Role of Anisotropic Interactions for Proteins and Patchy Nanoparticles

Protein-protein interactions are inherently anisotropic to some degree, with orientation-dependent interactions between repulsive and attractive or complementary regions or patches on adjacent proteins. In some cases it has been suggested that such patch-patch interactions dominate the thermodynamics of dilute protein solutions, as captured by the osmotic second virial coefficient (B-22), but delineating when this will or will not be the case remains an open question. A series of simplified but exactly solvable models are first used to illustrate that a delicate balance exists between the strength of attractive patch-patch interactions and the patch size, and that repulsive patchpatch interactions contribute significantly to B-22 for only those conditions where the repulsions are long-ranged. Finally, B-22 is reformulated, without approximations, in terms of the density of states for a given interaction energy and particleparticle distance. Doing so illustrates the inherent balance of entropic and energetic contributions to B-22. It highlights that simply having strong patch-patch interactions will only cause anisotropic interactions to dominate B-22 solution properties if the unavoidable entropic penalties are overcome, which cannot occur if patches are too small. The results also indicate that the temperature dependence of B-22 may be a simple experimental means to assess whether a small number of strongly attractive configurations dominate the dilute solution behavior.