Clustering instability in adhesive contact between elastic solids via diffusive molecular bonds

Motivated by experimental observations that cell–cell and cell–matrix adhesion often involves formation of discrete patches of dense molecular bonds, we consider the plane strain problem of two elastic half-spaces, each covered with a layer of lipid membrane, joined together by mobile molecular bonds that diffuse along the interface under the combined action of a thin layer of glycocalyx repellers and an externally applied tensile stress. We show that, for a range of bond density values with or without the applied stress, the state of a uniform distribution of bonds is intrinsically unstable with respect to perturbations in bond density distribution. This instability is found to be primarily driven by elastic deformation energies in the bulk and the membrane. The change in free energy associated with a cosine perturbation in bond density distribution indicates that there exists a critical wavelength beyond which the perturbation becomes unstable and a fastest growing wavelength that tends to dominate as the instability grows. These length scales have typical values in the order of a micrometer, in agreement with the general characteristic size of bond clusters observed in cell adhesion.