Willi Jäger: Multiscale modelling and simulation of processes in membranes and tissues – applications in life sciences

Modelling and simulation of processes in Life Sciences lead in general to complex model systems, caused by the system size, the arising nonlinearities, the range of scales involved, the stochastic nature of the processes and of the underlying geometry. It is a main challenge for Mathematics to reduce the complexity e.g. by analytic and numerical multiscale methods.

This lecture is concerned with processes in membranes and tissue, in particular with reactive flow, diffusion and transport through membranes, cell layers and tissue. Using multi-scale homogenization methods it is possible to derive reduced systems, e.g. replacing a membrane by an interface and requiring transmission conditions at this interface which take into account the processes in the membrane, A survey is given on the results, which could be achieved so far, assuming mesoscopic models for the processes in the membrane.

However the processes have to be modelled quite often on the molecular scale. Here we consider a population of ions penetrating a membrane through channels. A stochastic model for the dynamics, describing main features and characteristics of the interactions is developed and simulated. Both cases, nano- and micro-channels are considered. Nano-channels and ions are of the same size. Different nondimensionalization procedures, supported by numerical simulation, are discussed. This investigation has also to be considered as preparatory study for deriving an analytic scaling limit.