Virial Coefficient B2 Research Articles

Equations of state for the hard disk fluids

By using the published accurate new virial coefficients B11∼B14 for the hard disk fluids [C. Zhang and B.M. Pettitt, Mol. Phys., 2014, 112, 1427], we here propose a new updated version of Tian-Gui-Mulero equation of state [J.X. Tian, Y.X. Gui and A. Mulero, Phys. Chem. Chem. Phys., 2010, 12, 13597]. Compared with other proposals, the new version stands strongly to be the only which can reproduce the known virial coefficients B2∼B13 at the same time that can describe the relation of the compressibility factor versus the packing fraction for the hard disk fluids with high accuracy.

Studying Excipient Modulated Physical Stability and Viscosity of Monoclonal Antibody Formulations Using Small-Angle Scattering.

Excipients are substances that are added to therapeutic products to improve stability, bioavailability, and manufacturability. Undesirable protein-protein interactions (PPI) can lead to self-association and/or high solution viscosity in concentrated protein formulations that are typically greater than 50 mg/mL. Therefore, understanding the effects of excipients on nonspecific PPI is important for more efficient formulation development. In this study, we used National Institute of Standards and Technology monoclonal antibody (NISTmAb) reference material as a model antibody protein to examine the physical stability and viscosity of concentrated formulations using a series of excipients, by varying pH, salt composition, and the presence of cosolutes including amino acids, sugars, and nonionic surfactants. Small angle X-ray scattering (SAXS) together with differential scanning calorimetry (DSC), dynamic light scattering (DLS), and viscosity measurements were used to obtain various experimental parameters to characterize excipient modulated PPI and bulk solution viscosities. In particular, a good correlation was found between SAXS and DLS/SLS results, suggesting that the use of DLS/SLS is valid for predicting the colloidal stability of NISTmAb in concentrated solutions. Moreover, further analysis of effective structure factor S(q)eff measured from SAXS enabled the dissection of net PPI into hydrodynamic forces due to excluded volume as well as any additional attractive or repulsive interactions with the presence of excipients. It was found that although no denaturation or aggregation of NISTmAb was observed and that the net PPI were repulsive, the use of ionic excipients such as pH and salts leads to increased short-range attraction, whereas the nonionic excipients including sugars, amino acids, and polysorbate surfactants lead to increased repulsive PPI with increasing protein concentration. Results obtained from viscosity measurements showed that the use of excipients can lead to increased solution viscosities at high protein concentrations. The use of S(q)eff, interaction parameter kD, and second virial coefficient B22 as predictors for solution viscosity was also evaluated by comparing the predicted results with the measured viscosities. Although B22 and S(q)eff appeared to be better predictors than kD, disagreement between the predicted and measured results suggests other factors apart from PPI contribute to the bulk rheological properties of concentrated protein solutions.

Protein-protein interactions and water activity coefficients can be used to aid a first excipient choice in protein formulations.

With respect to all biopharmaceuticals marketed to date, monoclonal antibodies represent the largest fraction with more than 48% market share (2012). However, the development of biopharmaceutical formulations is a challenging task, and time-consuming and cost-intensive high-throughput screenings are still state-of-the-art in formulation design. These screening techniques are almost exclusively based on heuristic decisions thus the benefit in terms of mechanistic understanding is often unclear. It requires novel, physical-sound methods to enhance/optimize future formulation development, ideally by understanding molecular interactions in these complex solutions. A suitable and evaluated measure-of-choice to characterize protein-protein interactions in aqueous protein solutions is the second osmotic virial coefficient B22 which can be measured using static light scattering techniques. Furthermore B22 can be modeled/predicted via the extended mxDLVO model for protein-protein interactions in the presence of single excipients and excipient-mixtures. Building up on this approach, giving an additional insight into water-water and water-excipient interactions, the thermodynamic equation-of-state ePC-SAFT is used to calculate water activity coefficients in the presence of excipient-mixtures. Immunoglobulin G (IgG) was chosen as a model protein to predict B22-values for IgG in the presence of model excipient-mixtures (trehalose-NaCl, l-histidine-trehalose, l-histidine-NaCl). The combination of water activity coefficients and B22 allows to quickly identify a first guess on suitable formulation conditions that then can be further evaluated with existing methods/knowledge.

Second osmotic virial coefficients of therapeutic proteins in the presence of excipient-mixtures can be predicted to aid an efficient formulation design

Early- and late-stage formulation development for biopharmaceuticals (e.g. therapeutic proteins) remains a challenging task, especially when one has to select suitable excipients to solubilize and stabilize the protein in solution. State-of-the-art formulation development includes cost-intensive high-throughput screening methods to identify suitable excipients and formulation conditions. These methods often deliver a “working” formulation but unlikely the “optimal” formulation.Within this work, we developed a novel method that allows identifying suitable excipients based on the second osmotic virial coefficient B22 as the measure of choice for protein-protein interactions in aqueous solution in the presence of excipients (salts, sugars, amino acids, etc.). B22 is easily accessible by light scattering methods and advanced thermodynamic models like the extended mxDLVO model. Aqueous immunoglobulin G solutions including sugars, amino acids, surfactants and salts as excipients were used as model systems within this work. Applying the extended mxDLVO, the model allows predicting protein-protein interactions with high accuracy for mixtures of up to three excipients in one formulation. These measurements allow for quick identification of suitable excipients and their concentrations and are giving a greater insight in molecular interactions under the respective formulation conditions. An application of this method in future formulation development has the potential to reduce time-consuming and cost-intensive screening methods and finally lead to optimal formulations.

A Reappraisal of Sedimentation Nonideality Coefficients for the Analysis of Weak Interactions of Therapeutic Proteins

The study of weak or colloidal interactions of therapeutic proteins in different formulations allows prediction and optimization of protein stability. Various biophysical techniques have been applied to determine the second osmotic virial coefficient B2 as it reflects on the macromolecular distance distribution that governs solution behavior at high concentration. In the present work, we exploit a direct link predicted by hydrodynamic theory between B2 and the nonideality of sedimentation, commonly measured in sedimentation velocity analytical ultracentrifugation through the nonideality coefficient of sedimentation, kS. Using sedimentation equilibrium analytical ultracentrifugation for independent measurement of B2, we have examined the dependence of kS on B2 for model proteins in different buffers. The data exhibit the expected linear relationship and highlight the impact of protein shape on the magnitude of the nonideality coefficient kS. Recently, measurements of kS have been considerably simplified allowing higher throughput and simultaneous polydispersity assessment at higher protein concentrations. Thus, sedimentation velocity may offer a useful approach to compare the impact of formulation conditions on weak interactions and simultaneously on higher-order structure of therapeutic proteins.

Reentrant Phase Behavior in Protein Solutions Induced by Multivalent Salts: Strong Effect of Anions Cl- Versus NO3.

In this work, the effects of the two anions Cl- and NO3- on the phase behavior of bovine serum albumin (BSA) in solution with trivalent salts are compared systematically. In the presence of trivalent metal salts, negatively charged proteins such as BSA in solution undergo a reentrant condensation (RC) phase behavior, which has been established for several proteins with chlorides of trivalent salts. Here, we show that replacing Cl- by NO3- leads to a marked change in the phase behavior. The effect is investigated for the two different cations Y3+ and La3+. The salts are thus YCl3, Y(NO3)3, LaCl3, and La(NO3)3. The experimental phase behavior shows that while the chloride salts induce both liquid-liquid phase separation (LLPS) and RC, the nitrate salts also induce LLPS, but RC becomes partial with La(NO3)3 and disappears with Y(NO3)3. The observed phase behavior is rationalized by effective protein-protein interactions which arecharacterized using small-angle X-ray scattering. The results based on the reduced second virial coefficients B2/ B2HS and 1/ I( q → 0) demonstrate that the NO3- salts induce a stronger attraction than the Cl- salts. Overall, the effective attraction, the width of the condensed regime in the RC phase diagram, and the nature of LLPS follow the order LaCl3 < YCl3 < La(NO3)3 < Y(NO3)3. Despite the decisive role of cations in RC phase behavior, isothermal titration calorimetry measurements indicate that replacing anions does not significantly influence the cation binding to proteins. The experimental results observed are discussed based on an "enhanced Hofmeister effect" including electrostatic and hydrophobic interactions between protein-cation complexes.

Calculation of cross second virial coefficients using ab initio intermolecular potential energy surfaces for dimer H2-N2

The ab initio intermolecular pair potentials of dimer H2-N2 were calculated from all constructed orientations, using the level of the theory CCSD(T) and basis sets aug-cc-pVmZ (m = 2, 3, 4, 23, 24, 34). The complete basis sets limit aug-cc-pVmZ (m = 23, 24, 34) were extrapolated by ab initio interaction energies at the level of the theory CCSD(T) with correlation-consistent basis sets aug-cc-pVmZ (m = 2, 3, 4). Then the quantum mechanical results were used for developing two new 5-site intermolecular potential functions by fitting ab initio intermolecular energies of dimer H2-N2. The ab initio fitted intermolecular potentials are used in standard thermodynamic relationships to get the cross second virial coefficients B12(T) of the hydrogen-nitrogen mixture. The calculated results are in good agreement with the experimental data.

Accurate Experimental (p, ρ, and T) Data for the Introduction of Hydrogen into the Natural Gas Grid (II): Thermodynamic Characterization of the Methane–Hydrogen Binary System from 240 to 350 K and Pressures up to 20 MPa

Most of the experimental density data of the methane–hydrogen binary system available at the time of the development of the equation of state for natural gases and related mixtures, GERG-2008, at temperatures above 270 K were limited to hydrogen contents higher than 0.20 (amount-of-substance fraction). On the contrary, for mixtures with hydrogen contents lower than 0.20, experimental data were limited to temperatures below 270 K. This work intends to close the gap and provides accurate experimental (p, ρ, and T) data for three binary mixtures of methane and hydrogen, (0.95 CH4 + 0.05 H2), (0.90 CH4 + 0.10 H2), and (0.50 CH4 + 0.50 H2), at temperatures of 240, 250, 260, 275, 300, 325, and 350 K, thus extending the range of available experimental data to higher temperatures for mixtures with hydrogen contents lower than 0.20 and, accordingly, to lower temperatures for mixtures with hydrogen contents higher than 0.20. The density measurements were performed by using a single-sinker densimeter with magnetic suspension coupling at pressures up to 20 MPa. Experimental data were compared to the corresponding densities calculated from the GERG-2008 and the AGA8-DC92 equations of state, respectively. The experimental data are within the uncertainty of both equations of state, except at the lower temperatures of 240 and 250 K and pressures over 14 MPa for the mixtures with a hydrogen content of 0.05 and 0.10, respectively. The virial coefficients B(T, x) and C(T, x), as well as the second interaction virial coefficient B12(T) for the methane–hydrogen binary system, were also calculated from the experimental data set at temperatures from 240 to 350 K using the virial equation of state.

Theory for the Liquid-Liquid Phase Separation in Aqueous Antibody Solutions.

This study presents the theory for liquid-liquid phase separation for systems of molecules modeling monoclonal antibodies. Individual molecule is depicted as an assembly of seven hard spheres, organized to mimic the Y-shaped antibody. We consider the antibody-antibody interactions either through Fab, Fab' (two Fab fragments may be different), or Fc domain. Interaction between these three domains of the molecule (hereafter denoted as A, B, and C, respectively) is modeled by a short-range square-well attraction. To obtain numerical results for the model under study, we adapt Wertheim's thermodynamic perturbation theory. We use this model to calculate the liquid-liquid phase separation curve and the second virial coefficient B2. Various interaction scenarios are examined to see how the strength of the site-site interactions and their range shape the coexistence curve. In the asymmetric case, where an attraction between two sites is favored and the interaction energies for the other sites kept constant, critical temperature first increases and than strongly decreases. Some more microscopic information, for example, the probability for the particular two sites to be connected, has been calculated. Analysis of the experimental liquid-liquid phase diagrams, obtained from literature, is presented. In addition, we calculate the second virial coefficient under conditions leading to the liquid-liquid phase separation and present this quantity on the graph B2 versus protein concentration.

Higher virial coefficients of water

Virial coefficients B2 to B4 for the non-polarizable TIP4P/2005 model of water and the polarizable BK3 model were computed. An analysis and assessment of available experimental and pseudoexperimental data is carried out. Various forms of the virial expansion are examined with respect to their potential use for the description of steam (working agent in turbines). At higher temperatures (higher densities) the expansion starts exhibiting an irregular behavior indicating that at these conditions the vapor can likely not be viewed as a system made up only of individual molecules, and the occurrence of clusters of different size must be accounted for.

The effect of temperature on the colligative properties of food-grade konjac gum in water solutions.

This research paper presents the results of tests on the colligative properties of konjac gum chains in water solutions. For this purpose, the measurements of osmotic pressure and intrinsic viscosity of aqueous solutions, in the function of konjac gum concentration and temperature were carried out. The applied methods allowed for the determination of the second osmotic virial coefficients B2, which raised with the increase of temperature. It indicate that increase of temperature causes higher affinity of polysaccharide's chains to water. It was determined, that the osmotic average molecular mass of the konjac gum in non-purified solutions increases with temperature (1.07×105-3.80×105g×mol-1). Values of the reduced viscosity linearly increased in range 18-29dL×g for all temperatures. Received values of the Huggins constant (0.81-1.72) lead that water is poor solvent for konjac gum. The theta (θ) conditions were extrapolated for non-purified solutions - 325K (52°C), and interpolated for purified solutions - 307K (34°C). Based on the results of tests using the dynamic light scattering, the values of two main relaxation times (fast - 0.4-1.8ms and slow components - 4300-5500ms) were determined (the Kohlrausch-Williams-Watts). The obtained autocorrelation functions were characteristic for sol type systems or these which indicate a gel-like structure.

Influence of structure properties on protein-protein interactions-QSAR modeling of changes in diffusion coefficients.

Information about protein-protein interactions provides valuable knowledge about the phase behavior of protein solutions during the biopharmaceutical production process. Up to date it is possible to capture their overall impact by an experimentally determined potential of mean force. For the description of this potential, the second virial coefficient B22, the diffusion interaction parameter kD, the storage modulus G', or the diffusion coefficient D is applied. In silico methods do not only have the potential to predict these parameters, but also to provide deeper understanding of the molecular origin of the protein-protein interactions by correlating the data to the protein's three-dimensional structure. This methodology furthermore allows a lower sample consumption and less experimental effort. Of all in silico methods, QSAR modeling, which correlates the properties of the molecule's structure with the experimental behavior, seems to be particularly suitable for this purpose. To verify this, the study reported here dealt with the determination of a QSAR model for the diffusion coefficient of proteins. This model consisted of diffusion coefficients for six different model proteins at various pH values and NaCl concentrations. The generated QSAR model showed a good correlation between experimental and predicted data with a coefficient of determination R2 = 0.9 and a good predictability for an external test set with R2 = 0.91. The information about the properties affecting protein-protein interactions present in solution was in agreement with experiment and theory. Furthermore, the model was able to give a more detailed picture of the protein properties influencing the diffusion coefficient and the acting protein-protein interactions. Biotechnol. Bioeng. 2017;114: 821-831. © 2016 Wiley Periodicals, Inc.

All-dimensional H2-CO potential: Validation with fully quantum second virial coefficients.

We use a new high-accuracy all-dimensional potential to compute the cross second virial coefficient B12(T) between molecular hydrogen and carbon monoxide. The path-integral method is used to fully account for quantum effects. Values are calculated from 10 K to 2000 K and the uncertainty of the potential is propagated into uncertainties of B12. Our calculated B12(T) are in excellent agreement with most of the limited experimental data available, but cover a much wider range of temperatures and have lower uncertainties. Similar to recently reported findings from scattering calculations, we find that the reduced-dimensionality potential obtained by averaging over the rovibrational motion of the monomers gives results that are a good approximation to those obtained when flexibility is fully taken into account. Also, the four-dimensional approximation with monomers taken at their vibrationally averaged bond lengths works well. This finding is important, since full-dimensional potentials are difficult to develop even for triatomic monomers and are not currently possible to obtain for larger molecules. Likewise, most types of accurate quantum mechanical calculations, e.g., spectral or scattering, are severely limited in the number of dimensions that can be handled.

Second Virial Coefficient As Determined from Protein Phase Behavior.

We quantitatively link the macroscopic phase behavior of protein solutions to protein-protein interactions based on a coarse-grained colloidal approach. We exploit the extended law of corresponding states and apply the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory in order to infer the second virial coefficient b2, an integral measure of the interaction potential, from the phase behavior, namely, cloud-point temperature (CPT) measurements under conditions favoring protein crystallization. This determination of b2 yields values that quantitatively agree with the results of static light scattering (SLS) experiments. The strength of the attractions is quantified in terms of an effective Hamaker constant, which accounts for van der Waals attractions as well as non-DLVO forces, such as hydration and hydrophobic interactions. Our approach based on simple lab experiments to determine the CPT in combination with the DLVO theory is expected to facilitate further biophysical research on protein-protein interactions in complex solution environments.

Protein partition coefficients can be estimated efficiently by hybrid shortcut calculations

The extraction of therapeutic proteins like monoclonal antibodies in aqueous two-phase systems (ATPS) is a suitable alternative to common cost intensive chromatographic purification steps within the downstream processing. Thereby the protein partitioning can be selectively changed using a displacement agent (additional salt) in order to allow for a successful purification of the target protein. Within this work a new shortcut strategy for the calculation of protein partition coefficients in polymer-salt ATPS is presented. The required protein-solute (phase-forming component, displacement agent) interactions are covered by the cross virial coefficient B23 measured by composition gradient multi-angle light scattering (CG-MALS). Using this shortcut calculation allows for an efficient determination of the partition coefficients of the target protein immunoglobulin G (IgG) and the impurity human serum albumin (HSA) within PEG-citrate and PEG-phosphate ATPS independently on the protein concentration. We demonstrate that the selection of a suitable displacement agent allowing for a selective purification of IgG from HSA is accessible by B23. Based on the determination of the protein–protein interactions via CG-MALS covered by the second osmotic virial coefficient B22 a further optimization of ATPS preventing protein precipitation is enabled. The results show that our approach contributes to an efficient downstream processing development.

Relation Between Solvent Quality and Phase Behavior of Ternary Mixtures of Polymers and Two Solvents that Exhibit Cononsolvency

The phase boundaries of polymer solutions in mixed solvents can be extremely complex due to the many competing van der Waals and associative interactions that can arise in these ubiquitous and technologically important complex fluids. The present paper focuses specific attention on ternary solutions of polymers (B) dissolved in a mixture of two solvents (A, C) that competitively associate with the polymer. We are particularly concerned with explaining the origin of the peculiar phenomenon of cononsolvency in mixed solvents, where a mixture of two individually good solvents behaves effectively as a poor solvent. Our computations are based on a recently developed generalization of Flory-Huggins theory that incorporates the competitive solvation of a polymer by two associating solvents. On the basis of this framework, we evaluate the limit of polymer phase stability (spinodal curves) and the second osmotic virial coefficient [Formula: see text] as a function of temperature and the composition of the pure solvent mixture that is maintained in osmotic equilibrium with the ternary solution. The calculations provide new insights into the miscibility patterns of ternary A/B/C polymer solutions exhibiting cononsolvency.

Explicit-water theory for the salt-specific effects and Hofmeister series in protein solutions.

Effects of addition of salts on stability of aqueous protein solutions are studied theoretically and the results are compared with experimental data. In our approach, all the interacting species, proteins, ions, and water molecules, are accounted for explicitly. Water molecules are modeled as hard spheres with four off-center attractive square-well sites. These sites serve to bind either another water or to solvate the ions or protein charges. The ions are represented as charged hard spheres, and decorated by attractive sites to allow solvation. Spherical proteins simultaneously possess positive and negative groups, represented by charged hard spheres, attached to the surface of the protein. The attractive square-well sites, mimicking the protein-protein van der Waals interaction, are located on the surface of the protein. To obtain numerical results, we utilized the energy route of Wertheim's associative mean spherical approximation. From measurable properties, we choose to calculate the second virial coefficient B2, which is closely related to the tendency of proteins to aggregate and eventually crystalize. Calculations are in agreement with experimental trends: (i) For low concentration of added salt, the alkali halide salts follow the inverse Hofmeister series. (ii) At higher concentration of added salt, the trend is reversed. (iii) When cations are varied, the salts follow the direct Hofmeister series. (iv) In contrast to the colloidal theories, our approach correctly predicts the non-monotonic behavior of B2 upon addition of salts. (v) With respect to anions, the theory predicts for the B2 values to follow different sequences below and above the iso-ionic point, as also confirmed experimentally. (vi) A semi-quantitative agreement between measured and calculated values for the second virial coefficient, as functions of pH of solution and added salt type and concentration, is obtained.

Accurate experimental (p, ρ, T) values and virial coefficients for the (methane and helium) binary system

This work provides accurate experimental (p, ρ, T) values for three binary mixtures of methane with helium: (0.95 (amount-of-substance fraction) CH4+0.05 He) and (0.90 CH4+0.10 He) at temperatures of (240, 250, and 260)K and (0.50 CH4+0.50 He) from (240 to 400)K. This work is a continuation of a previous one which reported accurate experimental (p, ρ, T) values for the (0.95 CH4+0.05 He) and the (0.90 CH4+0.10 He) binary mixtures over the temperature range from (250 to 400)K. All density measurements were performed by using a single-sinker densimeter with magnetic suspension coupling at pressures up to 20MPa. Experimental values were compared with the corresponding densities calculated from the GERG-2008 and the AGA8-DC92 equations of state, respectively. Deviations from the GERG-2008 are much larger than from the AGA8-DC92 (up to −6.5%). These deviations increase with decreasing temperature, with increasing pressure, and with increasing helium fraction. In contrast, deviations from the AGA8-DC92 are within the 0.5% band. The experimental values were also used to calculate the second and the third virial coefficients, B(T,x) and C(T,x), as well as the second interaction virial coefficient B12(T) for this mixture.

Prediction of Protein Aggregation in High Concentration Protein Solutions Utilizing Protein-Protein Interactions Determined by Low Volume Static Light Scattering

The development of highly concentrated protein formulations is more demanding than for conventional concentrations due to an elevated protein aggregation tendency. Predictive protein-protein interaction parameters, such as the second virial coefficient B22 or the interaction parameter kD, have already been used to predict aggregation tendency and optimize protein formulations. However, these parameters can only be determined in diluted solutions, up to 20 mg/mL. And their validity at high concentrations is currently controversially discussed. This work presents a μ-scale screening approach which has been adapted to early industrial project needs. The procedure is based on static light scattering to directly determine protein-protein interactions at concentrations up to 100 mg/mL. Three different therapeutic molecules were formulated, varying in pH, salt content, and addition of excipients (e.g., sugars, amino acids, polysorbates, or other macromolecules). Validity of the predicted aggregation tendency was confirmed by stability data of selected formulations. Based on the results obtained, the new prediction method is a promising screening tool for fast and easy formulation development of highly concentrated protein solutions, consuming only microliter of sample volumes.

A semi empirical compact equation of state for hard sphere fluids at any density

We propose a new semi empirical expression of the equation of state for a hard sphere fluid which is valid in the disordered phases over the whole density range below and above the freezing point. Starting from the existing numerical results for the virial coefficients, we elaborate a compact expression for the equation of state which is compatible both with the low and medium density behaviour (disordered stable phase) and with the asymptotic high density behaviour (disordered metastable phase). The resulting equation of state has a compact form and exhibits a simple pole at the close random packing density and a double pole at a density equal to 1. That equation of state only depends on the following quantities: the virial coefficients B2, B3 ,B4, which can be exactly computed, the random packing density ξ0, which is imposed by statistical geometry, and the residue of the pole in ξ0. The results are in a fairly good agreement with the numerical data.