Linker-mediated phase behavior of DNA-coated colloids

TL;DR

This paper introduces a new approach to programmable self-assembly of DNA-coated colloids using linker oligonucleotides, revealing complex phase behavior and enabling precise control over colloidal interactions.

Contribution

It presents a novel linker-mediated paradigm for DNA colloid assembly, with a comprehensive phase diagram and a predictive theory matching experimental results.

Findings

Re-entrant melting transition observed with increasing linker concentration

Multiple linker species can be combined for complex interactions

Theoretical model accurately predicts phase behavior without fitting

Abstract

The possibility of prescribing local interactions between nano- and microscopic components that direct them to assemble in a predictable fashion is a central goal of nanotechnology research. In this article we advance a new paradigm in which self-assembly of DNA-functionalized colloidal particles is programmed using linker oligonucleotides dispersed in solution. We find a phase diagram that is surprisingly rich compared to phase diagrams typical of other DNA-functionalized colloidal particles that interact by direct hybridization, including a re-entrant melting transition upon increasing linker concentration, and show that multiple linker species can be combined together to prescribe many interactions simultaneously. A new theory predicts the observed phase behavior quantitatively without any fitting parameters. Taken together, these experiments and model lay the groundwork for future research in programmable self-assembly, enabling the possibility of programming the hundreds of specific interactions needed to assemble fully-addressable, mesoscopic structures, while also expanding our fundamental understanding of the unique phase behavior possible in colloidal suspensions.

Equations

$${\rho }_1(h)$$

$${\rho }_b(h)$$

$${\rho }_A(h)$$

$$\frac{\Delta H_0}{k_B{T}_m}-\frac{\Delta S_0}{k_B}=\ln \left[\frac{Z\pi R{v}_0}{2(\ln (\rho v_f)+1)}\frac{{\Psi }_A{\Psi }_B{C}_l^0}{C^{\circ }}\right],$$

$$C_{l,re}^0=\frac{Z\pi R{v}_0}{2(\ln (\rho v_f)+1)}{C}^{\circ }{\Psi }_A{\Psi }_B.$$

Full text

††preprint: APS/123-QED

DNA-coated colloids are one of the most promising systems for designing complex self-assembling materials Rogers et al. (2016); Di Michele and Eiser (2013); Jones et al. (2015). As in nature, the information required to specify the interactions and assembly pathways leading to a desired structure can be stored in the building blocks themselves. In the case of DNA-coated particles, this information is stored in the base sequences. In recent years, considerable progress has been made in using DNA to program the self-assembly of a variety of crystalline materials Macfarlane et al. (2011); Srinivasan et al. (2013); Macfarlane et al. (2014); Wang et al. (2015, 2017). However, experimental demonstrations of the addressable assembly of DNA-coated particles into fully prescribed structures have yet to be realized.

Recent theoretical work highlights the challenges of using DNA-coated particles for assembling prescribed materials, which need not be symmetric or periodic Hormoz and Brenner (2011); Zeravcic et al. (2014). To produce an arbitrary, complex structure with high yield from particles with specific, yet isotropic interactions, every particle must be different and have interactions chosen to favor the desired local configuration of the target structure Zeravcic et al. (2014). Furthermore, all favorable interactions must have comparable energies Hormoz and Brenner (2011). As a result, programming the assembly of even modest structures, which might contain only dozens of particles, requires specifying hundreds of unique binding interactions, all of which must have the same affinity.

In principle, DNA can encode these hundreds of interactions through careful design of the base sequences Zeravcic et al. (2014); Wu et al. (2012). In practice, however, this potential is nearly impossible to realize in systems of DNA-coated particles interacting through direct binding of their grafted strands: the steep temperature dependence of the interactions Biancaniello et al. (2005); Dreyfus et al. (2009); Rogers and Crocker (2011), the inherent uncertainty in predictions of the binding affinities SantaLucia (1998), and the inability to tune the relative interactions without resynthesizing the particles Kim et al. (2005), make matching hundreds of unique interactions intractable. While strategies have been explored to reduce the number of specific interactions, including adding directional binding and exploiting hierarchical pathways to assembly Halverson and Tkachenko (2013, 2017), these come with their own practical challenges.

An alternative approach is to design particles that interact through single-stranded DNA (ssDNA) oligomers dissolved in solution instead of through direct binding of grafted strands. Here the binding kinetics, the interaction strength, and even the interaction matrix itself could be tuned by changing the concentrations and sequences of the soluble linker strands. Furthermore, linker-mediated assembly could in principle enable the programming of large sets of specific interactions using considerably fewer unique sequences than required by systems interacting by direct hybridization. However, the practical limits of linker-based systems have not been determined and a quantitative understanding of how the linker sequences, concentrations, and surface densities of grafted molecules affect the emergent phase behavior is missing Biancaniello et al. (2005); Xiong et al. (2009, 2008); Tkachenko (2002).

In this article we combine experiments and theory to explore the phase behavior that emerges when interactions between DNA-grafted colloidal particles are encoded in soluble linker molecules. Our experiments reveal a rich phase diagram containing two previously unknown regions: (1) a re-entrant melting transition that occurs upon increasing linker concentration; and (2) a linker concentration at which coexistence between fluid and solid is stable over a wide range of temperatures. We show that the phase boundaries separating fluid and solid can be tuned by adjusting the grafting density, linker sequence, and concentration, and also demonstrate that a number of competing linker sequences can coexist in the same solution without interfering with one another, suggesting that it might be possible to encode hundreds of interactions simultaneously. Lastly, we develop a statistical-mechanical model that captures the unique phase behavior that we observe quantitatively, showing that we can predict and thus program the interactions required to direct assembly of prescribed, aperiodic structures.

Results and discussion

Linker-mediated phase behavior

Our experimental system consists of a binary mixture of micrometer-diameter colloidal particles coated with ssDNAKim et al. (2005). Each particle species bears 65-base-long, single-stranded DNA oligonucleotides which have a 54-thymine spacer and a unique 11 nucleotide (nt) “sticky end”, called either $A$ or $B$. The sticky ends are not directly complementary, but can be linked together by single-stranded oligonucleotides dissolved in solution, which are half complementary to $A$ and half complementary to $B$ (Fig. 1a). We call these strands “linkers”.

The linker-dependent interactions can be represented in a symmetric matrix (Fig. 1b), where each element of the matrix is encoded by a single linker sequence (e.g. $L_{ab}$ specifies the interactions between $A$ and $B$). As with binding due to direct hybridization, our linker-mediated interactions are temperature dependent: the particles aggregate when cooled and disaggregate when heated (Fig. 1c). We characterize the phase behavior of our system using the melting temperature $T_{m}$, which we define as the temperature at which half of the particles are completely unbound Dreyfus et al. (2009). See Supporting Information (SI) Section S1 for experimental details.

The concentration-dependent phase behavior that we find is unexpectedly rich, featuring three distinct regions upon increasing linker concentration (Fig. 2). At the lowest linker concentrations, particles do not aggregate even at room temperature (Fig. 2a, region I). At intermediate linker concentrations, the particles aggregate at low temperatures and disaggregate when heated (Fig. 2a, region II). Within this region of the phase diagram, the temperature at which the particles disassociate $T_{m}$ increases roughly monotonically with increasing linker concentration, increasing by roughly 10–15 degrees Centigrade upon increasing the linker concentration from about 100 nM to 100 $\mu$M. Above an even higher linker concentration the particles fail to aggregate completely over the entire experimental temperature range (Fig. 2a, region III). We call the linker concentration above which particles fail to aggregate the “re-entrant concentration” $C_{l,re}^{0}$.

We can understand the observed phase behavior qualitatively by considering the molecular-scale reactions that give rise to interactions between colloidal particles (see Fig. 2b). At the lowest linker concentrations, there are too few linkers to stabilize bridges linking particles together. At intermediate linker concentrations, linkers can form bridges between particles via the molecular-scale reaction $A+L+B\leftrightharpoons ALB$. Under these conditions, the melting temperature increases upon increasing linker concentration, since increasing the amount of linker shifts the local equilibrium toward the bridged conformation $ALB$. At the highest linker concentrations, we suspect that every grafted strand is bound to its own linker. Since the linkers cannot bind to other linkers, the particles become effectively passivated against assembly. We note that the state with two half bridges $AL$ and $BL$ has the same total number of base pairs as the bridged state $ALB$, thus we suspect that the solid phase within region II is actually stabilized by the entropy of the free linker strands dispersed in solution Rogers and Manoharan (2015).

The generic phase diagram that emerges illustrates two important features of linker-mediated self-assembly: 1) there is a wide ‘dynamic range’ of linker concentrations—spanning roughly 4 orders of magnitude—over which the melting temperature (and thus the interaction strength) can be tuned; and 2) there is a linker concentration above which colloids cannot self-assemble, irrespective of the temperature. We highlight that the re-entrant concentration is not simply the “equivalence point” at which there is one linker molecule for each grafted strand in the system (labeled “equivalence” in Fig. 2a). Indeed, for a grafting density of 2000 DNA/$\mu$m2 and a total particle volume fraction of 0.5$\%$, there are roughly 2000 linkers for each grafted DNA strand at the re-entrant concentration we find in experiment.

To explore these two features more fully, we perform similar experiments for different grafting densities and linker lengths (i.e. binding affinities). In both cases we find qualitatively similar phase behavior with respect to increasing linker concentration as before—the melting temperature first increases and then decreases rapidly—but the melting temperatures and boundaries between regions I, II, and III change (Fig. 3).

We find that increasing the linker affinity increases the melting temperature within region II, but does not affect the re-entrant concentration appreciably. We measure the melting temperature as a function of linker concentration for four linkers having different lengths: 17, 19, 21, and 23 nucleotides. We find that the melting temperature increases monotonically with increasing linker affinity, changing by about 20 degrees Centigrade between 17 nt and 23 nt (Fig. 3a). The re-entrant concentration, however, remains unchanged: all four linkers fail to aggregate above linker concentrations of roughly 200–300 $\mu$M (Fig. 3a).

Changing the grafting density, in contrast, has two effects: Both the melting temperature and the re-entrant concentration decrease with decreasing grafting density. Here we prepare different batches of colloids $A$ and $B$ with grafting densities ranging from roughly 20–2000 DNA/$\mu$m2 and measure their melting temperatures as a function of linker concentration (Fig. 3b). As grafting density decreases, we find that the melting temperature decreases monotonically by roughly 15 degrees Centigrade over the range we explore. We also find that the re-entrant concentration decreases with decreasing grafting density, shifting from roughly 300 $\mu$M to 90 nM (about a factor of $10^{4}$) over the 100-fold change in grafting density explored in our experiment, hinting at a squared dependence between the re-entrant concentration and grafting density.

Returning to our two observations concerning self-assembly from above, these experiments demonstrate that we can further increase the range over which we tune the interaction strength by adjusting the linker affinity, in addition to the linker concentration. We also find that we can adjust the range of workable linker concentrations by adjusting the grafting densities: higher grafting densities yield a wider dynamic range. Moreover, the the qualitative trends relating affinity and grafting density to the melting temperature and re-entrant concentration are consistent with our molecular-scale description of the phase behavior: Increasing linker affinity or grafting density (i.e. the concentrations of $A$ and $B$) should shift the equilibrium toward the bridged conformation $ALB$, stabilizing the solid phase and increasing the melting temperature.

Mean-field theory

To confirm our physical picture from above and to develop a quantitative link relating the experimental parameters to the effective interactions that emerge between colloids, we develop a mean-field theory of linker-mediated binding. Modeling the phase behavior involves two steps: 1) first we develop a model to relate the sequence, concentration, and grafting density to the multivalent free energy per particle; and 2) we relate that free energy per particle to the phase behavior of the system as a whole.

We adapt a recent approach developed in Ref. Di Michele et al. (2016), which models interactions between colloidal particles that result from multimeric ligand-receptor complexes to calculate the free energy per particle in the fluid and solid phases. Briefly, this involves computing the equilibrium densities of the different molecular species—bridges, half-bridges ($AL$ or $BL$), and unbound strands—and relating those densities to the free energy of multivalent binding between two particles Di Michele et al. (2016). In our specific case, we compute the surface densities of half bridges $\rho_{1}$ and full bridges $\rho_{b}$ from equations of local chemical equilibrium Rogers and Crocker (2011); Varilly et al. (2012), starting from the densities of unhybridized strands $A$ and $B$ ($\rho_{A}$ and $\rho_{B}$), and the concentration of free linkers ($C_{l}$):

[TABLE]

where $\Psi$ is the surface density of grafted strands, $\beta=1/k_{B}T$ is the reciprocal of the thermal energy $k_{B}T$, $\Delta G_{0}$ the hybridization free energy of pairing $A$ with $L$ or $B$ with $L$, $C^{\circ}=1$ M is the reference concentration at which $\Delta G_{0}$ is defined, and $K(h)$ is an effective area accounting for configurational costs associated with bridge Mognetti et al. (2012); Varilly et al. (2012). For simplicity, we assume that the hybridization free energies of reacting $A$ with $L$ and $B$ with $L$ are the same.

Because the grafted molecules are immobile and the densities are spatially nonuniform in the contact region between two particles, we evaluate the equilibrium densities locally. We define $h$ to be the local distance between the particles’ surfaces: $h$ is a minimum along the line of centers and extends to the maximum separation distance at which bridges can form. Since $\rho_{l}(h)$ and $\rho_{b}(h)$ are coupled for all values of $h$ by the concentration of free linkers $C_{l}=C^{0}_{l}-\rho[n_{1}+{n_{b}\over 2}]$, where $\rho$ is the density of colloids, the numbers of half-bridges $n_{1}$ and bridges $n_{b}$ per particle are calculated self-consistently by integrating $\rho_{1}$ and $\rho_{b}$ over the particles’ surfaces. See SI Section S1 for details.

Next we derive the phase boundaries by equating the chemical potential of the fluid to the chemical potential of the solid. We model the solid phase as a cluster of particles with coordination number $Z$. As is usually done for particles whose interaction range $\delta$ is much smaller than their radius ($\delta\ll R$), we use a cell model in which particles in the solid phase are assumed to move independently within a volume $v_{f}$ Sear (1999); Charbonneau and Frenkel (2007). Following the same arguments as Ref. Charbonneau and Frenkel (2007), we find that the melting temperature is given by $\Delta f_{\mathrm{coll}}/k_{B}T_{m}=\log\left(\rho v_{f}\right)+1$, where $\Delta f_{\mathrm{coll}}$ is the free energy per particle. It should be noted that the previous expression has been derived using pairwise square-well potentials with an attractive well-depth equal to $\Delta f_{\mathrm{coll}}$ and interaction range equal to $\delta$ with $v_{f}=(\delta/2)^{3}$ Sear (1999). In the present case, $\Delta f_{\mathrm{coll}}$ is not strictly pairwise since the concentration of free linkers $C_{l}$ depends on the distances between all interacting particles. However, we ignore this effect.

Our mean-field theory reproduces the phase diagram that we find in our experiments. Figure 4a shows a comparison between our experimentally measured melting temperatures and the predictions of our mean-field theory for the 19-nt linker and a grafting density of 2000 DNA/$\mu$m2. Both agree quantitatively above linker concentrations of roughly 10 nM: the melting temperature increases logarithmically from 40–50 degrees Centigrade upon increasing linker concentration up until roughly 300 $\mu$M, at which point the melting temperature plummets. We note a discrepancy between theory and experiment at the lowest linker concentrations ($<10$ nM). Here we find suspensions that are always disaggregated in experiment, whereas our theory predicts a melting temperature that decreases more slowly. We hypothesize that this disagreement is due to kinetic limitations and a difficulty for our system to equilibrate on experimental time scales.

Examining the number of multimeric complexes predicted by our mean-field theory confirms our molecular-scale description of the nature of the transitions between regions I, II, and III. Figure 4b shows the number of bridges and half-bridges ($AL$ or $BL$) in the contact region between two particles within the solid phase as a function of linker concentration for two distinct pathways through the phase diagram. Following a path at constant temperature (indicated by the blue arrow in Fig. 4a), we find that the number of bridges in equilibrium is $<1$ per contact at linker concentrations below roughly 1 nM: there are too few bridges to stabilize the solid phase. Upon increasing linker concentration, the number of bridges $ALB$ increases monotonically until linker concentrations of roughly 1 $\mu$M. Within this intermediate region, bridges greatly outnumber half-bridges. As the linker concentration is increased further, the number of bridges decreases as half-bridges take over and eventually saturate nearly all grafted molecules at the re-entrant concentration.

Taking an alternative path through the phase diagram, one which follows the phase boundary, highlights another unique feature of our system: the number of bridges at the melting transition is roughly constant above a certain linker concentration (again about $>10$ nM). Following the path indicated by the orange arrow in Figure 4a, our mean-field theory predicts that there are roughly four bridges per contact for all linker concentrations above 10 nM and below the re-entrant concentration (Fig. 4b). This prediction hints at the possibility that our experiments and theory occur in the so-called “weak binding limit”, in which the free energy per particle is approximated by the average number of bridges multiplied by the thermal energy, $\Delta f_{\mathrm{coll}}\approx-\langle n_{b}\rangle k_{B}T$ Biancaniello et al. (2005); Rogers and Crocker (2011, 2012).

Weak-binding limit: Melting temperature

We explore the possibility that our experiments might be described by a weak binding limit approximation. Specifically, we take the limit of our full theory as $\Delta G_{0}\rightarrow\infty$, resulting in a compact analytic expression for $T_{m}$:

[TABLE]

where $\Delta H_{0}$ is the enthalpy change of hybridization, $\Delta S_{0}$ is the entropy change of hybridization, $R$ is the particle radius, $\Psi_{i}$ is the grafting density of particle species $i$, and $v_{0}$ is a microscopic interaction volume defined as $v_{0}=\int K(h)dh$ (see SI Section S2 for details). Examining Equation 1, we find that the free energy of hybridization at the melting transition $\Delta G_{0}(T_{m})=\Delta H_{0}-T_{m}\Delta S_{0}$, which depends on the linker sequence, is balanced by an entropic term, which has a logarithmic dependence on the grafting densities and linker concentration—the other two independent variables in our experiment.

To test the predictions of our weak-binding limit model, we measure the melting temperatures for hundreds of unique combinations of linker lengths, linker concentrations, and particle grafting densities for the case where $\Psi_{A}=\Psi_{B}$. The data that we find shows melting temperatures ranging from roughly 25–65 degrees Centigrade, re-entrant concentrations spanning from 1 nM–300 $\mu$M, and follows the same basic trends as before: melting temperature decreases with decreasing linker concentration, decreasing grafting density, and decreasing linker length, whereas the re-entrant concentration only decreases with decreasing grafting density (see Fig. 5a).

Remarkably, we find that all 200+ measurements of the melting temperature collapse to a single master curve when rescaled according to Equation 1. Figure 5b shows the hybridization free energy evaluated at the melting temperature versus the right-hand side of Equation 1, as well as the predictions from the weak-binding-limit model (the $y=x$ line) and the full theory. We find that all data collapse to a narrow band which falls just below the weak-binding predictions. The spread in the rescaled data reflects a variation in the melting temperatures of roughly $\pm 1.5$ degrees Centigrade, which is consistent with our experimental precision of roughly $1^{\circ}$C. The offset indicates a minor discrepancy between the weak-binding model and our experimental measurements of about 3 degrees Centigrade. We note that the full theory is also offset slightly from the weak-binding limit, though by a smaller extent, and that the offset grows at the lowest grafting densities and linker concentrations explored. Here we would expect the weak binding approximation to break down since the depletion of free linkers—an effect not considered in the weak binding limit—becomes important.

Weak-binding limit: Re-entrant concentration

Next we explore the dependence of the re-entrant concentration on the grafting density in the weak binding limit. Specifically, we take the limit of our full theory when both the binding is weak and the concentration of linkers is large. Here we find the following expression (see SI Section S2 for details):

[TABLE]

We see immediately that this expression for the re-entrant concentration makes two important predictions: (1) it does not depend on the hybridization free energy and thus does not depend on the linker sequence; and (2) it scales as the product of the grafting densities of particles $A$ and $B$: higher grafting densities yield higher re-entrant concentrations. Both of these features are consistent with our observations from before (see Fig. 3), and the scaling of the re-entrant concentration with $\Psi_{A}\Psi_{B}$ is reminiscent of the squared dependence we saw previously.

To test the predictions of Equation 2 we perform two types of experiments: one in which we decrease the grafting densities of both particle species together from 2000–20 DNA/$\mu$m2 for each of the four linkers, and another in which we hold the grafting density of particle A at 2000 DNA/$\mu$m2 and decrease the grafting density of particle B for a single linker (19 nt). In both cases we measure the re-entrant concentration to a precision of roughly a factor of two.

We find that our measurements of the re-entrant concentration again collapse well when plotted against the predictions of Equation 2, confirming the dependence on the grafting densities. In both cases—either equal grafting densities or mixed grafting densities—we observe a scaling of the re-entrant concentration that goes as the product of the two densities over a range spanning roughly four orders of magnitude (Fig. 5c). The fact that our data collapses so well confirms the prediction that the re-entrant concentration depends not on the grafting density alone, but on the product of the grafting densities of the two particles. Furthermore, these predictions tell us directly the maximum linker concentration that we can use for a self-assembly experiment: it must be less than the re-entrant concentration in order for the particles to assemble. The predictions of the full theory show a similar scaling for grafting densities above 50 strands$/\mu$m2.

The weak-binding limit predictions of the melting temperatures and re-entrant concentrations (Eqs. 1 and 2) agree quantitatively with our experimental measurements and provide indispensable tools for programming self-assembly. Specifically, they provide simple, closed-form analytic expressions that predict the melting temperature and re-entrant concentration from the experimental inputs: linker sequence, linker concentration, and grafting density. Returning to our original motivation of fully addressible self-assembly, Equation 1 can be used to choose combinations of linker sequences and concentrations that would match the melting temperatures (and thus binding affinities) of dozens of pairs of interacting particles; Equation 2 can be used to prepare DNA-coated colloids with sufficiently high grafting densities such that the re-entrant concentration is higher than the intended linker concentrations.

Combining multiple linkers

Thus far we have considered a situation in which two particle species interact with one another via a single linker sequence. However, we ultimately aim to use combinations of many linker sequences to specify the complex interaction matrix between many different particle species simultaneously. For instance, to self-assemble a modest structure formed from only a few same-size particles—like the 19-particle dipyramid shown in Figure 6a—requires programming $>50$ specific interactions. Within the framework we propose, each particle species must then bind to multiple unique linker sequences—one for each pair interaction—which requires that some individual particles will need as many as 12 different linker sequences to specify their interactions with their neighbors in the final structure. This begs the question: Does adding multiple linker sequences, some of which bind to the same particle species and thus the same grafted sequence, interfere with binding or compromise the validity of our models developed above? For instance, does adding a linker sequence which binds A to C interfere with the binding of a linker which binds A to B?

We investigate the crosstalk between different linker sequences by designing a slight variant on our previously described experiments. Specifically, we modify our 19-nt linker so that it only binds to one grafted strand and replace the other half of the linker’s bases with a series of inert thymines. Thus, instead of forming bridges, our modified linkers can only form half-bridges, which inhibit assembly. We call these modified linkers “competitors.” In the context of complex self-assembly, the competitors act like other linker species that would bind particles A or B to other particle types in the same solution (Fig. 6b). Indeed, we design two of these: $L_{a}$ which binds to particle $A$ but not particle $B$; and $L_{b}$ which binds to particle $B$ but not $A$. We then mix these competitors together with our active linker $L_{ab}$—the only linker sequence that can form bridges—and investigate the effect of increasing competitor concentrations on the melting temperature of our colloidal suspension.

If the thermodynamics of linker-mediated binding were unaffected by the presence of other linkers in solution, we would not expect to observe a change in the melting temperature upon the addition of the competitors. Figure 6c shows the change in melting temperature of the binary mixture of colloids $A$ and $B$ upon addition of a single competitor species $L_{a}$ at different concentrations of the active linkers. We find that up to quite high concentrations of the competitor, the melting temperature is independent of the amount of competitor added (see Fig. 6c), even when the competitor is added in 100-fold excess of the active linker (see for example 100 nM active linker and 10 $\mu$M competitor). Indeed, up to competitor concentrations of roughly 10 $\mu$M we find that the melting temperature is constant for active linker concentrations ranging from 100 nM–100 $\mu$M. Above this concentration, the melting temperature decreases and the particles eventually melt by competitor concentrations of a few hundred $\mu$M. This behavior is independent of the amount of active linker $L_{ab}$ added to the suspension: we explore 100 nM, 1 $\mu$M, 10 $\mu$M, and 100 $\mu$M and find the same behavior in all cases.

A more stringent test of the competition between different linker species is to add both competitor linkers, $L_{a}$ and $L_{b}$, simultaneously. Here we find similar qualitative behavior: the melting temperature remains unchanged at low competitor concentrations and then decreases above some threshold until the particles no longer aggregate. Figure 6d shows the change in melting temperature versus increasing competitor concentration, rescaled by $(C_{l,cmp}^{0})^{2}/C_{l}^{0}$, where $C_{l,cmp}^{0}$ is the concentration of $L_{a}$ and $L_{b}$ (we keep the concentrations of both competitors the same). Below a rescaled concentration of roughly 10 $\mu$M, the melting temperature does not change. At higher concentrations the melting temperature decreases with increasing competitor concentration, with a transition that appears to depend weakly on the active linker concentration: the melting temperature of the highest active linker concentration (100 $\mu$M) decreases by roughly 5 degrees Centigrade at a scaled competitor concentration of about 10 $\mu$M, whereas the melting temperature of the 100-nM active linker case does not decrease until the scaled competitor concentration exceeds $>100$ $\mu$M.

At first blush, the results we find—that adding 10–100 fold excess competitor does not interfere with the assembly of $A$ to $B$ via linker $L_{ab}$—are counterintuitive. However, the predictions of our model provide an explanation. The change in the phase behavior that we observe in our competitor experiments can be understood as a competition between half-bridges, which could be formed by either active linkers or competitors, and bridges, which can only be formed by active linkers. The observation that the melting temperature is unchanged by modest concentrations of competitor suggests that bridges are considerably more thermodynamically stable than half-bridges. Indeed, returning to Figure 4b we see that bridges outnumber half-bridges up until linker concentrations of roughly 1 $\mu$M and that half-bridges do not occupy a majority of grafted sites until concentrations above 10 $\mu$M. In other words, at low competitor or linker concentrations the system would prefer to form bridges instead of half-bridges, again due to the entropy of the free linkers dispersed in solution (see SI Section S2). At higher concentrations, the situation changes and half-bridges become more thermodynamically stable than bridges. This same mechanism is responsible for the decrease in melting temperature that we observe in our competitor experiments, as well as for the re-entrant melting transition that we observed previously.

Most importantly, our findings demonstrate that multiple linker species can in fact be added together without interfering with one another, provided that the concentrations of linkers are below a threshold value. A back of the envelope estimation shows that linker-prescribed assembly of modest aperiodic structures can be accomplished using our scheme. Given that same-sized spherical particles are able to have at most 12 neighbors Hales (2005), there need not be more than 12 linker sequences that bind to the same particle type, corresponding to a competitor/linker ratio of 11/1. Our experimental data show that even a 100/1 ratio of competitor to linker does not change the melting temperature appreciably for linker concentrations in the range of 10 nM to $10~{}\mu$M. Thus we conclude that we could indeed prescribe the 60 total pair interactions needed to encode the 19-particle dipyramid in Figure 6 using 60 unique linker sequences, and then match all of their pair-interaction free energies using the theory we develop above.

Conclusions

In this work, we have shown that linker-mediated self-assembly has a number of interesting features and distinct advantages compared to self-assembly of DNA-coated colloids due to direct hybridization. First, many distinct linker sequences can be combined to specify and tune the hundreds of specific interactions needed to encode a prescribed, aperiodic structure as the only ground state in a complex mixture of same-sized colloids. Unlike DNA interactions due to direct binding, in which every orthogonal pair interaction must be specified by a different grafted sequence and the mutual interaction strengths are hardwired once the particles are synthesize, linker-mediated interactions can be tuned in situ by adjusting the linker concentrations. Furthermore, many specific interactions can be encoded between particles which are each grafted with a single sequence by creating cocktails of many linker sequences in the same solution: one linker sequence per pair interaction. Therefore, encoding every possible pair interaction between $P$ particle species in a linker-based system would require only $P$ distinct grafted sequences, whereas specifying the same number of interactions would require $P(P+1)/2$ grafted sequences in the direct-binding case.

This enhanced flexibility in linker-mediated binding results from additional degrees of freedom introduced to the system—the molar concentrations of each linker sequence—which modify the interaction free energy per particle in nontrivial ways. Importantly, we show that the influence of these new degrees of freedom can be modeled quantitatively using our mean-field theory, and can thus be programmed a priori. We stress that one technical hurdle to assembling prescribed structures from uniformly coated spheres remains: the structures must be assembled in systems containing only one of each particle species. Microfluidics-based methods have been developed to conduct experiments within such constraints Lee et al. (2014), but these considerations are beyond the scope of this article.

Second, the phase behavior of linker-mediated self-assembly is qualitatively different. Whereas the interactions between DNA-coated colloids due to direct hybridization increase monotonically with increasing DNA density, interactions between DNA-coated particles due to linker sequences dissolved in solution are non-monotonic: they first increase and then decrease upon increasing linker concentration, inducing a re-entrant melting transition in the phase diagram. Since this re-entrant transition is reproduced by our mean-field theory, which assumes local equilibrium at the molecular scale, we emphasize that the re-entrant melting transition should be generic to systems in which assembly is due to weak, multivalent binding mediated by free molecules in solution. Indeed qualitatively similar behavior is observed in a wide range of experimental systems, ranging from ‘squelching’ in gene expression Smith et al. (1996) to re-entrant condensation in proteins Zhang et al. (2008) and nucleic acids Nguyen et al. (2000) to self-assembly of virus particles Asor et al. (2017). Thus our model may find applications in a number of other settings.

Finally, while the current study examines the phase behavior that emerges in equilibrium, we highlight that linker-based systems could also be used to study non-equilibrium routes to self-assembly. For instance, our demonstration that linker-mediated phase behavior results from the local equilibrium of molecular-scale reactions opens the door to inclusion of complex DNA-based circuits and devices from DNA nanotechnology into colloidal self-assembly, such as catalytic amplifiers, cascaded circuits, and logic gates Zhang and Seelig (2011). The integration of such non-equilibrium devices, which break detailed balance, could yield schemes for error correction, adaptation, and other strategies exploited by biological systems to engineer an astonishing diversity of self-assembling materials Hopfield (1974); Hartwell et al. (1999).

Acknowledgements.

We thank Rees Garmann, Vinothan Manoharan, and Michael Hagan for fruitful discussions, and Michael Perlow and Kyra Hamel for their assistance in measuring the hybridization free energies. JL, GNP, and WBR acknowledge support from the National Science Foundation (NSF DMR-1710112) and the Brandeis MRSEC (NSF DMR-1420382). The work of BO and BMM is supported by an ARC (ULB) grant of the *Fédération Wallonie-Bruxelles * and by the F.R.S.-FNRS under grant n∘ MIS F.4534.17.