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.     

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.     

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).     

Analysis of self-associations from partition isotherms

Self-associations can be studied from the measurements of the partition of the self-associating solute between two immiscible liquids. The apparent partition coefficient, Kapp, is proportional to the ratio of the apparent weight fraction of monomer, fa, in each phase. If one assumes that the Adams-Fujita convention for the activity coefficients of the self-associating species applies, then fa is related to Mna and Mwa, the apparent values of the number and weight average molecular weights, respectively; and one can use previously developed methods to analyze the self-association. In order to use the method, one must make an independent study at the same temperature of one of the phases by an appropriate thermodynamic method, such as vapor pressure osmometry or sedimentation equilibrium. Then one can test the other phase for the type of self-association present and evaluate the equilibrium constant or constants (ki) and the nonideal term (BM1) from the partition data. One can also evaluate the partition coefficient (Kpar). From these measurements, one can obtain the free energy (delta G0) for the association in each phase and for the transfer between phases. Temperature-dependence studies will provide the enthalpy (delta H0) or entropy (delta S0) of self-association or transfer. This method should be quite useful for studying small molecules of biological importance.     

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).     

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.     

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.     

Analysis of interacting biopolymer systems by analytical ultracentrifugation

Many of the functions of biological macromolecules are based on specific interactions. Extended concentration dependent studies of sedimentation coefficients or molecular masses of biopolymers are highly useful for describing the different kinds of association phenomena. These studies allow one to determine the partial concentrations of monomers and associates or reactants and complexes in self-associating systems or heterologous associations, respectively. Furthermore, in combination with corresponding measurements of biological activity these data allow one to estimate the individual activity parameters of components involved in equilibrium processes. The study of self-association and heterologous association using analytical ultracentrifugation, some recent developments therein, and its application to different examples are outlined here. Accepted: 18 October 1996     

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.     

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.     

Chromatographic evidence of the self-association of oxyhemoglobin in concentrated solutions: its biological implications.

Expressions that take into account the effects of thermodynamic non-ideality, described in terms of a high-order virial expansion, are derived for the concentration-dependence of the weight-average partition coefficient in exclusion chromatography of a single solute and of a solute undergoing reversible self-association. Comparison of the concentration-dependences predicted by those expressions with results obtained for bovine and human oxyhemoglobins on CPG-10-120 porous glass beads in 0.156 I phosphate-chloride buffer, pH 7.3, shows that neither oxyhemoglobin conforms with the concept of it being a single alpha 2 beta 2 entity with Stokes radius of 3.13 nm, the experimental value. Previously published osmotic pressure and sedimentation equilibrium results are also shown to be inconsistent with this concept. On the other hand, both sets of exclusion chromatography results are consistent with the joint operation of thermodynamic non-ideality and reversible association of the alpha 2 beta 2 species. From the magnitude of the equilibrium constant, derived for either of two possible modes of association, it is calculated that only half of the oxyhemoglobin would be in the alpha 2 beta 2 states under conditions of oxygen saturation and a concentration of 320 g/liter, that pertaining in the red blood cell. The consequences of this association phenomenon are discussed in relation to the oxygen binding curves obtained by others in the presence and absence of 2,3-diphosphoglycerate (DPG). An explanation is provided of the observed dependence on hemoglobin concentration of oxygen-binding in the presence of DPG, and of the absence of such an effect in DPG-free solutions. It is concluded that the control of oxygen binding to hemoglobin in the physiological situation involves the joint operation of self-association and allosteric effects.     

Ultracentrifugal studies of the effect of molecular crowding by trimethylamine N-oxide on the self-association of muscle glycogen phosphorylase b.

The suitability of sedimentation equilibrium for characterizing the self-association of muscle glycogen phosphorylase b has been reappraised. Whereas sedimentation equilibrium distributions for phosphorylase b in 40 mM Hepes buffer (pH 6.8) supplemented with 1 mM AMP signify a lack of chemical equilibrium attainment, those in buffer supplemented additionally with potassium sulfate conform with the requirements of a dimerizing system in chemical as well as sedimentation equilibrium. Because the rate of attainment of chemical equilibrium under the former conditions is sufficiently slow to allow resolution of the dimeric and tetrameric enzyme species by sedimentation velocity, this procedure has been used to examine the effects of thermodynamic nonideality arising from molecular crowding by trimethylamine N-oxide on the self-association behaviour of phosphorylase b. In those terms the marginally enhanced extent of phosphorylase b self-association observed in the presence of high concentrations of the cosolute is taken to imply that the effects of thermodynamic nonideality on the dimer-tetramer equilibrium are being countered by those displacing the T<==>R isomerization equilibrium for dimer towards the smaller, nonassociating T state. Because the R state is the enzymically active form, an inhibitory effect is the predicted consequence of molecular crowding by high concentrations of unrelated solutes. Thermodynamic nonideality thus provides an alternative explanation for the inhibitory effects of high concentrations of glycerol, sucrose and ethylene glycol on phosphorylase b activity, phenomena that have been attributed to extremely weak interaction of these cryoprotectants with the T state of the enzyme.     

Component D of chicken hemoglobin and the hemoglobin of the embryonic Tammar wallaby (Macropus eugenii) self-associate upon deoxygenation: Effect on oxygen binding

Extensive measurements of oxygen binding by some vertebrate hemoglobins (Hbs) have suggested an unusually high degree of cooperativity with reported Hill coefficients, n(H), greater than 4.0. We have reexamined this possibility of "super-cooperativity" with chicken Hb components A (alpha(A) (2)beta(2)) and D (alpha(D) (2)beta(2)). Prior studies have shown that component D but not A self-associates to dimers of tetramers upon deoxygenation. This self-association is reflected in the oxygen equilibrium of Hb D which shows a maximal n(H), greater than 4.0 at approximately 4 mM heme concentration. In contrast, component A has maximal n(H) value below 3. The value of the maximal n(H) for Hb D increases linearly with the fraction of octamer present in the deoxy Hb. We anticipate that deoxygenation-dependent self-association will be shown to be a general property of Hb D from birds and reptiles. Neither oxygen equilibria nor sedimentation measurements show any evidence that components A and D interact to form a complex when deoxygenated. We have also reexamined the oxygen equilibria of Hbs of an embryonic marsupial, the wallaby. The equilibria in red cells have been reported to have Hill coefficients as high as 5-6. Although our oxygen equilibrium measurements of solutions of unfractionated wallaby Hb at a concentration of approximately 1 mM show no n(H) values greater than approximately 3.0, sedimentation velocity measurements provide clear evidence for deoxygenation-dependent self-association.     

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.     

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     

Self-association of band-protein from human erythrocyte membranes in aqueous solutions

Band 3, the main integral protein of the human erythrocyte membrane, was solubilized and purified in high concentrations of acetic acid. After removal of the organic solvent by dialysis, the self-association of the protein in aqueous solutions was studied by analytical ultracentrifugation. Sedimentation velocity and sedimentation equilibrium experiments clearly demonstrate that, under appropriate conditions of protein preparation, at protein concentrations c less than 200 micrograms/ml, ionic strengths 2 less than 10mM and pH values remote from the isoelectric pH of the protein, band 3 shows a monomer/dimer/tetramer-association equilibrium. With some preparations, as well as at higher values of c or I, hexamers and octamers contribute to the association equilibrium. The time needed for relaxation towards association equilibrium depends on the blood donor from whom the membranes were derived and varies between less than one minute and more than several hours. The results of analytical ultracentriguation, together with previously published data on the incorporation of band 3 into planar lipid bilayers, from chemical crosslinking and from electronmicroscopy suggest that band 3 will also show a monomer/dimer/tetramer-association equilibrium in the human erythrocyte membrane. This hypothesis contrasts the widely-held assumption that, in the membrane, band 3 is a stable dimer; however, it is consistent with nearly all known data on band 3-self-association.     

Effect of temperature on the self-assembly of the Escherichia coli ClpA molecular chaperone

Protein quality control pathways rely upon ATP-dependent proteases, such as Escherichia coli ClpAP, to perform maintenance roles in the cytoplasm of the cell. ATP-dependent proteases remove misfolded and partially synthesized proteins. This action is particularly important in situations where an unregulated accumulation of such proteins will have a deleterious effect on the cell. ClpAP is composed of a tetradecameric serine protease, ClpP (21.6 kDa monomer), and the ATPase/protein unfoldase ClpA (84.2 kDa monomer). ClpA also uses its protein unfolding activity to remodel proteins and protein complexes; thus, in the absence of the proteolytic component, ClpA is considered a molecular chaperone. Previous reports, by others, suggested that ClpA exists in a monomer-dimer equilibrium at 4 °C. In contrast, using a combination of sedimentation velocity, sedimentation equilibrium, and dynamic light scattering, we recently reported that ClpA exists in a monomer-tetramer equilibrium at 25 °C. Here we report an investigation of the effect of temperature on the self-association of the E. coli ClpA protein unfoldase using analytical ultracentrifugation techniques. The results of sedimentation velocity and sedimentation equilibrium experiments performed at multiple loading concentrations of ClpA over a range of temperatures from 3.9 to 38.2 °C are discussed. Sedimentation velocity experiments show a decrease in weight average s(20,w) at the extremes of temperature. This result, along with extensive sedimentation equilibrium data and analysis, suggests the presence of a dimeric intermediate of ClpA that is differentially populated as a function of temperature. Further, analysis of sedimentation equilibrium data as a function of temperature led us to propose a monomer-dimer-tetramer equilibrium to describe the temperature dependence of ClpA self-assembly in the absence of nucleotide.     

Modern analytical ultracentrifugation in protein science: a tutorial review

Analytical ultracentrifugation (AU) is reemerging as a versatile tool for the study of proteins. Monitoring the sedimentation of macromolecules in the centrifugal field allows their hydrodynamic and thermodynamic characterization in solution, without interaction with any matrix or surface. The combination of new instrumentation and powerful computational software for data analysis has led to major advances in the characterization of proteins and protein complexes. The pace of new advancements makes it difficult for protein scientists to gain sufficient expertise to apply modern AU to their research problems. To address this problem, this review builds from the basic concepts to advanced approaches for the characterization of protein systems, and key computational and internet resources are provided. We will first explore the characterization of proteins by sedimentation velocity (SV). Determination of sedimentation coefficients allows for the modeling of the hydrodynamic shape of proteins and protein complexes. The computational treatment of SV data to resolve sedimenting components has been achieved. Hence, SV can be very useful in the identification of the oligomeric state and the stoichiometry of heterogeneous interactions. The second major part of the review covers sedimentation equilibrium (SE) of proteins, including membrane proteins and glycoproteins. This is the method of choice for molar mass determinations and the study of self-association and heterogeneous interactions, such as protein-protein, protein-nucleic acid, and protein-small molecule binding.