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.     

Description of the thermodynamic incompatibility of the guar–dextran aqueous two-phase system by light scattering

The phase diagram of the guar–dextran aqueous two-phase system has been described on the basis of static light scattering measurements in the dilute regime. By determining the molar weight and second virial coefficient from the two single polymers and the second virial cross coefficient from mixtures at constant guar/dextran ratio (either 17/83 or 28/72), the thermodynamic models based on the virial expansion or the Flory–Huggins theory were successfully applied. The second virial coefficient of guar was difficult to estimate with enough accuracy by light scattering and therefore was obtained by adjustment using a simple criterion stating that the calculated spinodal passes through the experimental critical point. The obtained value was within the confidence interval given by light scattering. Virial expansion and Flory–Huggins approaches yielded quite similar theoretical phase diagrams that satisfactorily fitted the experimental one. The slight discrepancies observed on the position of binodals and critical points has been attributed to the polydispersity of guar or the difficulty in extrapolating from the dilute regime to the semidilute one. The slope of the tie-lines was predicted with a good accuracy, especially with the virial expansion model. The fact that both approaches gave such similar results is probably related to the fact that the expressions of chemical potentials are equivalent if the polymer concentrations are low enough. In this particular case, both models are based on excluded volume interactions and equally describe the phase behavior of the guar–dextran aqueous system.     

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.     

Graphical analysis of nonideal monomer N-mer, isodesmic, and type II indefinite self-associating systems by equilibrium ultracentrifugation.

A graphical procedure is described by which one can obtain in principle the monomer molecular weight, stoichiometry, equilibrium constant, and second virial coefficient of nonideal monomer N-mer, isodesmic, and type II indefinite self-associating systems. In addition, a method is presented for obtaining both the equilibrium constant and the second virial coefficient from the maximum in a plot of apparent molecular weight vs. concentration if the monomer molecular weight and stoichiometry are known. The usefulness and limitations of the methods are discussed, as well as the quality and range of data required for determination of the relevant parameters. The techniques described are applicable to analysis of self-associating systems by osmotic pressure and light scattering, as well as equilibrium ultracentrifugation measurements.     

Physicochemical studies on xylinan (acetan). II. Characterization by static light scattering

Laboratory-made samples of the polysaccharide xylinan, also called acetan, were studied in aqueous solution at various ionic strengths I (0.01 mol/L ≤ I ≤ 0.30 mol/L). The conditions for clarification (ultracentrifugation/membrane filtration) were studied. The Zimm procedure was used to obtain the average molar mass, the second virial coefficient, and the radius of gyration. The interpretation of the angular dependence of scattered light by fitting with “Master Curves” led to double-stranded wormlike chains with persistence lengths between 90 and 120 nm. The ionic strength had a strong effect on the thermodynamic second virial coefficient, but the overall structure remained unchanged. The ambiguity of the light scattering data was discussed assuming alternatively a two-component system instead of the wormlike chain model for the experimental scattering curves. The two-component system can be ruled out on the basis of model calculations. © 1996 John Wiley & Sons, Inc.     

Effect of alpha-crystallin on thermal denaturation and aggregation of rabbit muscle glyceraldehyde-3-phosphate dehydrogenase

The study of thermal denaturation of rabbit muscle glyceraldehyde-3-phosphate dehydrogenase (GAPDH) in the presence of alpha-crystallin by differential scanning calorimetry (DSC) showed that the position of the maximum on the DSC profile (T(max)) was shifted toward lower temperatures with increasing alpha-crystallin concentration. The diminishing GAPDH stability in the presence of alpha-crystallin has been explained assuming that heating of GAPDH induces dissociation of the tetrameric form of the enzyme into dimers interacting with alpha-crystallin. The dissociation of the enzyme tetramer was shown by sedimentation velocity at 45 degrees C. Suppression of thermal aggregation of GAPDH by alpha-crystallin was studied by dynamic light scattering under the conditions wherein temperature was elevated at a constant rate. The construction of the light scattering intensity versus the hydrodynamic radius (R(h)) plots enabled estimating the hydrodynamic radius of the start aggregates (R(h,0)). When aggregation of GAPDH was studied in the presence of alpha-crystallin, the start aggregates of lesser size were observed.     

Rapid measurement of protein osmotic second virial coefficients by self-interaction chromatography

Weak protein interactions are often characterized in terms of the osmotic second virial coefficient (B(22)), which has been shown to correlate with protein phase behavior, such as crystallization. Traditional methods for measuring B(22), such as static light scattering, are too expensive in terms of both time and protein to allow extensive exploration of the effects of solution conditions on B(22). In this work we have measured protein interactions using self-interaction chromatography, in which protein is immobilized on chromatographic particles and the retention of the same protein is measured in isocratic elution. The relative retention of the protein reflects the average protein interactions, which we have related to the second virial coefficient via statistical mechanics. We obtain quantitative agreement between virial coefficients measured by self-interaction chromatography and traditional characterization methods for both lysozyme and chymotrypsinogen over a wide range of pH and ionic strengths, yet self-interaction chromatography requires at least an order of magnitude less time and protein than other methods. The method thus holds significant promise for the characterization of protein interactions requiring only commonly available laboratory equipment, little specialized expertise, and relatively small investments of both time and protein.     

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.     

Structural changes in alpha-crystallin and whole eye lens during heating, observed by low-angle X-ray diffraction

Whole eye lens and alpha-crystallin gels and solutions were investigated using X-ray scattering techniques at temperatures ranging from 20 degrees C to 70 degrees C. In whole lens isolated in phosphate-buffered saline, the spacing of the dominant X-ray reflection seen with low-angle scattering was constant from 20 degrees C to 45 degrees C but increased at 50 degrees C from 15.2 nm to 16.5 nm. At room temperature, the small-angle X-ray diffraction pattern of the intact lens was very similar to the pattern of alpha-crystallin gels at near-physiological concentration (approximately 300 mg/ml), so it is reasonable to assume that the alpha-crystallin pattern dominates the pattern of the intact lens. Our results therefore indicate that in whole lens alpha-crystallin is capable of maintaining its structural properties over a wide range of temperature. This property would be useful in providing protection for other lens proteins super-aggregating. In the alpha-crystallin gels, a moderate increase in both the spacing and intensity of the reflection was observed from 20 degrees C to 45 degrees C, followed by an accelerated increase from 45 degrees C to 70 degrees C. Upon cooling, this effect was found to be irreversible over 11 hours. Qualitatively similar results were observed for alpha-crystallin solutions at a variety of lower concentrations.     

Random coil scission rates determined by time-dependent total intensity light scattering: hyaluronate depolymerization by hyaluronidase

A recently developed theory of the light scattering by random coils undergoing random scission is applied to the digestion of hyaluronate by hyaluronidase. The time dependence of the scattered light from solutions undergoing digestion was monitored. Working at a high angle with high molecular weight hyaluronate allowed the use of a powerful approximation for determining initial velocities and the Henri-Michaelis-menten coefficients, without explicit knowledge of the hyaluronate molecular weight, radius of gyration, second virial coefficient, or polydispersity. Effects due to a molecular weight dependent second virial coefficient and to non-Gaussian behavior are briefly considered. Assays were performed over nearly two orders of magnitude in substrate concentration. The initial velocities are compared with those obtained by a standard reducing sugar assay, which was performed on identical samples. The main advantages of the light scattering assay procedure over the more traditional assays are that many relatively high-precision data points can be quickly and automatically collected with simple apparatus, and that the technique is most sensitive for the initial period of digestion, where the other assays are least sensitive. The shapes of the scattering curves also provide evidence that hyaluronate in these solutions is not a stable double strand and that the hyaluronidase cleaves bond randomly. The curves also indicate that enzyme deactivation occurs, which accounts for the lower velocities yielded by the slower reductimetric assay, which is measured over longer initial periods.     

Nucleosome core particle structure and structural changes in solution

The radius of gyration, Rg, of chicken erythrocyte nucleosome core particles, was found to be 4.56 (+/- 0.07) nm by small-angle X-ray scattering, independent of particle concentration and of NaCl concentration between 0.1 M and 0.6 M-NaCl. The large, positive, second virial coefficient, A2, from particle concentration dependence (but independent of NaCl concentration) in small-angle X-ray scattering may indicate non-electrostatic repulsive ordering over large distances. Density contrast variation in equilibrium sedimentation with a small probe (sucrose) and a larger probe (gamma-cyclodextrin) yields good results for core particle hydration in the first instance, and for an estimate of the total particle volume in the second instance.     

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.     

Effects of Na+ on the persistence length and excluded volume of T7 bacteriophage DNA.

Total intensity, Rayleigh light scattering has been used to measure the rms radius, second virial coefficient, persistence length, and excluded volume of homogeneous T7 bacteriophage DNA as a function of Na+ concentration (0.005 to 3.0 M). All parameters decrease sharply as [Na+] increases, and tend to level off at high Na+. The variation of persistence length with [Na+] is consistent with predictions from counterion condensation theory.     

A light-scattering study of the effect of calcium chloride on the molecular weight of Busycon hemocyanin.

Solutions of Busycon canaliculatum have been studied by light scattering. In 0.05 M Trizma buffer +0.1 M NaCl at pH 7.0 at 14 degrees, the weight-average molecular weight is 8.9 X 10(6). In the presence of added CaCl2 (0.02 M), the molecular weight of the protein increases to 10.7 X 10(6), and the second virial coefficient is reduced. At pH 9.95, the molecular weights with and without 0.02 M CaCl2, are 3.7 X 10(6) and 1.3 X 10(6), respectively; and the effect of Ca++ in reducing the second virial coefficient is much greater than at pH 7.0. These results can be understood on the basis that at pH 7.0, ca++ increases the association of hemocyanin, by binding and intermolecular linkage through the carboxyl groups of protein side chains. At pH 9.95, amino groups are deprotonated and therefore also become available for Ca++ binding. The relative effect of Ca++ in enhancing the association of hemocyanin therefore becomes greater at the higher pH.     

The effect of ionic strength, temperature, and pressure on the interaction potential of dense protein solutions: from nonlinear pressure response to protein crystallization

Understanding the intermolecular interaction potential, V(r), of proteins under the influence of temperature, pressure, and salt concentration is essential for understanding protein aggregation, crystallization, and protein phase behavior in general. Here, we report small-angle x-ray scattering studies on dense lysozyme solutions of high ionic strength as a function of temperature and pressure. We show that the interaction potential changes in a nonlinear fashion over a wide range of temperatures, salt, and protein concentrations. Neither temperature nor protein and salt concentration lead to marked changes in the pressure dependence of V(r), indicating that changes of the water structure dominate the pressure dependence of the intermolecular forces. Furthermore, by analysis of the temperature, pressure, and ionic strength dependence of the normalized second virial coefficient, b2, we show that the interaction can be fine-tuned by pressure, which can be used to optimize b2 values for controlled protein crystallization.     

Evaluation of second osmotic virial coefficients from molecular simulation following scaled-particle theory

ABSTRACT

We report a scaled particle theory-based method for evaluation of second osmotic virial coefficients from molecular simulations of dilute species in solution. In this method, we evaluate the work associated with growing a cavity in solution that is perfectly permeable to the solvent but is completely impermeable to the solutes, thereby establishing an osmotic stress between the cavity interior and exterior. Extrapolating our results to determine the solute concentration in contact with a cavity with an infinite radius, we are able to evaluate the solute osmotic pressure and second osmotic virial coefficient. A finite size correction is introduced to account for the impact of effectively concentrating the solutes in the periphery of the simulation box with increasing cavity size. We demonstrate the utility of the proposed method by evaluating second osmotic virial coefficients for methane in water as a function of temperature. The approach proposed here provides a physically transparent route for calculation of second osmotic virial coefficients by direct interrogation of simulation configurations without having to explicitly evaluate the long-range integral over solute-solute correlations required following McMillan-Mayer theory.     

Accounting for thermodynamic non-ideality in the Guinier region of small-angle scattering data of proteins

Hydrodynamic studies of the solution properties of proteins and other biological macromolecules are often hard to interpret when the sample is present at a reasonably concentrated solution. The reason for this is that solutions exhibit deviations from ideal behaviour which is manifested as thermodynamic non-ideality. The range of concentrations at which this behaviour typically is exhibited is as low as 1–2 mg/ml, well within the range of concentrations used for their analysis by techniques such as small-angle scattering. Here we discuss thermodynamic non-ideality used previously used in the context of light scattering and sedimentation equilibrium analytical ultracentrifugation and apply it to the Guinier region of small-angle scattering data. The results show that there is a complementarity between the radially averaged structure factor derived from small-angle X-ray scattering/small-angle neutron scattering studies and the second virial coefficient derived from sedimentation equilibrium analytical ultracentrifugation experiments.     

Phospholipid interactions in model membrane systems. II. Theory.

We describe statistical thermodynamic theory for the lateral interactions among phospholipid head groups in monolayers and bilayers. Extensive monolayer experiments show that at low surface densities, PC head groups have strong lateral repulsions which increase considerably with temperature, whereas PE interactions are much weaker and have no significant temperature dependence (see the preceding paper). In previous work, we showed that the second virial coefficients for these interactions can be explained by: (a) steric repulsions among the head groups, and (b) a tilting of the P-N+ dipole of PC so that the N+ end enters the oil phase, to an extent that increases with temperature. It was also predicted that PE interactions should be weaker and less temperature dependent because the N+ terminal of the PE head-group is hydrophilic, hence, it is tilted into the water phase, so dipolar contributions among PE's are negligible due to the high dielectric constant of water. In the present work, we broaden the theory to treat phospholipid interactions up to higher lateral surface densities. We generalize the Hill interfacial virial expansion to account for dipoles and to include the third virial term. We show that to account for the large third virial coefficients for both PC and PE requires that the short range lateral attractions among the head groups also be taken into account. In addition, the third virial coefficient includes fluctuating head group dipoles, computed by Monte Carlo integration assuming pairwise additivity of the instantaneous pair potentials. We find that because the dipole fluctuations are correlated, the average triplet interactions do not equal the sum of the average dipole pair potentials. This is important for predicting, the magnitude and the independence of temperature of the third virial coefficients for PC. The consistency of the theory with data of both the second and the third virial coefficients extends the applicability of the head-group model to semiconcentrated monolayers, in agreement with the surface potential data in the foregoing paper.     

Predictive crystallization of ribonuclease A via rapid screening of osmotic second virial coefficients

Important progress has been made in recent years toward developing a molecular-level understanding of protein phase behavior in terms of the osmotic second virial coefficient, a thermodynamic parameter that characterizes pairwise protein interactions. Yet there has been little practical application of this knowledge to the field of protein crystallization, largely because of the difficult and time-consuming nature of traditional techniques for characterizing protein interactions. Self-interaction chromatography has recently been proposed as a highly efficient method for measuring the osmotic second virial coefficient. The utility of the technique is examined in this work by characterizing virial coefficients for ribonuclease A under 59 solution conditions using several crystallization additives, including PEG, sodium chloride, ammonium sulfate, and propanol. The virial coefficient measurements show some counterintuitive trends and shed light on the previous difficulties in crystallizing ribonuclease A. Crystallization experiments at the corresponding solution conditions were conducted by using ultracentrifugal crystallization. Using this methodology, ribonuclease A crystals were obtained under conditions for which the virial coefficients fell within the "crystallization slot." Crystallographic characterization showed that the crystals diffract to high resolution. Metastable crystals were also obtained for conditions outside, but near, the "crystallization slot," and they could also be frozen and used to collect structural information.