Multiscale Modelling in Biosciences- Ion Transport Through Membranes
Modelling and simulation of processes in biosciences lead in general to complex model systems This complexity is 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 or numerical multi-scale methods. Already in setting up models these techniques are necessary to include information about the real processes on different scales and to link the corresponding models. The effect of microscale processes on the macroscale behaviour has to be analysed.
As an important example we consider the transport of ions through membranes. Mathematical modelling and simulation of ion concentrations inside and outside living cells, in their cyto-plasma and their nucleus, separated by membranes, are decisive for a better understanding of the bio-system cell. Due to that fact that there is more and more information available about the processes on the micro-scale it is necessary to link model equations on micro-scale to a macroscopic description. It is important that model data can be computed using micro-scale information. Here two domains are considered separated by a membrane perforated by channels placed in periodically distributed cells. The thickness of the membrane and the diameter of the cells are of order. The transport of ions is modelled by the Nernst-Planck equations, properly scaled in the channels. Charges fixed to the channels are included modelling the influence of the channels and the changes of its conformation. Effective laws for the ion transport trough membranes are derived performing an asymptotic analysis with respect to the scale parameter. The effective model consists in the Nernst-Planck equations on both sides of the membrane together with appropriate transmission conditions for the ion concentrations and the electric potential across the cell membrane. These conditions are determined solving micro-problems for cell problems in the membrane.
New methods of homogenization have to be developed and applied in order to deal with the nonlinear model equations and the reduction of the membrane to a two dimensional interface.