Computational simulation of a new system modelling ions electromigration through biological membranes

The interest in cell membrane has grown drastically for their important role as controllers of biological functions in health and illness. In fact most important physiological processes are intimately related to the transport ability of the membrane, such as cell adhesion, cell signaling and immune defense. Furthermore, ion migration is connected with life-threatening pathologies such as metastases and atherosclerosis. Consequently, a large amount of research is consecrated to this topic. To better understand cell membranes, more accurate models of ionic flux are required and also their computational simulations.

This paper is presenting the numerical simulation of a more general system modelling ion migration through biological membranes. The model includes both the effects of biochemical reaction between ions and fixed charges. The model is a nonlinear coupled system. In the first we describe the mathematical model. To realize the numerical simulation of our model, we proceed by a finite element discretisation and then by choosing an appropriate resolution algorithm to the nonlinearities.

We give numerical simulations obtained for different popular models of enzymatic reaction which were compared to those obtained in literature on systems of ordinary differential equations. The results obtained show a complete agreement between the two modellings. Furthermore, various numerical experiments are presented to confirm the accuracy, efficiency and stability of the proposed method. In particular, we show that the scheme is unconditionally stable and second-order accurate in space.