Protein Transport in Curved Lipid Bilayer Membranes: An Extended Saffman-Delbruck Approach Incorporating Hydrodynamics in Curvatured Fluid Intefaces*

We develop fluctuating hydrodynamics approaches to extend Saffman-Delbruck theory to capture the collective drift-diffusion dynamics of proteins within curved lipid bilayer membranes. Our approach is at the level of fluid interfaces having any curved radial manifold shape. We take into account the two dimensional hydrodynamics of the two curved leaflets of the bilayer coupled with the three dimensional hydrodynamics of the surrounding bulk fluid. Using analytic and computational approaches, we show how Gaussian curvature can significantly impact dissipation within the curved two dimensional membrane fluid to augment the collective drift-diffusion dynamics of protein inclusions. We further show for the self-assembly of protein clusters that these effects contribute significant kinetic contributions giving differences with widely used non-hydrodynamic theories. We also present general results on the collective drift-diffusion dynamics when heterogeneous curved structures are present in the membrane geometry showing how these local Gaussian curvature effects influence hydrodynamic coupling in some interesting ways.