Effective rate models for receptors distributed in a layer above a surface: application to cells and Biacore

In the Biacore biosensor, a widely used tool for studying the kinetics of ligand/receptor binding, receptors are commonly localized to the sensor surface through attachment to polymers that extend from the surface to form a layer. The importance of the polymeric layer in analyzing data is controversial. The question of the effect of a binding layer also arises in the case of ligands interacting with binding sites distributed in the extracellular matrix of cells. To identify and quantify the effects of a binding layer on the estimation of association and dissociation rate constants, we derived effective rate coefficients. The expressions show that rate constants determined under the standard assumption that binding takes place on a two-dimensional surface underestimate the true reaction rate constants by a factor that depends on the ratio of the height of the layer to the mean free path of the ligand within the layer. We show that, for typical biological ligands, receptors, cells, and Biacore conditions, the binding layer will affect the interpretation of data only if transport of the ligand in the layer is slowed substantially--by one or two orders of magnitude--relative to transport outside the layer. From existing experiments and theory, it is not clear which Biacore experiments, if any, have transport within the dextran layer reduced to such an extent. We propose a method, based on the effective rate coefficients we have derived, for the experimental determination of ligand diffusion coefficients in a polymeric matrix.     

Improving biosensor analysis

The quality of optical biosensor data must be improved in order to characterize the mechanism and rate constants associated with molecular interactions. Many of the artifacts associated with binding data can be minimized or eliminated by designing the experiment properly, collecting data under optimum conditions and processing the data with reference surfaces. It is possible to globally fit high-quality biosensor data with simple bimolecular reaction models, which validates the technology as a biophysical tool for interaction analysis.     

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.     

Antigen binding to receptors on immunocompetent cells. I. Simple models and interpretation of experiments

We have developed simple mathematical models for treating the kinetics of binding of multivalent antigen to immune cell receptors when the binding may be in competition with nonspecific binding of antigen to cell surfaces and with the binding of hapten to the receptors. All three kinds of binding are treated as reversible bimolecular reactions. In general, the resulting equations must be solved numerically. When, however, the antigen and hapten concentrations are large compared to receptor concentrations, approximate algebraic solutions are found. It is shown that the most important effect of the hapten-receptor binding and of the nonspecific antigen binding is to slow down the antigen-receptor association; this may be viewed as a decrease in the antigen-receptor association rate constant.We have applied these models to analyze experiments of Davie and Paul on the binding of antigen to receptors on immunocompetent cells. Many difficulties have been found to arise from nonspecific binding. In particular, the association rate constant and equilibrium constant will appear reduced by nonspecific binding and the association rate constant will appear anomalously temperature dependent. We interpret hapten inhibition of antigen binding as a nonequilibrium effect in which hapten reduces the rate of antigen-cell association. In this way the concentration of hapten required to give 50% inhibition of antigen binding is found to decrease, as observed, with time after immunization. If equilibrium were to be achieved we predict that the required concentrations of hapten would be found to increase with time.     

Biosensor Architectures for High-Fidelity Reporting of Cellular Signaling

Understanding mechanisms of information processing in cellular signaling networks requires quantitative measurements of protein activities in living cells. Biosensors are molecular probes that have been developed to directly track the activity of specific signaling proteins and their use is revolutionizing our understanding of signal transduction. The use of biosensors relies on the assumption that their activity is linearly proportional to the activity of the signaling protein they have been engineered to track. We use mechanistic mathematical models of common biosensor architectures (single-chain FRET-based biosensors), which include both intramolecular and intermolecular reactions, to study the validity of the linearity assumption. As a result of the classic mechanism of zero-order ultrasensitivity, we find that biosensor activity can be highly nonlinear so that small changes in signaling protein activity can give rise to large changes in biosensor activity and vice versa. This nonlinearity is abolished in architectures that favor the formation of biosensor oligomers, but oligomeric biosensors produce complicated FRET states. Based on this finding, we show that high-fidelity reporting is possible when a single-chain intermolecular biosensor is used that cannot undergo intramolecular reactions and is restricted to forming dimers. We provide phase diagrams that compare various trade-offs, including observer effects, which further highlight the utility of biosensor architectures that favor intermolecular over intramolecular binding. We discuss challenges in calibrating and constructing biosensors and highlight the utility of mathematical models in designing novel probes for cellular signaling.     

Biosensor Architectures for High-Fidelity Reporting of Cellular Signaling

Understanding mechanisms of information processing in cellular signaling networks requires quantitative measurements of protein activities in living cells. Biosensors are molecular probes that have been developed to directly track the activity of specific signaling proteins and their use is revolutionizing our understanding of signal transduction. The use of biosensors relies on the assumption that their activity is linearly proportional to the activity of the signaling protein they have been engineered to track. We use mechanistic mathematical models of common biosensor architectures (single-chain FRET-based biosensors), which include both intramolecular and intermolecular reactions, to study the validity of the linearity assumption. As a result of the classic mechanism of zero-order ultrasensitivity, we find that biosensor activity can be highly nonlinear so that small changes in signaling protein activity can give rise to large changes in biosensor activity and vice versa. This nonlinearity is abolished in architectures that favor the formation of biosensor oligomers, but oligomeric biosensors produce complicated FRET states. Based on this finding, we show that high-fidelity reporting is possible when a single-chain intermolecular biosensor is used that cannot undergo intramolecular reactions and is restricted to forming dimers. We provide phase diagrams that compare various trade-offs, including observer effects, which further highlight the utility of biosensor architectures that favor intermolecular over intramolecular binding. We discuss challenges in calibrating and constructing biosensors and highlight the utility of mathematical models in designing novel probes for cellular signaling.     

Receptor aggregation by intermembrane interactions: a Monte Carlo study

The lateral organization of receptors on cell surfaces is critically important to their function; many receptors transmit transmembrane signals when redistributed into clusters, while the response of others is potentiated by their aggregation. Cell-cell contact can play a crucial role in receptor aggregation, even when the bonds between receptors on one cell and ligands on the other are monovalent. Monte Carlo simulations on a two-membrane model were carried out to determine whether weak enthalpic interactions among receptors in one membrane, and among ligands in another, can work synergistically to give large-scale clustering when the two membranes are brought into contact. The simulations give support to such a clustering mechanism. In addition, because clustering is a cooperative process akin to a phase separation, individual receptors and ligands may undergo repeated binding and unbinding while in a clustered "phase," and a single ligand could interact with multiple different receptor partners. The results suggest a resolution of the dichotomy between serial triggering and aggregation models of T cell activation.     

Is movement of mannose 6-phosphate-specific receptor triggered by binding of lysosomal enzymes?

Mannose 6-phosphate-specific receptors with an apparent molecular mass of 215,000 are present in fibroblasts at the cell surface and in intracellular membranes. The cell surface receptors mediate endocytosis of exogenous lysosomal enzymes and exchange with the intracellular receptors, which function in the sorting of endogenous lysosomal enzymes. In the present study, several methods independent of receptor ligands were designed in order to examine the exchange of receptors under conditions where receptor-ligand complexes do not dissociate (weak bases and monensin) or where receptor-ligand complexes are not formed due to absence of endogenous ligands as a result of inhibition of protein synthesis. Weak bases and monensin reduce the concentration of receptors at the cell surface by 20-30% and free cell surface receptors were replaced by occupied receptors. The latter continued to be exchanged with internal ligand-occupied receptors and the rates of the exchange were similar to the control values. The exchange of receptors between the cell surface and internal membranes was also not affected when the receptor ligands were depleted from the transport compartments by treating the cells with cycloheximide for up to 10 h. We conclude from these results that movement of mannose 6-phosphate-specific receptors along the endocytosis and sorting pathways is constitutive and not triggered by binding or dissociation of ligands.     

Solvent effects on ouabain binding to the (Na+,K+)-ATPase of rat brain

The effects of the solvents deuterated water (2H2O) and dimethyl sulfoxide (Me2SO) on [3H]ouabain binding to (Na+,K+)-ATPase under different ligand conditions were examined. These solvents inhibited the type I ouabain binding to the enzyme (i.e., in the presence of Mg2+ + ATP + Na+). In contrast, both solvents stimulated type II (i.e., Mg2+ + Pi-, Mg2+-, or Mn2+-dependent) binding of the drug. The solvent effects were not due to pH changes in the reaction. However, pH did influence ouabain binding in a differential manner, depending on the ligands present. For example, changes in pH from 7.05 to 7.86 caused a drop in the rate of binding by about 15% in the presence of Mg2+ + Na+ + ATP, 75% in the Mg2+ + Pi system, and in the presence of Mn2+ an increase by 24% under similar conditions. Inhibitory or stimulatory effects of solvents were modified as various ligands, and their order of addition, were altered. Thus 2H2O inhibition of type I ouabain binding was dependent on Na+ concentration in the reaction and was reduced as Na+ was elevated. Contact of the enzyme with the Me2SO, prior to ligands for type I binding, resulted in a greater inhibition of ouabain binding than that when enzyme was exposed to Na+ + ATP first and then to Me2SO. Likewise, the stimulation of type II binding was greater when appropriate ligands acted on enzyme prior to addition of the solvent. Since Me2SO and 2H2O inhibit type I ouabain binding, it is proposed that this reaction is favored under conditions which promote loss of H2O, and E1 enzyme conformation; the stimulation of type II ouabain binding in the presence of the solvents suggests that this type of binding is favored under conditions which promote the presence of H2O at the active enzyme center and E2 enzyme conformation. This postulation of a role of H2O in modulating enzyme conformations and ouabain interaction with them is in concordance with previous observations.     

A Simple Kinetic Model with Explicit Predictions for Nuclear Transport

Molecular exchange between the cell nucleus and cytoplasm is one of the most fundamental features of eukaryotic cell biology. The nuclear pores act as a conduit of this transport, both for cargo that crosses the pore autonomously as well as that whose translocation requires an intermediary receptor. The major class of such receptors is regulated by the small GTPase Ran, via whose interaction the nucleo-cytoplasmic transport system functions as a selective molecular pump. We propose a simple analytical model for transport that includes both translocation and receptor binding kinetics. The model is suitable for steady-state kinetics such as fluorescence recovery after photobleaching. Time constants appear as a combination of parameters whose effects on measured kinetics are not separable. Competitive cargo binding to receptors and large cytoplasmic volume buffer the transport properties of any particular cargo. Specific limits to the solutions provide a qualitative insight and interpretation of nuclear transport in the cellular context. Most significantly, we find that under realistic conditions receptor binding, rather than permeability of the nuclear pores, may be rate-limiting for nucleo-cytoplasmic exchange.     

Interrupting autocrine ligand-receptor binding: comparison between receptor blockers and ligand decoys.

Stimulation of cell behavioral functions by ligand/receptor binding can be accomplished in autocrine fashion, where cells secrete ligand capable of binding to receptors on their own surfaces. This proximal secretion of autocrine ligands near the surface receptors on the secreting cell suggests that control of these systems by inhibitors of receptor/ligand binding may be more difficult than for systems involving exogenous ligands. Hence, it is of interest to predict the conditions under which successful inhibition of cell receptor binding by the autocrine ligand can be expected. Previous theoretical work using a compartmentalized model for autocrine cells has elucidated the conditions under which addition of solution decoys for the autocrine ligand can interrupt cell receptor/ligand binding via competitive binding of the secreted molecules (Forsten, K. E., and D. A. Lauffenburger. 1992. Biophys. J. 61:1-12.) We now apply a similar modeling approach to examine the addition of solution blockers targeted against the cell receptor. Comparison of the two alternative inhibition strategies reveals that a significantly lower concentration of receptor blockers, compared to ligand decoys, will obtain a high degree of inhibition. The more direct interruption scheme characteristic of the receptor blockers may make them a preferred strategy when feasible.     

Affinity enhancement by multivalent lectin–carbohydrate interaction

The binding of simple carbohydrate ligands by proteins often requires affinity enhancement to attain biologically relevant strength. This is especially true for endocytotic receptors and the molecules that engage in the first-line of defense. For such purposes, nature often utilizes a mode of affinity enhancement that arises from multiple interactions between the binding proteins and the carbohydrate ligands, which we term glycoside cluster effect. In this review article we give a number of examples and describe important factors in the multi-valent interactions that govern the degree of affinity enhancement.     

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.     

Capture and reconstitution of G protein-coupled receptors on a biosensor surface

Surface plasmon resonance (SPR) biosensors offer a unique opportunity to study the binding activity of G protein-coupled receptors (GPCRs) in real time with minimal sample preparation. Using two chemokine receptors (CXCR4 and CCR5) as model systems, we captured the proteins from crude cell preparations onto the biosensor surface and reconstituted a lipid environment to maintain receptor activity. The conformational states of the receptors were probed using conformationally dependent antibodies, and by characterizing the binding properties of a native chemokine ligand (stromal cell-derived factor 1alpha). The results suggest that the detergent-solubilized receptors are active for ligand binding in the presence and absence of a reconstituted bilayer. There are three advantages to using this receptor-capturing approach: (1) there is no need to purify the receptor prior to immobilization on the biosensor surface, (2) the receptors are homogeneously immobilized through the capturing step, and (3) the receptors can be captured at high enough densities to allow the study of relatively low-molecular-mass ligands (2000-4000Da). We also demonstrated that the receptors are sensitive to the solubilizing conditions, which illustrates the potential for using SPR biosensors to rapidly screen solublization conditions for GPCRs.     

Transforming growth factor-beta (TGF-beta) binding to the extracellular domain of the type II TGF-beta receptor: receptor capture on a biosensor surface using a new coiled-coil capture system demonstrates that avidity contributes significantly to high affinity binding

Mature TGF-beta isoforms, which are covalent dimers, signal by binding to three types of cell surface receptors, the type I, II and III TGF-beta receptors. A complex composed of the TGF-beta ligand and the type I and II receptors is required for signaling. The type II receptor is responsible for recruiting TGF-beta into the heteromeric ligand/type I receptor/type II receptor complex. The purpose of this study was to test for the extent that avidity contributes to receptor affinity. Using a surface plasmon resonance (SPR)-based biosensor (the BIACORE), we captured the extracellular domain of the type II receptor (TbetaRIIED) at the biosensor surface in an oriented and stable manner by using a de novo designed coiled-coil (E/K coil) heterodimerizing system. We characterized the kinetics of binding of three TGF-beta isoforms to this immobilized TbetaRIIED. The results demonstrate that the stoichiometry of TGF-beta binding to TbetaRIIED was one dimeric ligand to two receptors. All three TGF-beta isoforms had rapid and similar association rates, but different dissociation rates, which resulted in the equilibrium dissociation constants being approximately 5pM for the TGF-beta1 and -beta3 isoforms, and 5nM for the TGF-beta2 isoform. Since these apparent affinities are at least four orders of magnitude higher than those determined when TGF-beta was immobilized, and are close to those determined for TbetaRII at the cell surface, we suggest that avidity contributes significantly to high affinity receptor binding both at the biosensor and cell surfaces. Finally, we demonstrated that the coiled-coil immobilization approach does not require the purification of the captured protein, making it an attractive tool for the rapid study of any protein-protein interaction.     

Sandmeyer reaction repurposed for the site-selective,non-oxidizing radioiodination of fully-deprotected peptides: Studies on the endogenous opioid peptide α-neoendorphin

Standard radioiodination methods lack site-selectivity and either mask charges (Bolton-Hunter) or involve oxidative reaction conditions (chloramine-T). Opioid peptides are very sensitive to certain structural modifications, making these labeling methods untenable. In our model opioid peptide, α-neoendorphin, we replaced a tyrosyl hydroxyl with an iodine, and in cell lines stably expressing mu, delta, or kappa opioid receptors, we saw no negative effects on binding. We then optimized a repurposed Sandmeyer reaction using copper(I) catalysts with non-redoxing/non-nucleophilic ligands, bringing the radiochemical yield up to around 30%, and site-selectively incorporated radioactive iodine into this position under non-oxidizing reaction conditions, which should be broadly compatible with most peptides. The ¹²⁵I- and ¹³¹I-labeled versions of the compound bound with high affinity to opioid receptors in mouse brain homogenates, thus demonstrating the general utility of the labeling strategy and of the peptide for exploring opioid binding sites.     

A reagentless optical biosensor based on the intrinsic absorption properties of peroxidase

During the reversible reaction between peroxidase (HRP) and H(2)O(2), several peroxidase intermediate species, showing different molecular absorption spectra, are formed which can be used for H(2)O(2) determination; when H(2)O(2) is generated in a previous enzymatic reaction, the substrate involved in this reaction can also be determined. On this basis, a new family of fully reversible reagentless optical biosensors containing HRP is presented; glucose determination is used as a model. The biosensor (which can be used for at least 6 months and/or more than 750 measurements) is prepared by HRP and glucose oxidase entrapment in a polyacrylamide gel matrix. A mathematical model (in which optical, kinetic and transport aspects are considered) relating the measured absorbance with the substrate concentration is also presented together with a simple methodology for characterization of this kind of biosensor. Regarding the optical model, the Kubelka-Mulk theory of reflectance does not give good results and the biosensors are better described by the Rayleigh theory of polymer solutions. Under working conditions, linear response ranges from 1.5x10(-6) to 3.0x10(-4)M glucose and CV was about 4%. This biosensor has been applied for glucose determination in fruit juices and synthetic serum samples without sample pretreatment.     

Transport and kinetic processes underlying biomolecular interactions in the BIACORE optical biosensor

The transport and kinetic processes describing biomolecular interactions in the BIACORE optical biosensor have been studied with the help of a mathematical model. In comparison to previous models, the model presented here couples, for the first time, transport phenomena in the flow channel with hindered diffusive transport and reactions inside the hydrogel. Simulated experiments based on this model, and two simpler models extant in the literature, are used to identify cases under which the detailed model is essential for accurate prediction of kinetic parameters. It is shown that this model can substantially improve the accuracy of kinetic parameter estimation when transport limitations in the flow channel and/or the hydrogel significantly influence the observed instrument response curves. The model can extend the range of the instrument's applicability to higher concentrations of immobilized species within the hydrogel. It can also be used for accurate design of experiments with the purpose of minimizing errors in the estimation of the kinetic parameters.     

Solubilization, stabilization, and purification of chemokine receptors using biosensor technology

Establishing solubilization conditions for membrane-associated receptors is often a tedious empirical process. Here we describe a novel application of SPR biosensor technology to screen solubilization conditions automatically and to assess receptor activity directly. We focus on two chemokine receptors, CXCR4 and CCR5, which are important in HIV cell invasion. The autosampler in Biacore 3000 permitted whole cells expressing C-terminally tagged receptors to be automatically lysed under a given solubilization condition and the lysates to be injected over an antibody surface. The total amount of solubilized receptor could be quantitated from the antibody capture level, whereas the amount of active receptor could be quantitated using a subsequent injection of conformationally sensitive antibody or protein. Using this approach, we identified detergent/lipid/buffer combinations that enhanced and maintained receptor activity. We also used the biosensor to demonstrate CD4-dependent binding of gp120 to solubilized CCR5 and to develop affinity chromatography-based purification methods that increased receptor activity more than 300%. Together, these results illustrate the benefits of using the biosensor as a tool for isolating functional membrane receptors and for analyzing ligand/receptor interactions.