On the use of Kolmogorov equations for determining ion channels characteristics

Abstract

The paper proposes a method for determination of the dynamic properties of conduction within voltage-gated ion channels with several sequentially located energy barriers named gate particles using a theory of Markov random processes in continuous time and with a discrete state space. The number of states was taken equal to the number of energy barriers in the channel plus one. If the hypothesis that random flows switching gate particles between an open and a closed are Poisson process, is true, then mathematical description of the system’s state of similar channels can be a system of Kolmogorov linear differential equations for state probabilities. Using this model and based on Volt-Clamp test results, published in open access journals, it is possible to find out how fixed values of membrane potential depend on the intensities of Poisson process for different types of potassium channels. Function parameters, that describe the intensity of ion transition across the channel, in the Kolmogorov equation can be restored using the generalized least squares method. This paper contains the examples of determining the intensities of transitions for two types of voltage-dependent potassium channels known as «delayed rectifiier potassium channels» (IKdr, two identical activation gate particles) and channels with fast activation and inactivation processes (IKa, three identical activation and one inactivation gate particles). It is shown that channel activation and deactivation is described by solving the general Kolmogorov equation.