  A theoretical approach has been undertaken in order to model the
thermodynamic equilibrium of a vesicle adhering to a flat substrate. The
vesicle is treated in a canonical description with a fixed number of sites. A
finite number of these sites are occupied by mobile ligands that are capable of
interacting with a discrete number of receptors immobilized on the substrate.
Explicit consideration of the bending energy of the vesicle shape has shown
that the problem of the vesicle shape can be decoupled from the determination
of the optimum allocation of ligands over the vesicle. The allocation of bound
and free ligands in the vesicle could be determined as a function of the size
of the contact zone, the ligand-receptor binding strength and the concentration
of the system constituents. Several approximate solutions for different regions
of system parameters are determined and in particular, the distinction between
receptor-dominated equilibria and ligand-dominated equilibria is found to be
important. The crossover between these two types of solutions is found to occur
at a critical size of the contact zone. The presented approach enables the
calculation of the effective adhesion strength of the vesicle and thus permits
meaningful comparisons with relevant experiments as well as connecting the
presented model with the proven success of the continuum approach for modeling
the shapes of adhering vesicles. The behavior of the effective adhesion
strength is analyzed in detail and several approximate expressions for it are
given.
