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.     

Characterization of the metal ion-binding domains from rat alpha- and beta-parvalbumins

We have examined the metal ion-binding domains from rat alpha and beta parvalbumin. We find that the CD-EF fragments differ markedly in their tendency to self-associate. Whereas Ca(2+)-free alpha CD-EF is monomeric, the Ca(2+)-free beta peptide dimerizes weakly (K(2) = 2400 +/- 200 M(-1)). In buffer containing 1.0 mM Ca(2+), the apparent dimerization constant for beta CD-EF (191,000 +/- 29,000 M(-1)) is more than 50 times that of alpha (3400 +/- 200 M(-1)). Alpha CD-EF binds two Ca(2+) with positive cooperativity. Titration calorimetry data afford binding constants of 3.7(0.1) x 10(3) M(-1) and 8.6(0.2) x 10(4) M(-1). Beta CD-EF also binds two Ca(2+) cooperatively but with lower affinity. Equilibrium dialysis yields Adair constants of 4.2(0.1) x 10(3) and 6.1(0.2) x 10(3) M(-1). Significantly, the difference in Ca(2+) affinity is substantially smaller than that observed for the full-length proteins-suggesting that the AB domain can modulate divalent ion affinity. Analysis of beta calorimetry data requires explicit consideration of the self-association behavior. Data collected at low CD-EF concentration are consistent with preferential occupation of the EF site, dimerization of singly bound monomers, and cooperative filling of the CD sites. At higher concentrations, apo-protein dimerization can apparently precede cooperative occupation of the EF sites. In the presence of Ca(2+), alpha CD-EF exhibits higher thermal stability, consistent with its higher Ca(2+) affinity. However, the beta melting temperature shows greater concentration dependence, consistent with its greater tendency to dimerize. Neither fragment exhibits a sigmoidal melting curve in the Ca(2+)-free state, suggesting that the apo-peptides are disordered.     

The dimerization of ferrihaems. I. The effect of buffer ions and specific cations on deuteroferrihaem dimerization.

The dimerization of dueteroferrihaem in aqueous solution has been investigated using a parameter, named the dimerization index (Robs). This is defined as the ratio of extinction coefficients at wavelengths corresponding to Soret band maxima for the monomeric and dimeric species, respectively. For solutions containing mainly monomeric species, Robs greater than 2, whereas for solutions containing mainly dimeric species Robs less than 1. A computer programme has been applied to determine values of the dimerization constant, K, defined as: K = [dimer] [H+]/[monomer]2. Phosphate buffer anions and Tris . HCl buffer enhanced dimerization. Monovalent and divalent cations also increased dimerization, but in a specific manner. The magnitudes of their effects increased in the order K+ less than Na+ less than Li+ less than Sr2+ less than Mg2+ approximately or equal to Ca2+. Values of K were determined for several concentrations of Na+ and Sr2+. These data are interpreted in terms of a stabilization of the ferrihaem dimer by the formation of ion triplets with the added cation 'sandwiched' between carboxyl residues of the adjacent ferrihaem monomeric units. General guidelines are recommended for the choice of conditions which minimize dimerization.     

Characterization of detergent-solubilized sarcoplasmic reticulum Ca2+-ATPase by high-performance liquid chromatography

Sarcoplasmic reticulum Ca2+-ATPase solubilized by the nonionic detergent octaethylene glycol monododecyl ether was studied by molecular sieve high-performance liquid chromatography (HPLC) and analytical ultracentrifugation. Significant irreversible aggregation of soluble Ca2+-ATPase occurred within a few hours in the presence of less than or equal to 50 microM Ca2+. The aggregates were inactive and were primarily held together by hydrophobic forces. In the absence of reducing agent, secondary formation of disulfide bonds occurred. The stability of the inactive dimer upon dilution permitted unambiguous assignment of its elution position and sedimentation coefficient. At high Ca2+ concentration (500 microM), monomeric Ca2+-ATPase was stable for several hours. Reversible self-association induced by variation in protein, detergent, and lipid concentrations was studied by large-zone HPLC. The association constant for dimerization of active Ca2+-ATPase was found to be 10(5)-10(6) M-1 depending on the detergent concentration. More detergent was bound to monomeric than to dimeric Ca2+-ATPase, even above the critical micellar concentration of the detergent. Binding of Ca2+ and vanadate as well as ATP-dependent phosphorylation was studied in monomeric and in reversibly associated dimeric preparations. In both forms, two high-affinity Ca2+ binding sites per phosphorylation site existed. The delipidated monomer purified by HPLC was able to form ADP-insensitive phosphoenzyme and to bind ATP and vanadate simultaneously. These results suggest that formation of Ca2+-ATPase oligomers in the membrane is governed by nonspecific forces (low affinity) and that each polypeptide chain constitutes a functional unit.     

Dimerization of the Polymeric Immunoglobulin Receptor Controls Its Transcytotic Trafficking

Binding of dimeric immunoglobulin (Ig)A to the polymeric Ig receptor (pIgR) stimulates transcytosis of pIgR across epithelial cells. Through the generation of a series of pIgR chimeric constructs, we have tested the ability of ligand to promote receptor dimerization and the subsequent role of receptor dimerization on its intracellular trafficking. Using the cytoplasmic domain of the T cell receptor-ζ chain as a sensitive indicator of receptor oligomerization, we show that a pIgR:ζ chimeric receptor expressed in Jurkat cells initiates a ζ-specific signal transduction cascade when exposed to dimeric or tetrameric IgA, but not when exposed to monomeric IgA. In addition, we replaced the pIgR’s transmembrane domain with that of glycophorin A to force dimerization or with a mutant glycophorin transmembrane domain to prevent dimerization. Forcing dimerization stimulated transcytosis of the chimera, whereas preventing dimerization abolished ligand-stimulated transcytosis. We conclude that binding of dimeric IgA to the pIgR induces its dimerization and that this dimerization is necessary and sufficient to stimulate pIgR transcytosis.     

Evaluation of equilibrium constants for the binding of N-acetyl-L-tryptophan to monomeric and dimeric forms of alpha-chymotrypsin

The binding of N-acetyl-tryptophan to the monomeric and dimeric forms of alpha-chymotrypsin in I = 0.2 acetate-chloride buffer, pH 3.86, has been studied quantitatively. Equilibrium sedimentation studies in the absence of inhibitor yielded a dimerization constant of 3.5 L/g. This value was confirmed by frontal gel chromatography of the enzyme on Bio-Gel P-30, which was also used to establish that N-acetyl-L-tryptophan binds preferentially to monomeric enzyme. From kinetic studies of competitive inhibition with N-acetyl-L-tryptophan ethyl ester as substrate, an equilibrium constant of 1300 M-1 was determined for the binding of N-acetyl-L-tryptophan to monomeric alpha-chymotrypsin. An intrinsic binding constant of 250 M-1 for the corresponding interaction with dimeric enzyme was calculated on the basis of these results and binding data obtained with concentrated (18.5 g/L) alpha-chymotrypsin. The present results refute earlier claims for exclusive binding of competitive inhibitors to monomer and also those for equivalence of inhibitor binding to monomeric and dimeric forms of alpha-chymotrypsin.     

Preferential ligand binding to multi-state acceptor systems: the unexplored paradox of acceptor self-association that is ligand-mediated but detrimental to ligand binding

Consideration is given to the interactions of ligand with self-associating acceptor systems for which preferential ligand binding is an ambiguous term, in that the acceptor species with greater affinity for ligand possesses relatively fewer binding sites. A paradoxical situation wherein ligand-mediated self-association is seemingly detrimental to ligand binding is shown to be the predicted outcome for a transient range of ligand concentrations. This outcome reflects the existence of a critical point in the dependence of the extent of acceptor self-association upon ligand concentration that coincides with a cross-over point of ligand-binding curves for different, fixed total concentrations of acceptor. By classical differentiation methods the conditions for the existence of these critical points are established not only for two-state acceptor systems but also for three-state acceptor systems in which the ligand-binding form of monomer also undergoes reversible isomerization to an inactive state. Similar procedures are used to comment upon the forms of binding curves for the three-state acceptor systems, the Scatchard representations of which may exhibit as many as three critical points (two maxima and a minimum). This delineation of quantitative expressions for critical points and other distinctive features associated with the conflicting interplay of ligand-binding and self-association behaviour should provide a more definitive means of characterizing systems with one acceptor state the preferred binding form on affinity grounds but with the other the preferred state from the viewpoint of binding-site numbers.     

The role of ecotin dimerization in protease inhibition

Ecotin is a homodimeric protein from Escherichia coli that inhibits many serine proteases of the chymotrypsin fold, often with little effect from the character or extent of enzyme substrate specificity. This pan-specificity of inhibition is believed to derive from formation of a heterotetrameric complex with target proteases involving three types of interface: the dimerization interface, a primary substrate-like interaction, and a smaller secondary interaction between the partner ecotin subunit and the protease. A monomeric ecotin variant (mEcotin) and a single-chain ecotin dimer (scEcotin) were constructed to study the effect of a network of protein interactions on binding affinity and the role of dimerization in broad inhibitor specificity. mEcotin was produced by inserting a beta-turn into the C-terminal arm, which normally exchanges with the other subunit. While the dimerization constant (K(dim)) of wild-type (WT) ecotin was found to be picomolar by subunit exchange experiments using FRET and by association kinetics, mEcotin was monomeric up to 1 mM as judged by gel filtration and analytical centrifugation. A crystal structure of uncomplexed mEcotin to 2.0 A resolution verifies the design, showing a monomeric protein in which the C-terminal arm folds back onto itself to form a beta-barrel structure nearly identical to its dimeric counterpart. The kinetic rate constants and equilibrium dissociation constants for monomeric and dimeric ecotin variants were determined with both trypsin and chymotrypsin. The effect of the secondary binding site on affinity was found to vary inversely with the strength of the interaction at the primary site. This compensatory effect yields a nonadditivity of up to 5 kcal/mol and can be explained in terms of the optimization of binding orientation. Such a mechanism of adaptability allows femtomolar affinities for two proteases with very different specificities.     

Regulated dimerization of the thyrotropin-releasing hormone receptor affects receptor trafficking but not signaling

To investigate the function of dimerization of the TRH receptor, a controlled dimerization system was developed. A variant FK506 binding protein (FKBP) domain was fused to the receptor C terminus and dimerization induced by incubating cells with dimeric FKBP ligand, AP20187. The TRH receptor-fusion bound hormone and signaled normally. Addition of dimerizer to cells expressing the receptor-FKBP fusion dramatically increased the fraction of receptor running as dimer on SDS-PAGE. AP20187 caused dimerization in a time- and concentration-dependent manner, acting within 1 min. Dimerizer had no effect on TRH receptors lacking the FKBP domain, and its effects were blocked by excess monomeric FKBP ligand. AP20187-induced dimerization did not cause receptor phosphorylation, inositol phosphate production, or ERK1/2 activation, and dimerizer did not alter signaling by TRH. Induced dimerization did, however, alter TRH receptor trafficking. TRH promoted greater receptor internalization in cells treated with AP20187 but not monomeric ligand, based on loss of surface binding sites and immunostaining. Dimerization increased the rate of internalization of TRH receptors and decreased the apparent rate of receptor recycling. AP20187 enhanced the small amount of TRH-induced receptor internalization when the receptor-FKBP fusion protein was expressed in cells lacking beta-arrestins. The results show that controlled dimerization of the TRH receptor potentiates hormone-induced receptor trafficking.     

Modulation of dimerization, binding, stability, and folding by mutation of the neurophysin subunit interface

Bovine neurophysins, which have typically served as the paradigm for neurophysin behavior, are metastable in their disulfide-paired folded state and require ligand stabilization for efficient folding from the reduced state. Studies of unliganded porcine neurophysin (oxytocin-associated class) demonstrated that its dimerization constant is more than 90-fold greater than that of the corresponding bovine protein at neutral pH and showed that the increased dimerization constant is accompanied by an increase in stability sufficient to allow efficient folding of the reduced protein in the absence of ligand peptide. Using site-specific mutagenesis of the bovine protein and expression in Escherichia coli, the functional differences between the bovine and porcine proteins were shown to be attributable solely to two subunit interface mutations in the porcine protein, His to Arg at position 80 and Glu to Phe at position 81. Mutation of His-80 alone to Arg had a relatively small impact on dimerization, while mutation to either Glu or Asp markedly reduced dimerization in the unliganded state, albeit with apparent retention of the positive linkage between dimerization and binding. Comparison of the peptide-binding constants of the different mutants additionally indicated that substitution of His-80 led to modifications in binding affinity and specificity that were independent of effects on dimerization. The results demonstrate the importance of the carboxyl domain segment of the subunit interface in modulating neurophysin properties and suggest a specific contribution of the energetics of ligand-induced conformational change in this region to the overall thermodynamics of binding. The potential utility to future studies of the self-folding and monomeric mutants generated by altering the interface is noted.     

Characterizing bathocuproine self-association and subsequent binding to Alzheimer's disease amyloid beta-peptide by NMR.

Aggregated amyloid beta-peptide (A beta) is the primary constituent of the extracellular plaques and perivascular amyloid deposits associated with Alzheimer's disease (AD). Deposition of the cerebral amyloid plaques is thought to be central to the disease progression. One such molecule that has previously been shown to 'dissolve' deposited amyloid in post-mortem brain tissue is bathocuproine (BC). In this paper 1H NMR chemical shift analysis and pulsed field gradient NMR diffusion measurements were used to study BC self-association and subsequent binding to A beta. The results show that BC undergoes self-association as its concentration increases. The association constant of BC dimerization, Ka, was estimated to be 0.64 mM(-1) at 25 degrees C from 1H chemical shift analysis. It was also found that dimerization of BC appeared to be essential for its binding to A beta. From the self-association constant of BC, Ka, the fraction of dimeric BC in the complex was obtained and the dissociation constant, Kd, of BC bound to A beta40 peptide was then determined to be approximately 1 mM.     

Dye binding probes of lipid-binding structures. An investigation of 2-p-toluidinylnaphthylene-6-sulfonate binding to human and bovine prothrombin and fragment 1 in the presence and absence of calcium and magnesium ions

TNS (2-p-toluidinylnaphthylene-6-sulfonate) binds to human and bovine prothrombin and Fragment 1 in the absence and presence of added Ca2+. The stoichiometry of TNS binding is 1:1 for human and bovine prothrombin and Fragment 1. The Ca2+-dependence of the fluorescence of TNS bound to bovine prothrombin Fragment 1 yields a modified Hill plot slope of 2.7, which is consistent with the slope obtained by monitoring the Ca2+ dependence of protein fluorescence quenching, CD changes and phospholipid binding. Mg2+ has have no effect on the fluorescence of TNS-prothrombin fluorescence. TNS binding to the amino-terminal region of prothrombin is the first relatively simple probe of the subtle and complex relationship which exists between protein structure and phospholipid binding.     

The nature of the slow metal ion-dependent conformational transition in bovine prothrombin.

Kinetic parameters characterizing the slow structural isomerization observed via metal ion-dependent intrinsic fluorescence quenching of bovine prothrombin Fragment 1 have been determined. From forward and reverse rate constants, an equilibrium constant of approx. 0.25 is calculated. This result is consistent with the hypothesis that there exists, in the absence of metal ions, an equilibrium between two forms of bovine Fragment 1, one of which can interact rapidly with Ca2+ and subsequently with phospholipid. The other form of Fragment 1 cannot interact with Ca2+ in a manner that yields a phospholipid-binding form of the protein. Interconversion of these two forms of Fragment 1 occurs and may involve the isomerization of a proline residue.     

Interactions between micellar ligand systems and acceptors: Forms of binding curves

A previously formulated expression describing the competitive binding to an acceptor of two states of a ligand, monomeric and polymeric, coexisting in equilibrium is examined in terms of the different forms of Scatchard plots which may arise in cases of exclusive and of preferential binding of the ligand states. It is shown by differentiation of the binding equation written in Scatchard format, and by numerical examples, that exclusive binding of the monomeric form of ligand leads to Scatchard plots that are either sigmoidal or convex to the abscissa, whereas exclusive binding of the polymeric form results in plots concave to the abscissa and exhibiting a maximum. Particular attention is given to Scatchard plots which possess two critical points, a situation which is shown to be possible when the polymeric form of ligand binds preferentially (but not exclusively) to the acceptor. The two-state ligand concept is especially pertinent to solutes capable of globular micelle formation and several examples are cited of binding studies which have been conducted with such micellar systems. Of these, the chlorpromazine-brain tubulin system is given detailed consideration in order to illustrate the use of the present theory in describing the binding results which exhibit two critical points when plotted in Scatchard format.     

The self-association of human apolipoprotein A-IV. Evidence for an in vivo circulating dimeric form

We have investigated the self-association properties of human apolipoprotein A-IV using several complementary physical techniques. Sedimentation equilibrium analysis demonstrated that human apolipoprotein A-IV formed oligomeric species in aqueous solution at physiologic pH. Computer analysis established that the best model of self-association is a monomer-dimer-tetramer scheme, with an unusually large monomer-dimer association constant of 2.9 X 10(5) liters/mol. Fluorescence spectroscopy and electrophoretic analysis demonstrated that the rate of monomer-oligomer interconversion is sufficiently slow that a stable population of dimeric protein exists in solution, even at low total protein concentrations, and that the extent of dimerization is minimally influenced by pH. Moreover, these techniques established that the dissociation of oligomeric forms and the unfolding of the monomeric form are discrete and sequential events. In experiments where apolipoprotein A-IV was incubated with human high density lipoproteins, fractionated by gradient gel electrophoresis, and localized by immunoblotting, dimer formation occurred, but very little binding to lipoproteins was observed. Immunoblots of human serum fractionated on acrylamide gradient gels and isopycnic density gradients demonstrated an apolipoprotein A-IV band of size and density consistent with a circulating dimeric form, unassociated with lipid. We conclude that human apolipoprotein A-IV undergoes high affinity self-association in aqueous solutions, and that such self-association likely occurs in vivo. Self-association may thus be important in determining the biologic behavior of human apolipoprotein A-IV by influencing both the kinetics and distribution of its association with plasma lipoproteins.     

Different affinities of glycosaminoglycan oligosaccharides for monomeric and dimeric interleukin-8: a model for chemokine regulation at inflammatory sites

The binding of interleukin-8 (IL-8) to heparan sulfate (HS) proteoglycans on the surface of endothelial cells is crucial for the recruitment of neutrophils to an inflammatory site. Fluorescence anisotropy measurements yielded an IL-8 dimerization constant of 120 nM. The binding affinities, obtained by isothermal fluorescence titration, of size-defined heparin and HS oligosaccharides to the chemokine were found to depend on the oligomerization state of IL-8: high affinity was detected for monomeric and low affinity was detected for dimeric IL-8, referring to a self-regulatory mechanism for its chemoattractant effect. The highest affinity for monomeric IL-8 was detected for the HS octamer with a K(d) < 5 nM whereas the dissociation constants of dimeric IL-8 were found in the medium micromolar range. No indication for increasing affinities for monomeric IL-8 with increasing oligosaccharide chain length was found. Instead, a periodic pattern was obtained for the dissociation constants of the GAG oligosaccharides with respect to chain length, referring to optimum and least optimum chain lengths for IL-8 binding. GAG disaccharides were identified to be the minimum length for chemokine binding. Conformational changes of the dimeric chemokine, determined using CD spectroscopy, were detected only for the IL-8/HS complexes and not for heparin, pointing to an HS-induced activation of the chemokine with respect to receptor binding. Thermal unfolding of IL-8 yielded a single transition at 56 degrees C which was completely prevented by the presence of undigested HS or heparin, indicating structural stabilization, thereby prolonging the biological effect of the chemokine.     

Effect of a specific inhibitor on the unfolding and refolding kinetics of dimeric triosephosphate isomerase: establishing the dimeric and similarly structured nature of the main transition states on the forward and backward reactions

2-Phosphoglycolate (PGA), a strong competitive inhibitor of the dimeric enzyme triosephosphate isomerase (TIM), brings about a large decrease in the unfolding rate constant of the protein. The data set of rate constants versus ligand concentration may be equally well explained by regarding either a monomeric or a dimeric transition state (TS). However, if the thermodynamics for binding of PGA to native TIM is taken into account, it becomes clear that a dimeric TS is the right assumption. Furthermore, by studying the effect of the ligand on the second-order refolding reaction, we found results indicating similar PGA-binding affinities to be present in the transition states for the rate-limiting steps of the forward and backward reactions. Most likely, therefore, both TS resemble each other in respect to the active site architecture. It should be mentioned, however, that our data do not rule out the possible occurrence of an unstable, (partially) folded monomeric intermediate, which would rapidly interconvert with the unfolded monomer.     

Determinants of intra versus intermolecular self-association within the regulatory domains of Rlk and Itk

A protein fragment from the Tec family member Rlk (also known as Txk) containing a single proline-rich ligand adjacent to a Src homology 3 (SH3) domain has been investigated by nuclear magnetic resonance (NMR) spectroscopy. Analysis of the concentration dependence of the chemical shifts, NMR linewidths and self-diffusion coefficients reveal that the Rlk fragment dimerizes in solution. Mutation of two critical prolines in the proline-rich ligand abolishes dimerization. Furthermore, analysis of the extrapolated chemical shifts at infinite dilution reveal that intramolecular binding of the proline-rich ligand to the SH3 domain is disfavored. This is in contrast to the corresponding fragment of Itk, for which the proline-rich ligand/SH3 interaction occurs exclusively in an intramolecular fashion and no intermolecular binding is observed. Comparison of the Itk and Rlk sequences reveals that Rlk contains five fewer residues than Itk in the linker region between the proline-rich ligand and the SH3 domain. To assess whether linker length is a molecular determinant of intra- versus intermolecular self-association, we varied the length of the linker in both Rlk and Itk and analyzed the resulting variants by NMR. Intramolecular binding in Itk is reduced by shortening the linker and conversely a longer linker between the proline-rich ligand and the SH3 domain in Rlk enhances intramolecular self-association. Association constants for the binding of peptides corresponding to the proline-rich ligand with their respective SH3 domains were also measured by NMR. The protein/peptide data combined with the association constants for binding of each proline-rich peptide to the corresponding SH3 domain provide an explanation for the opposing modes of self-association within the otherwise closely related Rlk and Itk proteins.     

Crystal structure of the cysteine-rich domain of scavenger receptor MARCO reveals the presence of a basic and an acidic cluster that both contribute to ligand recognition

MARCO is a trimeric class A scavenger receptor of macrophages and dendritic cells that recognizes polyanionic particles and pathogens. The distal, scavenger receptor cysteine-rich (SRCR) domain of the extracellular part of this receptor has been implicated in ligand binding. To provide a structural basis for understanding the ligand-binding mechanisms of MARCO, we have determined the crystal structure of the mouse MARCO SRCR domain. The recombinant SRCR domain purified as monomeric and dimeric forms, and their structures were determined at 1.78 and 1.77 A resolution, respectively. The monomer has a compact globular fold with a twisted five-stranded antiparallel beta-sheet and a long loop covering a single alpha-helix, whereas the dimer is formed via beta-strand swapping of two monomers, thus containing a large eight-stranded beta-sheet. Calculation of the surface electrostatic potential revealed that the beta-sheet region with several arginines forms a basic cluster. Unexpectedly, an acidic cluster was found in the long loop region. In the monomer, the acidic cluster is involved in metal ion binding. Studies with cells expressing various SRCR domain mutants showed that all of the arginines of the basic cluster are involved in ligand binding, suggesting a cooperative binding mechanism. Ligand binding is also dependent on the acidic cluster and Ca2+ ions whose depletion appears to affect ligand binding at least by modulating the electrostatic potential or relative domain orientation. We propose that the SRCR domain dimerization can contribute to the recognition of large ligands by providing a means for the MARCO receptor oligomerization.     

Magnesium-induced self-association of calf brain tubulin. II. Thermodynamics.

The thermodynamic parameters of the magnesium ion induced self-association of calf brain tubulin in pH 7.0, 0.01 M phosphate buffer containing 10(-4) M GTP, were determined from sedimentation velocity experiments. This reaction proceeds by an isodesmic mechanism terminated by the highly favored formation of a closed ring shaped polymer of degree of association 26 +/- 4. Analysis of the variation of the apparent dimerization constant in the isodesmic mechanism s,ows that this self-association is characterized by positive enthalpy, entropy, heat capacity, and molar volume changes, as well as the binding of one additional magnesium ion, which is probably not involved as a bridge between the protein molecules. The addition of the last monomeric subunit has a free energy which is about three times that of dimer formation. Under the conditions of these experiments, tubulin binds 48 +/- 5 magnesium ions with a free energy of --2.8 kcal/mol.