We study the problem of diffusive transport of biomolecules in the intercellular space, modeled as porous medium, and of their binding to the receptors located on the surface membranes of the cells. Cells are distributed periodically in a bounded domain. To describe this process we introduce a reaction-diffusion equation coupled with nonlinear ordinary differential equations on the boundary. We prove existence and uniqueness of the solution of this problem. We consider the limit, when the number of cells tends to infinity and at the same time their size tends to zero, while the volume fraction of the cells remains fixed. Using the homogenization technique of two-scale convergence, we show that the sequence of solutions of the original problem converges to the solution of the so-called macroscopic problem. To show the convergence of the nonlinear terms on the surfaces we use the unfolding method (periodic modulation). We discuss applicability of the result to mathematical description of membrane receptors of biological cells and compare the derived model with those previously considered.
Marciniak-Czochra, A., & Ptashnyk, M. (2008). Derivation of a macroscopic receptor-based model using homogenization techniques. SIAM Journal on Mathematical Analysis, 40(1), 215-237. https://doi.org/10.1137/050645269