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.     

The sulphatase of ox liver. XIX. On the nature of the polymeric forms of sulphatase A present in dilute solutions.

Weight-average elution volumes of sulphatase A (an arylsulphate sulphohydrolase, EC 3.1.6.1) from Sephadex G-200 have been determined as functions of protein concentration, pH, ionic strength and temperature. The results are used to calculate the apparent association equilibrium constants for tetramer formation and the associated standard-state thermodynamic parameters. While the apparent association constant decreased from 10(28) to 10(21) M-3 on increasing the pH from 4.5 to 5.6 at ionic strength 0.1, at any particular pH value studied it was relatively insensitive to temperature variation so that deltaH is close to zero and tetramer formation in solution is associated with a positive entropy change. At pH 5.0, increasing the ionic strength from 0.1 to 2 decreased the association constant by a factor of 100. Methylumbelliferone sulphate has no effect on the association of sulphatase A. The equilibrium results are used to define the degree of association of sulphatase A likely to encountered in experiments designed to elucidate its kinetic properties. In the liver lysosome, the tetramer is probably the dominant species. The monomer and tetramer of sulphatase A have similar, or identical, specific activities with nitrocatechol sulphate and 4-methylumbelliferone sulphate as substrates. With nitrocatechol sulphate, sulphatase A shows Michaelis kinetics under conditions where the monomer is the dominant species and non-Michaelis kinetics where the tetramer is dominant. There is apparently a negative cooperativity between the monomer units in the tetramer. In 2 mM sodium taurodeoxycholate and 0.035 M MnCl2, but not in 0.1 M NaCl, the tetramer shows Michaelis kinetics. This is not due to dissociation of the tetramer. The critical micellar concentration of sodium taurodeoxycholate is about 0.8 mM in both 0.1 M NaCl and 0.035 M McCl2 but the aggregation number is greater in the latter.     

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.     

The sulphatase of ox liver XIX. On the nature of the polymeric forms of sulphatase a present in dilute solutions

Weight-average elution volumes of sulphatase A (an arysulphate sulphohydrolase, EC 3.1.6.1) from Sephadex G-200 have been determined as functions of protein concentration, pH, ionic strength and temperature. The results are used to calculate the apparent association equilibrium constants for tetramer formation and the associated standard-state thermodynamic parameters. While the apparent association constant decreased from 10²⁸ to 10²¹ M⁻³ on increasing the pH from 4.5 to 5.6 at ionic strength 0.1 at any particular pH value studied it was relatively insensitive to temperature variation so that ΔH° is close to zero and tetramer formation in solution is associated with a positive entropy change. At pH 5.0, increasing the ionic strength from 0.1 to 2 decreased the association constant by a factor of 100. Methylumbelliferone sulphate has no effect on the association of sulphatase A.The equilibrium results are used to define the degree of association of sulphatase A likely to be encountered in experiments designed to elucidate its kinetic properties. In the liver lysosome, the tetramer is probably the dominant species.The monomer and tetramer of sulphatase A have similar, or identical, specific activities with nitrocatechol sulphate and 4-methylumbelliferone sulphate as substrates. With nitrocatechol sulphate, sulphatase A shows Michaelis kinetics under conditions where the monomer is the dominant species and non-Michaelis kinetics where the tetramer is dominant. There is apparently a negativity cooperative between the monomer units in the tetramer.In 2 mM sodium taurodeoxycholate and 0.035 M MnCl₂, but not in 0.1 M NaCl, the tetramer shows Michaelis kinetics. This is not due to dissociation of the tetramer. The critical micellar concentration of sodium taurodeoxycholate is about 0.8 mM in both 0.1 M NaCl and 0.035 M MnCl₂ but the aggregation number is greater in the latter.     

Hidden self-association of proteins

Sedimentation equilibrium measurements were carried out on solutions of bovine serum albumin, aldolase, and ovalbumin in phosphate-buffered saline, pH 7.2, at 10 degrees C. The data obtained for each protein were analyzed to yield the dependence of apparent weight-average molecular weight upon protein concentration, over a concentration range of ca 1-200 g/L. Using the approximate theory of Chatelier and Minton [1987) Biopolymers 26, 507-524), models are formulated for the dependence of apparent weight-average molecular weight upon concentration in non-ideal solutions containing proteins which may self-associate according to a monomer/n-mer or a monomer/dimer/tetramer scheme. The concentration dependence data for serum albumin may be accounted for, assuming either no self-association or weak monomer/dimer association. The data for aldolase may be accounted for assuming either weak monomer/dimer or weak monomer/trimer association. The data for ovalbumin may be accounted for assuming either weak monomer/trimer or weak monomer/dimer/tetramer association. The associations do not approach saturation at the highest concentrations studied, and the standard-state free energy changes accompanying self-association amount to less than 4 kcal/mol of intermolecular contacts, suggesting that non-specific clustering of protein molecules at high concentration rather than the formation of specific complexes is being observed.     

Human platelet factor 4 monomer-dimer-tetramer equilibria investigated by 1H NMR spectroscopy

As a function of protein concentration, proton NMR spectra of human platelet factor 4 (PF4) differ. Correlation with low-angle laser light scattering data has allowed identification of concentration-dependent NMR spectral changes to PF4 aggregation, with tetramers being the largest aggregates formed. Well-resolved aromatic ring proton NMR resonances were assigned to Tyr-60, His-I, and His-II in monomer, dimer, and tetramer states. Since Tyr-60 3.5 ring proton resonances are well resolved from state to state, estimation of fractional populations in each state was possible. By varying the PF4 concentration, changes in these populations when plotted according to the Hill equation show a bimolecular mechanism of aggregation which proceeds from monomers to tetramers through a dimer intermediate. Equilibrium constants for dimer association (KD) and tetramer association (KT) have been estimated as a function of pH and ionic strength. At pH 4, where KD and KT approach the same value, resonances associated with all three aggregate states are observed. Lowering the pH shifts the equilibrium to the monomer state, while raising the pH shifts the equilibrium to dimer and tetramer states. Analysis of the pH dependence of KD and KT suggests that electrostatic interactions, probably arising from Glu/Asp and Lys/Arg side chains, play a role in the binding process. Increasing the solvent ionic strength stabilizes the tetramer state especially at low pH, suggesting that intersubunit, repulsive electrostatic interactions probably between/among cationic side chains (Lys/Arg) attenuate the aggregation process. Information based primarily on histidine pKa values and photo-CIDNP 1H NMR data suggests that Tyr-60 and His-I, but not His-II, are significantly affected by the aggregation process.     

The temperature-dependent self-association of beta-lactoglobulin C in glycine buffers

The self-association of β-lactoglobulin C at low pH (ca. 2.5) in glycine buffers has been studied at four temperatures, 10, 16, 20, and 25 °C, by low- and high-speed sedimentation equilibrium experiments. One buffer had an ionic strength of 0.1 and the other an ionic strength of 0.2. With either buffer the concentration dependence of the apparent weight average molecular weight, M_wa, was characteristic of a nonideal self-association. Like its genetic variants, β-lactoglobulin A and B, the self-association of β-lactoglobulin C increased with decreasing temperature; however, at the same temperature the association was always stronger in the buffer having the higher ionic strength. Several models were used to test the self-association, and a monomer-dimer self-association seemed to describe the self-association best with either buffer. Values of the association equilibrium constant, K₂, and the second virial coefficient, BM₁, are reported at each temperature for both series of experiments. Values of the thermodynamic functions, ΔG °, ΔH °, and ΔS °, are also reported for these experiments.     

Standard thermodynamic formation properties for the adenosine 5'-triphosphate series.

The criterion for chemical equilibrium at specified temperature, pressure, pH, concentration of free magnesium ion, and ionic strength is the transformed Gibbs energy, which can be calculated from the Gibbs energy. The apparent equilibrium constant (written in terms of the total concentrations of reactants like adenosine 5'-triphosphate, rather than in terms of species) yields the standard transformed Gibbs energy of reaction, and the effect of temperature on the apparent equilibrium constant at specified pressure, pH, concentration of free magnesium ion, and ionic strength yields the standard transformed enthalpy of reaction. From the apparent equilibrium constants and standard transformed enthalpies of reaction that have been measured in the adenosine 5'-triphosphate series and the dissociation constants of the weak acids and magnesium complexes involved, it is possible to calculate standard Gibbs energies of formation and standard enthalpies of formation of the species involved at zero ionic strength. This requires the convention that the standard Gibbs energy of formation and standard enthalpy of formation for adenosine in dilute aqueous solutions be set equal to zero. On the basis of this convention, standard transformed Gibbs energies of formation and standard transformed enthalpies of formation of adenosine 5'-trisphosphate, adenosine 5'-diphosphate, adenosine 5'-monophosphate, and adenosine at 298.15 K, 1 bar, pH = 7, a concentration of free magnesium ions of 10(-3) M, and an ionic strength of 0.25 M have been calculated.     

Interactions of lysozyme in concentrated electrolyte solutions from dynamic light-scattering measurements

The diffusion of hen egg-white lysozyme has been studied by dynamic light scattering in aqueous solutions of ammonium sulfate as a function of protein concentration to 30 g/liter. Experiments were conducted under the following conditions: pH 4-7 and ionic strength 0.05-5.0 M. Diffusivity data for ionic strengths up to 0.5 M were interpreted in the context of a two-body interaction model for monomers. From this analysis, two potential-of-mean-force parameters, the effective monomer charge, and the Hamaker constant were obtained. At higher ionic strength, the data were analyzed using a model that describes the diffusion coefficient of a polydisperse system of interacting protein aggregates in terms of an isodesmic, indefinite aggregation equilibrium constant. Data analysis incorporated multicomponent virial and hydrodynamic effects. The resulting equilibrium constants indicate that lysozyme does not aggregate significantly as ionic strength increases, even at salt concentrations near the point of salting-out precipitation.     

Yeast hexokinase: substrate-induced association--dissociation reactions in the binding of glucose to hexokinase P-II.

A method is described for the purification of native hexokinases P-I and P-II from yeast using preparative isoelectric focussing to separate the isozymes. The binding of glucose to hexokinase P-II, and the effect of this on the monomer--dimer association--dissociation reaction have been investigated quantitatively by a combination of titrations of intrinsic protein fluorescence and equilibrium ultracentrifugation. Association constants for the monomer-dimer reaction decreased with increasing pH, ionic strength and concentration of glucose. Saturating concentrations of glucose did not bring about complete dissociation of the enzyme showing that both sites were occupired in the dimer. At pH 8.0 and high ionic strength, where the enzyme existed as monomer, the dissociation constant of the enzyme-glucose complex was 3 X 10(-4) mol 1(-1) and was independent of the concentration of enzyme. Binding to the dimeric form at low pH and ionic strength (I=0.02 mol 1(-1), pH less than 7.5) was also independent of enzyme concentration (in the range 10-1000 mug ml-1) but was much weaker. The process could be described by a single dissociation constant, showing that the two available sites on the dimer were equivalent and non-cooperative; values of the intrinsic dissociation constant varied from 2.5 X 10(-3) mol 1(-1) at pH 7.0 to 6 X 10(-3) at pH 6.5. Under intermediate conditions (pH 7.0, ionic strength=0.15 mol 1(-1)), where monomer and dimer coexisted, the binding of glucose showed weak positive cooperatively (Hill coefficient 1.2); in addition, the binding was dependent upon the concentration of enzyme in the direction of stronger binding at lower concentrations. The results show that the phenomenon of half-sites reactivity observed in the binding of glucose to crystalline hexokinase P-II does not occur in solution; the simplest explanation of our finding the two sites to be equivalent is that the dimer results from the homologous association of two identical subunits.     

Gel filtration studies on oxyhemerythrin. II. Effect of temperature and ionic strength on the association-dissociation equilibria.

The effects of temperature and ionic strength on the association of oxyhemerythrin have been studied. deltaH degrees and deltaS degrees for association at pH 7.0 are -2.6 kcal and +16.5 eu per mol of monomer. These values suggest that solvent adjacent to the surface of the protein undergoes rearrangement on association. Increasing ionic strength is observed to promote dissociation while decreasing the rate of attainment of equilibrium between monomers and octamers. Qualitatively similar results are observed on lowering the pH from 7.0 to 4.8, thereby linking the effects of increasing ionic strength to those of protonation of specific amino acid residues at the subunit contacts of hemerythrin. The apparent enthalpy of ionization of the amino acid residue controlling dissociation at acidic pH was found to be -1.9 to +2.1 kcal/mol. These values are consistent with a carboxyl group.     

The self-association of chicken-erythrocyte histones.

The self-association of the separate histone fractions isolated from chicken erythrocytes has been studied in solution at a number of different pH values and ionic strengths. The apparent molecular weights of the histones were determined over a range of macromolecular concentrations using the techniques of osmotic pressure and sedimentation equilibrium. Histone F2c (H5) did not associate under any of the conditions investigated whereas the other histone fractions all appeared to undergo self-association forming dimers, dimers of dimers, etc. The degree of association increased with the pH and ionic strength of the medium. The tendency to aggregate increased in the order; histone F2c (H5) (non-aggregating), histone F2b (H2B), histone F2a2 (H2A), histone F3 (H3), histone F2a1 (H4) (highly aggregating). In the case of histone F2a2 (H2A) at pH 3.0 and ionic strength 0.1, the apparent weight-average molecular weight was determined at a number of macromolecular concentrations at five different temperatures. The self-association was analysed according to the method of Adams (published by Beckman Instruments Inc. in 1967) and shown to be a monomer-dimer-tetramer equilibrium. The association constants were evaluated at each of the temperatures studied and from their variation with temperature the values of the enthalpy and entropy of association were calculated. The intermolecular association was characterised by only a small change in enthalpy but a large, positive, change in entropy. This suggests that the association of histones at acid pH is due to hydrophobic interactions between the relatively uncharged segments of like polypeptide chains.     

The polymerization pattern of zinc(II)-insulin at pH 7.0.

Sedimentation equilibrium experiments were conducted at pH 7.0 using solutions of bovine insulin containing 2 mol of zinc(II) ions per six base-mol of insulin. A detailed analysis of these results revealed the existence of a stable zinc-insulin hexamer together with linked polymerization reactions. Specifically these are a background polymerization of zinc-free insulin as previously described by Jeffrey et al. ((1976) Biochemistry 15, 4660--4665) and a slight tendency for the zinc-insulin hexamer to undergo indefinite self-association. Equilibrium constants governing these reactions are reported together with equations which permit calculation of the composition of the solution at any given total concentration. Comment is made on the possible biological significance of this linked polymerization pattern, and on the likely identity of the structure of the stable zinc-insulin hexamer with that previously reported from X-ray crystallographic studies.     

Physicochemical characterization of the 68 000-dalton protein of bovine neurofilaments

The 68 000-dalton protein from bovine neurofilaments was purified by a combination of chromatography on DEAE-cellulose and on hydroxylapatite in buffers containing 8 M urea. Although the separation of this protein from the other proteins of the neurofilament appeared to be hampered by a mixed association of the several components, a nearly homogeneous product was obtained for study. Sedimentation equilibrium experiments in buffers containing 8 M urea showed the molecule to be a monomer with a molecular weight of 70 600 +/- 2000. Circular dichroic spectra taken under the same conditions gave no evidence of residual alpha-helix. Molecular sieve chromatography in 8 M urea on controlled-pore glass showed that the molecule eluted at an unexpectedly small volume. The small elution volume did not depend significantly on protein concentration and is unlikely to be the result of intermolecular association. Rather, the monomer probably has a conformation more rigid or extended than a classical random coil. When dialyzed into 0.01 M tris(hydroxymethyl)aminomethane/1 mM ethylene glycol bis(beta-aminoethyl ether)-N,N,N',N'-tetraacetic acid/0.1 mM dithioerythritol, pH 8.5, the protein does not assemble into filaments. Sedimentation velocity reveals that under these conditions it consists mainly of a 4.8S molecular species, containing few large particles; sedimentation equilibrium shows that it is composed of oligomers, the smallest present in significant concentration having a molecular weight approximately that of a trimer. Circular dichroism measurements lead to the interpretation that the molecule has refolded in this buffer into a structure that has approximately 55% alpha-helix. Assembly into filamentous particles resembling neurofilaments occurs when the protein is dialyzed against 0.1 M 2-(N-morpholino)ethane-sulfonic acid/0.1% beta-mercaptoethanol/1 mM ethylene glycol bis(beta-aminoethyl ether)-N,N,N',N'-tetraacetic acid/0.17 M NaCl, pH 6.5. We suggest that the oligomeric species present in 0.01 M tris(hydroxymethyl)aminomethane may frequently be present in solubilized preparations of intermediate filaments and may represent an intermediate in the assembly process.     

Fusion of negatively charged phospholipid vesicles by insulin. Relationship with lipid fluidity

We have studied, by fluorescence methods, the association of insulin to liposomes, the modification of lipid fluidity, and the fusion of vesicles induced by insulin. All parameters showed a similar dependence on pH and ionic strength of the medium and on negative charges in liposomes. The influence of temperature indicated that the association of insulin to liposomes per se was not sufficient to produce a decrease in lipid fluidity and fusion of liposomes. The modification of lipid fluidity induced by insulin in biological membranes is discussed as a possible general event in the action of the hormone.     

Golgi β-glucuronidase of androgen-stimulated mouse kidney

The equilibrium constant of the phosphoglyceromutase reaction was determined over a range of pH (5.4-7.9), in solutions of different ionic strength (0.06-0.3) and in the presence of Mg(2+), at 30 degrees C and at 20 degrees C. The values obtained (8.65-11.65) differ substantially from previously published values. The third acid dissociation constants were redetermined for 2- and 3-phosphoglycerate, and in contrast with previous reports the pK values (7.03 and 6.97 respectively at zero ionic strength) were closely similar. The Mg(2+)-binding constants were measured spectrophotometrically and the values, 286mm(-1) and 255mm(-1) for 2- and 3-phosphoglycerate at pH7 and ionic strength 0.02, were also very similar. From the relative lack of effect of temperature, pH and ionic strength it is concluded that the equilibrium constant differs from unity largely because of entropic factors. At low ionic strength, in the neutral region, the pH-dependence can be attributed to the small difference in the acid dissociation constants, but the difference in dissociation constants does not explain the pH-dependence in the acid region or at high ionic strength. Within physiological ranges of pH, Mg(2+) concentration and ionic strength there will be little variation in equilibrium constant.     

Kinetic studies on the conformational isomerization reaction of the four diastereomers of beta,gamma-bidentate CrATP

The rates of the conformational isomerization reaction of the diastereomers of beta,gamma-bidentate CrATP were studied as a function of pH, buffer concentration, ionic strength, and temperature. The progress of the reaction was monitored by quenching the reaction at various times, and then isolating the individual diastereomers and quantitating the percent of each. This was accomplished using the reverse-phase high-performance liquid chromatography separation technique developed in this laboratory [K. J. Gruys, and S. M. Schuster, Anal. Biochem. 125, 66-73 (1982)]. The rate constants for this isomerization were then determined by obtaining the best computer fit of the data to a reversible binary mechanism (i.e., A in equilibrium B) using interative descent methods. The reaction rate was shown to be dependent on pH, temperature, and ionic strength, but independent of buffer concentration. Keq. constants were independent of all variables except ionic strength. The results from this study are interpreted in terms of a reaction mechanism involving a preequilibrium ionization of the diastereomers followed by a rate-limiting interconversion process.     

Low-affinity platelet factor 4 1H NMR derived aggregate equilibria indicate a physiologic preference for monomers over dimers and tetramers

Low-affinity platelet factor 4 (LA-PF4), unlike another related, sequentially homologous (about 50%) platelet-specific protein, platelet factor 4 (PF4), is an active mitogenic and chemotactic agent. PF4 exhibits a high binding affinity for heparin, while LA-PF4 does not. Both PF4 and LA-PF4 can exist in dimer and tetramer aggregate states. Equilibrium constants for PF4 aggregation have recently been estimated from fractional populations derived from proton nuclear magnetic resonance (NMR) integrals assigned to resonances in monomer, dimer, and tetramer states [Mayo & Chen (1989) Biochemistry 28, 9469]. On a 500-MHz NMR time scale, relatively slow exchange among LA-PF4 aggregate species has also allowed Tyr 15 ring proton resonances to be assigned for monomer, dimer, and tetramer states in LA-PF4. As a function of pH and ionic strength, equilibrium association constants for LA-PF4 dimer (KD) and tetramer (KT) formation have been estimated from Tyr 15 ring proton resonance integrals. At low ionic strength, KD reaches a minimum value of 12 M-1 at pH 3 where KT is at its maximum value of 1.6 x 10(5) M-1. At pH 4.1, KD and KT have the same value, 1.1 x 10(3) M-1, which is the minimum value for KT. KD plateaus off to its maximum value of 2.2 x 10(4) M-1 by pH 5.5. These values are significantly lower than those for PF4. Analysis of the pH dependence of KD and KT suggests that electrostatic interactions probably among Glu/Asp and Lys/Arg side chains form the predominant force in the monomer-monomer binding process, i.e., KD, while like-charge repulsion due to proximal, intersubunit Glu/Asp residues decreases KT as the pH is raised. At pH 7 and low ionic strength, the dimer state is highly favored over the tetramer state. Elevating the solvent ionic strength at pH 7 destabilizes the dimer state. Under these more physiologic conditions, i.e., pH 7 and 0.1-0.2 M NaCl, LA-PF4 monomers are highly favored over dimers and tetramers. For PF4 under similar solvent conditions, tetramers predominate. Differences in biological activities between these homologous platelet-specific proteins may be the result, at least in part, of differing aggregation properties. The biologically active state for PF4 is tetramer, while for LA-PF4 it is monomer. Quaternary structure may, therefore, account for strong heparin binding in PF4, most likely by presenting a more favorable structural matrix for effective glycosaminoglycan interactions.     

Self-association of rabbit muscle phosphofructokinase: effects of ligands

The effects of ligands on the self-association of rabbit muscle phosphofructokinase (PFK) were investigated by velocity sedimentation at pH 7.0 and 23 degrees C. The concentration dependence of the weight-average sedimentation coefficient was monitored in the presence of these ligands. The mode of association and equilibrium constants characterizing each association step were determined by theoretical fitting of the sedimentation data. The simplest mode of association for the PFK system is M in equilibrium M2 equilibrium M4 in equilibrium M16. Ligands and temperature would perturb the various equilibrium constants without altering the mode of association. The apparent equilibrium constants for the formation of tetramer, K4app, are increased in the presence of 0.1 mM ATP and 1.0 mM fructose 6-phosphate. The value of the sedimentation coefficient for the tetramer, S4 degrees, that would best fit the data is 12.4 S instead of 13.5 S determined in the absence of substrates, thus implying a structural change in the tetramer induced by substrates. Only an insignificant amount of dimer is present under the experimental conditions. The presence of activators, ADP or phosphate, enhances the formation of tetramers, and S4 degrees assumes a value of 13.5 S. Similar results are obtained with decreasing concentrations of proton. The presence of the inhibitor, citrate, however, favors the formation of dimers. The equilibrium constants determined as a function of ADP concentration were further analyzed by the linked-function theory derived by Wyman [Wyman, J. (1964) Adv. Protein Chem. 19, 224--285], leading to the conclusion that the formation of a tetramer involves the binding of two additional molecules of ADP per monomer. Similar analysis results in a conclusion that the formation of a dimer involves the binding of one additional molecule of citrate per phosphofructokinase subunit.