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.     

Measurement of thermodynamic nonideality arising from volume-exclusion interactions between proteins and polymers

The effective thermodynamic radii of 23 ribosomal proteins from the 50 S subunit have been determined by gel chromatography on Sephadex G-50, thereby supporting the contention that most of the proteins of the 50 S ribosomal unit exhibit reasonably globular structures. To investigate further the usefulness of modelling proteins as spheres, the second virial coefficient describing excluded volume interactions of some ribosomal proteins with two inert polymers, polyethylene glycol (PEG) and dextran, has been determined by gel chromatography and/or sedimentation equilibrium techniques. Protein-polymer excluded volumes obtained with PEG 20000 and Dextran T70 as the space-filling solute are shown to conform reasonably well with a quantitative expression describing interaction between an impenetrable sphere and an ideal Brownian path (K.M. Jansons and C.G. Phillips, J. Colloid Interface Sci., 137 (1990) 75).     

Protein sorption and recovery by hydrogels using principles of aqueous two-phase extraction

Use of the thermodynamic principles of aqueous two-phase extraction (ATPE) to drive protein into a crosslinked gel is developed as a protein isolation and separation technique, and as a protein loading technique for drug delivery applications. A PEG/dextran gel system was chosen as a model system because PEG/dextran systems are widely used in aqueous two-phase extraction and dextran gels (Sephadex(R)) are common chromatographic media. The effects of polymer concentrations and molecular weights, salts, and pH on the partitioning of ovalbumin matched ATPE heuristics and data trends. Gel partition coefficients (Cgel/Csolution) increased with increasing PEG molecular weight and concentration and decreasing dextran concentration (increased gel swelling). The addition of PEG to the buffer solution yielded partition coefficients more than an order of magnitude greater than those obtained in systems with buffer alone, or added salt. A combined salt/PEG system yielded an additional order of magnitude increase. For example, when ovalbumin solution (2.3 mg/mL) was equilibrated with Sephadex(R) G-50 at pH 6.75, the partition coefficients were 0.13 in buffer, 0.11 in buffer with 0.22M KI, 2.3 in 12 wt% PEG-10,000 and 32.0 in 12 wt% PEG-10, 000 with 0.22M KI. The effect of anions and cations as well as ionic strength and pH on the partitioning of ovalbumin also matched ATPE heuristics. Using the heuristics established above, partition coefficients as high as 80 for bovine serum albumin and protein recoveries over 90% were achieved. In addition, the wide range of partition coefficients that were obtained for different proteins suggests the potential of the technique for separating proteins. Also, ovalbumin sorption capacities in dextran were as high as 450 mg/g dry polymer, and the sorption isotherms were linear over a broad protein concentration range.     

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.     

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.     

An approach to the study of phase separation in ternary aqueous systems

1. Simple thermodynamic expressions are used to describe the properties of uncharged binary and ternary polymer solutions, in particular the sedimentation equilibrium of binary systems and the osmotic pressures and `incompatible' phase separations of ternary systems. 2. Sedimentation-equilibrium experiments were performed on four samples of dextran and two of polyethylene glycol. The critical points of the phase diagrams were determined for the mixed solutions of polyethylene glycol–dextran–water and of polyethylene glycol–bovine serum albumin–0·2m-sodium chloride solution. Osmotic pressures were measured on a single-phase mixed solution of a polyethylene glycol and a dextran. By use of the simple thermodynamic expressions consistent values of second virial and interaction coefficients for the materials used were obtained from these experiments. 3. The interpretation of the values of the second virial and interaction coefficients, on the basis of three models of molecular interaction, is discussed.     

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.     

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.     

Direct measurement of protein osmotic second virial cross coefficients by cross-interaction chromatography

The importance of weak protein interactions, such as protein self-association, is widely recognized in a variety of biological and technological processes. Although protein self-association has been studied extensively, much less attention has been devoted to weak protein cross-association, mainly due to the difficulties in measuring weak interactions between different proteins in solution. Here a framework is presented for quantifying the osmotic second virial cross coefficient directly using a modified form of self-interaction chromatography called cross-interaction chromatography. A theoretical relationship is developed between the virial cross coefficient and the chromatographic retention using statistical mechanics. Measurements of bovine serum albumin (BSA)/lysozyme cross-association using cross-interaction chromatography agree well with the few osmometry measurements available in the literature. Lysozyme/alpha-chymotrypsinogen interactions were also measured over a wide range of solution conditions, and some counterintuitive trends were observed that may provide new insight into the molecular origins of weak protein interactions. The virial cross coefficients presented in this work may also provide insight into separation processes that are influenced by protein cross-interactions, such as crystallization, precipitation, and ultrafiltration.     

Application of a thermodynamic model to the prediction of phase separations in freeze-concentrated formulations for protein lyophilization

Many of the compounds considered for use in pharmaceutical formulations demonstrate incompatibilities with other components at high enough concentrations, including pairs of polymers, polymers and salts, or even proteins in combination with polymers, salts, or other proteins. Freeze concentration can force solutions into a region where incompatibilities between solutes will manifest as the formation of multiple phases. Such phase separation complicates questions of the stability of the formulation as well as labile components, such as proteins. Yet, phase separation events are difficult to identify by common formulation screening methods. In this report, we use the osmotic virial expansion model of Edmond and Ogston (1) to describe phase-separating behavior of ternary aqueous polymer solutions. Second osmotic virial coefficients of polyethylene glycol 3350 (PEG) and dextran T500 were measured by light scattering. Assuming an equilibrium between ice and water in the freeze-concentrated solution, a degree of freeze concentration can be estimated, which, when combined with the phase separation spinodal, describes a "phase separation envelope" in which phase separation tendencies can be expected in the frozen solution. The phase separation envelope is bounded at low temperatures by the glass transition temperature of the freeze-concentrated solution. Scanning electron microscopic images and infrared spectroscopy of protein structure are provided as experimental evidence of the phase separation envelope in a freeze-dried system of PEG, dextran, and hemoglobin.     

Effects of inert volume-excluding macromolecules on protein fiber formation. I. Equilibrium models

The equilibrium Oosawa-Asakura model for nucleated assembly of rod-like protein fibers is recast in terms of dimensionless (scaled) quantities. The model is then generalized to treat arbitrarily large deviations from thermodynamic ideality arising from high fractional volume occupancy by an inert protein or polymer. Each state of association of the self-associating protein is modeled as an equivalent rigid convex particle (sphere or spherocylinder) and the crowding species is modeled either as an equivalent sphere or cylindrical rod. The resulting conservation of mass relation is readily solved to yield the fractional abundance of monomer, from which the entire equilibrium distribution of oligomeric species can be calculated, either directly or through the use of an additional scaling relationship. Results indicating the potential effect of volume occupancy on the equilibrium solubility of the self-associating protein and upon the equilibrium distribution of polymer size are presented. It is found that the fractional (logarithmic) change in both solubility and in the breadth of the polymer size distribution scale almost linearly with the fractional (logarithmic) change in the thermodynamic activity of monomer.     

Analysis of thermodynamic non-ideality in terms of protein solvation.

The effects of thermodynamic non-ideality on the forms of sedimentation equilibrium distributions for several isoelectric proteins have been analysed on the statistical-mechanical basis of excluded volume to obtain an estimate of the extent of protein solvation. Values of the effective solvation parameter delta are reported for ellipsoidal as well as spherical models of the proteins, taken to be rigid, impenetrable macromolecular structures. The dependence of the effective solvated radius upon protein molecular mass exhibits reasonable agreement with the relationship calculated for a model in which the unsolvated protein molecule is surrounded by a 0.52-nm solvation shell. Although the observation that this shell thickness corresponds to a double layer of water molecules may be of questionable relevance to mechanistic interpretation of protein hydration, it augurs well for the assignment of magnitudes to the second virial coefficients of putative complexes in the quantitative characterization of protein-protein interactions under conditions where effects of thermodynamic non-ideality cannot justifiably be neglected.     

Protein partitioning in weakly charged polymer-surfactant aqueous two-phase systems

The study includes partitioning of proteins in aqueous two-phase systems consisting of the polymer dextran and the non-ionic surfactant C₁₂E₅ (pentaethylene glycol mono-n-dodecyl ether). In this system a micelle-enriched phase is in equilibrium with a polymer-enriched phase. Charges can be introduced into the micelles by the addition of charged surfactants. The charge of the mixed micelles is easily varied in sign and magnitude independently of pH, by the addition of different amounts of negatively charged surfactant, sodium dodecyl sulphate (SDS), or positively charged surfactant dodecyl trimethyl ammonium chloride (DoTAC). A series of water-soluble model proteins (BSA, β-lactoglobulin, myoglobin, cytochrome c and lysozyme), with different net charges at pH 7.1, have been partitioned in non-charged systems and in systems with charged mixed micelles or charged polymer (dextran sulphate). It is shown that partition coefficients for charged proteins in dextran-C₁₂E₅ systems can be strongly affected by addition of charged surfactants (SDS, DoTAC) or polymer (dextran sulphate) and that the effects are directly correlated to protein net charge.     

Measurement of grouped intracellular solute osmotic virial coefficients

Models of cellular osmotic behaviour depend on thermodynamic solution theories to calculate chemical potentials in the solutions inside and outside the cell. These solutions are generally thermodynamically non-ideal under cryobiological conditions. The molality-based Elliott et al. form of the multi-solute osmotic virial equation is a solution theory which has been demonstrated to provide accurate predictions in cryobiological solutions, accounting for the non-ideality of these solutions using solute-specific thermodynamic parameters called osmotic virial coefficients. However, this solution theory requires as inputs the exact concentration of every solute in the solution being modeled, which poses a problem for the cytoplasm, where such detailed information is rarely available. This problem can be overcome by using a grouped solute approach for modeling the cytoplasm, where all the non-permeating intracellular solutes are treated as a single non-permeating “grouped” intracellular solute. We have recently shown (Zielinski et al., J Physical Chemistry B, 2017) that such a grouped solute approach is theoretically valid when used with the Elliott et al. model, and Ross-Rodriguez et al. (Biopreservation and Biobanking, 2012) have previously developed a method for measuring the cell type-specific osmotic virial coefficients of the grouped intracellular solute. However, the Ross-Rodriguez et al. method suffers from a lack of precision, which—as we demonstrate in this work—can severely impact the accuracy of osmotic model predictions under certain conditions. Thus, we herein develop a novel method for measuring grouped intracellular solute osmotic virial coefficients which yields more precise values than the existing method and then apply this new method to measure these coefficients for human umbilical vein endothelial cells.     

Optimization of bovine serum albumin sorption and recovery by hydrogels

Aqueous two-phase systems are composed of aqueous solutions of either two water-soluble polymers, usually polyethylene glycol (PEG) and dextran (Dx), or a polymer and a salt, usually PEG and phosphate or sulfate. Partitioning of proteins in such systems provides a powerful method for separating and purifying mixtures of biomolecules by extraction. If one of the phase forming polymers is a crosslinked gel, then the solution-controlled gel sorption may be considered as a modification of aqueous two-phase extraction. Since PEG/dextran systems are widely used in aqueous two-phase extraction and dextran gels (Sephadex) are common chromatographic media, we choose a PEG/dextran gel system as a model system in this study. The partitioning behavior of pure bovine serum albumin (BSA) in PEG/dextran gel systems is investigated to see the effects of variations in PEG and NaCl concentrations on the partition coefficient K. By making use of the Box-Wilson experimental design, K is shown to be maximized at 9.8 (%, w/w) PEG and 0.2 M NaCl concentrations, respectively, as 182.     

The indefinite self-association of lysozyme: consideration of composition-dependent activity coefficients

An improved iterative method for computing association constants from sedimentation equilibrium results obtained with self-interacting protein systems is presented which accounts for the composition-dependence of the activity coefficients of all oligomeric species. The method is based on the calculation of virial coefficients from covolume and charge considerations, the statistical mechanical basis of which is discussed in relation to the DLVO theory. The method is applied to results obtained with lysozyme in diethylbarbiturate buffer of pH 8.0 and ionic strength 0.15 at 15°C. It is shown that these results, encompassing a range of total solute concentration up to 19.7 g/liter are consistent with self-association patterns comprising either a monomer-dimer-trimer system or an isodesmic indefinite self-association of the monomer, the latter being favored. A firmer distinction between these possibilities is sought on the basis of the dependence of the weight-average partition coefficient, determined by frontal gel chromatography, on total solute concentration (up to 56.6 g/liter). This analysis accounts for the composition-dependence of the ratio of the activity coefficients of partitioning monomer in mobile and stationary phases. It is concluded that all results are consistent with an indefinite self-association of lysozyme governed by a single association constant of 4.61 × 10² liter/mole.     

Effect of sucrose on the dimerization of alpha-chymotrypsin. Allowance for thermodynamic nonideality arising from the presence of a small inert solute

The space-filling effects of sucrose on the dimerization of alpha-chymotrypsin have been investigated by sedimentation equilibrium studies on the enzyme in acetate-chloride buffer, pH 3.9, I 0.2. From the extent of enhancement of the apparent dimerization constant in the presence of 0.05-0.16 M sucrose, it is concluded that this effect of thermodynamic nonideality finds quantitative explanation in terms of excluded volume. However, the suggested approximation that the radius of an inert small solute would be sufficiently small to be neglected in the calculation of covolumes (D.J. Winzor and P.R. Wills, Biophys. Chem. 25 (1986) 243) has not withstood the more stringent test afforded by the present study of alpha-chymotrypsin dimerization. A value of 0.34 nm for the effective thermodynamic radius of sucrose was inferred from the covolume for self-interaction obtained by frontal gel chromatography on Sephadex G-10 under the conditions of the ultracentrifugal studies. Finally, results of sedimentation equilibrium experiments on alpha-chymotrypsin in the presence of 0.1 M glycerol were also shown to be consistent with interpretation in terms of the model of space-filling effects entailing complete exclusion of small solute from the hydrated protein domain.     

Immobilization of a (dextran-adamantane-COOH) polymer onto beta-cyclodextrin-modified silica

Adamantane-modified compounds are known to form stable complexes with beta-cyclodextrins (beta-CD) by host-guest interactions. In this study, the inclusion complex formed between beta-CD cavities and the adamantane group was evaluated for the elaboration of a cation-exchange support. The synthesis of the chromatographic supports involved three steps: (i) a polymer of beta-CD was grafted to diol-modified silica, (ii) a dextran polymer was modified by both adamantane groups and ionizable COOH functions, (iii) the dextran derivative (Ad-Dex-COOH) was bound to the chromatographic support by complexation between the adamantane groups of the dextran and beta-CD cavities of the support. The polymer immobilization on the beta-CD support was successful as the resulting support exhibited weak cation-exchange properties. The stationary phase was easy to prepare under mild conditions (aqueous media, room temperature) and was quite stable when using aqueous mobile phases. The chromatographic behaviour of model proteins was studied in isocratic elution by examining the effect of salt concentration in the buffer on retention. A mixed retention mode was found for lysozyme, revealing both electrostatic and hydrophobic interactions with the stationary phase.