The binding of a ligand to an acceptor undergoing indefinite self-association

Explicit expressions are derived which describe the binding of a univalent ligand to equivalent and independent sites on each state of an acceptor undergoing indefinite self-association that is governed by an isodesmic equilibrium constant KI. From considerations of systems in which the same site-binding constant kA applies to all acceptor-ligand interactions, the general forms of binding curves and Scatchard plots are deduced for situations in which binding sites are either created or lost at each monomer-monomer interface. Greater generality is then introduced into the model by allowing ligand interactions with polymeric acceptor states to be governed by a site-binding constant kp that differs in magnitude from that for monomeric acceptor kA. Finally, experimental results with the glutamate dehydrogenase-GTP and lysozyme-saccharide systems are used to illustrate ways in which the present quantitative expressions may be applied to the characterization of inteactions between a ligand and an indefinitely self-associating acceptor.     

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.     

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

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

Vincristine-induced self-association of calf brain tubulin

The vincristine-induced self-association of tubulin has been examined in a sedimentation velocity study as a function of free drug concentration in PG buffer (0.01 M NaPi and 10(-4) M GTP, pH 7.0) at 20 degrees C. Analysis of the weight-average sedimentation coefficient (S20,w) as a function of protein concentration showed a good fit with the model of an indefinite, isodesmic self-association mechanism. Analysis of the apparent association constants in terms of the Wyman linkage relations showed a good fit to mediation of the self-association by the binding of one ligand molecule. The intrinsic association constant for dimerization of the vincristine-liganded tubulin was found to be 3.8 X 10(5) M-1, and the intrinsic equilibrium constant for the binding of the self-association-linked vincristine molecule had a value of 3.5 X 10(4) M-1, consistent with that measured by fluorescence in our laboratory [Prakash, V., & Timasheff, S. N. (1983) J. Biol. Chem. 258, 1689-1697]. Both reactions are stronger in the presence of vincristine than of vinblastine, reflecting the oxidation of a -CH3 group to -CHO when going from the latter drug to the former one.     

Graphical analysis of nonideal monomer N-mer, isodesmic, and type II indefinite self-associating systems by equilibrium ultracentrifugation.

A graphical procedure is described by which one can obtain in principle the monomer molecular weight, stoichiometry, equilibrium constant, and second virial coefficient of nonideal monomer N-mer, isodesmic, and type II indefinite self-associating systems. In addition, a method is presented for obtaining both the equilibrium constant and the second virial coefficient from the maximum in a plot of apparent molecular weight vs. concentration if the monomer molecular weight and stoichiometry are known. The usefulness and limitations of the methods are discussed, as well as the quality and range of data required for determination of the relevant parameters. The techniques described are applicable to analysis of self-associating systems by osmotic pressure and light scattering, as well as equilibrium ultracentrifugation measurements.     

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.     

Self-association of myelin basic protein: enhancement by detergents and lipids

Self-association of basic protein has been proposed to be of functional significance in central nervous system myelin. In aqueous solution this protein self-associates, previous data being consistent with the formation of dimers, which then undergo an indefinite isodesmic self-association [Smith, R. (1980) Biochemistry 19, 1826-1831]. As this protein is membrane bound in vivo, we have now examined the effects of amphiphiles on the self-association equilibria. Contrary to the expected effects, at low molar ratios dodecyl sulfate, deoxycholate, Triton X-100, and lysophosphatidylcholine increased protein intermolecular attraction. The anionic detergents led to partial precipitation even at 1:1 protein:detergent molar ratios whereas the zwitterionic lipid and the nonionic detergent exerted less pronounced effects. Sedimentation velocity and equilibrium measurements have been used to define quantitatively the effects of lysophosphatidylcholine. The sedimentation coefficient increases up to a lipid:protein ratio of 4:1 and then remains constant up to a ratio of 12:1. The sedimentation equilibrium data suggest that the mode of protein-protein interaction is the same as in the absence of lipid but with substantially increased association constants. The dimerization constant is increased from 1.20 X 10(2) M-1 to 1.0 X 10(3) M-1 and the isodesmic association constant from 3.4 X 10(4) M-1 to 1.2 X 10(5) M-1. The effects of detergents on myelin basic protein are compared with the effects on other proteins, and the implications for the state of the protein with myelin are discussed.     

The self-association of zinc-free bovine insulin. A single model based on interactions in the crystal that describes the association pattern in solution at pH 2, 7 and 10

Sedimentation equilibrium studies are used to establish that a new pattern for the self-association of zinc-free insulin in solution is applicable over a wide range of conditions of pH, ionic strength and temperature. In this pattern, which is based on information from the existing literature on the X-ray crystal structure of insulin, the insulin monomer is viewed as having two distinct faces both capable of self-interaction. Sedimentation equilibrium experiments were analysed using expressions formulated for this association pattern that describe the dependence of weight average molecular weight and monomer concentration on total protein concentration. It has thereby been possible to obtain values for the two association constants which govern the system for each set of conditions studied, due allowance having been made for composition dependent non-ideality effects. Furthermore, by relating the pH, temperature and ionic strength dependence of the association constants with properties of various amino acid residues on the surface of the insulin monomer, it has also been possible to assign tentatively each constant to a particular reaction domain.     

Interaction of vinblastine with calf brain tubulin: multiple equilibria

The binding of the anticancer drug vinblastine to calf brain tubulin was measured by a batch gel filtration method in PG buffer (0.01 M NaPi, 10(-4) M GTP, pH 7.0) at three different protein concentrations. The Scatchard binding isotherms obtained were curvilinear. The binding of the first vinblastine molecule to each tubulin alpha-beta dimer (Mr 110,000) was enhanced by an increase in the protein concentration. Additional binding of vinblastine to the protein was independent of the protein concentration. Theoretical ligand binding isotherms were calculated for a ligand-induced macromolecule self-association involving various ligand stoichiometries and association schemes. Fitting of the experimental data to these isotherms showed that the system can be described best by a one-ligand-induced isodesmic indefinite self-association. The pathway giving the best fit consists of a ligand-mediated plus -facilitated self-association mechanism. The self-association-linked bound vinblastine binds specifically at a site with an intrinsic binding constant K1 = 4 X 10(4) M-1. Additional vinblastine molecules can bind less strongly to tubulin in probably nonspecific fashion, and the previous reports of two specific sites on alpha-beta tubulin for binding vinblastine are incorrect. The self-association constant K2 for liganded tubulin is 1.8 X 10(5) M-1. This analysis is fully consistent with the conclusions derived earlier from the linked function analysis of the vinblastine-induced tubulin self-association [Na, G. C., & Timasheff, S. N. (1980) Biochemistry 19, 1347-1354; Na, G. C., & Timasheff, S. N. (1980) Biochemistry 19, 1355-1365].     

Preferential ligand binding to multi-state acceptor systems: comparisons of the calcium-binding and dimerization characteristics of prothrombin and fragment 1.

Consideration is given to the interactions of a ligand with self-associating acceptor systems for which preferential binding is an ambiguous term in that ligand-mediated self-association does not necessarily imply a greater binding constant for polymeric acceptor--even in instances where binding sites are preserved in the self-association process. This dilemma is shown to arise in situations involving the binding of ligand to monomeric and polymeric forms of an acceptor that also coexist in equilibrium with inactive isomeric states. For example, the ten-fold increase in the measured dimerization constant for prothrombin Fragment 1 in the presence of a saturating concentration of Ca2+ ion may well reflect the existence of a 12% greater binding constant for the interaction of metal ion with dimeric acceptor. However, that result, as well as the detailed form of the sigmoidal binding curve, are also reasonably described by another extreme model in which the monomeric and dimeric forms of the acceptor possess equal affinities for Ca2+ ion. Likewise, the fact that the same experimental dimerization constant applies to prothrombin and its Ca(2+)-saturated complex does not preclude the possibility that the active form of dimeric zymogen exhibits a 12% greater affinity for metal ion. Numerical simulations have established that characterization of the dimerization behaviour as a function of free ligand concentration should allow greater discrimination between such models of the interplay between calcium binding and self-association of prothrombin and Fragment 1. Finally, by illustrating the likelihood that the disparity in self-association behaviour of prothrombin and Fragment 1 merely reflects minor differences in the relative magnitudes of isomerization constants and/or binding constants for monomeric and dimeric states of the two acceptors, the present investigation serves to allay concern about the validity of employing the proteolytic fragment as a model of the intact zymogen.     

Ligand-induced self-association of human luteinizing hormone. Negative cooperativity in the binding of 8-anilino-1-naphthalensulfonate.

The self-association of human luteinizing hormone (hLH) is enhanced in the presence of 8-anilino-1-naphthalenesulfonate (ANS). Sedimentation equilibrium measurements indicate that the hormone exists primarily as a dimer in the presence of excess ANS. It is shown that, for a self-associating protein system in which monomer and dimer have different affinities and/or capacities for ligand, both the shape and the position of the binding curve depend on protein concentration. Gel filtration and fluorescence measurements indicate that the hLH dimer has a single high affintiy site (K greater than 10(6) M-1) for ANS while binding to the monomer is too weak to be observed. This leads to negative cooperativity in the binding and to a shift of the binding curve to lower free ligand concentration with increasing concentration of the hormone.     

Self-association mode of a flavoenzyme D-amino acid oxidase from hog kidney. I. Analysis of apparent weight-average molecular weight data for the apoenzyme in terms of models

The self-association of D-amino acid oxidase apoenzyme in 0.1 M sodium pyrophosphate, pH 8.3, at 25 degrees C was studied by low-angle laser light scattering. The concentration (c) dependence of the apparent weight-average molecular weight (Mwapp) was determined over a wide concentration range of 0.04 to 6.1 mg/ml. The extrapolated Mwapp value, to zero enzyme concentration, corresponded to the Mr value of the monomer. The self-association mode of the apoenzyme was systematically explored with nonlinear least-squares analysis of the Mwapp versus c data. The simplest model that fitted the data well was a model of isodesmic indefinite self-association of the monomer with the isodesmic association constant of 0.467 +/- 0.034 liter/g. The monomer-dimer model proposed previously, but only in a low enzyme concentration range of less than 0.9 mg/ml at 5-20 degrees C (Henn, S. W., and Ackers, G. K. (1969) Biochemistry 8, 3829-3838), did not fit the Mwapp versus c data either in the limited low concentration range or in the whole concentration range examined at 25 degrees C. To test the validity of the chosen model, the observed sedimentation boundary profiles were compared with the idealized boundary profiles calculated for the better-fit models. The profile calculated with the model of the isodesmic indefinite self-association mechanism was qualitatively consistent with the observed ones. The utility of the nonlinear least-squares procedure for analyzing self-associating systems was demonstrated.     

The study of multiple polymerization equilibria by glass bead exclusion chromatography with allowance for thermodynamic non-ideality effects

Two related aspects are explored of the frontal exclusion chromatography of proteins employing controlled-pore glass beads as the stationary phase. First, it is shown theoretically that, despite the absence of osmotic shrinkage effects previously encountered with Sephadex matrices, the experimentally measurable partition coefficient of a single non-associating solute will be dependent on its concentration due to the differing ratios of activity coefficients in mobile and stationary phases at different total concentrations. The effect is demonstrated with results obtained using ovalbumin in phosphate buffer of pH 7.4, and is Shown to be consistent (up to a solute concentration of 5 $\text{g}\text{litre}$) with theoretical prediction formulated in terms of a single virial coefficient. Secondly, it is shown for self-associating systems that it is possible to determine the monomer concentration as a function of total concentration, provided the stationary phase is selected to ensure exclusion of all oligomeric species except monomer: the relation derived for this purpose accounts for the concentrationdependence of the partition coefficient of monomer, again as a first approximation involving one virial coefficient. Such information on the monomer concentration permits elucidation of the polymerization characteristics of the system in terms of the types of species present and the relevant equilibrium constants. The feasibility of the method, its likely sources of error and the relative contribution of the non-ideality effect are investigated using bovine glutamate dehydrogenase (up to a total concentration of 5.4 $\text{g}\text{litre}$) in phosphate buffer of pH 6.9. This system was selected since comparison was possible with results obtained by other methods, which have established the enzyme polymerization pattern as an isodesmic indefinite self-association. The isodesmic equilibrium constant of 1.5 ± 0.3 $\text{litre}\text{g}$ found in this work is in reasonable agreement with previous findings.     

The indefinite self-association of lysozyme: consideration of composition-dependent activity coefficients

An improved iterative method for computing association constants from sedimentation equilibrium results obtained with self-interacting protein systems is presented which accounts for the composition-dependence of the activity coefficients of all oligomeric species. The method is based on the calculation of virial coefficients from covolume and charge considerations, the statistical mechanical basis of which is discussed in relation to the DLVO theory. The method is applied to results obtained with lysozyme in diethylbarbiturate buffer of pH 8.0 and ionic strength 0.15 at 15°C. It is shown that these results, encompassing a range of total solute concentration up to 19.7 g/liter are consistent with self-association patterns comprising either a monomer-dimer-trimer system or an isodesmic indefinite self-association of the monomer, the latter being favored. A firmer distinction between these possibilities is sought on the basis of the dependence of the weight-average partition coefficient, determined by frontal gel chromatography, on total solute concentration (up to 56.6 g/liter). This analysis accounts for the composition-dependence of the ratio of the activity coefficients of partitioning monomer in mobile and stationary phases. It is concluded that all results are consistent with an indefinite self-association of lysozyme governed by a single association constant of 4.61 × 10² liter/mole.     

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

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

Osmotic pressure measurements on insulin: anomalous results indicate that the monomer is preferentially adsorbed

The concentration dependence of the number average molecular weight of insulin at pH 2, ionic strength 0.05, and 20 degrees C as determined by osmotic pressure measurements indicates that the hormone is a homogeneous protein of molecular weight close to that of the dimer. Since sedimentation equilibrium experiments confirm what is well known, namely that insulin is a self-associating protein dissociating to monomer under these conditions, an explanation for the anomaly was sought in the possible loss of protein from solution by adsorption. Analysis of the results strongly supports this conclusion and consideration of the adsorption properties of insulin in terms of hydrophobic interactions shows them to be consistent with the behaviour of insulin as a self-associating protein. The monomer appears to be the primary molecular species responsible for insulin adsorption.     

Computer stimulation of sedimentation in the ultracentrifuge. VII. Solutes undergoing indefinite self-association

The velocity sedimentation of solutes involved in self-association equilibria of the indefinite type was simulated using a computer model. The changes in boundary shape that resulted from varying the association constant and the molecular weight of the self-associating monomer were examined. Both ideally and nonideally sedimenting solutes were considered, and several alternative treatments of the variation of the frictional ratio with molecular size were used. All of the calculated boundaries were skewed, with the leading limb of the gradient profile steeper than the trailing limb. For relatively tightly self-associating solutes, the boundaries were very broad and strongly skewed. No shoulders or subsidiary peaks were observed for any of the model solutes used.     

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.     

Rapid quantitative characterization of protein interactions by composition gradient static light scattering

Two new applications of the recently developed technique of composition gradient static light scattering (CG-SLS) are presented. 1), The method is demonstrated to be capable of detecting and quantitatively characterizing reversible association of chymotrypsin and bovine pancreatic trypsin inhibitor in a solution mixture and simultaneously occurring reversible self-association of chymotrypsin at low pH in the same mixture. The values of equilibrium constants for both self- and heteroassociation may be determined with reasonable precision from the analysis of data obtained from a single experiment requiring <15 min and <1 mg of each protein. 2), Analysis of the results of a single CG-SLS experiment carried out on Ftsz, a protein that self-associates to form linear oligomers of indefinite size in the presence of guanosine diphosphate, yields the dependence of the equilibrium constant for monomer addition upon oligomer size.     

Characterization of the kinetic pathway for liberation of fibrinopeptides during assembly of fibrin

The time dependence of the release of fibrinopeptides from fibrinogen was studied as a function of the concentration of fibrinogen, thrombin, and Gly-Pro-Arg-Pro, an inhibitor of fibrin polymerization. The release of fibrinopeptides during fibrin assembly was shown to be a highly ordered process. Rate constants for individual steps in the formation of fibrin were evaluated at pH 7.4, 37 degrees C, gamma/2 = 0.15. The initial event, thrombin-catalyzed proteolysis at Arg-A alpha 16 to release fibrinopeptide A (kcat/Km = 1.09 X 10(7) M-1s-1) was followed by association of the resulting fibrin I monomers. Association of fibrin I was found to be a reversible process with rate constants of 1 X 10(6) M-1s-1 and 0.064 s-1 for association and dissociation, respectively. Assuming random polymerization of fibrin I monomer, the equilibrium constant for fibrin I association (1.56 X 10(7) M-1) indicates that greater than 80% of the fibrin I protofibrils should contain more than 10 monomeric units at 37 degrees C, pH 7.4, when the fibrin I concentration is 1.0 mg/ml. Association of fibrin I monomers was shown to result in a 6.5-fold increase in the susceptibility of Arg-B beta 14 to thrombin-mediated proteolysis. The 6.5-fold increase in the observed specificity constant from 6.5 X 10(5) M-1s-1 to 4.2 X 10(6) M-1s-1 upon association of fibrin I monomers and the rate constant for fibrin association indicates that most of the fibrinopeptide B is released after association of fibrin I monomers. The interaction between a pair of polymerization sites in fibrin I dimer was found to be weaker than the interaction of fibrin I with Gly-Pro-Arg-Pro and weaker than the interaction of fibrin I with fibrinogen.