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.     

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.     

Modulation of repulsive forces between neurofilaments by sidearm phosphorylation

Recent studies have advanced the notion that the axonal organization of neurofilaments (NFs) is based on mutual steric repulsion between the unstructured "sidearm" domains of adjacent NFs. Here, we present experimental evidence that these repulsive forces are modulated by the degree of sidearm phosphorylation. When NFs are sedimented into a gelatinous pellet, pellet volume falls with increasing ionic strength and enzymatic dephosphorylation; sedimentation of phosphorylated NFs in the presence of divalent cations also dramatically reduces pellet volume. Further, atomic force microscopy imaging of isolated mammalian NFs reveals robust exclusion of colloidal particles from the NF backbone that is reduced at high ionic strength and attenuated when the filaments are enzymatically dephosphorylated. Phosphate-phosphate repulsion on the NF sidearm appears to modulate NF excluded volume in a graded fashion, thereby controlling axonal NF organization through interfilament forces.     

Convection-free boundary sedimentation of dilute protein solutions

Use of the density gradient sedimentation velocity technique appears to be essential for the accurate determination of the mean sedimentation coefficients of dilute protein solutions. When performed on an analytical ultracentrifuge equipped with a photoelectric-scanning-absorption optical system, the density gradient sedimentation velocity technique has been shown to be particularly useful in studying the subunit association-dissociation equilibria of multisubunit enzyme systems. The time factor has been shown to be a major advantage of the density gradient sedimentation velocity technique, as opposed to the sedimentation equilibrium technique, in studying the subunit association-dissociation equilibria of multisubunit enzymes such as rabbit muscle apo-glyceraldehyde-3-phosphate dehydrogenase, which is very unstable in dilute solution.     

Osmotic pressure of DNA solutions and effective diameter of the double helix

A simple osmometer with nuclear filters (polymer films with pores of a preset diameter) were used to measure the osmotic pressure of Col E1 plasmid DNA solutions in the concentration range of 1-4 mg/ml DNA. Linear and open circular DNA forms proved to have the same osmotic pressure within the experimental accuracy. The results of the measurements were used for calculating the second virial coefficient A2 of the solution of DNA segments and the effective chain diameter d eff in the ionic strength range of 10(-2)-0.1 M. As the ionic strength is lowered from 0.1 to 10(-2) M the effective diameter of DNA increases from 80 to 220 A. The results are in rather good agreement with theory and with other experimental data.     

Beyond the second virial coefficient: Sedimentation equilibrium in highly non-ideal solutions

The general theory of sedimentation equilibrium (SE), applicable to mixtures of interacting sedimentable solutes at arbitrary concentration, is summarized. Practical techniques for the acquisition of SE data suitable for analysis are described. Experimental measurements and analyses of SE in concentrated protein solutions are reviewed. The method of non-ideal tracer sedimentation equilibrium (NITSE) is described. Experimental studies using NITSE to detect and quantitatively characterize intermolecular interactions in mixtures of dilute tracer species and concentrated proteins or polymers are reviewed.     

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.     

Analysis of various indefinite self-associations

Two methods have been developed for the analysis of four types of indefinite self-associations. Unlike previous treatments by others, the procedures can be applied to nonideal cases. The two methods were first tested with simulated data. and it was found that one could indeed distinguish between the four types of indefinite self-associations. For a more realistic test, sedimentation equilibrium experiments were performed on solutions of β-lactoglobulin A at 16°C in 0.15 ionic strength acetate buffer, pH 4.65. The self-association of the β-lactoglobulin A was best described by either method as a sequential indefinite self-association having two equilibrium constants and one second virial coefficient.     

Characterization of weak protein dimerization by direct analysis of sedimentation equilibrium distributions: the INVEQ approach

Closer scrutiny has been accorded a recently reported procedure for characterizing weak protein dimerization by sedimentation equilibrium (INVEQ) in which the equilibrium distribution is analyzed as a dependence of radial distance on solute concentration rather than of solute concentration on radial distance. By demonstrating theoretically that the fundamental parameter derived from the analysis is simply the difference between the dimerization constant and the osmotic second virial coefficient for monomer-monomer interaction, this investigation refutes the original claim that independent estimates of these two parameters can be obtained by nonlinear curve fitting of the sedimentation equilibrium distribution. This criticism also applies to conventional analyses of sedimentation distributions by the commonly employed Beckman Origin and NONLIN software. Numerically simulated distributions are then analyzed to demonstrate limitations of the procedure and also to indicate a means of improving the reliability of the returned estimate of the dimerization constant. These features are illustrated by applying the original and revised analytical procedures to a sedimentation equilibrium distribution for alpha-chymotrypsin (pH 4.0, I 0.05 M).     

Direct observation of the self-association of dilute proteins in the presence of inert macromolecules at high concentration via tracer sedimentation equilibrium: theory, experiment, and biological significance.

The technique of tracer sedimentation equilibrium [Rivas, G., et al. (1994) Biochemistry, 2341-2348 (1); Rivas, G., et al. (1996) J. Mol. Recognit. 9, 31-38 (2)] is utilized, together with an extension of the theory of sedimentation equilibrium of highly nonideal solutions [Chatelier and Minton, (1987) Biopolymers 26, 1097-1113 (3)], to characterize the thermodynamic activity and/or the state of association of a dilute, labeled macromolecular solute in the presence of an arbitary concentration of a second, unlabeled macromolecular solute. Experiments are performed on solutions of labeled fibrinogen (0.25-1 g/L) in bovine serum albumin (0-100 g/L) in the presence and absence of divalent cations (Ca(2+), Mg(2+)), and on solutions of labeled tubulin (0.2-0.6 g/L) in dextran (0-100 g/L). It is found that in the absence of the divalent cations, the large dependence of the thermodynamic activity of fibrinogen on BSA concentration is well accounted for by a simple model for steric repulsion. In the presence of the cations and sufficiently large concentrations of BSA (>30 g/L), fibrinogen appears to self-associate to a weight-average molar mass approximately twice that of monomeric fibrinogen. Tubulin appears to self-associate to an extent that increases monotonically with increasing dextran concentration, reaching a weight-average molar mass almost 3 times that of the alphabeta dimer in the presence of 100 g/L dextran. Possible biological ramifications are discussed.     

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.     

Sedimentation equilibrium in macromolecular solutions of arbitrary concentration. II. Two protein components

Relations describing sedimentation equilibrium in solutions containing two macromolecular solute components are derived for the following cases: (1) two nonassociating proteins at arbitrary concentration, (2) one dilute self-associating protein in the presence of a second inert protein at arbitrary concentration, and (3) two proteins at arbitrary concentration that can associate to form a single heterocomplex of arbitrary composition. As in earlier work (R. C. Chatelier and A. P. Minton (1987) Biopolymers, 26, 507–524), the relations are obtained by using scaled particle theory to calculate the thermodynamic activity of each species present at a given radial distance in the centrifuge. The results of numerical simulations of sedimentation equilibrium are presented as the dependence of apparent molecular weights, or apparent weight-average molecular weights, upon solution composition. Semiempirical methods are presented, by means of which the weight-average molecular weights of self- and heteroassociating proteins in highly nonideal solutions may be estimated from experimental data. It is found that the semiempirical methods yield reasonably accurate estimates of the true weight-average molecular weight over a broad range of experimental conditions, providing that the partial specific volumes of two components in a heteroassociating system do not differ by more than about 0.05 mL/g.     

Thermodynamics of antiparallel hairpin-double helix equilibria in DNA oligonucleotides from equilibrium ultracentrifugation

Five highly palindromic DNA dodecamers, four of which may form G-A or I-A purine-purine mispairs at either the 5.8 or 6.7 positions, have been studied at sedimentation equilibrium in the analytical ultracentrifuge. Each DNA oligonucleotide forms an equilibrium mixture of ordered antiparallel hairpin and double-stranded helical structures in solutions of 0.1 or 0.5 M NaCl between 5 and 40 degrees C. The dimeric duplex is favored by conditions of high salt and low temperature. The monomer-dimer equilibrium constants vary from 5 x 10(6) to 5 x 10(3) and are unique for each DNA dodecamer. Analysis of the temperature dependence of the equilibrium constants shows that the double helix to hairpin conversion is driven by a positive entropy change and is associated with an endothermic enthalpy change. The mispair substitutions at the 5.8 positions and the IA(6.7) mispair have the greatest tendency toward hairpin formation and exhibit significantly larger entropy changes than the nonmispaired dGGTACGCGTACC parent sequence and the thermodynamically similar GA(6.7) DNA. The consequences of such hairpin-double helix equilibria must be considered in the interpretation of other kinds of experiments carried out on oligonucleotides at different concentrations.     

Sedimentation equilibrium in macromolecular solutions of arbitrary concentration. I. Self-associating proteins

Relations describing sedimentation equilibrium in solutions of self-associating macromolecules at arbitrary concentration are presented. These relations are obtained by using scaled-particle theory to calculate the thermodynamic activity of each species present at a given radial distance. The results are expected to be valid for solutions of globular proteins under conditions such that interactions between individual solute molecules may be approximated by a hard-particle potential. Sedimentation equilibria in solutions containing either a nonassociating solute or a solute that self-associates according to several different schemes are simulated using the derived relations. The results of these simulations are presented in terms of the dependence of apparent weight-average molecular weight upon solute concentration. Simple empirical relations are presented for estimating the true weight-average molecular weight from the apparent weight-average molecular weight, without reference to any particular self-association scheme. The weight-average molecular weight estimated in this fashion is within a few percent of the true weight-average molecular weight at all experimentally realizable solute concentrations ( < 400 g/L).     

Osmotic Pressure of DNA Solutions and Effective Diameter of the Double Helix

Abstract

A simple osmometer with nuclear filters (polymer films with pores of a preset diameter) were used to measure the osmotic pressure of Col El plasmid DNA solutions in the concentration range of 1–4 mg/ml DNA. Linear and open circular DNA forms proved to have the same osmotic pressure within the experimental accuracy. The results of the measurements were used for calculating the second virial coefficient A ₂ of the solution of DNA segments and the effective chain diameter d _eff in the ionic strength range of 10^?2-0.1 M, As the ionic strength is lowered from 0.1 to 10^?2 M the effective diameter of DNA increases from 80 to 220 A. The results are in rather good agreement with theory and with other experimental data.     

Studies of solute self-association by sedimentation equilibrium: allowance for effects of thermodynamic non-ideality beyond the consequences of nearest-neighbor interactions

A sedimentation equilibrium study of alpha-chymotrypsin self-association in acetate-chloride buffer, pH 4.1 I 0.05, has been used to illustrate determination of a dimerization constant under conditions where thermodynamic non-ideality is manifested beyond the consequences of nearest-neighbor interactions. Because the expressions for the experimentally determinable interaction parameters comprise a mixture of equilibrium constant and excluded volume terms, the assignment of reasonable magnitudes to the relevant virial coefficients describing non-associative cluster formation is essential for the evaluation of a reliable estimate of the dimerization constant. Determination of these excluded volume parameters by numerical integration over the potential-of-mean-force is shown to be preferable to their calculation by approximate analytical solutions of the integral for this relatively small enzyme monomer with high net charge (+10) under conditions of low ionic strength (0.05 M).     

Negative second virial coefficients as predictors of protein crystal growth: evidence from sedimentation equilibrium studies that refutes the designation of those light scattering parameters as osmotic virial coefficients

The effects of ammonium sulphate concentration on the osmotic second virial coefficient (BAA/MA) for equine serum albumin (pH 5.6, 20 degrees C) have been examined by sedimentation equilibrium. After an initial steep decrease with increasing ammonium sulphate concentration, BAA/MA assumes an essentially concentration-independent magnitude of 8-9 ml/g. Such behaviour conforms with the statistical-mechanical prediction that a sufficient increase in ionic strength should effectively eliminate the contributions of charge interactions to BAA/MA but have no effect on the covolume contribution (8.4 ml/g for serum albumin). A similar situation is shown to apply to published sedimentation equilibrium data for lysozyme (pH 4.5). Although termed osmotic second virial coefficients and designated as such (B22), the negative values obtained in published light scattering studies of both systems have been described incorrectly because of the concomitant inclusion of the protein-salt contribution to thermodynamic nonideality of the protein. Those negative values are still valid predictors of conditions conducive to crystal growth inasmuch as they do reflect situations in which there is net attraction between protein molecules. However, the source of attraction responsible for the negative virial coefficient stems from the protein-salt rather than the protein-protein contribution, which is necessarily positive.     

Description of the thermodynamic incompatibility of the guar–dextran aqueous two-phase system by light scattering

The phase diagram of the guar–dextran aqueous two-phase system has been described on the basis of static light scattering measurements in the dilute regime. By determining the molar weight and second virial coefficient from the two single polymers and the second virial cross coefficient from mixtures at constant guar/dextran ratio (either 17/83 or 28/72), the thermodynamic models based on the virial expansion or the Flory–Huggins theory were successfully applied. The second virial coefficient of guar was difficult to estimate with enough accuracy by light scattering and therefore was obtained by adjustment using a simple criterion stating that the calculated spinodal passes through the experimental critical point. The obtained value was within the confidence interval given by light scattering. Virial expansion and Flory–Huggins approaches yielded quite similar theoretical phase diagrams that satisfactorily fitted the experimental one. The slight discrepancies observed on the position of binodals and critical points has been attributed to the polydispersity of guar or the difficulty in extrapolating from the dilute regime to the semidilute one. The slope of the tie-lines was predicted with a good accuracy, especially with the virial expansion model. The fact that both approaches gave such similar results is probably related to the fact that the expressions of chemical potentials are equivalent if the polymer concentrations are low enough. In this particular case, both models are based on excluded volume interactions and equally describe the phase behavior of the guar–dextran aqueous system.     

Second virial coefficients as a measure of protein--osmolyte interactions

The cytoplasm contains high concentrations of cosolutes. These cosolutes include macromolecules and small organic molecules called osmolytes. However, most biophysical studies of proteins are conducted in dilute solutions. Two broad classes of models have been used to describe the interaction between osmolytes and proteins. One class focuses on excluded volume effects, while the other focuses on binding between the protein and the osmolyte. To better understand protein--smolyte interactions, we have conducted sedimentation equilibrium analytical ultracentrifugation experiments using ferricytochrome c as a model protein. From these experiments, we determined the second virial coefficients for a series of osmolytes. We have interpreted the second virial coefficient as a measure of both excluded volume and protein--osmolyte binding. We conclude that simple models are not sufficient to understand the interactions between osmolytes and proteins.     

Analysis of solvent-free ethyl oleate enzymatic synthesis at equilibrium conditions

This work reports experimental equilibrium data for the esterification of pure oleic acid and a fatty acid mixture with ethanol, using an immobilized Candida antarctica B lipase as catalyst. Reactions are performed in a solvent-free system, containing a mixture of substrates and different amounts of distilled water. According to the initial amount of water and the extent of the reaction, one or two liquid phases are present. Therefore, when the equilibrium is achieved, the liquid–liquid and chemical reaction equilibria have to be simultaneously satisfied.

Several reports dealing with enzymatic reactions performed in two-phase systems have found that the value of the reaction equilibrium constant calculated from overall experimental concentrations varies not only with temperature but also with substrate ratio and water content. Although this approach is a valuable way to explore equilibrium shifts in biphasic systems, it is limited to ideal systems with constant partition coefficients. The aim of this work is to consider the biphasic nature of the reactive mixture through a computational procedure that simultaneously takes into account liquid–liquid and reaction equilibria. This approach enables the determination of a classical temperature-dependent thermodynamic equilibrium constant, which accurately fits experimental equilibrium conversions over a wide range of operating conditions.