McMillan–Mayer solution thermodynamics for a protein in a mixed solvent

McMillan–Mayer solution theory is used to calculate the thermodynamic properties of solute/solvent systems from the set of solvent-averaged interactions between solute particles. To match experimental data, the independent variables of the McMillan–Mayer framework (temperature, the set of solvent chemical potentials, and the set of solute concentrations) need to be converted to the Gibbs framework, where the independent variables are temperature, pressure, and the set of concentrations for all components. Previous conversions have been restricted to systems with a single solvent and one or more solutes. However, in the analysis of light-scattering data, it is sometimes necessary to consider multiple solvents. Here, we present the thermodynamics of McMillan–Mayer solution theory for a single protein solute and a mixed solvent of aqueous salt; we show that protein–salt preferential-interaction parameters can be readily determined from data reduction within the McMillan–Mayer framework. We provide a sample calculation of a liquid–liquid phase separation for protein solutions, wherein we convert the phase diagram from the McMillan–Mayer framework to the Gibbs framework. For the coexistence curve of the McMillan–Mayer phase diagram, the salt molality of a hypothetical outside solution is held constant, whereas for the coexistence curve for the Gibbs phase diagram, the salt molality of the light phase is held constant. The difference between the curves is determined by the preferential-interaction parameter. The two curves are identical only in the limiting case where the preferential-interaction parameter is zero, the solvent can be considered as a pseudo-one-component solvent.