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).     

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.     

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.     

Self-association of sodium cholate in isotonic saline solutions

The self-association of dialyzed solutions of sodium cholate in isotonic saline solutions has been studied by vapor pressure osmometry and sedimentation equilibrium. These studies were carried out at 25, 31 and 37 degrees C. In all experiments the self-association could be described as a two-equilibrium constant, indefinite self-association in which odd species beyond monomer were absent. The plots of M1/Mna or M1/Mwa vs. c were quite smooth with no sharp breaks; this suggested that there were no critical phenomena. The temperature dependence of the self-association was quite small. Our results are in accord with other studies on sodium cholate which indicate that the self-association involves several species, and that it is not a monomer-n-mer self-association.     

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.     

Self-association of rabbit muscle phosphofructokinase: effects of ligands

The effects of ligands on the self-association of rabbit muscle phosphofructokinase (PFK) were investigated by velocity sedimentation at pH 7.0 and 23 degrees C. The concentration dependence of the weight-average sedimentation coefficient was monitored in the presence of these ligands. The mode of association and equilibrium constants characterizing each association step were determined by theoretical fitting of the sedimentation data. The simplest mode of association for the PFK system is M in equilibrium M2 equilibrium M4 in equilibrium M16. Ligands and temperature would perturb the various equilibrium constants without altering the mode of association. The apparent equilibrium constants for the formation of tetramer, K4app, are increased in the presence of 0.1 mM ATP and 1.0 mM fructose 6-phosphate. The value of the sedimentation coefficient for the tetramer, S4 degrees, that would best fit the data is 12.4 S instead of 13.5 S determined in the absence of substrates, thus implying a structural change in the tetramer induced by substrates. Only an insignificant amount of dimer is present under the experimental conditions. The presence of activators, ADP or phosphate, enhances the formation of tetramers, and S4 degrees assumes a value of 13.5 S. Similar results are obtained with decreasing concentrations of proton. The presence of the inhibitor, citrate, however, favors the formation of dimers. The equilibrium constants determined as a function of ADP concentration were further analyzed by the linked-function theory derived by Wyman [Wyman, J. (1964) Adv. Protein Chem. 19, 224--285], leading to the conclusion that the formation of a tetramer involves the binding of two additional molecules of ADP per monomer. Similar analysis results in a conclusion that the formation of a dimer involves the binding of one additional molecule of citrate per phosphofructokinase subunit.     

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.     

Determination of the parameters of self-association by direct fitting of the omega function

Nonlinear regression is used to fit the omega function vs. protein concentration curves (first described by B.K. Milthorpe, P.D. Jeffrey and L.W. Nichol, Biophys. Chem. 3 (1975) 169) obtained from sedimentation equilibrium experiments on self-associating macromolecules. Nonlinear regression allows the direct fit of these curves with discrete or indefinite self-association reaction models in order to obtain values for the equilibrium constants and second virial coefficient. The method is independent of the choice of reference concentration and avoids the original method of extrapolating an omega function curve to zero concentration and then using the extrapolated value to construct a monomer activity curve used for analysis. This extrapolation can become very difficult for mild to strong self-associations where incorrectly extrapolated values lead to systematic error in the monomer activity curves. The method is applied to results from a mild, indefinite self-association, exemplified by the self-association of human spectrin, and to computer-simulated data of weak, mild and strong, indefinite self-associations.     

Self-association mode of a flavoenzyme D-amino acid oxidase from hog kidney. II. Stoichiometry of holoenzyme association and energetics of subunit association

The self-association pattern of D-amino acid oxidase holoenzyme in 0.1 M sodium pyrophosphate, pH 8.3, at 25 degrees C was examined by the low-angle laser light-scattering method. As to the results of nonlinear least-squares analysis of the apparent weight-average molecular weight (Mwapp) versus protein concentration (c) data, the following three models fitted equally well the data over the concentration range of 0.03-11.4 mg/ml: 1) the model of isodesmic indefinite self-association of the monomer where the dimerization constant differs from the isodesmic association constant, 2) the model which involves the dimerization of the monomer and isodesmic indefinite self-association of the dimer, and 3) the model which involves the trimerization of the monomer and isodesmic indefinite self-association of the trimer. In a more limited concentration range (0.3-11.4 mg/ml), a model of isodesmic indefinite self-association of the stable dimer where the dimer does not dissociate into the monomers cannot be excluded from the above three models. Measurements with the concentration range lowered to 0.03 mg/ml enabled us to exclude unequivocally the model involving such a stable dimer and to extrapolate the Mwapp data to the Mr of the monomer at infinite dilution as in the case of the apoenzyme. The observed sedimentation boundary profiles were qualitatively consistent with the idealized boundary profiles calculated with the model which involves the dimerization of the monomer and isodesmic indefinite self-association of the dimer, so this model is the most probable of the models examined. These results provide the first evidence that the association mode of the holoenzyme is different from that of the apoenzyme, i.e. isodesmic indefinite self-association of the monomer (Tojo, H., Horiike, K., Shiga, K., Nishina, Y., Watari, H., and Yamano, T. (1985) J. Biol. Chem. 260, 12607-12614). The overall linkage scheme, between binding of coenzyme FAD and subunit association, was considered, and the overall free energy change in each process in the scheme was calculated. The total stabilization energies of the intersubunit interaction in the holoenzyme relative to the apoenzyme were found to be -2.2 kcal/mol at the dimerization step and -0.5 kcal/mol at the step of the addition of the dimer to any 2i-mer (i = 1,2, ...).     

Reversible self-association of recombinant bovine factor B

The recombinant bovine factor B, obtained by a newly developed bacterial expression system, was found to exhibit features characteristic of a reversible self-associating system. Using size-sieving chromatography, distribution of the factor B species ranged from a monomer to a trimer, but not oligomers of higher molecular weights. At high protein concentrations, factor B migrated as a single band in a native gel. Cross-linking with the amino-reactive cross-linking reagent bis (sulfosuccinimidyl) suberate (BS), at a low cross-linker to protein ratio yielded cross-linked products identified as factor B dimer and trimer. The cross-linking pattern was shown to be a function of the protein and cross-linker concentrations. The range of sedimentation coefficients in a sedimentation velocity experiment suggested that the largest particle present in the distribution was more than twice as large as the smallest. The data obtained under multiple conditions in the sedimentation equilibrium experiments are best fit to a model describing a reversible self-association of a monomer-trimer of factor B species, with a dissociation constant Kd(1,3)=2.48x10(-10) M(2).     

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.     

A thermodynamic model for the self-association of human spectrin

The self-association of human spectrin at 28.8 degrees C in 0.11 M salt (pH 7.5) has been studied by means of sedimentation equilibrium. Coincidence of omega function plots as a function of total spectrin concentration (0-2 g/L) indicated that equilibrium was achieved and that no significant concentration of solute was incapable of participating in the self-association reaction. On the basis of the root-mean-square deviation of the fits and the randomness of the residuals, the behavior can be described equally well, either by a cooperative isodesmic model, in which K12 approximately 2 x 10(6) M-1 and all other K approximately 10(6) M-1, or by an attenuated scheme in which K(i-1)i approximately (3.5 x 10(6)/i M-1. The returned values of the second virial coefficient, B, for both these models fall within the range calculated from the charge and Stokes radius of spectrin. A mechanism for spectrin self-association consistent with both schemes is proposed in which spectrin heterodimers undergo a reversible opening at the self-association interface. These open heterodimers then undergo indefinite self-association to form a series of open-chain oligomers in dynamic equilibrium with closed-loop oligomers.     

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.     

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.     

Self-association of rabbit muscle phosphofructokinase at pH 7.0: stoichiometry

The self-association of rabbit muscle phosphofructokinase at pH 7.0 was investigated by velocity sedimentation. The process was demonstrated to be in a rapid, dynamic equilibrium. The concentration dependence of the weight-average sedimentation coefficient was monitored within the range of 10-750 microgram/mL. The sedimentation properties of phosphofructokinase were analyzed by theoretical simulations or an associating system in rapid equilibrium. In the absence of any ligands and at a temperature of 23 degrees C, the simplest computed model which gives the best fit between theoretical and experimental points can be described as progressive association of monomer in equilibrium or formed from tetramer in equilibrium or formed from 16-mer with apparent equilibrium constants K4 = 5.06 X 10(5) (mL/mg)3 and K16 = 3.25 X 10(23) (mL/mg)15. However, at 5 degrees C, the equilibrium was altered and can best be described as monomer in equilibrium or formed from dimer in equilibrium or formed from tetramer in equilibrium or formed from 16-mer.     

Self-association of sperm whale metmyoglobin

The solution behavior of sperm whale metmyoglobin in 0.15 I phosphate-chloride buffer, pH 7.2, has been examined by sedimentation equilibrium, frontal gel chromatography, and sedimentation velocity. Results obtained from all three studies are shown to be consistent with a self-association model in which dimerization of the myoglobin is governed by an association equilibrium constant of 0.068 liter/g (580 M-1) at 20 degrees C.     

Properties of talin from chicken gizzard smooth muscle

This paper describes the structural and biochemical characterization of talin, a protein localized to various cellular sites where bundles of actin filaments attach to the plasma membrane. By sodium dodecyl sulfate-polyacrylamide gel electrophoresis, the protein has a molecular mass of 225,000 +/- 5,000 daltons. Hydrodynamic measurements at protein concentrations less than 0.72 mg/ml indicate a monomeric protein with a native molecular mass of 213,000 +/- 15,000 daltons. Sedimentation equilibrium experiments indicate self-association at protein concentrations of 0.72 mg/ml and higher. The data suggest that this self-association is a simple monomer:dimer equilibrium over the range of concentrations observed. At low protein concentrations where talin is a monomer, the Stokes radius and sedimentation coefficient vary with ionic strength. Under low ionic strength conditions (5-20 mM NaCl), talin has a Stokes radius of 6.5 nm and a sedimentation value of 9.4, suggesting an asymmetric globular molecule; whereas under high ionic strength conditions (200 mM NaCl), the Stokes radius increases to 7.7 nm and the sedimentation coefficient decreases to 8.8, suggesting a more elongated protein. This conformation change is confirmed by electron microscopy which reveals a more globular protein at low ionic strength which unfolds to become an elongated flexible molecule as the ionic strength is increased to physiological and higher levels. The amino acid composition of talin indicates a low level of aromatic residues, consistent with its relatively low extinction coefficient, talin has an isoelectric point between pH 6.7 and 6.8 based on isoelectric focusing. The detailed purification of talin is described.     

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).     

Absorption and circular dichroic spectral studies on the self-association of daunorubicin

The self-association of daunorubicin in aqueous solution has been examined using visible absorption, fluorescence, and CD measurements. Spectral changes in the concentration range 10^?6 to 1.5 × 10^?3M have been interpreted in terms of a monomer–dimer equilibrium for daunorubicin. The data have been analyzed using a nonlinear curve-fitting technique. The results obtained in this study differ markedly from previously published results, and possible reasons for this difference are discussed. The effect of solvent, temperature, and ionic strength on the dimerization process is investigated.