Linker-mediated self-assembly of mobile DNA-coated colloids

Model and mean-field theory

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 i is coated with n_i 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 ΔG_bind (Fig. 1A). Free DNA linkers are modeled 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 (22–24), which has the contributions from both bonded linkers on mDNACCs and unbound free linkers in solution. If we assume that the excluded volume effect between the grafting sites of ssDNA receptors on the same particle can be approximated using the hard disk–like repulsion on a 2D surface, when the area fraction of the grafted ssDNA receptors on the particle is less than 5%, then the compressibility factor is less than 1.1 (25), and the interaction between the grafting sites can be approximated as an ideal gas. Moreover, we assume that the concentration of DNA linkers in solution is low, and except the binding between complementary ssDNAs, the interaction between them is negligible. These assumptions are typical in experiments. Thus, the grand canonical partition function for the bonded DNA linkers on mDNACCs isZ({mi,qij})=∑{mi,qij}W({mi,qij})ξa∑imiξb∑i∑j>iqijeβμ(∑imi+∑i∑j>iqij)(1)where m_i and q_ij are the numbers of linkers bonded with the receptors on particle i, with one free end and linkers bridging between particles i and j, respectively. Here, β = 1/k_BT with k_B the Boltzmann constant and T the temperature of the system, respectively, and μ = k_BT log ρ with ρ the concentration of free linkers in the reservoir. W({m_i, q_ij}) accounts for all possible combinations of hybridization of {m_i, q_ij} (see section S1)W({mi,qij})=∏ini!mi!(ni−mi−∑jqij)!∏j>iqij!(2)where 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 mDNACC with the other end unbound) and (ii) state b (two ends of the linker are bonded to two different mDNACCs). We introduce a reference linker state a^′ in the dilute limit of mDNACCs, in which particles are at infinite distance from each other, so that bridges cannot form, and the linker is in state a but not interacting with other mDNACCs. Its partition function is ξ_a^′ = V_a^′ exp (−βΔG_bind) with V_a^′ the configurational volume that the linker in state a′ can explore. At a finite colloidal concentration, the existence of neighboring 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(ΔG_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 linkers and otherwise computed with Monte Carlo (MC) simulations for semiflexible polymeric linkers (see section S2).

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Using the free energy of the bonded linkers ℱ({m_i, q_ij}), we can rewrite Eq. 1 into Z = ∑{m_i, q_ij} exp (−βℱ({m_i, q_ij})), and with the saddle point approximation ∂ℱ({m_i, q_ij})/∂{m_i, q_ij} = 0, we obtain{mi=n¯iξaeβμ,qij=n¯in¯jξbeβμ(3)with n¯i=ni−mi−∑iqij the number of unbound free ssDNA receptors on particle i. p¯i is the probability to find an unbound ssDNA receptor on particle i, andp¯i+mini+∑jqijni=1(4)

Combining Eqs. 3 and 4, we obtain a set of self-consistent equations1p¯i=1+ξaeβμ+∑jp¯jnjξbeβμ(5)

Then, the free energy of bonded DNA linkers at the saddle point isβF=∑i(nilog p¯i+12∑jqij)(6)where p¯i and q_ij are the solution to Eq. 5 (see section S3). This resulting effective potential shares the same form as the one in conventional mDNACCs (22), while the physics is different.

Besides the free energy contribution from the linkers bonded on mDNACCs, the unbound free linkers in solution also contributes a depletion effect U_dep (26, 27) (see section S2), and the resulting linker-mediated effective interaction between mDNACCs can be written asβUeff=∑i[nilog (p¯ip¯i′)+12∑jqij]+βUdep(7)where p¯i′ is the probability of finding an unbound ssDNA receptor on an isolated particle i, i.e., no bridge formed on particle i, in the reservoir of free linkers with chemical potential μ. We note that the mean-field approach used here is only meaningful if ΔG_bind is on the scale of a few k_BT (22–24); otherwise, kinetic effects need to be taken into account (28, 29). To check the validity of our mean-field results in experiments, one can compare the time it takes for a single colloid to diffuse its own radius t_diffusion, with the time scale of bond formation t_on and bond breaking t_off, and ensure t_diffusion ≫ t_on + t_off.

Numerical verification of the mean-field theory

We perform MC simulations with explicit DNA linkers to verify Eq. 7. 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 integrations (see Methods) 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 ssDNA receptors compared with the theoretical prediction of Eq. 7 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. 7 generally predicts a slightly stronger attraction than the one measured in direct simulations. The reason causing the discrepancy is twofold. First, it is known that the mean-field approach developed here generally predicts a slightly stronger attraction between mDNACCs compared with direct simulations with explicit linkers, as the mean-field theory assumes that the probability for any ssDNA receptor coated on colloids to be unbound is uncorrelated to the probability of an ssDNA tail on a free linker to be unbound (24, 30). The difference is visible when the number of ssDNA receptors or the free DNA linkers in the system is very small, which is not the case in our simulations. Second, 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. We believe that the overestimation of F_cnf is the major (but not the only) reason responsible for the overestimation on the amount of bridging linkers and stronger attraction (see section S2).

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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. 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 pib(2R+2rc)=qij/ni 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∑jqijmi=∑jn¯jξbξa(8)

When μ → ∞, all ssDNA receptors on mDNACCs are bonded, i.e., n¯j→0, and ξ_a and ξ_b are finite numbers and do not depend on μ, which implies ∑jn¯jξb/ξa→0. On the left side of Eq. 8, m_i changes with μ but remains finite leading to ∑_jq_ij → 0, which suggests that at very high linker concentration, there is no bridge formed between mDNACCs and explains the drop of pib(2R+2rc) at large μ. The physical explanation is that at μ → ∞, mDNACCs tend to absorb as many free linkers as possible to minimize the −μ(∑_im_i + ∑_i∑_{j > i}q_ij) term in the grand potential of the system by maximizing ∑_im_i + ∑_i∑_{j > i}q_ij, which drives every ssDNA receptor to bind with a free linker with no bridge formed between ssDNA receptors, i.e., q_ij = 0. This suggests a re-entrant melting of mDNACCs with increasing μ. To demonstrate this, we perform MC simulations with the effective interaction of Eq. 7 for an equimolar binary mixture containing N = 864 mDNACCs at the packing fraction η = 4NπR³/3V = 0.1 with various μ, where we regard a colloid as a singlet if there is no bridge formed on it. As shown in Fig. 3, one can see that with increasing μ, the singlet fraction first drops and then increases at large μ. It is known that in systems of mDNACCs, multibody effects are important (22), while the predicted re-entrant melting transition based on the effective pair potential between mDNACCs indeed exists in multiparticle systems. The reason is that Eq. 8 actually accounts for multiparticle systems, which already includes multibody effects. Similar re-entrant melting was also found in a recent work of linker-mediated immobile DNACCs, in which the ssDNA receptors are grafted on the colloids instead of being mobile on the particle surface (31). A local chemical equilibrium approach was used in (31) to treat the linker-mediated interaction between colloids, and it is related to the statistical mechanical description used here (24). Moreover, to account for the immobility of the grafting points of ssDNA receptors on colloids, a Derjaguin-like approach (32) adapted to the presence of free linkers in solution was used in (31), while as mentioned above, the essential physics driving the re-entrant melting of DNACCs remains the same and does not depend on the mobility of ssDNA receptors on colloidal surface.

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Entropic effects in the strong binding limit

Furthermore, we investigate the effective interaction between mDNACCs with changing the binding strength between DNA linkers and corresponding receptors, i.e., decreasing ΔG_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 ΔG_bind → −∞. Intriguingly, however, as shown in Fig. 4A, at ΔG_bind → −∞, 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 ΔG_bind and reach a plateau at the strong binding limit (Fig. 4B). To understand this counterintuitive phenomenon, we modify our mean-field theory as follows (see section S4). When ΔG_bind → − ∞, all receptors on mDNACCs are occupied, i.e., n¯i=0, and using the saddle point approximation, one can write down the free energy of the bonded DNA linkers asβFinf=∑i[nilog (miniξaeβ(ΔGbind+μ)) +12∑jqij+niβΔGbind](9)whereqij=mimjξbξa2eβμ=mimjVa′eβ(2Frep−Fcnf−μ)(10)

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Here, the “enthalpy” term ∑_in_iβΔG_bind does not depend on the colloidal configuration and can be neglected, and ξ_ae^{β(ΔG_bind + μ)} = V_a^′e^{β(μ − F_rep)} is the entropy part of the partition function and does not depend on ΔG_bind. Therefore, the resulting effective colloidal interaction at the strong binding limit isβUeff−∞=∑i[nilog (miniVa′eβ(μ−Frep))+12∑jqij]+βUdep(11)which solely depends on entropy. We plot the prediction of Eq. 11 as dashed lines in Fig. 4A, which quantitatively agree with the converged plateau of βU_eff at the strong binding limit. Moreover, in Fig. 4C, we plot βUeff−∞ as a function of μ for various n_i, and one can see that βUeff−∞ increases with increasing μ or decreasing n_i. To explain this, we consider the effective pair interaction between two fixed mDNACCs i and j with n_i = n_j (like in Fig. 2C) at ΔG_bind → −∞, where m_i = m_j, and m_i + q_ij = n_i. Equation 10 implies qij/mi2=ξb/[ξa2exp (βμ)], and ξ_b/a does not depend on μ or n_i. With increasing μ, ξb/[ξa2exp (βμ)] decreases, and at fixed n_i, q_ij becomes smaller, which implies less bridges formed between i and j, and the less negative βUeff−∞. Similarly, at fixed μ, qij/mi2=ξb/[ξa2exp (βμ)] is a constant, and the smaller n_i leads to the smaller q_ij and the less negative βUeff−∞.

It is known that for systems of short-range attractive particles, at the infinitely strong attraction limit, vibration entropy stabilizes the floppy crystals (21), which first nucleate from the dilute fluid. Using Eqs. 7 and 11, we perform isothermal-isobaric (NPT) MC simulations for the binary system of linker-mediated mDNACCs at βμ = 1.6 at the strong binding, i.e., βΔG_bind = −6, and the calculated equation of state (EOS) is shown in Fig. 5. Compared with the EOS at βΔG_bind → −∞, one can see that the random fluid first nucleates into CsCl at the pressure PR³/k_BT ≃ 1.1, and this is qualitatively different from the situation of conventional mDNACC systems, in which the crystallization pressure of the system approaches zero at the strong binding limit (21).

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Generalization to multicomponent systems

In this section, we generalize our mean-field theory to calculate the effective interaction in a multicomponent system consisting of N_c types of mDNACCs, in which the total number of mDNACCs is N. mDNACC i is coated with only one type of ssDNA receptors, e.g., type I, and we regard the mDNACCs coated with type I ssDNA receptors as the type I particles. The number of ssDNA receptors coated on mDNACC i is n_i. There are multiple types of free linkers in solution, which are described by a N_c × N_c connectivity matrix M, and if the free linkers connecting mDNACCs of type I and J exist in the solution, then M_IJ = 1; otherwise, M_IJ = 0. We assume that the lengths of all free linkers in solution and all ssDNA receptors coated on mDNACCs are l and r_c, respectively. Then, the grand canonical partition function of the bonded linker isZ({mi,qij})=∑{mi,qij}W({mi,qij})∏I∏ii∈NIξa,I∑imieβμI∑imi∏I,JMIJ=1∏ii∈NI∏j>ij∈NJξb,IJ∑ijqijeβμIJ∑ijqij(12)where m_i and q_ij are the number of linkers bonded on particle i with a free end and the number of bridges formed between particles i and j, respectively. N_I consists of all the particles of type I in the system. μ_IJ is the chemical potential of the linkers that can bridge particles of type I and J, and μI=∑JMIJ=1μIJ. ξ_{a, I} and ξ_{b, IJ} are the partition functions of linkers in the states of bonded on particle type I with a free end (state a) and bridging between particles of type I and J (state b), respectively, with ξ_{a, I} = V_a^′ exp [−β(ΔG_{bind, I} + F_rep)] and ξ_{a, I} = V_a^′ exp [−β(ΔG_{bind, I} + ΔG_{bind, J} + F_cnf)], where ΔG_{bind, I} is the binding free energy between an ssDNA receptor of type I coated on mDNACC and another ssDNA tail of complementary sequence on a free linker in solution. W({m_i, q_ij}) accounts for all possible combinations of hybridization of {m_i, q_ij}, and the form is the same as in Eq. 2. Using the free energy of the bonded linkers ℱ({m_i, q_ij}), we can rewrite Eq. 12 into Z = ∑_{{mi, qij}} exp (−βℱ({m_i, q_ij})), and with the saddle point approximation ∂ℱ({m_i, q_ij})/∂{m_i, q_ij} = 0, we obtain{mi=n¯iξa,IeβμIqij=n¯in¯jξb,IJeβμIJ(13)

With these, we further obtain a set of self-consistent equationsp¯i=1ξa,IeβμI+∑JMIJ=1∑jj∈NJnjp¯jξb,IJeβμIJ+1(14)where p¯i is the probability to find an unbound ssDNA receptor on particle i. Then, the free energy of bonded DNA linkers at the saddle point isβF=∑i(nilog p¯i+12∑jqij)(15)where p¯i and q_ij are the solution to Eq. 14. One can see that Eq. 15 shares the same form as Eq. 6, and using the same way as proving the positive definiteness of Eq. 5 (section S3), one can prove that Eq. 14 is also positive definite.

Similarly, when βΔG_{bind, I} → −∞ ,1 ≤ I ≤ N_c, one can obtain the free energy of the bonded DNA linkers asβFinf=∑i[nilog (miniVa′eβ(μI−Frep))+12∑jqij+niβΔGbind,I](16)withqij=mimjeβ(2Frep−Fcnf)Va′e−β(μI+μJ−μIJ)(17)where the enthalpy term ∑_in_iβΔG_{bind, I} does not depend on the colloidal configuration and can be neglected. Therefore, in the multicomponent system of linker-mediated mDNACCs, the effective colloidal interaction at the infinitely strong binding limit is not diverging and solely depends on the entropy of linkers.