Membrane fluctuations near a plane rigid surface.

We use analytical calculations and Monte Carlo simulations to determine the thermal fluctuation spectrum of a membrane patch of a few tens of nanometer in size, whose corners are located at a fixed distance d above a plane rigid surface. Our analysis shows that the surface influence on the bilayer fluctuations can be effectively described in terms of a uniform confining potential that grows quadratically with the height of the membrane h relative to the surface: V=(1/2)gammah2. The strength gamma of the harmonic confining potential vanishes when the corners of the membrane patch are placed directly on the surface (d=0), and achieves its maximum value when d is of the order of a few nanometers. However, even at maximum strength, the confinement effect is quite small and has a noticeable impact only on the amplitude of the largest bending mode.