On the thermodynamics of the McMillan–Mayer state function

The osmotic second virial coefficient is a key parameter in light scattering, protein crystallisation, self-interaction chromatography, and osmometry. The interpretation of the osmotic second virial coefficient depends on the solution theory. On the macroscopic level an expansion of the osmotic pressure is employed. A common statistical interpretation of the osmotic second virial coefficient of the expansion employs the McMillan–Mayer framework and the potential of mean force to characterise the solute–solute interaction. Supplementary to the statistical interpretation, it may be advantageous to develop the McMillan–Mayer framework in a classical thermodynamic context for which we develop the relationship between the state function of the McMillan–Mayer framework and the Helmholtz state function.     

Protein crystal nucleation: is the pair interaction potential the primary determinant of kinetics?

Fundamental understanding of protein crystal nucleation facilitates crystallization of biological macromolecules for structure determination and control of crystal size distribution. In the studies presented here, nucleation kinetics of hen egg-white lysozyme crystals were measured at solution conditions that exhibited equal solubility by adjusting pH, temperature, or sodium chloride concentration. It was observed that solution conditions that lead to equal solubility resulted in equal nucleation rates and hence kinetic parameters. Since the solubility of globular proteins correlates with the osmotic second virial coefficient, B(22), an integral measure of the protein pair interaction potential, this observation indicates that the protein pair interaction plays a key role in determining nucleation kinetic parameters.     

Protein—Protein Interactions in Aqueous Ammonium Sulfate Solutions. Lysozyme and Bovine Serum Albumin (BSA)

Osmotic pressures have been measured to determine lysozyme—lysozyme,BSA—BSA, and lysosyme—BSA interactions for protein concentrations to 100 g-L^–1in an aqueous solution of ammonium sulfate at ambient temperature, as a functionof ionic strength and pH. Osmotic second virial coefficients for lysozyme, forBSA, and for a mixture of BSA and lysozyme were calculated from theosmotic-pressure data for protein concentrations to 40 g-L^–1. The osmotic second virialcoefficient of lysozyme is slightly negative and becomes more negative withrising ionic strength and pH. The osmotic second virial coefficient for BSA isslightly positive, increasing with ionic strength and pH. The osmotic second virialcross coefficient of the mixture lies between the coefficients for lysozyme andBSA, indicating that the attractive forces for a lysozyme—BSA pair areintermediate between those for the lysozyme—lysozyme and BSA—BSA pairs. For proteinconcentrations less than 100 g-L^–1, experimental osmotic-pressure data comparefavorably with results from an adhesive hard-sphere model, which has previouslybeen shown to fit osmotic compressibilities of lysozyme solutions.     

Solubility of lysozyme in the presence of aqueous chloride salts: common-ion effect and its role on solubility and crystal thermodynamics

Understanding protein solubility is important for a rational design of the conditions of protein crystallization. We report measurements of lysozyme solubility in aqueous solutions as a function of NaCl, KCl, and NH4Cl concentrations at 25 degrees C and pH 4.5. Our solubility results are directly compared to preferential-interaction coefficients of these ternary solutions determined in the same experimental conditions by ternary diffusion. This comparison has provided new important insight on the dependence of protein solubility on salt concentration. We remark that the dependence of the preferential-interaction coefficient as a function of salt concentration is substantially shaped by the common-ion effect. This effect plays a crucial role also on the observed behavior of lysozyme solubility. We find that the dependence of solubility on salt type and concentration strongly correlates with the corresponding dependence of the preferential-interaction coefficient. Examination of both preferential-interaction coefficients and second virial coefficients has allowed us to demonstrate that the solubility dependence on salt concentration is substantially affected by the corresponding change of protein chemical potential in the crystalline phase. We propose a simple model for the crystalline phase based on salt partitioning between solution and the hydrated protein crystal. A novel solubility equation is reported that quantitatively explains the observed experimental dependence of protein solubility on salt concentration.     

A Model for Hydration Interactions between Apoferritin Molecules in Solution

We have studied how non-DLVO forces between molecules of the globular protein apoferritin in solution affect its osmotic second virial coefficient. A model explaining the effects of the solution ionic strength and pH on the interprotein interaction is developed, to give a physical interpretation of recently published experimental findings showing that the second virial coefficient of the protein apoferritin, supported by acetate buffer, goes through a minimum as a function of ionic strength. At low ionic strengths, the apoferritin second virial coefficient initially decreases with increasing sodium ion concentration, as DLVO theory predicts. However, non-DLVO hydration forces due to overlapping of the Stern layers of the protein molecules increase the second virial coefficient with further increase of sodium ion concentration, again as found experimentally at higher ionic strengths. The non-DLVO effect arises from ionic exchange between hydrogen and sodium ions at the protein surface. An adsorption shell of hydrated sodium ions forms around the protein molecules with increasing buffer concentration.     

The osmotic second virial coefficient and the Gibbs–McMillan–Mayer framework

The osmotic second virial coefficient is a key parameter in light scattering, protein crystallisation, self-interaction chromatography, and osmometry. The interpretation of the osmotic second virial coefficient depends on the set of independent variables. This commonly includes the independent variables associated with the Kirkwood–Buff, the McMillan–Mayer, and the Lewis–Randall solution theories. In this paper we analyse the osmotic second virial coefficient using a Gibbs–McMillan–Mayer framework which is similar to the McMillan–Mayer framework with the exception that pressure rather than volume is an independent variable. A Taylor expansion is applied to the osmotic pressure of a solution where one of the solutes is a small molecule, a salt for instance, that equilibrates between the two phases. Other solutes are retained. Solvents are small molecules that equilibrate between the two phases. The independent variables of the solvents are temperature, pressure, and chemical potentials. The derivatives in the Gibbs–McMillan–Mayer framework are transformed into derivatives in the Gibbs framework. This offers the possibility for an interpretation and correlation of the osmotic second virial coefficient using activity coefficient models.     

Membrane osmometer for use at moderate applied pressures

A membrane osmometer designed for use at pressures greater than 0.1 MPa and less than 6 MPa was employed to determine the pressure coefficient of the equilibrium osmotic pressure (?π/?P) of a dilute polystyrene/toluene solution. The pressure coefficient of the second virial coefficient (?A₂/?P), calculated from ?π/?P, was 6 (±4) × 10^?5 cm³ mol g^?2 MPa^?1, which was in reasonable agreement with the value obtained from pressure‐dependent light scattering. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 3064–3069, 2003     

Osmotic Coefficients of Aqueous Solutions of Some Poly(oxyethylene) Glycols at the Freezing Point of Solution

The freezing temperatures of dilute aqueous solutions of some poly(oxyethylene) glycols (PEG, HO–(CH₂CH₂O) _n –H, n varying from 4 to 117) were measured over a solute to solvent mass ratio from 0.0100 to 0.3900. The second and third osmotic virial coefficient (A ₂₂ and A ₂₂₂) of poly(oxyethylene) glycols in aqueous solution were determined. The molecular weight dependence of the second virial coefficient can be described by a simple relation A ₂₂=2×10^–5 M _n ^1.86, and the third virial coefficient is A ₂₂₂=0.038A ₂₂ ². The activity coefficients of the solute were calculated using the Gibbs-Duhem equation as applied by Bjerrum. From the osmotic and activity coefficients the excess Gibbs energies of solution, as well as the respective partial molar functions of solute and solvent and the virial pair interaction coefficients for the excess Gibbs energies were estimated. The second and the third osmotic virial coefficients are correlated with the Mc-Millan-Mayer virial coefficients.     

Molecular properties of poly(carboxybetaine) in solutions with different ionic strengths and pH values

Viscometry and dynamic and static light scattering are employed to study the molecular properties of water-soluble poly(carboxybetaine), that is, poly(2-(diallyl(methyl)ammonium) acetate). It is shown that, in solutions with pH 1, the polymer behaves as a polyelectrolyte. In media with pH 6 and 13, an increase in the concentration of sodium chloride increases the intrinsic viscosity of the polymer and the hydrodynamic radius of its macromolecules, thereby indicating the antipolyelectrolyte effect typical of polymer zwitterions. In water and 0.1 M NaOH, the second virial coefficient of the polymer is close to zero, while exponent ν, which relates the sizes of macromolecules to their molecular masses, is 0.5. In 1 M NaCl, the second virial coefficient becomes positive, while exponent increases to 0.58. The Kuhn segment lengths of poly(carboxybetaine) molecules are 6.3 and 6.6 nm in water and 1 M NaCl, respectively. An increase in the hydrodynamic radius of macromolecules with the ionic strength of the solution is due to the shielding of attraction between zwitterions belonging to polybetaine monomer units located far apart along a macromolecular chain.     

Self-interaction nanoparticle spectroscopy: a nanoparticle-based protein interaction assay

Solution conditions conducive to protein crystallization are identified mainly in an empirical manner using screening methods. Measurements of a dilute solution thermodynamic parameter, the osmotic second virial coefficient, have been shown to be useful in guiding this search, yet the measurement of this parameter remains difficult. In this work, a nanoparticle-based assay, self-interaction nanoparticle spectroscopy, is presented as an efficient alternative. The method involves adsorbing proteins on the surface of gold nanoparticles and adding the protein/gold conjugates to solutions of interest for crystallization. The optical properties of gold colloid, including macroscopic ones such as color, are sensitive to the interparticle separation distance, and they are demonstrated to correlate with the value of the second virial coefficient for BSA and ovalbumin. Serendipitously, the conditions that correspond to second virial coefficient values within the thermodynamic region ideal for protein crystallization lead to the maximum change in color of the gold suspensions. Given the remarkable efficiency of this method, it holds significant potential to aid in the crystallization of proteins that have not been crystallized previously. Moreover, this method may find utility in the analysis of weak homo- and heterotypic interactions involved in other biological applications, including preventing protein aggregation and formulating therapeutic proteins.     

Osmotic Coefficients of Aqueous Solutions of Some Poly(oxyethylene) Glycols at the Freezing Point of Solution

Summary. The freezing temperatures of dilute aqueous solutions of some poly(oxyethylene) glycols (PEG, HO–(CH₂CH₂O) _n –H, n varying from 4 to 117) were measured over a solute to solvent mass ratio from 0.0100 to 0.3900. The second and third osmotic virial coefficient (A ₂₂ and A ₂₂₂) of poly(oxyethylene) glycols in aqueous solution were determined. The molecular weight dependence of the second virial coefficient can be described by a simple relation A ₂₂=2×10^–5 M _n ^1.86, and the third virial coefficient is A ₂₂₂=0.038A ₂₂ ². The activity coefficients of the solute were calculated using the Gibbs-Duhem equation as applied by Bjerrum. From the osmotic and activity coefficients the excess Gibbs energies of solution, as well as the respective partial molar functions of solute and solvent and the virial pair interaction coefficients for the excess Gibbs energies were estimated. The second and the third osmotic virial coefficients are correlated with the Mc-Millan-Mayer virial coefficients.     

Gaussian charge polarizable interaction potential for carbon dioxide

A number of simple pair interaction potentials of the carbon dioxide molecule are investigated and found to underestimate the magnitude of the second virial coefficient in the temperature interval 220-448 K by up to 20%. Also the third virial coefficient is underestimated by these models. A rigid, polarizable, three-site interaction potential reproduces the experimental second and third virial coefficients to within a few percent. It is based on the modified Buckingham exp-6 potential, an anisotropic Axilrod-Teller correction, and Gaussian charge densities on the atomic sites with an inducible dipole at the center of mass. The electric quadrupole moment, polarizability, and bond distances are set to equal experiment. Density of the fluid at 200 and 800 bars pressure is reproduced to within some percent of observation over the temperature range 250-310 K. The dimer structure is in passable agreement with electronically resolved quantum-mechanical calculations in the literature, as are those of the monohydrated monomer and dimer complexes using the Gaussian charge polarizable model water potential. Qualitative agreement with experiment is also obtained, when quantum corrections are included, for the relative stability of the trimer conformations, which is not the case for the pair potentials.     

Hydrophobic interactions and osmotic second virial coefficients for methanol in water

A recent theory of the hydrophobic effect together with a simple model for an alcohol molecule is used to calculate the osmotic (McMillan-Mayer) second virial coefficientB ₂ for methanol dissolved in water. We use this calculation to study the validity of common arguments that try to draw microscopic structural information from experimental virial coefficient data. In disagreement with many workers, we find that the hydrophobic interaction between hard spheres in water is attractive and that its strength diminishes as temperature is raised. Models that have come to the opposite conclusions have neglected complications inherent to real solutes such as the role of the hydroxy groups in affecting the correlations between the apolar portions of neighboring alcohols. The calculations reported here indicate that this neglect is a poor approximation for methanol. Our calculations also show that osmotic virial coefficients are sensitive to subtle details in the potentials of mean force. Therefore, slowly varying (e.g., dispersion) interactions may also contribute significantly to the values of these coefficients without significantly changing the solvent structure near the solute molecules.     

Zeta potentials and Debye screening lengths of aqueous, viscoelastic surfactant solutions (cetyltrimethylammonium bromide/sodium salicylate system)

In a series of experiments, we studied the dynamic properties of aqueous surfactant solutions of cetyltrimethylammonium bromide (CTAB) at conditions after adding different amounts of sodium salicylate (NaSal). The aggregates, present in these solutions, are elongated, wormlike micelles, which tend to form entanglement networks. The viscoelastic, gel-like samples were analyzed by means of static, dynamic, and electrophoretic light scattering techniques. We separately investigated the effects of surfactant concentration and added salt on intermicellar interactions. The electrostatic interactions between the anisometric micelles were analyzed by considering the effective dimensions of the aggregates. We calculated the Debye-Huckel lengths from experimental data of the osmotic second virial coefficient and from the diffusion second virial coefficient. It turned out that the results were in good agreement with theoretically estimated values. We also measured the zeta potential and intensity of scattered light in a large range of different salt concentrations keeping the CTAB concentration constant. We observed an isoelectric point and charge reversal of the threadlike micelles at an excess salicylate concentration of about 100 mM. The observed decrease of the zeta potential points to striking processes of counterion condensation. In these solutions, the salicylate ion acts as a cosurfactant, due to its discrepancy between polar and hydrophobic groups. We also detected a simple linear correlation between the zeta potentials and the Debye screening lengths of the surfactant solutions.     

A statistical model for non-ionic micellar solutions and their phase diagrams

The micellar phase separation (cloud-point transition) of non-ionic surfactants is studied on the basis of a statistical theory, of liquids. The low value of the critical concentration is explained by the presence of an extended repulsive intermicellar interaction due to a region of structured water. Excellent agreement with experiment is found for the polyoxyethylene surfactants C₁₂E₈ and C₆E₃. Our model can give a closed solubility loop with a very different value of the concentration of the upper and lower consolution points. We study, also the osmotic compressibility and second virial coefficient and the correlation length.     

渗透压法对牛血清蛋白在NaCl水溶液中相互作用的研究

测量了牛血清蛋白不同pH值和不同离子强度Nacl水溶液中的渗透压,计算了渗透第二维里系数,按照Prausnitz提出的分子热学模型计算了蛋白质之间的静电排斥能、色散吸引能和离子排斥体积产生的吸引势能,并对这三种势能与溶液pH值和离子强度的变化关系进行讨论。     

Study on Electric Double Layer of a Cylindrical Particle with Functional Theoretical Approach

A new method, i.e. the iterative method in functional theory, was introduced to solve analytically the nonlinear Poisson-Boltzmann （PB） equation under general potential ψ condition for the electric double layer of a charged cylindrical colloid particle in a symmetrical electrolyte solution. The iterative solutions of ψ are expressed as functions of the distance from the axis of the particle with solution parameters： the concentration of ions c, the aggregation number of ions in a unit length m, the dielectric constant e, the system temperature T and so on. The relative errors show that generally only the first and the second iterative solutions can give accuracy higher than 97%. From the second iterative solution the radius and the surface potential of a cylinder have been defined and the corresponding values have been estimated with the solution parameters, Furthermore, the charge density, the activity coefficient of ions and the osmotic coefficient of solvent were also discussed,     

Nonconformal interaction models and thermodynamics of polar fluids

In this work, we develop a simple potential model for polar molecules which represents effectively and accurately the thermodynamics of dilute gases. This potential models dipolar interactions whose nonpolar part is either spherical, as in Stockmayer (SM) molecules, or diatomic, as for 2-center Lennard-Jones molecules (2CLJ). Predictions of the second virial coefficient for SM and polar 2CLJ fluids for various dipole moments and elongations agree very well with results of recent numerical calculations by C. Vega and co-workers (Phys. Chem. Chem Phys. 2002, 4, 3000). The model is used to predict the critical temperature of Stockmayer fluids for variable dipole moment and is applied to HCl as an example of a real polar molecule.