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Protein PEGylation Decreases Observed Target Association Rates via a Dual Blocking Mechanism

Department of Biochemistry (S.K., C.A.S., A.P.), and Department of Oncology (S.K.), University Hospital, University of Zürich, Zürich, Switzerland

Received May 18, 2005; accepted August 11, 2005

	Abstract

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PEGylation is an attractive strategy to enhance the therapeutic efficacy of proteins with a short serum half-life. It can be used to extend the serum persistence and to reduce the immunogenicity of proteins. However, PEGylation can also lead to a decrease in the functional activity of the molecule to which it is applied. We constructed site-specifically PEGylated variants of anti-p185^HER-2 antibody fragments in the format of a monovalent single-chain variable fragment and a divalent miniantibody and characterized the antigen binding properties in detail. Mass-transport limited BIAcore measurements and binding assays on HER-2-overexpressing cells demonstrated that the immunoreactivity of the antibody fragments is fully maintained after PEGylation. Nevertheless, we found that the attachment of a 20-kDa polyethylene glycol (PEG) moiety led to a reduction in apparent affinity of approximately 5-fold, although in both formats, the attachment site was most distal to the antigen binding regions. This decrease in affinity was observed in kinetic BIAcore measurements as well as in equilibrium binding assays on whole cells. By analysis of the binding kinetics, we could pinpoint this reduction exclusively to slower apparent on rates. Through both experimental and computational analyses, we demonstrate that these reduced on-rates do not arise from diffusion limitations. We show that a mathematical model accounting for both intramolecular and intermolecular blocking mechanisms of the PEG moiety can robustly explain the observed binding kinetics. The results suggest that PEGylation can significantly alter the binding-competent fraction of both ligands and receptors and may help to explain some of the beneficial effects of PEGylation in vivo.

However, reduction or even loss of functional activity can be an unintended side effect of PEGylation if the polymer strand sterically hinders the binding of the conjugate to the target. Many PEGylated antibody fragments of the first generation encountered this problem, because at that time the PEG conjugation was performed via random attachment, most commonly through lysine residues (Chapman, 2002; Weir et al., 2002). Thereafter, site-specific PEGylation techniques were developed (Harris and Chess, 2003; King et al., 1994), in which the PEG molecule is attached to the protein at a single unpaired cysteine residue that can be engineered at a position distal to the target-binding region of the protein. Successful applications of this technique have been reported (Chapman et al., 1999; Lee et al., 1999; Weir et al., 2002), showing that PEG tails from 2 kDa up to 40 kDa can be coupled to antibody fragments (scFv or Fab). However, conflicting conclusions about changes in affinity have been reached (see below).

Because of its great potential for increasing serum half-life and decreasing immunogenicity, we chose site-specific PEGylation as a strategy to improve the pharmacokinetic behavior of antibody fragments that were generated for use in tumor targeting (S. Kubetzko, E. Balic, R. Waibel, U. Zangemeister-Wittke, and A. Plückthun, manuscript in preparation). We constructed PEGylated versions of the monovalent scFv 4D5 and the bivalent miniantibody 4D5-dhlx, which consists of the scFv 4D5 fused via a hinge region to the self-associating dimerization peptide dhlx (for review, see Willuda et al., 2001) (Fig. 1). These antibody fragments were derived from the humanized antibody 4D5 (Carter et al., 1992; for review, see Willuda et al., 2001), which binds specifically and with high affinity to the extracellular domain of p185^HER-2, a transmembrane glycoprotein that is overexpressed in 25 to 30% of breast and ovarian carcinomas (Slamon et al., 1989).

View larger version (18K): [in this window] [in a new window]   Fig. 1. Schematic representation of the molecular set up of the PEGylated antibody formats. A, PEGylated monomeric miniantibody 4D5-PEG20. The scFv 4D5, which is in the VL-linker-VH orientation, was site-specifically conjugated with a 20-kDa maleimide-PEG moiety (PEG20) at a single engineered cysteine residue, placed at the C terminus of the scFv fragment. Coupling was obtained by formation of a thioether bond between the free cysteine and the maleimide residue. B, PEGylated dimeric miniantibody 4D5-dhlx-PEG20. The antibody fragment 4D5-dhlx consists of the scFv 4D5 and the synthetic dhlx peptide (Hill and deGrado, 1998), which is C-terminally fused to the scFv via a hinge peptide, forms an antiparallel helix-turn-helix motif and mediates dimerization by self-association. For PEGylation, the construct carries a C-terminal cysteine and the same strategy was used as for the monomeric construct, resulting in a dimeric miniantibody with two PEG molecules attached. C, gene constructs of the modified 4D5 miniantibodies. For periplasmic expression in E. coli, the lac promoter and the ompA signal-peptide sequence were used. The monomeric construct (I) starts with an N-terminal short FLAG tag (F), followed by the scFv 4D5, a myc tag, and a His6 tag. The construct terminates in a single cysteine residue, separated by a short glycine (Gly2) linker from the His6 tag. Instead of the myc tag, the gene construct of the dimeric antibody fragment (II) contains the dimerization domain dhlx, flanked by a murine IgG3 hinge and a GGSGGAP spacer sequence (Willuda et al., 2001). Here, the C-terminal cysteine is separated from the His6 tag by four glycine residues.

The conflicting views in the literature about the effect of PEGylation on binding affinity prompted us to investigate this effect in more detail. We assessed the antigen-binding properties of the different constructs on tumor cells as well as by kinetic BIAcore measurements. To be able to interpret the impact of PEG on the association and dissociation rates independently, a special effort was undertaken to determine the percentage of active molecules. Furthermore, we analyzed and compared hydrodynamic parameters (size and diffusion coefficient) of the PEGylated and the unmodified antibody fragments. This allowed us to differentiate the effects of slower diffusion from those of steric hindrance by the long mobile PEG moiety, leading to a lower percentage of successful collisions that result in binding. All together, the results of this study should help to clarify the molecular factors responsible for the change in apparent affinity upon PEGylation and their respective contributions.

	Materials and Methods

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Tumor Cell Line and Recombinant Antigen. For the binding experiments, we used the ovarian carcinoma cell line SK-OV-3 (HTB 77, ECACC, Salisbury, Wilts, UK). Culturing and harvesting of the cells was performed as described previously (Willuda et al., 2001). The purified recombinant antigen p185^HER-2-ECD was a kind gift from Genentech Inc. (South San Francisco, CA).

Design, Expression, and Purification of the Constructs. The construction of the cysteine mutants, the expression of the monovalent scFv and the divalent miniantibody, and the purification of these proteins are described in the Supplemental Materials.

PEGylation of the 4D5 Miniantibodies. Purified protein samples of the monomeric (scFv 4D5-Cys) and dimeric (4D5-dhlx-Cys) miniantibodies were concentrated to approximately 0.3 to 1 mg/ml by centrifugation at 2000g and 4°C, using a 10-kDa cutoff microconcentrator (Ultrafree-MC low protein binding; Millipore, Billerica, MA). To enable site-specific PEGylation, the C-terminal cysteine residue was selectively reduced before incubation with maleimide-PEG20 (Nektar, Huntsville, AL). The reducing conditions had to be mild to prevent breakage of the internal disulfide bonds in the V_L and V_H domains. Therefore, the concentrated protein solution was incubated with 3 mM dithiothreitol (final concentration) for 30 min at 37°C. The excess reducing agent was then immediately removed by desalting over a Sephadex G-25 column (PD-10 or NAP-5; GE Healthcare, Little Chalfont, Buckinghamshire, UK). In this step, the buffer was also exchanged to the one used in the PEGylation reaction (100 mM citric acid, 100 mM NH₄Ac, 2 mM EDTA, pH 6.0, filtered, degassed and saturated with N₂). The PEGylation reaction was then carried out by addition of maleimide-PEG20 in 5- to 10-fold molar excess over freshly reduced protein, followed by incubation at 37°C for 2 h. The efficiency of PEGylation was analyzed by SDS-PAGE (12% gel, Coomassie-staining) and size exclusion chromatography on a Superdex 200 column with an ÄKTAexplorer system (GE Healthcare, Little Chalfont, Buckinghamshire, UK). The same chromatography system was used to purify the PEGylated proteins from both the native antibody fragments and the unreacted free PEG.

Size Exclusion Chromatography. Analytical gel filtration analysis of the unmodified and PEGylated antibody fragments was performed with an ÄKTAexplorer chromatography system at 4°C and a flow rate of 0.5 ml/min, using a Superdex-200 column (24-ml bed volume). The column was equilibrated with filtered and degassed phosphate-buffered saline (PBS), containing 1 M NaCl and 0.05% Tween 20. For calibration, five protein standards were used: -amylase (200 kDa), alcohol dehydrogenase (150 kDa), bovine serum albumin (BSA; 66 kDa), carbonic anhydrase (29 kDa), and cytochrome c (12.5 kDa). Samples of the different constructs were injected at a concentration between 500 µg/ml and 1.2 mg/ml in a volume of 100 µl. The absorption was recorded at 280, 260, and 230 nm. If a low PEGylation yield (less than 80%) or impurities were detected, preparative size exclusion chromatography under the same conditions was used as an additional purification step.

Static Light Scattering Analysis. The molar masses of the different 4D5-miniantibodies were determined by multiangle static light scattering analysis and compared with the theoretically expected values. We used the tri-angle (45°, 90°, 135°) light scattering detector miniDAWN (Wyatt Technology Corporation, Santa Barbara, CA) in combination with the interferometric refractometer OPTILAB (Wyatt Technology Corporation), which were serially connected between the UV and conductivity detectors of the ÄKTAexplorer chromatography system. Thus, it was possible to perform size exclusion chromatography and determine the molar mass of every single protein peak online. The same conditions as for the gel filtration analysis were used, including filtration and degassing of the buffers, flow rate (0.5 ml/min), injection volume (100 µl) and concentration of the injected protein samples (0.5–1.2 mg/ml, sterile-filtered). Before the measurements were started, the detector system of the miniDAWN was equilibrated with the running buffer (PBS, containing 1 M NaCl and 0.05% Tween 20) for at least 2 h to ensure stable baseline signals. During the measurements, the laser scattering (690 nm), the UV absorption (280 nm), and the refractive index (690 nm) of the protein solutions were recorded. Data were evaluated with the Wyatt software ASTRA. For calculations of the molar masses, we set the refractive index of the buffer solution to 1.33, and the refractive index increment (dn/dc) of the proteins either to 0.166 ml/g for the PEGylated miniantibodies or to 0.185 ml/g for the unPEGylated ones.

Dynamic Light Scattering Analysis. To determine the diffusion coefficients and the hydrodynamic sizes of the PEGylated and unPEGylated miniantibodies, dynamic light scattering (DLS) analyses were performed. We used the one-angle (90°, laser wavelength 826 nm) DLS-instrument DynaPro (Viscotek, Houston, TX) and protein concentrations between 650 µg/ml and 1.3 mg/ml in PBS. Before measurements were started, the detector was equilibrated with ultra high-purity water and subsequently with PBS to ensure that the background scatter, caused by the solvent, was steady (fluctuation rate <10%) and at a low level. All solutions, including the analyzed protein samples, were filtered through a 0.1-µm filter (Whatman, Clifton, NJ) upon injection. The measurements were carried out in a 10-µl quartz cuvette at 20°C, following a schedule of 10 acquisition points in 10 min and repeated three times per analysis. Parameters for the data collection were set as follows: 10-s maximal acquisition time, a sensitivity of 100% avalanche photodiode bias (maximal intensity, 1.5 x 10⁶ photon counts/s) and signal-to-noise threshold ratio of 1. Data were evaluated with the software DYNAMICS version 4.0 (Viscotek), using a monomodal size distribution model.

Radioimmunoassay on Human SK-OV-3 Tumor Cells. The apparent affinities of the various 4D5-miniantibodies to the p185^HER-2 overexpressing tumor cells SK-OV-3 were determined by radioimmunoassays, which were carried out in essentially the same manner as described previously (Willuda et al., 2001), with the following modifications. Stock solutions of the ^99mTc(CO)₃-labeled antibody constructs were prepared at 10 different concentrations by 2-fold serial dilution, and a 20-µl aliquot of each of these solutions was incubated with 100 µl of an SK-OV-3 cell suspension (corresponding to 5 x 10⁵ cells in PBS, containing 0.5% BSA and 0.005% Tween 20) for 1 h at 4°C on a shaker. The final concentrations of active radiolabeled miniantibodies (Lindmo et al., 1984) were between 0.5 nM and 1 µM. All measurements were performed in triplicate.

Analysis of Binding Kinetics by BIAcore Measurements. The binding kinetics of the different 4D5-miniantibody formats were analyzed and compared by surface plasmon resonance measurements, using a BIAcore 3000 instrument (BIAcore AB, Uppsala, Sweden). A CM5-Sepharose chip was coated by standard amine coupling chemistry (Johnsson et al., 1991) with the recombinant extracellular domain (ECD) of the antigen p185^HER-2 to a density of 400 RU. This rather low coating density was chosen to minimize mass transfer and rebinding effects. Measurements were carried out at 25°C, using a flow rate of 30 µl/min with an association phase of 3 min after injection, followed by dissociation for 10 min. The miniantibodies were diluted in HBS-EP running buffer [10 mM HEPES, pH 7.4, 150 mm NaCl, 3 mM EDTA, 0.005% surfactant P20 (polyoxyethylene sorbitan), filtered and degassed] and injected at concentrations between 1 and 100 nM. For subtraction of bulk effects, caused by changes in the buffer composition or nonspecific binding, we performed double-referencing (Myszka, 1999). Therefore, all analyzed samples were additionally injected onto an uncoated reference surface, including a sample of the running buffer, which was also tested on the HER-2-coated flow cell. Data were evaluated with the BIAevaluation software (version 3.0), applying a simple 1:1 binding model. The obtained sensorgrams were fitted globally over the whole range of injected concentrations and simultaneously over the association and dissociation phase. Equilibrium dissociation constants were then calculated from the rate constants (K_D,obs = k_off/k_on).

Evaluation of the Immunoreactive Fraction on Cells. The percentage of immunoreactive molecules was determined by equilibrium binding assays on SK-OV-3 tumor cells, performed essentially as described by Lindmo et al. (1984). Triplicate samples with increasing numbers of cells (0.25 to 5 x 10⁶ cells in 100 µl of PBS containing 0.5% BSA) were mixed with constant amounts of ^99mTc(CO)₃-labeled miniantibodies (in 20 µl of PBS containing 0.5% BSA and 0.005% Tween 20). The final concentration of miniantibody molecules in these cell-suspensions was approximately 20 nM. The samples were incubated for 1 h at 4°C on a shaker. Then, cells were washed three times with PBS containing 0.5% BSA and 0.005% Tween 20, and the bound radioactivity in the cell pellets was determined by -scintillation counting. The obtained data were fit using a 1:1 binding model accounting for ligand depletion (see eq. A12 in Supplemental Materials with = 0 and = 0).

Comparison of the Association Behavior on Ni-NTA- or HER-2-Coated Surfaces. To assess the proportion of active molecules in the samples of the various miniantibody formats, we compared their association behavior on two differently coated chip surfaces in parallel by BIAcore measurements. The first was a CM5 chip, coated with the ECD of the antigen p185^HER-2 to a high density of 3700 RU; the second was an NTA chip saturated with Ni²⁺ ions. We used a slow flow rate of 5 µl/min, low analyte concentrations between 1 and 10 nM, and a short injection time of 2 min. These conditions should provide a huge excess of coated antigen over injected analyte and an association phase in which the binding of the antibody fragments on the HER-2-coated chip is mass-transport-limited and thus proportional to the amount of active molecules entering the flow cell. The Ni-NTA surface was used as a "reference cell" to determine the RU signals, according to the total amount of injected miniantibody molecules. On this surface, the constructs should be able to bind via their C-terminal His tag, whether they are denatured or in functional conformation. All measurements were performed at 25°C, using a running buffer composed of 10 mM HEPES, 150 mM NaCl, 0.005% surfactant P20, and 3 mM EDTA (HER-2 chip) or 0.05 mM EDTA (Ni-NTA chip). To evaluate the percentage of active molecules, the slopes of the sensorgrams as well as the absolute increases in response units during analyte injection determined on the HER-2 coated chip, were compared with the corresponding ones on the Ni-NTA chip.

Evaluation of the Concentration of Functional Molecules in BIAcore Measurements by Varying the Flow Rate. The concentration of functional molecules was evaluated by BIAcore analysis of the binding kinetics under partial mass transport limitation. Samples of the PEGylated and unmodified scFv 4D5 were injected at a protein concentration of 5 nM (determined by spectrophotometric measurements at 280 nm) for 2 min on a CM5 chip, coated with the ECD of p185^HER-2 to a high density of 3700 RU. Measurements were carried out at 25°C in 0.01 M HEPES, pH 7.4, 0.15 M NaCl, 3 mM EDTA, and 0.005% surfactant P20, using six different flow rates: 5, 10, 25, 50, 75, and 100 µl/min. The obtained sensorgrams were processed (subtraction of bulk effects) with the BIAevaluation software (version 3.0) and then exported into ClampXP (http://www.cores.utah.edu/interaction/clamp.html) to assess the concentration of active molecules. The sensorgrams were fitted with a 1:1 binding model under mass transport limitation:

(1)

(2)

Here, L_bulk is the analyte in the bulk phase, L_surface is the analyte near the chip surface containing antigen linked to the dextran matrix, k₊ is the mass transport coefficient, R is the immobilized antigen (the receptor HER-2, in this case), C is the analyte-antigen complex, k_a is the intrinsic association rate constant, and k_d is the intrinsic dissociation rate constant. When the antibody fragments are injected into the flow cell, they first have to diffuse from the bulk phase to the chip surface (eq. 1), where the antigens are immobilized and the chemical binding reaction takes place. The correlation between the diffusion properties of the analyte in the sample solution and the mass transport rate is given by

(3)

where k₊ is the transport rate constant (in seconds^-1), k_t is the transport velocity (in meters per second), h_b is the height of the diffusive boundary layer at the chip surface, D_t is the diffusion coefficient; f is the volumetric flow rate; and h, w, and l are the cell dimensions (height, width, and length, respectively) (BIAsimulation Software) (Christensen, 1997; Myszka et al., 1998). Fitted parameters were the intrinsic association rate constant (k_a), the intrinsic dissociation rate constant (k_d), the transport rate constant between the bulk and the surface (k₊), and the analyte concentration L_bulk. We used the diffusion coefficients that were independently determined by DLS. As starting points for the on- and off-rate fits, we used the k_a and k_d values that had been obtained in BIAcore measurements with a constant flow rate of 30 µl/min.

	Results

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In the present study, the binding properties of PEGylated monomeric scFvs and dimeric miniantibodies were characterized. To explore the effects of PEGylation on apparent affinity and related parameters, we pursued the following strategy. The PEGylation site of the antibody fragments was designed at a position as far away from the antigen binding region as possible. To ensure that the attachment of the PEG moiety did not result in a loss of functional protein, the percentage of active molecules was determined independently on tumor cells as well as in mass-transport-limited BIAcore assays. To understand the molecular basis of any effects of PEGylation on the binding properties of the antibody fragments, we analyzed the equilibrium dissociation constants, determined the on and off rates by kinetic BIAcore measurements, and evaluated the hydrodynamic sizes and diffusion coefficients by gel filtration and light scattering analyses.

Construction, Expression, and Purification of PEGylated scFv Fragments. We constructed site-specifically PEGylated variants of the scFv 4D5 and the miniantibody 4D5-dhlx (Willuda et al., 2001). The attachment site of the PEG moiety was placed in both constructs at the C terminus by introducing a single unpaired cysteine residue, separated by a glycine linker from the C-terminal His₆ tag. 4D5-dhlx contains the synthetic helix-turn-helix peptide dhlx (Hill and deGrado, 1998), which causes spontaneous dimerization of the fused proteins by self-association via hydrophobic interactions (for review, see Willuda et al., 2001). Thus, a monovalent scFv with one PEG molecule attached and a bivalent miniantibody with two PEG entities were generated (Fig. 1).

The monomeric and dimeric antibody fragments, with and without the additional cysteine residue, were all expressed in the periplasm of Escherichia coli SB536 and purified by two sequential affinity chromatography steps (see Materials and Methods and Supplemental Materials). We determined, by SDS-PAGE analysis, the purity of these proteins to be greater than 90%. For the monomeric scFv fragments and the unmodified miniantibody, we routinely obtained 2 to 3 mg/l (E. coli culture in shake flasks). The dimeric miniantibody with the free thiol group at the C terminus, however, yielded only approximately 500 µg/l. This reduction in yield of periplasmic proteins upon insertion of free thiols is not unexpected because of the interference with disulfide bond formation.

To prepare the antibody fragments for the PEGylation reaction, we concentrated them to approximately 0.3 to 1 mg/ml and reduced the C-terminal cysteines under mild conditions (see Materials and Methods) to prevent breakage of the internal disulfide bonds in the V_L and V_H domains. After removal of the reducing agent, the 20-kDa PEG polymer, containing a maleimide coupling group, was site-specifically attached to the C terminus of each antibody fragment. The conjugation yield was approximately 80 to 90%, as determined by SDS-PAGE and size exclusion chromatography (Fig. 2A). The retention of the internal disulfide bonds was verified by subjecting the unmodified scFv 4D5 to the same PEGylation procedure as described above. In this case, no attachment of the PEG polymer to the protein could be detected.

View larger version (38K): [in this window] [in a new window]   Fig. 2. Analysis of the size and format of the PEGylated and unPEGylated antibody fragments. The apparent hydrodynamic sizes and the actual molecular masses of the scFv 4D5 (theoretical, 29 kDa), the dimeric miniantibody 4D5-dhlx-SS (theoretical, 66 kDa), the PEGylated scFv 4D5-PEG20 (theoretical, 50 kDa), and the PEGylated dimeric miniantibody 4D5-dhlx-PEG20 (theoretical, 106 kDa) were investigated by gel filtration analysis (A) and static light scattering analysis (B). A, gel filtration analysis was carried out on an ÄKTAexplorer system with a Superdex-200 column (24-ml bed-volume). For calibration, the following molecular mass standards were used and their elution volumes are shown with vertical dashed lines: -amylase (200 kDa), alcohol dehydrogenase (150 kDa), bovine serum albumin (66 kDa), carbonic anhydrase (29 kDa), and cytochrome c (12.5 kDa). The elution volumes of the antibody fragments were 17.0 ml (scFv 4D5), 15.09 ml (4D5-dhlx-SS), 13.03 ml (4D5-p53-SS), 11.88 ml (4D5-PEG20), and 10.15 ml (4D5-dhlx-PEG20). B, dynamic light scattering analysis was performed with the tri-angle light scattering detector miniDAWN (Wyatt) in combination with the interferometric refractometer OPTILAB (Wyatt), which were both serially connected to the ÄKTAexplorer size exclusion chromatography system. We assessed molecular masses of 29.7 kDa for the scFv 4D5, 63.5 kDa for the dimeric miniantibody 4D5-dhlx-SS, and 61 kDa for the PEGylated scFv 4D5-PEG20. The molecular mass of the PEGylated dimeric miniantibody 4D5-dhlx-PEG20 could not be determined.

Static light scattering, which was performed online during gel filtration runs, confirmed the calculated molecular weights of the different constructs, rather than the apparent hydrodynamic sizes. Thus, it could be shown that the desired molecular species had indeed been prepared. We determined a size of 29.7 kDa for the scFv 4D5 (predicted, 29.2 kDa), 63.5 kDa for the dimer 4D5-dhlx-SS (predicted, 66.5 kDa), and 61 kDa for the PEGylated monomer 4D5-PEG20 (predicted, 50 kDa) (Fig. 2B). Only the mass of the PEGylated dimer 4D5-dhlx-PEG20 could not be determined reliably, because it eluted at a volume close to the exclusion volume of the Superdex 200 column, where it overlapped with a scatter peak caused by abrasion of the injection valve.

Determination of Diffusion Coefficients by Dynamic Light Scattering Analysis. Size exclusion chromatography analysis indicated that the 20-kDa PEG tail has a strong effect on the hydrodynamic volume of the attached proteins. To verify this conclusion and exclude any interference of the column material, we examined and compared the translational diffusion of the PEGylated and unmodified antibody fragments in solution by DLS. Measurements were performed with protein concentrations between 650 µg/ml and 1.3 mg/ml. For data evaluation, we used a monomodal size distribution model and, for each set of data, determined a translational diffusion coefficient representing the mean fraction of scatter and mass percentage.

We assessed a diffusion coefficient (D_t) of 8.4 x 10^-7 cm²/s for the monomeric scFv 4D5 and 6.0 x 10^-7 cm²/s for the dimer 4D5-dhlx. For each PEGylated construct, two mean scatter peaks were detected, corresponding to D_t values of 3.1 and 4.4 x 10^-7 cm²/s (Table 1). For the PEGylated scFv 4D5-PEG20, 70% of the data corresponded to a D_t value of 4.4 x 10^-7 cm²/s, whereas for the PEGylated dimer 4D5-dhlx-PEG20 only 35% corresponded to this value. The majority of the data (50%) for this PEGylated dimer corresponded to a D_t value of 3.1 x 10^-7 cm²/s. Comparing the PEGylated species with their unmodified counterparts, it seems that PEGylation decreased the diffusion coefficient of the antibody fragments by approximately 2-fold. Based on these values, we calculated the apparent molecular weights of the constructs when treated as globular proteins. For the unconjugated scFv fragments, we determined sizes of 27 kDa (monomeric scFv 4D5) and 61 kDa (dimeric miniantibody 4D5-dhlx), which are consistent with the predicted molecular weights. The diffusion coefficients of these PEGylated constructs, however, correspond to sizes of 133 and 309 kDa, respectively. These values are clearly above their actual molecular mass and completely consistent with the findings of the gel filtration analysis. Most important for the evaluation of the binding properties was the observation that PEGylation did decrease the diffusion coefficients of the antibody fragments, as expected, but only by approximately 2-fold.

Comparison of Binding Kinetics by Surface Plasmon Resonance. The apparent affinities of the 4D5-derived antibody fragments to their target antigen p185^HER-2 were examined by radioimmunoassays on SK-OV-3 tumor cells and by BIAcore measurements. We found a 5-fold decrease in apparent affinity upon attachment of the 20-kDa PEG moiety for both the monomeric and the dimeric antibody fragment (Table 2). As explained in the subsequent section, we can exclude a difference in the percentage of functional molecules as a possible cause.

To investigate this observation in more detail, we compared the binding kinetics of the different constructs by separate analysis of the association and dissociation rates using surface plasmon resonance (Fig. 3). The antigen was coated on a CM5 chip at a relatively low density of 400 RU, and measurements were performed at a high flow rate of 30 µl/min. This setup was chosen to minimize mass transport effects and rebinding of fully dissociated molecules, which both could compromise the measured kinetics. The determined k_on, k_off, and K_D,obs values are given in Table 3 and reveal that the reduction in functional affinity, caused by PEGylation of the antibody fragments, is due almost exclusively to a slower on rate, whereas the off rate is nearly unchanged. The PEGylated scFv 4D5-PEG20, for example, showed a k_on of 6.1 x 10⁴ M^-1 s^-1, which is approximately 5.5-fold smaller than that of the corresponding scFv 4D5, displaying a k_on of 3.4 x 10⁵ M^-1 s^-1. However, their dissociation rates, determined as 4.9 x 10^-5 s^-1 (4D5-PEG20) and 5.0 x 10^-5 s^-1 (scFv 4D5), are virtually the same. While this work was in progress, similar findings were also reported for other scFv fragments that had been site-specifically conjugated with PEG polymers of different sizes (Yang et al., 2003). When comparing the binding kinetics of these constructs with the corresponding binding kinetics of the unPEGylated scFvs, a modest effect was found if a small PEG molecule of 5 kDa was attached, whereas conjugates modified with a 20- or 40-kDa PEG tail displayed a reduction in on rate of as much as 100-fold. Consistent with our data, the dissociation rates of these conjugates were nearly equivalent to those of the unmodified scFvs.

View larger version (38K): [in this window] [in a new window]   Fig. 3. Comparison of the binding kinetics of the PEGylated and unPEGylated antibody fragments by BIAcore measurements. Association and dissociation kinetics of the monomeric scFv 4D5 (A), the PEGylated scFv 4D5-PEG20 (B), the dimeric miniantibody 4D5-dhlx (C), and the PEGylated dimeric miniantibody 4D5-dhlx-PEG20 (D) were compared by surface plasmon resonance measurements, using a BIAcore 3000 instrument. A CM5-Sepharose chip (BIAcore AB) was coated with p185HER-2-ECD antigen to a density of 400 RU. The constructs were injected at a high flow rate of 30 µl/min, using concentrations between 1 and 100 nM. Associations were monitored for 3 min and dissociations for 10 min. Data were evaluated with the BIAevaluation 3.0 software (BIAcore AB), applying a simple 1:1 binding model and a global fit. An overlay of experimental data and global curve fits is shown.

View this table: [in this window] [in a new window]   TABLE 4 Immunoreactivity of the PEGylated and unPEGylated anti-p185HER-2 antibody fragments on human SK-OV-3 tumor cells The antibody fragments were radioactively labeled with 99mTc(CO)3 and the percentage of active molecules was determined for each construct by equilibrium binding assays on SK-OV-3 cells as described by Lindmo et al. (1984). The data were fit using a 1:1 binding model accounting for ligand depletion (see eq. A12 in Supplemental Materials with = 0 and = 0). Immunoreactive fraction means binding to the p185HER-2 antigen with at least one binding site.

To better understand the molecular mechanism(s) by which the observed association rates of the PEGylated analogs are reduced, we experimentally tested and computationally simulated several possible hypotheses (Fig. 4).

View larger version (28K): [in this window] [in a new window]   Fig. 4. Diagram presenting potential reasons for the decrease in apparent on rates of the PEGylated molecules. There are several factors that could, in principle, lead to reduced on rates upon PEGylation of antibody fragments. A, the presence of permanently inactive molecules (Linactive) would result in a lower percentage of functional molecules (L) that are capable of binding reversibly to immobilized receptor (R) with forward and reverse rate constants ka and kd, respectively. B, a translational or rotational diffusion limitation (resulting from reduced transport rate constant k+,t or k+,r, respectively) would slow the delivery of bulk ligand (L) both to and from the surface. Surface-proximal ligand (Ls) could bind reversibly to R with forward and reverse rate constants ka and kd, respectively. C, the PEG moiety could intramolecularly block the binding region of the antibody, with rate constant k1, to form Lblock. Unblocked ligand (L) could be regenerated from Lblock with unblocking rate constant k-1. L could bind reversibly to R with forward and reverse rate constants ka and kd, respectively. D, L could bind reversibly to R with forward and reverse rate constants ka and kd, respectively. However, in the bound state, the PEG tail could hinder antibody binding to adjacent sites through intermolecular blocking; thus, a single PEGylated antibody would occupy (1 + ) sites on the surface.

We assessed their immunoreactivity on human SK-OV-3 tumor cells by applying the method described by Lindmo et al. (1984), where an increasing number of cells were used to saturate all antibody molecules with antigen. We determined the percentage of active molecules to be approximately 85 to 94% for all constructs, without observing a significant difference caused by PEGylation of the antibody fragments (Table 4).

In addition, we examined the binding reactivity of the 4D5-miniantibodies by surface plasmon resonance measurements on a BIAcore 3000 instrument under mass-transport limitation, using two different approaches. First, we compared their association behavior on two different chips in parallel (Fig. 5). One was a CM5 chip, coated to a high density (3700 RU) with the target antigen, the ECD of HER-2, and the other was a Ni-NTA chip. The HER-2-coated chip surface served as the "measuring cell" to determine the fraction of active molecules in the injected protein samples that is capable of antigen binding. The Ni-NTA chip was used as the "reference cell" to estimate the RU values corresponding to 100% of the molecules present. On this nickel-saturated surface, the antibody fragments should be able to associate via their C-terminal His tags, whether they are denatured or in an active conformation. In all measurements, a slow flow rate (5 µl/min), low analyte concentrations (1–10 nM), and a short injection time (2 min) were used. By employing these conditions we wanted to approach the situation where analyte binding is mass-transport-limited and thus proportional to the concentration of active molecules, resulting in linear association slopes. In accordance with the data of the cell-binding assays, we determined a fraction of 90 to 100% active molecules for all constructs.

View larger version (27K): [in this window] [in a new window]   Fig. 5. Comparison of binding rates on a HER-2-coated surface and a Ni-NTA-coated surface by BIAcore measurements to assess the percentage of active molecules. Samples of the various miniantibody formats were injected onto a CM5 chip, containing HER-2 antigen at a high density (3700 RU) (A), and onto an NTA chip, saturated with Ni2+ ions (B). A slow flow rate of 5 µl/min, low analyte concentrations between 1 and 10 nM, and a short injection time of 2 min were used. The proportion of functional molecules was evaluated by comparison of the association rates on the HER-2-coated chip with the corresponding ones on the Ni-NTA chip. Here, only the sensorgrams of the unmodified scFv 4D5 and the PEGylated monomeric scFv 4D5-PEG20 are shown.

View larger version (25K): [in this window] [in a new window]   Fig. 6. Evaluation of the concentration of functional molecules by BIAcore measurements with varied flow rates. The binding kinetics of the scFv 4D5 (A) and its PEGylated counterpart 4D5-PEG20 (B) were analyzed under partial mass transport limitation. Low concentrated samples (approximately 5 nM) of the constructs were injected onto a CM5 chip densely coated with the antigen HER-2 (3700 RU). Association was followed for 2 min at six different flow rates (5, 10, 25, 50, 75, and 100 µl/min). Sensorgrams were exported into ClampXP (http://www.cores.utah.edu/interaction/clamp.html) and data were evaluated by applying a 1:1 binding model under mass transport limitation (see text, eqs. 1 and 2), setting the analyte concentration as parameter to fit.

(4)

where k₊ is the transport rate constant (in seconds^-1) that depends on the respective diffusion coefficient and the geometry of the system (e.g., eqs. 3 and A8 in Supplemental Materials) (Lauffenburger and Linderman, 1993). Note, however, that the basic mathematical formulation is identical for translational and rotational diffusion, and for the purposes of this analysis, we do not need to consider the explicit form of k₊. The intrinsic association and dissociation rate constants are k_a (in molar^-1 seconds^-1) and k_d (in seconds^-1), respectively. When the concentration of the intermediate L_s is small and this species is short-lived, we can invoke the pseudo–steady-state approximation:

(5)

where h_b is the height of the diffusive boundary layer at the surface, and the transport velocity k_t = h_bk₊ (in meters per second). The rate of change in complexes is given by

(6)

In eq. 6, the first equality gives the rate of change in terms of the intrinsic rate constants and L_s, whereas the second equality gives it in terms of the experimentally measured quantities k_on (in molar^-1 seconds^-1), k_off (in seconds^-1), and L. Plugging the result for L_s from eq. 5 into the first equality in eq. 6 gives

(7)

Direct comparison of the second equality in eq. 6 with eq. 7 reveals:

(8)

In other words, transport limitations that occur when k_t is not far greater than k_aR would reduce the apparent association and dissociation rate constants by the same percentage, thus giving an apparent equilibrium dissociation constant K_D,obs (= k_off/k_on) identical to the intrinsic K_D (= k_d/k_a). This is not consistent with the experimental data, because only the observed association rate constant is reduced, resulting in a 5-fold higher K_D,obs (5-fold lower affinity). In addition, the BIAcore data for each PEGylated molecule were fit with both a simple 1:1 binding model and the model accounting for mass-transport limitations (see Materials and Methods), and both models gave the same values for k_on and k_off for a given molecule. This suggests that, in our BIAcore experimental setup, k_t >> k_aR; therefore, we can rule out the effect of any diffusion limitations in our kinetic measurements.

For translational diffusion limitations on cells, it can be shown that k_t = D_t/r_c (Smoluchowski, 1917), where D_t is the translational diffusion coefficient and r_c is the radius of the cell (see Supplemental Materials). In this case, we can see from eq. 8 that

(9)

where Da is the Damköhler number, quantifying the ratio of the reaction (binding) velocity to the transport (diffusion) velocity. When diffusion is very fast compared with binding, then Da << 1 and the intrinsic kinetics are observed experimentally. The particular form of this solution is simply a specialized case of the general result described above (eq. 8). Thus, the observed K_D on cells should also be unaffected by diffusion limitations. Because the PEGylated species have lower equilibrium affinities than their unPEGylated counterparts (Table 3), this suggests that mechanisms other than slower diffusion contribute to the binding of the PEGylated molecules.

This model with spherical cells was used to simulate the binding kinetics that may be observed on cells expressing 20,000 and 2,000,000 receptors (Fig. 7). For HER-2, these values roughly correspond to the expression levels seen in normal breast tissue and in breast cancer cells, respectively. On normal cells (Fig. 7, A and B), diffusion limitations in observed binding kinetics become significant only at very high k_a values (>10⁷ M^-1 s^-1) for ligands with D_t values similar to those in the current study (10^-7 to 10^-6 cm²/s; Table 1). However, various experimental studies summarized by Northrup and Erickson (1992) (and further analyzed computationally by these authors) suggest that the intrinsic association rate constant for protein-protein interactions in normal salt conditions does not normally exceed 5 x 10⁶ M^-1 s^-1. On the other hand, in tumor cells with high receptor numbers, k_aR increases Da significantly, and thus the observed kinetics are slowed compared with intrinsic kinetics (Fig. 7, C and D). These numbers particularly apply to ligands with properties similar to those of the dimer and dimer-PEG species in the current study. Thus, in the absence of rapid internalization or degradation processes, the ligands are predicted to have longer mean residence times on the surfaces of cancer cells than on those of normal cells, which may be a desirable effect for sustained, localized delivery of radionuclides or other agents to cancer cells. The decrease in the observed association rate constant compared with the intrinsic one can be rationalized as an increase in competition, under slow delivery of ligand, for binding of each receptor by neighboring receptors; likewise, the decrease in the observed dissociation rate constant can be linked to greater rebinding effects at higher receptor densities relative to diffusion of ligand away from the surface.

View larger version (35K): [in this window] [in a new window]   Fig. 7. Mathematical model of the influence of translational diffusion coefficient, intrinsic association rate constant, and receptor overexpression on observed binding kinetics. A, observed association rate constant (kon) as a function of translational diffusion coefficient (Dt) and intrinsic association rate constant (ka) at a receptor expression level of 20,000 per cell, which, for HER-2, is comparable with that seen in normal human breast tissue. For units of moles/area, , where rc is the cell radius (10 µm) and NA is Avogadro's number. The values of ka in the simulation are multiples of 4 from 5 x 104 M-1 s-1 to 5.12 x 107 M-1 s-1. B, observable rate constants [as fraction of intrinsic (maximal) kinetic rate constants (kon/ka and koff/kd)] as a function of Dt and kon. The Damköhler number (Da), equal to kaR/(Dt /rc), gives the ratio of the binding velocity to the diffusion velocity. C and D, similar to A and B, respectively. Here, however, the receptor number is 2,000,000 per cell, which, for HER-2, is comparable with the overexpression level found in human breast cancer cells. Symbols represent the predicted behaviors of the scFv (), scFv-PEG (), dimer (), and dimer-PEG () from this study, based on their experimentally measured Dt values.

Effect of Intramolecular and Intermolecular Blocking on Observed Binding Kinetics and Equilibrium Affinities. We finally explore the hypotheses depicted in Fig. 4, C and D. Although the PEG moiety was chemically linked to a position as far as possible from the antigen binding site, it is still possible that the flexible polymer sterically blocks the binding interface. Based on the hydrodynamic radius of the monomeric scFv (see Table 1), its half-circumference is 7.8 nm. The Flory radius (R_F aN^0.6) of a PEG molecule with a molecular mass of 20 kDa (a = 0.35 nm, length of a monomer; N 450 units) is 14 nm in aqueous solution; in fact, previous work with PEG tethers suggests that the average end position may lie even further (R_e aN^0.64) from the attachment point (Jeppesen et al., 2001). Regardless, the PEG would sample conformations up to its fully extended length, which would be 160 nm for a 20-kDa moiety. Based on these length scales, the polymer chain could easily access the site of the protein most distal to its attachment point. Thus, one possibility is that the PEG moiety acts intramolecularly to dynamically block the antigen-binding site on the antibody itself (Fig. 4C).

Another major consequence of PEGylation is that it greatly increases the effective size of the molecule. As far as total collisions are concerned, they would be expected to be independent of size in the spherical approximation, because the larger radius of the protein increases the target size but also reduces diffusivity, such that these two effects exactly cancel (Smoluchowski, 1917; Janin, 1997). The fraction of successful collisions among all collisions, however, is proportional to the fraction of surface area comprising the binding site (Janin, 1997), and thus the observed association rate constant should decrease if the area of the binding site is held constant but the total surface area of the ligand is increased by PEGylation. PEGylation may also indirectly affect the binding properties of the ligand via interactions that change the plasticity or surface charge distribution of the molecule (Kerwin et al., 2002). Here, we use the term `intramolecular blocking' to encompass all of these indistinguishable effects that the PEG moiety may have on the molecule to which it is coupled.

A second possibility is that, once a PEGylated antibody molecule binds to its antigen on a surface, the polymer tail acts intermolecularly to hinder binding of antibodies to adjacent antigen molecules (Fig. 4D). This is analogous to the `parking problem' in adsorption kinetics (Evans, 1993; O'Shannessy and Winzor, 1996). This latter mechanism is also plausible, because the average distance between receptor molecules, assuming uniform receptor density, was calculated to be in the range of 20 to 30 nm in both the BIAcore setup and on the SK-OV-3 cells used in our experiments; in reality, it is likely that the receptors are clustered, thereby reducing this intermolecular spacing.

If these two blocking modes are the only major factors affecting the binding kinetics, the relevant processes are:

(10)

The parameter describes the degree of intramolecular blocking of ligand L to give blocked ligand L_block (Fig. 4C) and is equal to the equilibrium constant between the unblocked and blocked states of the ligand. Free receptors (R) can be bound by L to give complexes (C). The parameter describes the degree of intermolecular blocking and is equal to the effective number of additional receptors sterically blocked by a bound ligand (Fig. 4D). Thus, the total number of receptors (R₀) is the sum of unbound, accessible receptors (R), bound receptors (C), and unbound receptors blocked by bound receptors (C). It should be noted that is an intrinsic property of the ligand and thus independent of the experimental setup; however, may depend on the receptor density or clustering.

In BIAcore, there is a continuous flow of fresh buffer, so it is reasonable to assume that the concentrations of ligand (L and L_block) in the flow cell do not change appreciably from those in the buffer entering the flow cell. Thus, the total ligand concentration in the flow cell is:

(11)

The change in complexes with respect to time can be described using mass-action kinetics:

(12)

Solving this result for the association phase [C(0) = 0] (see Supplemental Materials) and arranging terms:

(13)

From eq. 13, it is clear that intramolecular blocking ( > 0) can reduce the apparent association rate constant [k_on = (1 + )k_a/(1 + )] and increase the apparent equilibrium dissociation constant [K_D,obs = (1 + )K_D/(1 + )], because k_off would be unaffected. Conversely, intermolecular blocking ( > 0) has the opposite effect on both of these parameters and, furthermore, decreases the apparent number of binding sites [R_0,obs = R₀/(1 + )]. The counterintuitive result of having a higher k_on and a lower K_D,obs as a result of intermolecular blocking is explained later in this section.

The result in eq. 13 is simulated in Fig. 8A for different values of and . Here, it can be seen that the initial rate of binding is dependent on , but actually independent of . When very few antigens are bound, there is not a pronounced effect of intermolecular blockage; consequently, intramolecular blockage limits the rate of association. Because the number of ligand-receptor complexes at the beginning of any time course is zero, we can evaluate eq. 12 at C(0) = 0 to obtain a mathematical expression for the initial rate of change in complexes:

(14)

Note that this expression is indeed independent of (because of the cancellation of its effect on k_on and R_0,obs at t = 0) and corresponds to the initial, linear portion of an association binding curve. At intermediate times, both types of blockage significantly affect the binding profile in Fig. 8A. As the binding reaction reaches equilibrium (t in eq. 13), the value of C_eq, the equilibrium number of complexes formed, also depends on both and . However, if L₀ >> (1 + )K_D/(1 + ):

(15)

It should be noted that the dissociation phase (L₀ = 0 in eq. 12) is unaffected by either blocking mechanism.

View larger version (18K): [in this window] [in a new window]   Fig. 8. Mathematical model of the influence of intramolecular () and intermolecular () blocking on observed binding kinetics. A, the fraction of occupied binding sites (C/R0) over time is simulated for several values of and . The dependence of complex formation on and can be analyzed in three phases: early-time kinetics are independent of ; intermediate kinetics depend on both and ; and the equilibrium value for large L0 [specifically L0 >> (1 + )KD/(1 + )] is independent of . B, simulated profiles of scFv-like and PEG-scFv-like ligands based on and values consistent with experimental data. The dashed line at C/R0 = 1/3 represents the fraction of binding sites occupied by the PEG-scFv-like ligand at equilibrium [= 1/(1 + ) where = 2; cf. eq. 15]. The kinetic parameters of the scFv monomer were used as the intrinsic rate constants for all simulations (ka = 3.4 x 105 M-1 s-1 and kd = 5.0 x 10-5 s-1); L0 was 78 nM [= 100 x (1 + )KD/(1 + ) where = 15 and = 2].

With eqs. 14 and 15, we have independent methods for estimating and , respectively. The simplest way to test experimentally whether intermolecular blocking influences the kinetics in BIAcore is to perform measurements at very high ligand concentrations and allow the flow cell to reach equilibrium. After proper referencing, the observed signal RU_max = (mR_o)(MW)/(1 + ), where m is a proportionality constant and MW is the molecular weight of the ligand. For an unmodified antibody, = 0; therefore, RU_max is directly proportional to (R₀)(MW). Performing the same analysis with the PEGylated analog should directly yield . To determine whether this blocking is significant for our PEGylated constructs, both unmodified and PEGylated monomer were passed, at very high concentrations (1 µM), over an antigen-coated BIAcore chip—importantly, the same chip used for the kinetic analyses because coating density affects —and RU_max/MW values were determined. For the PEGylated monomer, this value was approximately three times smaller than that for the unmodified monomer (Table 5), which corresponds to an value of approximately 2. This suggests that the PEG chain of each bound scFv fragment can hinder the association of additional scFv-PEG molecules to approximately two neighboring antigens, as spaced in this experiment.

View this table: [in this window] [in a new window]   TABLE 5 Summary of monomer BIAcore kinetic data and mathematical models See Results for details on model formulation and parameter explanation. The model parameters that are consistent with the experimental values are bold ( > 0 and > 0). The diffusion and intramolecular blocking models cannot simultaneously explain both the kon ratio and the initial association rate (dC/dtt = 0) ratio, which are observed to be different experimentally. Therefore, the latter ratio is not highlighted in either case. For the combination model, independent estimates of and can be obtained from the R0,obs ratio and the dC/dtt = 0 ratio, respectively, and the experimental KD,obs and kon ratios can be accurately predicted with these estimates.

We have determined experimentally that the observed association rate of the PEGylated monomer is 5.4-fold slower than that of the unmodified monomer, and this corresponds directly to a 5.4-fold decrease in equilibrium affinity. For the model to capture this observation [i.e., (1 + )/(1 + ) = 5.4], the value of must be 15.

Is the experimentally measured ratio of initial association rates smaller than the k_on ratio, as predicted by the model? And, if so, does the value of obtained from the experimental K_D,obs values match well with the calculated from the initial association rates? From Table 5, we see that the answer to both of these questions is yes. It should be noted that, a priori, the experimental K_D,obs, R_0,obs, and dC/dt_t = 0 ratios in Table 5 would be expected to be independent. However, the fact that we can successfully fit all three ratios with only two independent parameters, and , suggests that they are actually dependent and that the model may capture the basic principles of the mechanism.

The model and experiments suggest that, at equilibrium, the concentration of intramolecularly blocked ligands is approximately 15-fold that of unblocked ligands, a surprising result. The decrease in the association rate constant caused by the increase in nonbinding surface area (Janin, 1997) would scale as the square of the hydrodynamic radius (R_h). Based on the R_h values given in Table 1, this would correspond to approximately a 4-fold decrease for spherical ligands. The remainder of the 15-fold decrease is probably caused by the physical blocking of the binding site and indirect effects of PEG interactions. This implies that less than 7% of the total scFv-PEG is capable of binding the antigen at any given point in time, a rather counterintuitive result. However, binding to the antigen immediately displaces the rapid equilibrium; thus, all of the ligand can eventually bind to the receptor, albeit with a slower observed on rate. The off rate is identical, so a reduced affinity results. This reduced on rate resulting from intramolecular blocking is balanced by the fact that intermolecular blocking increases the apparent on rate by decreasing the apparent number of binding sites. In other words, there are far more receptors actually available for binding than the maximally observed number. For example, if = 2, the ligand molecule will bind to one antigen and then block two antigens, thus "occupying" three antigens. However, in that initial binding step, the ligand can actually bind to any of those three antigens, thus increasing the apparent association rate and consequently the apparent affinity by 3-fold, compared with a system where R_0,obs = R₀. A comparison of an unmodified monomer and a PEGylated scFv with = 15 and = 2 is shown in Fig. 8B.

In equilibrium cell-binding experiments, we also observed a 5-fold decrease in apparent affinity (Table 3). As mentioned previously, this effect cannot be caused by diffusion limitations, because K_D,obs would still equal K_D at equilibrium (see eq. 8). Thus, we propose a similar dual blocking model for the binding experiments on cells. However, in this setup, the assumption of constant ligand concentration is not necessarily valid. We must modify eq. 11 to account for depletion through binding:

(16)

Combining eqs. 10 and 16, we can derive an expression for C at equilibrium (see Supplemental Materials):

(17)

A comparison of eqs. 17 and 13 clearly shows that and have the same effects on K_D,obs [= (1 + )K_D/(1 + )] and R_0,obs [= R₀/(1 + )] as in the kinetic model of association. This is expected, because neither parameter has any effect on dissociation. Eq. 17 can easily be solved explicitly for C (see Supplemental Materials). Although the same parameter ranges proposed above for and may also readily explain the observed equilibrium cell-binding assays (Table 2), we should mention that the observed maximum receptor numbers obtained in the cell-binding assays were not conclusive; furthermore, the difference in assay temperature— 4°C for cell-binding assays versus room temperature for BIAcore assays— could also affect both and without grossly affecting the ratio (1 + )/(1 + ), thus matching the experimental value of 5.

From our experimental and computational analyses, it seems that neither a reduction in