Adaptive numerical simulation of intracellular calcium dynamics using domain decomposition methods

doi:10.1016/j.apnum.2007.10.003

Applied Numerical Mathematics

Volume 58, Issue 11, November 2008, Pages 1658-1674

https://doi.org/10.1016/j.apnum.2007.10.003 Get rights and content

Abstract

In this paper we present adaptive numerical simulations of intracellular calcium dynamics using domain decomposition methods. The modeling of diffusion, binding and membrane transport of calcium ions in cells leads to a system of reaction–diffusion equations. We describe the modulation of cytosolic and ER calcium concentration close to the membrane of the cell organelle. This leads to a two-dimensional model. Local grid refinement is adjusted to the strongly local effects of the calcium dynamics. A finite element method is used for spatial discretization and a Rosenbrock solver for time integration which is third-order accurate and very suitable for nonlinear parabolic problems. We present sequential and parallelized numerical results. The latter are based on domain decomposition methods.

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Stochastic models of intracellular calcium signals
2014, Physics Reports
Cellular signaling operates in a noisy environment shaped by low molecular concentrations and cellular heterogeneity. For calcium release through intracellular channels–one of the most important cellular signaling mechanisms–feedback by liberated calcium endows fluctuations with critical functions in signal generation and formation. In this review it is first described, under which general conditions the environment makes stochasticity relevant, and which conditions allow approximating or deterministic equations. This analysis provides a framework, in which one can deduce an efficient hybrid description combining stochastic and deterministic evolution laws. Within the hybrid approach, Markov chains model gating of channels, while the concentrations of calcium and calcium binding molecules (buffers) are described by reaction–diffusion equations. The article further focuses on the spatial representation of subcellular calcium domains related to intracellular calcium channels. It presents analysis for single channels and clusters of channels and reviews the effects of buffers on the calcium release. For clustered channels, we discuss the application and validity of coarse-graining as well as approaches based on continuous gating variables (Fokker–Planck and chemical Langevin equations). Comparison with recent experiments substantiates the stochastic and spatial approach, identifies minimal requirements for a realistic modeling, and facilitates an understanding of collective channel behavior. At the end of the review, implications of stochastic and local modeling for the generation and properties of cell-wide release and the integration of calcium dynamics into cellular signaling models are discussed.
Adaptive space and time numerical simulation of reaction-diffusion models for intracellular calcium dynamics
2012, Applied Mathematics and Computation
Citation Excerpt :
A both space and time adaptive strategy will further improve the simulation efficiency. The space and time adaptivity for the deterministic simulation of calcium dynamics is well presented in Nagaiah et al. [24] where the coupling of stochastic channel transition was not considered. The spatial grid adaptivity plays an important role in the hybrid numerical simulations and leads to a good improvement in the CPU time.
Adaptivity in space and time for the numerical simulation of stochastic and deterministic equations for intracellular calcium dynamics is presented. The modeling of diffusion, reaction and membrane transport of calcium ions in cells leads to a system of reaction–diffusion equations. We describe the modulation of cytosolic and ER calcium concentrations close to the membrane of the cell organelle.
A conforming piecewise linear finite element method is used for the spatial discretization. Linearly implicit methods of Rosenbrock type are used for the time integration. We adopt a hybrid algorithm to solve the stochastic part. The space grid is adjusted to the strong localization of the calcium release following stochastic channel transitions. By automatically adapting the spatial meshes and time steps to the proper scales during the transition of channel states, the method accurately resolves the evolution of intracellular calcium concentrations as well as buffer concentrations. This article emphasizes adaptive and efficient hybrid numerical simulations in two space dimensions. The presented work establishes the basis for future simulations in a realistic 3D geometry.
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Our numerical method consists of coupled solvers for the deterministic set of partial differential equations and the stochastic equations. In view of the multiple space- and timescales, we employ a conforming finite element method for the spatial discretization and an adaptive linear implicit time-stepping for the deterministic part (26). The stochastic channel gating is incorporated by using the recently devised hybrid method (27).
We study Ca²⁺ release through single and clustered IP₃ receptor channels on the ER membrane under presence of buffer proteins. Our computational scheme couples reaction-diffusion equations and a Markovian channel model and allows our investigating the effects of buffer proteins on local calcium concentrations and channel gating. We find transient and stationary elevations of calcium concentrations around active channels and show how they determine release amplitude. Transient calcium domains occur after closing of isolated channels and constitute an important part of the channel's feedback. They cause repeated openings (bursts) and mediate increased release due to Ca²⁺ buffering by immobile proteins. Stationary domains occur during prolonged activity of clustered channels, where the spatial proximity of IP₃Rs produces a distinct [Ca²⁺] scale (0.5–10 μM), which is smaller than channel pore concentrations (>100 μM) but larger than transient levels. While immobile buffer affects transient levels only, mobile buffers in general reduce both transient and stationary domains, giving rise to Ca²⁺ evacuation and biphasic modulation of release amplitude. Our findings explain recent experiments in oocytes and provide a general framework for the understanding of calcium signals.
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Adaptive numerical simulation of intracellular calcium dynamics using domain decomposition methods

Abstract

Sobolev Spaces

Discrete stochastic modeling of calcium channel dynamics

Phys. Rev. Lett.

UG: A flexible software toolbox for solving partial differential equations

Computing and Visualization in Science

Elementary and global aspects of calcium signalling

J. Physiol.

Calcium oscillations

J. Biol. Chem.

Inositol trisphosphate and calcium signalling

Nature

Calcium—a life and death signal

Nature

The versatility and universality of calcium signalling

Nature Rev. Mol. Cell Biol.

Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics

Increased frequency of calcium waves in Xenopus laevis oocytes that express a calcium-atpase

Science

Hierarchical organization of calcium signals in hepatocytes: from experiments to models

Biochim. Biophys. Acta

Buffers and oscillations in intracellular Ca2+ dynamics

Biophys. J.

On the role of stochastic channel behavior in intracellular Ca2+ dynamics

Biophys. J.

Reading the patterns in living cells—the physics of Ca2+ signaling

Adv. Phys.

Stochastic spreading of intracellular Ca2+ release

Phys. Rev. E

A pi stepsize control for the numerical solution of ordinary differential equations

BIT

Solving Ordinary Differential Equations II

Buffers and oscillations in intracellular Ca²⁺ dynamics

On the role of stochastic channel behavior in intracellular Ca²⁺ dynamics

Reading the patterns in living cells—the physics of Ca²⁺ signaling

Stochastic spreading of intracellular Ca²⁺ release