Optimal control of metabolite transport across cell membranes driven by the membrane potential

The movement of the metabolites crossing biological membranes is essentially controlled by the action of the membrane potential. In biophysics, the interest is on the way in which the membrane potential drives the particle motion and makes more probable its translocation, i.e., the transit through the channel and the escape through the opposite side with respect to the entrance. Our investigation will focus on the determination of the membrane potential which allows the particle escape from the channel and may possibly limit the particle lifetime inside it. In mathematical terms, the purpose of this work is to study some mathematical nonlinear control problems, consisting in the determination of an optimal potential which makes the channel more efficient in a certain sense. Some physically significant variations of this basic problem are also investigated and a discussion of the possible extension of the model to time-dependent situations is made. Finally, numerical results are presented.