Homogenization of fast diffusion on surfaces with a two-step method and an application to T-cell signaling

In the context of periodic homogenization based on two-scale convergence, a nonlinear system of six coupled partial differential equations is homogenized. The system describes the process of signaling in a T cell (thymus lymphocyte) including the dynamics of calcium and of the molecule Stim1. Two of the six equations are defined on the finely structured surface of the endoplasmic reticulum and to make global diffusion after homogenization possible, we extend the existing theoretical convergence results and introduce the two-step method. Therefore the membrane of the endoplasmic reticulum is given an extent in normal direction such that it has a volume with width 0<δ≪1. For convergence of the functions defined on the membrane we can now use well-known two-scale convergence results and obtain fast diffusion after homogenization. To come back to the original shape of the surface, δ tends to zero in the reference cell, if some compactness results are satisfied, which leads to a non-standard cell problem, and we obtain global diffusion on the surface of the endoplasmic reticulum. The results justify a model for signaling in a T-cell recently proposed heuristically.