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The structural and thermodynamic factors responsible for the singly and doubly occupied saturation states of the gramicidin channel are investigated with molecular dynamics simulations and free energy perturbation methods. The relative free energy of binding of all of the five common cations Li+, Na+, K+, Rb+, and Cs+ is calculated in the singly and doubly occupied channel and in bulk water. The atomic system, which includes the gramicidin channel, a model membrane made of neutral Lennard-Jones particles and 190 explicit water molecules to form the bulk region, is similar to the one used in previous work to calculate the free energy profile of a Na+ ion along the axis of the channel. In all of the calculations, the ions are positioned in the main binding sites located near the entrances of the channel. The calculations reveal that the doubly occupied state is relatively more favorable for the larger ions. Thermodynamic decomposition is used to show that the origin of the trend observed in the calculations is due to the loss of favorable interactions between the ion and the single file water molecules inside the channel. Small ions are better solvated by the internal water molecules in the singly occupied state than in the doubly occupied state; bigger ions are solvated almost as well in both occupation states. Water-channel interactions play a role in the channel response. The observed trends are related to general thermodynamical properties of electrolyte solutions.

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An approximate equation for the treatment of atom–atom collision problems is proposed, based on the assumption that in the closed channels nuclear kinetic energy is negligible compared with the electronic energy. It is shown that this approximation yields upper bounds on the exact scattering phase shifts when the collision energy lies below the lowest nonadiabatic resonance level. To test the quality of this approximation, three model problems, all of them involving two coupled channels one of which is closed, are introduced. Each of these models is solvable exactly, solvable in the ordinary adiabatic approximation (which provides lower bounds on the phase shifts) and solvable within the new approximation. The accuracy of the results obtained is discussed with special reference to its dependence upon the form and strength of the interactions involved.