  In biological settings membranes typically interact locally with other
membranes or the extracellular matrix in the exterior, as well as with internal
cellular structures such as the cytoskeleton. Characterization of the dynamic
properties of such interactions presents a difficult task. Significant progress
has been achieved through simulations and experiments, yet analytical progress
in modelling pinned membranes has been impeded by the complexity of governing
equations. Here we circumvent these difficulties by calculating analytically
the time-dependent Green's function of the operator governing the dynamics of
an elastically pinned membrane in a hydrodynamic surrounding and subject to
external forces. This enables us to calculate the equilibrium power spectral
density for an overdamped membrane pinned by an elastic, permanently-attached
spring subject to thermal excitations. By considering the effects of the finite
experimental resolution on the measured spectra, we show that the elasticity of
the pinning can be extracted from the experimentally measured spectrum.
Membrane fluctuations can thus be used as a tool to probe mechanical properties
of the underlying structures. Such a tool may be particularly relevant in the
context of cell mechanics, where the elasticity of the membrane's attachment to
the cytoskeleton could be measured.
