  Adhesion processes of biological membranes that enclose cells and cellular
organelles are essential for immune responses, tissue formation, and signaling.
These processes depend sensitively on the binding constant K2D of the
membrane-anchored receptor and ligand proteins that mediate adhesion, which is
difficult to measure in the 'two-dimensional' (2D) membrane environment of the
proteins. An important problem therefore is to relate K2D} to the binding
constant K3D} of soluble variants of the receptors and ligands that lack the
membrane anchors and are free to diffuse in three dimensions (3D). In this
article, we present a general theory for the binding constants K2D and K3D of
rather stiff proteins whose main degrees of freedom are translation and
rotation, along membranes and around anchor points 'in 2D', or unconstrained
'in 3D'. The theory generalizes previous results by describing how K2D depends
both on the average separation and thermal nanoscale roughness of the apposing
membranes, and on the length and anchoring flexibility of the receptors and
ligands. Our theoretical results for the ratio K2D/K3D of the binding constants
agree with detailed results from Monte Carlo simulations without any data
fitting, which indicates that the theory captures the essential features of the
'dimensionality reduction' due to membrane anchoring. In our Monte Carlo
simulations, we consider a novel coarse-grained model of biomembrane adhesion
in which the membranes are represented as discretized elastic surfaces, and the
receptors and ligands as anchored molecules that diffuse continuously along the
membranes and rotate at their anchor points.
