Surface reactions on the cytoplasmatic membranes—Mathematical modeling of reaction and diffusion systems in a cell

A human cell consists schematically of an outer cellular membrane, a cytoplasm containing a large number of organelles (mitochondria, endoplasmatic reticulum etc.), a nuclear membrane and finally the cellular nucleus containing DNA. The organelles create a complex and dense system of membranes or sub-domains throughout the cytoplasm. The mathematical description leads to a system of reaction–diffusion equations in a complex geometrical domain, dominated by thin membranous structures with similar physical and chemical properties. In a previous model, we considered only spatially distributed reaction and diffusion processes. However, from experiments it is known that membrane bound proteins play an important role in the metabolism of certain substances. In the present paper we develop a homogenization strategy which includes both volume and surface reactions. The homogenized system is a reaction–diffusion system in the cytoplasm which is coupled to the surrounding cell components by correspondingly modified transfer conditions. The approach is verified by application to a system modeling the cellular uptake and intracellular dynamics of carcinogenic polycyclic aromatic hydrocarbons.