Effects of thermodynamic nonideality on protein interactions. Equivalence of interpretations based on excluded volume and preferential solvation

Published results on the stabilization of proteins by sucrose (J.C. Lee and S.N. Timasheff, J. Biol. Chem. 256 (1981) 7193) have been reexamined and interpreted in terms of thermodynamic nonideality. The composition dependence of activity coefficients may be accounted for on a statistical-mechanical basis using the concept of excluded volume. An expression is derived in which the effect of sucrose on determination of the partial specific volume of a protein, previously interpreted in terms of preferential protein solvation, is also seen to be attributable to excluded volume. Gel chromatographic studies of the reversible unfolding of alpha-chymotrypsin are presented which demonstrate temperature- and sucrose-mediated changes in the effective volume of the enzyme. These measurements support the quantitative interpretation of the stabilization in terms of thermodynamic nonideality arising from the difference between covolumes for sucrose and the two isomeric states of alpha-chymotrypsin. By establishing the equivalence of the two approaches that have been used to account for the effects of inert solutes on protein transitions, the present investigation eliminates the need for any distinction between such solutes on the basis of molecular size; and also enhances greatly the potential sensitivity of thermodynamic nonideality as a means of probing protein isomerizations, since greater displacement of the equilibrium position may be effected by small rather than by macromolecular solutes present at the same weight concentrations.     

Effects of thermodynamic nonideality in kinetic studies

Experimental evidence is presented for concentration dependence of the pseudo-firstorder rate constant describing the rate of inversion of sucrose by 2 m HCl; and also of the increase in maximal velocity for the catalytic reduction of pyruvate by lactate dehydrogenase that results from addition of the inert macromolecular solutes bovine serum albumin, ovalbumin, and Dextran T70. These somewhat unusual and seemingly diverse observations are examined in terms of a theory formulated on the basis of two equilibrium reactions, the first describing complex formation between two reactants, and the second isomerization of that complex to an activated state prior to product formation. This formulation permits consideration of activity coefficient ratios relevant to the equilibria and the expression of these ratios as power series in total solution composition. Quantitative assessment of the experimental results is made possible in these terms by estimating the magnitudes of the constant coefficients of the virial expansions as excluded volumes. It is concluded that the result observed in the sucrose inversion study finds rational explanation in thermodynamic nonideality factors governing the overall equilibrium between the reactants and the activated complex of sucrose and hydronium ion. For the enzyme-catalyzed reaction the same general equation applies but particular attention is given to the simplified form that is relevant to high substrate concentrations, where, in the absence of inert compounds, the conventional maximal velocity is approached. In this region an increase in velocity observed upon addition of an inert macromolecular component may be considered explicitly in terms of excluded volume effects related to a shape change in the isomerization between enzyme-substrate complex and its activated state.     

Effects of thermodynamic nonideality on the kinetics of ester hydrolysis by alpha-chymotrypsin: a model system with preexistence of the isomerization equilibrium

The effects of a small inert solute, sucrose, on the kinetics of hydrolysis of N-acetyl-tryptophan ethyl ester by bovine alpha-chymotrypsin have been investigated. In studies at pH 7 and 20 degrees C the presence of 0.5 M sucrose in assay mixtures caused no discernible change in kinetic parameters, a result consistent with existence of the enzyme in a single conformational state under those conditions. However, at pH 3.5 and 50 degrees C, conditions under which the enzyme comprises an equilibrium mixture of compact and expanded isomeric states, inclusion of the inert solute led to a considerable decrease in Michaelis constant (0.84 to 0.61 mM) but no significant change in maximal velocity. These results were shown to be amenable to quantitative interpretation in terms of thermodynamic nonideality effects on catalysis by an enzyme undergoing reversible isomerization in the absence of substrate. For that analysis, which required experimental estimates of the equilibrium constant for preexisting isomerization of enzyme and the activity coefficient of substrate, the magnitude of the former (0.3) was obtained by difference spectroscopy: liquid-liquid partition studies with bromobenzene as organic phase were used to determine the effect of sucrose on the activity coefficient of N-acetyltryptophan ethyl ester. Such agreement between experimental kinetic findings and theoretical predictions based on considerations of excluded volume points to the possible use of the space-filling effects of small solutes for delineating the gross extent of conformational changes associated with reversible isomerization of proteins, and hence to the potential of thermodynamic nonideality as a probe for studying protein denaturation mechanisms as well as substrate-mediated changes associated with enzyme reaction mechanisms.     

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.     

Effects of thermodynamic nonideality in ligand binding studies

Effects of thermodynamic nonideality are considered in relation to the quantitative characterization of the interaction between a small ligand. S, and a macromolecular acceptor. A, by two types of experimental procedure. The first involves determination of the concentration of ligand in dialysis equilibrium with the acceptor/ligand mixture, and the second, measurement of the concentration of unbound ligand in the reaction mixture by ultrafiltration or the rate of dialysis method. For each situation explicit expressions are formulated for the appropriate binding function with allowance for composition-dependent nonideality effects expressed in terms of molar volume, charge-charge interaction and covolume contributions. The magnitudes of these effects are explored with the aid of experimental studies on the binding of tryptophan and of methyl orange to bovine serum albumin. It is concluded for experiments conducted utilizing either equilibrium dialysis or frontal gel chromatography that, provided a correction is made for any Donnan redistribution of ligand, theoretically predicted acceptor-concentration dependence is likely to be negligible and that use of the conventional binding equation written for an ideal system is appropriate to the analysis of the results. Use of ultrafiltration or the rate of dialysis method requires examination of the assumption that the activity coefficient ratio y(A)y(s)/y(AS) for the reaction mixture approximates unity; but again reassurance is provided that nonideality manifested as a dependence of the binding function on acceptor concentration is unlikely to be significant.     

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.     

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.     

Analysis of thermodynamic non-ideality in terms of protein solvation.

The effects of thermodynamic non-ideality on the forms of sedimentation equilibrium distributions for several isoelectric proteins have been analysed on the statistical-mechanical basis of excluded volume to obtain an estimate of the extent of protein solvation. Values of the effective solvation parameter delta are reported for ellipsoidal as well as spherical models of the proteins, taken to be rigid, impenetrable macromolecular structures. The dependence of the effective solvated radius upon protein molecular mass exhibits reasonable agreement with the relationship calculated for a model in which the unsolvated protein molecule is surrounded by a 0.52-nm solvation shell. Although the observation that this shell thickness corresponds to a double layer of water molecules may be of questionable relevance to mechanistic interpretation of protein hydration, it augurs well for the assignment of magnitudes to the second virial coefficients of putative complexes in the quantitative characterization of protein-protein interactions under conditions where effects of thermodynamic non-ideality cannot justifiably be neglected.     

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.     

A Hilly path through the thermodynamics and statistical mechanics of protein solutions

The opus of Don Winzor in the fields of physical and analytical biochemistry is a major component of that certain antipodean approach to this broad area of research that blossomed in the second half of the twentieth century. The need to formulate problems in terms of thermodynamic nonideality posed the challenge of describing a clear route from molecular interactions to the parameters that biochemists routinely measure. Mapping out this route required delving into the statistical mechanics of solutions of macromolecules, and at every turn mathematically complex, rigorous, general results that had previously been derived previously, often by Terrell Hill, came to the fore. Central to this work were the definition of the “thermodynamic activity”, the pivotal position of the polynomial expansion of the osmotic pressure in terms of molar concentration and the relationship of virial coefficients to details of the forces between limited-size groups of interacting molecules. All of this was richly exploited in the task of taking account of excluded volume and electrostatic interactions, especially in the use of sedimentation equilibrium to determine values of constants for molecular association reactions. Such an approach has proved relevant to the study of molecular interactions generally, even those between the main macromolecular solute and components of the solvent, by using techniques such as exclusion and affinity chromatography as well as light scattering.     

Thermodynamic nonideality as a probe of reversible protein unfolding effected by variations in pH and temperature: studies of ribonuclease

Thermodynamic nonideality arising from the space-filling effect of added sucrose is employed to confirm that the reversible unfolding of ribonuclease A effected by acid may be described as an equilibrium between native and unfolded states of the enzyme. However, the extent of the volume change is far too small for the larger isomer to be the fully expanded state, a result signifying that the acid-mediated unfolding of ribonuclease does not conform with the two-state equilibrium model of protein denaturation. Although the thermal denaturation of ribonuclease A is characterized by a larger increase in volume, quantitative reappraisal of published results on the effects of glycerol on this transition at pH 2.8 (Gekko, K., and Timasheff, S. N., 1981 Biochemistry 20, 4677-4686) leads to an estimated volume increase that is much smaller than that inferred from hydrodynamic studies--a disparity attributed to the dual actions of glycerol as a space-filling solute and as a ligand that binds preferentially to the thermally unfolded form of the enzyme. Even in this unfavorable circumstance the fact that glycerol exerts a net excluded volume effect at least confirms that the thermal unfolding of ribonuclease A is an equilibrium transition between two discrete states. The strengths and limitations of using thermodynamic nonideality as a probe of the two-state equilibrium model of protein denaturation are discussed in the light of these findings.     

Analysis of sedimentation equilibrium distributions reflecting nonideal macromolecular associations

A rigorous statistical-mechanical approach is adopted to derive general quantitative expressions that allow for the effects of thermodynamic nonideality in equilibrium measurements reflecting interaction between dissimilar macromolecular reactants. An analytical procedure based on these expressions is then formulated for obtaining global estimates of equilibrium constants and the corresponding reference thermodynamic activities of the free reactants in each of several sedimentation equilibrium experiments. The method is demonstrated by application to results from an ultracentrifugal study of an electrostatic interaction between ovalbumin and cytochrome c (Winzor, D. J., M. P. Jacobsen, and P. R. Wills. 1998. Biochemistry. 37:2226-2233). It is demonstrated that reliable estimates of relevant thermodynamic parameters are extracted from the data through statistical analysis by means of a simple nonlinear fitting procedure.     

The use of analytical sedimentation velocity to extract thermodynamic linkage

For 25 years, the Gibbs Conference on Biothermodynamics has focused on the use of thermodynamics to extract information about the mechanism and regulation of biological processes. This includes the determination of equilibrium constants for macromolecular interactions by high precision physical measurements. These approaches further reveal thermodynamic linkages to ligand binding events. Analytical ultracentrifugation has been a fundamental technique in the determination of macromolecular reaction stoichiometry and energetics for 85 years. This approach is highly amenable to the extraction of thermodynamic couplings to small molecule binding in the overall reaction pathway. In the 1980s this approach was extended to the use of sedimentation velocity techniques, primarily by the analysis of tubulin-drug interactions by Na and Timasheff. This transport method necessarily incorporates the complexity of both hydrodynamic and thermodynamic nonideality. The advent of modern computational methods in the last 20 years has subsequently made the analysis of sedimentation velocity data for interacting systems more robust and rigorous. Here we review three examples where sedimentation velocity has been useful at extracting thermodynamic information about reaction stoichiometry and energetics. Approaches to extract linkage to small molecule binding and the influence of hydrodynamic nonideality are emphasized. These methods are shown to also apply to the collection of fluorescence data with the new Aviv FDS.     

Excluded volume as a determinant of macromolecular structure and reactivity

The effect of excluded volume on the thermodynamic activity of globular macromolecules and macromolecular complexes in solution is studied in the hard-particle approximation. Activity coefficients are calculated as a function of the fraction of total volume occupied by macromolecules using relations obtained from scaled particle and lattice models. Significant and readily observable effects are predicted to occur as the fraction of volume occupied by globular macromolecules increases, including the following: (i) Compact quasi-spherical macromolecular conformations become increasingly energetically favored over extended anisometric conformations. (ii) Self- and heteroassociation processes are enhanced, particularly those leading to the formation of compact quasi-spherical aggregates. (iii) Depending upon the details of the reaction mechanism, the rate of an enzyme-catalyzed reaction may monotonically decrease, go through a maximum, or exhibit more complex behavior. A given degree of volume occupancy by larger macromolecules is predicted to have less effect on the structure and self-association of smaller macromolecules than the same degree of volume occupancy by smaller macromolecules has on the structure and self-association of larger macromolecules.     

Macromolecular crowding and its role as intracellular signalling of cell volume regulation.

Macromolecular crowding has been proposed as a mechanism by means of which a cell can sense relatively small changes in volume or, more accurately, the concentration of intracellular solutes. According to the macromolecular theory, the kinetics and equilibria of enzymes can be greatly influenced by small changes in the concentration of ambient, inert macromolecules. A 10% change in the concentration of intracellular proteins can lead to changes of up to a factor of ten in the thermodynamic activity of putative molecular regulatory species, and consequently, the extent to which such regulator(s) may bind to and activate membrane-associated ion transporters. The aim of this review is to examine the concept of macromolecular crowding and how it profoundly affects macromolecular association in an intact cell with particular emphasis on its implication as a sensor and a mechanism through which cell volume is regulated.     

Effects of inert volume-excluding macromolecules on protein fiber formation. II. Kinetic models for nucleated fiber growth

A sequential model for nucleated protein fiber formation is proposed that is similar in broad outline to models proposed previously (Thermodynamics of the Polymerization of Protein, Academic Press, New York, (1975); Biophys. J. 50 (1986) 583) but generalized to allow for thermodynamic nonideality resulting from a high degree of volume occupancy by inert macromolecular cosolutes (macromolecular crowding). The effect of volume occupancy on the rate of fiber formation is studied in the transition-state rate-limited regime through systematic variation of rate-limiting step (prenuclear oligomer formation, nucleus formation or fiber growth), shape of prenuclear oligomer, size of nucleus, extent of reversibility, nature of inert cosolute (hard globular particle or random coil polymer) and size of inert cosolute relative to that of fiber-forming protein. It is found that crowding can accelerate the rate of fiber formation by as much as several orders of magnitude. The extent of acceleration for a given degree of volume occupancy depends upon several factors, the most conspicuous of which is the stoichiometry of the nucleus. In contrast, the rate of redistribution of fiber length, which occurs on a much slower time scale than polymer formation, is found to be insensitive to the extent of crowding.     

Elimination of self-association as the source of the thermodynamic nonideality in aqueous proline solutions

The effect of high concentrations of proline on the diffusion coefficient of water has been examined to assess the extent to which the resulting thermodynamic nonideality could be explained on the statistical-mechanical basis of excluded volume. In fact, such a space-filling role not only accounts for the proline concentration-dependence of the diffusion coefficient of water but it also accounts for the nonideality of proline in freezing point depression and isopiestic measurements. These findings refute the conclusion (Schobert, B. and Tschesche, H. (1978) Biochim. Biophys. Acta 541, 270–277) that the stabilization of enzyme structure by high concentrations of proline stems from self-association of the imino acid via intermolecular hydrogen bonding; and thereby support the concept that the protective effect of proline on enzyme stability must reside mainly in its action as an inert, space-filling solute.