Membrane surrounded by viscoelastic continuous media: Anomalous diffusion and linear response to force

We study the dynamics of a nearly flat membrane bilayer that is surrounded by two semi-infinite viscoelastic fluids, and its response to local transverse forces. By modeling the surrounding fluids as continuous media with frequency dependent shear moduli G₁(ω) and G₂(ω) we derive the dispersion relation for undulations. We deduce the frequency-dependent transverse mean square displacement of a membrane segment and find that it is proportional to κ^-1/3[G₁(ω ) + G₂(ω)]^-2/3, where κ is the membrane bending modulus. We then consider the linear response of a membrane to external forces. Possible implications are elucidated for experiments probing the viscoelasticity of cells and vesicles encapsulating and/or embedded in viscoelastic fluids, for the dynamic structure factor of such systems, and for lamellipodia dynamics.