DESIGN: computerized optimization of experimental design for estimating Kd and Bmax in ligand binding experiments. I. Homologous and heterologous binding to one or two classes of sites

We have developed a versatile computer program for optimization of ligand binding experiments (e.g., radioreceptor assay system for hormones, drugs, etc.). This optimization algorithm is based on an overall measure of precision of the parameter estimates (D-optimality). The program DESIGN uses an exact mathematical model of the equilibrium ligand binding system with up to two ligands binding to any number of classes of binding sites. The program produces a minimal list of the optimal ligand concentrations for use in the binding experiment. This potentially reduces the time and cost necessary to perform a binding experiment. The program allows comparison of any proposed experimental design with the D-optimal design or with assay protocols in current use. The level of nonspecific binding is regarded as an unknown parameter of the system, along with the affinity constant (Kd) and binding capacity (Bmax). Selected parameters can be fixed at constant values and thereby excluded from the optimization algorithm. Emphasis may be placed on improving the precision of a single parameter or on improving the precision of all the parameters simultaneously. We present optimal designs for several of the more commonly used assay protocols (saturation binding with a single labeled ligand, competition or displacement curve, one or two classes of binding sites), and evaluate the robustness of these designs to changes in parameter values of the underlying models. We also derive the theoretical D-optimal design for the saturation binding experiment with a homogeneous receptor class.     

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.     

Autocrine ligand binding to cell receptors. Mathematical analysis of competition by solution "decoys".

Autocrine ligands have been demonstrated to regulate cell proliferation, cell adhesion, and cell migration in a number of different systems and are believed to be one of the underlying causes of malignant cell transformation. Binding of these ligands to their cellular receptors can be compromised by diffusive transport of ligand away from the secreting cell. Exogenous addition of antibodies or solution receptors capable of competing with cellular receptors for these autocrine ligands has been proposed as a means of inhibiting autocrine-stimulated cell behavioral responses. Such "decoys" complicate cellular binding by offering alternative binding targets, which may also be capable of aiding or abating transport of the ligand away from the cell surface. We present a mathematical model incorporating autocrine ligand production and the presence of competing cellular and solution receptors. We elucidate effects of key system parameters including ligand diffusion rate, binding rate constants, cell density, and secretion rate on the ability of solution receptors to inhibit cellular receptor binding. Both plated and suspension cell systems are considered. An approximate analytical expression relating the key parameters to the critical concentration of solution "decoys" required for inhibition is derived and compared to the numerical calculations. We find that in order to achieve essentially complete inhibition of surface receptor binding, the concentration of decoys may need to be as much as four to eight orders of magnitude greater than the equilibrium disociation constant for ligand binding to surface receptors.     

Minimax D-optimal designs for the logistic model

We propose an algorithm for constructing minimax D-optimal designs for the logistic model when only the ranges of the values for both parameters are assumed known. Properties of these designs are studied and compared with optimal Bayesian designs and Sitter's (1992, Biometrics, 48, 1145-1155) minimax D-optimal kk-designs. Examples of minimax D-optimal designs are presented for the logistic and power logistic models, including a dose-response design for rheumatoid arthritis patients.     

Mixed mode of ligand-DNA binding results in S-shaped binding curves

S-shaped binding curves often characterize interactions of ligands with nucleic acid molecules as analyzed by different physico-chemical and biophysical techniques. S-shaped experimental binding curves are usually interpreted as indicative of the positive cooperative interactions between the bound ligand molecules. This paper demonstrates that S-shaped binding curves may occur as a result of the "mixed mode" of DNA binding by the same ligand molecule. Mixed mode of the ligand-DNA binding can occur, for example, due to 1) isomerization or dimerization of the ligands in solution or on the DNA lattice, 2) their ability to intercalate the DNA and to bind it within the minor groove in different orientations. DNA-ligand complexes are characterized by the length of the ligand binding site on the DNA lattice (so-called "multiple-contact" model). We show here that if two or more complexes with different lengths of the ligand binding sites could be produced by the same ligand, the dependence of the concentration of the complex with the shorter length of binding site on the total concentration of ligand should be S-shaped. Our theoretical model is confirmed by comparison of the calculated and experimental CD binding curves for bis-netropsin binding to poly(dA-dT) poly(dA-dT). Bis-netropsin forms two types of DNA complexes due to its ability to interact with the DNA as monomers and trimers. Experimental S-shaped bis-netropsin-DNA binding curve is shown to be in good correlation with those calculated on the basis of our theoretical model. The present work provides new insight into the analysis of ligand-DNA binding curves.     

Single-drug multiligand conjugates: synthesis and preliminary cytotoxicity evaluation of a paclitaxel-dipeptide "scorpion" molecule

To improve the targeting properties of receptor-directed drug-peptide conjugates, a multiligand approach was proposed and a model "scorpion" conjugate (6, Figure 1), consisting of two peptide "claws" and a paclitaxel (PTX) "tail", was synthesized. The cell surface receptor-directed peptide used in this single-drug multiligand (SDML) model was a segment of the amphibian peptide bombesin (BBN) which had the Y6Q7W8A9V10G11H12L13M14-NH2 sequence, designated here as BBN[6-14] (2, Figure 2). Due to the lipophilic nature of both PTX and BBN[6-14], compound 6 had a low water solubility. To enhance the solubility, PEG derivatives of this conjugate were prepared with the polymer inserted either in the claws or in the tail regions. In a preliminary random screening, conjugate 6 showed superior cytotoxic activity in several GRPR-positive human cancer cell lines as compared to free PTX and two single-drug single-ligand (SDSL) conjugates. In a receptor blocking experiment, addition of excess unconjugated BBN[6-14] ligand reduced the cytotoxicity of conjugate 6, indicating the receptor-mediated mechanism of drug delivery. The PEG-derived conjugates showed activities which were intermediate between SDSL and the SDML congeners. Also, an increase in the number of the PEG segments lowered cytotoxicity, possibly due to steric hindrance against ligand-receptor binding. Taken together, these results demonstrate the potential of the multiligand approach in the design of receptor-targeting conjugates for tumor-specific drug delivery.     

Efficient design for estimation of median lethal dose and quantal dose-response curves

The results of quantal dose-response experiments are often summarized by an estimate of the "median lethal dose," denoted LD50, and many sequential designs have been proposed for efficient estimation of LD50. These designs strive to produce a sequence of trials at dose levels that get closer and closer to LD50. Consequently, they may not provide very good estimates of the overall shape of the dose-response curve. In this paper we propose guidelines for the design of experiments that estimate LD50 fairly efficiently and that also allow for efficient global estimation of the curve.     

EQUIL: Simulation and data analysis of binding reactions with arbitrary chemical models

We have developed an algorithm for simulation and analysis of arbitrary chemical systems in equilibrium, with emphasis on ligand binding reactions. The program EQUIL can treat reactions involving multiple ligands, multiple binding sites, ternary complex models, allosteric effectors, competitive and noncompetitive binding, conformational changes, cooperativity, and generally any scheme that can be represented as a set of chemical equations. EQUIL is based on a general thermodynamic model of chemical equilibria; it does not involve nonlinear transformation of experimental data, but it does require the user to define the model of interaction between ligands and receptors by writing down the appropriate chemical reactions. EQUIL contains features of particular importance to ligand binding experiments: variable binding capacities, nonspecific binding, and the ability to simultaneously analyze data from different types of experiments. Furthermore, the simulation feature of EQUIL allows the user to investigate the feasibility of experiments that could possibly distinguish between different reaction models. We illustrate the use of this program on personal computers to analyze and simulate simple and complicated interactions between ligands and receptors.     

A danger of low copy numbers for inferring incorrect cooperativity degree

Background  

A dose-response curve depicts the fraction of bound proteins as a function of unbound ligands. Dose-response curves are used to measure the cooperativity degree of a ligand binding process. Frequently, the Hill function is used to fit the experimental data. The Hill function is parameterized by the value of the dissociation constant and the Hill coefficient, which describes the cooperativity degree. The use of Hill's model and the Hill function has been heavily criticised in this context, predominantly the assumption that all ligands bind at once, which resulted in further refinements of the model. In this work, the validity of the Hill function has been studied from an entirely different point of view. In the limit of low copy numbers the dynamics of the system becomes noisy. The goal was to asses the validity of the Hill function in this limit, and to see in what ways the effects of the fluctuations change the form of the dose-response curves.     

Protein hormones and their receptors.

Technological advances in the isolation and characterization of novel receptors have led to a significant increase in our understanding of protein-ligand binding to receptors and the means by which responses are triggered. Hormones and their receptors are composed of structurally conserved domains, and several ligands appear to use similar surface regions for receptor binding. A key event in signal transduction is the aggregation by the ligand of one or more receptor subunits, and this can include the sharing of subunits between different ligands. These findings have allowed the design of ligands with receptor-antagonist properties.     

Ligand binding to proteins: the binding landscape model.

Models of ligand binding are often based on four assumptions: (1) steric fit: that binding is determined mainly by shape complementarity; (2) native binding: that ligands mainly bind to native states; (3) locality: that ligands perturb protein structures mainly at the binding site; and (4) continuity: that small changes in ligand or protein structure lead to small changes in binding affinity. Using a generalization of the 2D HP lattice model, we study ligand binding and explore these assumptions. We first validate the model by showing that it reproduces typical binding behaviors. We observe ligand-induced denaturation, ANS and heme-like binding, and "lock-and-key" and "induced-fit" specific binding behaviors characterized by Michaelis-Menten or more cooperative types of binding isotherms. We then explore cases where the model predicts violations of the standard assumptions. For example, very different binding modes can result from two ligands of identical shape. Ligands can sometimes bind highly denatured states more tightly than native states and yet have Michaelis-Menten isotherms. Even low-population binding to denatured states can cause changes in global stability, hydrogen-exchange rates, and thermal B-factors, contrary to expectations, but in agreement with experiments. We conclude that ligand binding, similar to protein folding, may be better described in terms of energy landscapes than in terms of simpler mass-action models.     

Binding analyses between Human PPARgamma-LBD and ligands.

The binding characteristics of a series of PPARgamma ligands (GW9662, GI 262570, cis-parinaric acid, 15-deoxy-Delta(12,14)-prostaglandin J(2), LY171883, indomethacin, linoleic acid, palmitic acid and troglitazone) to human PPARgamma ligand binding domain have been investigated for the first time by using surface plasmon resonance biosensor technology, CD spectroscopy and molecular docking simulation. The surface plasmon resonance biosensor determined equilibrium dissociation constants (KD values) are in agreement with the results reported in the literature measured by other methods, indicating that the surface plasmon resonance biosensor can assume a direct assay method in screening new PPARgamma agonists or antagonists. Conformational changes of PPARgamma caused by the ligand binding were detected by CD determination. It is interesting that the thermal stability of the receptor, reflected by the increase of the transition temperature (T(m)), was enhanced by the binding of the ligands. The increment of the transition temperature (DeltaT(m)) of PPARgamma owing to ligand binding correlated well with the binding affinity. This finding implies that CD could possibly be a complementary technology with which to determine the binding affinities of ligands to PPARgamma. Molecular docking simulation provided reasonable and reliable binding models of the ligands to PPARgamma at the atomic level, which gave a good explanation of the structure-binding affinity relationship for the ligands interacting with PPARgamma. Moreover, the predicted binding free energies for the ligands correlated well with the binding constants measured by the surface plasmon resonance biosensor, indicating that the docking paradigm used in this study could possibly be employed in virtual screening to discover new PPARgamma ligands, although the docking program cannot accurately predict the absolute ligand-PPARgamma binding affinity.     

Activation of EphB2 and its ligands promotes vascular smooth muscle cell proliferation.

EphB2 and its ligands regulate interactions between endothelial and mesenchymal cells in developing arteries. In adult arteries, the relationship between smooth muscle cells and overlying intact endothelium is responsible for maintaining the health of the vessel. Heparin inhibits vascular smooth muscle cell growth in culture and intimal hyperplasia following endothelial denudation. Using gene microarrays, we identified the tyrosine kinase receptor EphB2 as being differentially expressed in response to continuous intravenous heparin administration in the rabbit model of arterial injury. EphB2 protein levels increased in cultured bovine vascular smooth muscle cells following serum stimulation and were decreased in a dose-dependent fashion by heparin. Fc chimeras of the binding domain of the EphB2 ligands blocked the formation of the EphB2 ligand-receptor complex and reduced growth of serum-stimulated vascular smooth muscle cells in a dose-dependent fashion. Activation of the ligand by an Fc chimera to EphB2 followed a parabolic dose-response growth curve, indicating growth stimulation until the chimera begins to compete with native receptors. Co-administration of EphB2/Fc chimera with heparin shifted the dose-response curve to the right. These data indicate a possible new route of Heparin's antiproliferative effect and a role of EphB2 and its ligands in vascular smooth muscle cell proliferation.     

Multi-random binding molecular interactions: a general kinetic model

A kinetic model is suggested to account for the interactions of several ligands with a target whose molecule possesses several independent equivalent receptor sites for each ligand (multiligand multisite model). To analyse the problem, we shall derive solutions for three elementary situations: (a) interactions of a ligand with a mono-receptor site target molecule (monosite model); (b) interactions of several ligands with a target whose molecule possesses one receptor site for each ligand involved (multiligand model); (c) interactions of a ligand with a target whose molecule possesses several receptor sites of the same kind for this ligand (multisite model). Throughout this study, every ligand molecule is assumed to offer one binding site to the target. The main implications of the corresponding analytical solutions are discussed from a molecular point of view. The results cover a great many well-known aspects of the molecular interactions in various fields such as enzymology, endocrinology, radio-immunology and saturation analysis. As suggested by the inhibition patterns obtained, this model may therefore provide a new point of view to interpret the relevant phenomena. Furthermore, a kinetic approach to the generalized mass action law can be deduced from this model, and experimental conditions in which the isotopic dilution law applies are examined.     

Mixed Mode of Ligand-DNA Binding Results in S-Shaped Binding Curves

Abstract

S-shaped binding curves often characterize interactions of ligands with nucleic acid molecules as analyzed by different physicochemical and biophysical techniques. S-shaped experimental binding curves are usually interpreted as indicative of the positive cooperative interactions between the bound ligand molecules. This paper demonstrates that S-shaped binding curves may occur as a result of the “mixed mode” of DNA binding by the same ligand molecule. Mixed mode of the ligand-DNA binding can occur, for example, due to 1) isomerization or dimerization of the ligands in solution or on the DNA lattice, 2) their ability to intercalate the DNA and to bind it within the minor groove in different orientations. DNA- ligand complexes are characterized by the length of the ligand binding site on the DNA lattice (so-called “multiple-contact” model). We show here that if two or more complexes with different lengths of the ligand binding sites could be produced by the same ligand, the dependence of the concentration of the complex with the shorter length of binding site on the total concentration of ligand should be S-shaped. Our theoretical model is confirmed by comparison of the calculated and experimental CD binding curves for bis-netropsin binding to poly(dA-dT) poly(dA-dT). Bis-netropsin forms two types of DNA complexes due to its ability to interact with the DNA as monomers and trimers. Experimental S-shaped bis-netropsin-DNA binding curve is shown to be in good correlation with those calculated on the basis of our theoretical model. The present work provides new insight into the analysis of ligand-DNA binding curves.     

Selective immobilization of multivalent ligands for surface plasmon resonance and fluorescence microscopy

Cell surface multivalent ligands, such as proteoglycans and mucins, are often tethered by a single attachment point. In vitro, however, it is difficult to immobilize multivalent ligands at single sites due to their heterogeneity. Moreover, multivalent ligands often lack a single group with reactivity orthogonal to other functionality in the ligand. Biophysical analyses of multivalent ligand-receptor interactions would benefit from the availability of strategies for uniform immobilization of multivalent ligands. To this end, we report the design and synthesis of a multivalent ligand that has a single terminal orthogonal functional group and we demonstrate that this material can be selectively immobilized onto a surface suitable for surface plasmon resonance (SPR) experiments. The polymeric ligand we generated displays multiple copies of 3,6-disulfogalactose, and it can bind to the cell adhesion molecules P- and L-selectin. Using SPR measurements, we found that surfaces displaying our multivalent ligands bind specifically to P- and L-selectin. The affinities of P- and L-selectin for surfaces displaying the multivalent ligand are five- to sixfold better than the affinities for a surface modified with the corresponding monovalent ligand. In addition to binding soluble proteins, surfaces bearing immobilized polymers bound to cells displaying L-selectin. Cell binding was confirmed by visualizing adherent cells by fluorescence microscopy. Together, our results indicate that synthetic surfaces can be created by selective immobilization of multivalent ligands and that these surfaces are capable of binding soluble and cell-surface-associated receptors with high affinity.     

Spatio-temporal dependence of the signaling response in immune-receptor trafficking networks regulated by cell density: a theoretical model

Cell signaling processes involve receptor trafficking through highly connected networks of interacting components. The binding of surface receptors to their specific ligands is a key factor for the control and triggering of signaling pathways. In most experimental systems, ligand concentration and cell density vary within a wide range of values. Dependence of the signal response on cell density is related with the extracellular volume available per cell. This dependence has previously been studied using non-spatial models which assume that signaling components are well mixed and uniformly distributed in a single compartment. In this paper, a mathematical model that shows the influence exerted by cell density on the spatio-temporal evolution of ligands, cell surface receptors, and intracellular signaling molecules is developed. To this end, partial differential equations were used to model ligand and receptor trafficking dynamics through the different domains of the whole system. This enabled us to analyze several interesting features involved with these systems, namely: a) how the perturbation caused by the signaling response propagates through the system; b) receptor internalization dynamics and how cell density affects the robustness of dose-response curves upon variation of the binding affinity; and c) that enhanced correlations between ligand input and system response are obtained under conditions that result in larger perturbations of the equilibrium ligand + surface receptor [Please see text] ligand - receptor complex. Finally, the results are compared with those obtained by considering that the above components are well mixed in a single compartment.     

Cross-linking reconsidered: binding and cross-linking fields and the cellular response.

We analyze a model for the reversible cross-linking of cell surface receptors by a collection of bivalent ligands with different affinities for the receptor as would be found in a polyclonal anti-receptor serum. We assume that the amount of cross-linking determines, via a monotonic function, the rate at which cells become activated and divide. In addition to the density of receptors on the cell surface, two quantities, the binding field and the cross-linking field, are needed to characterize the cross-linking curve, i.e., the equilibrium concentration of cross-linked receptors plotted as a function of the total ligand site concentration. The binding field is the sum of all ligand site concentrations weighted by their respective binding affinities, and the cross-linking field is the sum of all ligand site concentrations weighted by the product of their respective binding and cross-linking affinity and the total receptor density. Assuming that the cross-linking affinity decreases if the binding affinity decreases, we find that the height of the cross-linking curve decreases, its width narrows, and its center shifts to higher ligand site concentrations as the affinities decrease. Moreover, when we consider cross-linking-induced proliferation, we find that there is a minimum cross-linking affinity that must be surpassed before a clone can expand. We also show that under many circumstances a polyclonal antiserum would be more likely than a monoclonal antibody to lead to cross-linking-induced proliferation.     

Minimal models of multi-site ligand-binding kinetics

Mathematical models based on the current understanding of co-operativity in ligand binding to the (macro) molecule and relating the dose-response (saturation) curve of the (macro) molecule ligation to intrinsic dissociation constants characterizing the affinities of ligand for binding sites of both unliganded and partly liganded (macro) molecule have been developed. The simplified models disregarding the structural properties and considerations concerning conformational changes of the (macro) molecule retain the ability to yield sigmoid curves of ligand binding and reflect the co-operativity. Model 1 contains only three parameters, parameter κ (a multiplier characterising the change in the affinity) reflects also the existence and type of co-operativity of ligand binding: κ<1 corresponds to positive co-operativity, κ>1 to the negative and κ=1 to the absence of any co-operativity. Model 2 contains an extra parameter, ω, equilibrium constant for the T₀↔R₀ transition but fails to produce dose-response, which would suggest negative co-operativity. For any fixed n>1, the deviation of the dose-response (saturation) curve from the Henri hyperbola depends either solely on parameter κ (Model 1) or also on parameter ω (Model 2). The (macro) molecule being a receptor, both models yield a diversity of dose-response curves due to possible variety of efficacies of the (macro) molecule. The models may be considered as extensions of the Henri model: in case the dissociation constants remain unchanged, the proposed models are reduced to the latter.