We introduce a mathematical model to study the transport of ions through ion channels. The system is derived in the framework of the energetic variational approach, taking into account the coupling between electrostatics, diffusion, and protein (ion channel) structure. The geometric constraints of the ion channel are introduced through a potential energy controlling the local maximum volume inside the ion channel. A diffusive interface (labeling) description is also employed to describe the geometric configuration of the channels. The surrounding bath and channel are smoothly connected with the antechamber region by this label function. A corresponding modified Poisson-Nernst-Planck channel system for ion channels is derived using the variational derivatives of the total energy functional. The functional consists of the entropic free energy for diffusion of the ions, the electrostatic potential energy, the repulsive potential energy for the excluded volume effect of the ion particles, and the potential energy for the geometric constraints of the ion channel. For the biological application of such a system, we consider channel recordings of voltage clamp to measure the current flowing through the ion channel. The results of one-dimensional numerical simulations are presented to demonstrate some signature effects of the channel, such as the current output produced by single-step and double-step voltage inputs.
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