Discontinuous Galerkin Methods for problems of mass transfer through semi-permeable membranes

We present an Interior-Penalty Discontinuous Galerkin method for semi-linear systems of pdes on partitioned domains with nonlinear conditions at the interfaces. The problem considered models mass transfer of a number of solutes through a semi-permeable membrane. The presentation will focus on the discretization of the nonlinear Kadem-Katchalsky transmission conditions. We will show that DG methods are particularly suited to mass transfer problems, as the transmission conditions are naturally imposed by choosing the appropriate numerical fluxes. In particular, no iteration between the sub-domains is necessary.
We will present an application to intracellular biochemical signaling. The cell physiology is regulated by highly complex signaling networks. Signaling molecules (e.g. tumor suppressor proteins) flow inside the cell while they undergo a cascade of chemical reactions, and finally cross biological membranes to reach the target cellular compartment. The goal of mathematical modeling and simulation of such complex systems is the discovery of rate limiting steps in the signaling cascade as potential targets of pharmacological therapy.