Statistical mechanics of an elastically pinned membrane: Static profile and correlations

The relation between the mechanical properties of a nonspecifically adherent, free membrane and its thermal fluctuations is well studied, both theoretically and experimentally. However, understanding this relationship is significantly more challenging for the biologically relevant case of membranes pinned by proteins to scaffolds such as the cytoskeleton, the extracellular matrix or another cell. For a free membrane, an expansion of the membrane profile into plane waves allows for analytical tractability and the calculation of the fluctuation properties. For a pinned membrane, the difficulty lies in the coupling of the plane wave modes, and analytic approaches to the problem were not successful so far. Here we calculate the mode coupling coefficients for the plane wave expansion, as well as the orthonormal fluctuating modes and the Green's function for the system. We show that the mean membrane shape is linearly related to the spatial correlation function of the free membrane as well as of the pinned membrane, the latter two differing only by a constant factor. Most importantly, we provide a set of tools which can be used in the future to address biologically relevant questions like the interaction of cellular membranes with internal and external scaffolds.