We consider porous media consisting of different components separated by interfaces. Inside of each component diffusion and nonlinear reaction processes take place. The exchange of substances at interfaces is controlled by the concentrations on both sides of the interface via nonlinear transmission conditions. At interfaces also nonlinear reaction and surface diffusion can occur. We start from microscopic descriptions, where the geometry of medium, the processes inside the components, and the transmission conditions at the interfaces are are resolved. Then, under suitable conditions on the distribution of the components inside the medium (e.g. periodic, locally periodic or stochastic distribution), we derive effective approximations by using techniques of multiscale analysis and homogenization. Hereby, existing techniques have to be further developed to cope with the special feature of our models, e.g. nonlinear transmission conditions on microscopic interfaces.
Our models are applied to the mathematical modeling of metabolic and regulatory processes in living cells, where biochemical species are exchanged between organelles (like mitochondria or plastids) and cytoplasm through organellar membranes, or are attached to membranes, where they undergo enzymatic reactions. In this context, the nonlinearities are given by kinetics corresponding to multi-species enzyme catalyzed reactions, which are generalizations of the classical Michaelis-Menten kinetics for multi-species reactions.
Knowledge concerning metabolic reaction networks and spatial enzyme organization, as well as experimental data are provided by our collaboration partners Uwe Sonnewald and Lars Voll (Biochemistry Department, University Erlangen-Nuremberg). Based on these experimental information, the effective models are simulated numerically. A main aim for our investigations is the identification of the impact of metabolic channeling on the carbon partitioning between starch, sucrose and respiration in Arabidopsis leaves.