  Besides direct molecular interactions, proteins and nanoparticles embedded in
or adsorbed to membranes experience indirect interactions that are mediated by
the membranes. These membrane-mediated interactions arise from the membrane
curvature induced by the particles and can lead to assemblies of particles that
generate highly curved spherical or tubular membranes shapes, but have mainly
been quantified for planar or weakly curved membranes. In this article, we
systematically investigate the membrane-mediated interactions of arc-shaped
particles adsorbed to a variety of tubular and spherical membrane shapes with
coarse-grained modelling and simulations. We determine both the pairwise
interaction free energy, with includes entropic contributions due to rotational
entropy loss at close particle distances, and the pairwise interaction energy
without entropic components from particle distributions observed in the
simulations. For membrane shapes with small curvature, the membrane-mediated
interaction free energies of particle pairs exceed the thermal energy kT and
can lead to particle ordering and aggregation. The interactions strongly
decrease with increasing curvature of the membrane shape and are minimal for
tubular shapes with membrane curvatures close to the particle curvature.
