Abstract
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
Highlights
The model considered in this article is based upon the finite channel gating model discussed and simulated in [9]
The deterministic model for the infinite channel has a fundamental drawback, since it allows the channel to remain permanently in a stationary state, corresponding to a fixed point, which is not observed for this class of voltage-gated channels
The critical term which separates this model from the standard Poisson-NernstPlanck systems (PNP) model is the nonlinear representation for the charge density of the protein, expressed as a function of mobile positive ion concentration p and electric field E
Summary
The model considered in this article is based upon the finite channel gating model discussed and simulated in [9]. The infinite channel model was introduced earlier in [8] In both of these articles, rectangular current pulses–the analog of gating–were obtained, via the implementation of a nonlinear charge response. The critical term which separates this model from the standard PNP model is the nonlinear representation for the charge density of the protein, expressed as a function of mobile positive ion concentration p and electric field E. This incorporates the characterization of the protein as undergoing a conformation change, permitting the gating of the channel. Among the open problems discussed at the conclusion of the article is that of extending the analysis to the corresponding states when the electric field decreases
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