Growth and shape stability of a biological membrane adhesion complex in the diffusion-mediated regime

We examine the process of expansion of a focal adhesion complex by which a biological membrane containing mobile binders adheres to a substrate with complementary binders. Attention is focused on the situation, common among living cells, in which the mean mobile binder density is insufficient to overcome generic resistance to close approach of the membrane to its substrate. For the membrane to adhere, binders must be recruited from adjacent regions to join an adhesion patch of density adequate for adhesion, thereby expanding the size of the patch. The specific configuration examined is the expansion of a circular adhesion zone for which diffusive binder transport driven by a chemical potential gradient is the mechanism of binder recruitment. An aspect of the process of particular interest is the stability of the circular shape of the expanding front. It is found that the adhesion front radius increases as radical(t), where t is the time elapsed since nucleation, and that the circular shape becomes unstable under sinusoidal perturbations for radii large compared with the nucleation size, as observed in recent experiments.