Ligand-induced changes in membrane-bound acetylcholine receptor observed by ethidium fluorescence. 2. Stopped-flow studies with agonists and antagonists.

The kinetics of cholinergic ligand binding to membrane-bound acetylcholine receptor from Torpedo californica have been followed in a stopped-flow photometer, by using the fluorescent probe ethidium. The overall reaction amplitude, as a function of ligand concentration, can be fit to the law of mass action for both agonist and antagonists. All agonists show at least biphasic kinetics, and the concentration dependence of the kinetic parameters is fit by a common mechanism involving sequential binding of ligands with increasingly lower affinity. The receptor-ligand precomplexes isomerize to different noninterconvertible final complexes depending on the number of ligands bound. In contrast, the kinetics observed with antagonists cannot be fit to a common model. These kinetics are always much slower than those observed with agonists, and the relaxation rates depend only weakly on antagonist concentration.     

Calculation of diffusion-limited kinetics for the reactions in collision coupling and receptor cross-linking

Both enzyme (e.g., G-protein) activation via a collision coupling model and the formation of cross-linked receptors by a multivalent ligand involve reactions between two molecules diffusing in the plasma membrane. The diffusion of these molecules is thought to play a critical role in these two early signal transduction events. In reduced dimensions, however, diffusion is not an effective mixing mechanism; consequently, zones in which the concentration of particular molecules (e.g., enzymes, receptors) becomes depleted or enriched may form. To examine the formation of these depletion/ accumulation zones and their effect on reaction rates and ultimately the cellular response, Monte Carlo techniques are used to simulate the reaction and diffusion of molecules in the plasma membrane. The effective reaction rate at steady state is determined in terms of the physical properties of the tissue and ligand for both enzyme activation via collision coupling and the generation of cross-linked receptors. The diffusion-limited reaction rate constant is shown to scale with the mean square displacement of a receptor-ligand complex. The rate constants determined in the simulation are compared with other theoretical predictions as well as experimental data.     

Residence time of receptor-ligand complexes and its effect on biological function

The formation and duration of binary receptor-ligand complexes are fundamental to many physiologic processes. Most often, the effectiveness of interaction between a receptor and its ligand is quantified in terms of closed system, equilibrium affinity measurements, such as IC50 and Kd. In the context of in vivo biology, however, the extent and duration of responses to receptor-ligand interactions depend greatly on the time period over which the ligand is in residence on its receptor. Here we define receptor-ligand complex residence time in quantitative terms and describe its significance to biological function. Examples of the importance of residence time are presented for natural ligands of different receptor types. The impact of residence time on the optimization of potential ligands as drugs for human medicine is also described.     

Analysis of the rate-limiting step in a ligand-cell receptor interaction: the immunoglobulin E system

Theory predicts that the kinetics of simple interactions between a ligand and a receptor bound on the surface of a cell will be affected by the occupancy of receptors on the same cell. In a diffusion-limited reaction the effect will be on the rate of dissociation but not on the rate of association until the cell is virtually saturated with ligand. If the rate of reaction is not diffusion limited, then the opposite holds; i.e., the forward velocities will be proportional to the concentration of vacant receptors, but the reverse reactions will not be. We examined the kinetics of reaction between immunoglobulin E (IgE) and its receptor and clearly demonstrated that the reaction is not diffusion controlled. The substantial (congruent to 30-fold) increase in the forward rate constant observed for the reaction of IgE with solubilized receptors as opposed to cell-bound receptors is therefore not an artifact of calculation. Since the reverse rate constants show little difference, we postulate that the presence of other surface components (rather than conformational differences in the receptor) affects the reaction with the cells. As an aid to the analysis, the theory has been extended so that not only the rate constants but also the entire course of the reaction of ligand with cell receptors can be predicted for diffusion-limited vs. non-diffusion-limited interactions.     

Analysis of intracellular receptor/ligand sorting. Calculation of mean surface and bulk diffusion times within a sphere.

Cell surface receptors bind extracellular ligand molecules and transport those ligands into the cell by a process termed receptor-mediated endocytosis. Receptor and ligand molecules are sorted from one another after endocytosis, apparently within a structure consisting of intracellular vesicles and connected thin tubules. The experimental observation is that most free (unbound) ligand molecules are found in the lumen of the vesicles and receptors are located primarily within the tubules. Because equilibrium and geometric considerations do not explain this segregation, a kinetic scheme involving the passive diffusion of molecules from a vesicle into a tubule is investigated. Two possible sorting mechanisms are considered: first, that receptors are able to move into tubules more rapidly than ligand molecules due to an advantage in dimensionality and, second, that receptors diffusing into tubules are trapped there while ligands are not. Mean diffusion times for receptor and ligand movement into a tubule are calculated by solving Poisson's equation in two and three dimensions, respectively, on the surface of and within a sphere. Using estimated parameter values, we found that only the second scheme is able to account for the experimentally observed sorting. An estimate is obtained for the length of time a tubule and vesicle must be connected in order to remove a significant number of receptors into a tubule. The fraction of free ligand that is "mis-sorted" with the recycling receptor population and thus exocytosed is also determined.     

Monte Carlo study of single molecule diffusion can elucidate the mechanism of B cell synapse formation

B cell receptors have been shown to cluster at the intercellular junction between a B cell and an antigen-presenting cell in the form of a segregated pattern of B cell receptor/antigen complexes known as an immunological synapse. We use random walk-based theoretical arguments and Monte Carlo simulations to study the effect of diffusion of surface-bound molecules on B cell synapse formation. Our results show that B cell synapse formation is optimal for a limited range of receptor-ligand complex diffusion coefficient values, typically one-to-two orders of magnitude lower than the diffusion coefficient of free receptors. Such lower mobility of receptor-ligand complexes can significantly affect the diffusion of a tagged receptor or ligand in an affinity dependent manner, as the binding/unbinding of such receptor or ligand molecules crucially depends on affinity. Our work shows how single molecule tracking experiments can be used to estimate the order of magnitude of the diffusion coefficient of receptor-ligand complexes, which is difficult to measure directly in experiments due to the finite lifetime of receptor-ligand bonds. We also show how such antigen movement data at the single molecule level can provide insight into the B cell synapse formation mechanism. Thus, our results can guide further single molecule tracking experiments to elucidate the synapse formation mechanism in B cells, and potentially in other immune cells.     

Consequences of ligand bivalency in interactions involving particulate receptors: equilibrium and kinetic studies with Sephadex-concanavalin A, butylagarose-phosphorylase b, and Fc receptor-IgG dimer interactions as model systems

Theoretical consideration is given to the interaction of a bivalent ligand with particulate receptor sites, not only from the viewpoint of quantitatively describing the binding behavior but also from that of the kinetics of ligand release upon infinite dilution of a receptor-ligand mixture. In the latter regard, a general expression is derived that describes the time dependence of the amount of ligand bound as a function of two rate constants for the stepwise dissociation of cross-linked ligand-receptor complex and a thermodynamic parameter expressing the initial ratio of singly linked to doubly linked ligand-receptor complexes. An experimental study of the interaction between Sephadex and concanavalin A is then used to illustrate application of this recommended theoretical approach for characterizing the binding behavior and dissociation kinetics of a bivalent ligand for a system in which all ligand-receptor interactions may be described by a single intrinsic association constant. Published results on the interaction of phosphorylase b with butylagarose are also shown to comply with this simplest model of the bivalent ligand hypothesis; but those for the interaction between immunoglobulin G (IgG) dimers and Fc receptors require modification of the model by incorporation of different intrinsic association constants for the successive binding of receptor sites to the bivalent ligand. These results emphasize the need to consider ligand bivalency as a potential phenomenon in studies of interactions between protein ligands and particulate receptors and illustrate procedures by which the effects of ligand bivalency may be identified and characterized.     

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.     

Morphology of cell-substratum adhesion. Influence of receptor heterogeneity and nonspecific forces.

Many cell types modulate growth, differentiation, and motility through changes in cell substrate adhesion, including regulation of focal contact formation. Clustering of cell surface adhesion receptors is an essential early step in the development of focal contacts, and thus may influence cell physiology. In this paper, we present a theoretical framework to examine how cell surface chemistry affects receptor clustering. Our one-dimensional tape-peeling model couples the equations of mechanical equilibrium for a cell membrane with kinetic receptor-ligand binding relations. We considered two distinct model scenarios: Adhesion mediated by multiple receptor-ligand interactions of different length and specific binding of a single receptor type occurs in the presence of van der Waals attraction and nonspecific repulsion. In each case, nonuniform (wave-like) membrane morphologies are observed in certain parameter ranges that support the clustering of adhesion receptors. The formation of these morphologies is described in terms of a balance of membrane stresses; when cell-surface potential as a function of separation distance is symmetric between two potential energy minima, nonuniform morphologies are obtained. Increases in the chemical binding energy between receptor and ligand (e.g., increases in ligand density) or decreases in the membrane rigidity result in smaller wavelengths for nonuniform interfaces. Additionally, we show wave-like geometries appear only when the mechanical compliance of receptor-ligand bonds is within an intermediate range, and examine how the mobility of "repellers"--glycocalyx molecules that exert a nonspecific repulsive force--influences membrane morphology. We find fully mobile repellers always redistribute to prevent nonuniform morphologies.     

Morphology of cell-substratum adhesion

Many cell types modulate growth, differentiation, and motility through changes in cell substrate adhesion, including regulation of focal contact formation. Clustering of cell surface adhesion receptors is an essential early step in the development of focal contacts, and thus may influence cell physiology. In this paper, we present a theoretical framework to examine how cell surface chemistry affects receptor clustering. Our one-dimensional tape-peeling model couples the equations of mechanical equilibrium for a cell membrane with kinetic receptor-ligand binding relations. We considered two distinct model scenarios: Adhesion mediated by multiple receptor-ligand interactions of different length and specific binding of a single receptor type occurs in the presence of van der Waals attraction and nonspecific repulsion. In each case, nonuniform (wave-like) membrane morphologies are observed in certain parameter ranges that support the clustering of adhesion receptors. The formation of these morphologies is described in terms of a balance of membrane stresses; when cell-surface potential as a function of separation distance is symmetric between two potential energy minima, nonuniform morphologies are obtained. Increases in the chemical binding energy between receptor and ligand (e.g., increases in ligand density) or decreases in the membrane rigidity result in smaller wavelengths for nonuniform interfaces. Additionally, we show wave-like geometries appear only when the mechanical compliance of receptor-ligand bonds is within an intermediate range, and examine how the mobility of “repellers”—glycocalyx molecules that exert a nonspecific repulsive force—influences membrane morphology. We find fully mobile repellers always redistribute to prevent nonuniform morphologies.     

Sorting and recycling of cell surface receptors and endocytosed ligands: the asialoglycoprotein and transferrin receptors

With few exceptions, receptor-mediated endocytosis of specific ligands is mediated through clustering of receptor-ligand complexes in coated pits on the cell surface, followed by internalization of the complex into endocytic vesicles. During this process, ligand-receptor dissociation occurs, most probably in a low pH prelysosomal compartment. In most cases the ligand is ultimately directed to the lysosomes, wherein it is degraded, while the receptor recycles to the cell surface. We have studied the kinetics of internalization and recycling of both the asialoglycoprotein receptor and the transferrin receptor in a human hepatoma cell line. By employing both biochemical and morphological/immunocytochemical approaches, we have gained some insight into the complex mechanisms which govern receptor recycling as well as ligand sorting and targeting. We can, in particular, explain why transferrin is exocytosed intact from the cells, while asialoglycoproteins are degraded in lysosomes. We have also localized the intracellular site at which endocytosed receptor and ligand dissociate.     

Two-step isomerization of quinuclidinyl benzilate-muscarinic receptor complex

A kinetic analysis was made of the dissociation reaction of the muscarinic receptor-l-[³H]quinuclidinyl benzilate complex at 25°C in 0.05 M K-phosphate buffer. The course of the reaction was followed by the decrease in the concentration of the membrane-bound radiolabelled antagonist while rebinding was prevented by the excess of nonradioactive quinuclidinyl benzilate. It was found that both bi- and mono-exponential kinetic curves of the process can be observed, depending on the time moment when the dissociation reaction is started. If the receptor-ligand complex had been incubated for a sufficiently long time before the excess of the nonradioactive ligand was added to “displace” the radioactive ligand from the complex, the dissociation reaction followed the first-order kinetics. The bi-exponential kinetics of the dissociation process was obtained if the displacement was started within a short time interval after the complex formation between the receptor and l-[³H]quinuclidinyl benzilate. The data obtained were analysed within the framework of a reaction scheme containing two consecutive isomerization steps of the receptor-antagonist complex. The “isomerized” receptor-ligand complexes differ in their dissociation rate and therefore their interconversion changes the observed kinetic behaviour of the dissociation reaction of the receptor-ligand complex.     

Monotonicity of interleukin-1 receptor-ligand binding with respect to antagonist in the presence of decoy receptor

We consider the interaction between interleukin-1 IL-1, its receptor IL-1RI, the receptor antagonist IL-1Ra and a decoy receptor (or trap) that binds both with the ligand and the antagonist. We study how the interaction between IL-1Ra and the decoy receptor influences the effect of either reagent on reducing the equilibrium concentration of the receptor-ligand complex. We obtain that, given a certain relationship among the equilibrium constants and the total concentrations of solutes, IL-1Ra can reverse the effect of the decoy receptor of decreasing the equilibrium concentration of the receptor-ligand complex. This finding derives from a mathematical result applicable to any reversible chemical reaction system comprising four species arranged in a square such that each species binds its two immediate neighbors. The result gives the monotonicity of the equilibrium concentrations of the complex species as functions of the total concentrations of the simple species.     

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.     

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.     

Monte Carlo Simulation of Buffered Diffusion into and out of a Model Synapse

Buffered diffusion occurs when ligands enter or leave a restricted space, such as a chemical synapse, containing a high density of binding sites. This study used Monte Carlo simulations to determine the time and spatial dependences of buffered diffusion without a priori assumptions about kinetics. The synapse was modeled as a box with receptors on one inner face. The exterior was clamped to some ligand concentration and ligands diffused through two sides. Onset and recovery simulations were carried out and the effects of receptor density, ligand properties and synapse geometry were investigated. This study determined equilibration times for binding and the spatial gradient of unliganded receptors. Onset was characterized by a high spatial gradient; equilibration was limited by the time needed for sufficient ligands to enter the synapse. Recovery showed a low spatial gradient with receptor equilibration limited by ligand rebinding. Decreasing ligand association rate or increasing ligand diffusion coefficient reduced the role of buffered diffusion and decreased the spatial gradient. Simulations with irreversible ligands showed larger, persistent spatial gradients. These simulations identify characteristics that can be used to test whether a synaptic process is governed by buffered diffusion. They also indicate that fundamental differences in synapse function may occur with irreversible ligands.     

Making sense of the diverse ligand recognition by NKG2D

NKG2D recognizes multiple diverse ligands. Despite recent efforts in determining the crystal structures of NKG2D-ligand complexes, the principle governing this receptor-ligand recognition and hence the criteria for identifying unknown ligands of NKG2D remain central issues to be resolved. Here we compared the molecular recognition between NKG2D and three of the known ligands, UL16 binding protein (ULBP), MHC class I-like molecule, and retinoic acid early inducible gene as observed in the ligand-complexed crystal structures. The comparison shows that while the receptor uses a common interface region to bind the three diverse ligands, each ligand forms a distinct, but overlapping, set of hydrogen bonds, hydrophobic interactions, and salt bridges, illustrating the underlying principle of NKG2D-ligand recognition being the conservation in overall shape complementarity and binding energy while permitting variation in ligand sequence through induced fit recognition. To further test this hypothesis and to distinguish between diverse recognition and promiscuous ligand binding, four ULBP3 interface mutations, H21A, E76A, R82M, and D169A, were generated to each disrupt a single hydrogen bond or salt bridge. All mutant ULBP3 displayed reduced receptor binding, suggesting a specific, rather than promiscuous, receptor-ligand recognition. Mutants with severe loss of binding affect the receptor interactions that are mostly buried. Finally, a receptor-ligand recognition algorithm was developed to assist the identification of diverse NKG2D ligands based on evaluating the potential hydrogen bonds, hydrophobic interactions, and salt bridges at the receptor-ligand interface.     

Synthetic multivalent ligands in the exploration of cell-surface interactions

Processes such as cell-cell recognition and the initiation of signal transduction often depend on the formation of multiple receptor-ligand complexes at the cell surface. Synthetic multivalent ligands are unique probes of these complex cell-surface-binding events. Multivalent ligands can be used as inhibitors of receptor-ligand interactions or as activators of signal transduction pathways. Emerging from these complementary applications is insight into how cells exploit multivalent interactions to bind with increased avidity and specificity and how cell-surface receptor organization influences signaling and the cellular responses that result.