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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
J Am Chem Soc. Author manuscript; available in PMC 2013 January 11.
Published in final edited form as:
PMCID: PMC3272676

Dependence of Avidity on Linker Length for a Bivalent Ligand-Bivalent Receptor Model System


This paper describes a synthetic dimer of carbonic anhydrase, and a series of bivalent sulfonamide ligands with different lengths (25 to 69 Å between the ends of the fully extended ligands), as a model system to use in examining the binding of bivalent antibodies to antigens. Assays based on analytical ultracentrifugation and fluorescence binding indicate that this system forms cyclic, noncovalent complexes with a stoichiometry of one bivalent ligand to one dimer. This dimer binds the series of bivalent ligands with low picomolar avidities (Kdavidity = 3 – 40 pM). A structurally analogous monovalent ligand binds to one active site of the dimer with Kdmono = 16 nM. The bivalent association is thus significantly stronger (Kdmono/Kdavidity ranging from ~500 to 5000 unitless) than the monovalent association. We infer from these results, and by comparison of these results to previous studies, that bivalency in antibodies can lead to associations much tighter than monovalent associations (although the observed bivalent association is much weaker than predicted from the simplest level of theory—predicted Kdavidity of ~ 0.002 pM and Kdmono/Kdavidity ~ 8 × 106 unitless).


Although antibodies are centrally important components of the immune systems of vertebrates1 we understand little about the thermodynamics of these proteins binding to their antigens.2 We know surprisingly little about the reasons antibodies have the structures they do; in particular, the reasons that the binding regions of all antibodies are dimers (i.e., two Fab domains), or multiples of dimers. Why dimers? Why not a tightly binding monomer? Or a trimer? There is no clear molecular rationale that leads to the conclusion that two Fab regions provide the best Darwinian solution to the problems in “molecular defense” posed to the immune system.

In trying to understand why all antibodies have at least two binding sites, we want to understand, quantitatively, the thermodynamics of association between antibodies and their antigens (both monovalent and polyvalent). Understanding the mechanism and thermodynamics that determines the free energy of the association between antibodies and multivalent antigens could, we believe, serve three purposes: i) it could help to answer the biochemical question of the reason for bi- or oligovalency in antibodies,3 ii) it could contribute broadly to understanding oligovalency in molecular biology,4,5 iii) it could help in the design of inhibitors of oligovalent interactions, including those involving antibodies.6,7 In particular, we are interested in the advantage, if any, that bivalency in antibodies affords to the thermodynamics of binding to antigen over that of a monovalent interaction—that of Fab to antigen.

This paper describes a system—a synthetic dimer of carbonic anhydrase (which we call a “dimer of CA”, or “(CA)2”) and a series of bivalent benzenesulfonamides connected by oligo-sarcosine linkers (LRL where L is a derivative of benzenesulfonamide)—that we use to model the thermodynamics of association between bivalent antibodies and multivalent antigens (Scheme 1). This synthetic dimer of CA is inspired by the structure of immunoglobulins, and formed by linking two site-specific mutants of human carbonic anhydrase II (HCAII, EC (Figure 1). We test the influence of the length of the oligo-sarcosine linker joining the two sulfonamide moieties (LRL) on the strength of the interaction between the synthetic dimer of CA and these sulfonamide ligands.

Figure 1
Crystal structures and molecular models of an IgG, and of (CA)2 with and without bivalent ligands. All structures are rendered at the same scale. a) X-ray crystal structure of a monoclonal murine IgG1 specific for phenolbarbital depicted as a multi-colored ...
Scheme 1
Thermodynamic schemes describing the binding of monovalent ligands (L) to monovalent carbonic anhydrase CA and bivalent benzensulfonamide ligands (LRL) to a synthetic dimer of carbonic anhydrase (CA)2. a) The binding of CA to a monovalent ligand in solution ...

We recently reported that dimers of CA bound less strongly to ligands immobilized on a surface than would be expected from a simple theory.3 This theory predicts the change in free energy (ΔG°avidity) for the bivalent association of (CA)2 to bivalent ligands would be twice the change in enthalpy for the monovalent interaction (ΔH°mono) plus the change in entropy for a single monovalent interaction (TΔS°mono) (eq 1).8 In this work, we designed and synthesized a series of bivalent ligands (LRL) with

equation M1

the goal of maximizing the strength of association between LRL and (CA)2. Surprisingly, but also consistent with our previous findings, the strength of association of these ligands to (CA)2G°avidity ranging from −14 to −16 kcal mol−1) is again much less than the simplest theory predicts (predicted ΔG°avidity = −19 kcal mol−1). Additionally, we found that varying the length of the linker connecting the arylsulfonamides led only to small, but significant, differences in the free energies of bivalent binding of LRL to (CA)2: the free energies of binding for six bivalent ligands to (CA)2G°avidity ranging from −14 to −16 kcal mol−1) were all within 2 kcal mol−1 of each other, despite an end-to-end distance ranging from 31 Å to 69 Å (21 to 41 rotatable bonds).

Schweitzer-Stenner and coworkers quantified the association between monoclonal anti-DNP IgE and a series of bivalent DNP ligands of different lengths (14 Å to 160 Å) using fluorescence titrations.9 They inferred that the longer ligands (145 Å and 160 Å) were capable of binding bivalently, although weakly, to IgE (enhancements, defined as Kdmono/Kdavidity, of 7 to 20 unitless), while the shorter ligands (<132 Å) formed cyclic complexes comprising two antibodies and two ligands with enhancements of 100 to 600. In a series of papers, the Baird group quantified the interaction of antibodies and multivalent ligands often using bivalent DNP ligands, anti-DNP IgE, and cells.1012 In one of these studies, Das and coworkers reported the synthesis of a series of bivalent DNP ligands with poly(ethylene glycol) (PEG) linkers (1, 3, and 10 kDa) and characterized their association to monoclonal anti-DNP IgE.13 They found that cyclic, bivalent complexes can form from the combination of one bivalent ligand and one IgE and observed enhancements of 10 to 300. These results are consistent with those of Schweitzer-Stenner: “long” ligands can bridge the binding sites of single antibody and form intramolecularly-bound, cyclic complexes of one antibody and one L2. In related work, we characterized the complexes formed from mixtures of monoclonal anti-DNP IgG and a synthetic trivalent DNP ligand.14 We characterized the stability of these species and found that complexes of two trivalent ligands (L3) and three antibodies can form in a yield of ~ 90% with an enhancement of ~50.

Bivalent Benzenesulfonamide Ligands LRL and (CA)2 Model Bivalency in Antibodies

We are interested in defining the advantages that bivalent antibodies have over their monovalent counterparts, the Fab domain, both in vivo and in vitro, because we wish to modulate (to inhibit or to enhance) their binding to antigens.15 We intend the combination of (CA)2 and LRL to serve as a model with which to examine the biophysics and physical-organic chemistry of bivalency in antibodies. This combination of (CA)2 and bivalent sulfonamides is particularly well-suited as a model for bivalent interactions for at least seven reasons: i) The carbonic anhydrase dimer has the essential structural features of IgG: The dimer consists of two identical binding domains linked by a flexible chain. IgG consists of two identical binding domains linked by two flexible chains to an Fc domain. We can easily change the nature of the linkers (length and flexibility) connecting the binding sites in the dimers.3 These sorts of changes would require major efforts in mutagenesis to carry out on any IgG. ii) The carbonic anhydrases (HCA I (28.8 kDa), HCA II (29.2 kDa), and bovine carbonic anhydrase BCA II (29.1 kDa), which differ only by 382 Da) are similar in size to the Fabs of IgG (~50 kDa). iii) HCA is structurally stable and readily available.1619 HCA is available in quantities of hundreds of milligrams from E. coli with the investment of a few days of work and a minimal cost of supplies. Commercially available monoclonal antibodies (for example, monoclonal anti-DNP antibodies) cost in the range of ~100$ per 0.1 mg. At that cost, conducting a panel of experiments with different ligands—and with enough repeats to provide estimates of uncertainties—would be very expensive (>$10,000). iv) HCA is easily modified by site-directed mutagenesis. Antibodies are available through site-directed mutagenesis, but their expression from yeast, insect, or mammalian cells is much more challenging and available in fewer academic groups (especially those of physical-organic chemists) than the expression and purification of proteins from E. coli. v) Numerous arylsulfonamides of the general structure R-Ar-SO2NH2 bind to HCA in a conserved and well characterized geometry.20 The geometry of the sulfonamide ligands in the active site of CA is as well defined as that of any ligand for any protein; there is no corresponding information on what would happen to a ligand in the binding site of a Fab, especially if we changed the structure of the ligand.21 We can think of no monoclonal antibody that has a set of ligands with properties as varied and as well characterized as the ones that bind to CA. vi) Synthesis of sulfonamide ligands which bind to CA is tractable, as is their incorporation in bi- and oligovalent systems. vii) The literature in assays for CA—from enzymatic through fluorescence to surface plasmon resonance and capillary electrophoresis—is much richer for CA than for IgGs. We believe HCA is an excellent model protein—the best we know—for physical-organic studies of protein-ligand binding.21

The CA-Arylsulfonamide Interaction

The carbonic anhydrases bind a wide range of ligands including, arylsulfonamides, with dissociation constants (Kdmono) in the range of micromolar to picomolar.20,21 The equilibrium for a monovalent ligand (L), a monomeric carbonic anhydrase (CA), and a CA-ligand complex (CA·L) is characterized by the monovalent affinity of the interaction, Kdmono (eq 2 and Scheme 1a).

equation M2
equation M3

Intramolecular Binding of a Tethered Ligand-CA System

In studies aimed at understanding intramolecular interactions, we characterized the thermodynamics of binding for the association between the active site of CA and an arylsulfonamide ligand that was covalently tethered to the surface of HCA by oligo(ethylene glycol) linkers of different lengths (EGn, n = 0, 2, 5, 10, and 20; Chart 1 and Scheme1b).22 We characterized the strength of the intramolecular bond using a unitless dissociation constant Kdintra —the ratio of the concentrations of CA–Llin to CA–Lcyc (eq 4, where the hyphen represents the covalent attachment between L and CA).

equation M4
equation M5

In that system, the relationship between the intramolecular dissociation constant (values of Meff ranging from 0.8 to 26 mM) and the length of the flexible linker (ranging from 25 to 105 Å in extended length between the nitrogen of the sulfonamide and the disulfide bond connecting ligand to the protein) was quantitatively well-explained by a model that describes the linker as a random-coil polymer (eq 6 and Table 1). We inferred

Table 1
Dissociation constants (Kd) and free energies (ΔG°) for the association of monovalent and bivalent ligands to CA and covalent dimers of CA, (CA)2’s. This table also contains entries from the literature.

equation M6

from calorimetric studies that the length of the linker influenced the thermodynamics of binding exclusively entropically.22

Dimers of CA Bind to Mixed SAMs Presenting Benzenesulfonamides

In studies aimed at understanding the binding of bivalent proteins to ligands on a surface, we developed five synthetic dimers of carbonic anhydrase and studied their interaction with benzenesulfonamide ligands presented at the surface of mixed self-assembled monolayers (SAMs) (Scheme 1c).3 We prepared three dimers of CA having different lengths of the oligo(ethylene glycol) linker that joined the two molecules of CA, and two dimers with different points of attachment of the linker to the protein. We found that the dimers of CA bound bivalently to the SAM with low nanomolar avidities (Kdavidity, surf = 1 − 3 nM, Meff = 3 to 20 μM) while monomeric carbonic anhydrase (CA) bound to the ligands of the SAM less strongly (Kdsurf = 89 nM). The bivalent binding of these dimers to the SAM represents a modest ~50-fold enhancement of bivalent over monovalent association. The avidity of these bivalent proteins were unexpectedly insensitive to the structure of the linker connecting them: changing the length of the linker with end-to-end distances between 9 and 18 Å, or the point of attachment between the molecules of CA (either near the active site (C133) or distal to the active site (C185)) had virtually no effect on the avidity of these dimers for the SAM.

Bivalent Interactions in Protein-Ligand Systems

In this paper, we focus on differences in the free energy (ΔG°) of binding of mono- and bivalent ligands to CA, and to a dimer of CA (Scheme 1a and 1d). Scheme 1d diagrams the association of (CA)2 and LRL as a process that involves two steps. Equation 7 (by analogy to equation 3) describes the equilibrium constant for the association of LRL to (CA)2. A dimer of CA is

equation M7

statistically four times more likely to bind a bivalent ligand as monomeric CA is to bind a monovalent ligand because (CA)2 has two binding sites and LRL has two ligand moieties that can bind to CA.23 The value of Kdinter determines the change in free energy for the monovalent binding of (CA)2 to LRL (eq 8).

equation M8

A dimer of CA can, in principle, form an intramolecular noncovalent bond with a bivalent ligand in which the unoccupied active site of (CA)2·LRLlin binds the second sulfonamide of LRL and forms (CA)2·LRLcyc (Scheme 1d). At equilibrium, 2 Kdintra equals the concentration of (CA)2·LRLlin divided by the concentration of (CA)2·LRLcyc (eq 9).24

equation M9

The overall strength of association between (CA)2 and LRL is characterized by the avidity of this bivalent interaction (Kdavidity, M) (eq 10 and Scheme 1d).

equation M10

It follows from equations 2 and 5 that the change in free energy for the formation of a cyclic, bivalently bound structure (CA)2·LRLcycG°avidity) is given by equation 11.

equation M11

The enhancement (β) is the ratio of the monovalent dissociation constant to the avidity, and can describe an increase or decrease in the strength of binding between (CA)2 and LRL relative to that of CA and L (eq 12).15 Values of β > 1 indicate that

equation M12

(CA)2 binds to LRL more strongly than (CA)2 binds to L.


To characterize the binding of the dimer of CA to the series of bivalent ligands, we estimated values of ΔG°avidity, because comparison of ΔG°avidity with ΔG°mono describes the energetic consequence of tethering together two benzenesulfonamides (here, with an oligo-sarcosine linker). In particular, we wanted to determine the dependence of Δ G°avidity—our metric for the strength of bivalent binding—on the length of the linker joining the two benzenesulfonamides of the ligand.

We synthesized bivalent derivatives of benzenesulfonamides with oligo-sarcosine linkers with 0, 2, 4, 6, 8, 10 sarcosine units (LSarnL in Chart 1). We chose oligo-sarcosines as the linkers because they are typically more soluble in aqueous solutions than other peptides.25 Oligomers of sarcosine show little non-specific binding to proteins because the N-methyl group prevents hydrogen bonding between the nitrogen of the amide and the carbonyl oxygens of the protein.26,27 We selected the number of sarcosine units comprising the different linkers (n = 0, 2, 4, 6, 8, 10) to provide lengths (31 to 69 Å) that range from “similar to”, to “larger than”, the minimum distance required to bridge the active sites of the dimer of CA (~30 Å). We also employed a short bivalent ligand (25 Å) with a tri(ethylene glycol) linker (LEG3L in Chart 1). We hypothesized that LEG3L would not bind bivalently to the dimer of CA, because this ligand is shorter than the minimum distance between the binding sites of the dimer. We used LEG3L as a control of monovalent binding to which we compared the longer ligands of LSarnL.

Numerous complexes defined by different stoichiometries—represented by ((CA)2)n·(LRL)m—can, in principle, form from mixtures of (CA)2 and LRL.28 The tendency of multivalent systems of receptors and ligands to form complexes of different stoichiometry in the same solution, coupled with the difficulties in determining which complexes are formed, represents a significant barrier in the determination of the free energies of association of multivalent interactions. We conducted two assays (analytical ultracentrifugation (AUC) and a kinetic assay based on observing the dissociation of LRL from (CA)2) that allowed us to determine the stoichiometry of ((CA)2)n· (LRL)m complexes. We used this information to construct a mathematical model that we fit to the data from a fluorescence displacement assay, ultimately extracting values of ΔG°avidity for the series of LRL binding to (CA)2.

Analytical ultracentrifugation (AUC) is a solution technique that provides an estimate of the molecular weights of proteins and their complexes with ligands.29 We used AUC to estimate the molecular weights of CA, (CA)2, and mixtures of (CA)2 and LRL and inferred from these measurements the coefficient n in ((CA)2)n·(LRL)m complexes. We then used the estimate of n to select a set of equations—a mathematical model—to describe the fluorescence displacement assay.

We developed a kinetic assay to follow the rate of dissociation of bivalent ligands from (CA)2 in the presence of the monovalent inhibitor ethoxzolamide, which competes with LRL for the binding sites of (CA)2. This kinetic assay makes it possible to determine the valency of the interaction between (CA)2 and LRL; that is, to determine the presence of intramolecular noncovalent bonds in these complexes. A dependence of the rate of dissociation of LRL from (CA)2 on the concentration of ethoxzolamide corresponds to the presence of intramolecular non-covalent bonds in the (CA)2 -LRL complexes, vide infra.

We followed the binding of (CA)2 to bivalent ligands in the presence of the monovalent ligand dansylamide using a fluorescence competition assay. Dansylamide (DNSA) is weakly fluorescent in buffered solution, but becomes fluorescent upon binding to CA.30 The intensity of the fluorescence of the solution is therefore proportional to the concentration of dimers with DNSA bound in their active sites. There are three advantages to this assay over alternative techniques: i) it is applicable to a wide variety of ligands that bind to the active site of CA, ii) it is applicable to all iso-types of CA and its mutants, and iii) it can be described by a mathematical model—i.e., an equation can be used to describe the equilibria and mass balance of the system—and can thus use the data (fluorescence binding isotherms) to provide estimates for the dissociation constants and the free energies.22 Comparison of free energies binding of ligands with different lengths of linker to (CA)2 is the focus of this paper.


Synthesis and Characterization of Dimers of CA

We previously described the preparation of the synthetic dimer of human carbonic anhydrase II ((CA)2) from a double mutant of HCAII.3

Preparation of Monovalent and Bivalent Sulfonamides

We prepared a series of bivalent benzenesulfonamide ligands (LRL) with oligo-sarcosine linkers (n = 0, 2, 4, 6, 8, 10 sarcosine units) using standard techniques of solid-phase peptide synthesis, a bivalent ligand with a tri(ethylene glycol) linker, and the monovalent benzenesulfonamide ligand L. We present many of the experimental results as a function of the distance between the two nitrogen atoms of the bivalent ligands when the ligands are in unstrained conformations that maximize this distance. We call this maximum distance the “extended length of the bivalent ligands”, or Γ. We calculated Γ from molecular models of ligands using the software package Chem3D Pro by Cambridgesoft.

(CA)2 and LRL form complexes containing only one (CA)2

Analytical ultracentrifugation (AUC) is a solution technique that provides an estimate for the molecular weights of proteins and their complexes with ligands. AUC is an optical technique based on the absorbance of light by the solution of protein. We required a concentration of (CA)2 (925 nM) that would provide a suitable ratio for the signal to noise in the instrument, and adjusted the concentration of LRL to maximize ligand-induced oligomerization. Dembo and Goldstein reported that the total concentration of LRL that maximizes the concentration of complexes with two or more (CA)2’s (that is, [LRL]0, max) is a function of the total concentration of bivalent receptors, [R2]0, and the dissociation constant Kdinter (eq 13).31 Substituting 16 nM for Kdinter and

equation M13

925 nM for [(CA)2]0 gives 960 nM for the total concentration of LRL ([LRL]0, max); this concentration of LRL (~1 μM) is the one we used in these experiments.

We carried out sedimentation equilibrium experiments on a Beckman XL-I ultracentrifuge at rotor speeds of 8,000, 10,000 and 12,000 rpm, at 25 °C in order to verify that in mixtures of 925 nM (CA)2 and 1 μM LRL, only complexes containing one molecule of (CA)2 are formed at equilibrium (Figure 2). It is reasonable to assume that under these conditions the ligands are bound to active sites of (CA)2. This procedure yielded an estimated molecular weight of 58 ± 4 kDa for (CA)2 (Table S1). This result is indistinguishable from the value of 58.385 kDa for (CA)2 determined by mass spectrometry. The results of the AUC experiments support the conclusion that only complexes that contain one (CA)2 and one LRL are observed in mixtures of (CA)2 and LRL, when the concentration of (CA)2 is 925 nM or less.

Figure 2
Sedimentation equilibrium experiments of CA and (CA)2 with and without added bivalent ligand as observed at 280 nm and 25 °C. a) The hollow circles are experimental data for 925 nM (CA)2 without added LRL, and the line is the fit for a single ...

(CA)2 and LRL dissociate rapidly in the presence of high concentrations of ethoxzolamide

AUC experiments cannot distinguish between complexes of similar molecular weight; that is, AUC does not tell us if LRL binds to (CA)2, or if a mixture of (CA)2·LRLlin and (CA)2·LRLcyc are both present at equilibrium. We require additional structural information on the complexes present at equilibrium in order to construct the mathematical model that we use to estimate the free energy of association from the fluorescence displacement assay.

We developed a kinetic assay based on fluorescence spectroscopy and confirmed the presence of (CA)2·LRLcyc by following the rate of dissociation of LSar4L from (CA)2 in the presence of the soluble monovalent inhibitor, ethoxzolamide, that competes with LRL for the binding sites of (CA)2.3 This assay consists of four steps (Figure 3a). i) We allowed a solution of (CA)2 and LRL to equilibrate in a stirred quartz cuvette while the temperature was maintained at 25 ± 1 °C, and observed the fluorescence of the tryptophan residues of the protein (λex = 280 nm, λem = 340 nm). The benzenesulfonamides of LRL do not quench or enhance the fluorescence of the tryptophan residues when bound to (CA)2. ii) After the contents of the cuvette reached equilibrium (fluorescence signal did not change with time), we added a solution of ethoxzolamide (Kdmono = 0.2 nM) to the cell. iii) We observed a decrease in the fluorescence as mixing took place over a period of ~10 sec. This initial decrease in fluorescence is a result of the inner-filter effect—the more ethoxzolamide present in the sample the less light reaches the detector. iv) During and after the ~10 sec mixing period, the fluorescence signal decreases as LRL dissociates from the active sites of (CA)2 and is replaced by ethoxzolamide, which, unlike benzenesulfonamide, quenches the fluorescence of tryptophan residues of CA when bound in the active site.

Figure 3
Change in fluorescence (λex = 280 nm, λem = 340 nm) as benzenesulfonamide ligands dissociate from the active sites of (CA)2 (16.7 nM) and are replaced by ethoxzolamide (ethox, Kd = 0.2 nM) which quenches the fluorescence of (CA)2. a) This ...

The rates of dissociation of LSar4L from (CA)2 increase as the concentration of ethoxzolamide increases (Figure 3d). We infer that LSar4L binds bivalently to (CA)2. We propose a mechanism consisting of several steps for LRL to dissociate from (CA)2 in the presence of ethoxzolamide (Scheme 2). Ethoxzolamide competes with LRL for the unoccupied binding site of (CA)2·LRLlin. Hence, as the concentration of ethoxzolamide increases, the rate at which ethoxzolamide binds to the unoccupied binding site of (CA)2·LRLlin (characterized by konethox [ethox], Scheme 2) becomes larger than the rate at which the unbound binding moiety of LRL binds to the unoccupied binding site of (CA)2·LRLlin (characterized by konintra). The combination of ethoxzolamide with the binding site of (CA)2 is effectively irreversible, and increases the rate of dissociation of LRL from (CA)2. When the product (konethox· [ethox]) is much larger than konintra, rebinding of (CA)2·LRLlin to form (CA)2·LRLcyc is prevented and LRL rapidly dissociates from (CA)2.

Scheme 2
Kinetic scheme describing the dissociation of a bivalent ligand (LRL) from (CA)2 in the presence of ethoxzolamide ([triangle]).

We also analyzed the dissociation of L and LEG3L—which we hypothesized to be too short to span the distance between binding sites of the dimer—from (CA)2 using the same assay. The presence of ethoxzolamide did not affect the rate of dissociation L and LEG3L from (CA)2 (Figure 3b and c). A rate of dissociation that is independent of the concentration of ethoxzolamide is compatible with a mechanism for dissociation that involves a single dissociation event of the ligand from the active site—in other words, these results indicate that once L dissociates from (CA)2, it does not rebind. Based on these results we include (CA)2·LRLcyc in the mathematical model that we fit to data from the fluorescent displacement assay to obtain the free energy of bivalent binding in the following section.

Determination of Values of Kdavidity for the Association of LRL’s with (CA)2 Using a Fluorescence Displacement Assay

We used a competitive fluorescence-based assay to estimate values of Kdavidity for binding of bivalent sulfonamide ligands to (CA)2 by competition with dansylamide (DNSA; Kdmono ~240 nM) for the active sites of (CA)2 (Figure 4). We began with a concentration of DNSA (5 μM) sufficient to saturate 95% of the active sites of (CA)2 (25 nM), titrated this solution with bivalent ligand (LRL), and followed the disappearance of the fluorescence as the total concentration of LRL increased.

Figure 4
Titration of (CA)2 with monovalent benzenesulfonamide L and bivalent benzenesulfonamide ligands LRL. (CA)2 (25 nM) was equilibrated with 5 μM dansylamide 2 in 20 mM sodium phosphate buffer (pH 7.4) and different amounts of L or LRL. The fluorescence ...

The titration is described by a thermodynamic scheme containing four (CA)2-ligand complexes (i.e., (CA)2· (DNSA)2, (CA)2·LRL·DNSA, (CA)2·LRLcyc, and (CA)2· (LRL)2) and three equilibria that connect them (K1, K2, and K3) (Figure 5a). We fit the mathematical description of Figure 5a (eq S21 in Supporting Information) to the titration data using Kdintra as the sole adjustable parameter (Table 1). The Supporting Information details the mathematical description of Figure 5 and the derivation of equation S21. We calculated values of Kdavidty from Kdintra (eq 10) and found that LSar4L was the tightest binding bivalent ligand (Figure 6).

Figure 5
Thermodynamic scheme and simulations describing the model used to analyze the fluorescence titration experiments. a) The thermodynamic scheme comprises three equilibria characterized by three equilibrium constants (K1, K2, and K3) and four species that ...
Figure 6
Plot of ΔG°inter and ΔG°avidity for the binding of monovalent (L) and bivalent benzenesulfonamide ligands (LRL) to a synthetic dimer of CA (CA)2 as a function of the extended length of the linker (Γ, Å). ...

Values of ΔG°avidity lie within a small range, but indicate that LSar4L is the tightest binder

The plot of ΔG°avidity as a function of the extended length shows two trends: the strength of binding increases as Γ increases from 30 to 45 Å, the strength reaches a maximum at Γ equal to 45 Å, and then the strength decreases as Γ increases from 45 to 70 Å (Figure 6). Lundquist and Toone have speculated that the length of the linker between the binding moieties of a bivalent ligand should determine the free energy of its interaction with a bivalent protein.32 In their view, bivalent ligands of less than ideal length can span the distance between two sites of a bivalent receptor but are too short to allow the binding moieties of the bivalent ligand to achieve simultaneously the optimal orientation in the binding sites. They hypothesized that these “short” bivalent ligands will lead to diminished avidity because the sum of the enthalpies of the two interactions will be less than 2ΔH°mono. Lundquist and Toone also suggested that the strength of the bivalent interaction should be at a maximum when the length of the linker is “exactly” the appropriate length (but not longer) to allow both binding moieties of the ligand and both subunits of the receptor to achieve simultaneously optimal orientations. Longer linkers, if the rotatable bonds in the linker are restricted upon the formation of the cyclic complex, diminish the strength of binding because of the unfavorable entropy of restricting rotatable bonds in the linker domain. In our previous work with a covalently tethered system, we found that the free energy of association decreased as the length of tether connecting ligand to protein increased.22 The behavior we observe, in this system, for increasing Γ beyond 45 Å is consistent with our previous work. Without characterizing the enthalpic and entropic contributions to binding, however, we cannot evaluate the validity of the model of Lundquist and Toone;32 although the trends we observe are consistent with their model.

Mixtures of (CA)2 and LSar4L form insoluble aggregates at concentrations needed for isothermal titration calorimetry

We attempted to measure the enthalpy of interaction between LRL and (CA)2 using isothermal titration calorimetry (ITC).33 We found that precipitates—which we detected as a milky-white suspension following the titration—are formed when (CA)2 (5 μM, required for ITC) is titrated with a solution of LRL (50 μM). We infer from the formation of precipitates that intermolecular binding occurs at these concentrations to form protein-ligand complexes containing more than one (CA)2, although we cannot determine from the formation of precipitates alone the degree to which intermolecular binding competes with the intramolecular binding. Since estimating values of Kd and ΔH° requires a thermodynamic model (e.g., one bivalent ligand binding to one—and only one—bivalent protein), it is impossible to use the data from ITC in this system to determine the enthalpic and entropic contributions to the bivalent interaction. The formation of these precipitates is consistent with the range of effective molarities (2 to 20 μM) we estimated by fluorescence displacement assay; that is, when the total concentration of (CA)2 approaches the value of Meff, intermolecular interactions become increasingly favored over intramolecular interactions.


(CA)2 is an excellent model for studying bivalency in antibodies

This paper describes a model bivalent receptor inspired by the structure of bivalent antibodies. Dimers of CA are versatile models of bivalency in IgG and IgE because they access a wealth of background information and assays available for CA that is not available for antibodies, and because dimers of CA are readily available. This model system allowed us to compare the thermodynamics of binding of bivalent ligands to our previous work with a tethered ligand-CA system 22 and to dimers of CA binding to SAMs.3 We plan to synthesize dimers of CA incorporating Fc mimics to explore the role of Fc in the thermodynamics of binding. We believe additional synthetic dimers, in addition to dimers of carbonic anhydrase, can be developed to explore in greater detail the structure-function relationships for antibodies.

ΔG°avidity is weaker than simple theory predicts

The free energy of bivalent binding (ΔG°avidity) for the tightest binding bivalent ligand, LSar4L, is much lower than predicted based on two monovalent interactions of binding. The simplest level of theory predicts that the change in free energy for the bivalent association of (CA)2 to bivalent ligands (ΔG°avidity) would be twice the change in enthalpy for the monovalent interaction (ΔH°mono) plus TΔS°mono for a single monovalent interaction (eq 1).8 The values of the enthalpy and entropy of binding for monomeric CA to benzenesulfonamide 6 are ΔH°mono = −9.2 kcal mol−1 and TΔS°mono = 0.4 kcal mol−1.34 This simple theory estimates a value of ΔG°avidity = −18.8 kcal mol−1 for (CA)2 and LSar4L; a value that is ~ 4 kcal mol−1 more favorable that the value we observed. The linker could, in principle, influence (favorably or unfavorably) the free energy of the bivalent interaction by interacting with the surface of the protein, although we have established in previous studies that oligomers of sarcosine do not affect the free energy of interaction of arylsulfonamides by associating with the surface of HCA.26 Unfortunately, we have not been able to obtain a value for the enthalpy for the binding of these ligands to (CA)2, because precipitates are formed at the concentrations required for calorimetry. The thermodynamic basis for the difference between the measured values of ΔG°avidity and those predicted by equation 1, therefore, has not yet been defined.

Variation in ΔG°avidity with the length of the linker is surprisingly small

The free energy of the bivalent interaction becomes more favorable as the length of the linker increases until it reaches a minimum at n = 4. For ligands with longer linkers, the free energy monotonically increases (i.e., becomes less favorable) with increasing length of the linker.

The values of ΔG°avidity for the bivalent ligands (LRL) differ by less than 2 kcal mol−1 between the tightest binding bivalent ligand (LSar4L) and the weakest binding bivalent ligand (LSar10L). The weak dependence of the strength of association on the length of the linker is consistent with our previous studies of intramolecular binding, where variations in oligo(ethylene glycol) linkers from n = 0 to 20 resulted in changes of binding of ~ 2 kcal mol−1. Interestingly—albeit, perhaps, coincidentally—the variation of the values of ΔG°avidity with increasing length of the sarcosine linker of the ligand is compatible with the thermodynamic analysis of Lundquist and Toone.32 The length of the linker affects the strength of binding, but length alone cannot account for the ~4 kcal mol−1 difference between the value of ΔG°avidity predicted by theory (−19 kcal mol−1) and the values of ΔG°avidity we measured (−15 kcal mol−1).

Tight-binding multivalent ligands can be synthesized with “long”, flexible linkers

Consistent with our previous work and the work of others, the results of this paper suggest that flexible bivalent ligands that are longer than the spacing between the binding sites of the receptors can still bind tightly to multivalent proteins.35 A bivalent ligand (LSar4L) with a linker of “optimal” length, in this system, gave ~2 kcal mol−1 more favorable free energy than a bivalent ligand 25 Å longer. The effort required to “optimize” the length of the linker may not be worth the effort to obtain an increase in the strength of binding of ~2 kcal mol−1.


General Methods

Chemicals we purchased from Aldrich, Alfa Aesar, and Novabiochem. NMR experiments were carried out on a Varian INOVA 500 MHz spectometer. Isothermal titration calorimetry (ITC) was performed using a VP-ITC microcalorimeter (MicroCal). UV-vis spectroscopy was conducted on a Hewlett- Packard 8453 spectrophotometer. Analytical HPLC was run on a Varian instrument with a C18 column, 5 μm (4.6 × 250 mm), from Vydac using a linear gradient of water with 0.1% TFA (A) followed by acetonitrile containing 0.08% TFA (B), at a flow rate of 1 mL min−1 (UV detection at 218 and 280 nm). Preparative reverse-phase HPLC was performed using a Varian HPLC instrument equipped with a C18 column, 5 μm (22 × 250 mm), from Vydac at a flow rate of 15 mL min−1 with UV detection at 218 and 280 nm. Fluorescence measurements were performed on Perkin Elmer LS 50 B Fluorimeter (kinetic assay) and on a Molecular Devices SpectraMax Gemini XS instrument for detection at 460 nm (for DNSA binding assays).

Synthesis of Monovalent and Bivalent Sulfonamides

All ligands, except LEG3L, were synthesized using standard Fmoc chemistry by stepwise solid-phase methodology, according to published procedures.36 The crude mixtures containing ligands were purified by preparative reverse-phase HPLC and the appropriate fractions lyophilized. Purity and molecular weight were consistent with the proposed structure as assessed by analytical RP-HPLC and ESI-HRMS. L: ESI-HRMS m/z found 400.1655 Da; calcd 400.1672 Da. LEG3L: ESI-HRMS m/z found 537.1115 Da; calcd 537.1090 Da. LSar0L: ESI-HRMS m/z found 640.2209 Da; calcd 640.2223. LSar2L: ESI-HRMS m/z found 782.2980 Da; calcd 782.2966 Da. LSar4L: ESI-HRMS m/z found 924.3718 Da; calcd 924.3708 Da. LSar6L: ESI-HRMS m/z found 1066.44 Da; calcd 1066.44 Da. LSar8L: ESI-HRMS m/z found 1208.52 Da; calcd 1208.52 Da. LSar10L: ESI-HRMS m/z found 1350.59 Da; calcd 1350.59 Da.

Quantitation of Stock Solutions of Arylsulfonamide Ligands

Solutions were prepared to ~20 mM by weighing the solid bivalent ligand then adding the appropriate amount of DMSO-d6. Stock solutions were diluted 1:10 with 2.00 mM maleic acid standard in DMSO-d6, which were prepared gravimetrically. Proton resonances from the arylsulfonamide were normalized relative to those from maleic acid (allowing a 10 s delay between pulses) to determine the concentration of the stock solutions accurately.

Analytical Ultracentrifugation (AUC)

The molecular weight of (CA)2 with and without added LRL in solution was estimated by sedimentation equilibrium on a Beckman XL–I ultracentrifuge. Concentrations of (CA)2 were fixed at 0.925 μM. Samples were dissolved in 10 mM phosphate buffer (pH 7.40), containing 137 mM NaCl and 2.7 mM KCl. The samples were centrifuged at 6,000, 9,000 and 12,000 rpm for 22 hours at 25 °C before absorbance scans were performed. Data obtained at 25 °C were fit globally to an equation that describes the sedimentation of a homogeneous species using XL-A/XL-I Data Analysis software version 6.03 based on Origin software by OriginLab Software Inc.

Kinetic assay

A solution of (CA)2 (16.7 nM) and bivalent ligand (160 nM) was prepared in 20 mM PBS (pH 7.4) and centrifuged to remove traces of dust. A 2 mL portion of solution was transferred to a quartz cuvette and the cuvette placed in a Perkin Elmer LS 50 B Fluorimeter. The solution was stirred and temperature was maintained at 25 ± 1 °C. Following equilibration, the fluorescence of the tryptophan residues of the protein (λex = 280 nm, λem = 340 nm) was followed as a function of time. When the fluorescence signal did not change with time (a stable baseline was reached), 100 uL of a solution of ethoxzolamide (Kdmono = 0.2 nM in PBS at pH 7.4) was added to the stirred solution of (CA)2 and ligand. The time immediately following the brief mixing period (~10 sec) was taken as time zero (t = 0 sec.). The fluorescence signal was normalized from time zero to the end of the experiment.

Determination of Avidities for the Binding of LRL to (CA)2

To the wells of a black 96 microwell plate was added L or LRL (which was diluted serially across the 12 wells of the plate) followed by a solution of (CA)2 (25 nM final concentration) in a final volume of 200 μL of 20 mM sodium phosphate pH 7.4. The covered plate was incubated at 25 °C for more than 2 h. The intensity of the fluorescence was measured ~5 times (λex = 290 nm, λem = 460 nm, with a 455 nm cut-off filter). The intensities were fit to eq S21 (Supporting Information).

Crystallization of (CA)2

Crystals were grown by vapor diffusion from a reservoir solution of 2.0 M ammonium sulfate at pH 8.8 (bicine). The crystals appeared as thin plates and grew to their full size (approximately 0.4 mm × 0.4 mm × 0.1 mm) in about 2 weeks at 4 °C. A crystal was captured in a nylon loop, transferred to a 20% glycerol/85% reservoir cryoprotectant solution briefly, rapidly frozen in a liquid nitrogen bath, and finally mounted in a cryo-stream (Oxford Cyrosystems). The X-ray source was a rotating copper anode, and diffraction data were collected on an RAXIS IV (Rigaku) instrument. The data set used in this structure was measured from a single crystal. The data describing these crystals appears in the supporting information.

Molecular Modeling

Models of (CA)2 with bivalent ligands were prepared using the Molecular Operating Environment (MOE) program available from Chemical Computing Group, Inc. The models of the dimer-ligand complexes were constructed using the coordinates of the crystal structure of (CA)2 and bivalents ligands constructed within MOE. Two Zn-N bonds were created between the sulfonamides of the ligands and the zinc atoms in the active sites of the dimer. The internal coordinates of the CA monomers were fixed and the energy of the complex was minimized using the AMBER force field allowing the bonds within bivalent ligand and the linker joining the monomers of CA to rotate. A pdb file of the structure with the minimum energy was saved and rendered in Pymol.

Supplementary Material



This work was supported by the NIH through a research award (GM030367) and by the Wyss Institute for Biologically Inspired Engineering at Harvard. E.T.M. acknowledges the support of the NIH in the form of a postdoctoral fellowship (NRSA-GM076971). D.T.M. acknowledges support from an NIH postdoctoral fellowship (1 F32 AI068605-0). We thank Prof. Krishna Kumar (Tufts University) for his help with analytical ultracentrifugation experiments (NIH 1S10RR017948).



Supporting Information. Analytical ultracentrifugation experiments, derivation of eq S21, synthesis of (CA)2, and collection of X-ray data. This material is available free of charge via the Internet at


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