Subdiffusive-Brownian crossover in membrane proteins: a generalized Langevin equation-based approach

Abstract

In this work, we propose a generalized Langevin equation-based model to describe the lateral diffusion of a protein in a lipid bilayer. The memory kernel is represented in terms of a viscous (instantaneous) and an elastic (noninstantaneous) component modeled through a Dirac d function and a three-parameter Mittag-Leffler type function, respectively. By imposing a specific relationship between the parameters of the three-parameter Mittag-Leffler function, the different dynamical regimesnamely ballistic, subdiffusive, and Brownian, as well as the crossover from one regime to another-are retrieved. Within this approach, the transition time from the ballistic to the subdiffusive regime and the spectrum of relaxation times underlying the transition from the subdiffusive to the Brownian regime are given. The reliability of the model is tested by comparing the mean-square displacement derived in the framework of this model and the mean-square displacement of a protein diffusing in a membrane calculated through molecular dynamics simulations.

On the derivation of a Nonlinear Generalized Langevin Equation

“…Despite the drastic reduction of the degrees of freedom, the GLE has shown to provide a reliable description of the physical diffusive processes. In fact, it has been employed in many fields of research, for instance, in Biophysics, for studying anomalous diffusion of proteins in lipid membrane [4][5][6][7], in Hydrodynamics, for studying how probe particles diffuse in viscoelastic materials [8][9][10][11][12], in Quantum Mechanics, for explaining the dissipation in quantum system on a more fundamental and general level [13][14][15][16][17] as well as in case of specific system such as the Caldeira-Leggett model [18,19] and, in Chemistry, as a diffusion model for the chemical reactions [20,21]. Nevertheless, the accuracy of the physical description provided by the GLE strictly depends on how one models the parameters and the timedependence of the friction function.…”