  The adhesion of cells is mediated by receptors and ligands anchored in
apposing membranes. A central question is how to characterize the binding
affinity of these membrane-anchored molecules. For soluble molecules, the
binding affinity is typically quantified by the binding equilibrium constant
K3D in the linear relation [RL] = K3D [R][L] between the volume concentration
[RL] of bound complexes and the volume concentrations [R] and [L] of unbound
molecules. For membrane-anchored molecules, it is often assumed by analogy that
the area concentration of bound complexes [RL] is proportional to the product
[R][L] of the area concentrations for the unbound receptor and ligand
molecules. We show here (i) that this analogy is only valid for two planar
membranes immobilized on rigid surfaces, and (ii) that the thermal roughness of
flexible membranes leads to cooperative binding of receptors and ligands. In
the case of flexible membranes, the area concentration [RL] of receptor-ligand
bonds is proportional to the square of [R][L] for typical lengths and
concentrations of receptors and ligands in cell adhesion zones. The cooperative
binding helps to understand why different experimental methods for measuring
the binding affinity of membrane-anchored molecules have led to values
differing by several orders of magnitude.
