  In this paper, we propose a Generalized Langevin Equation (GLE)-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 (non
instantaneous) component modeled respectively through a Dirac delta function
and a three-parameter Mittag-Leffler type function. By imposing a specific
relationship between the parameters of the three-parameters Mittag-Leffler
function, the different dynamical regimes, namely 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 distribution 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 Squared Displacement
(MSD) derived in the framework of this model and the MSD of a protein diffusing
in a membrane calculated through molecular dynamics (MD) simulations.
