Adaptive space and time numerical simulation of reaction-diffusion models for intracellular calcium dynamics

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