronecker structured representations and benchmark stationary distribution calculations.

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). We present a Kronecker structured representation for calcium release site models and perform benchmark stationary distribution calculations using numerical iterative solution techniques that leverage this structure. In this context we find multi-level methods and certain preconditioned projection methods superior to simple Gauss-Seidel type iterations. Response measures such as the number of channels in a particular state converge more quickly using these numerical iterative methods than occupation measures calculated via Monte Carlo simulation.