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 (B22), 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 patch-patch interactions contribute significantly to B22 for only those conditions where the repulsions are long-ranged. Finally, B22 is reformulated, without approximations, in terms of the density of states for a given interaction energy and particle-particle distance. Doing so illustrates the inherent balance of entropic and energetic contributions to B22. It highlights that simply having strong patch-patch interactions will only cause anisotropic interactions to dominate B22 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 B22 may be a simple experimental means to assess whether a small number of strongly attractive configurations dominate the dilute solution behavior.