Simulation of Association Curves and ‘Scatchard’ Plots of Binding Reactions Where Ligand and Receptor are Degraded or Internalized

Abstract

Frequently during the course of binding to a receptor, ligand is degraded. In some preparations receptor is degraded. And with isolated cell preparations, ligand and/or receptor may be internalized. Here we present a mathematical model in which the binding and other reactions are combined. The resulting set of differential equations is solved numerically to simulate association curves and the resulting values of bound and free ligand are used to construct Scatchard plots. Where non-ideal conditions exist, the Scatchard plots are generally curvilinear. Dependence of this curvilinearity - on time of measurement of free and bound ligand, on degradation and internalization of ligand, and on degradation and internalization of receptor - is shown. Equilibrium constants derived from the Scatchard plots are generally incorrect but the derived receptor concentration is often correct. The simulations suggest experimental possibilities for distinction among the several side reactions in ligand-receptor binding systems.     

Simulation of Association Curves and ‘Scatchard’ Plots of Binding Reactions where Ligand and Receptor are Degraded or Internalized

Abstract

Frequently during the course of binding to a receptor, ligand is degraded. In some preparations receptor is degraded. And with isolated cell preparations, ligand and/or receptor are internalized. Here we present a mathematical model for the combined binding and other reactions which gives useful information about the behaviour of such systems. The set of differential equations is solved numerically to simulate association curves and the resulting values of bound and free ligand are used to construct Scatchard plots. Where non-ideal conditions exist, the Scatchard plots are generally curvilinear. Dependence of this curvilinearity on time of measurement of free and bound ligand, on degradation and internalization of ligand and on degradation and internalization of receptor is shown. Equilibrium constants derived from the Scatchard plots are generally incorrect but the derived receptor concentration is often correct. The simulations lead to possibilities for distinction among the several side reactions in ligand-receptor binding systems.     

Misuse of nonlinear Scatchard plots

Scatchard plots--plots of bound/free ligand vs bound ligand--are a common graphical presentation of binding data. They are often nonlinear. Despite examples of correct usage and several articles calling attention to incorrect treatment of Scatchard plots, erroneous interpretations of nonlinear Scatchard plots remain frequent; plots are resolved incorrectly into two or more linear components which have no relation to an acceptable binding model. Correct analysis requires determination, usually by computer, of numerical values of the binding parameters that give the best nonlinear fit to an appropriate model, examples of which are specified.     

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.     

Interaction of phenylthiolato-(2,2'',2"-terpyridine)platinum(II) cation with DNA.

The interaction between a novel aromatic thiolato derivative from the family of DNA-intercalating platinum complexes, phenylthiolato-(2,2',2"-terpyridine)platinum(II)-[PhS(ter py)Pt+], and nucleic acids was studied by using viscosity, equilibrium-dialysis and kinetic measurements. Viscosity measurements with sonicated DNA provide direct evidence for intercalation, and show that at binding ratios below 0.2 molecules per base-pair PhS(terpy)Pt+ causes an increase in contour length of 0.2 nm per bound molecule. However, helix extension diminishes at greater extents of binding, indicating the existence of additional, non-intercalated, externally bound forms of the ligand. The ability of PhS(terpy)Pt+ to aggregate in neutral aqueous buffers at a range of ionic strengths and temperatures was assessed by using optical-absorption methods. Scatchard plots for binding to calf thymus DNA at ionic strength 0.01 (corrected for dimerization) are curvilinear, concave upward, providing further evidence for two modes of binding. The association constant decreases at higher ionic strengths, in accord with the expectations of polyelectrolyte theory, although the number of cations released per bound unipositive ligand molecule is substantially greater than 1. Stopped-flow kinetic measurements confirm the complexity of the binding reaction by revealing multiple bound forms of the ligand whose kinetic processes are both fast and closely coupled. Thermal denaturation of DNA radically alters the shapes of binding isotherms and either has little effect on, or enhances, the affinity of potential binding sites, depending on experimental conditions. Scatchard plots for binding to natural DNA species with differing nucleotide composition show that the ligand has a requirement for a single G X C base-pair at the highest-affinity intercalation sites.     

An examination of the forms of scatchard plots of binding results outside domains of sigmoidality

General expressions are formulated for the first and second derivatives of the Scatchard function, r/[S], with respect to the binding function, r, from an equation that describes the binding of a ligand to a two-state acceptor system (either isomerizing or polymerizing). The expressions are utilized to determine the sign of the second derivative for particular systems under conditions where the first derivative is negative for all r. The work therefore correlates with previous studies, which stressed conditions of existence of critical points in Scatchard plots, by examining more fully possible forms of binding curves outside such domains of sigmoidality. Particular attention is given to the condition, d(r/[S])/dr < 0 and d²(r/[S])/dr² > 0 for all r (which defines a Scatchard plot convex to the r-axis). In agreement with previous findings it is proven that the isomerizing acceptor model cannot give rise to this form of plot and is therefore distinguished from negatively co-operative allosteric models. On the other hand, the polymerizing acceptor model may yield such a Scatchard plot, a feature demonstrated by formulating explicit conditions for its manifestation when ligand binding is exclusive to the polymeric state, and illustrated numerically for a system in which ligand binds to both oligomeric states. Distinction between such systems and those exhibiting negative co-operativity is possible on the basis of the Scatchard plots, which exhibit dependence on acceptor concentration in the case of a polymerizing acceptor; indeed, an example is provided where variation of acceptor concentration for a system characterized by fixed interaction parameters effects a conversion from sigmoidal binding behaviour to that typified by a Scatchard plot convex to the r-axis.     

DNA bis-intercalation: Theoretical calculation of binding curves

A statistical mechanical calculation of the binding properties of DNA bis-intercalators is presented, based on the sequence-generating function method of Lifson. The effects of binding by intercalation of one or both chromophores of a bifunctional intercalating agent are examined. The secular equation for a general model that includes the effects of neighbor (nearest and non-nearest) exclusion and/or cooperativity in the binding of both singly and doubly intercalated ligands is derived. Numerical results for binding curves are presented for a more restricted model in which each type of bound ligand rigorously excludes its nearest neighbor and the total number of sites covered by a doubly intercalated ligand is variable. At low values of free ligand concentration bis-intercalation dominates the binding process, while at high value of free ligand concentration, intercalation of only one chromophore per ligand becomes significant due to the unavailability of contiguous free sites required for bis-intercalation. Also, depending on the binding parameters, the free energy of the system can be lowered by a loss of doubly intercalated ligands in favor of singly intercalated ones. Corresponding to this transition in binding mode, the average number of sites occupied by a bound ligand decreases from that characteristic of bis-intercalation to that characteristic of mono-intercalation as free ligand concentration increases. An analysis of Scatchard plots describing bis-intercalation is presented.     

Apparent “negative cooperativity” kinetics in the absence of a nonlinear Scatchard plot of thyrotropin-receptor interaction in a human thyroid adenoma

A human thyroid adenoma (benign nodule) was identified which exhibited a linear Scatchard plot of ¹²⁵I-TSH binding, characteristic of a single class of binding site with high affinity (K_d = 0.5±0.1 nM) and low binding capacity (0.8±0.2 pmol/mg protein). In contrast, Scatchard analysis of binding to adjacent normal thyroid was nonlinear, suggesting the presence of high and low-affinity binding sites with K_d's of 0.4±0.2 and of 27.9±11.0 nM and capacities of 0.7±0.3 and 1.8±1.0 pmol/mg protein, respectively. Dissociation of bound ¹²⁵I-TSH from membranes of both adenoma and normal tissue revealed identical enhancement of dissociation in the presence of excess native hormone, thought to be evidence for the “negative cooperativity” model of hormone-receptor interaction. Furthermore, adenylate cyclase from both tissues was equally responsive to TSH. Thus, a thyroid adenoma which contains TSH-responsive adenylate cyclase still exhibited enhanced dissociation by native hormone, even though Scatchard analysis yielded a single, non-cooperative class of binding sites. This suggests that enhanced dissociation of bound hormone does not provide a demonstration of negatively-cooperative site-site interaction. Furthermore, nonlinear Scatchard plots, typical of TSH binding in normal thyroid, represent two classes of binding sites, of which the high affinity type is responsible for stimulation of adenylate cyclase.     

Estradiol- and estriol-binding in human benign prostatic hyperplasia.

Binding of the natural estrogens, estradiol and estriol, was investigated, in 34 samples of human benign prostatic hypertrophy (BPH) tissue, using Scatchard analysis and agar gel electrophoresis. Saturation binding analysis using a wide range of concentrations of both ligands resulted in curvilinear Scatchard plots. This confirmed the presence of two binding forms for estradiol: a true estrogen receptor, and a protein with lower affinity and higher capacity. Both binding species were also demonstrated and quantified with estriol. The electrophoretic process, after incubation at low and high ligand concentrations also resulted in separation, for both estrogens, of two binding peaks. They are probably two distinct forms of the low affinity, high capacity binding measured by Scatchard. The procedure used in our laboratory was not able to provide accurate determination of the concentrations of these binding forms. Possible modifications to alleviate these drawbacks are discussed.     

Use of binding site neighbor-effect parameters to evaluate the interactions between adjacent ligands on a linear lattice. Effects on ligand-lattice association.

A method using binding site "neighbor-effect" parameters (NEPs) is introduced to evaluate the effects of interaction between adjacent ligands on their binding to an infinite linear lattice. Binding site overlap is also taken into account. This enables the conditional probability approach of McGhee & von Hippel to be extended to more complex situations. The general equation for the isotherm is v/LF = SFKF, where v is the ratio of bound ligands to lattice residues, LF is the free ligand concentration, SF is the fraction of binding sites that are free, and KF is the average association constant of a free site. Solutions are derived for three cases: symmetric ligands, and asymmetric ligands on isotropic or anisotropic lattices. For symmetric ligands there is one NEP, E, which is the ratio of the average binding affinity of a free site if the status of the lattice residue neighboring one end of the site is unspecified (left to chance) to the affinity when this residue is free (holding the other neighbor constant). Thus KF is KE2, where K is the affinity of an isolated site. If a site is n residues long, SF is f ffn-1, where f = 1 - nv is the fraction of residues that are free and ff is the conditional probability that a free residue is bordered on a given side by another free residue. The expression for ff is 1/(1 + x/E), where x is v/f, E is (1 - x + [(1 - x)2 + 4x omega]1/2)/2, and omega is the co-operativity parameter. The binding of asymmetric ligands to an isotropic lattice is described by two NEPs; the last case involves four NEPs and a bound ligand orientation parameter. For each case, the expected length distribution of clusters of bound ligands can be calculated as a function of v. When Scatchard plots with the same intercepts and initial slope are compared, it is found that ligand asymmetry lowers the isotherm (relative to the corresponding symmetric ligand isotherm), whereas lattice anisotrophy raises it.     

Implications of epidermal growth factor (EGF) induced egf receptor aggregation.

To investigate the role of receptor aggregation in EGF binding, we construct a mathematical model describing receptor dimerization (and higher levels of aggregation) that permits an analysis of the influence of receptor aggregation on ligand binding. We answer two questions: (a) Can Scatchard plots of EGF binding data be analyzed productively in terms of two noninteracting receptor populations with different affinities if EGF induced receptor aggregation occurs? No. If two affinities characterize aggregated and monomeric EGF receptors, we show that the Scatchard plot should have curvature characteristic of positively cooperative binding, the opposite of that observed. Thus, the interpretation that the high affinity population represents aggregated receptors and the low affinity population nonaggregated receptors is wrong. If the two populations are interpreted without reference to receptor aggregation, an important determinant of Scatchard plot shape is ignored. (b) Can a model for EGF receptor aggregation and EGF binding be consistent with the "negative curvature" (i.e., curvature characteristic of negatively cooperative binding) observed in most Scatchard plots of EGF binding data? Yes. In addition, the restrictions on the model parameters required to obtain negatively curved Scatchard plots provide new information about binding and aggregation. In particular, EGF binding to aggregated receptors must be negatively cooperative, i.e., binding to a receptor in a dimer (or higher oligomer) having one receptor already bound occurs with lower affinity than the initial binding event. A third question we consider is whether the model we present can be used to detect the presence of mechanisms other than receptor aggregation that are contributing to Scatchard plot curvature. For the membrane and cell binding data we analyzed, the best least squares fits of the model to each of the four data sets deviate systematically from the data, indicating that additional factors are also important in shaping the binding curves. Because we have controlled experimentally for many sources of receptor heterogeneity, we have limited the potential explanations for residual Scatchard plot curvature.     

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 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.     

Analysis of a calcium ion binding system composed of two different sites

Using murexide (Mx), a metallochromic indicator, and a dual wavelength spectrophotometer with a high signal-to-noise ratio, the Ca⁺⁺ binding in a system containing two classes of binding sites was studied. Solutions with solute containing one or two classes of Ca⁺⁺ binding sites and without such solute were titrated with Ca⁺⁺ using Mx as an indicator of free Ca⁺⁺ concentration. Since curvilinear Scatchard plots are obtained from titration curves of solutes containing two classes of binding sites, a computer program was developed to resolve such plots into two linear partial plots, each corresponding to a single class of binding site. The validity of the procedure was examined with solutions of ethylene glycol bis(β-aminoethyl)-N-N′-tetraacetic acid, adenosine triphosphate (EGTA, ATP), or a mixture thereof. The method was also applied to biological material and it was found that a protein fraction isolated from rat skeletal muscle sarcotubular membranes, termed Fraction-2 (Fr-2), has two classes of binding sites for Ca⁺⁺; the association constants of the high affinity site and low affinity site are 4.3 × 10⁵ M^-1 and 9 × 10³ M^-1, respectively. The advantages and limitations of this methodology are discussed.     

Preparation of 125I-calmodulin with retention of full biological activity: Its binding to human erythrocyte ghosts

Of several methods employed for preparing ¹²⁵I-calmodulin, only the glucose oxidase-lactoperoxidase system under controlled conditions produced an iodinated derivative which retained complete biological activity. Unlabeled calmodulin and ¹²⁵I-calmodulin stimulated cyclic nucleotide phosphodiesterase from bovine brain interchangeably and both proteins displaced ¹²⁵I-calmodulin from high-affinity binding sites on human erythrocyte ghosts with equal effectiveness. This procedure yielded a labeling stoichiometry of 1.34. Scatchard plots of binding of ¹²⁵I-calmodulin to ghosts were consistent with the presence of a single class of high-affinity binding sites with the properties expected of (Ca²⁺ + Mg²⁺)-ATPase molecules. The binding showed positive cooperativity and occurred only in the presence of Ca²⁺. The maximum amount of binding seen in Scatchard plots corresponded to 4.1 × 10³ sites per ghost.     

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.     

An extended Scatchard analysis for competitively bound ligands and its application to cyclic adenosine-3',5'-monophosphate and cyclic 5'-amido-5'-deoxyadenosine-3', 5'-monophosphate binding to beef skeletal muscle protein.

A method for determining the dissociation constants of ligands and ligand analogs is described. It is based on competition binding studies in the presence of an isotope-labeled, or otherwise measurable, ligand and suitable analog concentrations.The steps used are determination of (1) the maximal amount of radioactive ligand that can be bound, (2) the slopes and intercepts from Scatchard plots at different analog concentrations and (3) the values for the dissociation constants of radioactive ligand and ligand analog from replots of the reciprocals of the slopes and intercepts obtained from the Scatchard plots. Application of the method to a cyclic AMP-binding protein from beef muscle is demonstrated, yielding dissociation constants of 2.10-9 M for cyclic (3H) AMP and cyclic AMP, and 3.10-5 M for cyclic 5'-amido-5'-deoxyadenosine-3', 5'-monophosphate.     

Cytochrome P-450-dependent 14 alpha-demethylation of lanosterol in Candida albicans.

Analysis of receptor-ligand binding characteristics can be greatly hampered by the presence of non-specific binding, defined as low-affinity binding to non-receptor domains which is not saturable within the range of ligand concentrations used. Conventional binding analyses, e.g. according to the methods described by Scatchard or Klotz, relate the amount of specific receptor-ligand binding to the concentration of free ligand, and therefore require assumptions on the amount of non-specific binding. In this paper a method is described for determining the parameters of specific receptor-ligand interaction which does not require any assumption or separate determination of the amount of non-specific binding. If the concentration of labelled free ligand is constant, a plot of Fu/(B0*-B*) versus Fu yields a linear relationship, in the case of a single receptor class, in which Fu is the concentration of unlabelled free ligand, B0* is the total amount of labelled bound ligand in the absence of unlabelled ligand and B* is the total amount of labelled bound ligand in the presence of an unlabelled ligand concentration Fu; all of these data are readily obtained from binding studies. This linear relationship holds irrespective of the amount of non-specific binding, and the values for receptor density, ligand dissociation constant and a constant for non-specific binding can be readily obtained from it. If the concentration of labelled free ligand is not a constant for all data points, data are first converted according to a straightforward normalization procedure to permit the use of this relationship. The presence of multiple receptor classes with dissociation constants in the range of the ligand concentrations used results in a negative deviation from this linearity, and therefore the presence of multiple receptor classes can be discriminated unequivocally from non-specific binding. Both theoretical and practical advantages of the present method are described. The method, which will be referred to as the linear subtraction method, is illustrated using the binding of tumour promoters and polypeptide growth factors to their specific cellular receptors.     

Binding of physostigmine to rat and human plasma and crystalline serum albumins

The binding of 3H-physostigmine (3H-Ph) to human and rat plasma proteins and crystalline serum albumin was studied by ultrafiltration technique. This study showed that the percentage of 3H-Ph bound to rat plasma slightly decreased from 49% to 41% whereas human plasma showed an increase in binding from 29% to 43% over a 50-fold increase in drug concentration. Human plasma samples which were collected in a bag coated with citrate phosphate dextrose adenine-1 solution bound 50% less 3H-Ph than samples collected with EDTA indicating a drug-drug interaction between 3H-Ph and anticoagulants. No significant change in binding was observed if the samples were frozen prior to use. Scatchard plots for binding of 3H-Ph resulted in a positive slope for human plasma and a negative slope for rat plasma; whereas curvilinear Scatchard plots with negative slopes were obtained for binding to human and rat crystalline serum albumin.     

Influence of liposome charge and composition on their interaction with human blood serum proteins

The roles of sulfhydryl and disulfide groups in the specific binding of synthetic cannabinoid CP-55,940 to the cannabinoid receptor in membrane preparations from the rat cerebral cortex have been examined. Various sulfhydryl blocking reagents including p-chloromercuribenzoic acid (p-CMB), N-ethylmaleimide (NEM), o-iodosobenzoic acid (o-ISB), and methyl methanethiosulfonate (MMTS) inhibited the specific binding of [³H]CP-55,940 to the cannabinoid receptor in a dose-dependent manner. About 80–95% inhibition was obtained at a 0.1 mM concentration of these reagents. Scatchard analysis of saturation experiments indicates that most of these sulfhydryl modifying reagents reduce both the binding affinity (K_d) and capacity (B_max). On the other hand, DL-dithiothreitol (DTT), a disulfide reducing agent, also irreversibly inhibited the specific binding of [³H]CP-55,940 to the receptor and about 50% inhibition was obtained at a 5 mM concentration. Furthermore, 5mM DTT was abelt to dissociate 50% of the bound ligand from the ligand-receptor complex. The marked inhibition of [³H]CP-55,940 binding by sulfhydryl reagents suggests that at least one free sulfhydryl group is essential to the binding of the ligand to the receptor. In addition, the inhibition of the binding by DTT implies that besides free sulfhydryl group(s), the integrity of a disulfide bridge is also important for [³H]CP-55,940 binding to the cannabinoid receptor.