Ion Flow through Narrow Membrane Channels: Part II

This paper is devoted to the study of the flow of ions through protein channels in physiological membranes. More specifically, it is concerned with the role of the electrical properties of the channel in determining that flow. For the case of long channels, it is shown that, when the channels have a small permittivity (compared to that of the aqueous solution), the potential down the channel is markedly altered. In particular, this potential $\Phi $ does not satisfy a one-dimensional Poisson-Boltzmann, but rather is a solution of a new equation, namely,\[ \begin{gathered} \frac{{d^2 \Phi }}{{dz^2 }} + \frac{{2\epsilon }}{{\alpha ^2 In\alpha }}(\Phi - (1 - z)\Delta ) \hfill \\ \qquad = - \lambda ^2 \sum\limits_i {Z_i \frac{{l_{ic_L } e^{Z_i \Delta } \int_z^1 {e^{Z_i \Phi } d\zeta + r_i c_R \int_0^z {e^{Z_i \Phi } d\zeta } } }} {{e^{Z_i \Phi } \int_0^1 {e^{Z_i \Phi } d\zeta } }}} , \hfill \\ \end{gathered} \] where $\alpha $ is the small aspect ratio and $\epsilon $ is the ratio of the permittivity of the channel protein to that of the aqueous solution. The meaning of the other variables are given in the text.