Translocation energy of ions in nano-channels of cell membranes

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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 Σ is characterized by a "geometric mean" screening Σ∝ e² / √εw εₘR (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 ≫ 2 x₀, and short channels for which L ≪ 2 x₀. The latter condition is satisfied by most realistic channels (e.g., gramicidin A where R ≈ 3 Å, 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.