Pinning of fluid membranes by periodic harmonic potentials

We analyze the thermal fluctuations of fluid membranes in the presence of periodic confining harmonic potentials. This is a simple model of the biologically important, inhomogeneous attachment of the cytoskeleton to the external, fluid membrane of the cell. We study a two-dimensional checkerboard potential as well as one-dimensional, sinusoidal and periodic highly localized, $\delta$ function potentials. The membranes are described by an energy functional that includes the curvature bending modulus of the membrane and the harmonic external potential. We predict the magnitude of the membrane shape fluctuations. The sinusoidal potentials give a spontaneous surface tension, and an emergent intermediate-range order in the membrane undulations. The $\delta$ function potentials induce a renormalization of the curvature modulus, with perfect pinning at the $\delta$ potential sites. After spatial averaging, the $\delta$-function potentials also give rise to an effective surface tension. Finally, we compare these results with measurements of the fluctuations of the red-blood cell membrane, which shows the effects of cytoskeleton attachment to the cellular membrane.