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 ∝ e2 / √εwεmR (R being the radius of the pore, and εm ∼ 2 and εw ∼ 80 the room temperature dielectric constants of membrane and water, respectively). There exists a crossover length, x0 ∝ R √εw/εm ∼ 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 >> 2x0, and short channels for which L << 2x0. The latter condition is satisfied by most realistic channels (e.g., gramicidin A where R ∼ 3 A, L ∼ 2.5 nm and 2x0 ∼ 3.8 nm whose translocation energy is therefore controlled by the part of the self-energy, Σ, arising from the 1D potential. On this basis, we derive an expression for Σ, with no fitting parameter, which applies to a generic nano-channel of length L and radius R. Our results are related to model-independent translocation properties of nano-scale ionic channels, they improve on previous, curve-fitting, formulas and agree to within 5% with estimates, resulting from numerical simulations, available in the literature on the subject. (author)