The effect of co-operativity on the equilibrium binding of symmetric bivalent ligands to antibodies: theoretical results with application to histamine release from basophils

A spatio-temporal model reveals self-limiting FcɛRI cross-linking by multivalent antigens

Aggregation of cell surface receptor proteins by multivalent antigens is an essential early step for immune cell signalling. A number of experimental and modelling studies in the past have investigated multivalent ligand-mediated aggregation of IgE receptors (FcɛRI) in the plasma membrane of mast cells. However, understanding of the mechanisms of FcɛRI aggregation remains incomplete. Experimental reports indicate that FcɛRI forms relatively small and finite-sized clusters when stimulated by a multivalent ligand. By contrast, modelling studies have shown that receptor cross-linking by a trivalent ligand may lead to the formation of large receptor superaggregates that may potentially give rise to hyperactive cellular responses. In this work, we have developed a Brownian dynamics-based spatio-temporal model to analyse FcɛRI aggregation by a trivalent antigen. Unlike the existing models, which implemented non-spatial simulation approaches, our model explicitly accounts for the coarse-grained site-specific features of the multivalent species (molecules and complexes). The model incorporates membrane diffusion, steric collisions and sub-nanometre-scale site-specific interaction of the time-evolving species of arbitrary structures. Using the model, we investigated temporal evolution of the species and their diffusivities. Consistent with a recent experimental report, our model predicted sharp decay in species mobility in the plasma membrane in response receptor cross-linking by a multivalent antigen. We show that, due to such decay in the species mobility, post-stimulation receptor aggregation may become self-limiting. Our analysis reveals a potential regulatory mechanism suppressing hyperactivation of immune cells in response to multivalent antigens.

A quantitative approach for studying IgE-FcepsilonRI aggregation

Aggregation of cell surface receptors is a ubiquitous means of initiating signal transduction in many cellular systems. In this manuscript, we describe a combined theoretical and experimental approach based on multiparameter flow cytometry for measuring the time course of ligand induced aggregation of IgE-FcepsilonRI on RBL cells. By fluorescently labeling both the ligand and surface IgE (sIgE), we have developed an assay that permits us to simultaneously measure both occupancy of sIgE combining sites and association of antigen with the cell surface. This allows for a direct calculation of the degree of receptor aggregation present on the cell. By employing new mixing technologies developed for flow cytometry, we are able to look at aggregation in the sub second time domain. To extend our work, we have synthesized a new set of chemically well defined ligands (of valences 1-3) to use as probes in our studies. We show that the magnitude of the cellular response is dramatically increased as the valence of our ligand is raised from two to three.

Quantifying aggregation of IgE-FcepsilonRI by multivalent antigen

Aggregation of cell surface receptors by multivalent ligand can trigger a variety of cellular responses. A well-studied receptor that responds to aggregation is the high affinity receptor for IgE (FcepsilonRI), which is responsible for initiating allergic reactions. To quantify antigen-induced aggregation of IgE-FcepsilonRI complexes, we have developed a method based on multiparameter flow cytometry to monitor both occupancy of surface IgE combining sites and association of antigen with the cell surface. The number of bound IgE combining sites in excess of the number of bound antigens, the number of bridges between receptors, provides a quantitative measure of IgE-FcepsilonRI aggregation. We demonstrate our method by using it to study the equilibrium binding of a haptenated fluorescent protein, 2,4-dinitrophenol-coupled B-phycoerythrin (DNP25-PE), to fluorescein isothiocyanate-labeled anti-DNP IgE on the surface of rat basophilic leukemia cells. The results, which we analyze with the aid of a mathematical model, indicate how IgE-FcepsilonRI aggregation depends on the total concentrations of DNP25-PE and surface IgE. As expected, we find that maximal aggregation occurs at an optimal antigen concentration. We also find that aggregation varies qualitatively with the total concentration of surface IgE as predicted by an earlier theoretical analysis.

A systematic analysis of average molecular weights and gelation conditions for branched immune complexes: the interaction between a multivalent antigen with distinct epitopes and many different types of bivalent antibodies

A systematic model, based on standard equilibrium expressions and probability theory, is presented to calculate average molecular weights and gelation conditions for the immune system consisting of a single type of antigens with three (or more) different epitopes and three (or more) types of bivalent antibodies. Molecular weights Mn, Mw, Mz, and any other higher average molecular weights of formed branched antigen-antibody complexes in such an immune system are calculated directly without determining the whole distribution. The conditions for the formation of gel complexes also can be determined by this model.

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.

The kinetics of bivalent ligand-bivalent receptor aggregation: ring formation and the breakdown of the equivalent site approximation

When bivalent ligands capable of bridging binding sites on two different receptors interact with bivalent receptors, aggregates form. The aggregates can be of two types: chains (open structures containing n receptors, n-1 doubly bound ligands and 0, 1, or 2 singly bound ligands) and rings (closed structures containing n receptors and n doubly bound ligands). Both types of aggregates have been detected experimentally. In general, to determine the time dependence of the concentration of any particular aggregate requires solving an infinite set of coupled ordinary differential equations (ODEs). Perelson and DeLisi [19] showed that great simplification results if all receptor binding sites are equivalent, i.e., the binding properties of a site on a receptor are independent of the size of the aggregate the receptor is in. If only chains form, the problem reduces to solving two coupled ODEs for the concentrations of singly and doubly bound ligands. From the solutions to these ODEs, the time dependence of the entire aggregate size distribution can be determined. We show that the equivalent site approximation as formulated by Perelson and DeLisi [19] is incompatible with ring formation. We then present a modified equivalent site approximation that is useful if chains of any size can form but rings above a certain size (k) cannot. We show how to reduce the resulting infinite set of coupled ODEs to a closed system of at most 4k + 2 ODEs for the ligand concentrations, the ring concentrations, and the concentrations of all chains up to size k. Although we can only predict the kinetics of aggregate formation for aggregates of size k or less, at equilibrium the modified equivalent site approximation yields the complete aggregate size distribution.

Size and structure of antigen-antibody complexes: thermodynamic parameters

The role of antigen-antibody (Ag-Ab) complexes in the immune response depends, in part, on the size of the complexes. Previously, we combined electron microscopy with classical and quasi-elastic light scattering to characterize the molecular weight distribution and the conformation of Ag-Ab complexes made from bovine serum albumin (BSA) and pairs of anti-BSA monoclonal antibodies at a single concentration and Ag:Ab molar ratio. In this report, the molecular weight distribution of Ag-Ab complexes was determined by classical light scattering at a single Ag:Ab ratio and over a range of concentrations, and binding of BSA to pairs of MAb was determined by radioimmunoassay at several Ag:Ab molar ratios. A thermodynamic model was developed for the equilibrium size distribution of Ag-Ab complexes formed between a pair of MAb, each with unique affinity and specificity, and an Ag containing a single epitope for each of the pair of MAb. The combined experimental data were used in conjunction with the model to determine the values of cyclization and polymerization constants. Successful determination of the parameters required data from both classical light scattering and electron microscopy. Cyclization constants were lower than those reported in other studies of Ag-Ab complexes; this may be attributable to our use of a protein Ag, as compared to a divalent hapten. In two out of three cases, cyclization constants increased with increasing number of Ab in the complex, in contrast to previous assumptions. The validity of the thermodynamic model was further shown by its ability, in combination with conformational and hydrodynamic model, to predict the hydrodynamic radius of the complexes over a wide range of experimental conditions.

Potency of hexavalent and decavalent anaphylactogens: affinity enhancement by epitope clusters

Two deca-L-lysine conjugates, one carrying ten, the other carrying six D-benzylpenicilloyl haptenic groups as epitope models, were used as elicitors of passive cutaneous anaphylaxis in guinea pigs. It was found that equal haptenic doses yielded virtually indistinguishable responses, i.e., the potencies were close to 10:6. Interpretation of these data appears possible by using established immunochemical concepts of preequilibria and equilibria in free solution as starting point. The discussion is simplified due to the fact that the two conjugates being decavalent and hexavalent, respectively, in free solution, are both hexavalent on cell membranes. It was found that the clusters of haptenic groups displayed by the conjugates enhance the affinity for antibody binding and that the haptenic densities are directly related to potencies. Epitope clusters of enhanced affinity interacting with cell-bound antibody may be significant and useful in a number of ways.