Designing multivalent probes for tunable superselective targeting

Experimental Model with Tunable Features.

We performed a quantitative characterization of the host–guest interactions between self-assembled monolayers (SAMs) functionalized with guests (ferrocene or adamantane) and HA modified with the host β-cyclodextrin (β-CD) (Fig. 1A). We designed this particular model system where hosts (receptors) are attached to multivalent polymers, whereas guests (ligands) are on surfaces, to suppress undesired nonspecific polymer–polymer and polymer–surface interactions (16). We varied the guest surface density systematically, and investigated the effect of several parameters on the superselectivity of multivalent binding: affinity (i.e., binding strength of individual β-CD–guest interactions), polymer valency (i.e., the number of β-CD moieties per polymer chain), polymer linker (i.e., the linker connecting the HA backbone with β-CD moieties), and polymer concentration (in the diluted regime).

To study the effect of polymer valency, we synthesized HA–β-CD derivatives with different degrees of substitution of HA hydroxyl groups by β-CD (DS_β-CD, determined as the fraction of β-CD–functionalized disaccharides). We used a two-step synthetic procedure based on the esterification of HA hydroxyl groups with pentenoic anhydride followed by the reaction with a β-CD–thiol derivative (Fig. 1 B, 1) (16, 22). This thiol–ene coupling method provides mild and efficient functionalization of polysaccharides, and the DS_β-CD can be tuned by varying the HA disaccharide–β-CD–thiol molar ratio (22). Thus, using 0.09 and 0.30 molar equivalents of β-CD–thiol with respect to the repeating disaccharide unit of HA, we obtained HA–β-CD derivatives with DS_β-CD = 3% and DS_β-CD = 21%. These are abbreviated in the following as HA_L–β-CD_0.03 and HA_L–β-CD_0.21, where the index L reflects the presence of the extended pentenoate linker (around 1 nm contour length) between HA and β-CD. To study the effect of the linker, we also synthesized a construct with a simple amide bond between HA and β-CD (HA_noL–β-CD_0.04), using an acid–amine coupling between HA carboxylic groups and β-CD–NH₂ (Fig. 1 B, 2).

To produce surfaces with different densities of sites for the guest grafting, we formed mixed azide-terminated pegylated SAMs on gold surfaces (Fig. 1C). To produce ferrocene-terminated SAMs (SAM-Fc), we subsequently attached ferrocene using an azide–alkyne click reaction (16, 23). The surface density of ferrocene (Γ_Fc; Fig. 1 C, 3) was tuned by changing the fraction of the azide-terminated thiol in solution, and quantified electrochemically from the anodic charge associated with the conversion of Fc to Fc⁺ (Fig. 2A, Inset) (16, 23). Using this approach, Γ_Fc was varied from 0.5 to 330 pmol/cm², corresponding to root-mean-square distances between neighboring ferrocenes (l_Fc) ranging from 18 to 0.7 nm.

To study the effect of the affinity, we prepared adamantane-terminated SAMs (SAM-AD) using click grafting of AD-alkyne (Fig. 1 C, 4) to the azide-terminated SAMs. We chose adamantane because its affinity to β-CD is significantly higher than that of ferrocene: in a phosphate buffer at pH 7, K_d^AD = 10 μM (24) and K_d^Fc = 200 μM (25). The reaction of the terminal azide group is expected to change the surface hydrophobicity, which can be followed using contact angle goniometry after the surface modification. We therefore performed comparative contact angle measurements on surfaces before and after grafting of adamantane and ferrocene, respectively (Fig. 2A). In parallel, SAM-Fc samples were characterized electrochemically to determine Γ_Fc. Fig. 2 shows that the gold surfaces, which exhibited an initial contact angle of 75 ± 2°, became more hydrophilic after the formation of the pegylated SAM. As expected, the contact angles obtained for the mixed SAMs were intermediate between those of the pure monolayers of each single component (23), and surfaces became more hydrophobic after their click functionalization with hydrophobic guests. Remarkably, the evolution of the contact angles measured after immobilization of adamantane followed exactly the same trend as that obtained for SAM-Fc (Fig. 2A). This indicates that the click reaction occurs with the same efficiency in both cases and that the adamantane surface density (Γ_AD) is similar to Γ_Fc for a given thiol ratio. The match in the contact angles for identical surface coverages of SAM-Fc and SAM-AD can be explained by structural (i.e., size) and functional (i.e., hydrophobicity) similarities between Fc and AD. Based on the results of the contact angle measurements, we assumed that Γ_AD = Γ_Fc = Γ_guest and determined Γ_guest of SAM-AD electrochemically using SAM-Fc samples prepared in parallel.

The nature of the binding of HA–β-CD derivatives to guest-coated surfaces was first characterized by quartz crystal microbalance (QCM-D; SI Appendix, Figs. S2 and S3). HA binding was initially fast and then slowed down progressively (SI Appendix, Fig. S2 B and C). After 3 h of incubation, either no further binding was observed or additional binding was very slow, suggesting that equilibrium had been attained or was approached. Analysis of the QCM-D data revealed that the HA–β-CD films are typically several tens of nanometers thick and soft (SI Appendix, Fig. S2D), indicating that the surface-bound polymers form loops and/or tails dangling into the solution. The thickness was smaller than or comparable to the polymer’s radius of gyration [R_g ∼ 45 nm (16)], as would be expected for the adsorption of polymers to surfaces (26). Bound HA–β-CD did virtually not desorb during 2 h of rinsing in buffer (SI Appendix, Fig. S2 B and C); in contrast, free β-CD desorbed completely from SAM-Fc and SAM-AD within a few minutes (SI Appendix, Fig. S3 A and B). With nonspecific interactions being absent (SI Appendix, Fig. S3 C and D), we conclude that the stable binding of HA chains is the result of specific, multivalent host–guest interactions. However, because the individual host–guest interactions are reversible, the chains can dynamically rearrange on the surface thus presumably facilitating equilibration.

To characterize the sensitivity of HA–β-CD derivatives to variations in Γ_guest, we quantified the surface density of bound polymers (Γ_HA–β-CD) by spectroscopic ellipsometry (SE) over the full range of guest surface coverages (SI Appendix, Fig. S4). Fig. 3A shows plots of Γ_HA–β-CD vs. Γ_guest for the different binding scenarios sketched in Fig. 3B, i.e., (i) HA_L–β-CD_0.03 on SAM-Fc (Fig. 3A, purple), (ii) HA_noL–β-CD_0.04 on SAM-Fc (green), (iii) HA_L–β-CD_0.03 on SAM-AD (orange), and (iv) HA_L–β-CD_0.21 on SAM-Fc (blue). These particular systems were chosen to study how the selectivity of polymer binding is affected by the polymer linker [pentenoate linker (i) vs. amide bond (ii)], affinity [K_d^Fc = 200 μM (i) vs. K_d^AD = 10 μM (iii)], and polymer valency [DS_β-CD = 3% (i) vs. DS_β-CD = 21% (iv)]. In addition, we studied the effect of polymer concentration (c_HA–β-CD) by comparing system (iv) with (v) HA_L–β-CD_0.21 binding to SAM-Fc at 10-times-reduced c_HA–β-CD (cyan). The molecular characteristics of HA derivatives are summarized in Fig. 3C.

Fig. 3A shows that all HA–β-CD derivatives discriminate sharply between surfaces with different guest densities. To evaluate the selectivity of binding toward surface coverage, we use the parameter α (see ref. 3) as a measure of the relative rate of change of the number of bound nanoobjects with the relative increase in the surface density of guests $\alpha \equiv \text{d}\ln {\Gamma }_{\mathrm{HA}-\beta -\mathrm{CD}}\hspace{0.17em}/\text{d}\ln {\Gamma }_{\mathrm{guest}}$. For systems exhibiting superselectivity, α can reach values higher than 1, thus causing a faster-than-linear change in the surface density of bound objects: ${\Gamma }_{\mathrm{HA}-\beta -\mathrm{CD}}\propto {\Gamma }_{\mathrm{guest}}^{\alpha }$. In the log–log plot in Fig. 3A, straight-line segments with different values of α are shown. As the figure shows, there are superselective regions (α > 1) in all studied cases, with similar curve shapes being observed for the different systems. A detailed analysis of steepest slopes (i.e., for Γ_HA–β-CD < 30 fmol/cm²; Fig. 3A, Inset) reveals comparable maximal α-values for systems (i), (ii), and (iv), with a weighted average of 3.4 ± 0.2. A significantly higher value, α_max = 4.6 ± 0.6, is obtained in the case of decreased polymer concentration [system (v)]. The value for system (iii) (α_max = 4.9 ± 1.4) is in the same range as the other systems, although a large standard error precludes detailed comparison. The experiments show that the superselectivity range (i.e., the range of Γ_guest where α is maximal) strongly depends on the parameters of the multivalent system. Specifically, changing either the polymer linker [system (ii)], the β-CD–guest affinity [system (iii)], or the polymer valency [system (iv)] can shift the superselectivity range by more than one order of magnitude. To our knowledge, this is the first experimental demonstration that the superselectivity range is tunable over a wide range of densities of surface binding sites by the molecular design of multivalent probes.

Analytical Model.

To gain physical understanding of the observed behavior, we developed an analytical model which is an extension of the theoretical approach developed for reference system (i) (16). The model is described in detail in SI Appendix. It assumes the surface to be covered by an array of cubic cells, each of volume $a^3=\left(4\pi /3\right)R_{\mathrm{g}}^3$ and containing $n_{\text{L}}$ ligands ($n_{\text{L}}={\Gamma }_{\mathrm{guest}}{N}_{\mathrm{A}}{a}^2$ guests in our experiments). One or more polymers can bind into a given cell, and the $n_{\text{R}}$ receptors (hosts) per polymer can bind independently to the ligands in the cell. The ligand–receptor binding free energy $F=\ln \left(K_{\mathrm{d}}{a}^3\hspace{0.17em}{N}_{\mathrm{A}}\right)k_{\text{B}}T+U_{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}}+\mathrm{\Delta }{U}_{\mathrm{link}}$, which determines the probability that a particular ligand–receptor complex is formed once a polymer is in the cell, contains terms taking care of (i) the host–guest affinity and the confinement of the receptor in the cell ($K_{\mathrm{d}}{a}^3{N}_{\mathrm{A}}$), and (ii) entropic penalties related to the reduced conformational space of the polymer and the linker upon formation of a bond $\left(U_{\mathrm{poly}}\right)$, where we explicitly consider effects originating from a modification of the linker ($\mathrm{\Delta }{U}_{\mathrm{link}}$; relative to a reference system). The free-energy penalty for i polymers in a given cell due to polymer–polymer and polymer–surface repulsion is approximated by $U_i=A{i}^{9/4}+0.83k_{B}Ti$, where A (a prefactor in a scaling approximation) is expected to be of order 1 k_BT. The model also explicitly considers the gains in combinatorial entropy with increasing guest surface density or polymer valency.

To test the model against an extended range of parameters and to check its predictive ability, we used it to analyze different configurations of the host–guest multivalent system schematized in Fig. 3B. The model contains the two parameters U_poly and A that we cannot determine experimentally. We thus first fitted the reference system (i) treating U_poly and A as the sole fitting parameters. With a now fully determined set of input parameters, the model was then used to predict the behavior of systems (iii), (iv), and (v), using values of K_d, n_R, and c_HA–β-CD according to the experimental conditions (Fig. 3C). The magnitude of ΔU_link upon variation of the polymer linker is a priori unknown, and system (ii) was thus fitted using ΔU_link as the sole fitting parameter.

The results (Fig. 4 A and B and SI Appendix, Fig. S5) demonstrate that the analytical model can quantitatively reproduce the superselective behavior of the reference system (i), with the two adjustable parameters U_poly = 4.6 k_BT and A = 0.35 k_BT having the expected magnitude (i.e., on the order of 1 k_BT). Importantly, the model reproduces all trends in the quality of the superselectivity (i.e., variations in the curve shape and in the maximal α) and in the position of the superselectivity range obtained experimentally for systems (iii), (iv), and (v). In particular, the shift in the superselectivity range toward lower Γ_guest upon increasing affinity [system (iii)] is quantitatively reproduced, whereas some discrepancy remains upon increasing polymer valency [system (iv)]. The enhanced superselectivity at reduced polymer concentration [system (v)] is also well captured by the model, demonstrating a sizable increase of α_max from 3.1 to 4.8 (Fig. 4B). System (ii) was also reproduced well, yielding ΔU_link = −1.9 k_BT. This shift in free energy between systems (i) and (ii), as expected on the order of 1 k_BT, is the quantitative manifestation of the effect of the linker on the bond formation and the ensuing shift in the superselectivity range; detailed consideration of the binding free energy F reveals that replacing the extended pentenoate linker by an amide bond is equivalent to a decrease of K_d by a factor of $\exp \left(\mathrm{\Delta }{U}_{\mathrm{link}}/k_{\mathrm{B}}T\right)=6.7$.

The match of theoretical and experimental results (except close to the maximal Γ_HA–β-CD) is remarkable. It demonstrates that the developed model, although simplified, captures essential features of the complex multivalent interactions between polymers and surfaces and thus can be used to predict the superselective binding behavior of real systems. Specifically, the parameters K_d, n_R, ΔU_link, and U_poly affect exclusively the superselectivity range, in a way that is effectively predicted through the scaling parameter $x_{\mathrm{S}}={\Gamma }_{\mathrm{guest}}{n}_{\text{R}}{K}_{\mathrm{d}}^{-1}{\rho }_0{e}^{-\left(\mathrm{\Delta }{U}_{\mathrm{link}}+U_{\mathrm{poly}}\right)/k_{\mathrm{B}}T}$ (${\rho }_0=1\hspace{0.5em}\text{M}$ is the standard concentration). x_S is derived from the analytical model and is expected to provide faithful predictions as long as the fractions of occupied ligands and receptors are low (see SI Appendix for details). Indeed, all experimental data sets for a given polymer concentration essentially merge into a single master curve when Γ_guest is replaced by x_S (Fig. 4C), illustrating that our experimental systems obey this condition. In contrast, the polymer concentration (c) and the polymer size (R_g) are predicted to affect the range as well as the quality of superselectivity. This combined effect cannot be captured by a simple scaling variable, yet quantitative predictions can readily be made using the full analytical model: qualitatively, α_max increases with decreasing R_g (SI Appendix, Fig. S6A) and c (Fig. 4B and SI Appendix, Fig. S6B).

Numerical Simulations.

The analytical model is attractive, because one can readily appreciate the effect of various parameters on superselective binding. However, simplifying assumptions about the polymeric nature of the superselective probe had to be made. In particular, the deformation of the polymer upon binding to the surface was considered exclusively through the constant and empirical parameter U_poly. How do these simplifications affect the predictions of the analytical model? To test this, we additionally performed grand canonical Monte Carlo simulations with a soft-blob polymer model that explicitly considers the polymeric properties of the multivalent probe. The simulation approach is described in detail in SI Appendix and only the main results are presented here.

The simulations revealed multivalent binding of polymers and the formation of loops and tails extending from the surface (SI Appendix, Fig. S7), in agreement with the experimental results based on QCM-D analysis (SI Appendix, Fig. S2). The polymer surface densities at equilibrium obtained through simulation (Fig. 4D) matched the experimental data at high polymer surface densities well. This lends further support to our earlier conclusion that polymer binding in the experiment reached equilibrium. It also confirms that the overestimation of the maximal Γ_HA–β-CD values by the analytical model seen in Fig. 4A arises from the underestimation of U_i at high i (SI Appendix, Eqs. S4a and S4b) due to the neglected polymer deformation, as we had previously proposed (16). We also note that the maximal slope of the simulated curves (Fig. 4D) tends to be somewhat larger than what is predicted by the analytical model (Fig. 4A). Detailed analysis of the simulation data revealed that U_poly decreases with the number of bonds formed per polymer (SI Appendix, Fig. S8), i.e., this discrepancy arises from the neglected correlations between hosts within a polymer in the analytical model. It may appear surprising that the analytical model reproduces the quality of superselectivity in the experimental data better than the simulations. We hypothesize that this is fortuitous, due to the weakening of the quality of superselectivity by the finite distribution of polymer molecular weights in the experiment, which was not considered in the analytical model or in the simulations. The simulations also showed that U_poly decreases weakly with $n_{\text{R}}$ (SI Appendix, Fig. S8), which explains why the analytical model slightly underestimated the shift in superselectivity range experimentally observed when switching from low to high polymer valency [Fig. 4A; equivalent to the correction of U_poly for system (iv) by −0.7 k_BT in Fig. 4C].

Nonetheless, predictions of the analytical model for the translation of the superselectivity range along the x axis are in perfect agreement with simulations. Indeed, the same shift in the superselectivity range is obtained when the simulated binding free energy F_sim is varied by 3 k_BT (from 1 to −2 k_BT; Fig. 4D), equivalent to the 20-fold change in K_d of the experiment (from 200 μM [system (i)] to 10 μM [system (iii); Fig. 4A]. In addition, the shape of the curves remains virtually unaffected over a wide range of F_sim, a behavior that is correctly reproduced by the analytical model. Taken together, the simulations thus confirm that the analytical model indeed provides a physically reasonable approximation of reality over the experimentally studied parameter space and beyond.