Analysis of the thermodynamic non-ideality of proteins by sedimentation equilibrium experiments

This paper presents a modified method to determine experimentally the second virial coefficient of protein solutions by sedimentation equilibrium experiments. The improvement is based on the possibility of fitting simultaneously up to seven radial concentration distribution curves of solutions with different loading concentrations. The possibility of precise determination of the second virial coefficient allows estimation of the net charge and the excluded volume of a monomeric protein. Application of the method is demonstrated for lysozyme and ovalbumin. In 0.1 M sodium acetate buffer, pH 4.5, the second virial coefficient of hen egg white lysozyme amounts to 24 +/- 1 ml/g. Analysis based on spherical particle theory yield an excluded volume of 3.5 ml/g and a charge dependent value of 20.5 ml/g which is induced by a net charge number of 14.1 +/- 1. Under low salt conditions self-association processes on lysozyme are unfavorable due to electrostatic repulsion. To overcome these repulsive contributions, either a shift to neutral pH or addition of at least 2% NaCl is necessary. In this way the charge dependent contribution decreases below the value responsible for the excluded volume and allows crystallization of the protein. Similar effects can be observed with ovalbumin. The high virial coefficient observed at pH 8.5 is induced by the high net charge number of 27 +/- 1.     

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.     

Interpretation of thermodynamic non-ideality in sedimentation equilibrium experiments on proteins

This investigation re-examines theoretical aspects of the allowance for effects of thermodynamic non-ideality on the sedimentation equilibrium distribution for a single macromolecular solute, and thereby resolves the question of the constraints that pertain to the definition of the activity coefficient term in the basic sedimentation equilibrium expression. Sedimentation equilibrium results for ovalbumin are then presented to illustrate a simple procedure for evaluating the net charge (valence) of a protein from the magnitude of the second virial coefficient in situations where the effective radius of the protein can be assigned. Finally, published sedimentation equilibrium results on lysozyme are reanalysed to demonstrate the feasibility of employing the dependence of the second virial coefficient upon ionic strength to evaluate both the valence and the effective radius of the non-interacting solute.     

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.     

Effect of secondary structure on the potential of mean force for poly-L-lysine in the alpha-helix and beta-sheet conformations

Because poly-L-lysine (PLL) can exist in the alpha-helix or beta-sheet conformation depending on solution preparation and solution conditions, PLL is a suitable candidate to probe the dependence of protein interactions on secondary structure. The osmotic second virial coefficient and weight-average molecular weight are reported from low-angle laser-light scattering measurements for PLL as a function of NaCl concentration, pH, and alpha-helix or beta-sheet content. Interactions between PLL molecules become more attractive as salt concentration increases due to screening of PLL charge by salt ions and at low salt concentration become more attractive as pH increases due to decreased net charge on PLL. The experimental results show that interactions are stronger for the beta-sheet conformation than for the alpha-helix conformation. A spherically-symmetric model for the potential of mean force is used to account for specific interactions not described by DLVO theory and to show how differences in secondary structure affect PLL interactions.     

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.     

Calculation of NMR relaxation, covolume, and scattering-related properties of bead models using the SOLPRO computer program

The hydrodynamic properties of macromolecules and bioparticles, represented by bead models, can be calculated using methods implemented in the computer routine HYDRO. Recently, a new computer routine, SOLPRO, has been presented for the calculation of various SOLution PROperties. These include (1) time-dependent electro-optic and spectroscopic properties related to rotational diffusion, (2) non-dynamic properties like scattering curves, and (3) dimensionless quantities that combine two or more solution properties in a form which depends on the shape of the macromolecule but not on its size. In the present work we describe the inclusion of more of those types of properties in a new version of SOLPRO. Particularly, we describe the calculation of relaxation rates in nuclear magnetic resonance (NMR). For dipolar coupling, given the direction of the dipole the program calculates values of the spectral density, from which the NMR relaxation times can be obtained. We also consider scattering-related properties, namely the distribution of distances for the bead model, which is directly related to the angular dependence of scattered intensity, and the particle's longest distance. We have devised and programmed a procedure to calculate the covolume of the bead model, related to the second virial coefficient and, in general, to the concentration dependence of solution properties. Various shape-dependent dimensionless quantities involving the covolume are calculated. In this paper we also discuss some aspects, namely bead overlapping and hydration, that are not explicitely included in SOLPRO, but should be considered by the user. Received: 25 May 1998 / Revised version: 30 July 1998 / Accepted: 30 July 1998     

The ELLIPS suite of macromolecular conformation algorithms

This paper describes a series of four programmes for the PC based on ellipsoidal representations of macromolecular shape in solution using Universal shape functions. ELLIPS1 is based on simple ellipsoid of revolution models (where two of the three axes of the ellipsoid are fixed equal to each other). If the user types in a value for a shape function from sedimentation or other types of hydrodynamic measurement, it will return a value for the axial ratio of the ellipsoid. ELLIPS2 is based on the more general triaxial ellipsoid with the removal of the restriction of two equal axes. The user enters the three semi-axial dimensions of the molecule or the equivalent two axial ratios and ELLIPS2 returns the value of all the hydrodynamic shape functions. It also works of course for ellipsoids of revolution. ELLIPS3 and ELLIPS4 do the reverse of ELLIPS2, that is they both provide a method for the unique evaluation of the triaxial dimensions or axial ratios of a macromolecule (and without having to guess a value for the so-called „hydration”) after entering at least three pieces of hydrodynamic information: ELLIPS3 requires EITHER the intrinsic viscosity with the second virial coefficient (from sedimentation equilibrium, light scattering or osmometry) and the radius of gyration (from light or x-ray scattering) OR the intrinsic viscosity with the concentration dependence term for the sedimentation coefficient and the (harmonic mean) rotational relaxation time from fluorescence depolarisation measurements. ELLIPS4 evaluates the tri-axial shape of a macromolecule from electro-optic decay based Universal shape functions using another Universal shape function as a constraint in the extraction of the decay constants. Accepted: 1 November 1996     

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.     

Sedimentation-equilibrium and other physicochemical studies on the lysine-rich fraction of calf thymus histones

1. The lysine-rich fraction (Ia+Ib, or f1) of calf thymus histones was isolated as the sulphate by acid extraction. 2. Sedimentation-equilibrium measurements with interference optics showed that this fraction was monodisperse with a molecular weight of 19500±2000. 3. The `apparent molecular weight' calculated from the sedimentation-equilibrium studies varied markedly with concentration. The large second virial coefficient implied by such variation was attributed to the very high charge/mass ratio of this relatively small protein. Estimates of the charge were made from the values of this virial coefficient. 4. The very large value of the virial coefficient explains anomalies in the earlier reports of the molecular weight of this histone and also why the z-average molecular weight can appear to be lower than the weight-average molecular weight. 5. The differences of the specific refractive increments, and the partial specific volumes, between dialysed and undialysed solutions of this histone fraction could also be attributed to its high molecular charge, which was estimated from these differences and agreed, within the expected limits, with the value deduced from the second virial coefficient. 6. Sedimentation-velocity measurements combined with the known molecular weight imply that lysine-rich histone has a high frictional ratio and an extended shape. Optical-rotatory-dispersion measurements indicated that it had a low helical content.     

A General Method for Modeling Macromolecular Shape in Solution: A Graphical (II-G) Intersection Procedure for Triaxial Ellipsoids

A general method for modeling macromolecular shape in solution is described involving measurements of viscosity, radius of gyration, and the second thermodynamic virial coefficient. The method, which should be relatively straightforward to apply, does not suffer from uniqueness problems, involves shape functions that are independent of hydration, and models the gross conformation of the macromolecule in solution as a general triaxial ellipsoid. The method is illustrated by application to myosin, and the relevance and applicability of ellipsoid modeling to biological structures is discussed.     

Evaluation of second osmotic virial coefficients from molecular simulation following scaled-particle theory

ABSTRACT

We report a scaled particle theory-based method for evaluation of second osmotic virial coefficients from molecular simulations of dilute species in solution. In this method, we evaluate the work associated with growing a cavity in solution that is perfectly permeable to the solvent but is completely impermeable to the solutes, thereby establishing an osmotic stress between the cavity interior and exterior. Extrapolating our results to determine the solute concentration in contact with a cavity with an infinite radius, we are able to evaluate the solute osmotic pressure and second osmotic virial coefficient. A finite size correction is introduced to account for the impact of effectively concentrating the solutes in the periphery of the simulation box with increasing cavity size. We demonstrate the utility of the proposed method by evaluating second osmotic virial coefficients for methane in water as a function of temperature. The approach proposed here provides a physically transparent route for calculation of second osmotic virial coefficients by direct interrogation of simulation configurations without having to explicitly evaluate the long-range integral over solute-solute correlations required following McMillan-Mayer theory.     

The self-association of human spectrin at high concentration.

The self-association of purified human spectrin has been studied at sedimentation equilibrium over a wide range of concentration (0-20 g/L) at 30 degrees C and pH 7.5. Coincidence of apparent weight average molecular weight and omega (r) plots as a function of total spectrin concentration indicated that equilibrium was attained and that no significant concentration of solute was incapable of participating in the self-association reaction. Under these conditions, no significant dissociation of the heterodimer to component polypeptide chains could be detected. The behavior of spectrin between 0 and 20 g/L can be described reasonably well by a cooperative isodesmic model, in which the protomer for association is the alpha beta heterodimer. With this model, the equilibrium constant for the heterodimer-tetramer step, K24, is 2 x 10(6) M-1, and K(iso), the equilibrium constant describing all other steps, is approximately 0.2 x 10(6) M-1. The returned value of the second virial coefficient for this model, 1.0 x 10(-7) L mol g-2, is consistent with the lower limit of values calculated for the heterodimer from the charge and Stokes radius of spectrin. On the other hand, the attenuated indefinite association model fails to describe the self-association of spectrin adequately over the range 0-20 g/L. Systematic decreases in the estimates of the second virial coefficient and the equilibrium constants for association beyond the tetramer suggest that the assumption of a single value of the second virial coefficient may not be appropriate for spectrin, and that non-ideality would best be taken into account by consideration of the detailed solution composition.     

Weak interactions govern the viscosity of concentrated antibody solutions: high-throughput analysis using the diffusion interaction parameter

Weak protein-protein interactions are thought to modulate the viscoelastic properties of concentrated antibody solutions. Predicting the viscoelastic behavior of concentrated antibodies from their dilute solution behavior is of significant interest and remains a challenge. Here, we show that the diffusion interaction parameter (k(D)), a component of the osmotic second virial coefficient (B(2)) that is amenable to high-throughput measurement in dilute solutions, correlates well with the viscosity of concentrated monoclonal antibody (mAb) solutions. We measured the k(D) of 29 different mAbs (IgG(1) and IgG(4)) in four different solvent conditions (low and high ion normality) and found a linear dependence between k(D) and the exponential coefficient that describes the viscosity concentration profiles (|R| ≥ 0.9). Through experimentally measured effective charge measurements, under low ion normality where the electroviscous effect can dominate, we show that the mAb solution viscosity is poorly correlated with the mAb net charge (|R| ≤ 0.6). With this large data set, our results provide compelling evidence in support of weak intermolecular interactions, in contrast to the notion that the electroviscous effect is important in governing the viscoelastic behavior of concentrated mAb solutions. Our approach is particularly applicable as a screening tool for selecting mAbs with desirable viscosity properties early during lead candidate selection.     

Evidences of Changes in Surface Electrostatic Charge Distribution during Stabilization of HPV16 Virus-Like Particles

The stabilization of human papillomavirus type 16 virus-like particles has been examined by means of different techniques including dynamic and static light scattering, transmission electron microscopy and electrophoretic mobility. All these techniques provide different and often complementary perspectives about the aggregation process and generation of stabilized virus-like particles after a period of time of 48 hours at a temperature of 298 K. Interestingly, static light scattering results point towards a clear colloidal instability in the initial systems, as suggested by a negative value of the second virial coefficient. This is likely related to small repulsive electrostatic interactions among the particles, and in agreement with relatively small absolute values of the electrophoretic mobility and, hence, of the net surface charges. At this initial stage the small repulsive interactions are not able to compensate binding interactions, which tend to aggregate the particles. As time proceeds, an increase of the size of the particles is accompanied by strong increases, in absolute values, of the electrophoretic mobility and net surface charge, suggesting enhanced repulsive electrostatic interactions and, consequently, a stabilized colloidal system. These results show that electrophoretic mobility is a useful methodology that can be applied to screen the stabilization factors for virus-like particles during vaccine development.     

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.     

Predictive crystallization of ribonuclease A via rapid screening of osmotic second virial coefficients

Important progress has been made in recent years toward developing a molecular-level understanding of protein phase behavior in terms of the osmotic second virial coefficient, a thermodynamic parameter that characterizes pairwise protein interactions. Yet there has been little practical application of this knowledge to the field of protein crystallization, largely because of the difficult and time-consuming nature of traditional techniques for characterizing protein interactions. Self-interaction chromatography has recently been proposed as a highly efficient method for measuring the osmotic second virial coefficient. The utility of the technique is examined in this work by characterizing virial coefficients for ribonuclease A under 59 solution conditions using several crystallization additives, including PEG, sodium chloride, ammonium sulfate, and propanol. The virial coefficient measurements show some counterintuitive trends and shed light on the previous difficulties in crystallizing ribonuclease A. Crystallization experiments at the corresponding solution conditions were conducted by using ultracentrifugal crystallization. Using this methodology, ribonuclease A crystals were obtained under conditions for which the virial coefficients fell within the "crystallization slot." Crystallographic characterization showed that the crystals diffract to high resolution. Metastable crystals were also obtained for conditions outside, but near, the "crystallization slot," and they could also be frozen and used to collect structural information.     

Phospholipid interactions in model membrane systems. II. Theory.

We describe statistical thermodynamic theory for the lateral interactions among phospholipid head groups in monolayers and bilayers. Extensive monolayer experiments show that at low surface densities, PC head groups have strong lateral repulsions which increase considerably with temperature, whereas PE interactions are much weaker and have no significant temperature dependence (see the preceding paper). In previous work, we showed that the second virial coefficients for these interactions can be explained by: (a) steric repulsions among the head groups, and (b) a tilting of the P-N+ dipole of PC so that the N+ end enters the oil phase, to an extent that increases with temperature. It was also predicted that PE interactions should be weaker and less temperature dependent because the N+ terminal of the PE head-group is hydrophilic, hence, it is tilted into the water phase, so dipolar contributions among PE's are negligible due to the high dielectric constant of water. In the present work, we broaden the theory to treat phospholipid interactions up to higher lateral surface densities. We generalize the Hill interfacial virial expansion to account for dipoles and to include the third virial term. We show that to account for the large third virial coefficients for both PC and PE requires that the short range lateral attractions among the head groups also be taken into account. In addition, the third virial coefficient includes fluctuating head group dipoles, computed by Monte Carlo integration assuming pairwise additivity of the instantaneous pair potentials. We find that because the dipole fluctuations are correlated, the average triplet interactions do not equal the sum of the average dipole pair potentials. This is important for predicting, the magnitude and the independence of temperature of the third virial coefficients for PC. The consistency of the theory with data of both the second and the third virial coefficients extends the applicability of the head-group model to semiconcentrated monolayers, in agreement with the surface potential data in the foregoing paper.