Designing super selectivity in multivalent nano-particle binding Francisco J. Martinez-Veracoechea and Daan Frenkel1 Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
(MC) simulations (20, 21). However, a unified picture that explains why high selectivity is observed in some experimental realizations of multivalent systems but not in others is still lacking, even though some parts of this puzzle have received considerable attention (18, 20). Here we use a combination of numerical simulations and a simple analytical theory to understand the origin of the receptor-concentration threshold in the guest-host binding of multiligand nano-particles. To illustrate the nature of the phenomenon that we discuss, we first show a few typical snapshots obtained in our simulations (see Fig. 1). In Fig. 1 we compare the adsorption of monovalent and multivalent nano-particles to two host surfaces with receptor concentrations that differ by a factor three. Fig. 1 illustrates that the adsorption of nano-particles that bind to a surface through a single ligand increases slowly (less than a factor 2) with receptor concentration. Under the same conditions, the adsorption of multivalent nano-particles increases 10-fold as receptor concentration increases threefold. Below, we present a simple analytical model that accounts for these observations and that can be used to design multivalent nanoparticles. We then validate this model against the numerical simulations.
drug-delivery ∣ nano-medicine ∣ cancer treatment ∣ guest-host ∣ adsorption
Analytical Model As we are interested in the generic aspects of multivalency, we consider a simple but proto-typical model system, namely a solution of guest particles decorated with ligands that can bind reversibly to the receptors displayed on the surface of the host. We focus on the case where the host is much larger than the guest, which allows us to treat the surface of the host as effectively flat. For curved surfaces, the resulting behavior will be quantitatively different but qualitatively the same. For simplicity, we assume that the ligands on a given nanoparticle can only bind to receptors facing it. As different nanoparticles occupy different positions in space, they do not compete for the same receptors. In our model we account for this exclusion effect by dividing the surface of the host into N max cells, each one of which contains nR receptors. Every nano-particle has κ ligands that can bind to the surface receptors. Only one nano-particle at the time can bind to the receptors in a given cell. The number density of nano-particles in bulk solution is denoted by ρ. The binding free energy of a ligand-receptor bond is denoted by f B . In what follows we will always use the combination βf B , with β ¼ 1∕kT, where k is Boltzmann’s constant and T is the absolute temperature. For this simple model we can show (see SI Text) that the number of bound particles (N B ) or the fraction of bound guests (θ) can be written in the form:
O
ne of the key challenges in nano-medicine is to acquire the ability to design supramolecular constructs that can target surfaces that display a motif or receptor above a threshold concentration while leaving surfaces with lower coverage of such receptors unaffected (1–4). Experiments indicate that such selective behavior can be obtained using multivalency (5–8). During multivalent interactions a type of particle (henceforth referred to as the “guest”) uses multiple ligands to bind simultaneously to several of the receptors displayed by another type of particle or surface (the “host”) (9, 10). Mammen et al. (9) recognized the importance of this type of system more than ten years ago. Since then, the concept of multivalency has found numerous applications in cell biology (7, 11), supramolecular chemistry (10), nano-medicine (4, 6), immunology (12, 13), and cancer treatment (2, 3, 5, 14), to name but a few examples. The work of Davis et al. (3) provides a striking illustration of selective targeting achieved with multivalency: multivalent siRNA nano-particles administered to human patients were found to be selective in targeting cancer cells because the latter overexpress human-transferringprotein receptors. Similarly, Carlson et al. (5) exploited the multivalent binding properties of certain classes of antibodies to selectively kill tumor cells that expose high levels αv β3 integrin on their surface. There is a substantial body of theoretical work that aims to explain the pronounced enhancement in binding strength that certain multivalent systems can present in comparison with their monovalent counterparts (15–17). In particular, Kitov and Bundle (18) have pointed out that the strength of multivalent binding can be enhanced if there are many possible permutations in the binding pattern of receptors and ligands. The role of steric repulsion and conformational entropy in multivalent systems have been studied using molecular theories (19) and Monte Carlo www.pnas.org/cgi/doi/10.1073/pnas.1105351108
Author contributions: F.J.M.-V. and D.F. designed research; F.J.M.-V. performed research; F.J.M.-V. analyzed the data; and F.J.M.-V. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. 1
To whom correspondence should be addressed. E-mail:
[email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/ doi:10.1073/pnas.1105351108/-/DCSupplemental.
PNAS ∣ July 5, 2011 ∣ vol. 108 ∣ no. 27 ∣ 10963–10968
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A key challenge in nano-science is to design ligand-coated nanoparticles that can bind selectively to surfaces that display the cognate receptors above a threshold (surface) concentration. Nanoparticles 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.
CHEMISTRY
Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved May 24, 2011 (received for review April 4, 2011)
nR=2
nR=6
In the other limiting case, we assume that all the κ ligands on the guest are within reach of all the nR receptors of a “cell” (e.g., flexible ligands that are long compared with the interreceptor distance) (18). In this case the single-site bound-state partition function is given by:
Mono-valent
A
qðκ;nR ;βf B Þ ¼
minðκ;nR Þ
∑ λ¼1
κ!nR ! ; ðκ − λÞ!λ!ðnR − λÞ!
[3]
which for the limit nR ≫ κ can be given in closed formed as
B Multi-valent
qðκ;nR ;βf B Þ ≃ ð1 þ γÞκ − 1;
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 (nR ) 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.
θðκ;nR ;βf B ;zÞ ≡
NB z × qðκ;nR ;βf B Þ ; ¼ N max 1 þ z × qðκ;nR ;βf B Þ
[1]
where z measures the activity of the guest nano-particles and qðκ;nR ;βf B Þ is a partition function that describes the strength of binding between a single guest particle and a single cell of the host surface (explicit expressions are given in SI Text). In the practically important case of a dilute solution of guest particles, z ≈ ρ × vo , where vo is the volume that each nano-particle is allowed to explore while bound to a lattice site and can be easily fitted from experimental or simulation data. Note that the functional form of Eq. 1 is that of the well known Langmuir adsorption isotherm. However, this simplicity is deceptive: it is the function qðκ;nR ;βf B Þ that describes the interesting and nontrivial dependence of the adsorption probability on the surface density of receptors. As recognized by Kitov and Bundle (18) the enhancement in binding strength observed for a multivalent system stems from the increased degeneracy that bound states present when compared to the unbound state. This degeneracy determines the functional form of qðκ;nR ;βf B Þ. In the following, we consider two limiting cases and show that for both of them qðκ;nR ;βf B Þ increases much faster than linearly with the number of receptors. The derivation of the relevant expressions for qðκ;nR ;βf B Þ, together with a discussion about its dependence on geometry, is presented in the SI Text—here we only show the main results. In one limiting case, we assume that each ligand can only bind to one receptor; i.e., each ligand-receptor pair is independent from the rest. This situation can be found when rigid ligands/ receptors are small compared with their spacing. In addition, we are interested in the situation where nano-particles are coated with many ligands (i.e., κ ≫ nR ), such that the likelihood of finding more than one receptor within a ligand reach becomes vanishingly small. In such case qðκ;nR ;βf B Þ can be given in closed form as qðκ;nR ;βf B Þ ¼ exp½nR lnð1 þ e−βf B Þ − 1. 10964 ∣
e−βf B ×λ
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[2]
[4]
where we have defined the variable γ ¼ nR × expð−βf B Þ, which as we will show below is a convenient variable to use when plotting the results. In both limiting cases considered, qðκ;nR ;βf B Þ is a steeply increasing function of nR for κ > 1. Moreover, we observe (see SI Text) qualitatively similar “super selective” behavior when we consider either: nano-particles coated with a small number of long, flexible ligands, or with a large number of short ligands. In the following we will concentrate in the case of nano-particles coated with a small number of flexible ligands, though similar conclusions can be drawn from the other limiting case. Below, we validate the analytical predictions against MC simulations of nano-particles coated with a variable number (κ) of flexible ligands. Finally, we note that nR , the number of receptors per cell, may fluctuate. Such fluctuations are easily accounted for in the analytical model, if nR is Poisson distributed (see SI Text). Results and Discussion The selectivity of guest-host binding on hosts with different receptor densities can be related to the difference in binding free energy of the guest particles to the host surface (see SI Text). As we are interested in the conditions where the binding of guest particles is most sensitive to the variation in the concentration of host receptors, it is convenient to focus on the relative change in the number of bound particles with nR . Hence, we quantify selectivity with a parameter α defined as: α≡
d ln θ : d ln nR
[5]
In the SI Text we explain the relation between this quantity and the more conventional binding free energy. Super selectivity implies that the fraction of bound guest particles increases faster than linearly with the surface concentration of receptors. For nonselective adsorption, α never exceeds one and hence the fraction of bound particles varies slowly with receptor (surface) concentration over the whole range of concentrations. On the other hand, a highly selective or super selective system will display a radically different, nonmonotonic behavior: the parameter α will peak at a value larger than one around a certain threshold receptor coverage. Around this threshold value a slight change in nR will cause a rapid (nonlinear) change in the fraction of bound particles of about θ ∼ ðnR Þα . Thus, a large value of α reflects a high sensitivity of the degree of guest binding to the surface concentration of receptors. Model Predictions. Monovalent binding. In order to assess the effect of multivalent binding one should compare it to monovalent binding. In Fig. 2 we show the results obtained from our analytical model for the monovalent case (i.e., κ ¼ 1) and an activity z ¼ 0.003. Fig. 2A shows, in log-log form, θ as a function of nR . Fig. 2B shows α as function of nR . Note that, irrespective of the value of βf B , α is never larger than one, and it monotonically decreases with increasing nR . In other words, θ depends at Martinez-Veracoechea and Frenkel
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β f=-7 β f=-5
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nR Fig. 2. Results obtained from our analytical model for the monovalent case (i.e., κ ¼ 1) and activity z ¼ 0.003. (A) θ as a function of nR in a log-log scale. (B) α as function of nR . The parameter α is always less than one.
best linearly on nR . Note that it does not help to make the ligandreceptor bonds stronger (i.e., making βf B more negative): whilst this increases the absolute value of θ, it causes α to decrease even faster. In fact, it is well known that very strong ligand-receptor interactions make the guest-host binding less rather than more sensitive to the receptor-concentration. Hence, strongly binding, monovalent guest particles cannot be used for selective targeting of surfaces with a specific receptor concentration.
Origins of Selectivity. In order to understand the mechanisms by which multivalent binding systems can achieve super selectivity it is necessary to examine the role that the single-site bound-state partition function qðκ;nR ;βf B Þ plays in the fraction of bound
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β f=-1 β f=0 β f=2 β f=4 β f=7
2.5 2
α
1.5 1 0.5
Multivalent binding. Next, we consider the possibility to achieve
“super” selectivity (i.e., α > 1) with multivalent systems. In Fig. 3A, we show the parameter α for multivalent guests with a valence κ ¼ 10, as a function of the parameter γ ¼ nR × expð−βf B Þ, and for the activity z ¼ 0.003. Multivalent guests bind super selectively (i.e., α > 1) over a wide range of concentrations, displaying a nonmonotonic behavior, and reaching maximum values of up to α ∼ 2.8 at the conditions studied, though higher values can be achieved under suitable conditions. Interestingly, as the binding is made weaker (i.e., βf B more positive) the maxima are located at approximately the same value of the scaling variable γ. In addition, the peak value of α increases as binding strength is reduced, approaching a weak-binding limiting behavior. Therefore indicating that weak bonds are more selective than strong ones. The existence of a weak-binding limiting behavior is expected because as βf B becomes more positive, the location of the peak shifts to higher values of nR , thereby making relation 4 a better approximation. However, in practice, binding cannot be made arbitrarily weak as nonspecific interaction would become dominant. At low values of γ (i.e., low value of nR if βf B is fixed) θ varies linearly with concentration (α ∼ 1); however, at a certain threshold receptor coverage where the number of receptors accessible to a single guest particle becomes greater than one (i.e., γ becomes of order one), the number of bound particles increases rapidly with increasing receptor concentration, causing α to peak. After that, increasing γ leads to a rapid decrease of α, as the host
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surface becomes saturated with bound guests. This nonmonotonic behavior of α is an important result as it illustrates that multivalent guests can be designed to bind surfaces with high sensitivity to receptor concentration. In order to make this point clearer, we show in Fig. 3B a direct comparison between two monovalent guests (one that binds strongly and one that binds weakly) and a multivalent guest with weak individual bonds (i.e., βf B ¼ 2, and κ ¼ 10). The two monovalent guests provide little selectivity: their concentration varies slowly over several order of magnitude of nR . On the other hand, the multivalent guest nano-particles only bind significantly to surfaces with high values of nR , while leaving low-nR surfaces untouched. Thus, providing an almost “on-off” binding curve ideal for selective targeting applications. In contrast, the dependence of the adsorption on the bulk concentration of guest particles is not particularly sharp: at constant qðκ;nR ;βf B Þ, Eq. 1 describes normal Langmuir adsorption. This behavior, in fact, is good news: we wish the adsorption to be sensitive to the receptor surface concentration (and to the binding free energy), but less so to the bulk concentration of guest particles. The predicted Langmuir dependence of θ on the bulk concentration of multivalent guest particles has indeed been observed in experiments (11, 14, 16).
BIOPHYSICS AND COMPUTATIONAL BIOLOGY
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1 Mono-strong Mono-weak Multi
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Fig. 3. Results of the analytical model for multivalent guests with a valence κ ¼ 10 and activity z ¼ 0.003. (A) The parameter α as a function of the parameter γ ¼ nR × expð−βf B Þ. 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 “monoweak”) bind with βf B ¼ −7 and βf B ¼ −1, respectively. The multivalent guests (i.e., “multi”) binds with individually weak bonds (i.e., βf B ¼ 2), and displays an almost on-off behavior. PNAS ∣
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guests θ (see Eq. 1). The ability of multivalent guests to reach regimes of super selectivity is due to the strong increase of q with nR for large values of κ. Fig. 4A, shows the nR dependence of q for κ ¼ 1, 2, 5, and 10 for βf B ¼ 0. Fig. 4A clearly shows that even a small degree of multivalency makes q grow steeply with increasing nR . However, from the “Langmuir” Eq. 1 it is clear that, because q is in both the numerator and denominator of the expression, the change in θ (fraction of bound guests) can only benefit from the rapid variation of q if the product z × q itself is small at the onset point where q starts to grow steeply with nR . This onset occurs for nR ≥ 1 and hence super selective behavior will only be observed if z × qðκ;nR ¼ 1;βf B Þ ≪ 1. An obvious way to maintain z × qðκ;nR ¼ 1;βf B Þ ≪ 1 for large values of κ, is to decrease the bulk concentration (ρ ∼ z) of the guests. To observe super selective behavior (α > 1), q should be large while z should be small. Hence, explaining the increase in the peak value of α with decreasing z shown in Fig. 4B. Alternatively, if the bulk concentration of guest particles is fixed, the product z × q can be kept small by reducing the ligand-receptor
A 100 80 60
k=1 k=2 k=5 k=10
q 40 20
Rule of Thumb. On the basis of the discussion above, we can now
0
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C
2
1.5
α
1
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0 0.001
k=1 k=2 k=5 k=10 k=25 0.1
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nR Fig. 4. Role that the single-site bound-state partition function qðκ;nR ;βf b Þ in the phenomenon of super selectivity. (A) qðκ;nR ;βf b Þ as a function of the number of available receptors nR for κ ¼ 1, 2, 5, and 10 for βf B ¼ 0. (B) The parameter α as a function of nR for κ ¼ 10, βf B ¼ 1, and various values of the activity z. (C) The effect of varying the valence (κ) at βf B ¼ 0 and z ¼ 0.003. The maximum value of α is reached with κ ¼ 5, thus showing that increasing multivalency alone (without decreasing binding strength) not always improves selectivity. 10966 ∣
binding strength (i.e., by making βf B more positive). Indeed, Fig. 3 illustrates that weaker bonds lead to a more selective behavior. However, we stress that simply increasing the valency of the guest particle does not necessarily lead to super selectivity. In fact, as shown in Fig. 4C, increasing the value of κ at constant βf B and z may actually decrease the binding selectivity. In Fig. 4C, the most pronounced maximum in α is obtained for κ ¼ 5, while guests with κ ¼ 10 and κ ¼ 25 are less selective. This behavior shows that the design of a guest with a high sensitivity to the concentration of surface receptors is not simply a matter of increasing the multivalency. It is the subtle interplay of valency, binding strength, and guest bulk concentration that makes the current model a useful design tool. The fact that weaker bonds lead to a higher selectivity can be understood in terms of a simple physical argument. If individual binding is strong, nano-particles will bind to all receptors displayed on the surface regardless of their concentration and the surface will be saturated with bound particles before reaching the super selective regime (i.e., nR ∼ 1). Conversely, if bonds are weak and the receptor coverage is low (i.e., nR < 1), the probability of binding to the surface will be low because nano-particles can at most form one weak bond. However, as nR is increased, the number of bonds that can be formed simultaneously increases rapidly due to combinatorial reasons, thus making the overall binding strong even if the individual bonds are weak. As a consequence, we obtain a pronounced dependence of the number of bound particles with receptor coverage.
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formulate a simple rule of thumb for the design of guest particles that selectively target surfaces with a concentration of receptors greater than a critical value (i.e., CR > ccrit ). We start by noting that the variable nR in our model is directly related to the number of receptors present in an area (Ao ) that is accessible to the ligands of a single guest. Hence, we must design the guest such that its ligands can span an area Ao ∼ 1∕ccrit . In addition, we need to choose the remaining variables (κ, z, and βf B ) such that the product z × qðκ;nR ¼ 1;βf B Þ is lower than a given tolerance (e.g., ccrit where the function q will steeply grow. Finally, we note that in practice only those ligands on a guest particle that are exposed to the host surface can bind simultaneously. Therefore, the value of κ used in calculations should measure the number of ligands that can bind simultaneously to the host, rather than the total number of ligands per guest particle. Comparison Between Simulations and Analytical Model. In order to test the predictions of our model we performed MC simulations of a simple, coarse-grained model for guest-host interactions. Here we present the main features of the model: further details about the model and the simulations can be found in the SI Text (see also refs. 22, 23). The guest nano-particles are represented as hard spheres of diameter σ. Simulations were performed in a cubic box of linear dimension L ¼ 10σ and the bulk concentration of nano-particles was fixed at ρ ¼ 0.0038σ −3 . The number of ligands connected to the surface of each guest is denoted by κ. Importantly, the ligands must be flexible—they may, for instance, be grafted to the guest surface by a flexible, inert spacer. We do not model this spacer explicitly. Rather we represent it by an effective harmonic potential that tethers the ligands to the surface of the nano-particle while allowing them to freely move on the surface. The host is represented as a flat wall that cannot be penetrated by the guest nano-particles. Periodic boundary conditions are implemented in the directions parallel to the wall. The Martinez-Veracoechea and Frenkel
3
2
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A
β f=-1 (mod) β f= 0 (mod) β f= 1 (mod) β f=-1 (sim) β f=0 (sim) β f=-1 (sim)
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α
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β f= 1 β f= 0.5 β f= 0 β f=-0.5 β f=-1
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nR Fig. 5. Values of α as a function of nR 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: Nmax ¼ 73, ηeff ¼ 0.17, βf Bextra ¼ 1.53, and v 0 ¼ 0.58σ 3 .
receptors of the host are also tethered with a harmonic potential to the surface. Ligand-receptor bonds are also represented by harmonic springs. Using this model, we can simulate the adsorption of guest particles to a receptor-coated host surface. It is instructive to compare the analytical model with our simulation results. The mapping between model and simulation requires some care, as the present model assumes discrete adsorption sites, whereas the simulations make no such assumption. We therefore need to fit the model parameters. One of these parameters is N max , the maximum number of guest particles that can be adsorbed. In our model, this number is equal to the number of “cells.” The total number of receptors (N Rtot ) should be equal to the number of cells times the number of receptors per cell. However, in practice, not all receptors in a cell can bind simultaneously to a guest particle. We denote the fraction of available receptors by ηeff . Then nR¼ ηeff × N Rtot ∕N max . Also, we must account for the fact that βf B in the analytical model has to be an effective binding free energy that accounts for the loss in configurational entropy upon binding of tethered receptors and ligands: βf Bmodel ¼ βf Bsimul þ βf Bextra . In a more sophisticated model, we could estimate N max , ηeff , and βf Bextra directly—here we simply determine them from a fit of the theoretical expression to one set of numerical data. In addition, we have to fix the volume v0 defined as z∕ρ in Eq. 1. We perform this fit only once (namely for a system with κ ¼ 10 and βf Bsimul ¼ −0.5) obtaining the following values for the adjustable parameters: N max ¼ 73, ηeff ¼ 0.17, βf Bextra ¼ 1.53, and v0 ¼ 0.58σ 3 . We use these parameters to predict all other curves. In spite of the fact that our analytical model is highly simplified, it performs remarkably well. Fig. 5 shows a comparison of the predicted dependence of α on nR (solid line), we find good agreement with the simulations (symbols). This agreement suggests that our simple model captures the essential physics of super selective binding by multivalent interactions particles. An important feature of our model is that it predicts identical adsorption behavior of different systems with the same value of γ ¼ nR × expð−βf B Þ and this is precisely what is observed in Fig. 6. In Fig. 6, we show the simulation results for a system with κ ¼ 10. Fig. 6A shows N B as a function of γ, whilst Fig. 6B shows the corresponding plot of the selectivity parameter α. As predicted for weak binding, all the isotherms in Fig. 6A collapse onto a single curve and Fig. 6B shows that α always peaks around the same γ value. In practice, this result means that adsorption isotherms of multivalent particles with weak receptor-ligand bonds depend only on two parameters: the rescaled receptor surface concentration γ and the valence κ. Conclusions In this paper we have proposed and tested a simple analytical model to describe super selective behavior in systems with multi-
Martinez-Veracoechea and Frenkel
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Fig. 6. Plot of the simulation results for the κ ¼ 10 system, showing (A) NB and (B) the parameter α, as functions of the scaled receptor concentration γ ¼ nR × expð−βf B Þ. In (A) all the curves collapse. In (B) all the maxima are located roughly at the same position.
valent guest-host interactions. We found that monovalent guests bind with low selectivity to host surfaces regardless of the binding strength. Indeed, the adsorption of bound monovalent guests varies at best linearly with receptor concentration. In contrast, multivalent guest-host systems display super selective behavior: over a range of receptor coverage the adsorption varies (much) faster than linearly with the receptor concentration. In the limit of many ligands per guest particle, the adsorption approaches the on-off behavior that is ideal for selective targeting: the guests saturate surfaces with a receptor concentration above a given threshold value but do not significantly bind to surfaces with a subthreshold coverage of receptors. The threshold value of the surface concentration of receptors ccrit is determined by Ao , the effective cross section of the guest particles through Ao ∼ 1∕ccrit . A necessary condition for super selective behavior is that the single-site bound-state partition function qðκ;nR ;βf B Þ is a steeply growing function of nR . But this condition is not sufficient: super selectivity only occurs if the guest-particle activity z is such that the product z × qðκ;nR ¼ 1;βf B Þ is appreciable less than one. The predictions of the simple analytical model were tested in MC simulations of a coarse-grained off-lattice model for the multivalent guest-host system. We found that the simulations reproduced all the trends predicted by the analytical model. In summary, using the approach presented in this paper, in particular the simple rules that we formulate for the design of multivalent nano-particles, should facilitate the experimental design of multivalent nano-particles that exhibit pronounced super selective behavior. ACKNOWLEDGMENTS. This work was supported by the European Research Council (ERC) (Advanced Grant agreement 227758). D.F. acknowledges support from a grant of the Royal Society of London (Wolfson Merit Award). Research partially carried out at the Center for Functional Nanomaterials, Brookhaven National Laboratory (BNL), which is supported by the Department of Energy (DOE), Office of Basic Energy Sciences (BES), under Contract No. DE-AC02-98CH10886.
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