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Syk/Zap70 family kinases are essential for signaling via multichain immune-recognition receptors such as the tetrameric (αβγ2) FcεRI The simplest model assumes that Syk activation occurs throughcisbinding of its tandem SH2 domains to dual phosphotyrosines within immunoreceptor tyrosine-based activation motifs of individual γ chains. In this model, Syk activity is modulated by phosphorylation occurring between adjacent Syk molecules docked on γ homodimers and by Lyn molecules bound to FcεRIβ. However, the mechanistic details of Syk docking on γ homodimers are not fully resolved, particularly the possibility oftransbinding orientations and the impact of Y130 autophosphorylation within Syk interdomain A. Analytical modeling shows that multivalent interactions lead to increased WT Sykcis-oriented binding by three orders of magnitude. Molecular dynamics (MD) simulations show increased inter-SH2 flexibility in a Y130E phosphomimetic form of Syk, associated with reduced overall helicity of interdomain A. Hybrid MD/worm-like chain polymer models show that the Y130E substitution reducescisbinding of Syk. We report computational models and estimates of relative binding for all possiblecisandtrans2:2 Syk:FcεRIγ complexes. Calcium imaging experiments confirm model predictions thatcisbinding of WT Syk is strongly preferred for efficient signaling, whiletransconformations trigger weak but measurable responses.

Sugar recognition at the membrane is critical in various physiological processes. Many aspects of sugar-membrane interaction are still unknown. We take an integrated approach by combining conventional molecular-dynamics simulations with enhanced sampling methods and analytical models to understand the thermodynamics and kinetics of a di-mannose molecule in a phospholipid bilayer system. We observe that di-mannose has a slight preference to localize at the water-phospholipid interface. Using umbrella sampling, we show the free energy bias for this preferred location to be just −0.42 kcal/mol, which explains the coexistence of attraction and exclusion mechanisms of sugar-membrane interaction. Accurate estimation of absolute entropy change of water molecules with a two-phase model indicates that the small energy bias is the result of a favorable entropy change of water molecules. Then, we incorporate results from molecular-dynamics simulation in two different ways to an analytical diffusion-reaction model to obtain association and dissociation constants for di-mannose interaction with membrane. Finally, we verify our approach by predicting concentration dependence of di-mannose recognition at the membrane that is consistent with experiment. In conclusion, we provide a combined approach for the thermodynamics and kinetics of a weak ligand-binding system, which has broad implications across many different fields.

The high affinity interleukin-2 receptor is composed of three cell surface subunits, IL-2Ralpha, IL-2Rbeta, and IL-2Rgamma. Functional forms of the IL-2 receptor exist, however, that enlist only two of the three subunits. On activated T-cells, the alpha- and beta-subunits combine as a preformed heterodimer (the pseudo-high affinity receptor) that serves to capture IL-2. On a subpopulation of natural killer cells, the beta- and gamma-subunits interact in a ligand-dependent manner to form the intermediate affinity receptor site. Previously, we have demonstrated the feasibility of employing coiled-coil molecular recognition for the solution assembly of a heteromeric IL-2 receptor complex. In that study, although the receptor was functional, the coiled-coil complex was a trimer rather than the desired heterodimer. We have now redesigned the hydrophobic heptad sequences of the coiled-coils to generate soluble forms of both the pseudo-high affinity and the intermediate affinity heterodimeric IL-2 receptors. The properties of these complexes were examined and their relevance to the physiological IL-2 receptor mechanism is discussed.

Alefacept is a chimeric protein combining CD58 immunoglobulin-like domain 1 with human IgG1 Fc. Alefacept mediates adhesion by bridging CD2 on T cells to activating Fc receptors on effector cells, but the equilibrium binding parameters have not been determined. Alefacept mediated T cell killing by NK cells and adhesion between CD2- and CD16-expressing cells at an optimum concentration of 100 nM. We introduce novel measurements with supported planer bilayers, from which key two-dimensional and three-dimensional parameters can be determined by data fitting. Alefacept competitively inhibited cell bilayer adhesion mediated by the CD2-CD58 interaction. Alefacept mediated maximal adhesion of CD2(+) T cells to CD16B, an Fc receptor, in planar bilayers at 500 nM. A mechanistic model for alefacept-mediated cell-bilayer adhesion allowed fitting of the data and determination of two-dimensional binding parameters. These included the density of bonds in the adhesion area, which grew to maintain a consistent average bond density of 200 molecules/microm(2) and two-dimensional association constants of 3.1 and 630 microm(2) for bivalently and monovalently bound forms of alefacept, respectively. The maximum number of CD16 bound and the fit value of 4,350 CD2 per cell are much lower than the 40,000 CD2 per cell measured with anti-CD2 Fab. These results suggest that additional information is needed to correctly predict Alefacept-mediated bridge formation.

Cells may discriminate among ligands with different dwell times for receptor binding through a mechanism called kinetic proofreading in which the formation of an activated receptor complex requires a progression of events that is aborted if the ligand dissociates before completion. This mechanism explains how, at equivalent levels of receptor occupancy, a rapidly dissociating ligand can be less effective than a more slowly dissociating analog at generating distal cellular responses. Simple mathematical models predict that kinetic proofreading is limited to the initial complex; once the signal passes to second messengers, the dwell time no longer regulates the signal. This suggests that an assay for kinetic proofreading might be used to determine which activation events occur within the initial signaling complex. In signaling through the high affinity IgE receptor FcεRI, the transmembrane adaptor called linker for activation of T cells (LAT) is thought to nucleate a distinct secondary complex. Experiments in which the concentrations of two ligands with different dwell times are adjusted to equalize the level of LAT phosphorylation in rat basophilic leukemia 2H3 cells show that Erk2 phosphorylation, intracellular Ca2+, and degranulation exhibit kinetic proofreading downstream of LAT phosphorylation. These results suggest that ligand-bound FcεRI and LAT form a complex that is required for effective signal transmission.

The regulation of T-cell-mediated immune responses depends on the phosphorylation of immunoreceptor tyrosine-based activation motifs (ITAMs) on T-cell receptors. Although many details of the signaling cascades are well understood, the initial mechanism and regulation of ITAM phosphorylation remains unknown. We used molecular dynamics simulations to study the influence of different compositions of lipid bilayers on the membrane association of the CD3ϵ cytoplasmic tails of the T-cell receptors. Our results show that binding of CD3ϵ to membranes is modulated by both the presence of negatively charged lipids and the lipid order of the membrane. Free-energy calculations reveal that the protein-membrane interaction is favored by the presence of nearby basic residues and the ITAM tyrosines. Phosphorylation minimizes membrane association, rendering the ITAM motif more accessible to binding partners. In systems mimicking biological membranes, the CD3ϵ chain localization is modulated by different facilitator lipids (e.g., gangliosides or phosphoinositols), revealing a plausible regulatory effect on activation through the regulation of lipid composition in cell membranes.

There is a fundamental difference in complexity between signaling initiated by ligands on the surface of one cell binding to receptors on the surface of another cell and ligands in solution binding to these receptors. The fact that two cells must approach each other and form a number of intercellular bonds of different types, all within the restricted geometry of the intercellular contact region, introduces the possibility of complex spatio-temporal dynamics of surface receptors that is not present otherwise. Mathematical modelling of these dynamics is in its early stages. However, useful tools have emerged. The purpose of this paper is to describe some of these mathematical tools, indicating their strengths and shortcomings, and suggest some directions for future theoretical studies.

In many situations, cell-cell adhesion is mediated by multiple ligand-receptor pairs. For example, the interaction between T cells and antigen-presenting cells of the immune system is mediated not only by T cell receptors and their ligands (peptide-major histocompatibility complex) but also by binding of intracellular adhesion molecules. Interestingly, these binding pairs have different resting lengths. Fluorescent labeling reveals segregation of the longer adhesion molecules from the shorter T cell receptors in this case. Here, we explore the thermal equilibrium of a general cell-cell interaction mediated by two ligand-receptor pairs to examine competition between the elasticity of the cell wall, nonspecific intercellular repulsion, and bond formation, leading to segregation of bonds of different lengths at equilibrium. We make detailed predictions concerning the relationship between physical properties of the membrane and ligand-receptor pairs and equilibrium pattern formation, and suggest experiments to refine our understanding of the system. We demonstrate our model by application to the T cell/antigen-presenting-cell system and outline applications to natural killer cell adhesion.

To assess the roles of serial engagement and kinetic proofreading in T cell receptor (TCR) internalization, we have developed a mathematical model of this process. Our determination of TCR down-regulation for an array of TCR mutants, interpreted in the context of the model, has provided new information about peptide-induced TCR internalization. The amount of TCR down-regulation increases to a maximum value and then declines as a function of the half-life of the bond between the TCR and peptide-major histocompatibility complex (pMHC). The model shows that this behavior, which reflects competition between serial engagement and kinetic proofreading, arises only if it is postulated that activated TCRs remain marked for internalization after dissociation from pMHC. The model also predicts that because of kinetic proofreading, the range of TCR-pMHC-binding half-lives required for T cell activation depends on the concentrations and localization of intracellular signaling molecules. We show here that kinetic proofreading provides an explanation for the different requirements for activation observed in naïve and memory T cells.

Cholera toxin entry into mammalian cells is mediated by binding of the pentameric B subunit (CTB) to ganglioside GM(1) in the cell membrane. We used flow cytometry to quantitatively measure in real time the interactions of fluorescently labeled pentameric cholera toxin B-subunit (FITC-CTB) with its ganglioside receptor on microsphere-supported phospholipid membranes. A model that describes the multiple steps of this mode of recognition was developed to guide our flow cytometric experiments and extract relevant equilibrium and kinetic rate constants. In contrast to previous studies, our approach takes into account receptor cross-linking, an important feature for multivalent interactions. From equilibrium measurements, we determined an equilibrium binding constant for a single subunit of FITC-CTB binding monovalently to GM(1) presented in bilayers of approximately 8 x 10(7) M(-1) while that for binding to soluble GM(1)-pentasaccharide was found to be approximately 4 x 10(6) M(-1). From kinetic measurements, we determined the rate constant for dissociation of a single site of FITC-CTB from microsphere-supported bilayers to be (3.21 +/- 0.03) x 10(-3) s(-1), and the rate of association of a site on FITC-CTB in solution to a GM(1) in the bilayer to be (2.8 +/- 0.4) x 10(4) M(-1) s(-1). These values yield a lower estimate for the equilibrium binding constant of approximately 1 x 10(7) M(-1). We determined the equilibrium surface cross-linking constant [(1.1 +/- 0.1) x 10(-12) cm(2)] and from this value and the value for the rate constant for dissociation derived a value of approximately 3.5 x 10(-15) cm(2) s(-1) for the forward rate constant for cross-linking. We also compared the interaction of the receptor binding B-subunit with that of the whole toxin (A- and B-subunits). Our results show that the whole toxin binds with approximately 100-fold higher avidity than the pentameric B-subunit alone which is most likely due to the additional interaction of the A(2)-subunit with the membrane surface. Interaction of cholera toxin B-subunit and whole cholera toxin with gangliosides other than GM(1) revealed specific binding only to GD1(b) and asialo-GM(1). These interactions, however, are marked by low avidity and require high receptor concentrations to be observed.

In the context of cell signaling, kinetic proofreading was introduced to explain how cells can discriminate among ligands based on a kinetic parameter, the ligand-receptor dissociation rate constant. In the kinetic proofreading model of cell signaling, responses occur only when a bound receptor undergoes a complete series of modifications. If the ligand dissociates prematurely, the receptor returns to its basal state and signaling is frustrated. We extend the model to deal with systems where aggregation of receptors is essential to signal transduction, and present a version of the model for systems where signaling depends on an extrinsic kinase. We also investigate the kinetics of signaling molecules, “messengers,” that are generated by aggregated receptors but do not remain associated with the receptor complex. We show that the extended model predicts modes of signaling that exhibit kinetic discrimination for some range of parameters but for other parameter values show little or no discrimination and thus escape kinetic proofreading. We compare model predictions with experimental data.

The serial engagement model provides an attractive and plausible explanation for how a typical antigen presenting cell, exhibiting a low density of peptides recognized by a T cell, can initiate T cell responses. If a single peptide displayed by a major histocompatibility complex (MHC) can bind, sequentially, to different T cell receptors (TCR), then a few peptides can activate many receptors. To date, arguments supporting and questioning the prevalence of serial engagement have centered on the down-regulation of TCR after contact of T cells with antigen presenting cells. Recently, the existence of serial engagement has been challenged by the demonstration that engagement of TCR can down-regulate nonengaged bystander TCR. Here we show that for binding and dissociation rates that characterize interactions between T cell receptors and peptide-MHC, substantial serial engagement occurs. The result is independent of mechanisms and measurements of receptor down-regulation. The conclusion that single peptide-MHC engage many TCR, before diffusing out of the contact region between the antigen-presenting cell and the T cell, is based on a general first passage time calculation for a particle alternating between states in which different diffusion coefficients govern its transport.

Accurate estimation of biomolecular reaction rates from binding data, when ligands in solution bind to receptors on the surfaces of cells or biosensors, requires an understanding of the contributions of both molecular transport and reaction. Efficient estimation of parameters requires relatively simple models. In this review, we give conditions under which various transport effects are negligible and identify simple binding models that incorporate the effects of transport, when transport cannot be neglected. We consider effects of diffusion of ligands to cell or biosensor surfaces, flow in a BIAcore biosensor, and distribution of receptors in a dextran layer above the sensor surface. We also give conditions under which soluble receptors can be expected to compete effectively with surface-bound receptors.

Previous studies in our laboratory established that the symmetrical bivalent ligand, N,N'-bis-[[epsilon-(2,4-dinitrophenyl)amino]caproyl]-L-tyrosyl]-L-cystin e ((DCT)2-cys), stably cross-links anti-2,4-dinitrophenyl-immunoglobulin E (IgE) bound to high affinity receptors Fc epsilonRI on the surface of RBL-2H3 cells, forming mostly cyclic dimers containing two IgE-Fc epsilonRI and two (DCT)2-cys (Posner et al. (1995) J. Immunol. 155, 3601-3609). These cyclic dimers do not trigger Ca2+ or degranulation responses under a variety of conditions. However, we find that the linearly cross-linked IgE-Fc epsilonRI formed at higher concentrations of (DCT)2-cys do trigger degranulation in the presence of cytochalasin D, an inhibitor of actin polymerization. We further investigated stimulation by (DCT)2-cys of the earliest known events in the functional response, i.e., tyrosine phosphorylation of the beta and gamma subunits of Fc epsilonRI. At the higher (DCT)2-cys concentrations corresponding to linear dimers and maximal degranulation, tyrosine phosphorylation of both beta and gamma are observed. At lower (DCT)2-cys concentrations where cross-linking is maximal and cyclic dimers are overwhelmingly dominant, only gamma tyrosine phosphorylation is observed. Cytochalasin D does not affect these phosphorylation patterns, but instead appears to enhance coupling to downstream signaling events. Phosphorylation of Syk occurs at the higher (DCT)2-cys concentrations in parallel with beta phosphorylation but does not occur in its absence at the lower (DCT)2-cys concentrations. These results suggest that cyclic dimers of IgE-Fc epsilonRI are sterically restricted such that they stimulate tyrosine phosphorylation of gamma but not beta, and this is not sufficient for Syk binding and/or activation.

Previously, we demonstrated that aggregates of the high affinity receptor for IgE (FcϵRI), formed by the binding of chemically cross-linked oligomers of IgE, continue to signal early and late cellular responses long after the formation of new aggregates is blocked. In the present work, we explore quantitatively the relationship between aggregation of the receptors and one of the earliest biochemical changes this initiates. We compare the time course of aggregate formation, inferred from studies of the binding of dimers of IgE, and the time course of phosphorylation of tyrosines on receptor subunits when the receptors are aggregated. A simple model does not fit the data. It appears that aggregates formed late in the response are less effective signaling units than those formed initially. We propose new explanations for the persistence of the response and the unusual kinetics.

Numerous signaling proteins use multivalent binding to increase the specificity and affinity of their interactions within the cell. Enhancement arises because the effective binding constant for multivalent binding is larger than the binding constants for each individual interaction. We seek to gain both qualitative and quantitative understanding of the multivalent interactions of an adaptor protein, growth factor receptor bound protein-2 (Grb2), containing two SH3 domains interacting with the nucleotide exchange factor son-of-sevenless 1 (Sos1) containing multiple polyproline motifs separated by flexible unstructured regions. Grb2 mediates the recruitment of Sos1 from the cytosol to the plasma membrane where it activates Ras by inducing the exchange of GDP for GTP. First, using a combination of evolutionary information and binding energy calculations, we predict an additional polyproline motif in Sos1 that binds to the SH3 domains of Grb2. This gives rise to a total of five polyproline motifs in Sos1 that are capable of binding to the two SH3 domains of Grb2. Then, using a hybrid method combining molecular dynamics simulations and polymer models, we estimate the enhancement in local concentration of a polyproline motif on Sos1 near an unbound SH3 domain of Grb2 when its other SH3 domain is bound to a different polyproline motif on Sos1. We show that the local concentration of the Sos1 motifs that a Grb2 SH3 domain experiences is approximately 1000 times greater than the cellular concentration of Sos1. Finally, we calculate the intramolecular equilibrium constants for the crosslinking of Grb2 on Sos1 and use thermodynamic modeling to calculate the stoichiometry. With these equilibrium constants, we are able to predict the distribution of complexes that form at physiological concentrations. We believe this is the first systematic analysis that combines sequence, structure, and thermodynamic analyses to determine the stoichiometry of the complexes that are dominant in the cellular environment., Author Summary Many biochemical interactions are mediated by multivalent binding where signaling proteins use relatively weak promiscuous interactions to increase the strength and specificity of complex formation. For a bivalent adaptor protein binding to a multivalent ligand, the tethering of one of the adaptors binding sites to a motif on a multivalent ligand constrains the adaptors second binding site to a region with a high local concentration of ligand binding motifs. Intramolecular equilibrium constants associated with multivalency are difficult to measure. Typically, polymer models are utilized to estimate the enhancement in local concentration and, when the biomolecular equilibrium constants for the individual sites are known, to obtain intramolecular equilibrium constants. However, flexibility of structured regions in proteins that contain the binding motifs restricts the application of simple polymer models for many systems. Here, we develop a hybrid method combining molecular dynamics simulations and polymer models to estimate the intramolecular equilibrium constants. We apply this method to study the multivalent interactions between the widely expressed adaptor protein growth factor receptor bound protein-2 (Grb2) and the nucleotide exchange factor son of sevenless 1 (Sos1).

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 Fcγ 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.

Kinetic proofreading is an intrinsic property of the cell signaling process. It arises as a consequence of the multiple interactions that occur after a ligand triggers a receptor to initiate a signaling cascade and it ensures that false signals do not propagate to completion. In order for an aaive signaling complex to form after a ligand binds to a cell surface receptor, a sequence of binding and phosphorylation events must occur that are rapidly reversed if the ligand dissociates from the receptor. This gives rise to a mechanism by which cells can discriminate among ligands that bind to the same receptor but form ligand-receptor complexes with different lifetimes. We review experiments designed to test for kinetic proofreading and models that exhibit kinetic proofreading.

Rate constants that characterize the kinetics of binding and dissociation between biomolecules carry fundamental information about the biological processes these molecules are involved in. An instrument that is widely used to determine these rate constants is the Biacore. In a Biacore experiment, one of the reactants, which we will call the receptor, is immobilized on a sensor chip. During the binding phase of the experiment the other reactant flows past the chip. After binding, buffer alone is introduced into the flow cell and dissociation is monitored. Often surface-based binding assays are influenced by the transport of the reactant in solution, complicating the determination of the chemical rate constants from the observed binding kinetics. We propose a new way to determine the dissociation rate constant by adding soluble receptor during dissociation. The method is tested first on simulated data and then on Biacore experiments where the lac repressor protein binds and dissociates from a stretch of double stranded DNA containing the lac repressor binding site. With this method we find a dissociation rate constant k(d)=0.075 +/- 0.005s(-1), a value that is faster than previously obtained from Biacore experiments. In developing our method to analyze these experiments we obtain an expression for the transport limited rate constant for a Biacore experiment when soluble receptor is present during dissociation.

Many activities of cells are controlled by cell-surface receptors, which in response to ligands, trigger intracellular signaling reactions that elicit cellular responses. A hallmark of these signaling reactions is the reversible nucleation of multicomponent complexes, which typically begin to assemble when ligand-receptor binding allows an enzyme, often a kinase, to create docking sites for signaling molecules through chemical modifications, such as tyrosine phosphorylation. One function of such docking sites is the co-localization of enzymes with their substrates, which can enhance both enzyme activity and specificity. The directed assembly of complexes can also influence the sensitivity of cellular responses to ligand-receptor binding kinetics and determine whether a cellular response is up- or downregulated in response to a ligand stimulus. The full functional implications of ligand-stimulated complex formation are difficult to discern intuitively. Complex formation is governed by conditional interactions among multivalent signaling molecules and influenced by quantitative properties of both the components in a system and the system itself. Even a simple list of the complexes that can potentially form in response to a ligand stimulus is problematic because of the number of ways signaling molecules can be modified and combined. Here, we review the role of multicomponent complexes in signal transduction and advocate the use of mathematical models that incorporate detail at the level of molecular domains to study this important aspect of cellular signaling.

Signaling by the T cell receptor (TCR), and the related immunoreceptor Fc epsilon RI, is sensitive to ligand-receptor binding kinetics. Differences in the rate at which a ligand dissociates from a receptor cause disproportionate differences in signaling events and cellular responses to ligand-receptor engagement. Analysis of a simple mathematical model, developed by McKeithan (1995, Proc. Natl. Acad. Sci. USA, 92, 5042-5046), has indicated that such sensitivity to binding kinetics is expected if a bound receptor must complete a cascade of modifications before generating a productive signal. However, recent experiments show that some cellular responses mediated by immunoreceptors escape from the control of kinetic proofreading, in the sense that these responses do not exhibit the expected sensitivity to the lifetime of a ligand-receptor bond. Here, we use an extended form of the McKeithan model to investigate possible explanations for such exceptions to the kinetic proofreading rule. We examine cellular responses triggered by cytosolic messengers, which are activated by modified receptors, and responses triggered by receptors in intermediate states of modification, i.e., receptors that have not progressed through the full series of potential modifications. Receptor aggregation is also considered. We find that the expected relationship between ligand-receptor binding kinetics and cellular responses can change significantly when signal transduction depends on a messenger or a partially modified receptor. In particular, cellular responses triggered by a messenger, such as a transcription factor that translocates from the membrane to the nucleus after receptor-mediated activation, can be sensitive or insensitive to a change in the lifetime of a ligand-receptor bond, depending on the parameters that govern the activation and decay of a messenger.

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.

A few broadly neutralizing antibodies, isolated from HIV-1 infected individuals, recognize epitopes in the membrane proximal external region (MPER) of gp41 that are transiently exposed during viral entry. The best characterized, 4E10 and 2F5, are polyreactive, binding to the viral membrane and their epitopes in the MPER. We present a model to calculate, for any antibody concentration, the probability that during the pre-hairpin intermediate, the transient period when the epitopes are first exposed, a bound antibody will disable a trivalent gp41 before fusion is complete. When 4E10 or 2F5 bind to the MPER, a conformational change is induced that results in a stably bound complex. The model predicts that for these antibodies to be effective at neutralization, the time to disable an epitope must be shorter than the time the antibody remains bound in this conformation, about five minutes or less for 4E10 and 2F5. We investigate the role of avidity in neutralization and show that 2F5 IgG, but not 4E10, is much more effective at neutralization than its Fab fragment. We attribute this to 2F5 interacting more stably than 4E10 with the viral membrane. We use the model to elucidate the parameters that determine the ability of these antibodies to disable epitopes and propose an extension of the model to analyze neutralization data. The extended model predicts the dependencies of for neutralization on the rate constants that characterize antibody binding, the rate of fusion of gp41, and the number of gp41 bridging the virus and target cell at the start of the pre-hairpin intermediate. Analysis of neutralization experiments indicate that only a small number of gp41 bridges must be disabled to prevent fusion. However, the model cannot determine the exact number from neutralization experiments alone., Author Summary Most people who become infected with HIV generate a strong antibody response to the infecting virus population. Unfortunately, the protection offered by the antibody is short lived as the virus rapidly mutates and renders the antibodies impotent in preventing further infection. There are a few antibodies, however, that have been isolated from infected individuals that can block infection by many different viral strains. Among these are several that target sites on the HIV that are exposed only after the virus has attached to a cell. These antibodies have a brief window of time to prevent fusion of the virus and cell. They are special in that they bind both to the viral membrane and to sequences on the gp41 protein that lie along the viral surface. Here, we present a model that predicts the concentrations at which these antibodies effectively neutralize the virus. The model tells us what properties of antibody binding are key in determining efficient neutralization and what properties have little influence. A prediction of the model is that in a standard neutralization assay there are only a small number of attachments between virus and cell and disabling these is sufficient to prevent infection.

Important immune responses are linked to Src-mediated phosphorylation of cytoplasmic tyrosine-based motifs (ITAMs). However, the mechanism of how receptor ligation translates into ITAM phosphorylation remains elusive. Here, we use molecular dynamic simulations to explore the potential regulatory involvement of lipid membranes and their influence on the structure and behavior of the cytoplasmic portion of the CD3ɛ chain of the T-cell receptor. It has been hypothesized that the accessibility of ITAM motifs, such as those in the CD3ɛ cytoplasmic tail, can be blocked by ionic interactions between positively charged amino acids in receptor tails and negatively charged lipid head groups. This interesting hypothesis represents a previously unrecognized mechanism for control of receptor activation. Our simulations support the notion that the net charge of the lipids present in the membrane can affect peptide-membrane interactions. Results are consistent with experimental findings that show increase interfacial absorption in negatively charged lipid bilayer. Our simulations revealed the conformational variability of the disordered tail, which led to an additional focus on quantifying the interaction by free energy calculations, combined with long time-scale simulations using coarse-grained (CG) approaches. These studies will be extended to address how changes in ionic conditions can modulate phosphorylation of ITAM motifs and lead to regulation of activation of the TCR and other ITAM-bearing immunoreceptors.

It is often difficult to induce dissociation of multivalent antigens that are bound to cells by surface-associated immunoglobulin (sIg), even when high concentrations of monovalent hapten are used to fill all free binding sites. Here, Byron Goldstein and Carla Wofsy discuss the evidence that this appears to be due to a cellular response triggered by sIg crosslinking and propose that a receptor-desensitization process is responsible for the transition of multivalent antigens to a dissociation-resistant state.

Publisher Summary Interleukin-2 (IL-2) is a multifunctional cytokine that represents ligand recognition among the hematopoietin receptor family. To facilitate structure–function analysis of IL-2 and its receptor, the chapter describes a strategy of directed and stable solution assembly of receptor ectodomains. This strategy helps to prepare stable complexes of cytokine receptor ectodomains of defined composition and stoichiometry and that mimic the ligand binding characteristics of the equivalent cell surface receptor sites. The chapter demonstrates that it is possible to direct the assembly of cytokine receptor subunits in solution by employing coiled-coil molecular recognition. Heteromeric complexes can be prepared where the subunits function cooperatively to bind ligand. The use of idealized coiled-coil designs rather that naturally occurring sequences result in sub-nanomolar complex stabilities. The techniques of analytical ultracentrifugation and surface plasmon resonance help in quantitatively characterizing the complexes. Using these approaches and the present knowledge of coiled-coil molecular recognition, it should be possible to prepare soluble receptor complexes of defined stoichiometries and subunit composition.

We consider the problem of determining the time dependence of the bound ligand concentration for the reversible binding of a diffusing monovalent ligand to receptors uniformly distributed over the surface of a spherical cell. We start by formulating a boundary value problem that captures the essential physics of this situation. We then introduce a systematic approximation scheme based on the method of weighted residuals. By this means we convert the initial boundary value problem into a simpler problem that requires solving only a small number of ordinary differential equations. We show how, at the lowest order of approximation, the method can be used to obtain modified chemical rate equations where, in place of fundamental rate constants, effective rate coefficients appear. These rate coefficients are functions of the ligand diffusion coefficient, the cell radius, the receptor density and other variables. We compare exact and approximate solutions and discuss under what conditions the approximate equations can be used. We also apply the method of weighted residuals to obtain approximate descriptions of the binding kinetics when (1) there are two different cell surface receptor populations that bind the ligand and (2) the cell secretes a ligand that can bind back to receptors on the cell (autocrine binding).

The linker for activation of T cells (LAT), the linker for activation of B cells (LAB), and the linker for activation of X cells (LAX) form a family of transmembrane adaptor proteins widely expressed in lymphocytes. These scaffolding proteins have multiple binding motifs that, when phosphorylated, bind the SH2 domain of the cytosolic adaptor Grb2. Thus, the valence of LAT, LAB and LAX for Grb2 is variable, depending on the strength of receptor activation that initiates phosphorylation. During signaling, the LAT population will exhibit a time-varying distribution of Grb2 valences from zero to three. In the cytosol, Grb2 forms 1:1 and 2:1 complexes with the guanine nucleotide exchange factor SOS1. The 2:1 complex can bridge two LAT molecules when each Grb2, through their SH2 domains, binds to a phosphorylated site on a separate LAT. In T cells and mast cells, after receptor engagement, receptor phosphoyrlation is rapidly followed by LAT phosphorylation and aggregation. In mast cells, aggregates containing more than one hundred LAT molecules have been detected. Previously we considered a homogeneous population of trivalent LAT molecules and showed that for a range of Grb2, SOS1 and LAT concentrations, an equilibrium theory for LAT aggregation predicts the formation of a gel-like phase comprising a very large aggregate (superaggregate). We now extend this theory to investigate the effects of a distribution of Grb2 valence in the LAT population on the formation of LAT aggregates and superaggregate and use stochastic simulations to calculate the fraction of the total LAT population in the superaggregate.

The high affinity receptor for interleukin 2 (IL-2) is composed of at least two chains, a p55 chain that binds IL-2 with low affinity and a p75 chain that binds with intermediate affinity. Two molecular mechanisms have been proposed for the formation of the high affinity receptor-ligand complex. The affinity conversion model proposed by Honjo and coworkers (Saito et al., 1988) suggests that the high affinity receptor is formed via stepwise binding reactions in which IL-2 first binds to the p55 chain and the resulting complex then associates with the p75 chain to form a high affinity ternary complex. In the preformed heterodimer model the p55 and p75 chains form a non-covalently linked high affinity heterodimer in the absence of IL-2. We analyze these models theoretically and show that they can be distinguished on the basis of equilibrium binding experiments using cell lines expressing different numbers of p55 chains. We show that published IL-2 binding studies are consistent with a model in which a large concentration of preformed heterodimers is present on the cell surface and inconsistent with a model in which preformed heterodimers are absent from the cell surface.

The high affinity receptor for IL-2 is composed of at least two chains, a p55 chain that binds IL-2 with low affinity and a p75 chain that binds with intermediate affinity. Two molecular mechanisms have been proposed for the formation of the high affinity receptor-ligand complex: The affinity conversion model proposes that the high affinity receptor is formed via stepwise binding in which IL-2 first binds to the p55 chain and the resulting complex then associates with the p75 chain to form a high affinity ternary complex. In the performed heterodimer model the p55 and p75 chains form a non-covalently linked high affinity heterodimer in the absence of IL-2. We show that these two models can be distinguished on the basis of equilibrium binding experiments using cell lines expressing different numbers of p55 chains. To make this distinction we develop a general model for the interaction of IL-2 with its various receptors. We than analyze the case in which heterodimers exist in the absence of IL-2 and the case in which no preformed heterodimers exist. For both cases we predict the shape of equilibrium Scatchard plots. We then show that published IL-2 binding studies are consistent with a model in which a large concentration of preformed heterodimers is present on the cell surface and inconsistent with a model in which preformed heterodimers are absent from the cell surface. The models that we develop should have general applicability to the entire class of receptor systems in which low and intermediate affinity chains interact to constitute a high affinity receptor.

Fluorescence measurements are among the most powerful methods for investigating structure and structural changes on cell surfaces. The basic strength of this method is that nanomolar concentrations of fluorophores can be detected in the presence of a high level of background noise provided by the cells. An ideal fluorescent probe absorbs and emits at wavelengths not in common with the cellular components, has a high quantum yield, and can be placed specifically into a location such that it is sensitive to the structural aspect of interest. Recently there has been a great expansion in the commercial availability of fluorescent probes with a broad range of fluorescent properties and reactive groups for conjugation (e.g., Haugland, 1989). The requirement for specific placement on a macromolecular/cellular complex remains the most challenging experimentally. In this regard, specific ligands and specific monoclonal antibodies that can be fluorescently modified are valuable reagents. In this chapter we describe our use of quantitative fluorescence measurements to investigate the binding properties of cell surface receptors.

A theory for the diffusion of antibodies in a gel containing erythrocyte antigen is presented. We show that when t ⪢ (k2−1 (K2 is the reverse rate constant for the binding of an antibody to an erythrocyte binding site) the free antibody concentration appears to diffuse with a reduced diffusion coefficient D∗ = D2/(1 + Kϱ0). D2 is the antibody diffusion coefficient in the gel, ϱ0 is the density of binding sites, and K the equilibrium constant for binding of an antibody to an erythrocyte binding site. The theory is developed for experiments in which a circular well is cut from the gel and antiserum placed within the well. Expression for the free and bound antibody concentrations as a function of position and time are derived. We show how these expressions can be used to (1) determine diffusion coefficients and (2) gain information about the distribution of equilibrium constants which characterize the antiserum.

In the autoimmune disease, Systemic Lupus Erythematosus, an individual produces antibodies that bind to his or her own DNA. In this paper we consider a single, long DNA-like molecule in a solution containing bivalent antibodies that can bind to the DNA molecule at regularly spaced sites. The antibody can be attached to DNA by either one or two binding sites. We assume that, when an antibody molecule binds through both its sites, it spans a fixed number of free sites that remain accessible to antibody binding. In this model, antibody molecules can interdigitate along the DNA molecule. We allow steric hindrance within such interdigitating clusters of bound antibodies. We derive analytical expressions for the average number of free, monovalently bound and bivalently bound antibodies, and see how this distribution is influenced by steric hindrance and by the relative binding strengths of the singly and doubly bound antibody.

Bretscher (1983) has shown that on uniformly spread giant HeLa cells, the receptors for low density lipoprotein (LDL) and transferrin are concentrated toward the periphery of the cells. To explain these nonuniform distributions, he proposed that on giant HeLa cells, recycling receptors return to the cell surface at the cell's leading edge. Since the distribution of coated pits on these cells is uniform, Bretscher and Thomson (1983) proposed that there is a bulk membrane flow toward the cell centers. Here we present a mathematical model that allows us to predict the distribution of cell surface proteins on a thin circular cell, when exocytosis occurs at the cell periphery and endocytosis occurs uniformly over the cell surface. We show that on such a cell, a bulk membrane flow will be generated, whose average velocity is zero at the cell center and increases linearly with the distance from the cell center. Our model predicts that proteins that aggregate in coated pits will have concentrations that are maximal at the cell periphery. We fit our theory to the data of Bretscher and Thomson (1983) on the distribution of ferritin receptors for the following cases: the receptors move by diffusion alone; they move by bulk membrane flow alone; they move by a combination of diffusion and bulk membrane flow. From our fits we show that tau m greater than 3.5 tau p, where tau m and tau p are the lifetimes of the membrane and the ferritin receptor on the cell surface, and that tau pD less than 6.9 X 10(-7) cm2, where D is the ferritin receptor diffusion coefficient. Surprisingly, we obtain the best fits to the data when we neglect membrane flow. Our model predicts that for proteins that are excluded from coated pits, the protein concentration will be Gaussian, being maximal at the cell center and decreasing with the distance from the cell center. If on giant HeLa cells a protein with such a distribution could be found, it would strongly support Bretcher's proposal that there is an inward membrane flow.

We extend the mathematical theory of hemolytic plaque growth to include plaques produced by cells secreting antibodies in layers of finite thickness. Previous theories have assumed that the layer was either two-dimensional or of infinite thickness. By using the method of images we derive an equation for the plaque radius as a function of time for layers of any thickness. We show that at short times and at long times the equation reduces to the appropriate infinite three-dimensional and two-dimensional limiting forms, and obtain expressions for estimating the range of times for which these limiting results are valid. For the liquid monolayer technique we obtain a new limiting result. The equation for the plaque radius is a transcendental equation which we solve numerically for a number of cases of interest. These results illustrate a variety of different features of plaque growth associated with the finite thickness of the layer. Experimental studies are usually carried out in layers whose thicknesses are not standardized. In the assays commonly used the thickness h can vary more than six hundred fold, i.e. 1 × 10 −3 cm ⪅ h ⪅ 6.5 × 10 −1 cm. Such variation in h will cause widely different kinetics of plaque growth. For typical plaque experiments of one hour duration the two-dimensional limit is valid when h ⪅ 3 × 10 −3 cm while the infinite thickness limit is valid when h ⪆ 10 −1 cm. For thicknesses in between these values the finite layer results must be used.

Equations are presented which and can be used to describe the inhibition of γG plaques by hapten. Two types of experimental situations are considered. In the first the Red Blood Cell epitope density is large enough so that intramolecular reaction is favorable, and the reaction with antibody is considered irreversible during the time of the experiment. In the second case the epitopes are considered to be sparsely distributed so that only univalent attachment is possible. Since antibody site epitope interaction occurs rapidly ( ⪆1 sec ) on the time scale of diffusion, local equilibrium is assumed to apply in this case. We show that these models lead to different types of inhibition results. In the first instance a differential plot of the inhibition curve will reflect the affinity distribution as is often assumed. The reason involves the fact that only the interaction between antibody and free hapten is controlled by affinity, the interaction between antibody and Red Blood Cell (RBC) being irreversible. Consequently the interaction with RBC cannot effect the shape of the inhibition curve. In the second model, however, the interaction with RBC bound hapten and free hapten are both controlled by affinity. The model predicts that at a fixed secretion rate inhibition occurs abruptly, In this case, therefore, the breadth of the inhibition curve reflects the spread in the secretion rate distribution.

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.

We investigate the equilibrium binding of symmetric bivalent ligands to a heterogeneous population of symmetric bivalent cell surface receptors. The receptors are heterogeneous in their binding affinities (equilibrium binding constants) for the ligand. For any distribution of receptor binding affinities we show how to calculate the total concentration of receptors that are cross-linked by the ligand, i.e., the concentration of cell surface aggregates composed of two or more receptors, as well as the concentration of any given aggregate. We show that certain qualitative properties of cross-linking which hold for homogeneous antibody populations fail to hold in the heterogeneous case. We use our results to interpret certain in vitro experiments in which synthetic bivalent haptens are used to trigger histamine release from basophils which have on their surface antibody specific for the hapten.

The Zimm-Bragg theory is extended to treat the melting of the triple helix poly (A + 2U) for a solution with a 1 : 2 mole ratio of poly A to poly U. Only the case for long chains is considered. For a given set of parameters the theory predicts the fraction of segments in the triple helix, double helix, and random coil states as a function of temperature. Four nucleation parameters are introduced to describe the two order–disorder transitions (poly (A + 2U) ⇄ poly A + 2 poly U and poly (A + U) ⇄ poly A + poly U) and the single order–order transition (poly (A + 2U) ⇄ poly (A + U) + poly U). A relation between the nucleation parameters is obtained which reduces the number of independent parameters to three. A method for determining these parameters from experiment is presented. From the previously published data of Blake, Massoulie and Fresco8 for [Na+] = 0.04, we find σT = 6.0 × 10−4, σD = 1.0 × 10−3, and σσ* = 1.5 × 10−3. σT and σD are the nucleation parameters for nucleating a triple helix and double helix, respectively, from a random coil region. σσ* is the nucleation parameter for nucleating a triple helix from a double helix and a single strand. Melting curves are generated from the theory and compared with the experimental melting curves.

A simple model is introduced to investigate the stability of a sedimenting entanglement. The sedimenting entanglement is represented by a sedimenting sieve. Solvent can pass through it, but single-chain molecules that flow into it become entangled and their flow decreases or, if permanent entanglements form, ceases entirely. With this model we are able to find the conditions under which the mass of a sedimenting entanglement remains constant, grows or decays to a stable value, grows beyond limit, or decays to the mass of a single chain. The theory is applied to the sedimentation of small concentrations of large chain molecules in solutions of small chain molecules in solutions of small chain molecules for the case in which the entanglements are long-lived. Equations are derived which, (1) give the stable entanglement mass as a function of rotor speed and concentration and, (2) for a given concentration predict the rotor speed at which the entanglement mass grows without limit. Numerical results for small concentrations of T2 DNA sedimenting in solutions of T7 DNA are presented.

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.

For monovalent ligands interacting with cell surface receptors we have directly observed the functional dependence of the forward rate constant on the number of receptors per cell (N). The experimental system we studied consisted of monovalent ligand, 2,4-dinitrophenyl (DNP)-aminocaproyl-L-tyrosine (DCT), binding to bivalent, monoclonal anti-DNP immunoglobulin E (IgE) anchored to its high affinity receptor on rat basophilic leukemia (RBL) cells. To measure the fractional occupation of antibody combining sites by DNP we employed a recently developed fluorescence technique (Erickson, J., Kane, B. Goldstein, D. Holowka, and B. Baird, 1986, Mol. Immunol., 72:769-781). Our results are well fitted by the equation (Berg and Purcell, 1977, Biophys. J., 20:193-219) konc = 4 pi DaN kappa on/[4 pi Da + N kappa on] where konc is the forward rate constant for binding to the cell, D is the diffusion constant of the ligand, a is the radius of the cell, and kappa on is the intrinsic forward rate constant describing a single IgE combining site-DNP interaction. If D is fixed at 10(-5) cm2/s, the best fit of accumulated data predicts an average cell radius of approximately 4 microns and kappa on of approximately 1.8 x 10(-13) cm3/s [1.1 x 10(8)(M . s)-1]; both in excellent agreement with RBL cell size and the single-site forward rate constant for the binding of DCT to IgE in solution, respectively. We believe this is the first report of experimental evidence that directly illustrates the effect of surface density in determining the rates of binding for small molecules to membrane receptors.

Surface-based binding assays are often influenced by the transport of analyte to the sensor surface. Using simulated data sets, we test a simple two-compartment model to see if its description of transport and binding is sufficient to accurately analyze BIACORE data. First we present a computer model that can generate realistic BIACORE data. This model calculates the laminar flow of analyte within the flow cell, its diffusion both perpendicular and parallel to the sensor surface, and the reversible chemical reaction between analyte and immobilized reactant. We use this computer model to generate binding data under a variety of conditions. An analysis of these data sets with the two-compartment model demonstrates that good estimates of the intrinsic reaction rate constants are recovered even when mass transport influences the binding reaction. We also discuss the conditions under which the two-compartment model can be used to determine the diffusion coefficient of the analyte. Our results illustrate that this model can significantly extend the range of association rate constants that can be accurately determined from BIACORE.

Ligand-induced receptor aggregation is a well-known mechanism for initiating intracellular signals but oligomerization of distal signaling molecules may also be required for signal propagation. Formation of complexes containing oligomers of the transmembrane adaptor protein, linker for the activation of T cells (LAT), has been identified as critical in mast cell and T cell activation mediated by immune response receptors. Cross-linking of LAT arises from the formation of a 2:1 complex between the adaptor Grb2 and the nucleotide exchange factor SOS1, which bridges two LAT molecules through the interaction of the Grb2 SH2 domain with a phosphotyrosine on LAT. We model this oligomerization and find that the valence of LAT for Grb2, which ranges from zero to three, is critical in determining the nature and extent of aggregation. A dramatic rise in oligomerization can occur when the valence switches from two to three. For valence three, an equilibrium theory predicts the possibility of forming a gel-like phase. This prediction is confirmed by stochastic simulations, which make additional predictions about the size of the gel and the kinetics of LAT oligomerization. We discuss the model predictions in light of recent experiments on RBL-2H3 and Jurkat E6.1 cells and suggest that the gel phase has been observed in activated mast cells.

For certain cell types, the cross-linking of bivalent cell surface receptors by multivalent ligands is an important biochemical step in the transmission of information across the cell's membrane to its interior. The formation of cell surface receptor-ligand aggregates has been shown to "turn on" and "turn off" particular cell responses. It has been hypothesized that very large aggregates generate signals that small aggregates cannot. This hypothesis has not been rigorously tested as yet, in part because of a lack of quantitative information about aggregate sizes. Here we develop a general equilibrium theory for the clustering of bivalent receptors by trivalent ligands. In addition to predicting the concentrations of receptor-ligand aggregates of all possible sizes, we show that a range of ligand concentrations exists at which extremely large aggregates, i.e., superaggregates, form on the cell surface. The formation of a superaggregate corresponds to a sol-gel phase transition, and we study this transition in some detail. For the biologically interesting case of histamine release by basophils, we show, using realistic parameter values, that such transitions should occur when the cells are from highly allergic individuals. We prescribe in detail experimental conditions under which such transitions should occur. These conditions can be used as a guide to test whether or not large aggregates provide signals to cells that small aggregates do not.

Recent experiments focusing on the function of the immunological synapse formed between a T cell and an antigen-presenting cell raise many questions about its purpose. We examine the proposal that the close apposition of the cell membranes in the central region of the synapse acts to focus T-cell secretions on the target cell, thus reducing the effect on nearby cells. We show that the efficiency of targeted T-cell responses to closely apposed cells is only weakly dependent on the distance between the cells. We also calculate effective (diffusion-limited) rates of binding and unbinding for molecules secreted within the synapse. We apply our model to the stimulation of B cells by secreted interleukin-4 (IL-4), and find that very few molecules of IL-4 need be released to essentially saturate the IL-4 receptors on the B-cell surface.

We use the mathematical theory of plaque qrowth to determine if there is merit in performing a hemolytic plaque assay in the presence of an external electric field. In particular, we study the effects of an electric field on the transport of antibodies secreted by a single lymphocyte and on the size and shape of the plaques they produce. Our results indicate that in the presence of an applied electric field: (1) The mobility of the antibodies produced by the antibody forming cell can be determined from the plaque shape. (In the electric field the plaques are no longer circular, but cigar shaped.) (2) By changing the magnitude or direction of the applied electric field more than one plaque can be generated by a single AFC. Thus changes in mobility or the rate of antibody secretion can be assayed. (3) Plaques will reach a steady state size; for good emitters (cells that secrete antibodies at a high rate or that secrete high affinity antibodies) this steady state will be achieved rapidly. Equations are given which describe both the temporal development and steady state plaque size and shape. From the equations, computer generated plots of plaques produced by typical antibody forming cells are presented. These plots are then used to show how pictures of plaques formed in an electric field can be analyzed to determine the antibody mobility.

We present a model for the control of immunoglobulin E (IgE)-mediated histamine release from human basophils. We suggest that there is a calcium gating factor which interacts with crosslinked IgE to form a short-lived open calcium channel. After formation of the channel the activated gating factor rapidly decays to an inactive form. It is the loss of the active gating factor which causes the basophil to desensitize nonspecifically. We propose that the crosslinked IgE molecules are deactivated by a mechanism, such as endocytosis or shedding, which is independent of the mechanism which inactivates the calcium gating factor. This loss of functional IgE leads to specific desensitization. The mathematical formulation of the model explains the relationship of specific and nonspecific desensitization to the amount of specific IgE on the basophil surface; explains why there are two types of antigen excess inhibition; explains the relationship between antigen excess inhibition and desensitization; explains why, for a fixed antigen concentration, increasing the concentration of cell surface IgE increases histamine release until an optimal concentration is reached, then decreases histamine release; predicts the effects that changing the external calcium will have on the dose response curve; and predicts that increasing the amount of specific IgE on the cell surface will cause the dose response curve to undergo a transition from a curve with a single maximum to a curve with two maxima.

We present a kinetic model of histamine release from human basophils due to covalently linked IgE dimers. Comparison of theory with experiment shows that the model gives a good description of histamine release by IgE dimers and allows a number of the parameters of the model to be determined. Comparison with previous models of release by conventional antigens indicates that despite their covalent structure, IgE dimers are subject to the same laws governing inactivation as are antigen produced crosslinks. In addition, the kinetic equation which relates the rate of histamine release to the number of crosslinked Fc epsilon receptors per cell is the same for crosslinks formed by IgE dimers as for antigen induced crosslinks. Quantitative fitting of histamine release data also yields a value for the rate constant for crosslink formation by IgE dimer on the cell surface (rx approximately 5 x 10(-10) cm2/sec). This rate constant is remarkably high and indicates that the reaction is diffusion controlled.