Extraction of Thermodynamic Data from Ternary Diffusion Coefficients. Use of Precision Diffusion Measurements for Aqueous Lysozyme Chloride−NaCl at 25 °C To Determine the Change of Lysozyme Chloride Chemical Potential with Increasing NaCl Concentration Well into the Supersaturated Region

Abstract

For ternary systems, we present a method for using measured values of the four ternary diffusion coefficients and the Onsager reciprocal relations to extract derivatives of solute chemical potentials with respect to solute molar concentrations. The method is applicable to systems in which the molar concentration of one solute is very small compared to that of the other, and also small enough that an inverse concentration dependence dominates certain activity coefficient derivatives. These conditions apply to a large number of aqueous systems involving macromolecules of biological interest. Unlike other techniques, the present method can be used to study undersaturated and supersaturated solutions. The approach is illustrated for the lysozyme chloride−NaCl−H2O system at 25 °C, using data reported here for pH 6.0 at 0.60 mM (8.6 mg/mL) lysozyme chloride and 0.25, 0.50, 0.65, 0.90, and 1.30 M (1.4, 2.8, 3.7, 5.1, and 7.2 wt %) NaCl concentrations, and our earlier data for pH 4.5 at the same concentrations. We use these solute chemical potential derivatives to compute the protein cation charge approximately, and to construct a function approximating the derivative of the lysozyme chloride chemical potential with respect to NaCl concentration, which we integrate over a range of NaCl concentrations. This provides the change of the lysozyme chloride chemical potential with NaCl concentration well into the supersaturated region, and hence provides the driving force for nucleation and crystal growth of lysozyme chloride as a function of the extent of supersaturation. We also compute the diffusion Onsager coefficients (Lij)0 for each composition at pH 4.5 and 6.0. Binary diffusion coefficients of aqueous lysozyme chloride at 0.89 mM (12.7 mg/mL) for pH values from 4.0 to 6.0, and at pH 6.0 for concentrations from 0.25 to 1.95 mM (3.6−27.9 mg/mL) are also reported.