Discrete-State Stochastic Models of Calcium-Regulated Calcium Influx and Subspace Dynamics Are Not Well-Approximated by ODEs That Neglect Concentration Fluctuations

Abstract

Cardiac myocyte calcium signaling is often modeled using deterministic ordinary differential equations (ODEs) and mass-action kinetics. However, spatially restricted “domains” associated with calcium influx are small enough (e.g., 10−17 liters) that local signaling may involve 1–100 calcium ions. Is it appropriate to model the dynamics of subspace calcium using deterministic ODEs or, alternatively, do we require stochastic descriptions that account for the fundamentally discrete nature of these local calcium signals? To address this question, we constructed a minimal Markov model of a calcium-regulated calcium channel and associated subspace. We compared the expected value of fluctuating subspace calcium concentration (a result that accounts for the small subspace volume) with the corresponding deterministic model (an approximation that assumes large system size). When subspace calcium did not regulate calcium influx, the deterministic and stochastic descriptions agreed. However, when calcium binding altered channel activity in the model, the continuous deterministic description often deviated significantly from the discrete stochastic model, unless the subspace volume is unrealistically large and/or the kinetics of the calcium binding are sufficiently fast. This principle was also demonstrated using a physiologically realistic model of calmodulin regulation of L-type calcium channels introduced by Yue and coworkers.

Exact Stochastic Simulation of a Calcium Microdomain Reveals the Impact of Ca2+ Fluctuations on IP3R Gating

“…Our motivation to study the influence of Ca 2þ noise on the IP 3 R arose from the contemporary interest in the stochastic nature of complex Ca 2þ signals (11,20,26,41). Just recently, Thurley et al (13) showed that stimulus intensities of input signals are reliably encoded in stochastic sequences of random spikes, and the body of literature, emphasizing the functional relevance of noise in biological systems, is constantly growing (24,(48)(49)(50)(51)(52).…”

“…The influence of Ca 2þ buffers on Ca 2þ signaling has previously been studied, both theoretically (54,(60)(61)(62) and experimentally (63)(64)(65)(66). These studies mainly focused on the influence of mobile and immobile buffers on global Ca 2þ signals, whereas only a very few studies are concerned with the functional role of buffer-dependent Ca 2þ fluctuations (26). The expression profiles for Ca 2þ buffers with widely differing kinetic properties are highly variable among different cell types (35,46).…”

“…Morphologically predetermined structures, such as neuronal dendritic spines or the dyadic cleft of cardiac myocytes, are prominent examples for their vital roles in learning, memory (81), and heartbeat generation (82). Even though some recent work about stochastic fluctuations in the dyadic cleft exists (26), the question how such fluctuations influence the Ca 2þ signaling apparatus as a whole, remains unanswered.…”

“…Hybrid algorithms are able to cover the hierarchical structure of calcium signals, spanning several orders of magnitude in space and time by treating functionally important nonlinear components such as the IP 3 R stochastically, whereas passive bulk reactions and diffusion are treated deterministically (11,(18)(19)(20). Because noise-induced effects on nonlinear systems have received special attention over the last few years (21)(22)(23)(24)(25)(26), we chose a fully stochastic framework to investigate the effects of Ca 2þ noise, arising from buffer association/dissociation reactions and diffusion, on the nonlinear behavior of IP 3 Rs. Our model represents a Ca 2þ microdomain with a single nonconducting IP 3 R, integrating channel gating and Ca 2þ dynamics in a common stochastic simulation framework based on Gillespie's algorithm.…”

Ca2+-activation kinetics modulate successive puff/spark amplitude, duration and inter-event-interval correlations in a Langevin model of stochastic Ca2+release

“…One important observation is that correlations between the amplitude, duration, and interval-event intervals of simulated Ca 2+ puffs and sparks are strongly influenced by spark termination mechanism (i.e., Ca 2+ -dependent inactivation or stochastic attrition-like). Our formulation can be generalized to account for luminal depletion and/or regulation, both of which known to influence spark termination (Huertas and Smith, 2007), and finite system size effects (Weinberg and Smith, 2012).…”