A kinetic model for voltage-gated ion channels in cell membranes based on the path integral method

Abstract

A kinetic model of cell membrane ion channels is proposed based on the path integral method. From the Pauli-type master equations valid on a macroscopic time scale, we derive a first-order differential equation or the kinetic equation which governs temporal evolution of the channel system along the paths of extreme probability. Using known parameters for the batrachotoxin (BTX)-modified sodium channels in squid giant axon, the time dependence of the channel activation and the voltage dependence of the corresponding time constants ( τ ) are examined numerically. It is found that the channel activation relaxes to the steady (or equilibrium)-state values for a given membrane potential and the corresponding time constant reaches a maximum at a certain potential and thereafter decreases in magnitude as the membrane potential increases. A qualitative comparison between these results and the results of Hodgkin–Huxley theory, path probability method and thermodynamic models as well as the cut-open axon technique is presented. Good agreement is achieved.