Mathematics of Ions in Channels and Solutions

An important class of biological molecules—proteins called ionic channels—conduct ions (like Na+ , K+ , Ca2+ , and Cl− ) through a narrow tunnel of fixed charge (‘doping’). Ionic channels control the movement of electric charge and current across biological membranes and so play a role in biology as significant as the role of transistors in computers: a substantial fraction of all drugs used by physicians act on channels. Channels can be studied in the tradition of physical science. Poisson-Drift diffusion equations familiar in physics form an adequate model of current voltage relations in many types of channels under many conditions, and can be extended to describe ‘chemical’ phenomena like selectivity with some success by including the finite size of ions. Ionic channels are studied with the powerful techniques of molecular biology in hundreds of laboratories. Atoms can be modified a few at a time and the location of every atom can be determined. Ionic channels are one of the few biological systems of great importance whose biological function can be well described by a tractable set of equations.

The equations describing ions in channels and solutions are similar, but not identical to well known equations of mathematical physics. Mathematicians can help answer questions of great biological importance by studying and approximating these equations.