DESIGN: computerized optimization of experimental design for estimating Kd and Bmax in ligand binding experiments. II. Simultaneous analysis of homologous and heterologous competition curves and analysis blocking and of "multiligand" dose-response surfaces

We have developed a computer program, DESIGN, for optimization of ligand binding experiments to minimize the "average" uncertainty in all unknown parameters. An earlier report [G. E. Rovati, D. Rodbard, and P. J. Munson (1988) Anal. Biochem. 174, 636-649] described the application of this program to experiments involving a single homologous or heterologous dose-response curve. We now present several advanced features of the program DESIGN, including simultaneous optimization of two or more binding competition curves optimization of a "multiligand" experiment. Multiligand designs are those which use combinations of two (or more) ligands in each reaction tube. Such designs are an important and natural extension of the popular method of "blocking experiments" where an additional ligand is used to suppress one or more classes of sites. Extending the idea of a dose-response curve, the most general multiligand design would result in a "dose-response surface". One can now optimize the design not only for a single binding curve, but also for families of curves and for binding surfaces. The examples presented in this report further demonstrate the power and utility of the program DESIGN and the nature of D-optimal designs in the context of more complex binding experiments. We illustrate D-optimal designs involving one radioligand and two unlabeled ligands; we consider one example of homogeneous and several examples of heterogeneous binding sites. Further, to demonstrate the virtues of the dose-response surface experiment, we have compared the optimal surface design to the equivalent design restricted to traditional dose-response curves. The use of DESIGN in conjunction with multiligand experiments can improve the efficiency of estimation of the binding parameters, potentially resulting in reduction of the number of observations needed to obtain a desired degree of precision in representative cases.     

D-optimal design applied to binding saturation curves of an enkephalin analog in rat brain

The D-optimal design, a minimal sample design that minimizes the volume of the joint confidence region for the parameters, was used to evaluate binding parameters in a saturation curve with a view to reducing the number of experimental points without loosing accuracy in binding parameter estimates. Binding saturation experiments were performed in rat brain crude membrane preparations with the opioid mu-selective ligand [3H]-[D-Ala2,MePhe4,Gly-ol5]enkephalin (DAGO), using a sequential procedure. The first experiment consisted of a wide-range saturation curve, which confirmed that [3H]-DAGO binds only one class of specific sites and non-specific sites, and gave information on the experimental range and a first estimate of binding affinity (Ka), capacity (Bmax) and non-specific constant (k). On this basis the D-optimal design was computed and sequential experiments were performed each covering a wide-range traditional saturation curve, the D-optimal design and a splitting of the D-optimal design with the addition of 2 points (+/- 15% of the central point). No appreciable differences were obtained with these designs in parameter estimates and their accuracy. Thus sequential experiments based on D-optimal design seem a valid method for accurate determination of binding parameters, using far fewer points with no loss in parameter estimation accuracy.     

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.     

Optimal design of experiments for the estimation of precise hyperbolic kinetic and binding parameters

Optimal designs were evaluated for the estimation of precise parameters in kinetic and binding experiments in which the concentration dependence of the response (reaction velocity or bound ligand concentration) is characterized by a hyperbola. The designs were evaluated by maximizing the determinant of the appropriate information matrix. It is demonstrated that highest precision is obtained by replicating basic two-point designs. However, the calculated optimal designs should serve only as guidelines and not rules for experimentation.In the presence of constant variance, half of the observations should be obtained at the highest practically attainable concentration. In kinetic studies, the remaining measurements should yield half of the maximally attainable velocity. In binding experiments, the second half of observations for either bound (b) or free (f) ligand concentrations should be made at a total concentration of c = P + K (K is the dissociation constant, and P the binding capacity).With constant coefficient of variation (constant relative error), kinetic experiments should be performed, at equal frequency, at the highest and lowest attainable concentration. Half of the binding experiments should be made at their lowest precision: at the highest possible concentration when b is measured, and at the smallest feasible concentration when f is obtained. The other half of the readings should be taken at c = P ? K (if P > K) for the measurement of b, and at c = P + K for observations of f.Deviations from the calculated optimal designs result in diminished efficiency (increased variance) of the estimated parameters. The reduction can be substantial with some frequently used experimental designs, especially when the number of observations is not very small. Therefore, the first few readings could consider the validation of the model, but additional measurements should follow the guidelines presented here.     

Cooperation effects in binding of large ligands to DNA. II. Contact interactions between adsorbed ligands

Cooperative effects arising upon binding of biologically active ligands to DNA are considered. Equations are derived which enable one to describe the binding of two different ligands to DNA. We also consider the case when ligand can form two type of DNA complexes. The cooperative binding of the ligand in the vicinity of saturation level of binding can be described with a good accuracy by equation derived for the non-cooperative adsorption of the same ligand with some effective binding constant Keff. It is shown that cooperative effects arising upon binding of proteins and other ligands to DNA can be divided into two groups depending on the symmetry of interactions between the bound ligand molecules. In particular, if such interactions favor the formation of dimeric ligand species on the DNA, Keff approximately a1/2, where a is the ligand-ligand interaction constant. If cooperative interactions favor the formation of aggregates of unrestricted size, then Keff approximately aL+Y, where L is the size of the binding site for the ligand on DNA.     

Resolution of steroid binding heterogeneity by Fourier-derived affinity spectrum analysis (FASA)

A new mathematical method of analyzing radioreceptor assay data is presented. When there are many binding classes with different affinities, the probability-density function B(p) is described by the equation B(p) = (integral negative infinity to infinity) q(k)f(p-k)dk, where q(k) is the affinity spectrum (density of a particular binding class as a function of affinity) and f(p-k) is a probability function (probability that dissociation constants will fall between k and p-k, where p is the free ligand concentration). This equation is solved for q(k) and evaluated explicitly by Fourier transformation, namely, q(w) = b(w)/f(w), where w is frequency. Since division by f(w) can amplify and high frequency noise present in the experimental data, a Gaussian smoothing function is introduced thus: qs(w) = q(w)e(-w/W0)2, where W0 is a constant. This produces an affinity spectrum defined as a plot of the number of binding sites, qs(k), versus their respective dissociation constants, k. Using a FORTRAN computer program, we verify this algorithm using simulated data. We also apply the procedure to resolve heterogeneous populations of estrogen binders in human endometrium using [3H]estradiol as ligand. Two estrogen binder classes are revealed with dissociation constants approximately 2.5 natural logarithmic units apart. We identify one high-affinity (Kd = 0.18 nM)-low density (70 pM [or 72 fmol/mg protein]) subpopulation and one low affinity (Kd = 2.5 nM)-high density (101 pM [or 102 fmol/mg protein]) subpopulation of estradiol binders. The management of experimental error, sampling limitations, and nonspecific binding are discussed. This method directly transforms experimental data into an easily interpretable representation without mathematical modeling or statistical procedures.     

Assay development and data analysis of receptor-ligand binding based on scintillation proximity assay

In this paper, we described the optimization of a generic binding assay to measure ligand-receptor interactions for peroxisome proliferator-activated receptors (PPARs). The assay is based on scintillation proximity assay, in which a protein is coated on scintillant-incorporated beads, and a radiolabeled ligand stimulates the beads to emit a signal by binding to the immobilized protein. An intrinsic binding affinity of unlabeled ligands is determined by competitive displacement of the radioligand. The protein coating and ligand binding are achieved in one step by simply mixing ligands, protein and beads in sequence. No additional steps of pre-coating and washing of beads are required. Protein is captured on beads effectively by electrostatic interactions, thus no affinity labeling of protein is required. In data analysis, ligands are grouped into two classes based on their binding affinities. For tight binding ligands, an equation is derived to accurately determine the binding affinity. Otherwise a general equation applies. This quantitative and high throughput assay provides a tool to screen a large library of molecules in search of potent ligands.     

Experimental Artifacts and the Analysis of Ligand Binding Data: Results of a Computer Simulation

Abstract

Use of computerized analysis techniques for ligand binding data have recently become generally available, and are now used quite routinely. When used appropriately, these tools can improve the precision of the estimated parameters for binding affinity, K, and capacity, R. Furthermore, such programs can also calculate the uncertainty of the estimates e.g., as a percent coefficient of variation (%CV). However, because of unmeasured variability in specific activity, tracer purity, counting efficiency, counter background, efficiency of separation, etc., the actual uncertainty in the parameters K and R is usually much larger than stated. In an attempt to examine the effects of such artifacts, we have developed a computer program which simulates data arising from a number of commonly used experimental designs, and then intentionally distorted with each of these artifacts. Finally, the data are converted to B/F and B and plotted in the conventional Scatchard plot. Distortions revealed in this graph are indicative of the effect each artifact has on the parameter estimates. The computer program is generally written to simulate the binding of 2 or more ligands to one, two or many classes of independent or cooperative specific sites as well as to nonspecific sites. Thus, the program is applicable in a wide variety of situations. Results show that low tracer purity (“bindability”) or low filtration efficiency will significantly alter the measured R value. Poorly determined specific radioactivity may significantly alter the measured K value as well. Imprecise measurement of machine background may result in the specious appearance of positive cooperativity, or of additional high or low affinity classes of binding sites. Finally, under some circumstances, it is possible to detect and correct for the presence of these artifacts.     

On the need to consider kinetic as well as thermodynamic consequences of the parking problem in quantitative studies of nonspecific binding between proteins and linear polymer chains

Attention is drawn to a need for caution in the thermodynamic characterization of nonspecific binding of a large ligand to a linear acceptor such as a polynucleotide or a polysaccharide-because of the potential for misidentification of a transient (pseudoequilibrium) state as true equilibrium. The time course of equilibrium attainment during the binding of a large ligand to nonspecific three-residue sequences of a linear acceptor lattice has been simulated, either by numerical integration of the system of ordinary differential equations or by a Monte Carlo procedure, to identify the circumstances under which the kinetics of elimination of suboptimal ligand attachment (called the parking problem) create such difficulties. These simulations have demonstrated that the potential for the existence of a transient plateau in the time course of equilibrium attainment increases greatly (i) with increasing extent of acceptor saturation (i.e., with increasing ligand concentration), (ii) with increasing magnitude of the binding constant, and (iii) with increasing length of the acceptor lattice. Because the capacity of the polymer lattice for ligand is most readily determined under conditions conducive to essentially stoichiometric interaction, the parameter so obtained is thus likely to reflect the transient (irreversible) rather than equilibrium binding capacity. A procedure is described for evaluating the equilibrium capacity from that irreversible parameter; and illustrated by application to published results [M. Nesheim, M.N. Blackburn, C.M. Lawler, K.G. Mann, J. Biol. Chem. 261 (1986) 3214-3221] for the stoichiometric titration of heparin with thrombin.     

Rate constants for binding, dissociation, and internalization of EGF: effect of receptor occupancy and ligand concentration

We measured the kinetic parameters for interaction of epidermal growth factor (EGF) with fetal rat lung (FRL) cells under two sets of experimental conditions and applied sensitivity analysis to see which parameters were well-defined. In the first set of experiments (method 1), the kinetics of internalization and dissociation of radiolabeled EGF were measured with a temperature-shift protocol in medium initially devoid of free ligand. The initial concentration of radiolabeled EGF bound to the cell surface corresponded to levels of receptor occupancy ranging from approximately 200 receptors per cell to approximately 18,000 receptors per cell, a level at which EGF binding approaches saturation. In the second set of experiments (method 2), carried out at a constant temperature, we began with no surface-bound or internalized ligand. The initial free ligand concentration was varied from 0.2 to 50 ng/mL. In both sets of experiments, we measured surface-bound, internalized, and free 125I-EGF as functions of time and evaluated the parameters of a mathematical model of endocytosis. Sensitivity analysis showed that three rate constants were well-defined in this combination of two experimental approaches: ke, the endocytic rate constant; ka, the association rate constant; and kd, the dissociation rate constant. The endocytic parameter ke was found to be independent of initial surface receptor occupancy (method 1); there was some indication that it increased with initial free ligand concentration in method 2. Neither kd nor ka was found to change with extent of initial surface receptor occupancy or initial free ligand concentration, respectively, a finding of significance, since diffusion theory predicts these parameters will vary with surface receptor occupancy.(ABSTRACT TRUNCATED AT 250 WORDS)     

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.     

Optimization of the cost and sensitivity of receptor- and enzyme-based assays.

In detecting receptor antagonists or enzyme inhibitors, there are three parameters that often affect the outcome in a predictable quantitative manner: concentrations of the receptors (enzyme), labeled ligand (substrate), and antagonist (inhibitor). The usual goal of assay optimization is to maximize the ability of the assay to detect low concentrations of the analyte. Another question of practical importance, especially in screening of large numbers of samples, would be minimization of the reagent cost. Although the mathematical theory of optimization of the receptor binding assay was developed a long time ago, the resulting formulas (in the general case of unequal affinities of ligand and competitor) were not well suited for practical use. The current availability of computational programs, such as Mathematica, makes possible an efficient solution, both for receptor- and enzyme-based assays. We use a graphical approach to assay optimization and apply it to the following problems: (1) optimization of assay sensitivity, (2) optimization of the reagent cost, and (3) analysis of the entire range of the parameter values since the mathematically optimal values may sometimes be impractical. The computation is extremely simple and the problem can sometimes be solved in several minutes.     

Determination of protein-ligand binding constants at equilibrium in biological samples.

Protein-ligand complexes can be separated functionally into two classes. "Specific" binding is characterized, in relative terms, by a high affinity for the ligand and a low binding capacity. "Non-specific" binding is characterized by a low affinity and a very large capacity. The calculation of equilibrium binding constants for any specific protein-ligand interaction requires the exact determination of the unbound ligand concentration and the specifically bound ligand concentration. These determinations usually require corrections for the contribution of non-specific binding. The use of two correction terms, kn and f, is proposed: kn is the product of the affinity constant k times the number of binding sites n of the non-specific components, while f is the fraction of the non-specific binding included in the experimental estimates of bound ligand. Several theoretical solutions using these terms are proposed for the calculation of specific binding constants. The practical choice of the correction factor may be different when the simultaneous measurement of the affinity constant and maximum number of binding sites, or when only the latter, is desired. In the case of complex binding systesm containing more than one specific component, the individual constants can be determined by non-graphical methods, using computer-aided iterative statistical calculations. A complete solution is given for a system containing two specific plus non-specific interactions and actual experiments are reported for steroid hormone-receptro complexes.     

Iodination significantly influences the binding of human transferrin to the transferrin receptor

The human transferrin receptor (TfR) and its ligand, the serum iron carrier transferrin, serve as a model system for endocytic receptors. Although the complete structure of the receptor's ectodomain and a partial structure of the ligand have been published, conflicting results still exist about the magnitude of equilibrium binding constants, possibly due to different labeling techniques. In the present study, we determined the equilibrium binding constant of purified human TfR and transferrin. The results were compared to those obtained with either iodinated TfR or transferrin. Using an enzyme-linked assay for receptor-ligand interactions based on the published direct calibration ELISA technique, we determined an equilibrium constant of Kd=0.22 nM for the binding of unmodified human Tf to surface-immobilized human TfR. In a reciprocal experiment using soluble receptor and surface-bound transferrin, a similar constant of Kd=0.23 nM was measured. In contrast, covalent labeling of either TfR or transferrin with 125I reduced the affinity 3-5-fold to Kd=0.66 nM and Kd=1.01 nM, respectively. The decrease in affinity upon iodination of transferrin is contrasted by an only 1.9-fold decrease in the association rate constant, suggesting that the iodination affects rather the dissociation than the association kinetics. These results indicate that precautions should be taken when interpreting equilibrium and rate constants determined with covalently labeled components.     

Receptor binding of asialoerythropoietin.

The interaction of 125I-asialoerythropoietin (asialoepo) with receptors has been characterized both by binding assay and affinity cross-linking. Purified spleen cells from mice infected with the anemia strain of Friend virus (FVA cells) have receptors for 125I-asialoepo with two classes of affinity constant: one with Kd = 0.02-0.03 nM and 300-400 per cell, the other with lower affinity (Kd = 0.9-1.2 nM) and 1,000-1,200 per cell. The Kd value for the high affinity site is one-third of that for the binding of native 125I-erythropoietin (125I-epo) to the same FVA cells (Kd = 0.08-0.1 nM). Using 125I-asialoepo or 125I-epo affinity cross-linking methods, we find two components with apparent molecular weights of 88 kDa and 105 kDa in FVA cells, and in the transformed mouse cell lines, 201, IW32, and NN10, in agreement with earlier studies using 125I-epo. These results indicate that 125I-asialoepo binds to the same receptors as 125I-epo, but with greater affinity for the high affinity site. Since 201 cells contain only a single class of lower affinity receptors for erythropoietin (epo), finding the same two components as found for FVA cells by cross-linking experiment indicates that the two components do not represent the two classes of receptor.     

Intrinsic affinity of protein – ligand binding by differential scanning calorimetry

Differential scanning calorimetry (DSC) determines the enthalpy change upon protein unfolding and the melting temperature of the protein. Performing DSC of a protein in the presence of increasing concentrations of specifically-binding ligand yields a series of curves that can be fit to obtain the protein–ligand dissociation constant as done in the fluorescence-based thermal shift assay (FTSA, ThermoFluor, DSF). The enthalpy of unfolding, as directly determined by DSC, helps improving the precision of the fit. If the ligand binding is linked to protonation reactions, the intrinsic binding constant can be determined by performing the affinity determination at a series of pH values. Here, the intrinsic, pH-independent, affinity of acetazolamide binding to carbonic anhydrase (CA) II was determined. A series of high-affinity ligands binding to CAIX, an anticancer drug target, and CAII showed recognition and selectivity for the anticancer isozyme. Performing the DSC experiment in buffers of highly different enthalpies of protonation enabled to observe the ligand unbinding-linked protonation reactions and estimate the intrinsic enthalpy of binding. The heat capacity of combined unfolding and unbinding was determined by varying the ligand concentrations. Taken together, these parameters provided a detailed thermodynamic picture of the linked ligand binding and protein unfolding process.     

Design and analysis of protein binding experiments

The design and analysis of protein binding experiments for obtaining precise parameter estimates for a one-site and a two-site model treating fu, the fraction unbound as the experimentally determined quantity was investigated. Total drug concentrations were chosen at which the binding isotherm is determined to yield the most information about the parameters under study. The D-optimization information criterion was used to achieve this although other criteria are also discussed. For both the one-site and the two-site models the number of design points was always equal to the number of parameters being estimated. The results arrived at when dealing with constant variance and unconstrained total drug concentration were rather unique in that in most of the cases studied, all the optimal design points were away from the boundary conditions. For constant relative variance and unconstrained total drug concentrations, one of the design points was always placed at the smallest possible value of fu, the fraction unbound. For the one-site model the second point was always given by K(-1) + nP. The optimal designs arrived at lead to lower theoretical coefficients of variation in the parameters than the corresponding conventional ones. Simulated experiments supported these theoretical findings for both the one-site and the two-site models. For the one-site model, results from nonlinear regression were compared with Scatchard analysis and the optimal designs were also optimal in Scatchard space. We also show that using Scatchard analysis with the conventional strategy leads to poorly determined estimates particularly when the number of observations is low.     

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.     

A method for binding parameters estimation of A1 adenosine receptor subtype: a practical approach

Working with pig brain striatum in which A1 and A2 adenosine receptor subtypes coexist, we describe an uncomplicated method for unequivocally obtaining the equilibrium parameters (KD and binding capacity) of A1 receptor without interference from ligand binding to A2 receptor. Also, the equilibrium parameter estimation method we propose avoids the experimental determination of nonspecific binding by the inclusion of the corresponding unknown parameter in the function. This not only saves time but also avoids the use of expensive radioligands in saturation experiments. The method is suitable for any system with two different receptor subtypes for the same physiological ligand, and good estimates of the equilibrium parameters corresponding to the subtype displaying the higher affinity for the ligand can be obtained.