\subsection{Targeting selectivity strongly deteriorates in the presence of non-specific receptors}
\begin{figure}[h!]

\includegraphics[width=.5\textwidth]{JustClassicShort.pdf}
\includegraphics[width=.53\textwidth]{JustClassicAlphaMax}
\vspace{-1.5cm}
\caption{Top: Adsorption probability $\theta$ ( Eq.\,\ref{eq:adsorption}) as a function of the number of targeted ($\ensuremath{N_\mathrm{T} }$) and untargeted ($\ensuremath{N_\mathrm{U} }$) receptors. Results are reported for different number of ligands $\ensuremath{N_\mathrm{L} }$ and untargeted bonds energies $\ensuremath{\tilde{\Delta G}_\mathrm{U}}$ (the energy of a targeted bond is $\ensuremath{\tilde{\Delta G}_\mathrm{T}}=-3$). As the number of untargeted receptors increases, their effect becomes stronger and nanoparticles adsorb also when in the ideal case (rightmost panels), this would not occur. This problem increases with higher valencies and bond strengths (smaller $\ensuremath{\tilde{\Delta G}_\mathrm{U}}$). Bottom: Maximum selectivity $\alpha$ achievable, Eq.\,\ref{eq:alpha}, as a function of $\ensuremath{N_\mathrm{U} }$.}
\label{fig:real-binding}
\end{figure}

Fig\,\ref{fig:real-binding} illustrates the effect of non-specific interactions for nanoparticles differing in either the number of targeting ligands $\ensuremath{N_\mathrm{L} }$ or the bond strength with untargeted receptors. The rightmost panels in the top figure represent the ``ideal'' behaviour one would observe when ligands do not interact at all with untargeted receptors (see Fig.\,\ref{fig:cases}$a$). In this case, appreciable adsorption is observed only above a certain number of targeted receptors. 
When the strength of untargeted bonds increases, the effect of non-specific interactions rapidly increases too and particles also adsorb when this is not expected. In fact, above a certain number of untargeted receptors appreciable adsorption will occur even when the targeted receptors are completely absent, completely losing selectivity. 
As can be inferred by comparing nanoparticles with different number of ligands, this problem becomes more prominent as their valency increases. The observed behaviours can be qualitatively captured within a mean-field description, which provides the simple expression: 
\begin{equation}
q = \left( 1 + \ensuremath{\gamma_\mathrm{T}} + \ensuremath{\gamma_\mathrm{U}} \right)^{\ensuremath{N_\mathrm{L} }}-1; \quad \gamma_{\mathrm{x}} = \\
N_{\mathrm{x}} \exp\left(-\ensuremath{\tilde{\Delta G}_\mathrm{X}}\right).
\label{eq:mean-field-classic}
\end{equation}
Given the dependence of $\theta$ on $q$ (Eq.~\ref{eq:adsorption}) appreciable adsorption is expected whenever $q \gtrsim z^{-1}$. From Eq.~\ref{eq:mean-field-classic}, $q$ increases with both the strength of untargeted receptors and the number of ligands \ensuremath{N_\mathrm{L} }. Notably, this increase is super-linear with respect to the number of untargeted receptors whenever the valency is higher than one, predicting that selectivity will rapidly deteriorate for multivalent nanoparticles once interactions with untargeted receptors are accounted for. The same expression also shows that whenever the number of untargeted receptor is higher than an upper limit $\ensuremath{N_\mathrm{U} }^\mathrm{lim} = \exp(\beta\Delta G_\mathrm{U} ) \left[ (1/z+1)^{1 / \ensuremath{N_\mathrm{L} }} - 1\right]$, adsorption occurs even in the complete absence of {\it any} targeted receptor.\\
In most applications, multivalency have been exploited to increase the binding strength to a specific target using singularly weak-bonds. For this use, the binding strength enhancement has the positive effect to decrease the detection level of the target. However, what we show here is that exactly this same feature can provide enough binding strength to drive adsorption to the wrong targets due to the formation of many weak bonds with untargeted receptors. 
Hence, when considering targeting applications, care must be taken with multivalent architectures.\\  
The analysis of the selectivity parameter $\alpha$, reported in Fig\,\ref{fig:real-binding}, provides further insights into the extent of this problem. More precisely, for each value of $<\ensuremath{N_\mathrm{U} }>$ we consider the maximum $\alpha$ achievable within the $<\ensuremath{N_\mathrm{T} }>$ range, which provides an upper bound to the selectivity of the system.
When only targeted receptors are present, it was shown \cite{fran-pnas} that multivalent nanoparticles can achieve almost optimal on-off behaviour, i.e. $\alpha>1$ around some value of $\ensuremath{N_\mathrm{T} }$. In practice, this means that nanoparticles adsorb only when the number of receptors is higher that $\ensuremath{N_\mathrm{T} }$, and never below. This so-dubbed ``super-selectivity'' is a specific feature of multivalent particles that, importantly, monovalent ones can never achieve \cite{fran-pnas,tine-pnas,tine-jacs}. It is this advantage that suggest the use of multivalent nanoparticles for selective targeting, considering that in various diseases cells simply over-regulate the expression of certain receptors rather than expressing a mutated form that could be distinguished using an optimized ligand. 
Considering the effect of untargeted receptors, however, the picture changes. As shown in Fig\,\ref{fig:real-binding}, the super-selective regime can completely disappear, and indeed under broad conditions multivalent particles will perform worse then monovalent ones. As for the case of $\theta$, the observed trends are a decrease in selectivity with increasing valency and increasing strength of the bonds to untargeted receptors, as should be expected.\\
Given the problems previously discussed and the trends observed, a question arises: Can we simply optimize ligands to reduce non-specific interactions? The answer to this question is indeed positive, but one must solve a complex optimization problem. Ligands optimization is usually run to find the one with the highest possible binding strength for a given receptor \cite{review-translational}. This is a perfectly valid strategy if the aim is to decrease their detection threshold. This strategy, however, does not necessarily improve the ability of ligands to discern between different receptor types, which requires increasing the gap in the strength between the targeted receptor and {\it every other possible receptor type}, a much harder optimisation problem. Additionally, although increasing the strength of the bond with targeted receptors improve their detection
, it also negatively impacts the possibility to tell apart targets with different expression levels, since stronger bonds lead to appreciable adsorption for a broader range of receptors densities, as proved in \cite{fran-pnas}. In practice, ligands optimization is complicated by these contrasting requirements. 

The problems previously described arise from the very nature of multivalent interactions, and thus will affect multivalent constructs regardless of their exact specifics. 
Inspired by previous work on the self-assembly of ligand-receptor coated particles \cite{stefano-nature}, we propose here a functionalization scheme where competition between different receptors is exploited to achieve binding properties which are insensitive to the presence of untargeted receptors, and thus restore selectivity.\\
In this ``protected'' scheme, multivalent particles are functionalized with both ligands {\it and} receptors that can bind to them (see Fig.\ref{fig:cases} $d$). From here on, the latter will be called ``protecting'' receptors. %
\begin{figure}[h]
\includegraphics[width=.5\textwidth]{JustMineShort.pdf}\\
\includegraphics[width=.4\textwidth]{MineAlphaOnlyMax}
\caption{Top: As in Fig.~\ref{fig:real-binding} but for protected nanoparticles. Compared to normal particles, non-specific adsorption is strongly reduced. Bottom: Comparison of the maximum selectivity $\alpha_\mathrm{max}$ (Eq.\,\ref{eq:alpha}) between protected and normal nanoparticles, continuous and dashed lines, respectively. Colours denote different number of ligands (5, blue and 10, red). Protected nanoparticles show overall higher selectivity, and display super-selective behaviour, $\mathrm{max}(\alpha) > 1$, in regions where normal particles would perform even worse than monovalent ones. $\ensuremath{\tilde{\Delta G}_\mathrm{T}}=-3$, as in Fig.~\ref{fig:real-binding}, and $\ensuremath{\tilde{\Delta G}_\mathrm{P}}=\ensuremath{\tilde{\Delta G}_\mathrm{T}}$.}
\label{fig:myArchitecture}
\vspace{-.5cm}
\end{figure}
The effects of such design, reported in Fig.\,\ref{fig:myArchitecture}, are evident comparing with the previous results for normal nanoparticles in Fig.\,\ref{fig:real-binding}. First, the amount of untargeted receptors that protected particles can bear without affecting adsorption is greatly increased, and in fact ideal behaviour is observed for almost all receptors concentrations and binding strengths.
Moreover, super-selective adsorption ( $\alpha > 1$) is recovered in scenarios where the typical multivalent architecture shows sub-optimal behaviour even compared to monovalent particles. Furthermore, no strong deterioration of selectivity with increasing valency is observed for the protected design, whose properties are almost independent on valency.\\
Quantifying these effects requires a statistical mechanical description to properly account for the relative weight of all possible binding configurations, but they can be qualitatively understood using a simple microscopic picture.  
Ligands bound to a protecting receptor cannot concurrently bind those on the surface. However, the system is in a dynamic equilibrium where bonds continuously break and form, and on average bonds with surface receptors will occur if their formation is thermodynamically favourable compared to those with protecting receptors. This is typically so only for bonds with the targeted receptors for which ligands are optimised to have a high binding strength, not for others. It is indeed this mechanism that leads to high selectivity for this scheme.
The equilibrium behind it is controlled by the relative energies of these bonds, $\ensuremath{\tilde{\Delta G}_\mathrm{T}}-\ensuremath{\tilde{\Delta G}_\mathrm{P}}$ and $\ensuremath{\tilde{\Delta G}_\mathrm{U}}-\ensuremath{\tilde{\Delta G}_\mathrm{P}}$ ($P$ meaning protecting receptors). It also depends on the number of ligands, protecting, and surface receptors, since their quantity affect the relative entropy of binding. In fact, within a mean field model, one could simply regard the number of ligands or receptors as an additional energy contribution of $\ensuremath{\tilde{\Delta G}}_x = - \log N_x$ to be added to the true bond energy. 
The microscopic picture we presented easily describes other trends in this system, such as the increase in selectivity with increasing number of protecting receptors, and with stronger protecting bonds (see the Supplementary Information).
It also explains the increase in the selectivity parameter $\alpha$ for the protected scheme: protecting bonds shift the {\it effective} bond energy without reducing the difference between targeted and untargeted receptors. 
In this way, they boosts super-selectivity because they provide a mechanism to make targeted bonds effectively weaker and weaker without however increasing the contribution of non-specific interactions, which would instead come into play with a normal design. In practice, protecting receptors allow to reduce a realistic multi-receptor scenario to an ideal one where only targeted receptors are present.
 %




Before concluding, we discuss a possible practical realisation of this ``protected'' architecture. Although this scheme is very general and various solutions can be envisaged, we would like to focus on a specific system which we deem very promising, colloids coated with DNA (DNACCs). 
Protected DNACCs have already been experimentally implemented to study self-assembly \cite{mirjam-self,lorenzo-pnas1}. This system, where DNA acts both as ligand and receptor, represents in itself a very interesting and powerful solution. DNACCs are already under intense investigation in drug-delivery and biosensing \cite{mirkin}, but even more importantly DNA can bind to a large variety of molecules. Short DNA strands, so-called ``aptamers'', are widely used as targeting ligands for proteins, and a wide literature exists regarding their optimization \cite{review-translational}. 
Such knowledge combined with extensively validated models to calculate DNA-DNA binding energies \cite{santalucia,nupack-algo} can be used as a general and powerful platform for the design of protected targeting constructs. 
One simple way to implement the proposed scheme would be to functionalize nanoparticles with both the targeting aptamer and its complementary DNA sequence, which would act as its protecting receptor. DNA would also provide an easy way to tune the strength of protecting bonds, by simply changing the nucleotides sequence of protecting strands.\\

In conclusion, in this Letter we addressed selective targeting using multivalent nanoparticles via a statistical mechanical model. This was done by combining previous results \cite{fran-pnas,stefano-jcp} in a unified theory, allowing to unravel the physics behind non-specific binding arising from the presence of multiple receptor types.
We showed how multivalent particles are more sensitive to non-specific binding compared to monovalent ones, and that their selectivity rapidly deteriorates in presence of different receptors types.
To solve this problem, we proposed and justified a simple design principle: Beside targeting ligands, nanoparticles should also be coated with protecting receptors able to bind them, shielding their interactions with untargeted receptors. We showed that this mechanism prevents non-specific adsorption, and increases selectivity under conditions where typical nanoparticles would not work. Protecting receptors have been previously shown to enhance the self-assembly kinetics of DNACCs \cite{stefano-nature}, a very different problem from the one treated here.
They could also similarly prove beneficial for other soft-matter systems, e.g. in patchy particles on substrates when competing interactions are present \cite{patchy}. We finally stress that, in the context of nanomedical applications,
the problem and the solution proposed here do not apply only to nanoparticles, or to cells targeting. Instead, they emerge whenever multivalent constructs are used, e.g. in functionalized polymers or surfaces that recognize specific biomarkers via multiple binding sites.\\

I would like to acknowledge Daan Frenkel, Tine Curk, Bortolo Mognetti, Jure Dobnikar and Nicolas Tito for reading the manuscript and discussing multivalent targeting, and all members of the GCS group of the University of Milano-Bicocca for the feedback. I also acknowledge the BAIC-SME in Beijing for financial support.




