Hybrid stochastic and deterministic simulations of calcium blips

Abstract

Intracellular calcium release is a prime example for the role of stochastic effects in cellular systems. Recent models consist of deterministic reaction-diffusion equations coupled to stochastic transitions of calcium channels. The resulting dynamics is of multiple time and spatial scales, which complicates far-reaching computer simulations. In this article, we introduce a novel hybrid scheme that is especially tailored to accurately trace events with essential stochastic variations, while deterministic concentration variables are efficiently and accurately traced at the same time. We use finite elements to efficiently resolve the extreme spatial gradients of concentration variables close to a channel. We describe the algorithmic approach and we demonstrate its efficiency compared to conventional methods. Our single-channel model matches experimental data and results in intriguing dynamics if calcium is used as charge carrier. Random openings of the channel accumulate in bursts of calcium blips that may be central for the understanding of cellular calcium dynamics.

FIGURE 1

The stochastic model of channel gating is given by 26 possible transitions of each of the four subunits. The values p and c denote the concentrations of IP₃ and Ca²⁺, respectively.

FIGURE 2

The stationary Ca²⁺ concentration for an open channel directly at the ER membrane (solid) and perpendicular to the membrane (dashed) as calculated with the finite element discretization.

FIGURE 3

Open fraction (a), mean open (b), and mean close time (c) versus cytosolic [Ca²⁺] for [IP₃] = 10 μM. (Connected large dots) Hybrid simulations for runs of 100 s. (Small dots) Data from experimental work of Mak et al. (38). (Blue squares) Two-state Markovian method for runs of 10⁵ s.

FIGURE 4

The evolution of the number of subunits in state X_ACT for a simulation run with [IP₃] = 0.1 μM. The channel is open if X_ACT = 3 or 4. The resting concentration of Ca²⁺ was 0.05 μM.

FIGURE 5

The Ca²⁺ concentration at the channel mouth and the number of subunits in the active state, X_ACT, during the second burst shown in Fig. 4. After the last closing of the burst, residual Ca²⁺ decays in ∼100 ms.

FIGURE 6

Open fraction (a), mean open duration (b), and mean close duration (c) versus IP₃ concentration using calcium carrier. (Circles) Hybrid simulations (simulation time: 200 s); (squares) simulation with two-state Markovian method (150 s).