On Curvature Driven Rotational Diffusion of Protein on Membrane Surface

TL;DR

This paper models the curvature-driven rotational diffusion of membrane proteins, deriving a mathematical framework that predicts protein orientation dynamics on membranes, with proven existence, uniqueness, and an efficient numerical solution.

Contribution

It introduces a coupled energy functional and a generalized Smoluchowski equation for protein orientation, providing analytical proofs and a numerical method for stationary solutions.

Findings

Existence of stationary solutions under certain conditions

Uniqueness of solutions for biologically relevant parameters

An efficient Fourier-based numerical method for solutions

Abstract

Morphological dynamics of bilayer membrane is intrinsically coupled to the translational and orientational localization of membrane proteins. In this paper we are concerned with the orientational localization of membrane proteins in the absence of protein interaction and correlation. Entropic energy depending on the angular distribution function and the curvature energy depending on the principal curvature vectors are introduced to assemble an energy functional for the coupled system. Application of the Onsager's variational principle gives rise to a generalized Smoluchowskii equation governing the temporal and angular variations of the protein orientation. We prove the existence of the stationary solution of the equation as fixed points of a continuous nonlinear nonlocal map, and for biologically relevant conditions we obtain the uniqueness of the solution. To approximate the stationary solution in the Fourier space we construct an efficient numerical method that reduces the expansion and relates the coefficients to the modified Bessel functions of the first kind. Existence and uniqueness of the numerical solution are justified for biologically relevant conditions.