We consider a model for voltage-dependent gating of channels in which the gating charges are on the channel wall and move only a small distance. When this movement occurs across the closed gate, the charges move through the entire transmembrane potential, which is energetically equivalent to their moving across the entire membrane. The channel exists in two open states, O1 and O2, and two closed states, C1 and C2; each open and closed configuration is divided into two states because of the two possible positions of the gating charges. An unusual property of this model is that the electrical work in going from an open to a closed configuration (for example, in going from O1 to C2) is path dependent, and net work can result from going reversibly around a complete cycle. The model channel, like many biological channels, shows bursting activity. This flickering on and off of the channel enables the gate to sense the electric field and decide if it should be in the open or closed configuration. We prove here some general theorms concerning the electrical work associated with the movements of the walls of channels and the movements of charges on these walls.
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