Nonlinear Conformation Response in the Finite Channel: Existence
of a Unique Solution for the Dynamic PNP Model
By: Joseph W. Jerome
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
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
This article will appear in: Discrete and Continuous Dynamical Systems-B,
vol. 17, no. 7, October 2012 (pp. 2465--2482). doi:
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