Quantum Electrochemical Equilibrium: Quantum Version of the Goldman–Hodgkin–Katz Equation

Abstract

The resting membrane voltage of excitable cells such as neurons and muscle cells is determined by the electrochemical equilibrium of potassium and sodium ions. This voltage is calculated by using the Goldman–Hodgkin–Katz equation. However, from the quantum perspective, ions with significant quantum tunneling through closed channels can interfere with the electrochemical equilibrium and affect the value of the membrane voltage. Hence, in this case the equilibrium becomes quantum electrochemical. Therefore, the model of quantum tunneling of ions is used in this study to modify the Goldman–Hodgkin–Katz equation in such a way to calculate the resting membrane voltage at the point of equilibrium. According to the present calculations, it is found that lithium—with its lower mass—shows a significant depolarizing shift in membrane voltage. In addition to this, when the free gating energy of the closed channels decreases, even sodium and potassium ions depolarize the resting membrane voltage via quantum tunneling. This study proposes the concept of quantum electrochemical equilibrium, at which the electrical potential gradient, the concentration gradient and the quantum gradient (due to quantum tunneling) are balanced. Additionally, this concept may be used to solve many issues and problems in which the quantum behavior becomes more influential.