Designing super selectivity in multivalent nano-particle binding

Abstract

A key challenge in nano-science is to design ligand-coated nano-particles that can bind selectively to surfaces that display the cognate receptors above a threshold (surface) concentration. Nano-particles that bind monovalently to a target surface do not discriminate sharply between surfaces with high and low receptor coverage. In contrast, "multivalent" nano-particles that can bind to a larger number of ligands simultaneously, display regimes of "super selectivity" where the fraction of bound particles varies sharply with the receptor concentration. We present numerical simulations that show that multivalent nano-particles can be designed such that they approach the "on-off" binding behavior ideal for receptor-concentration selective targeting. We propose a simple analytical model that accounts for the super selective behavior of multivalent nano-particles. The model shows that the super selectivity is due to the fact that the number of distinct ligand-receptor binding arrangements increases in a highly nonlinear way with receptor coverage. Somewhat counterintuitively, our study shows that selectivity can be improved by making the individual ligand-receptor bonds weaker. We propose a simple rule of thumb to predict the conditions under which super selectivity can be achieved. We validate our model predictions against the Monte Carlo simulations.

Fig. 1.

Simulation snapshots comparing the targeting selectivity of monovalent and multivalent guest nano-particles. We compare the adsorption onto two host surfaces with receptor concentrations (n_R) that differ by a factor of three. (A) The monovalent guests provide little selectivity: increasing by three times the receptor coverage just increases the average number of bound guests by 1.8 (i.e., from 5.4 to 9.7 bound particles in average). (B) The multivalent nano-particles behave super selectively: an increase of three times in receptor coverage causes a 10-fold increase in the average number of adsorbed particles. (i.e., from 2.5 to 25.4 particles). The multivalent guests have ten ligands per particle. The individual bonds of the multivalent nano-particles are 5kT weaker than the monovalent ones.

Fig. 2.

Results obtained from our analytical model for the monovalent case (i.e., κ = 1) and activity z = 0.003. (A) θ as a function of n_R in a log-log scale. (B) α as function of n_R. The parameter α is always less than one.

Fig. 3.

Results of the analytical model for multivalent guests with a valence κ = 10 and activity z = 0.003. (A) The parameter α as a function of the parameter γ = n_R × exp(-βf_B). The parameter α shows a maximum where the number of bound particles is extremely sensitive to changes in receptor concentration (B) Direct comparison between two monovalent guests and a multivalent guest. The two monovalent guests (i.e., “mono-strong” and “mono-weak”) bind with βf_B = -7 and βf_B = -1, respectively. The multivalent guests (i.e., “multi”) binds with individually weak bonds (i.e., βf_B = 2), and displays an almost on-off behavior.

Fig. 4.

Role that the single-site bound-state partition function q(κ,n_R,βf_b) in the phenomenon of super selectivity. (A) q(κ,n_R,βf_b) as a function of the number of available receptors n_R for κ = 1, 2, 5, and 10 for βf_B = 0. (B) The parameter α as a function of n_R for κ = 10, βf_B = 1, and various values of the activity z. (C) The effect of varying the valence (κ) at βf_B = 0 and z = 0.003. The maximum value of α is reached with κ = 5, thus showing that increasing multivalency alone (without decreasing binding strength) not always improves selectivity.

Fig. 5.

Values of α as a function of n_R predicted by the model (solid line) and compared with the simulations (symbols). The agreement is good considering the simplicity of the model. The adjustable parameters have the following values: N_max = 73, η_eff = 0.17, βf_{B_extra} = 1.53, and v₀ = 0.58σ³.

Fig. 6.

Plot of the simulation results for the κ = 10 system, showing (A) N_B and (B) the parameter α, as functions of the scaled receptor concentration γ = n_R × exp(-βf_B). In (A) all the curves collapse. In (B) all the maxima are located roughly at the same position.