Osmotic virial coefficients for model protein and colloidal solutions: Importance of ensemble constraints in the analysis of light scattering data

Evaluation of Osmotic Virial Coefficients via Restricted Gibbs Ensemble Simulations, with Support from Gas-Phase Mixture Coefficients

We present a method for computing osmotic virial coefficients in explicit solvent via simulation in a restricted Gibbs ensemble. Two equivalent phases are simulated at once, each in a separate box at constant volume and temperature and each in equilibrium with a solvent reservoir. For osmotic coefficient BN, a total of N solutes are individually exchanged back and forth between the boxes, and the average distribution of solute numbers between the boxes provides the key information needed to compute BN. Separately, expressions are developed for BN as a series in solvent reservoir density ρ₁, with the coefficients of the series expressed in terms of the usual gas-phase mixture coefficients B_ij. Normally, the B_ij are defined for an infinite volume, but we suggest that the observed dependence of B_ij on system size L can be used to estimate L dependence of the BN, allowing them to be computed accurately at L → ∞ while simulating much smaller system sizes than otherwise possible. The methods for N = 2 and 3 are demonstrated for two-component mixtures of size-asymmetric additive hard spheres. The proposed methods are demonstrated to have greater precision than established techniques, for a given amount of computational effort. The ρ₁ series for BN when applied by itself is (for this noncondensing model) found to be the most efficient in computing accurate osmotic coefficients for the solvent densities considered here.

McMillan-Mayer theory of solutions revisited: simplifications and extensions

McMillan and Mayer (MM) proved two remarkable theorems in their paper on the equilibrium statistical mechanics of liquid solutions. They first showed that the grand canonical partition function for a solution can be reduced to one with an effectively solute-only form, by integrating out the solvent degrees of freedom. The total effective solute potential in the effective solute grand partition function can be decomposed into components which are potentials of mean force for isolated groups of one, two, three, etc., solute molecules. Second, from the first result, now assuming low solute concentration, MM derived an expansion for the osmotic pressure in powers of the solute concentration, in complete analogy with the virial expansion of gas pressure in powers of the density at low density. The molecular expressions found for the osmotic virial coefficients have exactly the same form as the corresponding gas virial coefficients, with potentials of mean force replacing vacuum potentials. In this paper, we restrict ourselves to binary liquid solutions with solute species A and solvent species B and do three things: (a) By working with a semi-grand canonical ensemble (grand with respect to solvent only) instead of the grand canonical ensemble used by MM, and avoiding graphical methods, we have greatly simplified the derivation of the first MM result, (b) by using a simple nongraphical method developed by van Kampen for gases, we have greatly simplified the derivation of the second MM result, i.e., the osmotic pressure virial expansion; as a by-product, we show the precise relation between MM theory and Widom potential distribution theory, and (c) we have extended MM theory by deriving virial expansions for other solution properties such as the enthalpy of mixing. The latter expansion is proving useful in analyzing ongoing isothermal titration calorimetry experiments with which we are involved. For the enthalpy virial expansion, we have also changed independent variables from semi-grand canonical, i.e., fixed {N(A), μ(B), V, T}, to those relevant to the experiment, i.e., fixed {N(A), N(B), p, T}, where μ denotes chemical potential, N the number of molecules, V the volume, p the pressure, and T the temperature.

Coarse-grained model for colloidal protein interactions, B(22), and protein cluster formation

Reversible protein cluster formation is an important initial step in the processes of native and non-native protein aggregation, but involves relatively long time and length scales for detailed atomistic simulations and extensive mapping of free energy landscapes. A coarse-grained (CG) model is presented to semiquantitatively characterize the thermodynamics and key configurations involved in the landscape for protein oligomerization, as well as experimental measures of interactions such as the osmotic second virial coefficient (B22). Based on earlier work (Grüenberger et al., J. Phys. Chem. B 2013, 117, 763), this CG model treats proteins as rigid bodies composed of one bead per amino acid, with each amino acid having specific parameters for its size, hydrophobicity, and charge. The net interactions are a combination of steric repulsions, short-range attractions, and screened long-range charge-charge interactions. Model parametrization was done by fitting simulation results against experimental value of B22 as a function of solution ionic strength for α-chymotrypsinogen A and γD-Crystallin (gD-Crys). The CG model is applied to characterize the pairwise interactions and dimerization of gD-Crys and the dependence on temperature, protein concentration, and ionic strength. The results illustrate that at experimentally relevant conditions where stable dimers do not form, the entropic contributions are predominant in the free-energy of protein cluster formation and colloidal protein interactions, arguing against interpretations that treat B22 primarily from energetic considerations alone. Additionally, the results suggest that electrostatic interactions help to modulate the population of the different stable configurations for protein nearest-neighbor pairs, while short-range attractions determine the relative orientations of proteins within these configurations. Finally, simulation results are combined with Principal Component Analysis to identify those amino-acids/surface patches that form interprotein contacts at conditions that favor dimerization of gD-Crys. The resulting regions agree with previously found aggregation-prone sites, as well as suggesting new ones that may be important.