Conditions for the existence of critical points in scatchard plots of binding results

An equation is presented in Scatchard format which describes the binding of a ligand to a two-state acceptor system (either isomerizing or polymerizing). It is used to formulate a general set of conditions for the existence of critical points in Scatchard plots and this set is explored in detail for particular cases. Explicit relations between the thermodynamic parameters governing the binding are thereby obtained which permit discussion of the boundaries of domains within which critical points may exist. Moreover, several items of evidence are presented which show that there is an exact correspondence between the appearance of critical points in Scatchard plots and points of inflexion in associated binding curves examples are presented where zero, one or more critical points and points of inflexion are observed. Finally, the effects on preferential binding of the variation of the environmental parameters, temperature and pressure (and for acceptor polymerization, the total acceptor concentration) are discussed in terms of the derived conditions of existence of critical points in Scatchard plots and their equivalent domains of sigmoidality of binding curves.     

The binding of an indefinitely associating ligand to acceptor: consideration of monovalent ligand species binding to a multivalent acceptor

Currently available binding theory is extended to incorporate the concept of indefinite self-association of the ligand. Binding equations are formulated in closed form for the case of the binding to a multivalent acceptor of a ligand capable of isodesmically indefinitely self-associating in a "head-to-tail" mode such that each ligand state bears one site capable of interacting with the acceptor. It is shown both mathematically and by way of numerical example that this system will give rise exclusively to binding curves convex to the r-axis in Scatchard format. Thus, the system provides another example of a binding mechanism capable of generating an apparent negatively co-operative binding response.     

Interaction of protein with a self-associating ligand. Deviation from a hyperbolic binding curve and the appearance of apparent co-operativity in the Scatchard plot

We derived a general formula to analyze a binding system in which a ligand self-associates, in terms of experimentally determinable quantities, i.e. r, the average number of bound ligands per protein molecule, and Lft, the total free ligand concentration, which are expressed as a ligand monomer unit. The limiting behaviors of the Scatchard plot (r/Lft vs r plot), that is, the intercepts on the r-axis and the r/Lft-axis, and the limiting slopes, are generally given. Three models that may be encountered are considered in detail. Numerical examples are also presented to illustrate how the self-association of a ligand affects the binding curves. The ligand self-association alone can cause deviation of the profile of the binding curve (r vs Lft plot) from a hyperbola, resulting in a nonlinear Scatchard plot. Therefore, analysis of the binding data without consideration of ligand self-association may lead to erroneous conclusions as to the numbers and classes of binding sites, co-operativity among the sites and binding parameter values.     

Binding equations for interacting systems comprising multivalent acceptor and bivalent ligand: application to antigen-antibody systems

Consideration is given to the reversible interaction of a bivalent ligand, B, with a multivalent acceptor, A (possessing f reactive sites) which leads to the formation of a series of complexes, A_iB_j, comprising networks of alternating acceptor and ligand molecules. A binding equation is derived on the basis of a site association constant, k, defined in terms of reacted site probability functions. This equation, which relates the binding function, r (the moles of ligand bound per mole of acceptor) to the concentration of unbound ligand, m_b, is used to show that plots of r vs. 2km_B constructed with fixed but different values of $\text{k}\text{m}{}_{\mathrm{A}}$ intersect at the point ($\text{m}{}_{\mathrm{B}}\text{=}\text{1}\text{2k}\text{, r =}\text{f}\text{2}$) where the extent of reaction and the concentrations of those complexes for which $\text{j}\text{i}\text{=}\text{f}\text{2}$ attain maximal values. Corresponding Scatchard plots are shown by numerical example to be non-linear, their second derivative being positive for all r. It follows that such deviations from linearity cannot be taken alone as evidence for site heterogeneity in cross-linking systems. The binding equation obtained directly is shown to be identical with that obtained with f = 2 by summation procedures involving the general expression for concentrations of complexes, m_AiBj, formulated in terms of appropriate statistical factors. In this way, previous findings on precipitation and gel formation in cross-linking systems are correlated with the present development of binding theory.     

Dependence of aggregation and ligand affinity on the concentration of the folate-binding protein from cow's milk

Ultracentrifugation and gel-filtration studies showed that the folate-binding protein from cow's milk possessed a remarkable aggregation tendency at pH 7.4. Aggregation was enhanced in the presence of folate which suggested an interrelationship between the mechanisms of ligand binding and polymerization. The degree of polymerization increased with increasing concentrations of binding protein. Thus, while the monomer prevailed at 1 nM, a polymer composed of more thn 32 monomers was formed at 130 microM. Two characteristics of folate binding, i.e., Scatchard plots that were convex upward and a ligand affinity that was inversely proportional to the concentration of binding protein, could be interpreted in terms of ligand binding to a polymerizing system in which the polymerization equilibria affect the ligand affinity.     

Irreversibility of the interaction of human growth hormone with its receptor and analysis of irreversible reactions in radioreceptor assays—Theoretical considerations

Kinetic studies of the binding and dissociation of [¹²⁵I]-human growth hormone to rabbit liver and mammary gland membrane receptors have showed that the binding of [¹²⁵I]-human growth hormone was largely irreversible to liver membrane receptors and completely to the solubilised mammary gland receptor. As Scatchard analysis assumes complete reversibility of the hormone-receptor interaction the validity of estimates of affinity and capacity of receptors derived by this analysis may be questionable. Theoretical considerations show that in unimolecular irreversible interactions of hormone and receptor, a nonlinear (concave) or a linear Scatchard plot can be obtained. In linear Scatchard plots the capacity of the receptor obtained by extrapolation represents an overestimation of true capacity. This overestimation correlates with the value of the intercept in the Scatchard plot.     

Use of mobility ratios to estimate binding constants of ligands to proteins in affinity capillary electrophoresis

This work evaluates the use of mobility ratios (M) to estimate binding constants of proteins to ligands using affinity capillary electrophoresis (ACE). This concept is demonstrated using two model systems: vancomycin (Van) from Streptomyces orientalis and carbonic anhydrase B (CAB, EC 4.2.1.1). A plot of change in M (ΔM) over the concentration of ligand [L] versus ΔM yields a more useful representation of the Scatchard plot in capillary electrophoresis (CE) than traditional plots of the change in mobility Δμ over [L] versus Δμ in a wide set of circumstances, especially when comparing electropherograms obtained in the presence of substantial variations in electroosmotic flow. Altering the voltage and/or capillary length of the CE system produced only small variations in M, but much larger changes in the more standard measures of migration used by the μ form of analysis. The use of M in the Scatchard analysis offers a new approach to estimating binding constants of ligands to proteins using ACE.     

Characteristics of β-adrenergic receptors in longitudinal muscle membranes of rat uterus: Changes in kinetic properties of the receptor during gestation

Binding characteristics of β-adrenergic receptors of longitudinal muscle membranes isolated from different stages of pregnant rat myometrium were investigated using [³H]dihydroalprenolol. Between Days 15 and 21 of gestation, the ratio of β₁- and β₂-adrenergic receptors of longitudinal membranes was constant. The membranes were found to be predominant in β₂-adrenergic receptors. The concentration of longitudinal muscle β-adrenergic receptors increased significantly during the last 7 days of gestation. Kinetic binding studies implied that the affinity of the membrane β-adrenergic receptors decreased through a slight decrease in the association rate and a large increase in the dissociation rate with progression of pregnancy. A Scatchard plot indicated that longitudinal muscle in β-adrenergic receptors on Days 15 and 18 constitute a single class of independent sites. By contrast, the dissociation kinetics, the convex downward curvature in a Scatchard plot and a Hill coefficient (h) of less than 1.00 of [³H] DHA binding to β-receptors of muscle on Day 21 suggested the existence of negatively cooperative multiple binding sites for β-adrenergic ligand. These results suggest that changes in the dynamics of uterus β-adrenergic receptors play an important role in the onset of labor.     

The positively cooperative binding of concanavalin a to corn starch: The isotherm of binding and a measure of the cooperativity

The binding of concanavalin A to corn starch was investigated by fluorimetric assay. The extent of binding varied linearly with the mass of ligand, and followed a hyperbolic law with respect to the mass of starch. This led to an isotherm of binding: r = 0.33A_oME_o^?0.88, where r is the extent of binding, A_o is the mass of concanavalin A present (both bound and unbound), and M_o is the mass of starch. These results, and Scatchard plots of the data, showed the binding to be positively cooperative. The exponent of the M_o term was shown to be a measure of cooperativity. The binding was dependent on the ionic strength of the dispersion medium, and this indicated that the binding may have an electrostatic component.     

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.     

Cooperative binding of calcium to glycerinated skeletal muscle fibers

The binding of ⁴⁵Ca²⁺ to glycerinated rabbit psoas fibers was measured by means of a double isotope technique. With 5 mM Mg²⁺ (no ATP) binding was half-maximal at 1.4 · 10^?6M Ca²⁺ and the maximal amount bound was 1.6 μmol/g protein. At < 50% saturation, the Scatchard plot had a positive slope and the Hill coefficient was 2.2. At greater than 50% saturation, the Scatchard plot was linear with a negative slope (K′ = 0.8 · 10⁶ M^?1) and the Hill coefficient was 1.0. In the absence of Mg²⁺, binding was half-maximal at 3 · 10^?7 M Ca²⁺ and the maximal amount bound was 2.9 μmol/g protein. The Scatchard plot indicated two classes of sites with K′ values of about 2 · 10⁷ and 2 · 10⁶ M^?1. The Hill coefficient in the mid-saturation range was approx. 0.6. The data indicate that in the presence of Mg²⁺ binding to about half of the total Ca²⁺ binding sites is suppressed and there is a strong positive cooperativity involving half of the remaining sites.     

Ligand binding characteristics of two molecular forms,one equipped with a hydrophobic glycosyl phosphatidylinositol tail,of the folate binding protein purified from human milk

Two molecular forms of the folate binding protein were isolated and purified from human milk by a combination of cation exchange- and affinity chromatography. One protein (27 kDa) was a cleavage product of the other 100 kDa protein as evidenced by N-terminal amino acid sequence homology and a reduction in the molecular size of the latter protein to 27 kDa after cleavage of its hydrophobic glycosylphosphatidylinositol tail by phosphatidylinositol-specific phospholipase C. High-affinity binding of [³H]folate was characterized by upward convex Scatchard plots and increasing ligand binding affinity with decreasing concentrations of both proteins. Downward convex Scatchard plots and binding affinities showing no dependence on the protein concentration were, however, observed in highly diluted solutions of both proteins. Radioligand binding was inhibited by folate analogs, and dissociation of radioligand was slow at pH 7.4 but rapid and complete at pH 5.0 and 3.5. Ligand binding quenched the tryptophan fluorescence of the 27 kDa protein suggesting that tryptophan is present at the binding site and/or ligand binding induces a conformation change that affects tryptophan environment in the protein. The 27 kDa protein representing soluble folate binding protein exhibited a greater affinity for ligand binding than the 100 kDa protein which possesses a hydrophobic tail identical to the one that anchors the folate receptor to the cell membrane.     

Abscisic acid binding to subcellular fractions from leaves of Vicia faba

A centrifugation binding assay has been used to demonstrate the binding of [³H] (±) abscisic acid to membrane-rich fractions prepared from leaves of Vicia faba L. Kinetic analysis of this binding shows evidence of saturation of binding sites with increasing concentration of ligand. Scatchard analysis of these data yields a biphasic plot possibly indicating the presence of two types of binding sites. The dissocation constant for the high affinity site has been calculated to be 3.5×10^-8 mol 1^-1.     

A simplified analysis of Scatchard plots for systems with two interacting binding sites

A simple graphical method for calculating stoichiometric and site binding constants for systems with two initially equivalent interacting sites is derived from a modified Scatchard equation. The binding constants can be calculated from Scatchard plots (r/[A] as a function of r) using the values of r/[A] (r is the molar ratio of bound ligands to total protein and [A] is the equilibrium concentration of free ligand) when r = 0 and r = 1 (half-saturation). The applicability of the method to the adsorption of bilirubin by peptide pendants immobilized on a polyacrylamide support is demonstrated.     

Properties of the binding of copper by bleomycin

The glycopeptide, bleomycin, binds metal ions including Cu²⁺. It is the copper complex of this material that is isolated from Streptomyces verticillus. Both free ligand and copper complex are excellent antitumor agents in animals. The biochemical and pharmacological relationship between these compounds has not been established. The present study begins an analysis of the chemistry and biochemistry of copper-bleomycin with structural and equilibrium properties of the complex. Potentiometric and fluorometric titrations of bleomycin confirm three acidic groups with pK_a values of 7.50, 4.93, and 2.72. The conjugate nitrogen bases of these groups, comprise three of the binding sites for Cu²⁺ according to similar titrations of copper-bleomycin. The fourth is a conjugate base of an acid with a very large pK_a that cannot be measured by these techniques. The participation of a fourth such group is inferred from both proton release studies of the binding of metal and ligand above pH 8 and from several studies of the thermodynamic stability of copper bleomycin. At low pH binding of copper to bleomycin occurs in two steps, as observed by several independent techniques which monitor either the metal or the ligand. Log stability constants for the reactions Cu²⁺ + H_kBlm ? CuH_k-nBlm + nH⁺ and CuH_k-nBlm ? CuH_k-n-rBlm + rH⁺ are 1.32 and ?4.31, respectively, with n of 2.21 in the first equation and r of 2.07 in the second equation. The derived logarithm of the pH independent stability constant for copper bleomycin multiplied by the protonation constant for the unknown fourth ligand in the binding site is 12.16. This agrees closely with values obtained from measurements of conditional formation constants. One of the groups which binds in the second reaction is the substituted pyrimidine.     

Equilibrium and kinetic studies on the binding of des-N-tetramethyltriostin A to DNA

The interaction between TANDEM (a des-methyl analogue of triostin A) and poly(dA-dT) results in extension of the helix by 6.8 Å for each ligand molecule bound, exactly as predicted for a bis-intercalation reaction. Cooperativity is evident in Scatchard plots for the interaction at ionic strengths of 0.2 and 1.0, where the binding constant is diminished compared to that which pertains at low salt concentration. Binding to a natural DNA (calf thymus), already considerably weaker than binding to poly(dA-dT), is also sensitive to increased ionic strength. With a self-complementary octanucleotide d(G-G-T-A-T-A-C-C) the binding curve indicates the presence of a single des-N-tetramethyltriostin A binding site per helical fragment with a non-cooperative association constant about 6·10⁶ M^?1. Detergent-induced dissociation of des-N-tetramethyltriostin A-poly(dA-dT) complexes results in a simple exponential decay at all levels of binding, but the time constant of decay is dependent upon the initial binding ratio. This behaviour cannot directly explain the cooperativity of equilibrium binding isotherms but suggests the occurrence of relatively long-lived perturbations of the helical structure by binding of the ligand. [Ala³, Ala⁷]des-N-tetramethyltriostin A, which has a more flexible octapeptide ring lacking the disulphide cross-bridge, dissociates from poly(dA-dT) much faster than des-N-tetramethyltriostin A. Dissociation of des-N-tetramethyltriostin A from calf thymus DNA is more rapid than dissociation of triostin A or other quinoxaline antibiotics, which may account for its low antimicrobial activity.     

Endogenous inhibitor of GABAB and GABAA receptors

An endogenous inhibitor of γ-aminobutyric acid (GABA) receptors was partially purified from bovine brain striatum. It was obtained as a low molecular weight fraction by gel filtration on Biogel P-2 and was adsorbed to Dowex AG 50W-X8, but not to Dowex AG 1-X8. It was ninhydrin-negative, basic, heat-stable substance. It caused dose-dependent inhibition of Na⁺-independent [³H]GABA bindings. Scatchard plot analysis of the [³H]GABA binding to GABA “B” receptor recognition site showed this inhibitor increased the K_d value (24.1 nM to 3.6 nM) without changing the B_max. On the other hand, Scatchard plot analysis of the [³H]GABA binding to GABA “A” receptor recognition site showed that the inhibitor decreased number of binding sites (706 fmol/mg protein to 494 fmol/mg protein) without affecting the K_d value. These results suggest that the endogenous inhibitor functions as a modulator for GABA_B and GABA_A receptors.     

Analysis of binding curves in multivalent antigen-heterogeneous antibody systems

The binding of antibodies to N-valent antigens can be utilized to gain information on the antibody affinity distribution, under the assumption, that the formation of each bond occurs as an independent event. The analysis of the most widely used plots of binding data (double reciprocal, Scatchard, Sips and the so-called “avidity” plot) leads to expressions which correlate asymptotical features of the binding curves to the antibody site concentration to the antigen valence and to the affinity moments <K^?1>, <K> and <K²>. Only in the “avidity” plot is the shape of the curve independent of the valence of the antigen, depending merely on the antibody affinity distribution.     

Effect of the incomplete separation of bound and free ligand on binding measurements

An incomplete separation of free and acceptor-bound ligand causes underestimation of specific binding even though the determinations are corrected with blank or nonspecific binding values. If the failure of the experimental procedure causes contamination of bound ligand with free ligand, Scatchard plots are linear, though their slopes and abscissa intercepts are different from the true ones. On the other hand, if the ligand-acceptor complex is incompletely recovered, Scatchard plots are curvilinear with downward concavity. These problems can be overcome if the separable fractions of free and bound ligand are measured and suitable corrections are applied.The separation of free and receptor-bound ¹²⁵I-labeled human growth hormone by a precipitation method is taken as an example of the procedure.     

The influence of aging on androgen dynamics in the male rat

Androgen assimilation was investigated in a variety of accessory sex organs (seminal vesicles and anterior, dorsal, lateral, and ventral prostates) and in several nonaccessory sex organs in male Wistar rats. After administration of a pulse dose of [³H]testosterone in vivo to intact young (3–4 months old) rats, [³H]testosterone was the primary radioactive steroid recovered from most organs examined, except for the secondary sex glands where the reduced metabolites, [³H]5α-dihydrotestosterone (DHT) and [³H]5α-androstanediol(s), predominated. At longer postinjection times, [³H]DHT was preferentially retained in the accessory sex glands, presumably reflecting intracellular metabolism of [³H]testosterone to this compound and subsequent specific binding of [³H]DHT to receptor proteins. At the longest postinjection interval investigated, the ventral prostate retained greater concentrations of [³H]DHT than the lateral prostate which in turn had a higher [³H]DHT concentration than the seminal vesicles or anterior or dorsal prostates. The latter three glands retained approximately equal concentrations of [³H]DHT. Scatchard plot analyses of cytosol binding in 24-h castrates indicated that with one exception, the level of high affinity DHT binding sites was generally correlated with the retention of [³H]DHT in vivo in intact rats. Specifically, while the affinity for DHT binding in all accessory sex organs was the same, the number of high affinity binding sites per mg wet tissue weight was on the order of ventral prostate > anterior prostate ≥ seminal vesicles ≥ dorsal prostate > lateral prostate. Studies of the influence of aging to 22–26 months revealed no apparent differences in the affinity of the DHT receptor for its ligand in any of the accessory sex glands from 24-h castrates when the receptors were present in levels sufficiently high to quantify. The concentration of available DHT receptors with advancing age remained constant in the anterior and dorsal prostates, increased in the seminal vesicles, and declined in the ventral and lateral prostates. The decreases observed in the ventral prostate were only partial, but the receptors of the lateral prostate declined to nondetectable levels.