An effective model for the cross-over from anomalous to Brownian dynamics of proteins based on a viscoelastic membrane description

In this paper we derive a model to describe the effective motion of a protein laterally diffusing in a lipid membrane. By postulating that the lipid membrane is a linear viscoelastic fluid and the protein a rigid body, we derive a continuum description of the system. Within this framework, the lipid membrane is modeled through the linearized Navier-Stokes equations complemented by a non Newtonian constitutive equation, and the protein by a (modified) Euler's law, where a noise term is introduced to account for the stochasticity of the system. Then, by eliminating the lipid membrane degrees of freedom, we obtain a Generalized Langevin Equation (GLE) describing the diffusive motion of the protein with a memory kernel that can be related to the response of the membrane encoded in the solution of the constitutive equation. By representing the viscoelastic behavior through the Prabhakar fractional derivative, one generates a memory kernel containing a three-parameter Mittag-Leffler function. An additional Dirac delta-function term in the memory kernel accounts for the instantaneous component of the system response. A comparison between 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 shows that the proposed model accurately describes all regimes of the diffusive process, namely, the ballistic, the subdiffusive (anomalous) and the Brownian one, as well as the transitions from one regime to another. The model further provides an estimation of the membrane viscosity, which is of the order of magnitude of the values found by rheology experiments on analogous systems.