ON CURVATURE DRIVEN ROTATIONAL DIFFUSION OF PROTEINS ON MEMBRANE SURFACES

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 induced by membrane curvature in the absence of protein interaction and correlation. Entropic free energy depending on the protein orientational distribution and the curvature free 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 Smoluchowski equation with nonlocal potential to govern the temporal and angular variations of the protein orientation. We proved the existence of the stationary solutions of the equation as fixed points of a continuous nonlinear map. The uniqueness of the stationary solution is obtained through approximation in the Fourier space, where the expansion is reduced and the coefficients are related to the modified Bessel functions of the first kind. An efficient numerical algorithm was developed to calculate these coefficients. Numerical examples reproduced the orientational preference of membrane proteins on bilayer tubules as indicated by the minimization of the energy functional.