Thermodynamic modeling of the partitioning of biomolecules in aqueous two-phase systems using a modified Flory–Huggins equation

A modified Flory–Huggins equation accounting for the solvation of polymer molecules by water molecules was used to model the phase behavior of aqueous two-phase systems (ATPS) formed by poly(ethylene glycol) (PEG) and dextran. The parameters of the equation were obtained by fitting experimental equilibrium data either accounting for or disregarding dextran polidispersity. The modified equation was subsequently applied to calculate partition coefficients of biomolecules in these systems. It was found that accounting for polidispersity did not affect significantly the calculated phase equilibrium, but increased the agreement of calculated partition coefficients with experimental data. Further improvement was obtained by using a size dependent interaction parameter for dextran pseudo-components.     

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.     

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.     

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.     

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.     

Characterization and intermolecular interactions of hydroxypropyl guar solutions

Aqueous solutions of guar galactomannan and hydroxypropyl guars (HPG) with different molar substitution (MS) levels were studied using dilute solution viscometry and gel permeation chromatography. When guar is modified to HPG, the added hydroxypropyl groups sterically block the hydrogen bonding sites on the guar backbone and reduce the hydrogen bonding attractions between guar molecules. The effects of molar substitution on the intermolecular interactions are inferred from measurements of the Huggins coefficients, which measure intermolecular interactions in dilute solution, and molecular volumes, which reflect intrachain associations. The behavior can be divided into three regimes: (1) at low MS levels (0 < MS < approximately 0.4), there is a sharp decrease in intermolecular interactions as a function of MS; (2) in the intermediate range ( approximately 0.4 < MS < approximately 1.0), interactions become independent of MS; (3) at high substitution levels (MS > approximately 1.0), the temperature dependence of inter- and intramolecular hydrophobic interactions produces a temperature dependence in the Huggins coefficient and molecular volumes that is not seen at lower substitutions. By acid hydrolysis, HPG samples with a range of molecular weights and consistent polydispersities were obtained. On the basis of these samples, the Mark-Houwink-Sakurada parameters and "characteristic ratio" C(infinity) were evaluated for HPG (MS approximately 0.6) and compared to the values for guar. The HPG chain stiffens as the degree of substitution increases.     

The effect of autoclaving on the dispersibility and stability of three neutral polysaccharides in dilute aqueous solutions

The dispersibility of three neutral polysaccharides, oat β-glucan, detarium xyloglucan and dextran in a dilute water–cadoxen mixture was studied by viscosity measurement. It was found that intrinsic viscosity measurement, with water–cadoxen mixtures as solvents, is a useful tool to distinguish polymer degradation from disruption of supramolecular aggregates. This approach, in conjunction with size exclusion chromatography, was used to study the effects of heat and pressure treatment on the dispersibility and stability of three polysaccharides in aqueous solutions. Autoclaving treatment at 121°C for 15 min may reduce the degree of aggregation. Following autoclaving in aqueous solution, the Huggins constants decreased from 0.66 to 0.42 for oat β-glucan and from 0.63 to 0.56 for detarium xyloglucan. It remains the same for dextran, indicating good solubility of this polymer in water. The current treatment did not cause evident changes in molecular weight and structures to detarium xyloglucan and dextran. However, degradation occurred with oat β-glucan. The Burchard–Stockmayer–Fixman approach was applied to estimate the unperturbed dimension of oat β-glucan and detarium xyloglucan on samples after autoclaving. The characteristic ratio C_∞ was found to be 7.3 for detarium xyloglucan and 4.7 for oat β-glucan, corresponding to the Kratky–Porod persistence lengths of 2.0 and 1.2 nm, respectively.     

Preparation and characterization of molecular weight fractions of guar galactomannans using acid and enzymatic hydrolysis

A procedure is described for the preparation of large amounts of guar galactomannan by acid hydrolysis that yields samples of various molecular weights (MW) with uniform polydispersity. This contrasts with preparation by enzymatic degradation that yields samples with a marked increase in polydispersity and a much broader molecular weight distribution (MWD). Acid hydrolyzed guar samples had a Mark-Houwink-Sakurada (MHS) relationship of [eta]=3.04x10(-4) M(w)(0.747) dl/g and a characteristic ratio of 11.87 as determined by gel permeation chromatography (GPC) and dilute solution viscometry. The Huggins coefficient for degraded guars is much smaller (approximately 0.4) than that of the native guar (approximately 0.79), suggesting a weakening of intermolecular association in guar prepared by acid hydrolysis.     

The factors affecting the compatibility of serum albumin and pectinate in aqueous medium

Compatibility is observed in aqueous solutions of serum albumin and pectin (degree of esterification 57%) at pH levels above the isoelectric point of protein. Both the variation in the pH values from 5 to 8 and the increase of ionic strength from 0·1 to 1·0 do not result in phase separation. These facts enable us to conclude that the affinity in this system is of a nonelectrostatic nature. The interaction of serum albumin and pectinate fractions with different degrees of esterification was studied by light scattering. The negative sign of A₂₄ (the second virial coefficient component of the mixture associated with the interaction between different polymer molecules) means that for any degree of esterification there is affinity between pectin and serum albumin. Information concerning excess thermodynamic functions was obtained from the temperature dependence of light scattering. Mixing microcalorimetry was used for precise measurement of enthalpy. The experimental results indicate that thermodynamic compatibility of serum albumin and pectin is controlled by increase of mixing entropy, which mainly stems from dehydration of biopolymer macromolecules during contact formation.     

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.     

Solution properties of the acrylamide-modified cellulose polyelectrolytes in aqueous solutions

A novel cellulose-based polyelectrolyte (AM-C) containing acylamino (DS = 0.625) and carboxyl (DS = 0.148) groups was homogeneously synthesized from cellulose with acrylamide in NaOH/urea aqueous solutions. Solution properties of AM-C in aqueous solutions were investigated by laser light scattering, rheometry, and viscometry. The results indicated that AM-C could form large aggregates spontaneously in water with or without the addition of salts by the strong hydrogen bonds and electrostatic interaction between acylamino and carboxyl groups. Steady-shear flow study showed a Newtonian behavior of the solutions in the dilute regime while a shear-thinning behavior as the concentration increases. The critical concentration (c_e) for transition from dilute to concentrated solution was determined to be 0.7 wt %. Aqueous solutions of AM-C displayed good thermo-stability, reversible liquid-like characters attributing to the chemical modification. The derivation from Cox-Merz rule at relatively low concentration was related to the co-existence of single chain and large aggregates of AM-C in dilute regime. As the polymer concentration increased, the AM-C system was transformed into a homogeneous entanglement structure, resulting in the disappearance of deviations from the Cox-Merz rule.     

On the molecular weight of pectic substances and its relation to their gel strengths

A study has been made on the possibility of determining the molecular weight M_w, the second virial coefficient A₂ and the radius of gyration by measuring the light scattering of pectin aqueous solutions. The experimental data were processed on a computer by using an algorithm, accounting for the effect of microgel formations in the pectic solution. A tendency towards a decrease in M_w when decreasing the degree of esterification for a certain type of pectin was established and was confirmed by the change in the intrinsic viscosity [η]. This fact is explained by degradation during deesterification.

The dependence of the gel strength of the pectic preparations on their degree of esterification, molecular weight M_w and purity (p%) was also studied as well as the dependence of the gel strength on the intrinsic viscosity [η], the degree of esterification and the purity p%. The passive multifactor experiment method was used to obtain regression models which were tested for an optimum in the factors influencing the gel strength. The optimal value for the degree of esterification was obtained at 57–58% which confirmed the view that within this range the pectin macromolecules possess maximum conformational flexibility.     

Aqueous solutions of native and hydrophobically modified polysaccharides: temperature effect

Amphiphilic polysaccharides, obtained by the attachment of various hydrocarbon groups onto dextran, are studied in aqueous solutions. The viscosity of their aqueous solutions is examined as a function of concentration and temperature in the range 25-65 degrees C. Varying polymer concentration, viscosity follows a polynomial development of Huggins equation in which the coefficients can be calculated from the Huggins constant determined in the dilute domain (Matsuoka-Cowman equation). For all polymers, the solution viscosity follows an Arrhenius-like variation with temperature. The activation energy of the aqueous solutions is determined as a function of polymer concentration and structural characteristics (nature and amount of grafted hydrocarbon groups). The variation of activation energy with polymer concentration is shown to be consistent with predictions based on the Matsuoka-Cowman equation combined with the equation of Andrade. This conclusion is extended to other polysaccharides using data from the literature.     

Second virial coefficient of alpha-crystallin

Light scattering studies were performed on bovine alpha-crystallin measuring the scattering intensities as a function of scattering angle, concentration, and temperature. The data yielded the molecular weight, radius of gyration, and second virial coefficient of alpha-crystallin at different temperatures. The second virial coefficient increased with increasing temperature. Both the enthalpy and entropy of solution of alpha-crystallin are positive. The Flory theta temperature was found to be 271 K.     

pH control of the third virial coefficient of hyaluronate solutions calculated from van der Waals forces by means of the Lifshitz-McLachlan susceptibility theory

Positive third virial coefficients and osmotic coefficients have been calculated for human umbilical cord hyaluronic acid solutions at pHs 6.0, 6.5, 7.0, 7.5, 8.0, and 8.5 and constant ionic strength 0.1. The calculations are based on experimental axial flow birefringence and radial linear dichroism data previously reported and the Lifshitz-McLachlan field theory of van der Waals forces. The second virial coefficients are negative, according to both this analysis and light scattering evidence, and reflect the tendency of hyaluronic acid to associate. This negativity denies the assumption of force additivity required by virial expansion theory.The results are in reasonable agreement with those of light scattering studies, and indicate the extreme nonideality of hyaluronate solutions with a high degree of pH control of osmotic pressure. The data are explained within the context of statistical mechanical and field theories of van der Waals forces, and the osmotic pressure of a solution is related to its optical properties. The numerical method used offers a way of exploring the applicability of modern interparticle force theory to biological systems.     

Effect of sucrose on the thermodynamic incompatibility of different biopolymers

The influence of sucrose on the thermodynamic incompatibility of a number of biopolymers in aqueous solutions has been studied. Three pairs of the biopolymers were chosen: sodium caseinate-ovalbumin, 11S globulin from vicia faba-ovalbumin and sodium caseinate-sodium alginate.

The cosolubility of the biopolymers was investigated at different sucrose concentrations in solution (in the range 0–50% w/v). A big increase in the cosolubility of the biopolymers was observed with increasing sucrose concentration. It was established that the increasing cosolubility of the biopolymers occurs in accordance with the increase in the protein solubility in the aqueous medium on sucrose addition. It was supposed that the same factor provides the basis of both the increase in the solubility of proteins and the cosolubility of the biopolymers in the aqueous medium. The thermodynamic parameters of the different pair interactions (the second virial coefficients) were estimated using light scattering data in the binary and ternary aqueous solutions of the biopolymers without sucrose and on the addition of 25% w/v of sucrose.     

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.     

Rheology and ultrasound scattering from aggregated red cell suspensions in shear flow

The shear flow dynamics of reversible red cell aggregates in dense suspensions were investigated by ultrasound scattering, to study the shear disruption processes of Rayleigh clusters and examine the effective mean field approximation used in microrheological models. In a first section, a rheo-acoustical model, in the Rayleigh scattering regime, is proposed to describe the shear stress dependence of the low frequency scattered power in relation to structural parameters. The fractal scattering regime characterizing the anisotropic scattering from flocs of size larger than the ultrasound wavelength is further discussed. In the second section, we report flow-dependent changes in the low-frequency scattering coefficient in a plane-plane flow geometry to analyze the shear disruption processes of hardened or deformable red cell aggregates in neutral dextran polymer solution. Rheo-acoustical experiments are examined on the basis of the rheo-acoustical model and the effective medium approximation. The ability of ultrasound scattering technique to determine the critical disaggregation shear stress and to give quantitative information on particle surface adhesive energy is analyzed. Lastly, the shear-thinning behavior of weakly aggregated hardened or deformable red cells is described.     

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.