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.     

Forward rate constants for receptor clusters. Variational methods for upper and lower bounds

We are interested in the effect of receptor clustering on k+, the diffusion-limited forward rate constant for the binding of a ligand to a cell surface receptor. Here we estimate the reduction in k+ when receptors are clustered in various configurations. We obtain two alternative expressions for the flux of ligands into receptors distributed on a surface. Next we show through a variational principle that these provide both upper and lower bounds on the flux when evaluated for trial concentration functions which satisfy only the boundary conditions of the Laplace equation. We use an analogy with electrostatics to calculate rigorous bounds within approx. 10% of the exact result for a variety of planar clusters of hemispherical receptor sites. We also obtain an exact result for the flux into a spheroidal receptor and use this result to obtain bounds on the flux into certain receptor clusters.     

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.     

Clustering of ligand on the surface of a particle enhances adhesion to receptor-bearing cells

Human leukocytes express a receptor that mediates the binding of cells and particles coated with C3bi, a fragment of the third component of complement. Previous data indicate that the capacity of this receptor to mediate binding is regulated by changes in its aggregation state. Randomly distributed receptors bind ligand very inefficiently, but stimulation of polymorphonuclear leukocytes with phorbol esters causes a ligand-independent clustering of the receptors in the membrane, and the clustered receptors avidly bind C3bi-coated cells (1). We examined whether the aggregation state of surface-bound ligands also affects the efficiency of binding between receptors and ligands. We found that erythrocytes bearing C3bi in clusters were bound by both macrophages and polymorphonuclear leukocytes far more avidly than erythrocytes bearing the same number of ligands in random array. We made similar observations with erythrocytes coated with C3b, a ligand that is recognized by a separate receptor. Our observations show that the ability of a receptor-bearing cell to bind particles coated with the corresponding ligands is dramatically affected by the distribution of ligand on the surface of the particle. Cell-cell interactions may thus be regulated by alterations in the two-dimensional distribution of receptors and ligands on opposing cell surfaces.     

Competition between solution and cell surface receptors for ligand. Dissociation of hapten bound to surface antibody in the presence of solution antibody.

We present a joint theoretical and experimental study on the effects of competition for ligand between receptors in solution and receptors on cell surfaces. We focus on the following experiment. After ligand and cell surface receptors equilibrate, solution receptors are introduced, and the dissociation of surface bound ligand is monitored. We derive theoretical expressions for the dissociation rate and compare with experiment. In a standard dissociation experiment (no solution receptors present) dissociation may be slowed by rebinding, i.e., at high receptor densities a ligand that dissociates from one receptor may rebind to other receptors before separating from the cell. Our theory predicts that rebinding will be prevented when S much greater than N2Kon/(16 pi 2D a4), where S is the free receptor site concentration in solution, N the number of free surface receptor sites per cell, Kon the forward rate constant for ligand-receptor binding in solution, D the diffusion coefficient of the ligand, and a the cell radius. The predicted concentration of solution receptors needed to prevent rebinding is proportional to the square of the cell surface receptor density. The experimental system used in these studies consists of a monovalent ligand, 2,4-dinitrophenyl (DNP)-aminocaproyl-L-tyrosine (DCT), that reversibly binds to a monoclonal anti-DNP immunoglobulin E (IgE). This IgE is both a solution receptor and, when anchored to its high affinity Fc epsilon receptor on rat basophilic leukemia (RBL) cells, a surface receptor. For RBL cells with 6 x 10(5) binding sites per cell, our theory predicts that to prevent DCT rebinding to cell surface IgE during dissociation requires S much greater than 2,400 nM. We show that for S = 200-1,700 nM, the dissociation rate of DCT from surface IgE is substantially slower than from solution IgE where no rebinding occurs. Other predictions are also tested and shown to be consistent with experiment.     

Revising Berg-Purcell for finite receptor kinetics

From nutrient uptake to chemoreception to synaptic transmission, many systems in cell biology depend on molecules diffusing and binding to membrane receptors. Mathematical analysis of such systems often neglects the fact that receptors process molecules at finite kinetic rates. A key example is the celebrated formula of Berg and Purcell for the rate that cell surface receptors capture extracellular molecules. Indeed, this influential result is only valid if receptors transport molecules through the cell wall at a rate much faster than molecules arrive at receptors. From a mathematical perspective, ignoring receptor kinetics is convenient because it makes the diffusing molecules independent. In contrast, including receptor kinetics introduces correlations between the diffusing molecules because, for example, bound receptors may be temporarily blocked from binding additional molecules. In this work, we present a modeling framework for coupling bulk diffusion to surface receptors with finite kinetic rates. The framework uses boundary homogenization to couple the diffusion equation to nonlinear ordinary differential equations on the boundary. We use this framework to derive an explicit formula for the cellular uptake rate and show that the analysis of Berg and Purcell significantly overestimates uptake in some typical biophysical scenarios. We confirm our analysis by numerical simulations of a many-particle stochastic system.     

Analysis of the steady state binding, internalization, and degradation of human interferon-alpha2

Scatchard analyses of the equilibrium binding of radiolabeled human interferon-alpha2 (huIFN-alpha2) to Madin-Darby bovine kidney cells previously exposed to subsaturating concentrations of IFN-alpha showed approximately a 50% decrease in the number of cell surface receptors and no change in the apparent dissociation constant, Kd, compared with cells not exposed to interferon. The steady state equations describing the interaction of polypeptide ligands with cell surface receptors under physiological conditions (Wiley, H.S., and Cunningham, D.D. (1981) Cell 25, 433-440) have allowed us to determine, under steady state conditions, the rate of insertion of receptors into the cell membrane, the endocytic rate constant of occupied receptors, the rate constant of turnover of unoccupied receptors, and the rate of hydrolysis of internalized ligand. Our results indicate that occupied and unoccupied interferon receptors are cleared from the cell surface at approximately the same rate. This suggests that the down-regulation of the huIFN-alpha2 receptor on Madin-Darby bovine kidney cells by huIFN-alpha2 differs from that of several other surface receptors for polypeptide hormones and growth factors analyzed on cultured cells in that the binding of huIFN-alpha2 to its receptor does not increase the rate of receptor endocytosis.     

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.     

Identification of CC chemokine receptor 7 residues important for receptor activation

The binding pocket of family A GPCRs that bind small biogenic amines is well characterized. In this study we identify residues on CC chemokine receptor 7 (CCR-7) that are involved in agonist-mediated receptor activation but not in high affinity ligand binding. The mutations also affect the ability of the ligands to induce chemotaxis. Two of the residues, Lys3.33(137) and Gln5.42(227), are consistent with the binding pocket described for biogenic amines, while Lys3.26(130) and Asn7.32(305), are found at, or close to, the cell surface. Our observations are in agreement with findings from other peptide and chemokine receptors, which indicate that receptors that bind larger ligands contain contact sites closer to the cell surface in addition to the conventional transmembrane binding pocket. These findings also support the theory that chemokine receptors require different sets of interactions for high affinity ligand binding and receptor activation.     

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.     

Chemoreception by a cell with age-dependent receptor binding sites

The authors determine the time-dependent ligand current into a spherical cell that is covered with a large number of age-dependent receptors. These receptors can be in either of two states: active (i.e., available for ligand binding) or inactive. An active receptor turns inactive upon binding a ligand, and it can reappear as active at some later time. The transition inactive----active is treated as a probabilistic process. The ligand distribution around the cell is determined analytically in terms of this distribution at the cell surface. A set of nonlinear integral equations is derived for the distribution at the cell surface, which is solved numerically. In this way the time-dependent ligand current into the cell as well as the average active receptor population at the cell surface are determined.     

Recycling of the asialoglycoprotein receptor: biochemical and immunocytochemical evidence

One of the best documented systems of receptor-mediated endocytosis is the clearance of asialoglycoproteins (ASGP) from the blood plasma by liver parenchymal cells. There are 200 000-500 000 ligand binding sites per cell, which makes this system favourable for molecular studies of receptor function. By using both biochemical and immunocytochemical approaches, we have obtained evidence for receptor recycling. We have also localized the intracellular site at which the endocytosed receptor and ligand dissociate. The human hepatoma cell Hep G2 contains abundant ASGP receptors (approximately 225 000 per cell). In growing cells approximately 85% of the functional receptors are on the cell surface and the remaining 15% are internal. The maximal rate of ligand uptake in this cell system at 37 degrees C is approximately 30 000 molecules per cell per minute. Each functional receptor can therefore bind and internalize more than 50 ligand molecules during a 6 h period (in the absence of new receptor synthesis), or one ligand each 8 min. To follow both ligand and receptor during their common endocytosis and to visualize the compartment in which the dissociation of ligand from receptor occurs, we have used our recently developed double-labelling immunocytochemical electron microscopic techniques with purified antibodies against ASGP ligand and ASGP receptor. In normal rat hepatocytes, both ligand and receptor are taken up from the sinusoidal cell surface in clathrin-coated vesicles. Both receptor and ligand are associated with the membrane of small clathrin-coated vesicles close to the cell surface. Larger vesicles, farther removed from the surface, contain ligand accumulated within the lumen. The membranes of these larger vesicles contain little receptor, but receptor was concentrated in detached vesiculotubular extensions, which were largely free of ligand. These vesicles represent the compartment of uncoupling of receptor and ligand (CURL) during their common endocytosis. Ligand contained within the vesicle lumen is then transferred to multivesicular bodies and lysosomes; the tubular extensions may carry receptor back to the cell surface.     

A dynamical model for receptor-mediated cell adhesion to surfaces in viscous shear flow

We present a dynamical model for receptor-mediated cell adhesion to surfaces in viscous shear flow when the surfaces are coated with ligand molecules complementary to receptors in the cell membrane. This model considers the contact area between the cell and the surface to be a small, homogeneous region that mediates the initial attachment of the cell to the surface. Using a phase plane analysis for a system of nonlinear ordinary differential equations that govern the changes in free receptor density and bond density within the contact area with time, we can predict the conditions for which adhesion between the cell and the surface will take place. Whether adhesion occurs depends on values of dimensionless quantities that characterize the interaction of the cell and its receptors with the surface and its ligand, such as the bond formation rate, the receptor-ligand affinity, the fluid mechanical force, the receptor mobility, and the contact area. A key result is that there are two regimes in which different chemical and physical forces dominate: a rate-controlled high affinity regime and an affinity-controlled low affinity regime. Many experimental observations, including the effects of temperature and receptor mobility on adhesiveness, can be explained by understanding which of these regimes is appropriate. We also provide simple approximate analytical solutions, relating adhesiveness to cell and surface properties as well as fluid forces, which allow convenient testing of model predictions by experiment.     

A two-step selection approach for the identification of ligand-binding determinants in cytokine receptors

We have developed a novel cell-based method for the isolation and selection of mutant cytokine receptors with defects in ligand binding and applied it to the human interleukin-4 receptor. The experimental procedure is based upon the functional heterologous expression of receptor mutants in eukaryotic cells followed by a two-step selection procedure. Positive selection for cells that express receptor variants is achieved by means of an agonistic antibody that mediates cell survival through receptor dimerization. An IL-4-coupled toxin is subsequently used to select against cells expressing wild-type receptors. Cells expressing mutant receptors that are unable to bind the cytotoxic ligand survive and can be amplified. The procedure allows the isolation of rare receptor variants from cell pools containing predominantly wild-type cells. This method, which should be equally applicable to similar receptor systems, was used to demonstrate the importance of a critical charged amino acid residue in the human IL-4 receptor alpha-subunit for IL-4-induced receptor activation.     

A dynamical model for receptor-mediated cell adhesion to surfaces.

We present a dynamical model for receptor-mediated adhesion of cells in a shear field of viscous fluid to surfaces coated with ligand molecules complementary to receptors in the cell membrane. We refer to this model as the "point attachment model" because it considers the contact area between the cell and the surface to be a small, homogeneous region that mediates the initial attachment of the cell to the surface. Using a phase plane analysis of a system of nonlinear ordinary differential equations which govern the changes in free receptor density and bond density within the contact area with time, we can predict the conditions for which adhesion between the cell and the surface will take place. Whether adhesion occurs depends on values of dimensionless quantities that characterize the interaction of the cell and its receptors with the surface and its ligand, such as the bond formation rate, the receptor-ligand affinity, the fluid mechanical force, the receptor mobility, and the contact area. A key result is that there are two regimes in which different chemical and physical forces dominate: a rate-controlled high affinity regime and an affinity-controlled low-affinity regime. Many experimental observations can be explained by understanding which of these regimes is appropriate. We also provide simple approximate analytical solutions, relating adhesiveness to cell and surface properties as well as fluid forces, which allow convenient testing of model predictions by experiment.     

Regulation by phorbol esters of asialoglycoprotein and transferrin receptor distribution and ligand affinity in a hepatoma cell line

We have investigated the simultaneous regulation of cell surface distribution and ligand binding of the asialoglycoprotein (ASGP) receptor and the transferrin receptor in a hepatoma cell line by phorbol esters. One hour exposure to phorbol esters causes a redistribution of both receptors to the cell interior as shown by radioligand binding at 4 degrees C and selective immunoprecipitation from the plasma membrane. This effect is temperature- and dose-dependent and is not seen with 4-alpha-phorbol, an inactive tumor promoter. The mechanism and kinetics of the ASGP receptor response to phorbol esters appears to differ from that of the transferrin receptor in this cell line. Within the first 10 min there is a decrease in binding of iodinated ligands for both receptors to the HepG2 cell surface. For the transferrin receptor this results from a net internalization of receptor molecules from the plasma membrane pool, while for the ASGP receptor this decrease is accounted for by a 3.5-fold reduction in ligand binding affinity (6.6 X 10(-8) M to 24.0 X 10(-8) M), with essentially no change in the number of ASGP receptors recoverable from the plasma membrane pool by immunoprecipitation. The altered affinity of the ASGP-R is transient; the Kd returns to control levels by 20 min of continued exposure to the agent. The transferrin receptor shows no change in binding affinity during the course of exposure to phorbol esters. ASGP receptors in cells exposed to phorbol esters for 1 h maintain their competence to deliver exogenous ligand to intracellular sites of degradation and to participate in the recycling pathway of receptor-mediated endocytosis, although at a lower rate than in control cells. We conclude that under identical conditions phorbol esters modulate the binding capacity of two receptors at the cell surface by separate mechanisms. Furthermore, the transient nature of the altered ASGP-R binding affinity suggests that at least two mechanisms, receptor redistribution as well as decreased binding affinity, are operative in the modulation of ASGP-R cell surface binding during the first hour of exposure to the phorbol esters.     

Interpretation of Scatchard plots for aggregating receptor systems.

Aggregation of cell surface receptors, with each other or with other membrane proteins, occurs in a variety of experimental systems. The list of systems where receptor aggregation appears to be important in understanding ligand binding and cellular responses is growing rapidly. In this paper we explore the interpretation of equilibrium binding data for aggregating receptor systems. The Scatchard plot is a widely used tool for analyzing equilibrium binding data. The shape of the Scatchard plot is often interpreted in terms of multiple noninteracting receptor populations. Such an analysis does not provide a framework for investigating the role of receptor aggregation and will be misleading if there is a relation between receptor aggregation and ligand binding. We present a general model for the equilibrium binding of a ligand with any number of aggregating receptor populations and derive theoretical expressions for observable Scatchard plot features. These can be used to test particular models and estimate model parameters. We develop particular models and apply the general results in the cases of six aggregating receptor systems where ligand binding and receptor aggregation are related: cross-linking of monovalent cell surface proteins by monoclonal antibodies, cross-linking of cell surface antibodies by bivalent ligand, antibody-induced co-cross-linking of cell surface antibodies and Fc gamma receptors, ligand-enhanced aggregation of identical epidermal growth factor receptors, aggregation of heterologous receptors for interleukin 2 to form a high-affinity receptor, and association of receptors, including those for interleukins 5 and 6, with nonbinding accessory proteins that influence receptor affinity or effector function.     

Ligand-receptor interaction rates in the presence of convective mass transport.

The rate of binding of a ligand to receptors on the cell surface can be diffusion limited. We analyze the kinetics of binding, diffusion-limited in a stationary liquid, in the presence of convective mass transport. We derive a formula that expresses the reaction kinetics in terms of the mass transfer coefficient. A moderately transport-limited kinetics is not readily recognizable from the shape of the binding curve and may lead to erroneous estimates of the rate coefficients. We apply our results to practically important cases: a cell suspension in a stirred volume of liquid and a confluent cell colony under a laminar stream. Using typical numbers characterizing the ligand-receptor interactions, we show that stirring and perfusion can be important factors determining the reaction rates. With the confluent colony, the early reaction kinetics requires a different treatment, and we provide it for the case of low receptor occupancy. We show that, even with a fast perfusion, a cell monolayer can transiently generate a zone of depletion of the ligand, and that would affect the early stages of the reaction. Our results are expressed in a simple analytical form and can be used for the design and interpretation of experimental data.     

Analytical expressions for the homotropic binding of ligand to protein dimers and trimers

Cooperative binding of a ligand to multiple subsites on a protein is a common theme among enzymes and receptors. The analysis of cooperative binding data (either positive or negative) often relies on the assumption that free ligand concentration, L, can be approximated by the total ligand concentration, L(T). When this approximation does not hold, such analyses result in inaccurate estimates of dissociation constants. Presented here are exact analytical expressions for equilibrium concentrations of all enzyme and ligand species (in terms of K(d) values and total concentrations of protein and ligand) for homotropic dimeric and trimeric protein-ligand systems. These equations circumvent the need to approximate L and are provided in Excel worksheets suitable for simulation and least-squares fitting. The equations and worksheets are expanded to treat cases where binding signals vary with distinct site occupancy.