Inverse Problems Related to Ion Channel Selectivity

Ion channels control many biological processes in cells, and, consequently, a large amount of research is devoted to this topic. Great progress in the understanding of channel function has been made recently using advanced mathematical modeling and simulation. This paper investigates another interesting mathematical topic, namely inverse problems, in connection with ion channels. We concentrate on problems that arise when we try to determine ("identify") one of the structural features of a channel--its permanent charge--from measurements of its function, namely current-voltage curves in many solutions. We also try to design channels with desirable properties--for example with particular selectivity properties--using the methods of inverse problems. The use of mathematical methods of identification will help in the design of efficient experiments to determine the properties of ion channels. Closely related mathematical methods will allow the rational design of ion channels useful in many applications, technological and medical. We also discuss certain mathematical issues arising in these inverse problems, such as their ill-posedness and the choice of regularization techniques, as well as challenges in their numerical solution. The L-type Ca channel is studied with the methods of inverse problems to see how mathematics can aid in the analysis of existing ion channels and the design of new ones.