  Cell membranes interact via anchored receptor and ligand molecules. Central
questions on cell adhesion concern the binding affinity of these
membrane-anchored molecules, the mechanisms leading to the receptor-ligand
domains observed during adhesion, and the role of cytoskeletal and other active
processes. In this review, these questions are addressed from a theoretical
perspective. We focus on models in which the membranes are described as elastic
sheets, and the receptors and ligands as anchored molecules. In these models,
the thermal membrane roughness on the nanometer scale leads to a cooperative
binding of anchored receptor and ligand molecules, since the receptor-ligand
binding smoothens out the membranes and facilitates the formation of additional
bonds. Patterns of receptor domains observed in Monte Carlo simulations point
towards a joint role of spontaneous and active processes in cell adhesion. The
interactions mediated by the receptors and ligand molecules can be
characterized by effective membrane adhesion potentials that depend on the
concentrations and binding energies of the molecules.
