On Curvature Driven Rotational Diffusion of Proteins on Membrane Surfaces

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.

A new approach to constrained total variation solvation models and the study of solute-solvent interface profiles

“…All of the physical properties of interest, including electrostatic free energies, biomolecular surface areas, molecular cavitation volumes and pK a values are very sensitive to the interface definition [26,59,63]. Variational implicit solvation models (VISM) stand out as a successful approach to compute the disposition of an interface separating the solute and the solvent [8,16,17,21,22,28,28,65,71]. In a VISM, the desired interface profile is obtained by minimizing a solvation energy functional coupling the discrete description of solute and the continuum description of solvent.…”

“…Numerical simulations show that diffuse-interface models can significantly improve the accuracy and efficiency of solvation energy computation [8,16,17,21,22,28,28,45,65,71]. In contrast, on a theoretic level, there are several open questions concerning model (1).…”

“…Indeed, most of the solvation energy functionals, regardless of sharp or diffuse interfaces, only predict local minimizers, cf. [8,16,17,21,22,28,45,65,71]. As a consequence, solutions of the corresponding Euler-Lagrange equations may not correctly depict the energy minimizing interface profile.…”

A Computational Model of Protein Induced Membrane Morphology with Geodesic Curvature Driven Protein-Membrane Interface

“…These asymmetrical deformations can be characterized by using the two distinct principal curvatures κ 1 = κ 2 . These two curvatures are associated with the complex shape and specific orientation of membrane proteins, and are indispensable for the generation of negative Gaussian curvature required by all endocytosis and exocytosis processes of living cells [19,20]. Determination of proteinmembrane interface, i.e., the contact curves between the protein and the two membrane surfaces, is at the center of the continuum or hybrid modelling of protein-membrane interactions [21,9].…”