The Membrane Potential and its Representation by a Constant Electric Field in Computer Simulations

A theoretical framework is elaborated to account for the effect of a transmembrane potential in computer simulations. It is shown that a simulation with a constant external electric field applied in the direction normal to the membrane is equivalent to the influence of surrounding infinite baths maintained to a voltage difference via ion-exchanging electrodes connected to an electromotive force. It is also shown that the linearly-weighted displacement charge within the simulation system tracks the net flow of charge through the external circuit comprising the electromotive force and the electrodes. Using a statistical mechanical reduction of the degrees of freedom of the external system, three distinct theoretical routes are formulated and examined for the purpose of characterizing the free energy of a protein embedded in a membrane that is submitted to a voltage difference. The W-route is constructed from the variations in the voltage-dependent potential of mean force along a reaction path connecting two conformations of the protein. The Q-route is based on the average displacement charge as a function of the conformation of the protein. Finally, the G-route considers the relative charging free energy of specific residues, with and without applied membrane potentials. The theoretical formulation is illustrated with a simple model of an ion crossing a vacuum slab surrounded by two aqueous bulk phases and with a fragment of the voltage-sensor of the KvAP potassium channel.