A quasichemical approach for protein-cluster free energies in dilute solution

Reversible formation of protein oligomers or small clusters is a key step in processes such as protein polymerization, fibril formation, and protein phase separation from dilute solution. A straightforward, statistical mechanical approach to accurately calculate cluster free energies in solution is presented using a cell-based, quasichemical (QC) approximation for the partition function of proteins in an implicit solvent. The inputs to the model are the protein potential of mean force (PMF) and the corresponding subcell degeneracies up to relatively low particle densities. The approach is tested using simple two and three dimensional lattice models in which proteins interact with either isotropic or anisotropic nearest-neighbor attractions. Comparison with direct Monte Carlo simulation shows that cluster probabilities and free energies of oligomer formation (DeltaG(i) (0)) are quantitatively predicted by the QC approach for protein volume fractions approximately 10(-2) (weight/volume concentration approximately 10 g l(-1)) and below. For small clusters, DeltaG(i) (0) depends weakly on the strength of short-ranged attractive interactions for most experimentally relevant values of the normalized osmotic second virial coefficient (b(2) (*)). For larger clusters (i"2), there is a small but non-negligible b(2) (*) dependence. The results suggest that nonspecific, hydrophobic attractions may not significantly stabilize prenuclei in processes such as non-native aggregation. Biased Monte Carlo methods are shown to accurately provide subcell degeneracies that are intractable to obtain analytically or by direct enumeration, and so offer a means to generalize the approach to mixtures and proteins with more complex PMFs.