Translocation energy of ions in nano-channels of cell membranes

Abstract

Translocation properties of ionic channels are investigated, on the basis of classical electrostatics, with an emphasis on asymptotic formulas for the potential and field associated with a point charge in the channel. Due to image charges in the membrane, we show that ions in an infinite length channel interact via a one-dimensional (1D) Coulomb potential. The corresponding electrostatic barrier $\Sigma$ is characterized by a "geometric mean" screening $\Sigma \propto e^2 / \sqrt{\epsilon_w \epsilon_m}R$ ($R$ being the radius of the pore, and $\epsilon_m \approx 2$ and $\epsilon_w \approx 80$ the room temperature dielectric constants of membrane and water, respectively). There exists a crossover length, $x_0 \propto R \sqrt{\epsilon_w / \epsilon_m} \sim 6.3 R$, below which the 1D potential governs the electrostatics and beyond which the three-dimensional (3D) Coulomb potential screened by the membrane takes over. Knowledge of this length enables us to discriminate between long channels, the length $L$ of which satisfies: $L \gg 2 x_0$, and short channels for which $L \ll 2 x_0$. The latter condition is satisfied by most realistic channels ({\it e.g.}, gramicidin A where $R \approx 3 {\mathrm{\AA}}$, $L \approx 2.5 {\mathrm{nm}}$ and $2x_0 \approx 3.8 {\mathrm{nm}}$) whose translocation energy is therefore controlled by the part of the self-energy, $\Sigma$, arising from the 1D potential. On this basis, we derive an expression for $\Sigma$, with no fitting parameter, which applies to a generic nano-channel of length $L$ and radius $R$.