ANALYTICAL
BIOCHEMISTRY
190, 1-6 (1990)
REVIEW Quantitative Associations Allen
Characterization of Reversible Molecular via Analytical Centrifugation’
P. Minton
Section on Pharmacology, Laboratory of Biochemical Pharmacology, National Institute of Diabetes Diseases, National Institutes of Health, Building 8, Room 226, Bethesda, Maryland 20892
The determination of the strength and stoichiometry of reversible associations between interacting species has long been, and will remain for the foreseeable future, one of the most commonly encountered objectives in biochemical research. Consequently, many techniques have been devised for such determinations, and the literature devoted to the subject of binding measurement is correspondingly large. Other minireviews in this series are devoted to techniques for measuring binding to immobilized substrates; centrifugal methods are necessarily limited to the study of interactions between soluble species.’ The purpose of the present minireview is threefold: to illustrate, by example, the several different types of binding measurements that may be carried out using the ultracentrifuge; to emphasize that binding measurements carried out using the ultracentrifuge are in many cases free from artifacts and ambiguities of interpretation that complicate other types of measurement; and to call attention to some new techniques that promise unprecedented ability to discriminate between possible alternative association mechanisms. Methods developed prior to the mid-1970’s are reviewed in more detail in Fujita (1) and Van Holde (2). In general, measurements of reversible association may be divided into two categories, termed direct and indirect. In a direct measurement of the association of two species (for example, ligand and acceptor) one attempts to individually measure the amount of uncomplexed and complexed forms of either or both of the two species, preferably while they remain in equilibrium with each other. In an indirect measurement one quan-
i This review is the first in a series “Quantitative Analysis of Protein Interactions with Ligands” that has been organized by T. William Hutchens of Baylor College of Medicine, Houston, Texas. ’ Binding assays employing low speed centrifugation to accelerate filtration or to effect pelleting as a means of separating free from membrane-bound ligands are not reviewed here. 0003-2697/90 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
and Digestive
and Kidney
tifies an average property of the solution as a whole that varies with the extent of binding or association. Composition-dependent variations in spectroscopic, hydrodynamic, thermodynamic, or functional properties are then interpreted in the context of models relating fractional changes in the measured property to fractional changes in association. Below we describe the use of the ultracentrifuge for both direct and indirect measurements of association. Each method of measurement described will be illustrated by reference to a single published application of the method. INSTRUMENTATION
All ultracentrifugal methods share in common the requirement that the distribution of solute(s) in a centrifuge cell or tube be determined quantitatively at a minimum of one point in time following application of a sedimenting force. This may be accomplished by a variety of means. Traditionally, the most careful measurements have been obtained through the use of an analytical ultracentrifuge (for example, the Beckman Model E) incorporating optical systems for measurement of absorbance or refractive index as a function of position within a rotating cell (3). The planned release in 1991 of a newly designed analytical ultracentrifuge by Beckman Instruments (4) promises to make such determinations much easier, more accurate, and more precise than ever before. Alternatively, within the last 10 years methods for quantitation of absorbance gradients formed in preparative ultracentrifuges have been developed that can provide data of quality comparable to that obtained from the analytical ultracentrifuge (5). In addition, gradients of radiolabel may now be obtained with precision and resolution comparable to that previously obtained via optical measurements, permitting ultracentrifugal analysis of associations in solutions that are so dilute or concentrated that optical methods are inapplicable (6). 1
2 DIRECT
ALLEN MEASUREMENT
OF
BINDING
The techniques of equilibrium dialysis and ultrafiltration have been frequently used to measure the equilibrium association of a variety of small molecule ligands with a variety of macromolecular acceptors (7). The success of these methods depends upon the ability of a given dialysis membrane or filter to permit unhindered transport of free ligand, to quantitatively inhibit transport of bound ligand, and to be inert with respect to (i.e., not bind significant amounts of) either ligand or acceptor. The range of application of these methods is limited by the fact that no membrane or filter is perfect in all respects, and that, in general, some (small) amount of ligand bound to acceptor will pass through the barrier during the time course of the experiment. The determination will only be valid to the extent that the amount of bound ligand “leaked” into the compartment nominally containing only free ligand is negligible with respect to the amount of free ligand in the compartment. The complications of possible membrane or filter artifacts are absent in the direct determination of free and bound ligand using the ultracentrifuge. Consider a centrifuge cell initially containing uniform (known) concentrations of a homogeneous macromolecular acceptor and a small molecule ligand. Following high speed centrifugation for a limited period of time the upper3 part of the centrifuge cell will be depleted of all acceptor. Prior to complete pelleting, a region of uniform concentration of acceptor, called the plateau region, will exist in the lower part of the centrifuge cell. The concentration of acceptor in the plateau will be equal to the loading concentration of acceptor if the cell has constant cross-sectional area, or will decrease with time in a readily calculable manner because of radial dilution if the cell is sector-shaped (1). The sedimentation velocity of freeligand will in many cases be negligible, but in any event should be so small relative to that of the acceptor that much, if not all, of the region lying above the plateau will contain a constant and readily determined concentration of free ligand. The same concentration of free ligand is maintained within the plateau region of acceptor as well, where it is in equilibrium with bound ligand. Thus the difference between ligand concentrations within and above the plateau regions of acceptor corresponds to the concentration of bound ligand. This method was introduced by Steinberg and Schachman (8), who used it to measure the binding of methyl orange to serum albumin. If the presence of acceptor interferes with direct determination of ligand within the plateau region, or if ac3 In the present work, the terms upper and lower will be used to refer to regions of the centrifuge tube or cell that are, respectively, nearer to and farther from the center of rotation.
P. MINTON
ceptor is pelleted, the amount of ligand (free + complexed) within the region containing acceptor may be calculated from the depletion of ligand above the acceptor plateau (or pellet) by application of mass conservation relations. The indirect measurement of bound ligand via the depletion of free ligand is less precise than the direct measurement of bound ligand described above (8), but requires fewer manipulations and is readily performed using commonly available centrifuge tubes and preparative ultracentrifuges, and hence has been more widely used. A typical example is the characterization of the equilibrium association of cyclodextrin and pancreatic amylase by Mora et al. (9). With slight modification, the method of Steinberg and Schachman (8) may be used to characterize the binding of macromolecular ligands to macromolecular acceptors, providing that the sedimentation coefficient of any complex of ligand and acceptor is both substantially greater than that of free ligand and essentially independent of the amount of bound ligand. An illustration of this method is provided by the study by Revzin and Woychik (10) of equilibrium association between RNA polymerase holoenzyme and double-helical DNA. Since holoenzyme and DNA have considerably different absorbance spectra in the near uv, the individual concentration gradients of protein and DNA may be obtained through linear combination of absorbance scans of the centrifuge cell at two wavelengths (230 and 260 nm). In Fig. 1, scans at the two wavelengths are shown preceding and during sedimentation at high speed. The unambiguous separation between free holoenzyme and the more rapidly sedimenting holoenzyme-DNA complexes evident in the lower right panel of this figure permits accurate quantitation of free and bound holoenzyme. Each cell containing ligand and acceptor at a specified loading composition (total ligand, total acceptor concentrations) contributes one point to a binding isotherm. INDIRECT
MEASUREMENT
OF
BINDING
Sedimentation Velocity The sedimentation velocity of a macromolecular solute depends upon its mass, buoyancy (density relative to that of solvent), and frictional coefficient (2). Thus measurement of sedimentation velocity may be used to characterize associations, in principle, when association results in significant changes in any combination of these three properties, as is usually the case.4 Sedimen’ For example, a dimeric complex formed from two quasispherical macromolecules of approximately equal size and mass would have a mass larger than that of either uncomplexed reactant by a factor of two, and a frictional coefficient greater than that of monomers by a factor of about 1.1-1.2. Moreover, the association of species having substantially different densities, such as protein and lipid, would yield complexes having intermediate densities.
ASSOCIATION
(b)
AND
BINDING
t=t
FIG.
1. Schematic diagram of high-speed centrifugation experiment for the separation of free and DNA-associated RNA polymerase. (a) Distribution of species and absorbance scans at 230 and 260 nm prior to sedimentation. (b) Distribution of species and absorbance scans at 230 and 260 nm following sedimentation at high rotor velocity for time t. The 230~nm scan reveals a more rapidly sedimenting boundary corresponding to DNA-associated polymerase and a more slowlysedimentingboundarycorrespondingtouncomplexedpolymerase. Figure reproduced from Ref. (10) with permission.
tation velocity experiments on reversibly associating systems are most readily interpreted when the rates of interconversion of macromolecular species are either very slow or very rapid on the time scale of a velocity experiment (l-3 h). A variety of approximate theoretical models have been put forward for the interpretation of sedimentation velocity profiles obtained from systems of associating species with intermediate rates of interconversion; the problem is not analytically soluble in the general case, and unambiguous interpretation of experimental data is difficult (11). In order to validly interpret concentration-dependent changes of sedimentation velocity in an interacting system, it is first necessary to characterize any concentration-dependent changes in sedimentation velocity that may exist in solutions of individual reactant species in the absence of potential interaction partners (11). In the limit of slow interconversion, each of the macromolecular species present sediment independently, even though more slowly sedimenting species will no longer be in equilibrium with more rapidly sedimenting species after partial separation occurs. The complex multicomponent boundary thus formed will reflect contributions from each species in the proportions characterizing the mixture prior to sedimentation. An example of an experimental study performed by Creeth and Nichol on a slowly interconverting system is the characterization of three distinct species of urease, tentatively identified as monomer, dimer, and trimer (12); oligomers are presumably crosslinked by slowly exchangeable disulfide bonds.
MEASUREMENTS
3
In the limit of rapid interconversion of macromolecular species, only a single boundary is observed, and the sedimentation coefficient calculated from the velocity of that boundary will be a weight average of those of the contributing species (11). An example of an experimental study performed in the limit of rapid interconversion is the characterization by Lakatos of associations between subcomponents of complement factor Cl (13). In this study the species Clq was radiolabeled. The sedimentation velocity of the boundary containing a constant amount of radiolabel was found to vary with the concentration of each of several unlabeled species added to labeled Clq in a manner consistent with simple models for rapid equilibrium association between labeled and unlabeled species. It has been pointed out elsewhere (14) that models for calculation of weight-average sedimentation coefficients in solutions containing more than a single equilibrium complex require the specification of sedimentation coefficients of intermediate complexes. These cannot be measured independently and must be treated as variable parameters in the context of model fitting, thus contributing to uncertainty in the determination of association stoichiometries and equilibrium constants. Sedimentation Equilibrium When a solution containing interacting macromolecular solutes is sedimented to equilibrium, each solute species is not only at sedimentation-diffusion equilibrium with respect to itself at all radial positions within the cell, but also at chemical equilibrium with respect to all other solute species at the same radial position (2). Thus quantitation of the gradient or gradients of interacting species in a single centrifuge cell at sedimentation equilibrium can, in principle, provide thermodynamically rigorous information about equilibrium association of macromolecules over the entire range of solution compositions within the cell. Sedimentation equilibrium is most straightforwardly analyzed to obtain quantitative information about macromolecular associations by one of two methods, which we respectively term concentration analysis and molecular weight analysk5 Concentration analysis proceeds by construction of a model, simultaneously incorporating both association and sedimentation equilibrium conditions, that calculates one or more experimentally measurable concentration-dependent equilibrium properties (such as absorbance) as a function of radial position at fixed rotor velocity, temperature, etc. The model (or, preferably, each of several possible alternate models) is then fit directly to experimental data by nonlinear 5 For the sake of completeness we cite, but do not describe, a third method of analysis (15) that seems to have been used exclusively by L. W. Nichol and collaborators.
4
ALLEN
P. MINTON
least-squares methods to determine (a) whether a given model is consistent with the data, and if so, (b) how we11 the parameters of the model (stoichiometric coefficients, equilibrium association constants) are determined by the data. An example of concentration analysis is the study by Lewis and Youle (16) of the reversible association of the A and B subunits of ricin. Data are plotted together with the best fit of the preferred model in Fig. 2. Molecular weight analysis of sedimentation equilibrium data requires that the equilibrium gradient of each component be determined experimentally.6*7 The apparent weight-average molecular weight of each component present is then calculated as a function of the concentration of all components via the following relation (lJ9): &f(i)
=
wp
RT (1 - u,p)w,r
dwi - dr
’
where Mgfa is the apparent weight average weight of component i, wi is the w/v concentration of component i at radius r, vi is the partial specific volume of component i (assumed to be independent of the state of association of component i), and p is the density of the solution, which may be calculated to a good approximation as a linear function of the concentrations of all solutes (20). If the solution is sufficiently dilute in all macromolecular solutes so that each solute species is sedimenting ideally, then M$ = M$, the true weight-average molecular weight of component is (1). The set of primary data, consisting of the concentration of each component as a s Components are distinguished from species as follows. A solution containing an arbitrary number of oligomeric forms of substance X in equilibrium with monomeric X is said to contain a single component, namely the substance X, while each presumed association state of X as a separate species. A solution (e.g., X, X9, X3, - . -) is counted containing an arbitrary number of complexes of substances X and Y in equilibrium with monomeric X and monomeric Y is said to contain two components, namely substances X and Y, while each presumed association state of X and/or Y (e.g., X, X,, Y, Y2, XY, X,Y, . . .) is counted as a separate species. 7 This may be accomplished optically if different components have sufficiently different absorbance spectra, or if an absorbance difference is created by labeling one component with an optically distinguishing chromophore (17). Alternately, one of the components may be radiolabeled (6), or both components may be labeled with different radioisotopes. ’ lf, at any point in the centrifuge cell, the concentration of one or more species becomes so large that nonspecific repulsive (so-called nonideal) interactions between solute molecules affect the sedimentation equilibrium, resort must be made to approximate models in order to estimate the true weight average molecular weight from the apparent weight average molecular weight (1). Quasispherical particle models for the effect of nonideal interactions upon sedimentation equilibrium (19,21) in solutions containing only globular proteins, and compact complexes thereof, have been found to be semiquantitatively valid over a broad range of concentrations, and permit realistic calculation of large deviations from ideal behavior. Unfortunately, equiva-
6.7
6.6
6.9 RADIUS
7.0
7.1
7.2
(CM)
FIG. 2. Absorbance scans at 280 nm from cells containing equimolar amounts of ricin A and B chains at three loading concentrations. Solid curves represent the best simultaneous fit of a l-l association model to all three data sets. Figure reproduced from Ref. (16) with permission.
function of radius, is thus transformed into an equivalent set of secondary data, consisting of the weightaverage molecular weight of each component as a function of the concentrations of all components. A model for association that calculates the weight-average molecular weight of each component as a function of the concentrations of all components is then constructed (2,19). In contrast to models formulated for the concentration analysis of primary data, a model formulated for molecular weight analysis of secondary data contains no reference to the method (i.e., sedimentation equilibrium) by which the data were obtained.s lent hard particle models for nonideal interactions in solutions containing molecular species of significantly differing shapes (22) may only be validly used to calculate small deviations from ideal behavior in relatively dilute solutions. ’ For this reason it might appear that, given a certain scheme for self- or heteroassociation, the corresponding model for molecular weight analysis would be simpler than the corresponding model for concentration analysis. This is true for certain simple selfassociation schemes where the concentrations of individual species may be obtained as analytical functions of total component concentrations, but not true in general. In many cases, particularly those involving heteroassociation, models for concentration analysis may be considerably simpler than those for molecular weight analysis (Hsu and Minton, in preparation). It should also be noted that conversion of the primary data to secondary data involves evaluation of either dwJdr or d In wi/dr from data on wi as a function of r that inevitably contains noise. Hence the secondary data will ordinarily be less precise, and may be considerably less precise, than the corresponding primary data.
ASSOCIATION
AND
BINDING
MEASUREMENTS
An example of molecular weight analysis of an associating system containing one component is provided by the study of glutamate dehydrogenase self-association by Reisler et al. (23) together with the reanalysis of Chatelier (24). Combined data obtained from several equilibrium experiments are plotted together with the best fit of the preferred model in Fig. 3. TRACER
METHODS
The characterization of macromolecular self- and heteroassociations via analysis of sedimentation equilibrium data has, in the past, been hampered by the relatively small gradients of absorbance or refractive index that can be measured accurately with the optical systems currently available for the analytical ultracentrifuge.l’ This shortcoming has been alleviated to a great extent by the introduction of tracer methods. A tracer is created by labeling a chemical species with a chromophoric or radioisotope label. The labeled species qualifies as a true tracer only after it is ascertained via careful experimental testing that the labeled species behaves in a fashion that is chemically and physically indistinguishable from the parent unlabeled compound under the conditions of the binding measurementl’ (6,17). A series of solutions are prepared by adding different amounts of unlabeled parent species to a small, fixed amount of tracer sufficient to provide an appropriate level of signal (absorbance, radioactivity). Since the tracer, in principle, distributes uniformly throughout the unlabeled species, gradients of concentration, and hence apparent weight-average molecular weights, may be measured accurately over a very broad range of concentrations. This technique was employed by Muramatsu and Minton (25) to detect and characterize selfassociations in proteins over a concentration range exceeding two orders of magnitude. Heteroassociations may be detected and characterized quantitatively by analyzing changes in the gradient of a tracer label brought about by addition of either the parent unlabeled species or a second unlabeled component with which the parent species is presumed to associate. An example of such an analysis is the study by Servillo et al. (26) of the heteroassociation of apolipoproteins A-II and C-l. ” Performance specifications for the absorbance scanner in the new analytical centrifuge announced by Beckman Instruments are expected to significantly exceed those of the current Model R scanner with respect to the range of absorbance that can be measured with confidence. ‘I One must be particularly concerned with the possibility that the label somehow alters the interaction to be studied. Radiolabeled tracers are preferable to chromophorically labeled tracers in this regard, since radiolabels can be introduced by exchanging one or more atoms of the parent species with a radioisotope, with little or no alteration of chemical structure.
0
0.4
0.2
0.6
0.8
WT wo
FIG. 3. Plot of the ratio of apparent weight-average molecular weight to the molecular weight of monomer as a function of total protein concentration. Points, data from Ref. (23). Curve, best-fit of indefinite isoenthalpic association model. Figure reproduced from Ref. (24) with permission.
SUMMARY
AND
CONCLUSION
The ultracentrifuge provides several techniques for the quantitative characterization of reversible small molecule-macromolecule and macromolecule-macromolecule interactions in solution. The nature of the association to be studied determines the preferred technique. High speed centrifugation is the method of choice for characterizing reversible heteroassociations between species of greatly different mass (i.e., sedimentation coefficient). This technique provides a relatively rapid, artifact-free, and thermodynamically rigorous means of quantifying the amount of nonsedimenting or slowly sedimenting free ligand in equilibrium with rapidly sedimenting acceptor-bound ligand at one particular solution composition. Results obtained over a broad range of ligand and/or acceptor concentrations lead to model-independent binding isotherms that may subsequently be analyzed in the context of models for ligandacceptor association. Lower speed centrifugation to sedimentation equilibrium is the method of choice for characterizing reversible selfassociations and for characterizing heteroassociations between components that cannot be well separated on the basis of sedimentation velocity. In the dilute limit, this technique can provide model-free information about the dependence of weight-average molecular weight of each component upon solution composition, which can subsequently be analyzed in the context of equilibrium models for self- or heteroassociation. At higher concentrations, the models must be generalized to allow for the effect of nonspecific (nonideal) interactions upon sedimentation and association. The use of tracers provides a means for greatly ex-
6
ALLEN
P. MINTON
tending the range of solute concentrations and solution compositions over which both types of measurements may be applied, providing enhanced ability to discriminate between alternative proposed mechanisms for selfor heteroassociations. REFERENCES 1. Fujita, H. (1975) ley, New York.
Foundations
of Ultracentrifugal
2. Van Holde, K. E. (1975) in The Proteins R. L., Eds.), 3rd ed., Vol. 1, pp. 225-291, York. 3. Chervenka, Ultracentrifuge, 4. Schachman, 5. Attri, 152. 6. Attri, 328.
(Neurath, Academic
Wi-
H. and Hill, Press, New
C. (1969) A Manual of Methods for the Analytical Beckman, Palo Alto, California. H. K. (1989) Nature (London) 341,259-260.
A. K., and Minton,
A. P. (1983)
Anal.
Biochem.
133,
A. K., and Minton,
A. P. (1986)
Anal.
Biochem.
136,319-
7. Connors, K. A. (1987) New York. 8. Steinberg, 3728-3747.
Binding
I. Z., and Schachman,
9. Mora, S., Simon, 205-209. 10. Revzin,
Analysis,
I., and Elodi,
A., and Woychik,
Constants,
pp.
H. K. (1966) P. (1974)
R. P. (1981)
11. Nichol, L. W., and Winzor, D. J. (1972) Systems, Clarendon, Oxford, England.
Mol.
305-311,
142-
Wiley,
Biochemistry
5,
Cell. Biochem.
4.
Biochemistry
20,250-256.
Migration
of Interacting
12. Creeth, 13. Lakatos, 384. 14. Minton,
J. M., and Nichol, L. W. (1960) Biochem. J. 77,230-239. S. (1987) B&hem. Biophys. Res. Commun. 149, 378A. P. (1989)
15. Milthorpe, Chem. 19,
Anal.
B. K., Jeffrey, 169-176.
Biochem.
176,
P. D., and Nichol,
209-216. L. W. (1975)
Biophys.
16. Lewis, M. S., and Youle, R. J. (1986) J. Biol. Chem. 261, 11,57111,577. 17. Osborne, J. C., Jr., Powell, G. M., and Brewer, H. B., Jr. (1980) Biochim. Biophys. Acta 619,559-571. 18. Attri, A. K., and Minton, A. P. (1987) Anal. Biochem. 419. 19. Chatelier, R. C., and Minton, A. P. (1987) Biopolymers 1113.
162,40926,1097-
20. Kupke, D. W. (1973) in Physical Principles and Techniques of Protein Chemistry (Leach, S. J., Ed.), Part C, pp. l-75, Academic Press, New York. 21. Chatelier, R. C., and Minton, A. P. (1987) Biopolymers 26,507524. 22. Nichol, L. W., Jeffrey, P. D., and Winzor, D. J. (1976) J. Phys. Chem. 80,648-649. 23. Reisler, E., Pouyet, 3095-3102.
J., and Eisenberg,
24. Chatelier,
Biophys.
25. Muramatsu, 171. 26. Servillo, Biophys.
R. (1987)
N., and Minton,
Chem.
H. (1970) 28,
A. P. (1989)
L., Brewer, H. B., Jr., Chem. 13,29-38
and
Biochemistry
9,
121-128. J. Mol.
Osborne,
Recogn. J. C., Jr.
1,166(1981)
BIOCHEMISTRY
190, 1-6 (1990)
REVIEW Quantitative Associations Allen
Characterization of Reversible Molecular via Analytical Centrifugation’
P. Minton
Section on Pharmacology, Laboratory of Biochemical Pharmacology, National Institute of Diabetes Diseases, National Institutes of Health, Building 8, Room 226, Bethesda, Maryland 20892
The determination of the strength and stoichiometry of reversible associations between interacting species has long been, and will remain for the foreseeable future, one of the most commonly encountered objectives in biochemical research. Consequently, many techniques have been devised for such determinations, and the literature devoted to the subject of binding measurement is correspondingly large. Other minireviews in this series are devoted to techniques for measuring binding to immobilized substrates; centrifugal methods are necessarily limited to the study of interactions between soluble species.’ The purpose of the present minireview is threefold: to illustrate, by example, the several different types of binding measurements that may be carried out using the ultracentrifuge; to emphasize that binding measurements carried out using the ultracentrifuge are in many cases free from artifacts and ambiguities of interpretation that complicate other types of measurement; and to call attention to some new techniques that promise unprecedented ability to discriminate between possible alternative association mechanisms. Methods developed prior to the mid-1970’s are reviewed in more detail in Fujita (1) and Van Holde (2). In general, measurements of reversible association may be divided into two categories, termed direct and indirect. In a direct measurement of the association of two species (for example, ligand and acceptor) one attempts to individually measure the amount of uncomplexed and complexed forms of either or both of the two species, preferably while they remain in equilibrium with each other. In an indirect measurement one quan-
i This review is the first in a series “Quantitative Analysis of Protein Interactions with Ligands” that has been organized by T. William Hutchens of Baylor College of Medicine, Houston, Texas. ’ Binding assays employing low speed centrifugation to accelerate filtration or to effect pelleting as a means of separating free from membrane-bound ligands are not reviewed here. 0003-2697/90 $3.00 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
and Digestive
and Kidney
tifies an average property of the solution as a whole that varies with the extent of binding or association. Composition-dependent variations in spectroscopic, hydrodynamic, thermodynamic, or functional properties are then interpreted in the context of models relating fractional changes in the measured property to fractional changes in association. Below we describe the use of the ultracentrifuge for both direct and indirect measurements of association. Each method of measurement described will be illustrated by reference to a single published application of the method. INSTRUMENTATION
All ultracentrifugal methods share in common the requirement that the distribution of solute(s) in a centrifuge cell or tube be determined quantitatively at a minimum of one point in time following application of a sedimenting force. This may be accomplished by a variety of means. Traditionally, the most careful measurements have been obtained through the use of an analytical ultracentrifuge (for example, the Beckman Model E) incorporating optical systems for measurement of absorbance or refractive index as a function of position within a rotating cell (3). The planned release in 1991 of a newly designed analytical ultracentrifuge by Beckman Instruments (4) promises to make such determinations much easier, more accurate, and more precise than ever before. Alternatively, within the last 10 years methods for quantitation of absorbance gradients formed in preparative ultracentrifuges have been developed that can provide data of quality comparable to that obtained from the analytical ultracentrifuge (5). In addition, gradients of radiolabel may now be obtained with precision and resolution comparable to that previously obtained via optical measurements, permitting ultracentrifugal analysis of associations in solutions that are so dilute or concentrated that optical methods are inapplicable (6). 1
2 DIRECT
ALLEN MEASUREMENT
OF
BINDING
The techniques of equilibrium dialysis and ultrafiltration have been frequently used to measure the equilibrium association of a variety of small molecule ligands with a variety of macromolecular acceptors (7). The success of these methods depends upon the ability of a given dialysis membrane or filter to permit unhindered transport of free ligand, to quantitatively inhibit transport of bound ligand, and to be inert with respect to (i.e., not bind significant amounts of) either ligand or acceptor. The range of application of these methods is limited by the fact that no membrane or filter is perfect in all respects, and that, in general, some (small) amount of ligand bound to acceptor will pass through the barrier during the time course of the experiment. The determination will only be valid to the extent that the amount of bound ligand “leaked” into the compartment nominally containing only free ligand is negligible with respect to the amount of free ligand in the compartment. The complications of possible membrane or filter artifacts are absent in the direct determination of free and bound ligand using the ultracentrifuge. Consider a centrifuge cell initially containing uniform (known) concentrations of a homogeneous macromolecular acceptor and a small molecule ligand. Following high speed centrifugation for a limited period of time the upper3 part of the centrifuge cell will be depleted of all acceptor. Prior to complete pelleting, a region of uniform concentration of acceptor, called the plateau region, will exist in the lower part of the centrifuge cell. The concentration of acceptor in the plateau will be equal to the loading concentration of acceptor if the cell has constant cross-sectional area, or will decrease with time in a readily calculable manner because of radial dilution if the cell is sector-shaped (1). The sedimentation velocity of freeligand will in many cases be negligible, but in any event should be so small relative to that of the acceptor that much, if not all, of the region lying above the plateau will contain a constant and readily determined concentration of free ligand. The same concentration of free ligand is maintained within the plateau region of acceptor as well, where it is in equilibrium with bound ligand. Thus the difference between ligand concentrations within and above the plateau regions of acceptor corresponds to the concentration of bound ligand. This method was introduced by Steinberg and Schachman (8), who used it to measure the binding of methyl orange to serum albumin. If the presence of acceptor interferes with direct determination of ligand within the plateau region, or if ac3 In the present work, the terms upper and lower will be used to refer to regions of the centrifuge tube or cell that are, respectively, nearer to and farther from the center of rotation.
P. MINTON
ceptor is pelleted, the amount of ligand (free + complexed) within the region containing acceptor may be calculated from the depletion of ligand above the acceptor plateau (or pellet) by application of mass conservation relations. The indirect measurement of bound ligand via the depletion of free ligand is less precise than the direct measurement of bound ligand described above (8), but requires fewer manipulations and is readily performed using commonly available centrifuge tubes and preparative ultracentrifuges, and hence has been more widely used. A typical example is the characterization of the equilibrium association of cyclodextrin and pancreatic amylase by Mora et al. (9). With slight modification, the method of Steinberg and Schachman (8) may be used to characterize the binding of macromolecular ligands to macromolecular acceptors, providing that the sedimentation coefficient of any complex of ligand and acceptor is both substantially greater than that of free ligand and essentially independent of the amount of bound ligand. An illustration of this method is provided by the study by Revzin and Woychik (10) of equilibrium association between RNA polymerase holoenzyme and double-helical DNA. Since holoenzyme and DNA have considerably different absorbance spectra in the near uv, the individual concentration gradients of protein and DNA may be obtained through linear combination of absorbance scans of the centrifuge cell at two wavelengths (230 and 260 nm). In Fig. 1, scans at the two wavelengths are shown preceding and during sedimentation at high speed. The unambiguous separation between free holoenzyme and the more rapidly sedimenting holoenzyme-DNA complexes evident in the lower right panel of this figure permits accurate quantitation of free and bound holoenzyme. Each cell containing ligand and acceptor at a specified loading composition (total ligand, total acceptor concentrations) contributes one point to a binding isotherm. INDIRECT
MEASUREMENT
OF
BINDING
Sedimentation Velocity The sedimentation velocity of a macromolecular solute depends upon its mass, buoyancy (density relative to that of solvent), and frictional coefficient (2). Thus measurement of sedimentation velocity may be used to characterize associations, in principle, when association results in significant changes in any combination of these three properties, as is usually the case.4 Sedimen’ For example, a dimeric complex formed from two quasispherical macromolecules of approximately equal size and mass would have a mass larger than that of either uncomplexed reactant by a factor of two, and a frictional coefficient greater than that of monomers by a factor of about 1.1-1.2. Moreover, the association of species having substantially different densities, such as protein and lipid, would yield complexes having intermediate densities.
ASSOCIATION
(b)
AND
BINDING
t=t
FIG.
1. Schematic diagram of high-speed centrifugation experiment for the separation of free and DNA-associated RNA polymerase. (a) Distribution of species and absorbance scans at 230 and 260 nm prior to sedimentation. (b) Distribution of species and absorbance scans at 230 and 260 nm following sedimentation at high rotor velocity for time t. The 230~nm scan reveals a more rapidly sedimenting boundary corresponding to DNA-associated polymerase and a more slowlysedimentingboundarycorrespondingtouncomplexedpolymerase. Figure reproduced from Ref. (10) with permission.
tation velocity experiments on reversibly associating systems are most readily interpreted when the rates of interconversion of macromolecular species are either very slow or very rapid on the time scale of a velocity experiment (l-3 h). A variety of approximate theoretical models have been put forward for the interpretation of sedimentation velocity profiles obtained from systems of associating species with intermediate rates of interconversion; the problem is not analytically soluble in the general case, and unambiguous interpretation of experimental data is difficult (11). In order to validly interpret concentration-dependent changes of sedimentation velocity in an interacting system, it is first necessary to characterize any concentration-dependent changes in sedimentation velocity that may exist in solutions of individual reactant species in the absence of potential interaction partners (11). In the limit of slow interconversion, each of the macromolecular species present sediment independently, even though more slowly sedimenting species will no longer be in equilibrium with more rapidly sedimenting species after partial separation occurs. The complex multicomponent boundary thus formed will reflect contributions from each species in the proportions characterizing the mixture prior to sedimentation. An example of an experimental study performed by Creeth and Nichol on a slowly interconverting system is the characterization of three distinct species of urease, tentatively identified as monomer, dimer, and trimer (12); oligomers are presumably crosslinked by slowly exchangeable disulfide bonds.
MEASUREMENTS
3
In the limit of rapid interconversion of macromolecular species, only a single boundary is observed, and the sedimentation coefficient calculated from the velocity of that boundary will be a weight average of those of the contributing species (11). An example of an experimental study performed in the limit of rapid interconversion is the characterization by Lakatos of associations between subcomponents of complement factor Cl (13). In this study the species Clq was radiolabeled. The sedimentation velocity of the boundary containing a constant amount of radiolabel was found to vary with the concentration of each of several unlabeled species added to labeled Clq in a manner consistent with simple models for rapid equilibrium association between labeled and unlabeled species. It has been pointed out elsewhere (14) that models for calculation of weight-average sedimentation coefficients in solutions containing more than a single equilibrium complex require the specification of sedimentation coefficients of intermediate complexes. These cannot be measured independently and must be treated as variable parameters in the context of model fitting, thus contributing to uncertainty in the determination of association stoichiometries and equilibrium constants. Sedimentation Equilibrium When a solution containing interacting macromolecular solutes is sedimented to equilibrium, each solute species is not only at sedimentation-diffusion equilibrium with respect to itself at all radial positions within the cell, but also at chemical equilibrium with respect to all other solute species at the same radial position (2). Thus quantitation of the gradient or gradients of interacting species in a single centrifuge cell at sedimentation equilibrium can, in principle, provide thermodynamically rigorous information about equilibrium association of macromolecules over the entire range of solution compositions within the cell. Sedimentation equilibrium is most straightforwardly analyzed to obtain quantitative information about macromolecular associations by one of two methods, which we respectively term concentration analysis and molecular weight analysk5 Concentration analysis proceeds by construction of a model, simultaneously incorporating both association and sedimentation equilibrium conditions, that calculates one or more experimentally measurable concentration-dependent equilibrium properties (such as absorbance) as a function of radial position at fixed rotor velocity, temperature, etc. The model (or, preferably, each of several possible alternate models) is then fit directly to experimental data by nonlinear 5 For the sake of completeness we cite, but do not describe, a third method of analysis (15) that seems to have been used exclusively by L. W. Nichol and collaborators.
4
ALLEN
P. MINTON
least-squares methods to determine (a) whether a given model is consistent with the data, and if so, (b) how we11 the parameters of the model (stoichiometric coefficients, equilibrium association constants) are determined by the data. An example of concentration analysis is the study by Lewis and Youle (16) of the reversible association of the A and B subunits of ricin. Data are plotted together with the best fit of the preferred model in Fig. 2. Molecular weight analysis of sedimentation equilibrium data requires that the equilibrium gradient of each component be determined experimentally.6*7 The apparent weight-average molecular weight of each component present is then calculated as a function of the concentration of all components via the following relation (lJ9): &f(i)
=
wp
RT (1 - u,p)w,r
dwi - dr
’
where Mgfa is the apparent weight average weight of component i, wi is the w/v concentration of component i at radius r, vi is the partial specific volume of component i (assumed to be independent of the state of association of component i), and p is the density of the solution, which may be calculated to a good approximation as a linear function of the concentrations of all solutes (20). If the solution is sufficiently dilute in all macromolecular solutes so that each solute species is sedimenting ideally, then M$ = M$, the true weight-average molecular weight of component is (1). The set of primary data, consisting of the concentration of each component as a s Components are distinguished from species as follows. A solution containing an arbitrary number of oligomeric forms of substance X in equilibrium with monomeric X is said to contain a single component, namely the substance X, while each presumed association state of X as a separate species. A solution (e.g., X, X9, X3, - . -) is counted containing an arbitrary number of complexes of substances X and Y in equilibrium with monomeric X and monomeric Y is said to contain two components, namely substances X and Y, while each presumed association state of X and/or Y (e.g., X, X,, Y, Y2, XY, X,Y, . . .) is counted as a separate species. 7 This may be accomplished optically if different components have sufficiently different absorbance spectra, or if an absorbance difference is created by labeling one component with an optically distinguishing chromophore (17). Alternately, one of the components may be radiolabeled (6), or both components may be labeled with different radioisotopes. ’ lf, at any point in the centrifuge cell, the concentration of one or more species becomes so large that nonspecific repulsive (so-called nonideal) interactions between solute molecules affect the sedimentation equilibrium, resort must be made to approximate models in order to estimate the true weight average molecular weight from the apparent weight average molecular weight (1). Quasispherical particle models for the effect of nonideal interactions upon sedimentation equilibrium (19,21) in solutions containing only globular proteins, and compact complexes thereof, have been found to be semiquantitatively valid over a broad range of concentrations, and permit realistic calculation of large deviations from ideal behavior. Unfortunately, equiva-
6.7
6.6
6.9 RADIUS
7.0
7.1
7.2
(CM)
FIG. 2. Absorbance scans at 280 nm from cells containing equimolar amounts of ricin A and B chains at three loading concentrations. Solid curves represent the best simultaneous fit of a l-l association model to all three data sets. Figure reproduced from Ref. (16) with permission.
function of radius, is thus transformed into an equivalent set of secondary data, consisting of the weightaverage molecular weight of each component as a function of the concentrations of all components. A model for association that calculates the weight-average molecular weight of each component as a function of the concentrations of all components is then constructed (2,19). In contrast to models formulated for the concentration analysis of primary data, a model formulated for molecular weight analysis of secondary data contains no reference to the method (i.e., sedimentation equilibrium) by which the data were obtained.s lent hard particle models for nonideal interactions in solutions containing molecular species of significantly differing shapes (22) may only be validly used to calculate small deviations from ideal behavior in relatively dilute solutions. ’ For this reason it might appear that, given a certain scheme for self- or heteroassociation, the corresponding model for molecular weight analysis would be simpler than the corresponding model for concentration analysis. This is true for certain simple selfassociation schemes where the concentrations of individual species may be obtained as analytical functions of total component concentrations, but not true in general. In many cases, particularly those involving heteroassociation, models for concentration analysis may be considerably simpler than those for molecular weight analysis (Hsu and Minton, in preparation). It should also be noted that conversion of the primary data to secondary data involves evaluation of either dwJdr or d In wi/dr from data on wi as a function of r that inevitably contains noise. Hence the secondary data will ordinarily be less precise, and may be considerably less precise, than the corresponding primary data.
ASSOCIATION
AND
BINDING
MEASUREMENTS
An example of molecular weight analysis of an associating system containing one component is provided by the study of glutamate dehydrogenase self-association by Reisler et al. (23) together with the reanalysis of Chatelier (24). Combined data obtained from several equilibrium experiments are plotted together with the best fit of the preferred model in Fig. 3. TRACER
METHODS
The characterization of macromolecular self- and heteroassociations via analysis of sedimentation equilibrium data has, in the past, been hampered by the relatively small gradients of absorbance or refractive index that can be measured accurately with the optical systems currently available for the analytical ultracentrifuge.l’ This shortcoming has been alleviated to a great extent by the introduction of tracer methods. A tracer is created by labeling a chemical species with a chromophoric or radioisotope label. The labeled species qualifies as a true tracer only after it is ascertained via careful experimental testing that the labeled species behaves in a fashion that is chemically and physically indistinguishable from the parent unlabeled compound under the conditions of the binding measurementl’ (6,17). A series of solutions are prepared by adding different amounts of unlabeled parent species to a small, fixed amount of tracer sufficient to provide an appropriate level of signal (absorbance, radioactivity). Since the tracer, in principle, distributes uniformly throughout the unlabeled species, gradients of concentration, and hence apparent weight-average molecular weights, may be measured accurately over a very broad range of concentrations. This technique was employed by Muramatsu and Minton (25) to detect and characterize selfassociations in proteins over a concentration range exceeding two orders of magnitude. Heteroassociations may be detected and characterized quantitatively by analyzing changes in the gradient of a tracer label brought about by addition of either the parent unlabeled species or a second unlabeled component with which the parent species is presumed to associate. An example of such an analysis is the study by Servillo et al. (26) of the heteroassociation of apolipoproteins A-II and C-l. ” Performance specifications for the absorbance scanner in the new analytical centrifuge announced by Beckman Instruments are expected to significantly exceed those of the current Model R scanner with respect to the range of absorbance that can be measured with confidence. ‘I One must be particularly concerned with the possibility that the label somehow alters the interaction to be studied. Radiolabeled tracers are preferable to chromophorically labeled tracers in this regard, since radiolabels can be introduced by exchanging one or more atoms of the parent species with a radioisotope, with little or no alteration of chemical structure.
0
0.4
0.2
0.6
0.8
WT wo
FIG. 3. Plot of the ratio of apparent weight-average molecular weight to the molecular weight of monomer as a function of total protein concentration. Points, data from Ref. (23). Curve, best-fit of indefinite isoenthalpic association model. Figure reproduced from Ref. (24) with permission.
SUMMARY
AND
CONCLUSION
The ultracentrifuge provides several techniques for the quantitative characterization of reversible small molecule-macromolecule and macromolecule-macromolecule interactions in solution. The nature of the association to be studied determines the preferred technique. High speed centrifugation is the method of choice for characterizing reversible heteroassociations between species of greatly different mass (i.e., sedimentation coefficient). This technique provides a relatively rapid, artifact-free, and thermodynamically rigorous means of quantifying the amount of nonsedimenting or slowly sedimenting free ligand in equilibrium with rapidly sedimenting acceptor-bound ligand at one particular solution composition. Results obtained over a broad range of ligand and/or acceptor concentrations lead to model-independent binding isotherms that may subsequently be analyzed in the context of models for ligandacceptor association. Lower speed centrifugation to sedimentation equilibrium is the method of choice for characterizing reversible selfassociations and for characterizing heteroassociations between components that cannot be well separated on the basis of sedimentation velocity. In the dilute limit, this technique can provide model-free information about the dependence of weight-average molecular weight of each component upon solution composition, which can subsequently be analyzed in the context of equilibrium models for self- or heteroassociation. At higher concentrations, the models must be generalized to allow for the effect of nonspecific (nonideal) interactions upon sedimentation and association. The use of tracers provides a means for greatly ex-
6
ALLEN
P. MINTON
tending the range of solute concentrations and solution compositions over which both types of measurements may be applied, providing enhanced ability to discriminate between alternative proposed mechanisms for selfor heteroassociations. REFERENCES 1. Fujita, H. (1975) ley, New York.
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of Ultracentrifugal
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