  Internalization of particles by cells plays a crucial role for adsorbing
nutrients and fighting infection. Endocytosis is one of the most important
mechanisms of the particles uptake which encompass multiple pathways. Although
endocytosis is a complex mechanism involving biochemical signaling and active
force generation, the energetic cost associated to the large deformations of
the cell membrane wrapping around the foreign particle is an important factor
controlling this process, which can be studied using quantitative physical
models. Of particular interest is the competition between membrane -
cytoskeleton and membrane - target adhesion. Here, we explore the wrapping of a
lipid membrane around a long cylindrical object in the presence of a substrate
mimicking the cytoskeleton. Using discretization of the Helfrich elastic energy
that accounts for the membrane bending rigidity and surface tension, we obtain
a wrapping phase diagram as a function of the membrane-cytoskeleton and the
membrane-target adhesion energy that includes unwrapped, partially wrapped and
fully wrapped states. We provide an analytical expression for the boundary
between the different regimes. While the transition to partial wrapping is
independent of membrane tension, the transition to full wrapping is very much
influenced by membrane tension. We also show that target wrapping may proceed
in an asymmetric fashion in the full wrapping regime.
