Translocation energy of ions in nano-channels of cell membranes
  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 $R$. 
Sofian Teber