  Many of the cell membrane vital functions are achieved by the
self-organization of the proteins and biopolymers embedded in it. The protein
dynamics are in part determined by its drag. A large number of these proteins
can polymerize to form filaments. In-vitro studies of protein-membrane
interactions often involve using rigid beads coated with lipid bilayers, as a
model for the cell membrane. Motivated by this, we use slender-body theory to
compute the translational and rotational resistance of a single filamentous
protein embedded in the outer layer of a supported bilayer membrane and
surrounded on the exterior by a Newtonian fluid. We first consider the regime,
where the two layers are strongly coupled through their inter-leaflet friction.
We find that the drag along the parallel direction grows linearly with the
filament length and quadratically with the length for perpendicular and the
rotational drag coefficients. These findings are explained using scaling
arguments and by analyzing the velocity fields around the moving filament. We,
then, present and discuss the qualitative differences between the drag of a
filament moving in a freely suspended bilayer and a supported membrane as a
function of the membrane inter-leaflet friction. Finally, we briefly discuss
how these findings can be used in experiments to determine membrane rheology.
In summary, we present a formulation that allows computing the effects of
membrane properties (its curvature, viscosity, and inter-leaflet friction), and
the exterior and interior 3D fluids depth and viscosity on the drag of a
rod-like or filamentous protein, all in a unified theoretical framework.
