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Linker-mediated self-assembly of mobile DNA coated colloids

An immense challenge in materials sciences is to find a way to construct materials with absolute control over the placement of each building block in order to tailor properties for given applications. DNA coated colloids offer the possibility of realizing programmable self-assembly, which in principle can assemble almost any structure in equilibrium, while remains challenging experimentally. Here we propose a new system of linker-mediated mobile DNA coated colloids (linker-mediated mDNACCs), in which the interaction between mDNACCs is induced by the bridging of free DNA linkers in solution, whose two single stranded DNA tails can bind with specific single stranded DNA receptors of complementary sequence coated on the colloids. We formulate a mean field theory to efficiently calculate the effective interaction between mDNACCs mediated by free DNA linkers, in which the entropy of DNA linkers plays a non-trivial role. Especially, when the binding between free DNA linkers in solution and the corresponding receptors on mDNACCs is very strong, the linker-mediated colloidal interaction is determined by the linker entropy, which depends on the concentration of free DNA linkers. As the concentration of free DNA linkers can be precisely controlled in experiments, this suggests a new way for experimentally addressable self-assembly of DNA coated colloids.

The ultimate goal of self-assembly is programming many distinct building blocks, and each of them occupies a specific location within a self-assembled structure [1]. The recent development of DNA nanotechnology offers possibilities of programmable self-assembly using the specific hybridization between single stranded DNAs (ss-DNAs) [2][3][4][5]. This works very well in programmable self-assembly of DNA bricks, in which a variety of designed superstructures consisting of thousands of preprogrammed DNA bricks were fabricated [6][7][8]. However, similar ideas were not well applied in the designed selfassembly of DNA coated colloids (DNACCs). One of the major challenges is that typically each colloid is coated with many DNA linkers, and the effective colloidal interaction mediated by DNA hybridization changes abruptly with temperature, which makes the system difficult to reach the equilibrium ordered state [1]. Thus, only a few groups were able to obtain 3D crystals of DNACCs [9][10][11][12][13][14][15][16]. This is particularly detrimental for designed selfassembly of colloidal superstructures, where the temperature window for high-yield self-assembly narrows down quickly with the increasing structure size [17]. In this work, we propose a new system of linker-mediated mobile DNA coated colloids (linker-mediated mDNACCs). Experimentally, one can fabricate mDNACCs by grafting DNA linkers onto the lipid-bilayer coated on the colloids [18,19], which makes the grafted linkers mobile on colloidal surface. It was found that the phase diagram of conventional mDNACCs is not qualitatively different from the corresponding immobile DNACCs, where the freezing colloidal density drops to zero quickly with increasing the binding strength between ssDNAs [20]. Here we find that if the ssDNAs grafted on different colloids do not bind with each other directly, but rather through the bridging of free DNA linkers in solution, the induced attraction does not diverge at the strong binding limit, i.e., low temperature limit. The reason is due to the special entropic effect in the strong binding limit making the linker-mediated attraction between mDNACCs finite and solely depending on the concentration of free DNA linkers in solution, which can be well controlled over orders of magnitude experimentally.
We consider an equimolar AB-type binary system of volume V consisting of N mDNACCs (hard spheres) of radius R in the solution containing free DNA linkers of chemical potential µ. Each mDNACC is coated with n α A or B type mobile ssDNA receptors of length r c , which can bind to the ssDNA tail of complementary sequence on a free DNA linker in solution with the binding free energy ∆G bind (Fig. 1a). Free DNA linkers are modelled as infinitely thin hard rods of length l with two ssDNA ends of length r c (Fig. 1b). Here we assume R l r c , and the free DNA linkers in solution and the mobile ssDNA receptors move much faster than colloids. During the motion of colloids, all linkers and receptors reach equilibrium quickly. One can write down the partition function and calculate the free energy of the linker system using the saddle point approximation. This can be used as the effective interaction between mDNACCs mediated by linkers [21][22][23], which has the contributions from both bonded linkers on mDNACCs and unbound free linkers in solution. We assume that the coating density of ss-DNA receptors and the concentration of DNA linkers in solution are low, and except the binding between complementary ssDNAs, the interaction between them is negli- gible. The partition function for the bonded DNA linkers on mDNACCs is where m α and x αγ are the numbers of linkers bonded with the receptors on particle α with one free end and linkers bridging between particles α and γ, respectively. Here β = 1/k B T with k B the Boltzmann constant and T the temperature of the system, respectively. W ({m α , x αγ }) accounts for all possible combinations of hybridization of {m α , x αγ } (see Supplementary Information): (2) ξ a and ξ b are the partition functions for the states of linkers only bonded to one mDNACC and bridging between two mDNACCs, respectively. Linkers bonded to mDNACCs can stay in two different states: (i) state a, (only one end of the linker is bonded to an mD-NACC with the other end unbound); (ii) state b (two ends of the linker are bonded to two different mD-NACCs). Using the free energy of the bonded linkers F({m α , x αγ }), we can re-write Eq. 1 into Z = {mα,xαγ } exp (−βF({m α , x αγ })), and with the saddle point approximation ∂F({m α , x αγ })/∂{m α , x αγ } = 0, we obtain m α =n α ξ a e βµ , withn α = n α − m α − γ x αγ the number of unbound free ssDNA receptors on particle α. The free energy of bonded DNA linkers at the saddle point is wheren α and x αγ are the solution to Eq. 3. We introduce a reference linker state a in the dilute limit of mDNACCs, in which the linker is in state a but not interacting with other mDNACCs, and its partition function is ξ a = V a exp (−β∆G bind ) with V a the configurational volume that the linker in state a can explore. At a finite colloidal concentration, the existence of neighbouring colloids influences the free volume of linkers in state a, which induces a repulsive free energy F rep , and ξ a = ξ a exp(−βF rep ). Similarly, for the bridging linkers in state b, ξ b = ξ a exp[−β(∆G bind + F cnf )], where F cnf is the conformational free energy of the linker bridging between two mDNACCs. Here F rep and F cnf can be calculated exactly at r c → 0 for systems of rigid DNA Besides the free energy contribution from the linkers bonded on mDNACCs, the unbound free linkers in solution also contributes a depletion effect U dep [24,25] (see Supplementary Information), and the resulting linkermediated effective interaction between mDNACCs can be written as wheren α is the number of unbound receptors on an isolated particle α, i.e., no bridge formed on α, in the reservoir of free linkers with chemical potential µ. We perform simulations with explicit DNA linkers to verify Eq. 5. As the system spans multiple length scales, direct simulations of many mDNACCs with explicit DNA linkers are prohibitively expensive, and we choose to perform thermodynamic integration (see Supplementary Information) to calculate the linker-mediated effective interaction between two different mDNACCs (like in Fig. 2c). We plot the calculated linker-mediated effective interaction βU eff (r) between two colloids coated with the same number but different types of mobile ss-DNA receptors compared with the theoretical prediction of Eq. 5 in Fig. 1c for various µ. One can see that the numerically calculated effective interaction between two mDNACCs agrees well with the theoretical prediction, and Eq. 5 generally predicts a slightly stronger attraction than the one measured in direct simulations. The reason is that the analytical form of F cnf is only exact at r c → 0, while for finite small r c , it slightly overestimates the conformational entropy for bridging linkers. This leads to the overestimation on the amount of bridging linkers and stronger attraction (see Supplementary  Information).
As shown in Fig. 1c, βU eff (r) is negative and attractive at 2R < r < 2R + l with the minimum located around 2R+2r c , whose magnitude indicates the strength of attraction between two mDNACCs. Interestingly, as shown in Fig. 2a, with increasing βµ from −10 to about 0, the attraction between two mDNACCs first becomes stronger, i.e., βU eff (2R + 2r c ) becomes more negative, while further increasing βµ makes the attraction weaker. Simultaneously, the probability of forming bridges between two colloids p b α (2R + 2r c ) = x αγ /n α first increases and then decreases with increasing µ (Fig. 2b). Typical snapshots from direct simulations at various µ (Fig. 2c) show that at both very low and high linker concentrations, there are very few bridges formed between colloids, while many bridges form at certain intermediate linker concentration. It is easy to understand that very few bridges form at small µ, as there are limited linkers available to bridge between mDNACCs, while it is not trivial that the number of bridges decreases at large µ.
To understand this, we refer to Eq. 3, which implies When µ → ∞, all ssDNA receptors on mDNACCs are bonded, i.e.,n γ → 0, and ξ a and ξ b are finite numbers independent with µ, which implies γn γ ξ b /ξ a → 0. On the left side of Eq. 6, m α changes with µ but remains finite leading to γ x αγ → 0, which suggests that at very high linker concentration, there is no bridge formed between mDNACCs and explains the drop of p b α (2R + 2r c ) at large µ. This suggests a re-entrant melting of mDNACCs with increasing µ. To demonstrate this, we perform MC simulations with the effective interaction of Eq. 5 for an equimolar binary mixture containing N = 864 mDNACCs at the packing fraction η = 4N πR 3 /3V = 0.1 with various µ, where we regard a colloid as a singlet if there is no bridge formed on it. One can see that with increasing µ, the singlet fraction first drops then increases at large µ. Similar reentrant melting was also found in a recent work of linkermediated immobile DNACCs [26], which suggests that re-entrant melting is a general phenomenon in linkermediated DNACCs and independent with the mobility of DNA receptors on colloidal surface. Furthermore, we investigate the effective interaction between mDNACCs with changing the binding strength between DNA linkers and corresponding receptors, i.e., decreasing ∆G bind . One might think that the stronger binding between DNA linkers and receptors would naturally make the attraction between mDNACCs stronger and eventually diverge at ∆G bind → −∞. Intriguingly, however, as shown in Fig. 4a, at ∆G bind → −∞, it seems that βU eff (2R + 2r c ) does not diverge but reaches a plateau depending on µ. The MC simulations for systems of many mDNACCs also show that the singlet fraction drops with decreasing ∆G bind and reach a plateau at the strong binding limit (Fig. 4b). To understand this counter-intuitive phenomenon, we modifies our mean field theory as follows (see Supplementary Information). When ∆G bind → −∞, all receptors on mDNACCs are occupied, i.e.,n α = 0, and using the saddle point approximation, one can write down the free energy of the bonded DNA linkers as βF inf = α n α log m α n α ξ a e β(∆G bind +µ) where Here the 'enthalpy' term α n α β∆G bind is independent with colloidal configuration and can be neglected, and ξ a e β(∆G bind +µ) = V a e β(µ−Frep) is the entropy part of the partition function, and independent with ∆G bind . Therefore, the resulting effective colloidal interaction at strong binding limit is which solely depends on entropy. We plot the prediction of Eq. 9 as dashed lines in Fig. 4a, which quantitatively agrees with the converged plateau of βU eff at strong binding limit. Moreover, in Fig. 4c, we plot βU −∞ eff as a function of µ for various n α , and one can see that βU −∞ eff increases with increasing µ or decreasing n α . To explain this, we consider the effective pair interaction between two fixed mDNACCs α and γ with n α = n γ (like in Fig. 2c) at ∆G bind → −∞, where m α = m γ , and m α +x αγ = n α . Eq. 8 implies x αγ /m 2 α = ξ b /[ξ 2 a exp(βµ)], and ξ b/a is independent with µ and n α . With increasing µ, ξ b /[ξ 2 a exp(βµ)] decreases, and at fixed n α , x αγ becomes smaller, which implies less bridges formed between α and γ, and the less negative or larger βU −∞ eff . Similarly, at fixed µ, x αγ /m 2 α = ξ b /[ξ 2 a exp(βµ)] is a constant, and the smaller n α leads to the smaller x αγ and the larger βU −∞ eff .
In conclusion, we have proposed a linker-mediated mD-NACC system, in which the interaction between mD-NACCs is bridged by the free DNA linkers in solution. We formulate a mean field theory to calculate the effective interaction between mDNACCs, which well agrees with numerical simulations with explicit DNA linkers. The mean field theory can be further used to construct MC simulations for efficiently simulating the collective self-assembly of linker-mediated mDNACCs. Furthermore, combined with analytic theories and numerical simulations, we find novel entropic effects in linker-mediated mDNACC systems. First, with increasing the concentration of free DNA linkers from zero, the linker-mediated effective interaction between mDNACCs changes non-monotonically, and the strongest colloidal interaction appears at some intermediate free linker concentration, which induces a re-entrant melting in linkermediated mDNACCs. Moreover, at fixed free linker concentration, with increasing the binding strength between free linkers in solution and the receptors on mDNACCs, we find that the linker-mediated attraction between mD-NACCs becomes stronger and reaches a plateau at the strong binding limit, i.e., ∆G bind → −∞. This is due to the fact that at the strong binding limit, all receptors on mDNACCs are bonded. Therefore, whether forming bridges between mDNACCs does not change the 'enthalpy' of the system, and the linker-mediated interaction between mDNACCs is dominated by the entropy of DNA linkers, which depends on the concentration of free linkers in solution. As the concentration of free DNA linkers can be well controlled in experiments, this suggests a new way to precisely tune the colloidal interaction for addressable assembly of DNA coated colloids to avoid the abrupt change of colloidal interaction with temperature in conventional systems of DNACCs. Compared with existing entropy driven methods for addressable assembly of DNACCs [27,28], the advantage of using linker-mediated mDNACCs is that to encode all possible specific interactions between N different mDNACCs would need only N distinct grafted sequences [26] instead of N (N − 1)/2 in the other systems, which is particularly important for large N [29], and each specific interaction can be easily switched on or off in situ by introducing or removing the corresponding free DNA linkers. Moreover, the mobile feature of the receptors coated on colloids makes it possible to derive the close form for effective interactions between colloids (like Eq. 5 and 9) to efficiently simulate, investigate and design the collective self-assembly, for which direct simulations with explicit linkers are pro-hibitively expensive.