# American Institute of Mathematical Sciences

October  2012, 17(7): 2465-2482. doi: 10.3934/dcdsb.2012.17.2465

## Nonlinear conformation response in the finite channel: Existence of a unique solution for the dynamic PNP model

 1 Department of Mathematics, Northwestern University, Evanston, IL 60208-2730, United States

Received  September 2011 Revised  February 2012 Published  July 2012

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. Eisen- berg suggested a model for the net charge density in the infinite channel, which has connections to stochastic dynamical systems, and which predicted rectan- gular 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. Significant open problems remain and are discussed
Citation: Joseph W. Jerome. Nonlinear conformation response in the finite channel: Existence of a unique solution for the dynamic PNP model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2465-2482. doi: 10.3934/dcdsb.2012.17.2465
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