Negative second virial coefficients as predictors of protein crystal growth: evidence from sedimentation equilibrium studies that refutes the designation of those light scattering parameters as osmotic virial coefficients

The effects of ammonium sulphate concentration on the osmotic second virial coefficient (BAA/MA) for equine serum albumin (pH 5.6, 20 degrees C) have been examined by sedimentation equilibrium. After an initial steep decrease with increasing ammonium sulphate concentration, BAA/MA assumes an essentially concentration-independent magnitude of 8-9 ml/g. Such behaviour conforms with the statistical-mechanical prediction that a sufficient increase in ionic strength should effectively eliminate the contributions of charge interactions to BAA/MA but have no effect on the covolume contribution (8.4 ml/g for serum albumin). A similar situation is shown to apply to published sedimentation equilibrium data for lysozyme (pH 4.5). Although termed osmotic second virial coefficients and designated as such (B22), the negative values obtained in published light scattering studies of both systems have been described incorrectly because of the concomitant inclusion of the protein-salt contribution to thermodynamic nonideality of the protein. Those negative values are still valid predictors of conditions conducive to crystal growth inasmuch as they do reflect situations in which there is net attraction between protein molecules. However, the source of attraction responsible for the negative virial coefficient stems from the protein-salt rather than the protein-protein contribution, which is necessarily positive.     

Interpretation of thermodynamic non-ideality in sedimentation equilibrium experiments on proteins

This investigation re-examines theoretical aspects of the allowance for effects of thermodynamic non-ideality on the sedimentation equilibrium distribution for a single macromolecular solute, and thereby resolves the question of the constraints that pertain to the definition of the activity coefficient term in the basic sedimentation equilibrium expression. Sedimentation equilibrium results for ovalbumin are then presented to illustrate a simple procedure for evaluating the net charge (valence) of a protein from the magnitude of the second virial coefficient in situations where the effective radius of the protein can be assigned. Finally, published sedimentation equilibrium results on lysozyme are reanalysed to demonstrate the feasibility of employing the dependence of the second virial coefficient upon ionic strength to evaluate both the valence and the effective radius of the non-interacting solute.     

Determination of the second virial coefficient of bovine serum albumin under varying pH and ionic strength by composition-gradient multi-angle static light scattering

Composition-gradient multi-angle static light scattering (CG-MALS) is an emerging technique for the determination of intermolecular interactions via the second virial coefficient B₂₂. With CG-MALS, detailed studies of the second virial coefficient can be carried out more accurately and effectively than with traditional methods. In addition, automated mixing, delivery and measurement enable high speed, continuous, fluctuation-free sample delivery and accurate results. Using CG-MALS we measure the second virial coefficient of bovine serum albumin (BSA) in aqueous solutions at various values of pH and ionic strength of a univalent salt (NaCl). The systematic variation of the second virial coefficient as a function of pH and NaCl strength reveals the net charge change and the isoelectric point of BSA under different solution conditions. The magnitude of the second virial coefficient decreases to 1.13 x 10⁻⁵ ml*mol/g² near the isoelectric point of pH 4.6 and 25 mM NaCl. These results illuminate the role of fundamental long-range electrostatic and van der Waals forces in protein-protein interactions, specifically their dependence on pH and ionic strength. Electronic supplementary material The online version of this article (doi:10.1007/s10867-014-9367-7) contains supplementary material, which is available to authorized users.     

Osmotic second virial cross‐coefficient measurements for binary combination of lysozyme,ovalbumin, and α‐amylase in salt solutions

Interactions measurement is a valuable tool to predict equilibrium phase separation of a desired protein in the presence of unwanted macromolecules. In this study, cross‐interactions were measured as the osmotic second virial cross‐coefficients (B₂₃) for the three binary protein systems involving lysozyme, ovalbumin, and α‐amylase in salt solutions (sodium chloride and ammonium sulfate). They were correlated with solubility for the binary protein mixtures. The cross‐interaction behavior at different salt concentrations was interpreted by either electrostatic or hydrophobic interaction forces. At low salt concentrations, the protein surface charge dominates cross‐interaction behavior as a function of pH. With added ovalbumin, the lysozyme solubility decreased linearly at low salt concentration in sodium chloride and increased at high salt concentration in ammonium sulfate. The B₂₃ value was found to be proportional to the slope of the lysozyme solubility against ovalbumin concentration and the correlation was explained by preferential interaction theory. © 2013 American Institute of Chemical Engineers Biotechnol. Prog., 29:1203–1211, 2013     

Nonequivalence of second virial coefficients from sedimentation equilibrium and static light scattering studies of protein solutions

Experimental data for ovalbumin and lysozyme are presented to highlight the nonequivalence of second virial coefficients obtained for proteins by sedimentation equilibrium and light scattering. Theoretical considerations confirm that the quantity deduced from sedimentation equilibrium distributions is B(22), the osmotic second virial coefficient describing thermodynamic nonideality arising solely from protein self-interaction. On the other hand, the virial coefficient determined by light scattering is shown to reflect the combined contributions of protein-protein and protein-buffer interactions to thermodynamic nonideality of the protein solution. Misidentification of the light scattering parameter as B(22) accounts for published reports of negative osmotic second virial coefficients as indicators of conditions conducive to protein crystal growth. Finally, textbook assertions about the equivalence of second virial coefficients obtained by sedimentation equilibrium and light scattering reflect the restriction of consideration to single-solute systems. Although sedimentation equilibrium distributions for buffered protein solutions are, indeed, amenable to interpretation in such terms, the same situation does not apply to light scattering measurements because buffer constituents cannot be regarded as part of the solvent: instead they must be treated as non-scattering cosolutes.     

Protein interactions in solution characterized by light and neutron scattering: comparison of lysozyme and chymotrypsinogen.

The effects of pH and electrolyte concentration on protein-protein interactions in lysozyme and chymotrypsinogen solutions were investigated by static light scattering (SLS) and small-angle neutron scattering (SANS). Very good agreement between the values of the virial coefficients measured by SLS and SANS was obtained without use of adjustable parameters. At low electrolyte concentration, the virial coefficients depend strongly on pH and change from positive to negative as the pH increases. All coefficients at high salt concentration are slightly negative and depend weakly on pH. For lysozyme, the coefficients always decrease with increasing electrolyte concentration. However, for chymotrypsinogen there is a cross-over point around pH 5.2, above which the virial coefficients decrease with increasing ionic strength, indicating the presence of attractive electrostatic interactions. The data are in agreement with Derjaguin-Landau-Verwey-Overbeek (DLVO)-type modeling, accounting for the repulsive and attractive electrostatic, van der Waals, and excluded volume interactions of equivalent colloid spheres. This model, however, is unable to resolve the complex short-ranged orientational interactions. The results of protein precipitation and crystallization experiments are in qualitative correlation with the patterns of the virial coefficients and demonstrate that interaction mapping could help outline new crystallization regions.     

Precipitation of egg white proteins below their isoelectric points by sodium dodecyl sulphate and temperature.

The precipitating of effect of sodium dodecyl sulphate (SDS) on the egg white proteins ovalbumin, conalbumin and lysozyme was studied at 25 degrees C and at different pH values. The proteins precipitated below their respective isolectric points, provided the detergent to protein ratio was appropriate. The pH profile of precipitation was different for the three proteins reflecting net charge differences. The binding of SDS to the proteins was studied with [35S]-labelled SDS and for ovalbumin a ratio of 21--28 SDS molecules/protein molecule, dependent on the concentration of SDS initially used, seem to be required for precipitation at pH 4.5. This number, however, is dependent on pH and increases with an increased positive net charge of the protein. The precipitating effect of SDS was identical for ovalbumin, conalbumin and lysozyme when compared on a gram to gram basis (0.1--0.15 g SDS/g precipitated protein). The precipitated protein was denatured as measured by differential scanning calorimetry, but was also completely redissolved if pH was increased to above the isoelectric point. The precipitating effecto f SDS was also examined at elevated temperatures. The two-phase systems of the proteins induced by SDS at 25 degrees C were heated from 25 degrees C to 90 degrees C at a rate of 1.25 degrees C/min. The precipitation behaviour was similar for the three proteins upon heating. When the SDS concentration was increased the precipitation curves were transferred towards lower temperatures and the courses of precipitation became less sharp. The synergistic effect of SDS and heat on protein precipitation was differentiated by denaturation measurements and radioactive labelling. The ratio SDS to precipitated protein gradually diminished towards higher temperatures but no purely thermal precipitation was found.     

Interactions of lysozyme in concentrated electrolyte solutions from dynamic light-scattering measurements

The diffusion of hen egg-white lysozyme has been studied by dynamic light scattering in aqueous solutions of ammonium sulfate as a function of protein concentration to 30 g/liter. Experiments were conducted under the following conditions: pH 4-7 and ionic strength 0.05-5.0 M. Diffusivity data for ionic strengths up to 0.5 M were interpreted in the context of a two-body interaction model for monomers. From this analysis, two potential-of-mean-force parameters, the effective monomer charge, and the Hamaker constant were obtained. At higher ionic strength, the data were analyzed using a model that describes the diffusion coefficient of a polydisperse system of interacting protein aggregates in terms of an isodesmic, indefinite aggregation equilibrium constant. Data analysis incorporated multicomponent virial and hydrodynamic effects. The resulting equilibrium constants indicate that lysozyme does not aggregate significantly as ionic strength increases, even at salt concentrations near the point of salting-out precipitation.     

Effects of pH on protein-protein interactions and implications for protein phase behavior

The effects of pH on protein interactions and protein phase behavior were investigated by measuring the reduced second osmotic virial coefficient (b2) for ovalbumin and catalase, and the aggregate and crystal solubilities for ovalbumin, beta-lactoglobulin A and B, ribonuclease A and lysozyme. The b2 trends observed for ovalbumin and catalase show that protein interactions become increasingly attractive with decreasing pH. This trend is in good agreement with ovalbumin phase behavior, which was observed to evolve progressively with decreasing pH, leading to formation of amorphous aggregates instead of gel bead-like aggregates, and spherulites instead of needle-like crystals. For both acidic and basic proteins, the aggregate solubility during protein salting-out decreased with decreasing pH, and contrary to what is commonly believed, neither aggregate nor crystal solubility had a minimum at the isoelectric point. beta-Lactoglobulin B was the only protein investigated to show salting-in behavior, and crystals were obtained at low salt concentrations in the vicinity of its isoelectric point. The physical origin of the different trends observed during protein salting-in and salting-out is discussed, and the implications for protein crystallization are emphasized.     

Interactions of lysozyme in guanidinium chloride solutions from static and dynamic light-scattering measurements

The interactions of partially unfolded proteins provide insight into protein folding and protein aggregation. In this work, we studied partially unfolded hen egg lysozyme interactions in solutions containing up to 7 M guanidinium chloride (GdnHCl). The osmotic second virial coefficient (B(22)) of lysozyme was measured using static light scattering in GdnHCl aqueous solutions at 20 degrees C and pH 4.5. B(22) is positive in all solutions, indicating repulsive protein-protein interactions. At low GdnHCl concentrations, B(22) decreases with rising ionic strength: in the absence of GdnHCl, B(22) is 1.1 x 10(-3) mLmol/g(2), decreasing to 3.0 x 10(-5) mLmol/g(2) in the presence of 1 M GdnHCl. Lysozyme unfolds in solutions at GdnHCl concentrations higher than 3 M. Under such conditions, B(22) increases with ionic strength, reaching 8.0 x 10(-4) mLmol/g(2) at 6.5 M GdnHCl. Protein-protein hydrodynamic interactions were evaluated from concentration-dependent diffusivity measurements, obtained from dynamic light scattering. At moderate GdnHCl concentrations, lysozyme interparticle interactions are least repulsive and hydrodynamic interactions are least attractive. The lysozyme hydrodynamic radius was calculated from infinite-dilution diffusivity and did not change significantly during protein unfolding. Our results contribute toward better understanding of protein interactions of partially unfolded states in the presence of a denaturant; they may be helpful for the design of protein refolding processes that avoid protein aggregation.     

COVOL: an interactive program for evaluating second virial coefficients from the triaxial shape or dimensions of rigid macromolecules

An interactive program is described for calculating the second virial coefficient contribution to the thermodynamic nonideality of solutions of rigid macromolecules based on their triaxial dimensions. The FORTRAN-77 program, available in precompiled form for the PC, is based on theory for the covolume of triaxial ellipsoid particles [Rallison, J. M., and S.E Harding. (1985). J. Colloid Interface Sci. 103:284-289]. This covolume has the potential to provide a magnitude for the second virial coefficient of macromolecules bearing no net charge. Allowance for a charge-charge contribution is made via an expression based on Debye-Hückel theory and uniform distribution of the net charge over the surface of a sphere with dimensions governed by the Stokes radius of the macromolecule. Ovalbumin, ribonuclease A, and hemoglobin are used as model systems to illustrate application of the COVOL routine.     

Exclusion in hyaluronate gels.

Osmotic pressures of solutions of hyaluronate (HA) (mol wt 117,000) and mixtures of HA and bovine serum albumin (BSA) in phosphate-buffered saline, pH 7.2 were measured with a membrane osmometer. The data were fit with a virial expansion in integral powers of total nondiffusible solute concentration. Values of number average molecular weight were calculated for HA and the mixtures from the first virial coefficients. The excluded volume of HA in the single nondiffusible solute solution was calculated from the second virial coefficient extracted from the data on the HA solution. The excluded volume of HA with respect to BSA was estimated from the "osmotic parameters" of HA and BSA by an approach developed in 1976 by Shaw. The resulting excluded volume of HA with respect to BSA was compared with those obtained from a lightly cross-linked HA gel and from solutions of HA (mol wt 1.5 x 10(6)) studied in 1964 by Laurent. The development of this cross-linked HA gel and its subsequent calibration are described.     

Effect of alcohols on aqueous lysozyme-lysozyme interactions from static light-scattering measurements

Alcohols have been widely used as protein denaturants, precipitants and crystallization reagents. We have studied the effect of alcohols on aqueous hen-egg lysozyme self-interactions by measuring the osmotic second virial coefficient (B22) using static light scattering. Addition of alcohols increases B22, indicating stronger protein-protein repulsion or weaker attraction. For the monohydric alcohols used in this study (methanol, ethanol, 1-propanol, n-butanol, iso-butanol and trifluoroethanol), B22 for lysozyme reaches a common plateau at approximately 5% (v/v) alcohol, while glycerol increases B22 more than monohydric alcohols. For a 0.05 M NaCl hen-egg lysozyme solution at pH 7, B22 increases from 2.4 x 10(-4) to 4.7 x 10(-4) ml mol/g2 upon addition of monohydric alcohols and to 5.8 x 10(-4) ml mol/g2 upon addition of glycerol. We describe the alcohol effect using a simple model that supplements the DLVO theory with an additional alcohol-dependent term representing orientation-averaged hydrophobic interactions. In this model, the increased lysozyme repulsive forces in the presence of monohydric alcohols are interpreted in terms of adsorption of alcohol molecules on hydrophobic sites on the protein surface. This adsorption reduces attractive hydrophobic protein-protein interactions. A thicker lysozyme hydration layer in aqueous glycerol solution can explain the glycerol-increased lysozyme-lysozyme repulsion.     

Studies of solute self-association by sedimentation equilibrium: allowance for effects of thermodynamic non-ideality beyond the consequences of nearest-neighbor interactions

A sedimentation equilibrium study of alpha-chymotrypsin self-association in acetate-chloride buffer, pH 4.1 I 0.05, has been used to illustrate determination of a dimerization constant under conditions where thermodynamic non-ideality is manifested beyond the consequences of nearest-neighbor interactions. Because the expressions for the experimentally determinable interaction parameters comprise a mixture of equilibrium constant and excluded volume terms, the assignment of reasonable magnitudes to the relevant virial coefficients describing non-associative cluster formation is essential for the evaluation of a reliable estimate of the dimerization constant. Determination of these excluded volume parameters by numerical integration over the potential-of-mean-force is shown to be preferable to their calculation by approximate analytical solutions of the integral for this relatively small enzyme monomer with high net charge (+10) under conditions of low ionic strength (0.05 M).     

Second virial coefficients as a measure of protein--osmolyte interactions

The cytoplasm contains high concentrations of cosolutes. These cosolutes include macromolecules and small organic molecules called osmolytes. However, most biophysical studies of proteins are conducted in dilute solutions. Two broad classes of models have been used to describe the interaction between osmolytes and proteins. One class focuses on excluded volume effects, while the other focuses on binding between the protein and the osmolyte. To better understand protein--smolyte interactions, we have conducted sedimentation equilibrium analytical ultracentrifugation experiments using ferricytochrome c as a model protein. From these experiments, we determined the second virial coefficients for a series of osmolytes. We have interpreted the second virial coefficient as a measure of both excluded volume and protein--osmolyte binding. We conclude that simple models are not sufficient to understand the interactions between osmolytes and proteins.     

Hydrophobic forces between protein molecules in aqueous solutions of concentrated electrolyte

Protein-protein interactions have been measured for a mutant (D101F) lysozyme and for native lysozyme in concentrated solutions of ammonium sulfate at pH 7 and sodium chloride at pH 4.5. In the mutant lysozyme, a surface aspartate residue has been replaced with a hydrophobic phenylalanine residue. The protein-protein interactions of D101F lysozyme are more attractive than those of native lysozyme for all conditions studied. The salt-induced attraction is correlated with a solvation potential of mean force given by the work required to desolvate the part of the protein surfaces that is buried by the protein-protein interaction. This work is proportional to the aqueous surface-tension increment of the salt and the fractional non-polar surface coverage of the protein. Experimental measurements of osmotic second virial coefficients validate a proposed potential of mean force that ascribes the salt-induced attraction between protein molecules to an enhancement of the hydrophobic attraction. This model provides a first approximation for predicting the protein-protein potential of mean force in concentrated aqueous electrolyte solutions; this potential is useful for determining solution conditions favorable for protein crystallization.     

Protein-protein and protein-salt interactions in aqueous protein solutions containing concentrated electrolytes

Protein-protein and protein-salt interactions have been obtained for ovalbumin in solutions of ammonium sulfate and for lysozyme in solutions of ammonium sulfate, sodium chloride, potassium isothiocyanate, and potassium chloride. The two-body interactions between ovalbumin molecules in concentrated ammonium-sulfate solutions can be described by the DLVO potentials plus a potential that accounts for the decrease in free volume available to the protein due to the presence of the salt ions. The interaction between ovalbumin and ammonium sulfate is unfavorable, reflecting the kosmotropic nature of sulfate anions. Lysozyme-lysozyme interactions cannot be described by the above potentials because anion binding to lysozyme alters these interactions. Lysozyme-isothiocyanate complexes are strongly attractive due to electrostatic interactions resulting from bridging by the isothiocyanate ion. Lysozyme-lysozyme interactions in sulfate solutions are more repulsive than expected, possibly resulting from a larger excluded volume of a lysozyme-sulfate bound complex or perhaps, hydration forces between the lysozyme-sulfate complexes.     

Net charge affects morphology and visual properties of ovalbumin aggregates

The effect of ovalbumin net charge on aggregate morphology and visual properties was investigated using chromatography, electrophoresis, electron microscopy, and turbidity measurements. A range of differently charged ovalbumin variants (net charge ranging from -1 to -26 at pH 7) was produced using chemical engineering. With increasing net charge, the degree of branching and flexibility of the aggregates decreased. The turbidity of the solutions reflected the aggregate morphology that was observed with transmission electron microscopy. Increasing the stiffness of the aggregates transformed the solutions from turbid to transparent. Artificially shielding the introduced net charge by introducing salt in the solution resulted in an aggregate morphology that was similar to that for low-net-charge variants. The morphology of heat-induced aggregates and the visual appearance of the solutions were significantly affected by net charge. We also found that the morphology of ovalbumin aggregates can be rapidly probed by high-throughput turbidity experiments.     

Anomalous temperature dependence of the thermal expansion of proteins

The temperature dependence of the apparent expansibility of lysozyme and ovalbumin in solution has been measured as a function of pH. This temperature dependence is explained in terms of suppressed fluctuations in bound water due to the protein. It is shown that the thermal expansion coefficient of bound water is different from bulk water. The pH dependence can be explained by increased hydration of side chains at lower pH. The amount in volume of hydration water in a typical protein-water system varies from 0.16 to 0.7. How the intrinsic thermal expansion coefficient of proteins can be derived from the apparent quantity is discussed. Intrinsic values of the thermal expansion coefficient for lysozyme at room temperature are between 1.7 and 4.4 x 10(-4) K-1 for a 10% solution.     

Second virial coefficient studies of cosolvent-induced protein self-interaction

Protein self-interaction is important in protein crystal growth, solubilization, and aggregation, both in vitro and in vivo, as with protein misfolding diseases, such as Alzheimer's. Although second virial coefficient studies can supply invaluable quantitative information, their emergence as a systematic approach to evaluating protein self-interaction has been slowed by the limitations of traditional measurement methods, such as static light scattering. Comparatively, self-interaction chromatography is an inexpensive, high-throughput method of evaluating the osmotic second virial coefficient (B) of proteins in solution. In this work, we used self-interaction chromatography to measure B of lysozyme in the presence of various cosolvents, including sucrose, trehalose, mannitol, glycine, arginine, and combinations of arginine and glutamic acid and arginine and sucrose in an effort to develop a better fundamental understanding of protein self-interaction in complex cosolvent systems. All of these cosolvents, alone or in combination, increased B, indicating a reduction in intermolecular attraction. However, the magnitude of cosolvent-induced changes in B was found to be largely dependent on the ability to control long-range electrostatic repulsion. To the best of our knowledge, this work represents the most comprehensive virial coefficient study to date focusing on complex cosolvent-induced effects on the self-interaction of lysozyme.