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A theoretical approach has been undertaken in order to model the thermodynamic equilibrium of a 3D 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 can 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 …