  We study the microrheology of nanoparticle shells [Dinsmore et al. Science
298, 1006 (2002)] and viral capsids [Ivanovska et al. PNAS 101, 7600 (2004)] by
computing the mechanical response function and thermal fluctuation spectrum of
a viscoelastic spherical shell that is permeable to the surrounding solvent. We
determine analytically the damped dynamics of the shear, bend, and compression
modes of the shell coupled to the solvent both inside and outside the sphere in
the zero Reynolds number limit. We identify fundamental length and time scales
in the system, and compute the thermal correlation function of displacements of
antipodal points on the sphere and the mechanical response to pinching forces
applied at these points. We describe how such a frequency-dependent antipodal
correlation and/or response function, which should be measurable in new
AFM-based microrheology experiments, can probe the viscoelasticity of these
synthetic and biological shells constructed of nanoparticles.
