Direct analysis of solute self-association by sedimentation equilibrium

An improved procedure is described for the characterization of solute self-association by sedimentation equilibrium. Whereas previous statistical-mechanical approaches to allowance for the effects of thermodynamic nonideality have entailed tedious iteration because of their specification of activity coefficients in terms of the equilibrium concentrations of all species, such reliance upon knowledge of the solution composition is avoided by the adaptation of an alternative statistical-mechanical formulation [T. L. Hill and Y. D. Chen (1973) Biopolymers, Vol. 12, pp. 1285–1312] in which thermodynamic nonideality is expressed in terms of total solute concentration. The development of an analysis in terms of a relationship with total solute concentration as the experimental variable allows this attribute of the Adams-Fujita approach to be retained without sacrifice of statistical-mechanical rigor. Its use is illustrated by application to Rayleigh interferometric records of sedimentation equilibrium distributions reflecting α-chymotrypsin dimerization and lysozyme self-association. © 1996 John Wiley & Sons, Inc.     

The determination of density and molecular weight distributions of lipoproteins by sedimentation equilibrium.

A method is presented by which an experimental record of total concentration as a function of radial distance, obtained in a sedimentation equilibrium experiment conducted with a noninteracting mixture in the absence of a density gradient, may be analyzed to obtain the unimodal distributions of molecular weight and of partial molar volume when these vary concomitantly and continuously. Particular attention is given to the caracterization of classes of lipoproteins exhibiting Gaussian distributions of these quantities, although the analysis is applicable to other types of unimodal distribution. Equations are also formulated permitting the definition of the corresponding distributions of partial specific volume and of density. The analysis procedure is based on a method (employing Laplace transforms) developed previously, but differs from it in that it avoids the necessity of differentiating experimental results, which introduces error. The method offers certain advantages over other procedures used to characterize and compare lipoprotein samples (exhibiting unimodal distributions) with regard to the duration of the experiment, economy of the sample, and, particularly, the ability to define in principle all of the relevant distributions from one sedimentation equilibrium experiment and an external measurement of the weight average partial specific volume. These points and the steps in the analysis procedure are illustrated with experimental results obtained in the sedimentation equilibrium of a sample of human serum low density lipoprotein. The experimental parameters (such as solution density, column height, and angular velocity) used in the conduction of these experiments were selected on the basis of computer-simulated examples, which are also presented. These provide a guide for other workers interested in characterizing lipoproteins of this class.     

Effect of sucrose on the dimerization of alpha-chymotrypsin. Allowance for thermodynamic nonideality arising from the presence of a small inert solute

The space-filling effects of sucrose on the dimerization of alpha-chymotrypsin have been investigated by sedimentation equilibrium studies on the enzyme in acetate-chloride buffer, pH 3.9, I 0.2. From the extent of enhancement of the apparent dimerization constant in the presence of 0.05-0.16 M sucrose, it is concluded that this effect of thermodynamic nonideality finds quantitative explanation in terms of excluded volume. However, the suggested approximation that the radius of an inert small solute would be sufficiently small to be neglected in the calculation of covolumes (D.J. Winzor and P.R. Wills, Biophys. Chem. 25 (1986) 243) has not withstood the more stringent test afforded by the present study of alpha-chymotrypsin dimerization. A value of 0.34 nm for the effective thermodynamic radius of sucrose was inferred from the covolume for self-interaction obtained by frontal gel chromatography on Sephadex G-10 under the conditions of the ultracentrifugal studies. Finally, results of sedimentation equilibrium experiments on alpha-chymotrypsin in the presence of 0.1 M glycerol were also shown to be consistent with interpretation in terms of the model of space-filling effects entailing complete exclusion of small solute from the hydrated protein domain.     

Polymerization pattern of insulin at pH 7.0.

Sedimentation equilibrium results, obtained with bovine zinc-free insulin (with and without a component of proinsulin) at pH 7.0, I o.2, 25 degrees C, and up to a total concentration of 0.8 g/l., are shown to be consistent with three different polymerization patterns, all involving an isodesmic indefinite self-association of specified oligomeric species. The analysis procedure, based on closed solutions formed by summing infinite series, yields for each pattern a set of equilibrium constants, It is shown that a distinction between the possible patterns can be made by analyzing sedimentation equilibrium results obtained in a higher total concentration range (up to 4 g/1.) with insulin freed of zinc and proinsulin, account being taken of the composition dependence of activity coefficients. The favored pattern, which differs from that previously reported in the literature, involves the dimerization of monomeric insulin (mol wt 5734), governed by a dimerization constant of 11 X 10(4) M-1 and the isodesmic indefinite self-association of the dimer, described by an association constant of 1.7 X 10(4) M-1. This polymerization pattern is also shown to be consistent with the reaction boundary observed in sedimentation velocity experiments.     

Analyses of sedimentation equilibrium data

A numerical procedure is presented which can quite adequately compute the molecular weight averages as a function of solute concentration from sedimentation equilibrium data for homogeneous systems and for monomer-dimer associating systems with a possible extension to heterogeneous systems where monotonic variation in the weight average molecular weight is observed such as in weakly associating or dissociating systems. The procedure utilizes the method of orthogonal polynomials for curve fitting which allows for a rapid determination of best fit with minimal round off error. The procedure is particularly applicable in cases where the concentration of solute at the meniscus can be considered to be neither appreciable and reasonably well determined as in low speed sedimentation equilibrium experiments, nor essentially zero as in high speed sedimentation equilibrium experiments where the calculations become somewhat more simplified. The use of moderate speed sedimentation equilibrium has the advantage of providing a more broad concentration distribution in the centrifuge cell which yields more extensive information concerning dissociating systems yet still provides results at low solute concentrations where most solutes can be considered to be behaving ideally.     

A parameterized overspeeding method for the rapid attainment of low-speed sedimentation equilibrium

The approach of a solution of dilute, monodisperse, globular macromolecules to low-speed sedimentation equilibrium in an ultracentrifuge is simulated by numerical integration of the Lamm equation. Various combinations of overspeed time and angular velocity are used to assess the conditions needed to minimize the time it takes the solution to attain sedimentation equilibrium. The optimal overspeeding time and angular velocity are determined over a wide range of values of the molecular weight (relative molar mass) of the solute and the radial distance between the meniscus and base of the solution. The results may be expressed as simple functions which allow facile calculation of (a) the optimal overspeeding time and velocity, and (b) the time required to reach sedimentation equilibrium. The results are in reasonable agreement with previous analytical solutions which were based on several simplifying assumptions. The parameterized overspeeding procedure is shown to be robust over a wide range of conditions, and typically leads to a greater than 5-fold reduction in centrifugation time.     

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).     

The sedimentation equilibrium of heterogeneously associating systems and mixtures of non-interacting solutes: analysis without determination of molecular weight averages

Sedimentation equilibrium is first considered of a system in which a ligand of any size binds to an acceptor at p sites, the experimental result, obtained with either interference or absorption optics, being a distribution of total solute concentration as a function of radial distance. Theory illustrated by a numerical example, is presented which shows that this distribution may be analysed to give the activity of the unbound ligand as a function of total weight concentration. It is shown that this information may be used together with conservation of mass equations written in terms of the initial mixing composition to evaluate the equilibrium constant(s) relevant to the system. Correlation with composition evaluation by use of absorption optics (when possible) is also discussed. The procedure does not involve solution of simultaneous equations which are sums of exponentials nor differentiation of experimental results to obtain apparent weight-average molecular weights. It is general in that it leads to the evaluation of the activity of the species characterized by the smallest M(l-$\text{v}$π) product and, accordingly, is shown to be useful in the analysis of non-interacting as well as of interacting systems.     

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.     

Self-association of alpha-chymotrypsin at low ionic strength in the vicinity of its pH optimum.

The self-association of alpha-chymotrypsin and its di-isopropyl phosphoryl derivative in in I0.03 sodium phophate buffer, pH7,9, was investigated by velocity sedimentation, equilibrium sedimentation and difference gel chromatography. No differences between the native and chemically modified enzyme were observed in the ultracentrifuge studies, and only a marginal (0.6%) difference in weight-average elution volume was detected by difference gel chromatography of 5g/litre solutions on Sephadex G-75. From quantitative analyses of sedimentation velocity and sedimentation-equilibrium distributions obtained with iPr2P (di-isopropylphosphoryl)-chymotrypsin, the polymerizing system is postulated to involve an indefinite association of dimer (with an isodesmic association constant of 0.68 litre/g) that is formed by a discrete dimerization step with equilibrium constant 0.25 litre/g. In addition to providing the best fit of the experimental results, this model of chymotrypsin polymerization at low ionic strength is also consistent with an earlier observation that dimer formation is a symmetrical head-to-head phenomenon under conditions of higher ionic strength (I0.29, pH7.9) where association is restricted to a monomer-dimer equilibrium. It is proposed that the dimerization process is essentially unchanged by variation in ionic strength at pH7.9, and that higher polymers are formed by an entirely different mechanism involving largely electrostatic interactions between dimeric species.     

Interactions of lens proteins. Self-association and mixed-association studies of bovine alpha-crystallin and gamma-crystallin

Concentrated solutions of calf alpha-crystallin (up to 45 g/l) and gamma-crystallin (up to 67 g/l) were subjected to frontal exclusion chromatography at pH 7.3, ionic strength 0.17 and 20 degrees C. The experimental concentration dependence of the weight-average partition coefficient was compared with theoretical expressions, which include considerations of thermodynamic non-ideality effects, for the concentration dependence of a single solute and of a solute undergoing reversible self-association. Two types of association pattern were examined, discrete dimerization and indefinite self-association. The partition chromatography results are consistent with an indefinite self-association of gamma-crystallin, governed by an isodesmic association constant of 6.7 X 10(-3) l/g. alpha-Crystallin appears to self-associate either very weakly, with a maximal association constant of 0.9 X 10(-3) l/g, or not at all; the distinction depends on the assessment of the non-ideality coefficients. The consequences of excluded volume effects on these self-association equilibria at high total protein concentration are discussed. Mixtures of alpha-crystallin and gamma-crystallin were analyzed by frontal exclusion chromatography (up to 14 g/l) and sedimentation velocity (up to 115 g/l): no interaction was observed.     

Comparison of Methods for Characterizing Nonideal Solute Self-Association by Sedimentation Equilibrium

We have examined in detail analytical solutions of expressions for sedimentation equilibrium in the analytical ultracentrifuge to describe self-association under nonideal conditions. We find that those containing the radial dependence of total solute concentration that incorporate the Adams-Fujita assumption for composition-dependence of activity coefficients reveal potential shortcomings for characterizing such systems. Similar deficiencies are shown in the use of the NONLIN software incorporating the same assumption about the interrelationship between activity coefficients for monomer and polymer species. These difficulties can be overcome by iterative analyses incorporating expressions for the composition-dependence of activity coefficients predicted by excluded volume considerations. A recommendation is therefore made for the replacement of current software packages by programs that incorporate rigorous statistical-mechanical allowance for thermodynamic nonideality in sedimentation equilibrium distributions reflecting solute self-association.     

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.     

Allowance for effects of thermodynamic nonideality in sedimentation equilibrium distributions reflecting protein dimerization

This reexamination of a high-speed sedimentation equilibrium distribution for α-chymotrypsin under slightly acidic conditions (pH 4.1, I(M) 0.05) has provided experimental support for the adequacy of nearest-neighbor considerations in the allowance for effects of thermodynamic nonideality in the characterization of protein self-association over a moderate concentration range (up to 8 mg/mL). A widely held but previously untested notion about allowance for thermodynamic nonideality effects is thereby verified experimentally. However, it has also been shown that a greater obstacle to better characterization of protein self-association is likely to be the lack of a reliable estimate of monomer net charge, a parameter that has a far more profound effect on the magnitude of the measured equilibrium constant than any deficiency in current procedures for incorporating the effects of thermodynamic nonideality into the analysis of sedimentation equilibrium distributions reflecting reversible protein self-association.     

Low temperature solution behaviour of Methylophilus methylotrophus electron transferring flavoprotein: a study by analytical ultracentrifugation

The solution behaviour of electron transferring flavoprotein (ETF) from Methylophilus methylotrophus was investigated at low temperature (4 °C) by analytical ultracentrifugation. The concentration dependence of the apparent weight average molecular weight, M_w,app, established the existence of the protein in heterodimeric state (M = 63,700 Da), but also signified the possible dissociation of the heterodimer at lower concentrations into its constituent subunits (M = 28,900 Da and 33,700 Da, together with FAD and AMP cofactors of collective M = 1120 Da). This similarity in subunit size allows approximate quantification of the dissociation in terms of expressions for a monomer-dimer equilibrium. The dissociative behaviour was confirmed by determination of the point average molecular weight, M_w,app(r), as a function of the ETF concentration, c(r), throughout the sedimentation equilibrium distributions obtained with loading concentrations of 0.4 and 0.7 mg/ml. By means of the recently formulated ``psi' procedure for direct analysis of solute self-association a value of (1.5 ± 0.1) μM has been obtained for the dissociation constant K_d. Sedimentation velocity experiments yielded an estimate of the heterodimer sedimentation coefficient, s⁰ _20,w, of (4.5 ± 0.2) S which for M = 63,700 Da suggests a globular structure. Received: 29 November 1996 / Accepted: 2 December 1996     

Effect of large refractive index gradients on the performance of absorption optics in the Beckman XL-A/I analytical ultracentrifuge: an experimental study

An analytical centrifuge cell was modified to detect refraction of light transmitted through the cell caused by refractive index gradients formed by sedimenting solute during centrifugation. Sedimentation velocity and sedimentation equilibrium experiments were carried out in this cell on solutions containing high concentrations of protein and polysaccharide in a Beckman-Coulter XLA analytical ultracentrifuge. Analysis of the results indicates that in the absence of an optical artifact easily recognized as a "black band," the dependence of apparent absorbance upon radial position reported by the instrument may be considered a reliable measure of the solute concentration gradient.     

Boundary analysis in sedimentation transport experiments: a procedure for obtaining sedimentation coefficient distributions using the time derivative of the concentration profile.

A procedure is described for computing sedimentation coefficient distributions from the time derivative of the sedimentation velocity concentration profile. Use of the time derivative, (delta c/delta t)r, instead of the radial derivative, (delta c/delta r)t, is desirable because it is independent of time-invariant contributions to the optical baseline. Slowly varying baseline changes also are significantly reduced. An apparent sedimentation coefficient distribution (i.e., uncorrected for the effects of diffusion), g*(s), can be calculated from (delta c/delta t)r as [formula: see text] where s is the sedimentation coefficient, omega is the angular velocity of the rotor, c0 is the initial concentration, r is the radius, rm is the radius of the meniscus, and t is time. An iterative procedure is presented for computing g*(s)t by taking into account the contribution to (delta c/delta t)r from the plateau region to give (delta c/delta t)corr. Values of g*(s)t obtained this way are identical to those of g*(s) calculated from the radial derivative to within the roundoff error of the computations. Use of (delta c/delta t)r, instead of (delta c/delta r)t, results in a significant increase (greater than 10-fold) in the signal-to-noise ratio of data obtained from both the uv photoelectric scanner and Rayleigh optical systems of the analytical ultracentrifuge. The use of (delta c/delta t)r to compute apparent sedimentation coefficient distributions for purposes of boundary analysis is exemplified with an antigen-antibody system.     

On the analysis of protein self-association by sedimentation velocity analytical ultracentrifugation

Analytical ultracentrifugation is one of the classical techniques for the study of protein interactions and protein self-association. Recent instrumental and computational developments have significantly enhanced this methodology. In this paper, new tools for the analysis of protein self-association by sedimentation velocity are developed, their statistical properties are examined, and considerations for optimal experimental design are discussed. A traditional strategy is the analysis of the isotherm of weight-average sedimentation coefficients s(w) as a function of protein concentration. From theoretical considerations, it is shown that integration of any differential sedimentation coefficient distribution c(s), ls-g(*)(s), or g(s(*)) can give a thermodynamically well-defined isotherm, as long as it provides a good model for the sedimentation profiles. To test this condition for the g(s(*)) distribution, a back-transform into the original data space is proposed. Deconvoluting diffusion in the sedimentation coefficient distribution c(s) can be advantageous to identify species that do not participate in the association. Because of the large number of scans that can be analyzed in the c(s) approach, its s(w) values are very precise and allow extension of the isotherm to very low concentrations. For all differential sedimentation coefficients, corrections are derived for the slowing of the sedimentation boundaries caused by radial dilution. As an alternative to the interpretation of the isotherm of the weight-average s value, direct global modeling of several sedimentation experiments with Lamm equation solutions was studied. For this purpose, a new software SEDPHAT is introduced, allowing the global analysis of several sedimentation velocity and equilibrium experiments. In this approach, information from the shape of the sedimentation profiles is exploited, which permits the identification of the association scheme and requires fewer experiments to precisely characterize the association. Further, under suitable conditions, fractions of incompetent material that are not part of the reversible equilibrium can be detected.     

Adair was right in his time

The subunit molar mass of hemoglobin was established in the 19th century by chemical analysis, the tetramer structure by osmotic pressure determination in 1924 and by the newly developed analytical ultracentrifuge in 1926, which became a powerful tool for biological macromolecule molar mass determinations. The Svedberg equation was derived by eliminating the translational friction coefficient relating to sedimentation and diffusion in the ultracentrifuge in a strictly solute/solvent vanishing concentration two-component system analysis. A differential equation describing the radial equilibrium concentration distribution in the ultracentrifuge was also derived, both yielding the buoyant molar mass (1-nu2rho)M2 term. Many years later it was realized that solutions of biological macromolecules are multicomponent systems and the two-component analysis leads to minor or major erroneous results. Thermodynamic derivation of an equation for multicomponent systems redefines the buoyant molar mass terms by (deltarho/deltac2)muM2, leading to correct molar mass (g/mol) values following determination of the density increment at constant chemical potentials of diffusible solutes, and powerfully connects the analytical sedimentation equation to the osmotic pressure concentration derivative and, in a broad complementary sense, to light, X-ray and neutron scattering experiments. Macromolecular interactions can be studied with high precision and solute-solvent interactions yield powerful information relating to "thermodynamic" hydration, closely related to hydration derived from X-ray diffraction, as well as solute-cosolute interactions. A series of examples is given to demonstrate the correctness and usefulness of the thermodynamic multicomponent system approach. It is a strange fact that in current analytical ultracentrifugation analysis the elegant and powerful multicomponent solution technology is almost totally disregarded and the classical limited validity Svedberg approach is used uniquely.     

Analysis of protein self-association under conditions of the thermodynamic non-ideality

Analysis of protein-protein interactions in highly concentrated solutions requires a consideration of the non-ideality in such solutions which is expressed by the virial coefficients. Different equations are presented to estimate effects of the thermodynamic non-ideality on the macromolecular interaction of self-associating proteins in sedimentation equilibrium experiments. Usually the influence of thermodynamic non-ideal behavior are described by concentration power series. The convergence of such power series is limited at high solute concentration. When expressing the thermodynamic non-ideality by an activity power series this disadvantage can be minimized. The developed centrifuge equations are the basis for a global analysis to estimate equilibrium constants and the corresponding thermodynamic activities of the reactants. Based on fit analysis of synthetic concentration profiles it was established that marked deviations from the expected association constants are observed for proteins with strong association forces between solute molecules. Considerable differences were also observed in weakly interacting systems. This was due to the excluded volume of the protein which is similar in magnitude to the binding constant. For interactions with moderate affinities values extremely close to the true binding values were obtained, as confirmed by experimental results with concanavalin A.