Homogenization of the Cell Cytoplasm: The Calcium Bidomain Equations

All previous models of the dynamics of intracellular calcium concentration have either made the ad hoc assumption that the cytoplasm and the endoplasmic reticulum (ER) coexist at every point in space or have explicitly separated the cytoplasm and the ER into different spatial domains. The former approach is unjustified, and the dependence on the diffusion coefficients on the geometry of the ER is unclear; the latter approach leads to extreme computational difficulties. To avoid the disadvantages of these approaches, we derive a bidomain model of calcium concentration inside the ER network, and outside it, in the cytosol. The homogenized macroscopic behavior is described in a two-concentration field model, a formula is derived for the effective diffusion coefficients of calcium in the ER and in the cytoplasm, and the effective diffusion coefficients are numerically computed for different ER geometries.