The use of non-linear regression analysis and the F test for model discrimination with dose-response curves and ligand binding data

Computer fitting of binding data is discussed and it is concluded that the main problem is the choice of starting estimates and internal scaling parameters, not the optimization software. Solving linear overdetermined systems of equations for starting estimates is investigated. A function, Q, is introduced to study model discrimination with binding isotherms and the behaviour of Q as a function of model parameters is calculated for the case of 2 and 3 sites. The power function of the F test is estimated for models with 2 to 5 binding sites and necessary constraints on parameters for correct model discrimination are given. The sampling distribution of F test statistics is compared to an exact F distribution using the Chi-squared and Kolmogorov-Smirnov tests. For low order modes (n less than 3) the F test statistics are approximately F distributed but for higher order models the test statistics are skewed to the left of the F distribution. The parameter covariance matrix obtained by inverting the Hessian matrix of the objective function is shown to be a good approximation to the estimate obtained by Monte Carlo sampling for low order models (n less than 3). It is concluded that analysis of up to 2 or 3 binding sites presents few problems and linear, normal statistical results are valid. To identify correctly 4 sites is much more difficult, requiring very precise data and extreme parameter values. Discrimination of 5 from 4 sites is an upper limit to the usefulness of the F test.     

Exact analysis of competition ligand binding by displacement isothermal titration calorimetry

A rigorous method for the least-squares nonlinear regression analysis of displacement isothermal titration calorimetric data is presented. The method can fit the binding isotherm of a ligand which is competitively inhibited in its binding by another bound ligand to a molecule with n identical and independent binding sites. There are no other assumptions for the method and no approximations. Analysis of previously published data of the strong binding of acarbose to glucoamylase is presented as an example. The regression equations have been programmed for the Origin software supplied with the widely used titration calorimeters from Microcal, Inc., and an Origin Function Definition File with instructions is freely available from the author upon e-mail request.     

Analysis of ligand binding curves in terms of species fractions

The ligand binding curve for a macromolecular system presents the average number or ligand molecules bound per macromolecule as a function of the chemical potential or the logarithm of the ligand concentration. We show that various observable properties of this curve, for example its asymptotes and derivatives, are expressible in terms of linear combinations of the mole fractions α_i of macromolecules binding i molecules of ligand. Whenever enough such properties of the binding curve are known, the linear equations in α_i can be solved to give the mole fractions of each of the various macromolecular species. An application of these results is that a Hill plot for hemoglobin-ligand equilibrium where the asymptotes approach unit slope can be made to yield the four Adair constants by a simple algebraic method. A second use is that a knowledge of the first and second derivatives of the binding curve at points along the curve can yield the species fractions as functions of the degree of saturation without direct knowledge of the ligand binding constants. These methods are illustrated by some numerical examples.     

Misuse of thermodynamics in the interpretation of isothermal titration calorimetry data for ligand binding to proteins

Isothermal titration calorimetry (ITC) has given a mass of data on the binding of small molecules to proteins and other biopolymers, with particular interest in drug binding to proteins chosen as therapeutic indicators. Interpretation of the enthalpy data usually follows an unsound protocol that uses thermodynamic relations in circumstances where they do not apply. Errors of interpretation include incomplete definitions of ligand binding and equilibrium constants and neglect of the non-ideality of the solutions under study, leading to unreliable estimates of standard free energies and entropies of binding. The mass of reported thermodynamic functions for ligand binding to proteins estimated from ITC enthalpies alone is consequently of uncertain thermodynamic significance and utility. ITC and related experiments to test the protocol assumptions are indicated. A thermodynamic procedure avoiding equilibrium constants or other reaction models and not requiring protein activities is given. The discussion draws attention to the fundamental but neglected relation between the thermodynamic activity and bioactivity of drugs and to the generally unknown thermodynamic status of ligand solutions, which for drugs relates directly to effective therapeutic dosimetry.     

Cooperative equilibrium curves generated by ordered ligand binding to multi-site molecules

The sigmoid shape of equilibrium curves in normal axes and Hill coefficients higher than unity, are indexes of cooperativity or homotropic allostery where the affinity for the ligand increases as saturation progresses. The mathematical transformation of the Adair scheme of equilibria in the Hill plot, reveals that sigmoid binding curves can also be generated by ordered ligand binding to a receptor with multiple binding sites of identical microscopic association constants. This mechanism only based on the law of mass action, could participate to some extent to certain cooperative effects observed in non-biological systems and perhaps in the physiological binding of oxygen to heme proteins.     

Analysis of ligand binding curves on basis of mean intrinsic thermodynamic quantities

The mean intrinsic thermodynamic quantity can be defined by considering the relative population of complex species in the solution and the value of intrinsic thermodynamic quantity corresponds to each step of ligation. In the present study a new method is introduced for analysis of experimental ligand binding data on basis of mean intrinsic thermodynamic quantities. In this regard, a deviation parameter was defined by comparing the non-interacting system with the cooperative interactive one. This parameter can be calculated just by estimation of the first binding constant. A set of relations between this deviation parameter and other binding characteristics, such as mean intrinsic Gibbs free energy of binding and mean Gibbs free energy of site-site interaction, have been developed. This model presents binding mechanism in a unified way that is simple, yet stringent, more straightforward, more reliable and informative. This analyzing method has been successfully applied for evaluation of various systems such as oxygen binding to hemoglobin, laurate and warfarin binding to human serum albumin, and reveals some new biological features of these binding systems.     

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.     

Analysis of dose-dependent antibody titration curves

Several types of dose-response titration curves were considered. It was demonstrated that the use of the so-called coordinates of dilution suggested earlier by us allows one to analyze the titration curves, obtained either by ELISA, or by agglutination. Theoretical curves, obtained by the developed theory are very similar to those obtained in experiments. It was shown, that the analysis of the titration curves could give important information concerning antibody-blocking factors in titration sera or other samples of studied antibodies.     

Investigation of ligand binding to the multidrug resistance protein EmrE by isothermal titration calorimetry

Escherichia coli multidrug resistance protein E (EmrE) is an integral membrane protein spanning the inner membrane of Escherichia coli that is responsible for this organism's resistance to a variety of lipophilic cations such as quaternary ammonium compounds (QACs) and interchelating dyes. EmrE is a 12-kDa protein of four transmembrane helices considered to be functional as a multimer. It is an efflux transporter that can bind and transport cytoplasmic QACs into the periplasm using the energy of the proton gradient across the inner membrane. Isothermal titration calorimetry provides information about the stoichiometry and thermodynamic properties of protein-ligand interactions, and can be used to monitor the binding of QACs to EmrE in different membrane mimetic environments. In this study the ligand binding to EmrE solubilized in dodecyl maltoside, sodium dodecyl sulfate and reconstituted into small unilamellar vesicles is examined by isothermal titration calorimetry. The binding stoichiometry of EmrE to drug was found to be 1:1, demonstrating that oligomerization of EmrE is not necessary for binding to drug. The binding of EmrE to drug was observed with the dissociation constant (K(D)) in the micromolar range for each of the drugs in any of the membrane mimetic environments. Thermodynamic properties demonstrated this interaction to be enthalpy-driven with similar enthalpies of 8-12 kcal/mol for each of the drugs in any of the membrane mimetics.     

The insulin receptor. Structural basis for high affinity ligand binding

Treatment of the soluble insulin receptor from human placenta with 1.25 mM dithiothreitol and 75 mM Tris at pH 8.5 results in complete reduction of interhalf disulfide bonds (class 1 disulfides) and dissociation of the tetrameric receptor into the dimeric alpha beta form. The alpha beta receptor halves exhibit a reduced affinity for insulin binding (B?ni-Schnetzler, M., Rubin, J. B., and Pilch, P. F. (1986) J. Biol. Chem. 261, 15281-15287). Kinetic experiments reveal that reduction of class 1 disulfides is a faster process than the loss of affinity for ligand, indicating that events subsequent to reduction of interhalf disulfides are responsible for the affinity change. We show that a third class of alpha subunit intrachain disulfides is more susceptible to reduction at pH 7.6 than at pH 8.5 and appears to form part of the ligand binding domain. Reduction of the intrachain disulfide bonds in this part of the alpha subunit leads to a loss of insulin binding. Modification of this putative binding domain by dithiothreitol can be minimized if reduction is carried out at pH 8.5. When the insulin receptor in placental membranes is reduced at pH 8.5, the receptor's affinity for insulin is not changed when binding is measured in the membrane. However, the Kd for insulin binding is reduced 10-fold when alpha beta receptor halves are subsequently solubilized. Scatchard analysis of insulin binding to reduced or intact receptors in the membrane and in soluble form together with sucrose density gradient analysis of soluble receptors suggests that alpha beta receptor halves remain associated in the membrane after reduction, but they are dissociated upon solubilization. We interpret these results to mean that the association of two ligand binding domains, 2 alpha beta receptor halves, is required for the formation of an insulin receptor with high affinity for ligand.     

A gradient titration apparatus for determining spectrophotometric binding isotherms.

The construction of an automatic gradient titration apparatus using a multichannel peristaltic pump and a recording spectrophotometer is described. The ability of the apparatus to faithfully generate continuous spectrophotometric binding isotherms was tested in experiments studying the interaction of DNA with neutral red. The method has been shown to require low volumes of reactants, and complete binding curves can be produced in less than 15 min. The apparatus was also used to perform automatically the method of continuous variations in experiments determining the binding stoichiometry of calmagite and magnesium ion.     

EPSP synthase: binding studies using isothermal titration microcalorimetry and equilibrium dialysis and their implications for ligand recognition and kinetic mechanism.

Isothermal titration calorimetry measurements are reported which give important new binding constant (Kd) information for various substrate and inhibitor complexes of Escherichia coli EPSP synthase (EPSPS). The validity of this technique was first verified by determining Kd's for the known binary complex with the substrate, shikimate 3-phosphate (S3P), as well as the herbicidal ternary complex with S3P and glyphosate (EPSPS.S3P.glyphosate). The observed Kd's agreed very well with those from previous independently determined kinetic and fluorescence binding measurements. Further applications unequivocally demonstrate for the first time a fairly tight interaction between phosphoenolpyruvate (PEP) and free enzyme (Kd = 390 microM) as well as a correspondingly weak affinity for glyphosate (Kd = 12 mM) alone with enzyme. The formation of the EPSPS.PEP binary complex was independently corroborated using equilibrium dialysis. These results strongly suggest that S3P synergizes glyphosate binding much more effectively than it does PEP binding. These observations add important new evidence to support the hypothesis that glyphosate acts as a transition-state analogue of PEP. However, the formation of a catalytically productive PEP binary complex is inconsistent with the previously reported compulsory binding order process required for catalysis and has led to new studies which completely revise the overall EPSPS kinetic mechanism. A previously postulated ternary complex between S3P and inorganic phosphate (EPSPS.S3P.Pi, Kd = 4 mM) was also detected for the first time. Quantitative binding enthalpies and entropies were also determined for each ligand complex from the microcalorimetry data. These values demonstrate a clear difference in thermodynamic parameters for recognition at the S3P site versus those observed for the PEP, Pi, and glyphosate sites.     

Isothermal titration calorimetry reveals differential binding thermodynamics of variable region-identical antibodies differing in constant region for a univalent ligand

The classical view of immunoglobulin molecules posits two functional domains defined by the variable (V) and constant (C) regions, which are responsible for antigen binding and antibody effector functions, respectively. These two domains are thought to function independently. However, several lines of evidence strongly suggest that C region domains can affect the specificity and affinity of an antibody for its antigen (Ag), independent of avidity-type effects. In this study, we used isothermal titration calorimetry to investigate the thermodynamic properties of the interactions of four V region-identical monoclonal antibodies with a univalent peptide antigen. Comparison of the binding of IgG1, IgG2a, IgG2b, and IgG3 with a 12-mer peptide mimetic of Cryptococcus neoformans polysaccharide revealed a stoichiometry of 1.9-2.0 with significant differences in thermodynamic binding parameters. Binding of this peptide to the antibodies was dominated by favorable entropy. The interaction of these antibodies with biotinylated peptides manifested greater enthalpy than for native peptides indicating that biotin labeling affected the types of Ag-Ab complexes formed. Our results provide unambiguous thermodynamic evidence for the notion that the C region can affect the interaction of the V region with an Ag.     

Reversible and irreversible changes in hydrogen ion titration curves of melanins.

Hydrogen ion titration curves obtained on melanins prepared and modified differently show recognizable differences. Melanin prepared with NaOH exhibits irreversible loss of functional groups in the base range once titrated acidic. Melanin prepared with NH4OH shows two main branches selected by incubation at an extreme pH for 24 h. Longer exposure reveals other curves but they disappear once a 24-h schedule of titration is resumed. The existence of two major pH branches appears consistent with the oxidation-reduction properties of melanins. These results demonstrate the ability to obtain a reproducible feature in the hydrogen ion titration curves. Thus it is finally possible to quantitate CO2 adsorbed to melanin in solution at basic pH.