Reversible formation of protein oligomers or small clusters is a key step in processes such as protein polymerization and fibril formation, and protein phase separation. A relatively simple, statistical mechanical approach to accurately calculate cluster free energies in dilute solution under conditions of attractive osmotic second virial coefficients (b₂₂^*) was developed using a cell-based, quasi-chemical (QC) approximation for the canonical partition function of proteins in an implicit solvent. The inputs to the model are the protein potential of mean force and the corresponding subcell degeneracies for small independent subcells, up to a relatively low subcell density. For subcell degeneracies too large or complicated to calculate via direct enumeration, the degeneracies are approximated by an approach using the Wang Landau method for the energy probability and a random walk method for subcell cluster configurations at a constant subcell density. A comparison of the approximate degeneracy calculation to direct enumeration shows excellent quantitative agreement.

The QC approximation to the partition function was tested using simple 2d and 3d lattice models in which proteins interact with hard core repulsions and either isotropic or anisotropic nearest-neighbor attractions. Comparison with direct Monte Carlo simulation shows cluster probabilities and free energies of oligomer formation (ΔG_i⁰) are quantitatively predicted by the QC approach for protein concentrations less than approximately 65 mg/mL (assuming a partial specific volume of 0.75 cm3/g-protein) . For small clusters, ΔG_i⁰ depends weakly on the strength of short-ranged attractive interaction for most experimentally relevant values of the b₂₂^*. For larger clusters (i>>2) there is a significant b₂₂^* dependence. In both cases, ΔG_i⁰ is sensitive to the size (excluded volume) of the monomer, indicative of the strong contributions of configurational entropy to ΔG_i⁰ in dilute solutions. The results suggest increased inter-protein hydrophobic attractions for nonnative relative to native proteins are unlikely to account for more than a small fraction of experimentally observed increases in protein polymerization or fibrillogenesis rates upon unfolding unless the nucleus is large, and that the effects of changes in monomer excluded volume upon unfolding cannot be neglected. The QC with WL approach is cast in a form that is easily generalized to continuous and more complicated inter-protein potentials of mean force.

(1) F. Wang and D.P. Landau. "Efficient, Multiple-Range Random Walk Algorithm to Calculate Density of States". Physical Review Letters 86, p.2050 (2001)