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We developed a 3D finite element simulator interface for the numerical simulation of stochastic and deterministic equations for single and multiple clusters of Ca^(2+) releasing channels. Mathematically, diffusion, reactions and membrane transport of calcium ions in cells are represented by a coupled system of reaction-diffusion equations. We adopted a hybrid algorithm to address the coupling of the deterministic reaction-diffusion equations for Ca^(2+) and Ca^(2+) buffers and Markovian dynamics of IP_3 R channel gating. Using highly unstructured meshes, our method bridges many orders of magnitude to represent accurately the Ca^(2+)distribution from the single channel to entire cells with multiple clusters of channels. To save computational time, a conforming finite element method is employed for the spatial discretization and adaptive and higher order linearly implicit methods, Rosenbrock type methods, are used for the time integration. This allows an efficient representation of inhomogeneous intra-cluster Ca^(2+) distribution at the nanometer scale even for whole-cell simulations with multiple clusters. Numerical results are demonstrated for different fine spatial resolution meshes as well as different higher order time integrators to insure the numerical convergence of schemes which we apply to study the long time behavior. The parallelization is shown to be essential by the numerical study of long time behavior of calcium concentration. We further present the parallel scalability of the deterministic equations for different arrangements of clusters. The main emphasis is on large scale and long time behavior of the studied equations that capture the detailed local dynamics as well as the temporal hierarchy of dynamical processes. Our approach thus extends our earlier simulations of release from single channels and clusters of channels and systematically integrates stochasticity on all scales of a cell’s calcium dynamics.
Intracellular calcium signals, Gillespie method, finite elements, parallel computations.
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