Dependence of effective molarity on linker length for an intramolecular protein-ligand system

Abstract

This paper reports dissociation constants and "effective molarities" (M(eff)) for the intramolecular binding of a ligand covalently attached to the surface of a protein by oligo(ethylene glycol) (EG(n)) linkers of different lengths (n = 0, 2, 5, 10, and 20) and compares these experimental values with theoretical estimates from polymer theory. As expected, the value of M(eff) is lowest when the linker is too short (n = 0) to allow the ligand to bind noncovalently at the active site of the protein without strain, is highest when the linker is the optimal length (n = 2) to allow such binding to occur, and decreases monotonically as the length increases past this optimal value (but only by a factor of approximately 8 from n = 2 to n = 20). These experimental results are not compatible with a model in which the single bonds of the linker are completely restricted when the ligand has bound noncovalently to the active site of the protein, but they are quantitatively compatible with a model that treats the linker as a random-coil polymer. Calorimetry revealed that enthalpic interactions between the linker and the protein are not important in determining the thermodynamics of the system. Taken together, these results suggest that the manifestation of the linker in the thermodynamics of binding is exclusively entropic. The values of M(eff) are, theoretically, intrinsic properties of the EG(n) linkers and can be used to predict the avidities of multivalent ligands with these linkers for multivalent proteins. The weak dependence of M(eff) on linker length suggests that multivalent ligands containing flexible linkers that are longer than the spacing between the binding sites of a multivalent protein will be effective in binding, and that the use of flexible linkers with lengths somewhat greater than the optimal distance between binding sites is a justifiable strategy for the design of multivalent ligands.

Figure 1

(A) A monovalent binding event between a receptor and ligand is characterized by a dissociation constant (K_d^inter) with units of concentration (M). (B) When the ligand is tethered to the receptor, the dissociation constant (K_d^intra) is now dimensionless and is related to K_d^inter by the effective molarity, M_eff (eq 1). (C) Analogous to (A) for a binding event between a protein and ligand. (D) Analogous to (B) for an intramolecular protein-ligand binding event. In this case, however, the distance (d) between the two ends of the linker in the bound state is non-zero (unlike the case in (B)). (E) Dissociation of a bivalent ligand from a bivalent receptor. This process can be conceptualized as occurring in two steps: the first step (with dissociation constant 2K_d^intra*) is intramolecular, and the second step (with dissociation constant ¼K_d^inter) is intermolecular. The equation given provides a means of estimating M_eff from the observed dissociation constant (K_d^avidity) and the intermolecular dissociation constant (K_d^inter) by assuming that the dissociation constant for the first step is equivalent to the product of a statistical factor of 2 and K_d^intra (defined in (D); i.e., this procedure assumes that K_d^intra* = K_d^intra). This assumption is often quite poor because of complications arising from the influence of binding at one site on binding at the other site (e.g., cooperativity between binding sites, enthalpy/entropy compensation).

Figure 2

Possible models to explain the binding of a ligand, covalently attached to the surface of a protein by the “linker”, to the active site of that protein. When the ligand is bound at the active site, the linker: (A) has low conformational mobility (i.e., free rotations of the bonds of the linker are significantly restricted), (B) has significant conformational mobility, or (C) and (D) makes stabilizing contacts with the surface of the protein. In (D), the linker makes contacts with the surface of the protein even when the ligand is not bound at the active site.

Figure 3

Random-coil polymer with the number (n) of repeat units (represented as open circles) equal to 45. (A) The polymer forms a closed loop: the distance (〈r²〉^1/2) between the ends of the polymer is near zero. (B) The polymer bears a physical pole of length (d) attached at one end. The distance (〈r²〉^1/2) between the ends of the polymer is close to this defined distance (d).

Figure 4

Model for the interaction of p-H₂NSO₂C₆H₄CONH(CH₂CH₂O)₂CH₂CH₂NH₃⁺ (ArEG₃NH₃⁺) with HCA based on the deposited X-ray crystallographic coordinates (PDB: 1CNX). The arylsulfonamide ligand is rendered as a ball-and-stick model in CPK color scheme. HCA is depicted as a light blue ribbon diagram with the catalytically essential Zn²⁺ cofactor shown as a green sphere. Lys-133 of HCA II is represented as a red ball-and-stick model. The distance (d) between the last glycol unit of the ligand and the γ-CH₂ group of Lys-133 (corresponding to the thiol in the Lys→Cys HCA** mutant) is indicated by the dashed line.

Figure 5

Schematic diagram depicting the strategy for the measurement of K_d^intra for HCA**-SSEG_nSA using the observed dissociation constant (K_d^comp,Ethox) for a competing ligand, Ethox (shown as black triangle). For this thermodynamic cycle, K_d^intra = K_d^Ethox / K_d^comp,Ethox (eq 5).

Figure 6

Decrease in fluorescence of modified HCA** proteins (HCA**-SSEG_nSA, n = 0, 2, 5, 10, or 20) and HCA** (incubated with two equiv of SA-OMe) with increasing concentration of ethoxzolamide (Ethox). The intrinsic fluorescence of 200 nM of protein in 20 mM sodium phosphate pH 7.5 at 298 K with different concentrations of the fluorescence quencher Ethox was measured (excitation wavelength = 290 nm, emission wavelength = 340 nm). The data are shown after background subtraction, correction for the inner filter effect, and normalization to a maximum signal of unity (see Experimental Section). The solid curves are fits to the data using the full quadratic equation for binding (eq 7; this equation does not make the assumption that [Ethox]_free ≈ [Ethox]_total), and using a value for the minimum in fluorescence intensity of all samples of 0.3 (Figure S.1B). Error bars represent the maximum variation of an independent measurement from the mean of four experiments (duplicates from two independent experiments). Inset: The data on a logarithmic scale for the x-axis. The solid curves are fits to the data using a single-site Langmuir binding model (eq 8), which makes the assumption that [Ethox]_free ≈ [Ethox]_total — a poor assumption when n = 0 or 20 (see Experimental Section).

Figure 7

Variation of effective concentration (C_eff) and effective molarity (M_eff) with linker length (n; related to the root-mean-squared distance between the ends of the linker 〈r²〉^1/2, see eq 6) for HCA**-SSEG_nSA. The points are empirical values of M_eff (eq 1) from this study (Table 3). The solid curve is a fit to the data using the definition of C_eff (eq 4) with estimates of 〈r²〉^1/2 from eq 6. The best fit to the data was obtained with a value for d (the distance between the site of covalent attachment of the ligand to the protein and the active site of the protein; Figure 4) of 0.82 ± 0.03 nm, and for p of 0.12 ± 0.009.