  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.
