Multivalent interactions play a crucial role in a variety of biological processes. (1−6) They provide an “on–off” binding at a threshold receptor density, creating a biological barcode, targeting surfaces that have a receptor density above the threshold while leaving others untouched. As a result, the multivalent binding strategy is also widely used in many bio-related applications, particularly in drug delivery (7−11) and biosensing. (12−14)
The Martinez-Veracoecha and Frenkel (MF) model provides a selectivity parameter α = d ln θ/d ln n R quantifying the dependence of targeted adsorption θ on the receptor density n R. (15) Generally, the maximum of the selectivity parameter α max, where the targeted adsorption grows fastest, is defined as the selectivity employed to characterize the overall selectivity, indicating the onset of guest nanoparticle binding and clustering. (15−17) If the selectivity α max> 1, the binding of nanoparticles is superselective, which is a signature of multivalent binding. The MF model predicts that α max increases as the binding strength becomes weaker when neglecting non-specific interactions, (15,18) which was also observed in recent experimental systems including DNA-coated colloids, (19−21) multivalent guest–host polymers, (16,22−24) and influenza virus particles. (25)
All studies mentioned above assume that the receptors grafted on the host substrate follow the Poisson distribution, considering they are random and spatially uncorrelated. However, due to the complex environment on cell membranes, the receptors are heterogeneously distributed and correlated, (26) the effect of which remains unknown. Additionally, recent breakthroughs in DNA nanotechnology offer the possibility to precisely design the spatial distribution of receptors on a substrate. (27) Here we investigate how the uniformity of receptor distribution affects the selectivity in multivalent nanoparticle binding by focusing on the hyperuniform, Poisson, and anti-hyperuniform distributions. (28) We find that the more uniformly distributed receptors lead to higher selectivity α max, and intriguingly, the maximum selectivity appears at a certain intermediate binding energy for hyperuniform distributions, which is qualitatively different from the Poisson distribution and anti-hyperuniform distributions, with α max approaching the upper bound at the infinitely weak binding energy limit. Moreover, the highest selectivity obtained for receptors of hyperuniform distributions can be significantly larger than the upper bound in the Poisson and anti-hyperuniform distributions, where the relatively large number fluctuation of receptors masks the effect and causes the selectivity to increase monotonically with decreasing binding strength.
As shown in Figure 1a, we consider that immobile receptors are grafted on a host substrate. The nanoparticles are controlled by an activity z = v 0 exp(βμ), with μ the chemical potential of nanoparticles and β = 1/k B T, where v 0 is the volume that each particle can explore when bound on the substrate, and k B and T are the Boltzmann constant and temperature of the system, respectively. Each nanoparticle is coated with κ mobile ligands, which can bind to the receptors reversibly with the binding free energy f B. The binding free energy f B is determined by both the equilibrium constant of ligand–receptor binding in solvent K a and the configurational entropy penalty due to the constraint of tethering Δ S conf: β f B = −log K a – k B–1 Δ S conf. (29) Assuming that the adsorption of each guest particle is independent, we divide the substrate into N max sites, each of which can bind with one guest particle at most. The fraction of sites that are occupied by particles with at least one bond formed is θ(z,n R)=z q(n R)1+z q (1) Using the unbound site as the reference state, the single-site bound state partition function q(n R) with n R receptors can be written as q(n R)=∑λ=1 min(κ,n R)Q(λ,n R) (2) where λ is the number of bonds formed and Q(λ,n R)=e−λ β f B κ!n R!(κ−λ)!λ!(n R−λ)! (3) Then the fraction of bound sites or adsorption is ⟨θ⟩=⟨z q 1+z q⟩⟨n R⟩ (4) where ⟨·⟩⟨n R⟩ calculates the average over the receptor number distribution with the mathematical estimate ⟨n R⟩. The selectivity parameter is defined as α=d ln⟨θ⟩d ln⟨n R⟩ (5)
Figure 1. Multivalent nanoparticle binding. (a) Schematic representation of the prototypical multivalent adsorption model, in which the ligands (blue) on the particles (white) can bind with the immobile receptors (pink) on the substrate (gray) reversibly. (b) Illustration of the κ-μ VT Monte Carlo simulation, in which some receptors (red) are bound with implicit ligands on the nanoparticles (blue) while the others (black) are unbound. In the simulation, the bonds are implicit. (c) Part of typical snapshots of receptors following various distributions. The global typical snapshots of receptors can be found in the SI.
Since the higher selectivity usually appears at small activity, (15,30,31) the time for the substrate to exchange nanoparticles with the reservoir to reach equilibrium is very long, which makes the direct Monte Carlo (MC) simulations in 3D systems expensive and inefficient. Here we propose a κ-μ VT MC simulation method with implicit ligands and bonds in 2D, which enables us to efficiently sample in the additional bond number dimension (see simulation methods in the Supporting Information (SI)). As shown in Figure 1b, we model the multivalent nanoparticles as hard disks of diameter σ and volume s hd = πσ 2/4 controlled by the chemical potential μ. We assume that one receptor can only bind with the particle covering it; i.e., the center-to-center distance between the receptor and ligand is less than σ/2. The total number of sites N max = L 2/s hd, and the average number of receptors per site ⟨n R⟩ = N R s hd/L 2. The activity z = s hd exp(βμ)/Λ 2, with Λ the de Broglie wavelength. The distribution p(n R) is numerically sampled by the number of receptors within a 2D spherical window of radius σ/2. One can see that the κ-μ VT MC simulation essentially simulates a monolayer of nanoparticles near the host substrate, where nanoparticles can bind with receptors on the substrate, and the system exchanges nanoparticles with a bulk (3D) reservoir of chemical potential μ above. The advantage of the κ-μ VT model is that one does not need to explicitly simulate the exchange of nanoparticles between the host substrate and bulk reservoir through diffusion, which could be very computationally expensive at small activity.
We consider four different types of receptor distributions: anti-hyperuniform (Anti-HU) distributions, (32) the Poisson distribution, stealthy hyperuniform (SHU) distributions (33,34) and a square lattice (Figure S1). In equilibrium, Anti-HU can describe systems close to a critical point, and SHU describes the disordered systems with long-range correlations. (32) All those distributions are statistically homogeneous point processes and follow the central limit theorem, i.e., they can be approximated by Gaussian distributions at the large ⟨n R⟩ limit. (28) The spatial uniformity of a receptor distribution at given ⟨n R⟩ can be characterized by the relative local number variance σ n R 2/⟨n R⟩. For the Poisson distribution in 2D, σ n R 2/⟨n R⟩ = 1. For a perfect square lattice, σ n R 2/⟨n R⟩ ∼ ⟨n R⟩–1/2, which essentially implies that the square lattice is more uniform than the Poisson distribution.
SHU distributions follow the same scaling with the square lattice. The configurations are generated by minimizing Φ(r N) = ∑|k|On the contrary, in Anti-HU structures, σ n R 2 increases faster than ⟨n R⟩, and here we choose configurations that exhibit σ n R 2/⟨n R⟩ ∼ ⟨n R⟩1/2. The Anti-HU configurations are generated using the algorithm detailed in ref (36). Specifically, we use the limited-memory BFGS algorithm to minimize ∑|k|In Figure 2, we plot the average bound fraction ⟨θ⟩ and the selectivity parameter α as functions of ⟨n R⟩ for various receptor distributions. One can see that eqs 4 and 5 agree quantitatively with computer simulations when ⟨θ⟩ < 0.5 (indicated by the dotted horizontal line), and at very large ⟨n R⟩, the theoretically predicted ⟨θ⟩ is larger. This discrepancy is due to the fact that the excluded volume effect between nanoparticles is not considered in the mean field theory, which overestimates the adsorption at high density. When ⟨n R⟩ is small, with increasing ⟨n R⟩, less uniform distributions lead to larger value of ⟨θ⟩. This is because that ⟨θ⟩ ≈ z⟨q⟩ according to eq 4, where the average bound state partition function over the receptor distribution ⟨q⟩ is a convex function (see SI); hence, ⟨q⟩ increases with increasing the variance of the distribution σ n R 2. As shown in Figure 2, with increasing the uniformity, i.e., from Anti-HU to the Poisson, SHU structures, and square lattice, the selectivity α max increases monotonically, and for SHU structures and square lattice, α max is even larger than κ = 4. This is intriguing as it has been accepted that weaker binding energy enhances selectivity, of which the upper bound of α max is κ. (15)Figure 2Figure 2. Receptor uniformity enhances selectivity. Average bound fraction ⟨θ⟩ (a) and selectivity parameter α (b) as a function of ⟨n R⟩ with κ = 4, β f B = −2, and βμ = −10 for various receptors distributions. The solid curves are the theoretical predictions of eqs 4 and 5, and the symbols are obtained from simulations.Achieving the Highest Selectivity by Tuning Binding EnergyIn Figure 3, we plot ⟨θ⟩ and α as functions of ⟨n R⟩ for binding energy from strong (β f B = −6) to weak (β f B = 4) of various receptor distributions. For the receptor structures of the Poisson and Anti-HU distributions, α max increases monotonically with increasing β f B, namely weaker binding enhances selectivity, while for SHU structures with χ = 0.48 and square lattice, α max reaches the maximum at about β f B = −4. To understand this, we start with the selectivity parameter α zv = d ln θ/d ln n R in the zero variance scenario, which is the uniform limit of receptors with σ n R 2 = 0. As the superselective adsorption of nanoparticles of interest mostly occurs at low activity, according to eq 1, when zq→0, θ(n R) ≈ zq(n R) and α zv ≈ d ln q/d ln n R. We define the selectivity parameter at the low activity limit as α zv,0 = d ln q/d ln n R. (37) We plot the probability of forming λ bonds on the guest nanoparticle in the bound state in Figure 4a. For weak binding β f B = 4 and small n R, one can see q(n R) ≈ Q(⟨λ⟩bound,n R), with the most probable bond number ⟨λ⟩bound ≈ 1 (yellow region in upper panel of Figure 4a). This implies that there is only one bond formed for the particle bound on the substrate, and the ligands on each particle cannot bind cooperatively. As shown in Figure 4b, this leads to a linear dependence q ≈ n R κ e–β f B and α zv,0 ≈ 1 with no superselectivity. With increasing n R, ⟨λ⟩bound approaches κ at the large n R limit due to the restriction from the number of ligands available on nanoparticles, and this leads to a power-law dependence of q on n R (see SI): q≈Q(λ=κ,n R)=e−κ β f B n R!(n R−κ)!≈(n R e−β f B)κ (6) and α zv,0 ≈ κ. Here, κ ligands on each particle bind with crowded receptors together, and the emergent combinatorial entropy induces the power-law dependence.Figure 3Figure 3. Achieving the highest selectivity by tuning binding free energy. ⟨θ⟩ and α as a function of ⟨n R⟩ for various binding free energy β f B for typical receptors structures: Anti-HU a = 10, Poisson, SHU χ = 0.48, and square lattice. The solid curves are the theoretical predictions of eqs 4 and 5, and the symbols are obtained from simulations. In all simulations, κ = 4 and βμ = −10.Figure 4Figure 4. Multivalent binding of nanopar
On the contrary, in Anti-HU structures, σ n R 2 increases faster than ⟨n R⟩, and here we choose configurations that exhibit σ n R 2/⟨n R⟩ ∼ ⟨n R⟩1/2. The Anti-HU configurations are generated using the algorithm detailed in ref (36). Specifically, we use the limited-memory BFGS algorithm to minimize ∑|k|In Figure 2, we plot the average bound fraction ⟨θ⟩ and the selectivity parameter α as functions of ⟨n R⟩ for various receptor distributions. One can see that eqs 4 and 5 agree quantitatively with computer simulations when ⟨θ⟩ < 0.5 (indicated by the dotted horizontal line), and at very large ⟨n R⟩, the theoretically predicted ⟨θ⟩ is larger. This discrepancy is due to the fact that the excluded volume effect between nanoparticles is not considered in the mean field theory, which overestimates the adsorption at high density. When ⟨n R⟩ is small, with increasing ⟨n R⟩, less uniform distributions lead to larger value of ⟨θ⟩. This is because that ⟨θ⟩ ≈ z⟨q⟩ according to eq 4, where the average bound state partition function over the receptor distribution ⟨q⟩ is a convex function (see SI); hence, ⟨q⟩ increases with increasing the variance of the distribution σ n R 2. As shown in Figure 2, with increasing the uniformity, i.e., from Anti-HU to the Poisson, SHU structures, and square lattice, the selectivity α max increases monotonically, and for SHU structures and square lattice, α max is even larger than κ = 4. This is intriguing as it has been accepted that weaker binding energy enhances selectivity, of which the upper bound of α max is κ. (15)Figure 2Figure 2. Receptor uniformity enhances selectivity. Average bound fraction ⟨θ⟩ (a) and selectivity parameter α (b) as a function of ⟨n R⟩ with κ = 4, β f B = −2, and βμ = −10 for various receptors distributions. The solid curves are the theoretical predictions of eqs 4 and 5, and the symbols are obtained from simulations.Achieving the Highest Selectivity by Tuning Binding EnergyIn Figure 3, we plot ⟨θ⟩ and α as functions of ⟨n R⟩ for binding energy from strong (β f B = −6) to weak (β f B = 4) of various receptor distributions. For the receptor structures of the Poisson and Anti-HU distributions, α max increases monotonically with increasing β f B, namely weaker binding enhances selectivity, while for SHU structures with χ = 0.48 and square lattice, α max reaches the maximum at about β f B = −4. To understand this, we start with the selectivity parameter α zv = d ln θ/d ln n R in the zero variance scenario, which is the uniform limit of receptors with σ n R 2 = 0. As the superselective adsorption of nanoparticles of interest mostly occurs at low activity, according to eq 1, when zq→0, θ(n R) ≈ zq(n R) and α zv ≈ d ln q/d ln n R. We define the selectivity parameter at the low activity limit as α zv,0 = d ln q/d ln n R. (37) We plot the probability of forming λ bonds on the guest nanoparticle in the bound state in Figure 4a. For weak binding β f B = 4 and small n R, one can see q(n R) ≈ Q(⟨λ⟩bound,n R), with the most probable bond number ⟨λ⟩bound ≈ 1 (yellow region in upper panel of Figure 4a). This implies that there is only one bond formed for the particle bound on the substrate, and the ligands on each particle cannot bind cooperatively. As shown in Figure 4b, this leads to a linear dependence q ≈ n R κ e–β f B and α zv,0 ≈ 1 with no superselectivity. With increasing n R, ⟨λ⟩bound approaches κ at the large n R limit due to the restriction from the number of ligands available on nanoparticles, and this leads to a power-law dependence of q on n R (see SI): q≈Q(λ=κ,n R)=e−κ β f B n R!(n R−κ)!≈(n R e−β f B)κ (6) and α zv,0 ≈ κ. Here, κ ligands on each particle bind with crowded receptors together, and the emergent combinatorial entropy induces the power-law dependence.Figure 3Figure 3. Achieving the highest selectivity by tuning binding free energy. ⟨θ⟩ and α as a function of ⟨n R⟩ for various binding free energy β f B for typical receptors structures: Anti-HU a = 10, Poisson, SHU χ = 0.48, and square lattice. The solid curves are the theoretical predictions of eqs 4 and 5, and the symbols are obtained from simulations. In all simulations, κ = 4 and βμ = −10.Figure 4Figure 4. Multivalent binding of nanopar
In Figure 2, we plot the average bound fraction ⟨θ⟩ and the selectivity parameter α as functions of ⟨n R⟩ for various receptor distributions. One can see that eqs 4 and 5 agree quantitatively with computer simulations when ⟨θ⟩ < 0.5 (indicated by the dotted horizontal line), and at very large ⟨n R⟩, the theoretically predicted ⟨θ⟩ is larger. This discrepancy is due to the fact that the excluded volume effect between nanoparticles is not considered in the mean field theory, which overestimates the adsorption at high density. When ⟨n R⟩ is small, with increasing ⟨n R⟩, less uniform distributions lead to larger value of ⟨θ⟩. This is because that ⟨θ⟩ ≈ z⟨q⟩ according to eq 4, where the average bound state partition function over the receptor distribution ⟨q⟩ is a convex function (see SI); hence, ⟨q⟩ increases with increasing the variance of the distribution σ n R 2. As shown in Figure 2, with increasing the uniformity, i.e., from Anti-HU to the Poisson, SHU structures, and square lattice, the selectivity α max increases monotonically, and for SHU structures and square lattice, α max is even larger than κ = 4. This is intriguing as it has been accepted that weaker binding energy enhances selectivity, of which the upper bound of α max is κ. (15)
Figure 2. Receptor uniformity enhances selectivity. Average bound fraction ⟨θ⟩ (a) and selectivity parameter α (b) as a function of ⟨n R⟩ with κ = 4, β f B = −2, and βμ = −10 for various receptors distributions. The solid curves are the theoretical predictions of eqs 4 and 5, and the symbols are obtained from simulations.
In Figure 3, we plot ⟨θ⟩ and α as functions of ⟨n R⟩ for binding energy from strong (β f B = −6) to weak (β f B = 4) of various receptor distributions. For the receptor structures of the Poisson and Anti-HU distributions, α max increases monotonically with increasing β f B, namely weaker binding enhances selectivity, while for SHU structures with χ = 0.48 and square lattice, α max reaches the maximum at about β f B = −4. To understand this, we start with the selectivity parameter α zv = d ln θ/d ln n R in the zero variance scenario, which is the uniform limit of receptors with σ n R 2 = 0. As the superselective adsorption of nanoparticles of interest mostly occurs at low activity, according to eq 1, when zq→0, θ(n R) ≈ zq(n R) and α zv ≈ d ln q/d ln n R. We define the selectivity parameter at the low activity limit as α zv,0 = d ln q/d ln n R. (37) We plot the probability of forming λ bonds on the guest nanoparticle in the bound state in Figure 4a. For weak binding β f B = 4 and small n R, one can see q(n R) ≈ Q(⟨λ⟩bound,n R), with the most probable bond number ⟨λ⟩bound ≈ 1 (yellow region in upper panel of Figure 4a). This implies that there is only one bond formed for the particle bound on the substrate, and the ligands on each particle cannot bind cooperatively. As shown in Figure 4b, this leads to a linear dependence q ≈ n R κ e–β f B and α zv,0 ≈ 1 with no superselectivity. With increasing n R, ⟨λ⟩bound approaches κ at the large n R limit due to the restriction from the number of ligands available on nanoparticles, and this leads to a power-law dependence of q on n R (see SI): q≈Q(λ=κ,n R)=e−κ β f B n R!(n R−κ)!≈(n R e−β f B)κ (6) and α zv,0 ≈ κ. Here, κ ligands on each particle bind with crowded receptors together, and the emergent combinatorial entropy induces the power-law dependence.
Figure 3. Achieving the highest selectivity by tuning binding free energy. ⟨θ⟩ and α as a function of ⟨n R⟩ for various binding free energy β f B for typical receptors structures: Anti-HU a = 10, Poisson, SHU χ = 0.48, and square lattice. The solid curves are the theoretical predictions of eqs 4 and 5, and the symbols are obtained from simulations. In all simulations, κ = 4 and βμ = −10.
Figure 4. Multivalent binding of nanopar
