Fokker–Planck analysis of the Langevin–Lorentz equation: Application to ligand-receptor binding under electromagnetic exposure

Abstract

The statistical properties of the solution of the Langevin–Lorentz equation are analyzed by means of the Fokker–Planck approach. The equation describes the dynamics of an ion that is attracted by a central field and is interacting with a time-varying magnetic field and with the thermal bath. If the endogenous force is assumed to be elastic, then a closed-form expression for the probability density of the process can be obtained, in the case of constant magnetic exposure and, for the time-varying case, at least asymptotically. In the general case, a numerical integration of the resulting set of differential equations with periodically time-varying coefficients has been implemented. A framework for studying the possible effects of low-frequency, low-intensity electromagnetic fields on biological systems has been developed on the basis of the equation. The model assumes that an exogenous electromagnetic field may affect the binding of a messenger attracted by the endogenous force field of its receptor protein. The results are applicable to the analysis of experiments, e.g., exposing a Petri dish, containing a biological sample, to a periodically time-varying magnetic field generated by a pair of Helmholtz coils, most widely used in the scientific literature. The proposed model provides a theoretical mean for evaluating the biological effectiveness of low-frequency, low-intensity electromagnetic exposure.