Nonlinear Conformation Response in the Finite Channel: Existence of a Unique Solution for the Dynamic PNP Model

The standard PNP model for ion transport in channels in cell membranes has been widely studied during the previous two decades; there is a substantial literature for both the dynamic and steady models. What is currently lacking is a generally accepted gating model, which is linked to the observed conformation changes on the protein molecule. In [SIAM J. Appl. Math. 61 (2000), no.3, 792–802], C.W. Gardner, the author, and R.S. Eisenberg suggested a model for the net charge density in the infinite channel, which has connections to stochastic dynamical systems, and which predicted rectangular current pulses. The finite channel was analyzed by these authors in [J. Theoret. Biol. 219 (2002), no. 3, 291– 299]. The finite channel cannot, in general, be analyzed by a traveling wave approach. In this paper, a rigorous study of the initial-boundary value problem is carried out for the deterministic version of the finite channel; an existence/uniqueness result, with a weak maximum principle, is derived on the space-time domain under assumptions on the inital and boundary data which confine the channel to certain states. A significant open problem for the finite channel is the study of phase plane orbits, as exists for the infinite channel. Another open problem is the derivation of comparable existence/uniqueness results, under assumptions on the given data which allow for the complete set of states for the channel. AMS classification numbers: 34B15, 34B18, 35K20, 35M99, 76Z99, 92C05