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Copyright © 1999 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 76, Issue 5, 2421-2431, 1 May 1999

doi:10.1016/S0006-3495(99)77397-2

Biophysical Theory and Modeling

Quantifying Aggregation of IgE-FcϵRI by Multivalent Antigen

William S. Hlavacek^*, Alan S. Perelson^*, ,  , Bernhard Sulzer^*, Jennifer Bold^#, Jodi Paar^#, Wendy Gorman^§ and Richard G. Posner^#, ,  

^* Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
^# Department of Chemistry, Northern Arizona University, Flagstaff, Arizona 86011 USA
^§ Biology, Northern Arizona University, Flagstaff, Arizona 86011 USA

Address reprint requests to Dr. Alan S. Perelson, T-10, MS K710, Los Alamos National Laboratory, Los Alamos, NM 87545. Tel.: 505-667-6829; Fax: 505-665-3493.

Reprint requests may also be addressed to Dr. Richard G. Posner, Department of Chemistry, Northern Arizona University, Flagstaff, AZ 86011. Tel.: 520-523-4209; Fax: 520-523-8111.

Aggregation of cell surface receptors is a common mechanism involved in signal transduction across a cell membrane (Metzger, 1992). This mechanism is used, for example, by receptors that are intrinsic protein tyrosine kinases (Pazin and Williams, 1992,Fry et al), such as the epidermal growth factor receptor (Schreiber et al) and the platelet derived growth factor receptor (Heldin et al), and by multichain immune recognition receptors (Keegan and Paul, 1992), such as the high affinity receptor for IgE (FcϵRI) (Holowka and Baird, 1996) and the B-cell receptor (Kaye et al,Kaye and Janeway, 1984,Cambier and Ransom, 1987). For many of these receptors, early steps in the initiation of a signal are similar: multivalent interactions with a ligand lead to the aggregation of receptors and enhanced phosphorylation of tyrosines, which can be recognized by cytoplasmic regulatory molecules. The FcϵRI receptor, for example, is triggered when IgE-FcϵRI complexes are aggregated by multivalent antigen. Aggregation of FcϵRI, which is constitutively associated with the protein tyrosine kinase Lyn (Eiseman and Bolen, 1992), then leads to a series of events, which include recruitment of additional Lyn kinases (El-Hillal et al,Wofsy et al).

Signals generated by aggregation of FcϵRI, which can be negative or positive, depend on various properties of the aggregate structures that are formed on the cell surface. These properties include the overall number of receptors in aggregates on the cell surface, the size of aggregates (Fewtrell and Metzger, 1980,MacGlashan et al), the spacing of receptors in aggregates (Kane et al), and the time that individual receptors spend in aggregates (Torigoe et al). For example, it has been observed that IgE dimers are less effective than larger IgE oligomers at stimulating cellular responses (Fewtrell and Metzger, 1980) and that cellular responses are inhibited when an optimal degree of aggregation is exceeded (Becker et al,Menon et al,Seagrave and Oliver, 1990). Dependency of cellular responses on properties of receptor aggregates also has been observed for related receptors, such as the B-cell receptor (Dintzis et al,Dintzis et al) and T-cell receptor (Sloan-Lancaster et al,Madrenas et al,Lyons et al,Neumeister Kersh et al).

Because properties of ligand-induced receptor aggregates influence the signals that these aggregates generate, significant effort has been devoted to quantitative analysis of interactions between multivalent ligands and cell surface receptors, particularly in work with FcϵRI (Goldstein, 1988,Goldstein and Wofsy, 1994). A goal of these studies has been to measure or predict the number of receptors in aggregates on the cell surface so that this quantity then can be compared and correlated with cellular responses (Dembo et al,Dembo et al,Dembo and Goldstein, 1980,MacGlashan and Lichtenstein, 1983,MacGlashan et al). The number of receptors in aggregates is related to the number of receptor sites in aggregates, which in turn is related to the number of bound receptor sites in excess of the number of bound ligands. This difference, which can be interpreted as the number of receptor pairs in clusters (Perelson, 1981), is zero when ligand binding is monovalent and positive when ligand binding is multivalent because each bound ligand must engage at least one receptor site. Thus, we can quantify receptor aggregation if we can measure ligand and receptor site binding.

One method to determine both the number of ligands bound to receptors and the occupancy of receptor sites involves differentially labeled monovalent and multivalent ligands. This method has been used in experiments with chemically cross-linked oligomers of IgE. For example, by incubating rat basophilic leukemia (RBL) cells, which express FcϵRI, with ¹²⁵I-labeled dimers of IgE and then by adding ¹³¹I-labeled IgE to assay the number of Fc receptor sites left unbound by IgE dimers, one can determine the number of Fc receptor sites in dimer-induced aggregates (Segal et al). This method works well with IgE oligomers, because IgE-FcϵRI complexes are long lived (Kulczycki and Metzger, 1974,Sterk and Ishizaka, 1982). On the time scale of an experiment, the equilibrium between oligomeric IgE and FcϵRI is undisturbed by monomeric IgE. However, the method is difficult to apply when receptor sites have low affinity for sites on the multivalent ligand, as in a typical physiological situation, because introduction of monovalent ligand can now rapidly influence binding of the multivalent ligand to receptors.

Here, we develop a method that can be used to measure simultaneously the amount of multivalent ligand bound to receptors and the occupancy of receptor sites without the complication of introducing a monovalent ligand. The method combines approaches previously used to measure the amount of ligand bound to receptors (Seagrave et al) and the occupancy of receptor sites (Erickson et al). To demonstrate the method, we study the equilibrium binding of haptenated phycoerythrin to anti-hapten IgE, which is labeled with fluorescein isothiocyanate (FITC). By using two-color flow cytometry to measure fluorescence of phycoerythrin and FITC, we are able to estimate the amount of antigen on the surface of RBL cells and the occupancy of surface IgE combining sites. These measurements also allow us to estimate the number of FcϵRI pairs in antigen-induced clusters, i.e., the extent of receptor cross-linking.

Estimates of cross-linking are refined with the aid of a mathematical model, which we fit to data. The data are consistent with a model that accounts for cooperative effects, which arise, at least in part, for steric reasons. The model, which reduces to an equivalent site model (Perelson, 1981,Perelson, 1984,Macken and Perelson, 1985,Lauffenburger and Linderman, 1993,Sulzer and Perelson, 1996) in the absence of cooperative effects, allows us not only to refine our estimates of cross-linking but also to determine equilibrium binding parameters. Furthermore, the development of this model, together with the ability to test its predictions against experimental measurements of both ligand binding and receptor site binding, provide new insights into methods for quantifying multivalent ligand-receptor interactions.

To aid in analysis of experimental data, we develop a model for equilibrium binding of 2,4-dinitrophenol-coupled B-phycoerythrin (DNP₂₅-PE) to anti-DNP IgE-FcϵRI complexes on RBL cells. In this model, we treat IgE-FcϵRI as a bivalent cell surface receptor: each antibody combining site in an IgE-FcϵRI complex represents one of two potential binding sites per receptor for the ligand, DNP₂₅-PE.

The model is based on the reaction scheme shown in Figure 1A in which ligands bind and aggregate receptor sites through a series of reversible reactions. The initial reaction involves the binding of solution-phase ligand to a receptor site, and each subsequent reaction involves the addition of a receptor site to a ligand-receptor complex (Figure 1B). In this scheme, ligand-receptor complexes form without intramolecular rearrangement reactions, i.e., without either ring or network formation reactions (Perelson, 1984,Macken and Perelson, 1985). If these reactions were significant, then the scheme in Figure 1A would have to be modified.

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Figure 1

Reaction scheme. (A) Ligands bind receptor sites through a series of reversible reactions. L₀, L₁, L₂, and S indicate, respectively, a ligand in solution, a ligand bound to one receptor site, a ligand bound to two receptor sites, and a free receptor site. (B) A sequence of possible reactions is illustrated.

As indicated, the model is developed in terms of ligand states. Thus, variables in the model include the concentration of ligand in solution, which is denoted as L₀, the surface density of ligand that is bound to i receptor sites, which is denoted as L_i, and the surface density of free receptor sites, which is denoted as S. Related variables include the total concentration of ligand, which is denoted as L_T, and the total surface density of receptor sites, which is denoted as S_T.

To characterize the equilibria for the reactions in Figure 1A, we write

(1)

Sites on the ligand DNP₂₅-PE are chemically identical: each is a DNP group. Thus, the equilibrium constants in Eq. (1) are related, although K₁ through K_n−1 need not be identical. Differences among these equilibrium constants indicate functional nonequivalence of ligand sites (i.e., cooperativity), which can arise for a variety of physical reasons (Perelson, 1984). Sites on DNP₂₅-PE are likely to be functionally nonequivalent, at least in part, for steric reasons. The crystal structure of B-phycoerythrin (Ficner et al,Ficner and Huber, 1993) indicates that this molecule is cylindrical with a height of 6nm and a diameter of 11nm. Thus, the exposed surface area of DNP₂₅-PE is ∼200 square nm, which is insufficient to allow binding of more than a few DNP sites because the area covered by a bound antibody Fab arm is at least 30 square nm (Poljak et al,Padlan, 1994).

To account for functional nonequivalence of sites on DNP₂₅-PE, we introduce a cooperativity function ϕ(i) to relate the cross-linking equilibrium constants K₁ through K_n−1 (Cantor and Schimmel, 1980):

(2)

Because experiments are performed under conditions that inhibit recycling of FcϵRI, we treat the total numbers of ligands and receptor sites as conserved quantities. Conservation of ligand can be expressed as

(3)

(4)

Ligand-induced aggregation of receptors is quantified by the number of cross-links on the cell surface. A cross-link is defined as follows (Perelson, 1981). In the absence of intramolecular rearrangement reactions, as in the scheme of Figure 1A, a ligand bound at i sites is attached to i receptors. Thus, a ligand bound at two sites, as depicted in Figure 1B, forms a single cross-link, i.e., a cluster of two receptors. If we generalize this concept of a cross-link, then a ligand bound at i sites forms i−1 cross-links, i.e., i−1 clusters of receptor pairs. Consequently, in the absence of intramolecular rearrangement reactions, the number of cross-links is given by , which is equivalent to the number of bound receptor sites () in excess of the number of bound ligands ().

Mouse monoclonal anti-DNP IgE was isolated from hybridoma H1 26.82 (Liu et al) by affinity purification (Holowka and Metzger, 1982). Final steps in the purification process included ion exchange chromatography to remove bound DNP-glycine, then gel filtration to separate monomeric IgE from small amounts of IgE aggregates. Fluorescently labeled IgE (FITC-IgE) was prepared by attaching fluorescein-5-isothiocyanate (Molecular Probes, Eugene, OR) to IgE (Erickson et al). The fluorescent antigen DNP₂₅-PE, which is composed of 2,4-dinitrophenol (DNP) and B-phycoerythrin (PE), was custom synthesized by Molecular Probes. The molar ratio of DNP to PE is 25:1. The molecular weight of DNP₂₅-PE is ∼250,000.

RBL-2H3 cells (Barsumian et al) were grown adherent in 75cm² flasks. Cell cultures, which were used typically 5 days after passage, were maintained at 37°C and 5% CO₂. Culture media consisted of MEM 1X with Earle's salts without glutamine (Gibco BRL), 20% fetal bovine serum (HyClone, Logan, UT), 1% v/v l-glutamine, 1% v/v penicillin, and 1% v/v streptomycin (Gibco BRL). To harvest cells, we rinsed and then incubated the cells for 5 minutes at 37°C with trypsin-EDTA (Gibco BRL). Cells harvested for experiments were washed and resuspended in buffered salt solution (pH 7.7), which was freshly passed through a 0.22-μc filter. Buffered salt solution (BSS) consisted of 135mM NaCl, 5mM KCL, 1mM MgCl₂, 1.8mM CaCl₂, 5.6mM glucose, 0.1% gelatin, and 20mM Hepes. Cell suspensions in buffered salt solution were supplemented with 10mM sodium azide and 10mM 2-deoxy-d-glucose (Sigma, St. Louis, MO) to inhibit receptor recycling and cellular degranulation during binding experiments. To sensitize cells to DNP, we incubated cells overnight while cells were still in culture with excess (10μg) anti-DNP FITC-IgE. Sensitized cells were exposed to FITC-IgE for at least 12h prior to harvesting.

In each binding experiment, we incubated a suspension of sensitized cells at a density of 2.5×10⁵, 10⁶, or 4×10⁶ cells/ml, with DNP₂₅-PE at room temperature. The concentration of DNP₂₅-PE varied from 0.0001 to 100μg/ml. After cells were incubated with DNP₂₅-PE for at least 90min, we used a Becton Dickinson FACScan flow cytometer, which was controlled with Cell Quest software, to collect histograms of FITC and PE fluorescence. We determined that 90min was sufficient for binding to reach equilibrium at the relevant cell and ligand concentrations, because neither FITC nor PE fluorescence varied significantly as we varied the incubation time from 1 to 2 hours. Flow cytometric data were recorded as the mean FITC fluorescence (520nm) of the cell suspension, FL1, and as the mean PE fluorescence (550nm) of the cell suspension, FL2. To correct for nonspecific binding of DNP₂₅-PE to cells, we performed a control experiment in which cells lacked surface IgE. The difference between FL2 and the mean PE fluorescence for the control sample, ΔFL2, indicates the PE fluorescence due only to specific binding. To relate PE fluorescence to the surface density of DNP₂₅-PE, measurements of ΔFL2 were calibrated by using microspheres embedded with a known amount of B-phycoerythrin (Flow Cytometry Standards Corporation, San Juan, PR).

The fraction of surface IgE combining sites that are bound to DNP₂₅-PE, which we denote as σ, is related to variables in the model and, as established in earlier work (Erickson et al), to quenching of FITC-IgE fluorescence:

(5)

The ratio of cell-bound DNP₂₅-PE to surface IgE combining sites, which we denote as λ, is related to variables in the model and, as established in earlier work (Seagrave et al), to measurements of PE fluorescence:

(6)

Based on our earlier definition of a cross-link, the number of cross-links per IgE combining site, which we denote as χ, is related to σ and λ as follows:

(7)

Three sets of data were used to determine best-fit values for K₀, K₁, S_T, and a, where a is a parameter in the specified cooperativity function ϕ(i). We considered various one-parameter functional forms for ϕ(i), including ϕ(i)=a(i−1). Each data set consisted of measurements of FITC fluorescence (FL1) and PE fluorescence (ΔFL2) for a series of ligand concentrations. These measurements were taken at one of three cell densities: 2.5×10⁵, 10⁶, or 4×10⁶ cells/ml. Because data sets were collected with different instrument settings, it was necessary to determine best-fit scaling factors (FL1_min, FL1_max, and ΔFL2_max) for each set of FL1 and ΔFL2 measurements.

Best-fit parameter values were determined by using the FORTRAN subroutine DNLS1 from the SLATEC Common Mathematical Library (http://www.netlib.org/slatec), which implements a modified Levenberg-Marquardt algorithm for solving nonlinear least-squares problems.

Equilibrium states are calculated by solving the model equations, which is aided by combining Eqs. (1). By using Eq. (1) to express each L_i as a function of S and L₀, by using Eq. (2) to relate K₁ through K_n−1, and by using Eq. (3) to express L₀ as a function of S, we can rewrite Eq. (4) to obtain

(8)

(9)

When values for the parameters (n, C, L_T, S_T, K₀, and K₁) and a functional form for the cooperativity function ϕ(i) are specified, Eq. (8) is a nonlinear equation involving a single unknown: the fraction of free receptor sites S/S_T. To determine the fraction of free receptor sites at equilibrium, we solve this equation by using the method of bisection (Press et al). Once S/S_T is known, other states at equilibrium can be determined by using the relations L₀/L_T=1/[1+K₀CS_T] and L_i/S_T=(K₀L_T)(L₀/L_T)π(i), which are derived from Eqs. (1).

We use multiparameter flow cytometry to study the equilibrium binding of a multivalent ligand to a cell surface receptor. The ligand, DNP₂₅-PE, is haptenated phycoerythrin (PE): each molecule of PE is coupled to an average of 25 DNP molecules. The receptor for this ligand is FITC-labeled anti-DNP IgE that is bound to FcϵRI on the surface of RBL cells. Below, we first present qualitative features of DNP₂₅-PE binding to surface IgE, and we then illustrate how measurements of PE and FITC fluorescence can be used to quantify antigen-induced aggregation of IgE-FcϵRI complexes. The methods developed here allow us to determine how equilibrium cross-linking varies quantitatively with the total concentrations of ligand and receptor.

In our experiments, DNP₂₅-PE at various concentrations is added to suspensions of RBL cells that have been sensitized with FITC-IgE. Then, after equilibrium is reached, PE and FITC fluorescence are measured simultaneously in a flow cytometer. Fluorescence measurements for a series of experiments are shown in Fig. 2. As the concentration of DNP₂₅-PE increases, FITC fluorescence (FL1) decreases and PE fluorescence (ΔFL2) increases. A decrease in FITC fluorescence indicates an increase in the occupancy of IgE combining sites (Erickson et al), and an increase in PE fluorescence indicates an increase in association of DNP₂₅-PE with the cell surface (Seagrave et al).

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Figure 2

Flow cytometric measurements of PE and FITC fluorescence that monitor equilibrium binding of DNP₂₅-PE to FITC-IgE on the surface of RBL cells. The cell density is 10⁶ cells/ml.

As can be seen in Fig. 2, receptor sites saturate (i.e., FL1 reaches a lower plateau) at a ligand concentration of ∼0.1μg/ml, whereas ligand binding, as measured by ΔFL2, fails to reach an obvious plateau even at ligand concentrations greater than 10μg/ml. This effect cannot be explained by nonspecific binding of ligand to cells, which was assayed in control experiments, because ΔFL2 represents the PE fluorescence due only to specific binding. We expect that ΔFL2 continues to rise after FL1 reaches a plateau for the following reason. When FL1 first reaches a plateau, ligand binding is multivalent. Then, as the ligand concentration increases, the multiplicity of ligand binding decreases, which allows more ligand to bind to the cell surface. As a result, ΔFL2 increases. This explanation is consistent with the binding parameters that we later determine.

At ligand concentrations greater than 20μg/ml, measurements of PE fluorescence are unreliable due to light scatter. Thus, we are unable to measure PE fluorescence at saturation (i.e., we are unable to measure ΔFL2_max) as is required to directly relate measurements of PE fluorescence to the extent of ligand binding (Eq. (6)). Nevertheless, by using microspheres embedded with a known amount of PE to calibrate measurements of ΔFL2, we are able to estimate the extent of ligand binding. The calibration results indicate that the highest recorded value of ΔFL2 in Fig. 2 (1380) corresponds to ∼9×10⁵ PE molecules per cell and at least the same number of receptor sites per cell (each bound ligand engages at least one receptor site). Thus, ligand binding approaches saturation in the experiments of Fig. 2 because RBL cells only express 300,000 to 600,000 FcϵRI receptors per cell (Erickson et al). Based on this expected number of FcϵRI receptors per cell, we can estimate that the highest recorded value of ΔFL2 is within at least 75% of ΔFL2_max because the minimum number of receptor sites per cell indicated by bead calibration (9×10⁵) is 75% of the maximum number of surface IgE combining sites per RBL cell (1.2×10⁶). As we will see later, model-based analysis indicates that ligand binding actually approaches 90% saturation in the experiments of Fig. 2.

We quantify receptor aggregation by determining the number of bound receptor sites in excess of the number of bound ligands. To determine this quantity, which we interpret as the number of cross-links, we must measure or estimate the scaling factors in Eqs. (5) (FL1_min, FL1_max, and ΔFL2_max) and then use these equations to relate measurements of fluorescence (FL1 and ΔFL2) to quantities that characterize ligand and receptor binding (σ, λ, and χ).

The highest recorded value of ΔFL2 represents a lower bound on ΔFL2_max. By using this lower bound, we can place an upper bound on the extent of ligand binding and a lower bound on the extent of cross-linking, as can be seen by inspecting Eqs. (6). In Eq. (6), the number of bound ligands per receptor site λ is defined as ΔFL2/ΔFL2_max. Thus, an underestimate of ΔFL2_max leads to an overestimate of λ. In Eq. (7), the number of cross-links per receptor site χ is defined as σ−λ. Thus, an overestimate of λ leads to an underestimate of χ.

In Fig. 3, we show how the fluorescence measurements of Fig. 2 are related to biologically meaningful quantities. In Figure 3A, we plot receptor site occupancy σ, which is calculated by using Eq. (5), as a function of ligand concentration. The scaling factors FL1_min and FL1_max, which appear in Eq. (5), are readily determined from the fluorescence data (Fig. 2). In Figure 3B, we plot the extent of ligand binding λ, which is calculated by using Eq. (6), as a function of ligand concentration. The values of λ were calculated by using the highest recorded value of ΔFL2 for ΔFL2_max in Eq. (6). Because we have determined only that this value is within 75% of ΔFL2_max (on the basis of our bead calibration results), values of λ are uncertain to the extent indicated by the error bars. However, as illustrated, the fluorescence data directly indicate an upper bound on ligand binding. In Figure 3C, we plot the extent of cross-linking χ, which is calculated by using Eq. (7), as a function of ligand concentration. As illustrated, an upper bound on λ translates to a lower bound on χ. Note that the increasing portion of the cross-linking curve is insensitive to the estimated uncertainty in ΔFL2_max.

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Figure 3

Equilibrium binding characterized by measurements of PE and FITC fluorescence. Fluorescence measurements directly indicate (A) the fraction of receptor sites bound σ, (B) an upper bound on the number of bound ligands per receptor site λ, and (C) a lower bound on the number of cross-links per receptor site χ. The points were derived from the fluorescence data in Fig. 2 by using Eqs. (5) and the scaling factors: FL1_min=184, the lowest recorded value of FL1, FL1_max=450, the highest recorded value of FL1, and ΔFL2_max=1380, the highest recorded value of ΔFL2. The error bars in B and C indicate the uncertainty in λ and χ if the highest recorded value of ΔFL2 (1380) is within 75% of the actual ΔFL2_max, as suggested by bead calibration. The small solid points at the end of error bars are based on ΔFL2_max=1380/0.75. The cell density is 10⁶ cells/ml.

To aid in the analysis of fluorescence data, we developed a model for ligand-receptor binding (Eqs. (1)). We simultaneously fit this model to three data sets, one of which is that shown in Fig. 2. Each data set was collected at a different cell density. The following best-fit parameter values, which apply for all three data sets, were determined: K₀=4.4×10⁸ M⁻¹, K₁S_T=13, and S_T=9.6×10⁵ sites/cell. The fitting procedure also allowed us to specify a=0.43 for the cooperativity function ϕ(i)=a(i−1). We considered a variety of one-parameter functional forms for ϕ(i), but functions consistent with the data indicated essentially the same values for the cross-linking equilibrium constants K₁ through K_n−1. In addition to these parameter values, we also determined for each data set best-fit values for the scaling factors FL1_min, FL1_max, and ΔFL2_max. For example, we determined that ΔFL2_max=1500 for the data set of Fig. 2, which indicates that ligand binding in these experiments reached ∼90% saturation (1380/1500).

The extent to which the model fits the data is illustrated in Fig. 4. Both the fraction of bound receptor sites and the number of bound ligands per receptor site are plotted as a function of ligand concentration. The points are derived from the fluorescence data of Fig. 2, Eqs. (5), and the best-fit scaling factors. These plots meet two expectations. First, the fraction of bound receptor sites σ is greater than or equal to the number of bound ligands per receptor site λ. This result is expected because each bound ligand must engage at least one receptor site. Thus, σ must be greater than λ when ligand binding is multivalent, and σ must equal λ when ligand binding is monovalent. Second, σ and λ converge at high ligand concentrations, which indicates that ligand binding is predominantly monovalent at these concentrations. The quality of the fit illustrated in Fig. 4 is typical of the fits for the other two data sets.

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Figure 4

Comparison of best-fit theoretical binding curves with scaled data. The fraction of bound receptor sites σ (circles) and number of bound ligands per receptor site λ (squares) are plotted as a function of ligand concentration. The points were derived from the fluorescence data in Fig. 2 by using Eqs. (5), FL 1_min=184, FL1_max=450, and ΔFL2_max=1500. The broken and solid lines indicate the theoretical binding curves, which are based on the model equations (Eqs. (1)), Eqs. (5), n=25, S_T=9.6×10⁵ sites/cell, K₀=4.4×10⁸ M⁻¹, K₁S_T=13, and ϕ(i)=0.43 (i−1). The cell density is 10⁶ cells/ml.

The best-fit parameter values are consistent with independent data. The number of IgE combining sites per cell, 9.6×10⁵, is consistent with our bead calibration results, which indicate 9×10⁵ sites per cell, and with the number of FcϵRI receptors (300,000 to 600,000 per cell) that are expressed on RBL cells (Erickson et al). The estimated affinity of anti-DNP IgE for DNP on haptenated PE, 4.4×10⁸ M⁻¹, is comparable with the affinity of anti-DNP IgE for DNP on haptenated bovine serum albumin, 8.5×10⁸ M⁻¹ (Xu et al). As expected, for steric reasons, the cooperativity function ϕ(i)=0.43(i−1) is an increasing function of ligand site occupancy i. Also, as is consistent with the structures of Fab and PE, theoretical values for the ratio σ/λ, which indicates the number of bound receptor sites per bound ligand molecule, are much less than the chemical valence of DNP₂₅-PE. For example, σ/λ ≈ 4 at 0.1μg/ml in Fig. 4 is where the cross-linking curve peaks (Fig. 5). Cross-linking is indicated by the vertical distance between the two curves in Fig. 4 because χ=σ−λ (Eq. (7)).

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Figure 5

Best-fit estimates of cross-linking as a function of ligand concentration for three cell densities. The number of cross-links per receptor site χ is plotted as a function of ligand concentration for 2.5×10⁵ cells/ml (triangles), 10⁶ cells/ml (circles), and 4×10⁶ cells/ml (squares). The points were derived from fluorescence data and best-fit scaling factors. For example, the points indicated by circles were derived from the fluorescence data in Fig. 2 by using Eqs. (5), FL 1_min=184, FL1_max=450, and ΔFL2_max=1500. The lines indicate the theoretical binding curves, which are based on the model equations (Eqs. (1)), Eq. (7), n=25, S_T=9.6×10⁵ sites/cell, K₀=4.4×10⁸ M⁻¹, K₁S_T=13, and ϕ(i)=0.43(i−1).

The binding curves in Fig. 4, which are derived from our model-based analysis, can be compared with those in Fig. 3, which are derived directly from the fluorescence data. The two curves for receptor site binding are identical as can be seen by comparing Figure 3A with the corresponding plot in Fig. 4. The two curves for ligand binding also are similar. The binding curve in Fig. 4 falls within the range indicated in Figure 3B for ligand binding. In fact, this curve closely matches the upper bound on ligand binding indicated in Figure 3B. This suggests that our estimated upper bound on ligand binding and the resulting lower bound on cross-linking are tight and that the error bars shown in Fig. 3 overestimate the uncertainty in measurement of these quantities. The analysis-refined cross-linking curve that corresponds to Figure 3C is shown in Fig. 5.

In Fig. 5, we plot the extent of cross-linking χ as a function of ligand concentration for three cell densities. The points are derived from fluorescence measurements by using Eqs. (5) and best-fit scaling factors. Each curve indicates how cross-linking varies as a function of ligand concentration at a particular cell density or equivalently a particular receptor or receptor site concentration. Each of these cross-linking curves is bell shaped, as expected. Cross-linking increases with ligand concentration up to an optimal ligand concentration and then decreases as monovalent binding, because of excess ligand, begins to predominate.

The results shown in Fig. 5 indicate that cross-linking is influenced by the cell density. Three features are discernible. First, the location of the increasing portion of the cross-linking curve depends on the cell density. This portion of the curve, which corresponds to the regime where receptor sites are not saturated, shifts to the right as the cell density increases. Thus, at a fixed ligand concentration, cross-linking decreases as the cell density increases if receptor site binding is below saturation. Second, the peak of the cross-linking curve is independent of cell density, i.e., each curve has the same maximum height. Third, the location of the decreasing portion of the cross-linking curve, which corresponds to the regime where receptor sites are saturated, tends not to depend on the cell density. As can be seen, the three curves coincide at ligand concentrations greater than 1μg/ml. These qualitative features of cross-linking curves were predicted in an earlier theoretical analysis (Sulzer and Perelson, 1996).

In many antigen, hormone, and cytokine receptor systems, signal transduction is initiated by aggregation of cell surface receptors (Metzger, 1992). However, even in the well-studied FcϵRI system, the features of ligand-receptor binding that are critical for signaling have yet to be fully characterized, especially for cases in which the ligand has more than two binding sites. Here, we have developed and applied a flow cytometric method to monitor receptor site occupancy (Figure 3A and Figure 4A), ligand binding (Figure 3B and Figure 4B), and cross-linking (Figure 3C and Figure 5C). This study, like the recent work of Xu et al, represents an attempt to quantify the interactions of a multivalent antigen with IgE-FcϵRI.

The method that we have developed to quantify FcϵRI aggregation is significant for several reasons. First, the antigen is a haptenated protein (DNP₂₅-PE) that resembles a physiological allergen. Like an allergen, it elicits strong cellular responses (Seagrave et al). This is in contrast to bivalent haptens (Siraganian et al,Kane et al,Posner et al). Also like an allergen, DNP₂₅-PE cross-links FcϵRI via antigen-antibody reactions. This is in contrast to oligomers of IgE; the kinetics of IgE binding to FcϵRI differ significantly from typical antigen-antibody kinetics (Kulczycki and Metzger, 1974,Sterk and Ishizaka, 1982). Second, the method allows us to characterize reactions on the cell surface without disturbing these reactions. This is in contrast to another flow cytometric method that recently has been developed to study multivalent ligand-receptor binding (Woodard et al). In this method, which we discuss later, a labeled monovalent ligand is used to determine the number of receptor sites that are bound by a differentially labeled multivalent ligand. Third, our method can be applied not only in equilibrium binding studies, as we have demonstrated here, but also in kinetic studies (Posner et al). Kinetic binding studies may be important for understanding the temporal properties of FcϵRI aggregates that influence signaling (MacGlashan et al,Torigoe et al).

We quantify receptor aggregation by determining two quantities: the number of bound receptor sites and the number of bound ligands. These quantities reveal information about the state of receptor aggregation. For example, if the number of bound ligands equals the number of bound receptor sites, then each ligand is bound to a single receptor site and no receptors are aggregated. In the absence of intramolecular rearrangement reactions, the number of bound receptor sites in excess of the number of bound ligands can be interpreted as the number of cross-links, i.e., the number of clustered receptor pairs (Perelson, 1981). We assume that this interpretation is valid here, i.e., we say that a cross-link is formed each time two receptor sites are joined. However, our use of this assumption does not represent a limitation of the method because cross-linking can be determined more directly if bispecific chimeric IgE is available. In experiments with this reagent, receptor aggregation is equivalent to receptor site aggregation, which is indicated by the number of bound receptor sites in excess of the number of bound ligands.

One approach that has been used to quantify FcϵRI aggregation involves oligomers of IgE (Segal et al). The number of bound oligomers is determined by radiolabeling. Monomeric IgE labeled with a different isotope is then used to determine the number of free FcϵRI receptors. Because dissociation of IgE from FcϵRI is slow (Kulczycki and Metzger, 1974,Sterk and Ishizaka, 1982), it is possible to add oligomeric IgE, reach equilibrium, and then add excess monomeric IgE to fill empty Fc sites without disturbing oligomer binding on the time scale of the experiment. Although oligomers of IgE are useful tools for studying receptor aggregation and subsequent cellular responses in this system, they have a number of limitations (Goldstein, 1988). Aggregates can be no larger than the number of chemically cross-linked IgE molecules, so IgE oligomers are unable to produce signals that depend on large receptor aggregates. Binding of IgE to FcϵRI is slow, so it may be difficult to separate the kinetics of oligomer binding from the kinetics of the cellular response. Also, IgE oligomers produce long-lived cross-links, and thus their effects may be atypical because signaling events that require the continual formation of new cross-links will not be observed.

Recently, Woodard et al used a multivalent antigen to aggregate B-cell receptors and then used a differentially labeled monovalent antigen to count free receptor sites. Two-color flow cytometry was used together with FITC-DNP-l-papain, as a monovalent antigen, and TRITC-DNP-pol or TRITC-DNP-dextran, as a multivalent antigen, to determine the amount of monovalent antigen and the amount of multivalent antigen bound to DNP-specific B cells. Data obtained with this indirect method are difficult to interpret because the reactions are reversible (the typical dissociation rate constant for a bond between DNP and an antibody combining site is between 0.1 and 0.001s⁻¹). Thus, on the time scale of these experiments, the binding of multivalent antigen is disturbed when the monovalent antigen is added to the system. In comparison, our method of measurement does not require an additional monovalent probe. Thus, it avoids the complications that arise when different ligands compete for the same receptor. However, our approach requires that we label the receptors, which is impossible if the receptor is unavailable in a secreted form or cannot be reattached to the cell surface. The second condition is not fulfilled for many receptors, and consequently, an indirect method such as that used by Woodard et al is then required to measure aggregation.

Another approach used to study FcϵRI aggregation involves symmetric bivalent ligands, which represent the simplest type of ligand capable of aggregating receptors. Mathematical models have been developed for bivalent ligands interacting with bivalent cell surface receptors (Dembo and Goldstein, 1978,Perelson and DeLisi, 1980,Perelson, 1980,Wofsy and Goldstein, 1987,Posner et al). However, a major limitation of these ligands is their inability to activate strong cellular responses. A variety of evidence suggests that these ligands are poor activators of cellular responses because they aggregate receptors predominantly in the form of stable cyclic dimers (Kane et al,Schweitzer-Stenner et al,Erickson et al,Posner et al,Posner et al). Cyclic dimers, in which two ligand molecules connect two IgE-FcϵRI complexes to form a closed ring, prevent chain elongation and limit the size of ligand-induced aggregates. With ligands of larger valence, such as DNP₂₅-PE, ring formation does not necessarily prevent growth of receptor aggregates. Thus, multivalent ligands are capable of cross-linking many IgE molecules and typically initiate strong cellular responses. When interacting with surface IgE, these ligands can form chain-, ring-, tree-, and network-like structures, whereas bivalent ligands can form only chain- and ring-like structures. This advantage of multivalent ligands is also a disadvantage, because a theory that describes the binding of multivalent ligands to bivalent receptors has yet to be fully developed. However, in a variety of related reaction systems studied in polymer chemistry, it has been found that ignoring the intramolecular reactions that form rings and networks leads to results that adequately characterize most aggregates (Flory, 1953), except those that form rubber-like materials. For these reasons, we have used a theory of multivalent ligand-receptor interactions that accounts for chain- and tree-like but not ring- and network-like aggregate structures. The model based on this theory is consistent with our binding data (Figure 4 and Figure 5).

In our efforts to estimate parameter values, we observed that unique best-fit parameter values are difficult to determine. To address this problem, we used three independent data sets, each collected at a different cell density and each consisting of both FL1 and ΔFL2 measurements. We obtain different results if we fit only FL1 data, only ΔFL2 data, or only data at a single cell density (unpublished material). This suggests that models of multivalent ligand-receptor interactions should be tested with a global analysis of multiple data sets (Posner and Dembo, 1994; Myszka et al).

In summary, we have demonstrated a method by which the number of cross-links can be determined when both the number of receptor sites occupied and the number of ligands bound per receptor site are measured simultaneously. With this technique, we have confirmed that the equilibrium cross-linking curve is bell shaped (Figure 3C and Figure 5C). The results shown in Fig. 5 also confirm qualitative predictions concerning the influence of cell density on the cross-linking curve (Sulzer and Perelson, 1996). As predicted, we observe that an increase in cell density shifts the increasing portion of the cross-linking curve toward higher ligand concentrations, that the maximum height of the cross-linking curve is independent of cell density, and that the decreasing portion of the cross-linking curve also is independent of cell density. Thus, the equivalent site model studied by Sulzer and Perelson, 1996 apparently can be used to predict the qualitative features of cross-linking curves for real ligands. However, to obtain quantitative estimates of binding parameters and more accurate quantification of ligand-receptor aggregation, it was important here to consider a more complicated model that included negative cooperativity due to steric effects. In conclusion, we have developed new theoretical and experimental tools for studying multivalent ligand-receptor binding.