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.     

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.     

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.     

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.     

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.     

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.     

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 effect of ligand heterogeneity on the Scatchard plot. Particular relevance to lipoprotein binding analysis

Computer simulations of equilibrium binding studies of a mixture of two labeled ligands binding competitively to a single class of identical and independent sites (receptors) were performed to investigate how ligand heterogeneity affects the observed data in such studies. The simulated data are presented in Scatchard plots. Ligand heterogeneity was generally found to be indistinguishable from the case of a homogeneous ligand when usual experimental conditions applied (that is, Scatchard plots of the data were straight lines). Some factors that increased the probability of recognizing heterogeneity in the system were identified, however. These are 1) a large difference between the dissociation constants of the two ligands, 2) a high concentration of receptors relative to the dissociation constant of the higher-affinity ligand, 3) a high concentration of the lower-affinity ligand relative to that of the higher-affinity ligand, 4) a high specific activity of the lower-affinity ligand relative to that of the higher-affinity ligand, and 5) lack of experimental error. When ligand heterogeneity (under certain conditions) did cause curvilinearity in the Scatchard plot, the curve formed was always concave-downwards. Thus, ligand heterogeneity may occasionally mimic positive cooperativity, but never mimics negative cooperativity or multiple classes of binding sites. Implications of these findings for equilibrium binding studies involving lipoproteins (which are generally isolated as heterogeneous mixtures of particles) are discussed in detail. These findings are also relevant to equilibrium binding studies using ligands which are mixtures of stereoisomers or which contain chemical or radiochemical impurities.     

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.     

Heme reactivity of hemoglobins. Azide and fluoride binding equilibria of free and mercuriated ferri-gamma chains.

The free gamma chains, isolated from human foetal hemoglobin, are stable when oxidized and thus suitable for ligand binding and subunit equilibrium studies. The metaquo-ferri chains, with cysteine-F9 in the free state II ag gamma SH) possess several properties which are different from those of their p-mercuribenzoate derivative (III aq gammaSHgR); these are: stronger binding of a high-field ligand (N3- minus), altered spin equilibrium and an altered subunit equilibrium. A quantitative assessment of the free energy changes associated with all individual steps involved in changing the metaquo chains to their azide derivatives has been made. The results show that the higher apparent reactivity of III ag gammaSH (compared to IIIaq gammaSHgR) for the azide ion is not solely due to compensatory effects arising from differences of subunit dissociation or of spin equilibrium: other process(es) occurring in the ligand binding site have to be considered.     

Ligand-dependent aggregation and cooperativity: a critique

Ligand-dependent site-site (or subunit-subunit) interactions provide the basis for explaining cooperativity in chemical reactions. Even in the simplest possible nonaggregating system, interpretation of the interactions in terms of structural details requires an explicit assumption (or model) for the binding of the ligand to the sites when there are no interactions. This paper develops in detail the processes by which aggregation will yield ligand-dependent cooperativity. Two conceptually distinct free energy differences may contribute to cooperativity in an aggregation reaction. One is the free energy difference in ligand binding between the monomer and the aggregate. The other is derived from ligand-dependent interactions between the sites of the aggregate. In this analysis an explicit distinction is made between the experimentally accessible constants and those derived from assumed models. Experimental measurements of an aggregation cycle in which all of the species in equilibrium are defined do not allow for an evaluation of the energies of interaction without some model (or assumption). In the analysis presented, an explicit assumption is employed relating the constant for binding of the ligand to the isolated monomer and the constant for the binding of the ligand to aggregate under conditions where there are no ligand-dependent interactions.     

The influence of subterranean termites on the hydrological characteristics of a Chihuahuan desert ecosystem

Summary Rainfall simulation at an average intensity of 124 mm·h^-1 was used to compare infiltration and run off on arid areas where subterranean termites had been eliminated four years prior to the initiation of the study (termite free) with adjacent areas populated by subterranean termites (termites present). Infiltration rates on termite free plots with less than 5% perennial plant cover were significantly lower 51.3±6.8 mm·h^-1 than rates on comparable termites present plots 88.4±5.6 mm·h^-1. On plots centered on Larrea tridentata shrubs, there were no differences in infiltration rates with or without termites. Plots with shrub cover had the highest infiltration rates 101±6 mm·h^-1. Highest run-off volumes were recorded from termite free <5% grass cover plots and the lowest from plots with shrubs. There were no differences in suspended sediment concentrations from termites present and termite free plots. Average bed load concentration was more than three times greater from termite free, <5% cover plots than from termites present, <5% cover plots.The reduction in infiltration, high run-off volumes and high bedloads from termite free areas without shrub cover is related to increased soil bulk density resulting from the collapse of subterranean galleries of the termites that provide avenues of bulk flow into the soil. Subterranean termites affect the hydrology of Chihuahuan desert systems by enhancing water infiltration and retention of top soil. The presence of a shrub canopy and litter layer cancels any effect of subterranean termites on hydrological parameters. Since approximately 2/3 of the area is not under shrub canopies, subterranean termites are considered to be essential for the maintenance of the soil water characteristics that support the present vegetation.     

Free energy simulations for protein ligand binding and stability

Abstract

We summarize several computational techniques to determine relative free energies for condensed-phase systems. The focus is on practical considerations which are capable of making direct contact with experiments. Particular applications include the thermodynamic stability of apo- and holo-myoglobin, insulin dimerization free energy, ligand binding in lysozyme, and ligand diffusion in globular proteins. In addition to provide differential free energies between neighboring states, converged umbrella sampling simulations provide insight into migration barriers and ligand dissociation barriers and analysis of the trajectories yield additional insight into the structural dynamics of fundamental processes. Also, such simulations are useful tools to quantify relative stability changes for situations where experiments are difficult. This is illustrated for NO-bound myoglobin. For the dissociation of benzonitrile from lysozyme it is found that long umbrella sampling simulations are required to approximately converge the free energy profile. Then, however, the resulting differential free energy between the bound and unbound state is in good agreement with estimates from molecular mechanics with generalized Born surface area simulations. Furthermore, comparing the barrier height for ligand escape suggests that ligand dissociation contains a non-equilibrium component.     

Regulation of tryptophan synthase by temperature, monovalent cations, and an allosteric ligand. Evidence from Arrhenius plots, absorption spectra, and primary kinetic isotope effects

To investigate the linkage between enzyme conformation and catalysis, we have determined the effects of temperature on catalytic properties of the tryptophan synthase alpha(2)beta(2) complex and beta(2) subunit in the absence or presence of different monovalent cations (Cs(+), Na(+), and GuH(+)) and of an allosteric ligand, alpha-glycerol 3-phosphate. Arrhenius plots of the activity data between 5 and 50 degrees C are nonlinear in the presence of certain ligands but not others. The conditions that yield nonlinear Arrhenius plots also yield temperature-dependent changes in the equilibrium distribution of enzyme-substrate intermediates and in primary kinetic isotope effects. The results provide evidence that the nonlinear Arrhenius plots are caused by a temperature-dependent conformational change that precedes the rate-limiting step in catalysis. Thermodynamic analysis of the data associated with the conformational change shows that the activation energies are much higher at low temperatures than at high temperatures. We correlate the results with a model in which the enzyme is converted by increased temperature under certain conditions from a low-activity "open" conformation to a high-activity "closed" conformation. The allosteric ligand and different monovalent cations, including GuH(+), which also acts as a chaotropic agent, affect the equilibrium between the open and closed forms. The large positive entropy changes in the conformational conversion suggest that the closed conformation results from tightened hydrophobic interactions that exclude water from the active site of the beta subunit.     

The influence of cell surface receptor clustering on the thermodynamics of ligand binding and the kinetics of its dissociation.

We show that an equilibrium model for the clustering, by divalent ligand, of homogeneous plasma membrane-bound divalent receptors that do not change conformation predicts Scatchard plots with positive second derivatives (concave up). The result is thermodynamically indistinguishable from predictions based upon receptors that are heterogeneous for ligand, or that change conformation in a negatively cooperative way when ligand binds. A more general formulation of the theory, which allows application to dissociation kinetics, predicts that the dissociation of labeled ligand is accelerated in the presence of excess cold ligand and, moreover, that the accelerated dissociation has at least two components. The theory is briefly illustrated by fitting five different sets of data, at several temperatures for two different systems, with parameter values that are well within physically meaningful ranges. The model presented is the simplest and least ad hoc explanation thus far proposed as the basis for the kinetic phenomena, and raises the possibility that the numerous experiments in which such data are observed are primarily minifestations of cell surface clustering.     

Protein structural changes accompanying formation of enzymatic transition states: tryptophan environment in ground-state and transition-state analogue complexes of adenosine deaminase

The accessibility of protein tryptophan fluorescence to the quenching agent acrylamide has been studied in adenosine deaminase and in binary complexes of the enzyme with ground-state or transition-state analogues. Although the enzyme contains three tryptophan residues, Stern-Volmer plots are linear with all the fluorescence quenchable at high acrylamide concentrations. Tryptophan fluorescence is less easily quenched in the binary complexes than in the free enzyme, indicating a decrease in the accessibility of these residues. The greatest decrease in accessibility is found for the transition-state analogue complexes. Although the affinities of the transition-state analogues studied span a range of 10(6), the Stern-Volmer constants of the complexes are the same within experimental error. Thus, as measured by this technique, changes in enzyme conformation accompanying formation of these complexes are similar for all transition-state analogues. Resonance energy transfer from tryptophan as donor to ligand as acceptor successfully explains the differing abilities of ligands to quench the enzyme's intrinsic fluorescence upon formation of complexes in the absence of acrylamide. On the basis of Forster distance calculations, it is likely that the residues partially quenched upon formation of transition-state analogue complexes are distant from the active site.     

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.     

A simple theoretical model for fluorescence polarization binding assay development

Fluorescence polarization is a screening technology that is radioactivity free, homogeneous, and ratiometric. The signal measured with this technology is a weighted value of free and bound ligand. As a consequence, saturation curves are accessible only after calculation of the corresponding concentrations of free and bound ligand. To make this technology more accessible to assay development, the authors propose a simple mathematical model that predicts fluorescence polarization values from ligand and receptor total concentrations, depending on the corresponding dissociation constant. This model was validated using data of Bodipy-NDP-alphaMSH binding to MC(5), obtained after either ligand saturation of a receptor preparation or, conversely, receptor saturation of a ligand solution. These experimental data were also used to calculate the actual concentration of free and bound ligand and receptor and to obtain pharmacological constants by Scatchard analysis. A general method is proposed, which facilitates the design of fluorescence polarization binding assays by relying on the representation of theoretical polarization values. This approach is illustrated by the application to 2 systems of very different affinities.     

Theoretical calculations of the helix–coil transition of DNA in the presence of large,cooperatively binding ligands

Theoretical calculations are conducted on the helix–coil transition of DNA, in the presence of large, cooperatively binding ligands modeled after the DNA-binding proteins of current biological interest. The ligands are allowed to bind both to helx and to coil, to cover up any number of bases or base pairs in the complex, and to interact cooperatively with their nearest neighbors. The DNA is treated in the infinite homogeneous Ising model approximation, and all calculations are done by Lifson's method of sequence-generating functions. DNA melting curves are calculated by computer in order to expolore the effects on the transition of ligand size, binding constant, free activity, and ligand–ligand cooperativity. The calculations indicate that (1) at the same intrinsic free energy change per base pair of the complexes, small ligands, for purely entropic reasons, are more effective than are large ligands in shifting the DNA melting temperature; (2) the response of the DNA melting temperature to increased ligand binding constant K and/or free ligand activity L is adequately represented at high values of KL (but not at low KL) by a simple independent site model; (3) if curves are calculated with the total amount of added ligand remaining constant and the free ligand activity allowed to vary throughout the transition, biphasic melting curves can be obtained in the complete absence of ligand–ligand cooperativity. In an Appendix, the denaturation of poly[d(A-T)] in the presence of the drug, netropsin, is used to verify some features of the theory and to illustrate how the theory can be used to obtain numerical estimates of the ligand binding parameters from the experimental melting curves.