Virial Coefficient B2 Research Articles

Langmuir adsorption model to assess the impact of silane coupling on nano-dispersion of silica in SBR

Surface active agents are often used to improve dispersion of nanoparticles. Quantitative correlation between these surface-active molecules and nanoscale dispersion is absent from the literature partly because a quantitative measure of nanoscale dispersion does not exist. Recently, we have developed the Virial-van der Waals method to quantify dispersion in nanocomposites using virial coefficients. In this paper, the Langmuir adsorption model is used to quantify the influence of surface-active agents on nano-scale dispersion in terms of the effective second virial coefficient B2∗. The impact of silane coupling agent on the nano-dispersion and silica aggregate structure in precipitated silica/SBR nanocomposites is demonstrated. It is shown that the higher viscosity SBR matrix led to a greater silica aggregate structural breakup, while lower viscosity matrix improved surface silanization. The isomeric content of the SBR, which impacts the dielectric behavior, impacted whether the system could be modeled through a mean-field or specific interactions. We earlier showed that larger aggregates improve dispersion, and this is reaffirmed in these results. After account is made for aggregate size, nano-scale dispersion improves with the addition of silane coupling agent. The behavior is well modeled using Langmuir monolayer adsorption.

Modelling bi-specific antibodies in aqueous solution

This study presents theoretical results for physico-chemical properties of system of molecules modeling bi-specific antibodies, such as, dual-variable-domain monoclonal antibodies (DVD-Ig) and Fabs-In-Tandem Immunoglobulin (FIT-Ig). These molecules are representatives of the engineered proteins that combine the function and specificity of two monoclonal antibodies. Individual molecules are here depicted as an assembly of nine (or in case of the Fit-Ig eleven) hard spheres, organized to resemble the Y-shaped object. The effects of the increased size, asymmetry, and flexibility of individual molecules on measurable properties of such systems of molecules are investigated. We examined the liquid-liquid phase separation, the second virial coefficient B2, and viscosity under various experimental conditions. The calculations are compared with the data for regular monoclonal antibodies and discussed in view of the experimental results for DVD-Ig solutions available in literature.

Virialized equation of state for warm and dense stellar plasmas in proto-neutron stars and supernova matter

ABSTRACT We present microscopic molecular dynamics simulations including the efficient Ewald sum procedure to study warm and dense stellar plasmas consisting of finite-size ion charges immerse in a relativistic neutralizing electron gas. For densities typical of Supernova matter and crust in a proto-neutron star, we select a representative single ion composition and obtain the virialized equation of state (vEoS). We scrutinize the finite-size and screening corrections to the Coulomb potential appearing in the virial coefficients B2, B3, and B4 as a function of temperature. In addition, we study the thermal heat capacity at constant volume, CV, and the generalized Mayer’s relation i.e. the difference CP − CV with CP being the heat capacity at constant pressure, obtaining clear features signalling the onset of the liquid–gas phase transition. Our findings show that microscopic simulations reproduce the discontinuity in CV, whose value lies between that of idealized gas and crystallized configurations. We study the pressure isotherms marking the boundary of the metastable region before the gaseous transition takes place. The resulting vEoS displays a behaviour where effective virial coefficients include extra density dependence showing a generalized density–temperature form. As an application we parametrize pressure as a function of density and temperature under the form of an artificial neural network showing the potential of machine learning for future regression analysis in more refined multicomponent approaches. This is of interest to size the importance of these corrections in the liquid–gas phase transition in warm and dense plasma phases contributing to the cooling behaviour of early Supernova phases and proto-neutron stars.

Cross Second Virial Coefficients of the H2O-H2 and H2S-H2 Systems from First-Principles.

The cross second virial coefficients B12 for the interactions of water (H2O) with molecular hydrogen (H2) and of hydrogen sulfide (H2S) with H2 were obtained at temperatures in the range from 150 to 2000 K from new intermolecular potential energy surfaces (PESs) for the respective molecule pairs. The PESs are based on interaction energies determined for about 12 000 configurations of each molecule pair employing different high-level quantum-chemical ab initio methods up to coupled cluster with single, double, triple, and perturbative quadruple excitations [CCSDT(Q)]. Furthermore, the interaction energies were corrected for scalar relativistic effects. Both classical and semiclassical values for B12 were extracted from the PESs using the Mayer-sampling Monte Carlo approach. While our results for the H2O-H2 system validate the older first-principles results of Hodges et al. [J. Chem. Phys. 2004, 120, 710-720], B12 for the H2S-H2 system was, to the best of our knowledge, hitherto neither measured experimentally nor predicted from first principles.

An alternative approach to the osmotic second virial coefficient of protein solutions and its application to liquid-liquid phase separation.

The osmotic second virial coefficient B2 is an important parameter to describe the interactions and phase behavior of protein solutions, including colloidal systems and macromolecular solutions. Another key parameter to describe the driving force of the nucleation of a new phase is the supersaturation, which is used in the classical nucleation theory framework and is connected with the favorable contribution in the Gibbs free energy in the bulk solution. In this article, we establish a connection between B2 calculated from small angle x-ray scattering (SAXS) data and the values of B2 obtained from supersaturation measurements using thermodynamics considerations. The values of the second virial coefficient calculated employing this method agree with those determined via SAXS in the region near the liquid-liquid phase separation border for human serum albumin and bovine serum albumin. The general relations adopted are shown to be useful for the estimation of the second virial coefficient B2 for globular proteins, in the proximity of the binodal biphasic coexistent region.

The Relationship between Protein–Protein Interactions and Liquid–Liquid Phase Separation for Monoclonal Antibodies

Being able to predict and control concentrated solution properties for solutions of monoclonal antibodies (mAbs) is critical for developing therapeutic formulations. At higher protein concentrations, undesirable solution properties include high viscosities, opalescence, particle formation, and precipitation. The overall aim of this work is to understand the relationship between commonly measured dilute solution parameters, the reduced osmotic second virial coefficient b22 and the diffusion interaction parameter kD and liquid-liquid phase separation, which occurs at higher protein concentrations. For globular proteins such as lysozyme or γB-crystallin, the location of the liquid-liquid coexistence curve is controlled by the net protein-protein interaction, which is related to b22. Because many mAbs undergo reversible self-association due to forming highly directional interactions, it is not known if b22 can be used as a reliable predictor for LLPS since increasing the anisotropy in the interaction potential causes phase separation to occur at much stonger net protein-protein attractions or lower values of b22. Here, we map the coexistence curves for three mAbs, referred to as COE-01, COE-07, and COE-19, in terms of b22 and kD values. The measurements are carried out at a low salt condition near the pI, where protein-protein interactions are expected to be anisotropic due to the presence of electrostatic attractions, and under salting-out conditions at high ammonium sulfate concentrations, which is expected to reduce the anisotropy by screening electrostatic interactions. We also show that deviations from a linear correlation between b22 and kD can be used as an indicator of reversible self-association. Each of the mAbs under salting-out conditions follows the correlation supporting the hypothesis that protein-protein interactions are nonspecific, while deviations from the correlation occur for COE-01 and COE-19 under low salt conditions indicating the mAbs undergo reversible self-association. For five out of the six conditions, the onset of phase separation, as reflected by the reduced virial coefficient at the critical point b22c occurs in a small window -1.6 > b22c > -2.3, which is similar to what has been observed for lysozyme and for bovine γB-crystallin. Under low salt conditions, b22c ≈ -5.1 for COE-19, which we previously showed to self-associate into small oligomers. Our findings suggest that under conditions where mAb interactions are weakly anisotropic, such as occur at high salt conditions, phase separation will begin to occur in a small window of b22. Deviations from the window can occur when mAbs undergo reversible self-association, although this is not always the case and likely depends upon whether or not highly directional interactions are passivated in the oligomer formation. We expect fitting LLPS measurements to simplified interaction models for mAbs will provide additional insight into the nature of the protein-protein interactions and guide their development for calculating concentrated solution properties.

Cross Second Virial Coefficients of the H2O–H2S and H2O–SO2 Systems from First Principles

The cross second virial coefficients B12 for the interactions of water (H2O) with hydrogen sulfide (H2S) and of water with sulfur dioxide (SO2) were determined at temperatures from 200 to 2000 K employing new intermolecular potential energy surfaces (PESs) for the H2O–H2S and H2O–SO2 molecule pairs. The PESs were fitted to interaction energies calculated for more than 50 000 configurations of each molecule pair using quantum-chemical ab initio methods up to coupled cluster with single, double, and perturbative triple excitations [CCSD(T)] with consideration of relativistic effects. The B12 values were extracted from the PESs both classically and semiclassically using the Mayer-sampling Monte Carlo scheme. In addition, accurate correlations of the final B12 values were used to derive the dilute gas cross isothermal Joule–Thomson coefficients, ϕ12 = B12 – T(dB12/dT). For both investigated systems, Wormald provided the only experimentally based B12 and ϕ12 values available in the literature. While his B12 values, which he obtained from the ϕ12 data with the aid of model potentials, are not in satisfying agreement with the present B12 values, his ϕ12 data, after a reanalysis using more accurate pure-component ϕ values, agree very well with the calculated ϕ12 values of this work.

Excluded volume of the system of hard-core spheres revisited: New insights from computer simulations

The role of excluded volume in the equation of state of hard-sphere system is studied by analytical theory and computer simulations. The excluded-volume parameter b(ρ) introduced in the van der Waals equation is examined as a function of density. A number of approximations are considered and tested against Monte Carlo simulations. Parameter b(ρ) is seen to decline by a factor of 2 over the density range in which the system is still in the fluid state. Partial contributions of two- and three-sphere intersections to b(ρ) are analyzed as a function of density. The resulting series is seen to be converging very slowly. An approximation based on Padé approximants is seen to offer best accuracy at the lowest cost. It involves computation of the three lowest virial expansion coefficients B2,B3 and B4. Numerical tests reveal that this approximation is accurate in the case of ellipsoids and a number of hard-core polyhedra, supporting a potential application in phase equilibria studies of molecules with non-spherical shapes, such as cubes, pear-shaped molecules and so on.

Depletion interaction mediated by semiflexible polymers.

We present a simple mean-field theory to describe the polymer-mediated depletion attraction between colloidal particles that accounts for the polymer's chain stiffness. We find that for fixed polymer radius of gyration and volume fraction, the strength of this attraction increases with increasing chain stiffness in both dilute and semidilute concentration regimes. In contrast, the range of attraction monotonically decreases with chain stiffness in the dilute regime, while it attains a maximum in the semidilute regime. The obtained analytical expressions for the depletion interaction were compared with numerical self-consistent field lattice computations and shown to be in quantitative agreement. From the interaction potential between two spheres, we calculated the second osmotic virial coefficient B2, which appears to be a convex function of chain stiffness. A minimum of B2 as a function of chain stiffness was observed both in the numerical self-consistent field computations and the analytical theory. These findings help explain the general observation that semiflexible polymers are more effective depletants than flexible polymers and give insight into the phase behavior of mixtures containing spherical colloids and semiflexible polymers.

Universal effective interactions of globular proteins close to liquid-liquid phase separation: Corresponding-states behavior reflected in the structure factor.

Intermolecular interactions in protein solutions, in general, contain many contributions. If short-range attractions dominate, the state diagram exhibits liquid-liquid phase separation (LLPS) that is metastable with respect to crystallization. In this case, the extended law of corresponding states (ELCS) suggests that thermodynamic properties are insensitive to details of the underlying interaction potential. Using lysozyme solutions, we investigate the applicability of the ELCS to the static structure factor and how far effective colloidal interaction models can help to rationalize the phase behavior and interactions of protein solutions in the vicinity of the LLPS binodal. The (effective) structure factor has been determined by small-angle x-ray scattering. It can be described by Baxter's adhesive hard-sphere model, which implies a single fit parameter from which the normalized second virial coefficient b2 is inferred and found to quantitatively agree with previous results from static light scattering. The b2 values are independent of protein concentration but systematically vary with temperature and solution composition, i.e., salt and additive content. If plotted as a function of temperature normalized by the critical temperature, the values of b2 follow a universal behavior. These findings validate the applicability of the ELCS to globular protein solutions and indicate that the ELCS can also be reflected in the structure factor.

Effects of Monovalent Salt on Protein-Protein Interactions of Dilute and Concentrated Monoclonal Antibody Formulations.

In this study, we used sodium chloride (NaCl) to extensively modulate non-specific protein-protein interactions (PPI) of a humanized anti-streptavidin monoclonal antibody class 2 molecule (ASA-IgG2). The changes in PPI with varying NaCl (CNaCl) and monoclonal antibody (mAb) concentration (CmAb) were assessed using the diffusion interaction parameter kD and second virial coefficient B22 measured from solutions with low to moderate CmAb. The effective structure factor S(q)eff measured from concentrated mAb solutions using small-angle X-ray and neutron scattering (SAXS/SANS) was also used to characterize the PPI. Our results found that the nature of net PPI changed not only with CNaCl, but also with increasing CmAb. As a result, parameters measured from dilute and concentrated mAb samples could lead to different predictions on the stability of mAb formulations. We also compared experimentally determined viscosity results with those predicted from interaction parameters, including kD and S(q)eff. The lack of a clear correlation between interaction parameters and measured viscosity values indicates that the relationship between viscosity and PPI is concentration-dependent. Collectively, the behavior of flexible mAb molecules in concentrated solutions may not be correctly predicted using models where proteins are considered to be uniform colloid particles defined by parameters derived from low CmAb.

Comparison of Huggins Coefficients and Osmotic Second Virial Coefficients of Buffered Solutions of Monoclonal Antibodies.

The Huggins coefficient kH is a well-known metric for quantifying the increase in solution viscosity arising from intermolecular interactions in relatively dilute macromolecular solutions, and there has been much interest in this solution property in connection with developing improved antibody therapeutics. While numerous kH measurements have been reported for select monoclonal antibodies (mAbs) solutions, there has been limited study of kH in terms of the fundamental molecular interactions that determine this property. In this paper, we compare measurements of the osmotic second virial coefficient B22, a common metric of intermolecular and interparticle interaction strength, to measurements of kH for model antibody solutions. This comparison is motivated by the seminal work of Russel for hard sphere particles having a short-range “sticky” interparticle interaction, and we also compare our data with known results for uncharged flexible polymers having variable excluded volume interactions because proteins are polypeptide chains. Our observations indicate that neither the adhesive hard sphere model, a common colloidal model of globular proteins, nor the familiar uncharged flexible polymer model, an excellent model of intrinsically disordered proteins, describes the dependence of kH of these antibodies on B22. Clearly, an improved understanding of protein and ion solvation by water as well as dipole–dipole and charge–dipole effects is required to understand the significance of kH from the standpoint of fundamental protein–protein interactions. Despite shortcomings in our theoretical understanding of kH for antibody solutions, this quantity provides a useful practical measure of the strength of interprotein interactions at elevated protein concentrations that is of direct significance for the development of antibody formulations that minimize the solution viscosity.

Mitigation of liquid-liquid phase separation of a monoclonal antibody by mutations of negative charges on the Fab surface.

Some monoclonal antibodies undergo liquid–liquid phase separation owing to self-attractive associations involving electrostatic and other soft interactions, thereby rendering monoclonal antibodies unsuitable as therapeutics. To mitigate the phase separation, formulation optimization is often performed. However, this is sometimes unsuccessful because of the limited time for the development of therapeutic antibodies. Thus, protein mutations with appropriate design are required. In this report, we describe a case study involving the design of mutants of negatively charged surface residues to reduce liquid–liquid phase separation propensity. Physicochemical analysis of the resulting mutants demonstrated the mutual correlation between the sign of second virial coefficient B2, the Fab dipole moment, and the reduction of liquid–liquid phase separation propensity. Moreover, both the magnitude and direction of the dipole moment appeared to be essential for liquid–liquid phase separation propensity, where electrostatic interaction was the dominant mechanism. These findings could contribute to a better design of mutants with reduced liquid–liquid phase separation propensity and improved drug-like biophysical properties.

Thermodynamic characterization of the (CO2 + O2) binary system for the development of models for CCS processes: Accurate experimental (p, ρ, T) data and virial coefficients

Continuing our study on (CO2 + O2) mixtures, this work reports new experimental (p, ρ, T) data for two oxygen-rich mixtures with mole fractions x(O2) = (0.50 and 0.75) mol·mol−1, in the temperature range T = (250–375) K and pressure range p = (0.5–20) MPa, using a single-sinker densimeter. Experimental density data were compared to two well-established equation-of-state models: EOS-CG and GERG-2008. In the p, T-range investigated, the EOS-CG gave a better reproduction for the equimolar mixture (x(O2) = 0.5), whereas the GERG-2008 performed significantly better for the oxygen-rich mixture (x(O2) = 0.75). The EOS-CG generally overestimates the density, while the GERG-2008 underestimates it. This complete set of new experimental data, together with previous measurements, is used to calculate the virial coefficients B(T, x) and C(T, x), as well as the second interaction virial coefficient B12(T) for the (CO2 + O2) system.

Rapid identification of tailor-made aqueous two-phase systems for the extractive purification of high-value biomolecules

The industrial production of high-value biomolecules, such as monoclonal antibodies or industrially applicable enzymes (used as biocatalysts), has grown significantly within the last decades. Aqueous two-phase extraction (ATPE) using an aqueous two-phase system (ATPS) has long been shown to be a promising alternative to cost-intensive chromatography for purification of these potent biomolecules. However, state-of-the-art ATPS design is still based on “trial-and-error” screening approaches identifying un-optimized ATPS that often perform below requirements.Within this work, an optimized and enhanced method for the identification of a tailor-made ATPS for the ATPE of high-value biomolecules is presented. Through thermodynamic modeling using the ePC-SAFT equation-of-state, a first choice in phase-former candidates can be identified based on their molecular interactions in solution, leading to optimal water conditions for the biomolecule. Simultaneously, the conformational stability in presence of the phase-former candidates is investigated by measuring the unfolding temperature of the biomolecule to refine this selection. The aggregation propensity is estimated by determining the second osmotic virial coefficient B22, thus the colloidal stability of the biomolecule is taken into account. Based on these results, the partition coefficient of the biomolecule as well as a process window for the ATPE can be estimated. This method was applied to immunoglobulin G (IgG) and allows for a selection of phase-formers offering optimal working conditions regarding stability and minimized aggregation propensity. Moreover, a process window concerning temperature and phase-former concentration for the ATPE was defined allowing for a selective IgG purification.

Ab Initio Intermolecular Potential Energy Surfaces and Cross Second Virial Coefficients for the Dimer N2-NO.

Ab initio intermolecular potential energy surfaces (PES) of N2-NO have been constructed at the level of theory CCSD(T) with the augmented correlation-consistent basis sets aug-cc-pVmZ (with m = 2, 3, 4). The nitrogen in the closed-shell electronic configuration X1Σ+ and nitric oxide in the open-shell electronic configuration A2Σ+ were employed to calculate ab initio intermolecular interaction energies. The two new ab initio 5-site intermolecular pair potentials at the theoretical level CCSD(T)/aug-cc-pVmZ (with m = 4, 24) were developed appropriately and are suitable for N2-NO dimer by using the nonlinear least-squares fitting method combining MIO and Levenberg–Marquardt algorithms. The correlation quality of these two potentials was found to be very good with R2 values in the range of 0.98372 to 0.99775. The cross second virial coefficients B12(T) of the N2-NO dimer were calculated in the temperature range of 100 to 470 K using the two ab initio 5-site potentials. The discrepancies between the calculated results and the experimental data can be acceptable.

Quantifying Protein-Protein Interactions in Molecular Simulations.

Interactions among proteins, nucleic acids, and other macromolecules are essential for their biological functions and shape the physicochemcial properties of the crowded environments inside living cells. Binding interactions are commonly quantified by dissociation constants Kd, and both binding and nonbinding interactions are quantified by second osmotic virial coefficients B2. As a measure of nonspecific binding and stickiness, B2 is receiving renewed attention in the context of so-called liquid–liquid phase separation in protein and nucleic acid solutions. We show that Kd is fully determined by B2 and the fraction of the dimer observed in molecular simulations of two proteins in a box. We derive two methods to calculate B2. From molecular dynamics or Monte Carlo simulations using implicit solvents, we can determine B2 from insertion and removal energies by applying Bennett’s acceptance ratio (BAR) method or the (binless) weighted histogram analysis method (WHAM). From simulations using implicit or explicit solvents, one can estimate B2 from the probability that the two molecules are within a volume large enough to cover their range of interactions. We validate these methods for coarse-grained Monte Carlo simulations of three weakly binding proteins. Our estimates for Kd and B2 allow us to separate out the contributions of nonbinding interactions to B2. Comparison of calculated and measured values of Kd and B2 can be used to (re-)parameterize and improve molecular force fields by calibrating specific affinities, overall stickiness, and nonbinding interactions. The accuracy and efficiency of Kd and B2 calculations make them well suited for high-throughput studies of large interactomes.

Retraction

The development of a precipitation or crystallization step requires knowing the solubility of the target protein and its crystallization behavior in aqueous solutions at different pH, temperatures and in the presence of precipitating agents, especially salts. Within this work, a solubility model for proteins based on the second osmotic virial coefficient B22 is developed. For this, a relation between protein solubility and B22 was combined with the extended DLVO model. This solubility model was then used to model and also predict the protein solubility of lysozyme and monoclonal antibody for different salts, salt concentrations, and pH. The modeled as well predicted B22 and protein solubility data of lysozyme in the presence of sodium chloride and sodium p-toluenesulfonate and of a monoclonal antibody in the presence of ammonium sulfate at different pH shows good agreement with experimental data. This article is protected by copyright. All rights reserved.

Relationship of PEG-induced precipitation with protein-protein interactions and aggregation rates of high concentration mAb formulations at 5 °C

Native protein-protein interactions can play an important role in determining the tendency of monoclonal antibodies (mAbs) to aggregate under storage conditions. In this context, phase separation of mAb solutions induced by the addition of neutral polymers such as poly(ethylene glycol) (PEG) represents a simple method to assess the tendency of proteins to self-associate in the native state. Here, we investigated their relationships between PEG-induced phase separation, protein-protein interactions and long-term aggregation rate of several formulations of four mAbs at 100 mg/mL and 5 °C over 12 weeks of storage.We observed that the location of the phase boundary correlated well with the osmotic second virial coefficient B22 determined in absence of the polymer, indicating that for our solutions PEG primarily leads to depletion forces between protein molecules, which are additive to protein-protein interactions.However, limited correlation between aggregation rate at 5 °C and phase behavior was observed across different mAbs, pH values and ionic strengths, indicating that colloidal stability is not the only determinant of aggregation even at such low temperature and high protein concentration. Our results contribute to the growing realization that aggregation propensity in the context of antibody developability is a complex feature, which depends on a variety of biophysical properties rather than one single parameter.

An EOS for the Lennard-Jones fluid: A virial expansion approach

A large number (>30 000) of Monte Carlo simulations in range of 0.002–1.41 ρ* and T* ≤ 25 (* for reduced, dimensionless) was performed, producing a dense grid of state points for the internal energy U* and pressure p*. The dense grid in ρ* allows the direct integration to obtain the Helmholtz free energy F*. The results in U*, p*, and F* were used to fit an equations of state (EOS) for the Lennard-Jones fluid using the virial thermal coefficients B2–B6 taken from the literature and additional empirical coefficients (C7-C16), which correct the errors due to nonconverging behavior of virial thermal coefficients. Those additional coefficients have the same mathematical form as the virial thermal coefficients. The EOS allows an extrapolation to extreme conditions above T* > 100 and ρ* > 2.