Molecular origins of osmotic second virial coefficients of proteins.

The thermodynamic properties of protein solutions are determined by the molecular interactions involving both solvent and solute molecules. A quantitative understanding of the relationship would facilitate more systematic procedures for manipulating the properties in a process environment. In this work the molecular basis for the osmotic second virial coefficient, B22, is studied; osmotic effects are critical in membrane transport, and the value of B22 has also been shown to correlate with protein crystallization behavior. The calculations here account for steric, electrostatic, and short-range interactions, with the structural and functional anisotropy of the protein molecules explicitly accounted for. The orientational dependence of the protein interactions is seen to have a pronounced effect on the calculations; in particular, the relatively few protein-protein configurations in which the apposing surfaces display geometric complementarity contribute disproportionately strongly to B22. The importance of electrostatic interactions is also amplified in these high-complementarity configurations. The significance of molecular recognition in determining B22 can explain the correlation with crystallization behavior, and it suggests that alteration of local molecular geometry can help in manipulating protein solution behavior. The results also have implications for the role of protein interactions in biological self-organization.     

Measurement of osmotic second virial coefficients by zonal size-exclusion chromatography

Numerical simulation of protein migration reflecting linear concentration dependence of the partition isotherm has been used to invalidate a published procedure for measuring osmotic second virial coefficients (B₂₂) by zonal exclusion chromatography. Failure of the zonal procedure to emulate its frontal chromatographic counterpart reflects ambiguity about the solute concentration that should be used to replace the applied concentration in the rigorous quantitative expression for frontal migration; the recommended use of the peak concentration in the eluted zone is incorrect on theoretical grounds. Furthermore, the claim for its validation on empirical grounds has been traced to the use of inappropriate B₂₂ magnitudes as the standards against which the experimentally derived values were being tested.     

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.     

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.     

From osmotic second virial coefficient (B22) to phase behavior of a monoclonal antibody

Antibodies are complex macromolecules and their phase behavior as well as interactions within different solvents and precipitants are still not understood. To shed some light into the processes on a molecular dimension, the occurring self‐interactions between antibody molecules were analyzed by means of the osmotic second virial coefficient (B₂₂). The determined B₂₂ follows qualitatively the phenomenological Hofmeister series describing the aggregation probability of antibodies for the various solvent compositions. However, a direct correlation between crystallization probability and B₂₂ in form of a crystallization slot does not seem to be feasible for antibodies since the phase behavior is strongly dependent on their anisotropy. Kinetic parameters have to be taken into account due to the molecular size and complexity of the molecules. This is confirmed by a comparison of experimental data with a theoretical phase diagram. On the other hand the solubility is thermodynamically driven and therefore the B₂₂ could be used to establish a universal solubility line for the monoclonal antibody mAb04c and different solvent compositions by using thermodynamic models. © 2015 American Institute of Chemical Engineers Biotechnol. Prog., 31:438–451, 2015     

Application of the osmotic virial equation in cryobiology

The multisolute osmotic virial equation is the only multisolute thermodynamic solution theory that has been derived from first principles and can make predictions of multisolute solution behaviour in the absence of multisolute solution data. Other solution theories either (i) include simplifying assumptions that do not take into account the interactions between different types of solute molecules or (ii) require fitting to multisolute data to obtain empirical parameters. The osmotic virial coefficients, which are obtained from single-solute data, can be used to make predictions of multisolute solution osmolality. The osmotic virial coefficients for a range of solutes of interest in cryobiology are provided in this paper, for use with concentration units of both molality and mole fraction, along with an explanation of the background and theory necessary to implement the multisolute osmotic virial equation.     

Fundamental measure theory of hydrated hydrocarbons

To calculate the solvation of hydrophobic solutes, we have developed a method based on the fundamental measure treatment of density functional theory. This method allows us to carry out calculations of density profiles and the solvation energy for various hydrophobic molecules with high accuracy. We have applied the method to the hydration of various hydrocarbons (linear, branched and cyclic). The calculations of the entropic and enthalpic parts are also carried out. We have examined the question of the temperature dependence of the entropy convergence. Finally, we have calculated the mean force potential between two large hydrophobic nanoparticles immersed in water. Proceedings of “Modeling Interactions in Biomolecules II”, Prague, September 5th–9th, 2005.     

Effects of the kinetics of osmotic pressure variation on yeast viability

The variation rate of the osmotic pressure increase was found to have a great effect on the viability of yeasts subjected to hyperosmotic stress. A low intensity of the increase rate of osmotic pressure could maintain an important viability of the cells (about 90 to 100%) even for very high levels of osmotic pressure (about 10(8) Pa). The viability level was found to be highly dependent on the physiological state of the cells: Variations in the properties of the cell membrane were supposed to be involved in such a dependence. (c) 1992 John Wiley & Sons, Inc.     

Measurement of the second osmotic virial coefficient for protein solutions exhibiting monomer-dimer equilibrium

The second osmotic virial coefficient (B) is a measure of solution nonideality that is useful for predicting conditions favorable for protein crystallization and for inhibition of aggregation. Static light scattering is the technique most commonly used to determine B values, typically using protein concentrations less than 5 mg/mL. During static light scattering experiments at low protein concentrations, frequently the protein is assumed to exist either as a single nonassociating species or as a combination of assembly states independent of protein concentration. In the work described here, we examined the limit for ignoring weak reversible dimerization (Kd > or =1 mM) by comparing B values calculated with and without accounting for self-association. Light scattering effects for equilibrium dimer systems with Kd <20 mM and Kd <1 mM will significantly affect apparent B values measured for 20 and 150-kDa proteins, respectively. To interpret correctly light scattering data for monomer-dimer equilibrium systems, we use an expanded coefficient model to account for separate monomer-monomer (B(22)), monomer-dimer (B(23)), and dimer-dimer (B(33)) interactions.     

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.     

Correlation of second virial coefficient with solubility for proteins in salt solutions

In this work, osmotic second virial coefficients (B(22)) were determined and correlated with the measured solubilities for the proteins, α-amylase, ovalbumin, and lysozyme. The B(22) values and solubilities were determined in similar solution conditions using two salts, sodium chloride and ammonium sulfate in an acidic pH range. An overall decrease in the solubility of the proteins (salting out) was observed at high concentrations of ammonium sulfate and sodium chloride solutions. However, for α-amylase, salting-in behavior was also observed in low concentration sodium chloride solutions. In ammonium sulfate solutions, the B(22) are small and close to zero below 2.4 M. As the ammonium sulfate concentrations were further increased, B(22) values decreased for all systems studied. The effect of sodium chloride on B(22) varies with concentration, solution pH, and the type of protein studied. Theoretical models show a reasonable fit to the experimental derived data of B(22) and solubility. B(22) is also directly proportional to the logarithm of the solubility values for individual proteins in salt solutions, so the log-linear empirical models developed in this work can also be used to rapidly predict solubility and B(22) values for given protein-salt systems.     

A simpler analysis for the measurement of second virial coefficients by self-interaction chromatography

We describe a thermodynamic approach that supports the adoption of a simplified procedure for the determination of protein second virial coefficients (B(2)) by self-interaction chromatography. Its major advantage over the original method is a decrease in the number of parameters to which magnitudes must be assigned for the determination of B(2). Improved correlation of virial coefficients obtained by the chromatographic procedure with those obtained by light scattering is achieved by taking into account the twofold larger magnitudes of the former because of the experimental distinction between free and immobilized protein molecules in self-interaction chromatography.     

The protein concentration gradient within eye lens might originate from constant osmotic pressure coupled to differential interactive properties of crystallins

A protein concentration gradient exists from the center to the periphery of most lenses, the origin of which is still a matter of debate. The gradient, which contributes to the lens optical quality, seems to be accompanied by an uneven distribution of the crystallin classes, with the nucleus usually enriched in -and the cortex in -crystallins. Since the osmotic pressure within the lens seems to be constant and since a rather different interaction behaviour of -and -crystallins was demonstrated in previous studies, we propose that the maintenance of a constant osmotic pressure through the lens is sufficient to induce and stabilize a protein concentration gradient. The theoretical treatment has been worked out and the validity of the hypothesis has been demonstrated with colloidal osmotic pressure measurements of lens cortical and nuclear cytoplasmic extracts as a function of protein concentration. To account for the observed lens concentration gradient, however, a small additional concentration gradient in the opposite direction, involving an ion or small molecule, might be required.     

Extraction and analysis of sap from individual wheat leaf cells: the effect of sampling speed on the osmotic pressure of extracted sap

Abstract. A modification to the pressure probe is described which allows very rapid extraction of sap samples from single higher plant cells. The performance of this rapid-sampling probe was assessed and compared with the unmodified probe for cells of both wheat and Tradescantia. Under some conditions, the unmodified probe operated too slowly to avoid dilution of cell sap during the extraction process. This led to values for apparent sample osmotic pressures that were below the turgor pressures for the same cells. The problem was particularly acute in young wheatleaf epidermal cells which are small, elongate and have high turgor pressure. These exhibited rapid water influx when their turgor was depressed during the sampling of their contents (half-time for pressure recovery in wheat cells was less than 1 s while in Tradescantia cells it was 3–5 s). Dilution during sampling was apparently negligible when the rapid sampling probe was used. The study was complemented by a simple model of the way cells dilute during sampling. Quantitative predictions of the model were consistent with our observed findings. The model is used to assess the major factors which determine a cell's susceptibility to dilution during sampling.     

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     

Xylem pressure response in maize roots subjected to osmotic stress: determination of radial reflection coefficients by use of the xylem pressure probe

The absolute pressure in conducting xylem vessels of roots of 2-week-old, slowly transpiring intact maize plants (bathed in nutrition medium) was determined to be +0·024 ± 0·044 MPa using the xylem pressure probe. When the roots were subjected to osmotic stress (NaCI, KCI or sucrose), the xylem pressure decreased immediately and became more negative. However, the response of xylem pressure to osmotic stress was considerably attenuated, indicating that the radial reflection coefficients, σ₁₃ of the maize root for these solutes were rather low (between 0·2 and 0·4 depending on the concentration of the osmoticum). The low values of a, may be caused (partly) by unstirred layer effects. In repeated osmoticum/nutrition regimes a complex pattern of changes in xylem pressure was observed which was apparently linked to the interplay between transpiration and (passive and/or active) solute loading of the xylem. These processes were not observed when the roots were subjected to osmotic stress after excision. In this case, a biphasic response was observed comparable to that found for excised roots using the root pressure probe.     

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.     

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.     

Evolution of osmotic pressure in solid tumors

The mechanical microenvironment of solid tumors includes both fluid and solid stresses. These stresses play a crucial role in cancer progression and treatment and have been analyzed rigorously both mathematically and experimentally. The magnitude and spatial distribution of osmotic pressures in tumors, however, cannot be measured experimentally and to our knowledge there is no mathematical model to calculate osmotic pressures in the tumor interstitial space. In this study, we developed a triphasic biomechanical model of tumor growth taking into account not only the solid and fluid phase of a tumor, but also the transport of cations and anions, as well as the fixed charges at the surface of the glycosaminoglycan chains. Our model predicts that the osmotic pressure is negligible compared to the interstitial fluid pressure for values of glycosaminoglycans (GAGs) taken from the literature for sarcomas, melanomas and adenocarcinomas. Furthermore, our results suggest that an increase in the hydraulic conductivity of the tumor, increases considerably the intratumoral concentration of free ions and thus, the osmotic pressure but it does not reach the levels of the interstitial fluid pressure.     

Effects of accumulation of 3-O-methylglucose on growth and osmotic regulation in Chlorella emersonii

Abstract. The effect of accumulation of 3- O -methylglucose (MG) on growth and steady-stale concentrations of the endogenous osmotic solutes proline and sucrose was studied in Chlorella emersonii grown at external osmotic pressure (II) of 0.08-1.64 MPa. NaCL was used as osmoticum. The total solute content of the cells was manipulated by supplying 2 mol m⁻³ MG for 4 and 48 h. MG accumulated to 50–230 mol m⁻³ within 4h and was not metabolized. Uptake of MG resulted in decreases in concentrations of proline and sucrose, the two solutes mainly responsible for osmotic adaptation of C. emersonii to high II. After 4 or 48 h growth in the presence of MG, the decreases in concentration of proline and sucrose were as predicted from the contribution of MG to the total solute content of the cell.