A theoretical examination has been carried out of the steady-state properties of a homogeneous, ideally permselective, liquid ion-exchange membrane in which sites and counterions are incompletely dissociated. Dissociated species are assumed to be in chemical equilibrium with neutral ion pairs at every point in the membrane, their concentrations being interconnected by the law of mass action. The flux equations, which describe the complete behavior of the system, are derived by considering the free ions and their combined forms as separately flowing entities, and the boundary conditions are obtained by assuming the sites to be completely trapped in the membrane phase (although free to move within it) while the counterions are free to undergo ion exchange. In the present paper, under the restriction of zero membrane current, a general expression for the membrane potential is deduced in terms of external solution conditions and membrane parameters (e.g., mobilities, dissociation constants). From this expression, the factors governing the electrode properties of liquid ion-exchange membranes are discussed not only for the steady state but for certain transient situations as well. It is concluded that the steady-state expressions derived for convection-free systems describe the situations usually encountered with membrane electrodes made from liquid ion exchangers where the instantaneous values of successive potentials are measured with electrode systems which are not in the steady state and even when no precautions are taken to avoid convective mixing within the electrode. The parameters controlling electrode specificity are also discussed.
As a prototype for binding and interaction in biological Na and K channels, the single channel conductances for Li, Na, K, Rb, Cs, H, and Tl and the membrane potentials for Tl-K mixtures are characterized for gramicidin A over wider concentration rangers than previously and analyzed using an "equilibrium domain" model that assumes a central rate-determining barrier. Peculiarities in the conductance-concentration relationship for TlF, TlNO3, and TlAc suggest that anions bind to Tl-loaded channels, and the theory is extended to allow for this. For concreteness, the selectivity of cation permeation is characterized in terms of individual binding and rate constants of this model, with the conclusions that the strongest site binds Cs greater than Rb greater than K greater than Na greater than Li, while the next strongest binds Na greater than K greater than Li greater than Rb greater than Cs. However, because Schagina, Grinfeldt, and Lev's recent finding of single filing (personal communication) indicates that the channel sites in gramicidin cannot be at equilibrium with the solution, and work in progress with Hägglund and Enos (Biophys. J. 21:26a. [Abstr.]) indicates that the simplest model adequate to account for the observed concentration-dependences of flux-ratio, conductance, I--V characteristic, and permeability has three barriers and four sites, some implications of additional rate-determining barriers at the mouth of the channel are discussed. The results are summarized using phenomenological "experimental" parameters that provide a model-independent way to represent that data concisely and which can be interpreted physically in terms of any desired model.
A theory, recently developed by Sandblom, Eisenman and Neher (1977) for the conductance of single gramicidin A cha-nelspred icts three limiting behaviors of the relation between conductance and salt concentration. These are: (i) a saturating behavior resembling a simple adsorption isotherm at medium and high concentrations, (ii) a decrease in conductance at the highest obtainable concentrations and (iii) deviations from the isotherm at very low concentrations. Features i and ii have been described before. Experimental evidence for point iii is given here. The new feature points towards interactions among ions in the channel at ionic concentrations as low as 1--10 mM. Particular emphasis is given to the behavior at very low salt concentrations and the experimental problems encountered in this situation. In addition, mutual blocking effects among monovalent ions in symmetrical salt mixtures are characterized and found to be in satisfactory agreement with theoretical expectations, based upon the single salt conductance data presented here and zero-current potentials in salt mixtures to be described in a subsequent paper.
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