INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

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., 1993), such as the epidermal growth factor receptor (Schreiber et al., 1982) and the platelet derived growth factor receptor (Heldin et al., 1989), and by multichain immune recognition receptors (Keegan and Paul, 1992), such as the high affinity receptor for IgE (FcRI) (Holowka and Baird, 1996) and the B-cell receptor (Kaye et al., 1983; 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 FcRI receptor, for example, is triggered when IgE-FcRI complexes are aggregated by multivalent antigen. Aggregation of FcRI, 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., 1997; Wofsy et al., 1997).

Signals generated by aggregation of FcRI, 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., 1983), the spacing of receptors in aggregates (Kane et al., 1986), and the time that individual receptors spend in aggregates (Torigoe et al., 1998). 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., 1973; Menon et al., 1984; 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., 1976, 1983) and T-cell receptor (Sloan-Lancaster et al., 1994; Madrenas et al., 1995; Lyons et al., 1996; Neumeister Kersh et al., 1998).

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 FcRI (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., 1978, 1979; Dembo and Goldstein, 1980; MacGlashan and Lichtenstein, 1983; MacGlashan et al., 1985). 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 FcRI, 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., 1977). This method works well with IgE oligomers, because IgE-FcRI 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 FcRI 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., 1987) and the occupancy of receptor sites (Erickson et al., 1986). 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 FcRI 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, 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.

	MODEL

TOP ABSTRACT INTRODUCTION MODEL MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

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-FcRI complexes on RBL cells. In this model, we treat IgE-FcRI as a bivalent cell surface receptor: each antibody combining site in an IgE-FcRI complex represents one of two potential binding sites per receptor for the ligand, DNP₂₅-PE.

Reaction scheme

The model is based on the reaction scheme shown in Fig. 1 A 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 (Fig. 1 B). 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 Fig. 1 A would have to be modified.

View larger version (18K): [in this window] [in a new window]   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.

Equilibria

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

(1)

in which n is the chemical valence of the ligand (i.e., the number of DNP groups per PE molecule) and each K_i is an equilibrium constant. The equilibrium constant K₀ characterizes the affinity of a receptor site for a ligand site (i.e., a DNP group) when the ligand is in solution, and the cross-linking equilibrium constant K_i (i  1) characterizes the affinity of a receptor site for a ligand site when the ligand is bound to i receptor sites (Dembo and Goldstein, 1978; Perelson, 1981). If K₁ through K_n1 are identical, Eq. 1 reduces to an equivalent site model (Perelson, 1981, 1984; Macken and Perelson, 1985; Lauffenburger and Linderman, 1993; Sulzer and Perelson, 1996).

Cooperativity

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_n1 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., 1992; Ficner and Huber, 1993) indicates that this molecule is cylindrical with a height of 6 nm and a diameter of 11 nm. 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., 1973; Padlan, 1994).

To account for functional nonequivalence of sites on DNP₂₅-PE, we introduce a cooperativity function (i) to relate the cross-linking equilibrium constants K₁ through K_n1 (Cantor and Schimmel, 1980):

(2)

in which (1) = 0. The functional form of (i) is chosen to indicate how cooperative effects are expected to influence binding as a function of ligand site occupancy. If (i) = 0 for all i, sites are functionally equivalent and K₁ through K_n1 are identical. However, for steric reasons, we can expect the value of the cross-linking equilibrium constant K_i to approach zero as i approaches the effective valence of DNP₂₅-PE, which we expect to be much less than the chemical valence n. This suggests that we should specify (i) as an increasing function of i. The simplest such form for (i), with the required property that (1) = 0, is a(i  1), in which a is a positive constant.

Conservation

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

(3)

in which L_T is the total concentration of ligand and C is a factor that converts surface densities to concentrations. This factor is the cell concentration, expressed in the same units as L_T, if each surface density L_i is expressed in units of ligand molecules per cell. Conservation of receptor sites can be expressed as

(4)

in which S_T is the total surface density of receptor sites, which is twice the total surface density of receptors.

Cross-linking

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 Fig. 1 A, a ligand bound at i sites is attached to i receptors. Thus, a ligand bound at two sites, as depicted in Fig. 1 B, 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 _i=1ⁿ(i  1)L_i, which is equivalent to the number of bound receptor sites (_i=1ⁿiL_i) in excess of the number of bound ligands (_i=1ⁿL_i).

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MODEL MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Reagents

Mouse monoclonal anti-DNP IgE was isolated from hybridoma H1 26.82 (Liu et al., 1980) 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., 1986). 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.

Cells

RBL-2H3 cells (Barsumian et al., 1981) were grown adherent in 75 cm² 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 135 mM NaCl, 5 mM KCL, 1 mM MgCl₂, 1.8 mM CaCl₂, 5.6 mM glucose, 0.1% gelatin, and 20 mM Hepes. Cell suspensions in buffered salt solution were supplemented with 10 mM sodium azide and 10 mM 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 (10 µg) anti-DNP FITC-IgE. Sensitized cells were exposed to FITC-IgE for at least 12 h prior to harvesting.

Flow cytometric binding assays

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 100 µg/ml. After cells were incubated with DNP₂₅-PE for at least 90 min, 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 90 min 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 (520 nm) of the cell suspension, FL1, and as the mean PE fluorescence (550 nm) 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, FL2, indicates the PE fluorescence due only to specific binding. To relate PE fluorescence to the surface density of DNP₂₅-PE, measurements of FL2 were calibrated by using microspheres embedded with a known amount of B-phycoerythrin (Flow Cytometry Standards Corporation, San Juan, PR).

Data analysis

Relating measurements of fluorescence to binding

(5)

(6)

(7)

Estimating parameters by fitting the model to data

Solving the model equations

n1

(8)

(9)

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

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 FcRI 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-FcRI complexes. The methods developed here allow us to determine how equilibrium cross-linking varies quantitatively with the total concentrations of ligand and receptor.

Qualitative features of antigen binding to surface IgE

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 (FL2) increases. A decrease in FITC fluorescence indicates an increase in the occupancy of IgE combining sites (Erickson et al., 1986), and an increase in PE fluorescence indicates an increase in association of DNP₂₅-PE with the cell surface (Seagrave et al., 1987).

View larger version (26K): [in this window] [in a new window]   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.

Receptor binding saturates before ligand binding

Ligand binding approaches saturation

Quantifying antigen-induced aggregation of surface IgE sites

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. 5-7 (FL1_min, FL1_max, and FL2_max) and then use these equations to relate measurements of fluorescence (FL1 and FL2) to quantities that characterize ligand and receptor binding (, , and ).

Fluorescence measurements directly indicate a lower bound on cross-linking

View larger version (18K): [in this window] [in a new window]   FIGURE 3   Equilibrium binding characterized by measurements of PE and FITC fluorescence. Fluorescence measurements directly indicate (A) the fraction of receptor sites bound , (B) an upper bound on the number of bound ligands per receptor site , and (C) a lower bound on the number of cross-links per receptor site . The points were derived from the fluorescence data in Fig. 2 by using Eqs. 5-7 and the scaling factors: FL1min = 184, the lowest recorded value of FL1, FL1max = 450, the highest recorded value of FL1, and FL2max = 1380, the highest recorded value of FL2. The error bars in B and C indicate the uncertainty in  and  if the highest recorded value of FL2 (1380) is within 75% of the actual FL2max, as suggested by bead calibration. The small solid points at the end of error bars are based on FL2max = 1380/0.75. The cell density is 106 cells/ml.

Model-based analysis of fluorescence data yields refined estimates of receptor site occupancy, ligand binding, and cross-linking

1

n1

View larger version (26K): [in this window] [in a new window]   FIGURE 4   Comparison of best-fit theoretical binding curves with scaled data. The fraction of bound receptor sites  (circles) and number of bound ligands per receptor site  (squares) are plotted as a function of ligand concentration. The points were derived from the fluorescence data in Fig. 2 by using Eqs. 5 and 6, FL1min = 184, FL1max = 450, and FL2max = 1500. The broken and solid lines indicate the theoretical binding curves, which are based on the model equations (Eqs. 1-4), Eqs. 5 and 6, n = 25, ST = 9.6 × 105 sites/cell, K0 = 4.4 × 108 M1, K1ST = 13, and (i) = 0.43 (i  1). The cell density is 106 cells/ml.

1

1

View larger version (32K): [in this window] [in a new window]   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  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. 5-7, FL1min = 184, FL1max = 450, and FL2max = 1500. The lines indicate the theoretical binding curves, which are based on the model equations (Eqs. 1-4), Eq. 7, n = 25, ST = 9.6 × 105 sites/cell, K0 = 4.4 × 108 M1, K1ST = 13, and (i) = 0.43 (i  1).

Dependence of cross-linking on ligand and receptor concentration

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

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 FcRI 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 (Figs. 3 A and 4), ligand binding (Figs. 3 B and 4), and cross-linking (Figs. 3 C and 5). This study, like the recent work of Xu et al. (1998), represents an attempt to quantify the interactions of a multivalent antigen with IgE-FcRI.

The method that we have developed to quantify FcRI 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., 1987). This is in contrast to bivalent haptens (Siraganian et al., 1975; Kane et al., 1986; Posner et al., 1995a). Also like an allergen, DNP₂₅-PE cross-links FcRI via antigen-antibody reactions. This is in contrast to oligomers of IgE; the kinetics of IgE binding to FcRI 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., 1995). 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., 1998). Kinetic binding studies may be important for understanding the temporal properties of FcRI aggregates that influence signaling (MacGlashan et al., 1985; Torigoe et al., 1998).

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 FcRI aggregation involves oligomers of IgE (Segal et al., 1977). 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 FcRI receptors. Because dissociation of IgE from FcRI 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 FcRI 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. (1995) 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 0.001 s¹). 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. (1995) is then required to measure aggregation.

Another approach used to study FcRI 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., 1995b). 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., 1986; Schweitzer-Stenner et al., 1987; Erickson et al., 1991; Posner et al., 1991; Posner et al., 1995a). Cyclic dimers, in which two ligand molecules connect two IgE-FcRI 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 (Figs. 4 and 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 FL2 measurements. We obtain different results if we fit only FL1 data, only FL2 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., 1998).

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 (Figs. 3 C and 5). 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.