Self-assembly pathways towards floppy colloidal square lattices

Domain: physics
Published: 2025-12-26
Editorial outcome: minor_revision

Abstract
The ability to rearrange is crucial for the function of proteins and biopolymers and governs many properties of materials. While rearranging bonds are needed to create floppy structures, it has remained largely unexplored how they affect the self-assembly pathways. We here use a binary colloidal model system equipped with surface-mobile DNA-based bonds to study how regular structures with tunable flexibility and square network symmetry can be created through self-assembly. We find that reconfigurability during self-assembly leads to lattices with square symmetry which are inherently mechanically unstable and hence thermally floppy. By considering the role of size ratio, number ratio, and directionality induced by particle shape, we identify the pathways that maximize the yield and flexibility of these square lattices using a combination of experiments, analytical calculations, and simulations. Our study highlights the crucial role of reconfigurability in systems that are governed by enthalpic and entropic principles, from synthetic to biological, and might be useful for creating materials with novel or reconfigurable properties.

Introduction
Bond flexibility is crucial in materials and structures at all length scales. At the atomic and molecular scale, small rearrangements of the atoms in a material are key to heat and sound transport, and influence the mechanical, thermal and electronic properties of materials1–6. For DNA and nanoparticles, the flexibility provided by long linkers or flexible spacers can aid assembly into targeted crystalline structures7–10. In macroscopic materials, hinging between the constituent units has recently been employed to create a new generation of materials with designer properties, so-called mechanical metamaterials11. At the intermediate micrometer length scale of colloidal structures, however, rearranging bonds have long been elusive. Although theoretical and numerical studies typically assume rearrangements being possible after bonding, in experiments this is often not the case. Many colloidal bonds do not allow rearrangements due to surface inhomogeneities which trap their configuration in metastable, rigid states12, or, when they allow rearrangements also allow unbinding on the experimental timescale. This restricts an experimental comparison with theoretical predictions for example about the effect of flexibility on mechanical properties13 or the phase behavior6,14 and creating functional microscale structures that do work through conformational changes11,15.

Recently, two strategies have emerged to experimentally obtain rearranging yet strong bonds between colloidal particles. In the first approach, the temperature range and interaction strength is precisely controlled such that colloidal particles functionalized with DNA linkers covalently bound to the particles’ surface can rearrange4,16. This is made possible by creating weak, transient DNA bonds. The power of rearrangements have been demonstrated by showing that the particles can self-assemble into crystalline lattices that are otherwise not achievable by DNA-based interactions between colloidal particles4,16. An additional benefit is that the lattices become rigid and preserve their conformation after just a slight cooling.

The second strategy employs DNA linkers that are mobile on the colloid surface17–19. The linkers can be either placed on fluid droplets18 or inside lipid bilayers supported by solid colloidal particles to be laterally mobile over a wide temperature range17,19. Although mobile, the bond strength for surface mobile DNA linkers can be set by designing the DNA sequence to obtain any desired bond strength including strong, effectively irreversible bonds, which in addition can further be tuned by temperature in situ. This strategy has been employed to create a variety of flexible structures ranging from flexible chains to rings and colloidal molecules—with and without limited angular range—up to foldable structures, floppy square lattices, and mechanical metamaterials20–28. The latter two consisted of arrangements of colloidal particles obtained by assembling the particles one-by-one using optical tweezers27,28. Excitingly, the multivalent DNA-based bond was shown to act as a microscopic spring, making this a model system for thermal spring networks and allowing the extraction of the floppy and rigid normal modes27. However, the manual assembly strategy like the one adopted here is limited to small structures, with the largest structure achieved so far being a 7×7 square lattice.

A more powerful strategy for creating larger structures is self-assembly29. Self-assembly exploits the Brownian motion of the constituents and is driven by a lowering of the free energy. Bond flexibility has been leveraged for the self-assembly of flexible colloidal clusters23,25 and crystalline lattices4,16,30,31. Rearrangements played a crucial role: for the colloidal clusters lateral motion after binding enabled reaching the maximum geometrically possible cluster size and thus high yields23,25. For crystalline lattices rearrangements were important for obtaining highly ordered structures4,16,31. However, because the latter were created from surface-bound DNA linkers, their flexibility only persists for transient bonds and can therefore not be maintained outside a narrow temperature range. This makes it challenging to use them as a model system for understanding floppy modes in thermal systems and creating functional colloidal structures that, for example, execute work by changing their conformation.

Employing self-assembly in colloids with fully flexible bonds therefore has great potential to combine the advantages of both systems: on the one hand self-assembly may allow the creation of many large structures at the same time and control over their crystal symmetry through the design of the interactions, while at the same time maintain the bond flexibility and introduce floppiness in the assembled structures, at least when the bond network is appropriately designed. Monte Carlo simulations support this idea and predict a small region in phase space where floppy square lattices are stabilized by fluctuations32. However, these predictions have never been tested in experiments and the impact of rearranging bonds on the kinetics of lattice formation and how to guide self-assembly into floppy structures are areas of yet largely unexplored questions.

Here, we experimentally study the nucleation and growth dynamics of floppy 2D crystalline structures from a binary mixture of colloidal particles with flexible but irreversible bonds for a range of number and size ratios. We investigate the influence of bond flexibility for equal sized spherical particles and find that their ability to rearrange enhances the formation of two-dimensional square lattices. Utilizing in situ observations with confocal microscopy, we map out their evolution to gain a deeper understanding of the pathways leading to the formation of these flexible colloidal lattices. We leverage particle size ratio and shape to guide the self-assembly towards square lattices with high fidelity using simulations and analytical results in the presence of irreversible bonds. Our findings may serve as a tool to design innovative flexible materials with tailored properties that may be employed as sensors, actuators and in smart devices and machines.

Results
Self-assembly of equal sized spheres
To study the influence of rearranging bonds on nucleation and growth of floppy square lattices in 2D, we experimentally coat colloidal beads with a fluid lipid membrane into which DNA linkers are embedded17,19–25,27,28. This embedding allows the linkers to laterally move over the surface of the bead which permits rearrangements after bonding. Each bead is functionalized with one of two complementary DNA linkers, such that when they come in close contact the DNA linkers bind and the beads become connected by a multivalent bond, see Fig. 1a (see “Methods”, and SI for experimental details). Since the DNA linkers are able to freely move within the fluid membrane enveloping a bead, the angle between bound beads is able to freely change by thermal fluctuations, giving the bonds the required flexibility, see Fig. 1b (top row). The multivalent bond between two colloidal particles is mediated by the hybridization of multiple complementary DNA strands on their surfaces. While individual bonds between complementary linkers may form and break on the order of milliseconds due to the short DNA strands (free energy of hybridization of G0 = −6.2 kBT at 50 mM NaCl and for a reference concentration of 1 M33), the multivalent bond, which consists of many DNA linkers, only very rarely breaks on the experimental timescale34,35. We maintained the temperature at 23.0 ± 0.7 °C (melting temperature Tm = 25 °C), which balances bond stability and strength with flexibility20. Silica particles ensure quasi-2D motion due to gravitational confinement.

We first study the self-assembly of a system of two types of colloidal spheres with equal size (diameter d = 2.06 ± 0.05 μm) and concentration corresponding to a 2D area fraction ϕ ≈ 20%, but complementary DNA linker type; see Methods for experimental details. After mixing the two species in a temperature controlled setup, we observe their assembly in time by confocal microscopy, where different DNA functionalization can be identified by labeling with different dyes. Quickly after the start of the experiment, the beads assemble into small clusters which continue to grow through addition of single particles and cluster-cluster merging, see Fig. 1b (bottom row) and Movies S1 and S2. Interestingly, a high fraction of these structures resembles small A-B type square lattices, in which the oppositely functionalized beads form a chessboard like pattern, see Fig. 1b–d as well as Movie S2 and Fig. S2. Larger networks with square symmetry show relative motion of the particles due to the flexibility of the bonds and the arrangements of the particles. The resulting bond network is not mechanically stable with on average four neighbors, and therefore shows floppiness, see Fig. 1c (bottom), where traces of the particle positions in time are shown, as well as Movie S3.

We analyze the clusters that have formed after ≈ 2–16 h by calculating the coordination number, Zc, defined as the number of beads bound to a given sphere. For equal sized spheres, Zc takes a maximum value of Zc = 6 for close packed spheres and Zc = 4 for a square lattice in the bulk; at the boundary of a lattice the coordination number reduces to 3 at the edges and 2 at the corners. For a system of clusters each consisting of a lattice with coordination number Zlatc the mean coordination number is approximated by:

⟨Zc⟩ ≈ Zlatc (1 − γ) (1)

Where we have introduced the boundary ratio γ ∈ [0, 1] which represents the average proportion of spheres on the boundary of a cluster. This depends on the configurations of the clusters as well as their size; see SI for further discussion. In our experiments we observe a range of coordination numbers peaked around Zc = 3 and an estimated boundary ratio with a value of γ ≈ 0.6 which is compatible with square lattices (Zlatc = 4), see Fig. 1e. The presence of values smaller than four stems from the small size of the clusters where many particles are located at the boundary and hence have less than four neighbors.

To capture the degree to which the arrangement of particles have rotational symmetry of order 4, we introduce a square order parameter

ψ4 = ⟨| 1/ni ∑j exp(4iθij) |⟩ (2)

Where the sum runs over the ni particles bound to particle i and θij is the angle of the bond connecting them; angular brackets ⟨ ⟩ denotes an average over all particles i with ni > 1. Our experiments show large regions where ψ4 > 0.5 indicating a high degree of square order symmetry and a A-B lattice like arrangement, see Fig. 1f.

In a square lattice, each particle has on-average four neighbors. This is lower than the geometrically possible maximum of six and the entropically achievable maximum number of five bound particles which we previously identified23. It has been seen in colloids with complementary surface bound DNA linkers36, and the square arrangement is the 2D equivalent of the CsCl lattice previously found for 3D systems of equal sized nanoparticles and colloids with complementary surface bound DNA linkers4,9,16,31 and colloids with opposite charges37,38. In 2D, the appearance of square lattices has also been predicted for binary systems with short-range “sticky” interactions39, for DNA-coated nanoparticles36, and by Monte Carlo simulations of a binary system of colloids with complementary and mobile DNA linkers, albeit at significantly higher surface area fractions, around Φ ≈ 0.7332. The floppy modes and associated vibrational entropy were shown to be able to stabilize the square lattice structure32,40.

In our experiments, the probability for particles to unbind to optimize their position in the lattice and find the state of lowest energy is very low, i.e., the bonds are effectively irreversible. This implies that they cannot reach the thermodynamic ground state, as predicted by Monte Carlo simulations32, but only local minima in the free energy landscape. However, while maintaining the bond they are still able to move with respect to each other allowing an increase of the number of bonds and lowering of their free energy. For example, a chain of at least four particles can rearrange and bind to form a small square, see Fig. 1b. However, this process competes with the addition of another particle which changes the structure with possibly different order and new possibilities for rearrangements and internal bond formation. We therefore hypothesize that the formation of the square lattice stems from a combination of free energy minimization through bond formation at the one hand, and a competition between the timescales for adding another particle and reconfiguration of the existing structure on the other. This implies that the self-assembly pathways towards square lattices play an important role, in particular because they are unidirectional due to the irreversible nature of the bonds.

Self-assembly pathways of flexible colloidal lattices for spheres of equal size and number ratio
To illuminate the pathways that can lead to square lattices, we consider the first bonds created on the way to making colloidal clusters. To do so, we construct all possible colloidal aggregates of a certain size (N) in which no beads can be removed from the structure without breaking a bond. The number of possible structures starts small but quickly diverges with increasing N. Figure 2a shows examples of all possible structures containing N ≤ 6 for equal sized beads with equal number density. Note that the system is symmetric with respect to switching of the two populations of beads, and that we combined the two structures with opposite bead colors into one for clarity. By repeatedly simulating the assembly process, we can measure the relative likelihood of observing each structure, which is displayed as the size and color of the circle at each node of the diagram. Interestingly, there is only one possible aggregate of size N ≤ 6 that is incompatible with a square lattice: the “rosette” where a central A particle is bound to five B particles. However, this configuration occurs only in 3% of the cases for two reasons: first the space around the central A particle is almost full in this configuration meaning that the absorption of the fifth B particle has a low probability to occur23, second it requires a large imbalance of A and B type particles locally, which occurs with low probability for the equal number ratio’s present in the experiment. Moreover, we find that the square unit we hypothesized to appear in chains of four or more particles to be present in 90% of all clusters with size N = 6.

We can map out all possible pathways for formation of any specific aggregate; these pathways are shown as lines in Fig. 2a. The color of the lines indicates whether the bond represents the joining of two smaller clusters (blue) or an internal reconfiguration (red). From this information, we can construct the sub-network of pathways that accounts for over 95% of aggregates and bonding events, see Fig. 2b. The arrows indicate that these are unidirectional pathways, i.e., because of the irreversible nature of the bonds unbinding does not occur. Since each specific cluster can arise from a finite number of pathways, the likelihood of a cluster is directly connected to the likelihood of the relevant pathways, which are all driven by lowering of the free energy. All of these configurations have been observed in experiments, which are shown as confocal microscope images in Fig. 2b.

The relative rate of adding a new particle (blue lines) and internal reconfiguration (red lines) depends on the age of the system. When the system is young there are many small clusters and the distance between them is low, thus bonds between clusters are relatively common. When the system is old, it is comprised of larger aggregates, which diffuse slower and the distance between them is increased (Figs. S6 and S8). Therefore the timescale associated with the addition of beads increases over time and the structures have more time to reconfigure. For example, the most likely structures at N = 5 and N = 6 can only be formed by reconfiguration of less common structures, indicating that reconfiguration is fast compared to the addition of particles from an early stage. It is worthwhile to note that structures that are stable with respect to internal reconfiguration are the most common, because they can only be escaped from by addition of another particle or by clustering which is governed by a longer timescale.

Our results suggest that the nature of the early assembly process leads to square bond networks with high probability, large lattices appearing at a later time will have at least small regions with a square network. This is reflected by an initial increase in the global average value of ⟨ψ4⟩ for simulations of N = 100 particles with irreversible bonds, see Fig. 3a. Square lattices have increased flexibility as there is space between the particles in the aggregate and the average connectivity is below the mechanical stability threshold40.

At long times, however, we observe a decrease in ψ4 and an increase in six-fold order, as measured by the hexagonal order parameter ψ6 = ⟨| 1/ni ∑j exp(6iθij) |⟩ (see also Fig. S8). Figure 3b and c shows simulation and schematic snapshots of the system at early and long times (see also Fig. S8). We emphasize that this transition is not due to breaking and reformation of bonds as they are irreversible. Instead, this effect can be explained by the re-configurable nature of the bonds between the particles. A key feature of these floppy colloidal networks is the ability of the angles between the bonds to freely change, thus the square lattice has no resistance to shear deformations. These shear deformations can result in a hexagonal packing of the particles which can become fixed with the creation of a single additional bond. This is first possible for N = 7 with the configuration shown in Fig. 3c which has a predicted increase in both the coordination number Zc: 2.3 → 2.6 and sixfold order ψ6: 0.65 → 1.0 and a predicted decrease in four-fold order, ψ4: 0.79 → 0.21 and the flexibility of the cluster. These predictions are closely inline with simulated results shown Fig. 3d. The time taken for this process is exponentially distributed with a well-defined rate which indicates it arises from a Poisson process, see Fig. 3d (inset). Critically, this transition also corresponds to the cluster becoming completely rigid and all flexibility is lost as the system is left in a close packed state. Thermal noise and packing fraction further influence the kinetics of this transition, with a higher thermal noise and packing fraction reducing the transition time (see SI section 3.3 and Fig. S9).

To test these predictions, we return to our experiments of equal sized DNA-coated spheres and observe and quantify their structures after 16 h, Fig. 3e. We find that the previously floppy structures have turned into rigid and compact structures. This transition is accompanied by a clear increase in the number of neighbors and the hexagonal bond order parameter, see Fig. 3f, g, confirming the predictions made by our simulations. The transition from a floppy square lattice to hexagonally closed packed structures cannot be observed by Monte Carlo simulations of colloids with surface-mobile DNA linkers32 as they only identify the thermodynamic ground state and not the pathway towards it. Both the square and hexagonally closed packed structures were observed in these simulations separated by a 2% difference in surface area fraction. However, the densities where these structures occurred in MC simulations were significantly higher than the experimentally employed one sphere. In addition, the irreversible nature of the here considered bonds implies that detailed balance is broken and that the thermodynamic equilibrium is not reached, making it difficult to compare the results quantitatively.

The transformation from a square to a hexagonal lattice is reminiscent of the diffusionless transformation previously observed for binary systems of colloids functionalized with surface-bound DNA linkers when interactions between all colloids were present36,41,42. However, the origin is different, because in perfect n×n floppy square lattices without dangling particles such a transition has not been observed27. The transformation is thus a consequence of the formation of additional bonds of dangling particles which leads to a conformational change without a change of the topology of the underlying bond network. Indeed, in the absence of interactions between all particle types, as is the case here, square lattices have been predicted to be stable for colloids with surface bound DNA linkers36,42. The strong bonds employed here therefore compensate for the loss in entropy of the floppy square lattice while at the same time preventing compaction by surface rearrangements. Thus, we conclude that while rearrangements are at the heart of the square lattice symmetry initially, they also can lead to a loss of it by bond formation at long times.

Semi-flexible networks with cubic beads
To enhance the long-term stability by preserving the square symmetry we turn to cubic particles in the experiment, see Fig. 4. Because of the anisotropic cubic shape, spheres have a strong preference to locate on their flat surfaces after binding as more DNA linkers can bond there20,25. This thus imposes a four-fold symmetry for structures formed in 2D, which has been shown to allow selective assembly of clusters with four bounded spheres at highly asymmetric number ratios25 and lattices for nanoparticles10,43,44.

Upon mixing cubic particles (1.35 μm side-length) and spheres (1.66 μm diameter) at approximately equal number ratios, we once again see the development of small scale structures with square symmetry in the early stages of assembly (see Fig. 4a and Movies S4 and S5) for consecutive assembly steps (see Fig. 4d) for the results after ≈ 16 h. Clearly, the presence of the cubic particles enhances the appearance of square order in the system.

The cubic particle shape not only guides the spheres into a square arrangement but also limits the reconfigurability of the formed structure. Once a sphere is bound to a cube, the sphere is unable to transition to another face of the cube. This reinforces the square symmetry in the assembling structures, see Fig. 4a. While the cube can freely rotate around the sphere as it has no edges, the flexibility of larger structures is still reduced relative to structures assembled from spheres only, see Fig. 4a, b.

The effect of the cubic shape is also reflected in the quantitative measurements: we find the coordination number to be smaller and ψ4 to be higher than for aggregates with equal sized spheres, see Fig. 4d–f. The coordination number peaking below Zc = 4 stems in part from the fact that coordination by four spheres is the maximum possible value for a cubic particle. In addition, the reduced reconfigurability of the clusters containing cubic particles leads to them having more extended shapes and voids, resulting in more edges and corners and an estimated boundary ratio closer to γ ≈ 0.75. Yet, the square order parameter is higher, showing the strong guiding effect of the square particle shape. Indeed, simulations show there is no crossover to ψ6, see Fig. 4c and Fig. S4, again confirming that the cubic shape strongly guides the assembly into a square symmetry and prevents reconfiguration towards hexagonal symmetries.

Thus, by introducing cubic particles, it is possible to influence the system to adopt square lattices. However, this is achieved at the cost of restricting the flexibility of the lattices.

Effect of breaking symmetry between two particle types
The range of flexibility of a cluster depends on the bond network arrangement as well as the available space between the particles. The angular range of motion available for a perfect square lattice can be calculated using geometric arguments and depends on the aspect ratio R = Ra/Rb of the particles with radii Ra and Rb, see Fig. 5a (orange line). The square lattice is most flexible when the beads have equal size, but is susceptible to the transition towards a hexagonally symmetric structure, as outlined in Fig. 2d and shown as a red dot in Fig. 5a. Since hexagonal packing fills space optimally, there is no available space and flexibility is completely lost. However, when different particle sizes are used, the hexagonal lattice obtains a small range of motion. In contrast to the square lattice structures, such a hexagonal lattice is possible for a large range of size ratios, but has a smaller range of motion and requires a two to one ratio of beads to realize the AB2 arrangement with Zlatc = 5, see Fig. 5a (blue line). Therefore the flexibility and likelihood of a certain structure depend on the relative size and abundance of the constituent particles.

To understand how these two parameters can be tuned to target a certain lattice structure with a desired symmetry and flexibility, we again consider the assembly pathways that lead to different structures given the irreversible formation of bonds. The results presented in Fig. 2a correspond to systems that are symmetric with respect to exchange of the two different types of particle. If this symmetry is broken, for example by a size difference, the number of distinct combinations, and possible pathways between them, greatly increases.

Figure 5b shows the aggregation pathways for up to N = 4 when the A and B particles are not identical. The probability of taking any particular path is given by the product of the probability that an incident particle is of the correct type and that the particle can bind. The probability that the incident particle has the correct type is simply related to the relative abundance of the two particle types. For example, the top configurations in Fig. 5b contain more magenta particles, thus are more likely when magenta particles are more abundant than green ones. The probability an incident particle is able to bind depends on the current configuration and aspect ratio and can be estimated from geometric arguments23. In general, incident small particles are more likely to bind as the complementary large particles in a cluster present more surface area for them to bind to. For example, the top configurations in Fig. 5b all feature multiple large beads bound to a central smaller bead. Since there is less space on the surface of the smaller bead, this configuration is less likely than the inverse, in which multiple small beads bond to a central larger bead.

We measure the average cluster size, ⟨Nc⟩ after 107 simulation steps as a proxy for cluster growth rate. We see that clusters grow fastest when the spheres are of equal size and abundance, see Fig. 5c. The growth rate is lowest when there is an abundance of larger particles (Ra/Rb ≈ Na/Nb < 1). In this scenario incident particles to a cluster are more likely to be larger, which are less likely to find a place to successfully bind.

We next assess how the symmetry of the lattice structure in the long time limit depends on the number and size ratio of the beads by measuring ψ4 − ψ6, see Fig. 5d where positive values (red) indicate square structure and negative values (blue) indicate hexagonal structure. As we observed in Fig. 2c, for equal size and number ratio, in the long time limit the system adopts a predominantly 6-fold symmetric structure, see Fig. 5d, e (green diamond). When smaller beads are more abundant ψ6 dominates and we observe hexagonal structures, see Fig. 5d, e (yellow star). When larger beads are more abundant ψ4 dominates and we observe square structures, see Fig. 5d, e (blue circle). The regions with square structure show a corresponding drop in the coordination number (Fig. 5c, although this is also partly a function of the reduced cluster size which implies higher values of γ (see SI).

The mechanism by which ψ4 is selected for can be understood by considering Fig. 5b. When the larger beads are more abundant the number ratio favors taking the uppermost path, however, entropic effects stemming from the size ratio penalize taking it. Thus, pathways in the middle are preferred, where square lattices are more common. However, since the spheres are not equal in size, the square lattices are not susceptible to the formation of additional bonds which ridigify the structure upon thermal-fluctuations, similar to Fig. 3a–c. Thus by carefully controlling the size and number ratio of the beads, we can favor square lattices that retain a high degree of flexibility (see Fig. S7).

We now apply our understanding to generate flexible and stable square lattices in the experimental system. In line with our results from Fig. 5, we select a size ratio Ra/Rb = 1.67 and number ratio Na/Nb = 1.2 and overall area fraction of ≈ 20% to retain high bond flexibility and steer the assembly towards lattices with a high square symmetry. We employ complementary functionalized spheres with diameters 2.06 ± 0.05 μm and 1.25 ± 0.05 μm that can fully rearrange to form square lattices, see Fig. 6a.

Similar to previous experiments, we quickly observe the appearance of small structures with a square symmetry. These structures preserve flexibility as shown in Fig. 6b and Movie S9. After 15 min, we find that many small clusters have formed (see Movies S7 and S8) (Fig. 6d and Fig. S5). The small structures are indicative of the reduced growth rate of clusters when the larger beads are more abundant and leads to a highest estimated boundary ratio γ ≈ 0.8, see Movie S8. We also observe a reduced coordination number, Fig. 6e, which is a reflection of both the reduced cluster size and the predominantly square lattice structures and increased value of ψ4, see Fig. 6f. A simulation of the system with size ratio Ra/Rb = 1.67 and number ratio Na/Nb = 1.2 shows that ψ4 remains higher than ψ6 even in the long time limit, Fig. 6c and SI for details.

Methods
Reagents
Silica microspheres of diameters 1.25 ± 0.05 μm, 1.66 ± 0.05 μm, 2.06 ± 0.05 μm, in 5 wt/v% suspension were purchased from Microparticles GmbH. Sodium chloride, Ethanol, Sodium hydroxide, Ammoniun hydroxide (28–30 v/v%), Iron(III) Chloride Hexahydrate (FeCl3·6H2O), Tetraethyl orthosilicate (TEOS), 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid (HEPES), trimethoxysilyl propyl methacrylate (TPM), were purchased from Sigma-Aldrich and used as received. 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-[methoxy(polyethylene glycol)-2000] (DOPE-PEG2000), 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-(lissamine rhodamine B sulfonyl) (DOPE-Rhodamine) and dye 23(dipyrrometheneboron difluoride)-24-nor-cholesterol (TopFluor-Cholesterol) were obtained from Avanti Polar Lipids, Inc. Milli-Q water was used for all experiments.

DNA strands were purchased from Eurogentec. The sequences of DNA used were: Strand A: Double Stearyl-HEG-50-TT-TAT-CGC-TAC-CCT–TCG-CAC-AGT-CAC-CTT-CGC-ACA-GTC-ACA-TTC-AGA-GAG-CCC-TGT-CTA-GAG-AGC-CCT–GCC-TTA-CGA-G-T-A-G-A-A(30)-ATTO488, Strand B: Double Stearyl-HEG-50-TT-TAT-CGC-TAC-CCT–TCG-CAC-AGT-CAC-CTT-CGC-ACA-GTC-ACA–TTC-AGA-GAG-CCC-TGT-CTA-GAG-AGC-CCT–GCC-TTA-CGA-T-T-C-T-A-C(30)-Cy3, Strand C: 50-TCG-TAA-GGC-AGG-GCT-CTC-TAG-ACA-GGG–CTC-TCT-GAA-TGT-GAC-TGT-GCG-AAG-GTG–ACT-GTG-CGA-AGG-GTA-GCG-ATT-TT-30, Strand D: Double Stearyl-5TT-TAT-CGC-TAC-CCT-TCG-CAC-AGT-CAA-TCT-AGA-GAG-CCC-TGC-CTT-ACG-A and Strand E: TCG-TAA-GGC-AGG-GCT-CTC-TAG-ATT-GAC-TGT-GCG-AAG-GGT-AGC-GAT-TTT. The linker sequence in strand A and strand B is italicized.

Synthesis of silica cubes
Hematite cubes of edge length 1.08 ± 0.04 μm were synthesized following reference46 and coated with a 0.135 μm silica layer using the process described in ref.47 which resulted in hematite-silica core-shell cubes of edge length 1.35 ± 0.07 μm. In a typical synthesis of hematite cubes, 100 ml of aqueous 2 M FeCl3·6H2O were prepared in a 500 ml Pyrex bottle. Next, 100 ml of 5 M NaOH solution were added while stirring at 1000 rpm for 50 seconds. Then, the mixture was stirred continuously for another 10 minutes and subsequently placed in a preheated oven and left undisturbed at 100 °C for 8 days. The resulting hematite cubes were washed several times using centrifugation and redispersion in milliQ water. To coat them with a thick 0.135 μm layer of silica, 100 ml of ethanol and 0.6 g of synthesized cubes were mixed under ultrasonication and mechanical stirring in a 2-neck round-bottom flask at 50 °C. Subsequently, 5 ml of water, 15 ml of ammonium hydroxide solution and 0.6 ml of TEOS were poured into the reaction flask and allowed to coat the particles’ surface with silica layer for 5 hr under the same conditions. The resulting silica-coated cubic particles were first washed three times with ethanol and then three times with water to remove unreacted chemicals by repeated centrifugation and redispersion. To remove the hematite core, 5 ml of silica-coated hematite cubes was dispersed in 5 ml 37 wt/v% HCl solution. The hollow silica cubes exhibit characteristics of a superball shape, characterized by an m-value of 2.9 (see Fig. S1 for details)48.

Preparation of small unilamellar vesicles (SUVs)
Small unilamellar vesicles (SUVs) were prepared using a protocol described in ref.19. For the preparation of SUVs, we used 77 μl of 25 g/L DOPC, 7.34 μl of 10 g/L DOPE-PEG2000, and 2 μl of either 1 g/L dye DOPE-rhodamine or 2 μl of 1 g/L TopFluor-Cholesterol dissolved in chloroform were mixed together in a glass vial. Subsequently, the lipid mixture was dried for at least 2 h in a desiccator (Kartell) attached to a vacuum pump (KNF LABOPORT N816.3KT.18). Then, 1 ml of buffer solution consisting of 50 mM NaCl and 10 mM HEPES at pH 7.4 was added to the dried lipid. The prepared solution was vortexed for 30 min during which the solution became turbid indicating the formation of giant multilamellar vesicles. The dispersion of giant multilamellar vesicles was extruded with a Avanti Polar Lipids mini extruder 21 times through a 50 nm polycarbonate membrane supported with filter paper to achieve SUV formation. The prepared SUVs were stored in the fridge at 4 °C and used for up to 3 days.

DNA hybridization
We used double stranded DNA with a complementary 6 base-pair single stranded end to which we refer to as “linker" and inert double stranded DNA strands for bonding and stabilizing the colloids, respectively. Prior to use, single strand DNA was hybridized with the complementary backbone to obtain linker DNA. Strand A was hybridized with strand C to yield double-stranded linker DNA, strand B with strand C to obtain the complementary double-stranded linker DNA, and strand D with strand E to create double-stranded inert DNA (DNA strands are listed in the “Methods” section). For hybridization, we typically mixed 10 μl of 20 μM single strands and 10 μl of 20 μM complementary backbone in 90 μl buffer (200 mM NaCl, 10 mM HEPES, at pH 7.4) solution. The DNA solutions were placed in a preheated oven at 94 °C for 30 min. The oven then was switched off and allowed to cool slowly overnight. After cooling, the hybridized DNA strands were stored at 4 °C and used for up to 2 months.

Functionalization of colloidal particles with a lipid bilayer containing linker and inert DNA
To coat particles with a lipid bilayer we used a 25:1 surface ratio of SUVs to particles. We maintained the same surface area ratio of SUVs to particles when coating differently sized particles. Typically, for 2 μm particles, we use 100 μl of 0.25 wt/v% particles in Milli-Q water and mixed them with 37 μl SUVs. Then, the dispersion was rotated at 8 rpm for 1 h. During this period, SUVs collide, burst, spread and form a bilayer on the particles’ surface. Then, the coated particles were centrifuged at 800 rpm (=43g) for 2–5 min and the supernatant containing excess SUVs was removed using a micropipette. Subsequently, linker DNA was added to the particles at a concentration of 2 × 104 DNA strands/μm2 particle surface labeled with dye DOPE-rhodamine (magenta) and 4 × 104 DNA strands/μm2 labeled with dye TopFluor-Cholesterol (green), respectively. These concentrations ensure unlimited binding between particles up to the geometrically possible maximum23,49. We employed three combinations of particles: (1) equal sized spheres of 2.06 ± 0.05 μm diameter, (2) unequal sized spheres of 2.06 ± 0.05 μm and 1.25 ± 0.05 μm diameter, and (3) 1.66 ± 0.05 μm spheres and 1.35 ± 0.07 μm cubes. Inert DNA at a nominal concentration of 5 × 104 DNA strands/μm2 was added to all particles suspensions. The suspensions were then rotated for another 1 h. Thereafter, each suspension was centrifuged and washed 2 times with a 50 mM NaCl HEPES buffer. The repeated sedimentation cycles can introduce non-specific aggregation of a small percentage particles, as seen in the initial frames of Movies S2, S5, and S8. The repeated washing and integration of inert DNA strands suppressed short-range non-specific interactions such as van der Waals interactions and depletion interactions during self-assembly such that no further unspecific aggregation occurs.

Sample preparation and imaging
For all self-assembly experiments, 2.06 ± 0.05 μm spheres and 2.06 ± 0.05 μm spheres a number ratio of 1:1, 2.06 ± 0.05 μm spheres and 1.25 ± 0.05 μm spheres a number ratio of 1.2:1, 1.66 ± 0.05 μm spheres and 1.35 ± 0.07 μm cubes a number ratio of 1:1 was maintained. Polyacrylamide (PAA) coated coverslips were used as substrates and coated with PAA by following the protocol in ref.21. In a 0.5 ml vial, the amount of spheres that was calculated to yield an approximately 10% area fraction after moving to the sample chamber of 2.06 ± 0.05 μm DNA functionalized and 10% area fraction of complementary DNA functionalized spherical particles were mixed. Additionally, 200 μl of a 50 mM NaCl and 10 mM HEPES buffer, pH 7.4 was added to particles. The mixture was then transferred to a customized sample holder and allowed to self-assemble and while being captured with brightfield or confocal microscopy over time up to 2 h.

Due to the chosen particle material and size, the gravitational height of the suspended particles was smaller than their diameter, promoting a two-dimensional (2D) assembly. For the assembly of unequal sized spheres, the smaller gravitational heights of the smaller spheres allows for more fluctuations in the vertical direction compared to the larger spheres as visible in Movies S7–S9. The anisotropic shape of the cubic particles restricts their vertical motion when bound to spheres, as visible in Movies S4–6. During self-assembly the sample temperature was maintained at 23.0 ± 0.7 °C using a temperature controlled water bath (Julabo) equipped with a Pt100 sensor that was inserted into the sample cell.

Images and videos were captured with a Nikon inverted TI-E microscope equipped with an A1 confocal scan head and Prime BSI Express camera (Teledyne Photometrics) for capturing the sample evolution in brightfield mode. The images were taken with a 60x oil objective (N.A. 1.4) at frame rates of up to 25 fps.

Particles detection and image analysis. To extract the x and y coordinates from TIF images, the images were first processed by separating the green and magenta channels. A Hough Circle Detection technique was employed to identify particles appearing as bright circular rings within these channels. Following the detection of particle positions, the coordinates were stored in CSV files. These coordinates were subsequently used for analysis using custom Python scripts.

Simulation details
The simulation includes N beads in a square region with sides of length L and periodic boundary conditions. Each bead belongs to either population A or B which have complementary DNA tethers; there are NA (NB) beads in population A (B) such that N = NA + NB. The relative radii of the populations may also be distinct, with the radius of bead being described by Ri with Ri = RA (Ri = RB) for beads belonging to population A (B). The packing fraction of beads in the simulation is therefore given by ϕ = (NA RA2 + NB RB2)π/L2.

The location of bead i is described by the vector ri, the motion of which is described by the following overdamped Langevin equation

r_i = μ ∑j Fij + ηbξ (3)

where bξ is an uncorrelated randomly orientated unit vector capturing thermal fluctuations, η is the magnitude of the random movements due to thermal noise, μ is a mobility coefficient and Fij captures the forces exerted on particle i due to interactions with particle j. In principle, the coefficients could depend on which population particle i belongs to but here we keep them constant for all particles. Equation (3) is integrated according to ri(t+1) = Δri(t) to simulate the motion of the particles where Δ is the length of the timestep.

The force between two particles arises from two effects, collisions and bond formation, such that Fij = Fcij + Fbij.

The collision force between two particles is given by

Fcij = Kcbr ij (rij − (Ri + Rj)), if rij < Ri + Rj (4)

where we have introduced rij as the vector from particle i to particle j such that rij = −rji and rij = rji as the distance separating the two particles; ^ denotes a unit vector; Ri describes the radius of particle i; Kc is the elastic collision parameter.

The bonding force between two particles is given by

Fbij = KbAij br ij (rij − (Ri + Rj + Lbond)), if rij > Ri + Rj + Lbond (5)

Where we have introduced Lbond as the relaxed length of a pair of bound DNA tethers, Kb is the tether elastic coefficient and Aij is the adjacency matrix.

The adjacency matrix describes which particles are bound and hence defined as

Aij = 1, if particle i and j are bound
0, otherwise (6)

The adjacency matrix is updated dynamically as follows: If two particles with complementary tethers come into contact (rij < Ri + Rj + Lbond), a bond is formed between them. If two bound particles become sufficiently separated (rij > Ri + Rj + Lmax), the bond between them is broken. The adjacency matrix is symmetric and completely describes the graph of the bond network between the particles. Clusters in the simulation are connected sub-graphs of the whole system and can be represented by a submatrix of Aij. To identify a cluster, it is sufficient to generate the eigenvalues of its adjacency matrix. This can be used to track the existence of specific clusters in the simulation and generate the data shown in Fig. 2a.

Without loss of generality, we redefine η = η/Δ, Kc = Kcμ/Δ, Kb = Kbμ/Δ and set their values at η = 10−2 and Kb = Kc = 0.45. All simulations are performed with fixed packing fraction ϕ = 0.1 which defines the length of the periodic boundaries; this helps de-correlate the system from its initial conditions as the particles have space to move before they bind. We set Lbond = 0.04 to mirror the scales observed in experiments. Finally we set Lmax = 10Lbond to close the equations. Unless otherwise stated, all simulations are initialized with random, non-overlapping particle positions and run for T = 107 timesteps.

Results presented in Fig. 2a and c are based on 500 simulations with NA = NB = 50 particles with sizes RA = RB = 1.

Cluster probabilities in Fig. 2a give the likelihood that a random cluster of given size has a specific configuration over the duration of the simulation. This is a time-independent quantity, thus it includes both the likelihood of a cluster forming and its stability. Short lived structures such as for example the N=6 ring possess a low probability.

Pathway probabilities in Fig. 2a give the likelihood of each pathway relative to all other pathways that result in a cluster of the same size. This is also integrated over the duration of the simulation.

Figure 2e is based on a simulation of NA = 3, NB = 4, and RA = RB = 1. The simulation is started with the configurations shown on the left of Fig. 2d. Fig. 2e (inset) is based on 1000 realizations of this simulation.

Figure 4c is based on 500 simulations with NA = NB = 50 particles with sizes RA = RB = 1. This simulation gives the same average coordination number as that in Fig. 2c and Fig. 6c for direct comparison. Figure 5c is based on simulations with N = NA + NB = 100 and min{RA, RB} = 1. There are 25 simulations at each point in the parameter space. Figure 6c is based on 500 simulations with NA = 60, NB = 50, RA = 1.0, RB = 0.6. This simulation gives the same average coordination number as that in Fig. 2c and Fig. 4c for direct comparison.

Square particles with side length 2R are approximated by four discs each with radius R/2 arranged in a non-overlapping square around the center of the particle. The center of each disc (k) representing square particle i located at ri is given by

ck = ri + √2Rθbi(k)/2 (7)

where θbi(k) is a unit vector given by the equation

θbi(k) = [cos(θi + kπ/2), sin(θi + kπ/2)] (8)

for k ∈ {0, 1, 2, 3}. Here θi is the orientation of square particle i relative to the x axis.

When calculating the collision forces with a square particle, we consider overlaps of the constituent discs. We introduce Fcij(ki) to describe the force on the ki-th disc of the i-th square due to collisions with particle j. Each collision can introduce a translational and a rotational force on the square particle i, thus we must treat the two components differently.

The translational component is given by:

Fcij = ∑ki θbi(k):Fcij(ki) (9)

which can be simply introduced to equation (3). The resulting rotational component is given by:

τcij = √2R ∑ki θbi(k) × Fcij(ki)/2 (10)

where we have taken the out of plane component of the cross product.

This requires an additional overdamped Langevin equation to describe the evolution of the orientation of the square given by:

θ_i = μr ∑j τcij + ηr (11)

Where ηr is a random number sampled from a uniform distribution with range [−ω, ω] and μr is a rotational mobility coefficient. For the simulations in Fig. 4c we set ω = 10−2 and μr = 2.5.

Between a square i and a circular j particle the collision force must be calculated for each disc in the square, with the force on each disc being:

Fcij(ki) = Kcbr ij(ki)(rij(ki) − (Ri/2 + Rj)), if rij < Ri/2 + Rj (12)

0, otherwise

Where rij(ki) is now the vector from the center of particle j to the center of disc ki around particle i.

When calculating the force between two square particles, the collisions between all eight discs need to be considered. This is a simple extension of the previous case given by:

Fcij = ∑ki ∑kj Fcij(ki, kj) (13)

With the force on each disc being:

Fcij(ki, kj) = Kcbr ij(ki, kj)(rij(ki, kj) − (Ri/2 + Rj/2)), if rij < Ri/2 + Rj/2 (14)

0, otherwise

Where rij(ki, kj) is now the vector from the center of disc kj around particle j to the center of disc ki around particle i.

The bonding forces are not changed for the square particles. This recreates the greater bonding strength on the sides relative to the edges of the cubes.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript investigates the guided self-assembly of colloidal particles into flexible, reconfigurable lattices using DNA-functionalized building blocks. By employing binary mixtures of spherical and cubic colloids with mobile DNA linkers embedded in fluid lipid membranes, the authors demonstrate how bond flexibility and particle anisotropy influence lattice symmetry (square vs. hexagonal) and mechanical stability. This article brings a different perspective to an already existing article from the same group (Melio et al.), where the mechanical properties of floppy colloidal square lattices assembled by optical tweezers were established.\nThe drawback of the manuscript is it oversell and partially wrong statements. For example, in the abstract the statement ‘how large, flexible structures can be obtained from self-assembling particles is essentially unexplored.’ Is at best misleading. There is a lot of literature out there using DNA functionalized particles, their assembly and crystallization and how to sue them in dynamic structure. At the end of the days, the manuscript targets square lattices, a subsection of the assembled crystal lattices out there.\nThe strength of this manuscript lies mainly in creating a new perspective on flexible bonds and self-assembly dynamics. Enabling the formation of a square lattice in equal-sized spheres, and binary mixtures of spheres and cubes, which then transitions to rigid hexagonal packing over time due to the bonding of dangling or floppy particles. The work combines confocal microscopy, simulations, and analytical modeling to map assembly pathways and highlight the interplay between enthalpic (bond strength) and entropic (rearrangement) factors. Potential applications proposed were in sensors, actuators, and responsive devices with reconfigurable properties. Overall, the manuscript is an interesting read with potential but a few concerns (besides the oversell) are noted below.\n• On pg. 20, line 881 – 892: For the spherical beads with dangling particles that formed floppy square lattices with irreversible transition, it would be worthwhile to systematically analyze their dependence on temperature, concentration, or bond strength by conducting some time-resolved studies or simulations under these varied conditions to map phase boundaries and kinetic barriers as done by Nykypanchuk et al.\n• On pg. 22, line 969, the hematite cubes of edge length was reported as 1.080.04 µm. Is this an intentional ‘point-point release’ or a mistake that could be corrected for consistency of reported values?\n• The authors made no mention of the influence of non-specific interactions such as van der Waals and depletion forces.\n• It would be valuable to add the cluster growth rate and possible lattice defects to the analysis in addition to the time-stamped metric, of approximately 2-16 hours, as provided.\n• Line 200: DNA coverage should be quantified, the DNA density on the surface of a particle plays a vital role and should be clearly stated, including if the same density is kept within all particles. Can the distinction between coordination be attributed to the DNA density differences or due to DNA migration to maximize enthalpic interactions thus leaving vacancies that do not allow for higher coordination of 5 or due to depletion of free particle binding probability as stated in the manuscript.\n• SI movie 9, larger particles appear to be at a higher plane than smaller particles, different than SI movie 6. Why is this observed?\n• Line 185: “we experimentally coat colloidal beads with a fluid lipid membrane into which DNA linkers are embedded. [13, 15, 16]” There has been extensive work in this area and citations to coated colloidal particles with lipid membrane and DNA linkers should all be included here. References need to be updated to include the rest of the work accomplished utilizing this method.\n• Movie 4: There is a large aggregate of green particles. If it’s complementary DNA, it should not bind to each other unless another noncovalent interaction is occurring.", "reviewer_2": "In the manuscript by Shelve et al., the authors investigated the crystal formation pathway in DNA coated colloids (DNACCs). They first found that in system of equally sized DNACCs, floppy crystals of square lattice first form and slowly transform to hexagonal lattices. They later replace one type of spherical DNACCs with cubes, and they found that the square lattice can be stabilised. Lastly, they tried with binary mixture of DNACCs of different sizes, in which they also found stable square lattices. I believe that the experiments have been properly done, but I am not sure whether this work is important enough to be published in a high impact journal like Nature Communications. My questions are below.\n1. For systems of equally sized DNACCs, I think that the reason that they found square lattices first formed is simply due to the fact that square lattice is a metastable state, and its interfacial tension with the dilute phase is small, as they found many particles on the boundary. Therefore, I disagree with the authors that \"the formation of the square lattice stems from a combination of free energy minimization at the one hand and a competition between the timescales for adding another particle and reconfiguration of the existing structure on the other.\" If not, the authors must offer more data or evidence to support their claims.\n2. Similarly, in the mixture of spheres and cubes and the mixture of binary DNACCs of different sizes, the shape simply stabilised the square lattice thermodynamically, which lead to the formation of long lasting square lattice in experiments. If the authors wish to claim that they did just manipulate the pathway to guide formation, they have to offer convincing evidence. In my opinion, it is trivial that cubes stabilise the square lattice in their system instead of purely guiding the pathway of formation.\n3. Last thing I am not very sure about is the general implication of this work. The authors were essentially playing with very simple toy model systems in 2D, and it is unclear to me whether their finding can be generalised for fabrication of any functional materials. Therefore, I think that this work is better suitable for more specialised journals, e.g., Soft Matter.", "reviewer_3": "The work by Shelke et al studies, using combination of MC simulation and experiments, the assembly kinetics and distribution of different 2D DNA coated colloids. They look at ratio of different particles, number ratio, and particle shape on the target square lattice assembly.\nThe work is of interest to the general readership of the Nat Comm journal, and I recommend publication, provided the authors can address the following comments:\n1) Did the authors explore different temperature annealing protocols? Presumably, assembling at higher temperatures would also allow to escape some of low yields due to kinetic traps, and might lead to higher yield. Alternatively, it is also possible to shorten the DNA sequences to make binding reversible at room temperature. Have different annealing protocols been explored in MC simulations?\n2) Have the authors explored different lengths of poly-T spacers in the DNA strands? This is a commmon technique used in 3D assembly for large crystal lattices, and might also lead to potentially higher yield.\n3) It is not clear what is author's criterion for \"large, flexible\" self-assembled structures. There has been a lot of work dedicated to study of assembly of DNA coated particles (e.g. works by Oleg Gang, Chad Mirkin and Sharon Glotzer) where different lengths and the role of bond flexibility on the assembly has been discussed, eg. https://pubs.acs.org/doi/full/10.1021/jacs.3c03181 and https://doi.org/10.1126/science.adj1021. These strategies should be mentioned, and possibly dicsussed if insights from the role of shape and fleixibility in these works is relevant to the findings in this work."}, "author_response": "Response to Reviewers Comments\nResponse to Reviewer 1\nThis manuscript investigates the guided self-assembly of colloidal particles into flexible, re-configurable lattices using DNA-functionalized building blocks. By employing binary mixtures of spherical and cubic colloids with mobile DNA linkers embedded in fluid lipid membranes, theauthorsdemonstratehowbondflexibilityandparticleanisotropyinfluencelattice symmetry (square vs. hexagonal) and mechanical stability. This article brings a different perspective to an already existing article from the same group (Melio et al.), where the mechanical properties of floppy colloidal square lattices assembled by optical tweezers were established. The drawback of the manuscript is it oversell and partially wrong statements. For example, in the abstract the statement ‘how large, flexible structures can be obtained from self-assembling particles is essentially unexplored.’ Is at best misleading. There is a lot of literature out there using DNA functionalized particles, their assembly and crystallization and how to sue them in dynamic structure. At the end of the days, the manuscript targets square lattices, a subsection of the assembled crystal lattices out there.\nThe strength of this manuscript lies mainly in creating a new perspective on flexible bonds and self-assembly dynamics. Enabling the formation of a square lattice in equal-sized spheres, and binary mixtures of spheres and cubes, which then transitions to rigid hexagonal packing over time due to the bonding of dangling or floppy particles. The work combines confocal microscopy, simulations, and analytical modeling to map assembly pathways and highlight the interplay between enthalpic (bond strength) and entropic (rearrangement) factors. Potential applications proposed were in sensors, actuators, and responsive devices with reconfigurable properties. Overall, the manuscript is an interesting read with potential but a few concerns (besides the oversell) are noted below.\nWe thank the reviewer for their time to review our manuscript and their kind words regarding the strengths of our manuscript, particularly their recognition of it providing a perspective on bond flexibility, self-assembly dynamics, and the formation of reconfigurable lattices.\nWe agree that the sentence in the abstract cited by the reviewer might be misinterpreted, which was not our intention. We have now edited the abstract to better reflect our unique approach as follows: ”While rearranging bonds are needed to create floppy structures, it has remained largely unexplored how they affect the self-assembly pathways.”\nTo furthermore clarify the focus our work, we have sharpened the title to: ”Self-assembly pathways towards floppy colloidal square lattices”.\nWe thank the referee for their interesting suggestion. To address this question, we performed new simulations to probe the transition from four fold to six fold symmetry. The results have been added to a new section 3.3 in the SI and are being referred to in the main text. Below, we also summarize the approach and results.\nThe referee mentions three possible avenues to investigate this transition, which we address individually below: temperature, concentration and bond strength.\nBond strength (and high temperature). As the referee points out, the highlighted transition is significant because of its irreversible nature, which follows from the near irreversible bonding process observed in the experiments. If bond strength were reduced (or temperature sufficiently increased) until this wasn’t the case, we would be considering a fundamentally different system. In this scenario, it would be possible to break bonds and reconfigure indefinitely. This would ultimately result in the system reaching a “normal” equilibrium state that minimizes the free energy, similar to earlier work from Hu et al. (ref. 32). Therefore studying bond strength is not relevant to the mechanisms we outline in the manuscript, and studying temperature is only relevant in the regime where bonding remains irreversible.\nTemperature. To study the transition, we use a reduced simulation (as in Fig. 3c), since in both experiments and full simulations an infinite variety of clusters can undergo the transition in subsections, preventing systematic identification and direct comparison. In these simulations, we see that the characteristic transition time decreases when the magnitude of the thermal noise is increased, see section 3.3 in the SI. This is reflected in the fact that the global transition from ψ to ψ occurs sooner with increased thermal noise.\n4 6\nConcentration. As stated in the manuscript, the irreversible transition appears to take a well defined time to occur. This timescale depends on the mobility of the particles within the cluster, but not the concentration since the transition occurs for an already connected cluster. In contrast, we find that the global transition from ψ to ψ occurs sooner when the packing fraction is increased. We attribute this to the fact that at higher packing fractions thereisalreadyatendencytowardcreatingstructureswithψ orderduetoitshigherpacking\n6\nefficiency.\nOn pg. 22, line 969, the hematite cubes of edge length was reported as 1.080.04 µm. Is this an intentional ‘point-point release’ or a mistake that could be corrected for consistency of reported values?\nWe thank the referee for catching this typo, which we have corrected. The edge length should read 1.08 ± 0.04 µm.\nThe authors made no mention of the influence of non-specific interactions such as van der Waals and depletion forces.\nWe thank the reviewer for this important point. In our system, colloidal particles are coated with a fluid lipid bilayer and functionalized with both complementary DNA linkers\n2\n(∼30nm length) and inert (non-binding) DNA strands (∼20nm length). The inert strands are specifically incorporated to sterically stabilize the colloids and suppress short range non-specificinteractionssuchasvanderWaalsattractions(ref. 19)Thelinkerstrandsarefurther- more designed to be non-complementary between the same type of particles (e.g., cube–cube or sphere–sphere), enabling selective bonding only between different particle types. Thus, both inert and non-self-complementary linker DNA work together to minimize non-specific aggregation. In control experiments without inert DNA, or with self-complementary linkers, we do observe unintended binding.\nDepletion forces are also unlikely to contribute, as excess vesicles and unbound DNA that may work as depletant are thoroughly removed through 3–6 washing cycles following functionalization. Therefore, the observed assembly is governed almost entirely by specific DNA hybridization, with negligible influence from non-specific interactions.\nWe have added this information to the methods section as follows: “The washing and integration of inert DNA strands suppressed short-range non-specific interactions such as van der Waals interactions and depletion interactions.”\nIt would be valuable to add the cluster growth rate and possible lattice defects to the analysis in addition to the time-stamped metric, of approximately 2-16 hours, as provided.\nWe thank the referee for this suggestion. In addition to the average coordination number and cluster size as a function of time (which was already included in Figure S* for the simulations) we have now calculated the cluster growth rate and added it to as Figure S8c.\nQuantifying lattice defects is challenging, since the structures are initially quite small.\nWe have also added the average cluster size observed in experiments during the first hour, now presented in Figure S6 of the SI and below. However, we unfortunately do not have sufficiently time-resolved data to determine growth rates for longer times.\na b c\nFigure 1: (a) Average coordination number (cid:104)Z (cid:105), (b) cluster size (cid:104)N (cid:105), and (c) cluster growth\nc c\nrate d(cid:104)Nc(cid:105) for simulations used in main text Fig. 2c (blue), 3c (green), and 5c (red).\ndt\n3\n40\n30\n20\n10\n0 10 20 30 40 50 60\nT (min)\ncN\n(a)\nEqual Sized Spheres\nUnequal Sized Spheres\n2\n1\n0\n1\n2\n0 10 20 30 40 50 60\nT (min)\ntd/\ncNd\n(b)\nEqual Sized Spheres\nUnequal Sized Spheres\nFigure 2: (a) Average cluster size (cid:104)N (cid:105) and (b) cluster growth rate d(cid:104)N (cid:105)/dt as functions\nc c\nof time for systems of equal-sized spheres and unequal-sized spheres obtained from a 2 min\nrolling-average.\nLine 200: DNA coverage should be quantified, the DNA density on the surface of a particle plays a vital role and should be clearly stated, including if the same density is kept within all particles. Can the distinction between coordination be attributed to the DNA density differences or due to DNA migration to maximize enthalpic interactions thus leaving vacancies that do not allow for higher coordination of 5 or due to depletion of free particle binding probability as stated in the manuscript.\nWe agree with the referee that variations in DNA density differences can potentially limit the number of bound particles, as was found by McMullen et al. (PNAS 2021) and was independently in parallel confirmed by us (PhD thesis Ruben Verweij 2021). Since DNA linkers insert themselves with their hydrophobic tails, there is always a variation of DNA linkerconcentrationsacrossthesamplewithaboutanorderofmagnitudevariation. However, the DNA concentrations employed here have been chosen such that the coordination number are not limited by linker accumulation. The concentration of DNA linkers that was added duringexperimentswasalreadyspecifiedinthemethodssection,andwehavefurtherclarified that these are numbers per square micrometer of particle surface. We unfortunately do not have measurements of the precise number of linkers on the particles, only that they provide irreversible and unlimited bonding.\nIn addition, the agreement with simulation results support our hypothesis, that the kinetic pathways determine the number of bound neighbors rather than a linker depletion, as there are no limits to the maximum bonds per particle defined in the simulations. Finally, we note that a maximum binding number of 5 was not only observed by us (Chakraborty et al. ACS Nano (2022), but can also been seen in Fig. 3c of McMullen et al. PNAS (2021), where the maximum observed number of bound droplets is limited at 5.\nWe have added a sentence to the Materials Methods section / Functionalization to clarify that the concentration does not limit the number of bound particles.\nSI movie 9, larger particles appear to be at a higher plane than smaller particles, different than SI movie 6. Why is this observed?\nSince the movie is recorded from below (through the glass coverslip), objects closer to the focal plane appear brighter and sharper. That implies that smaller 1.25µm silica spheres\n4\nactually are higher above the substrate than the larger 2.12 µm spheres in SI Movie 9. This height difference arises from the different gravitational height of the particles. Specifically, the smaller 1.25µm spheres have a larger gravitational height (0.41µm) than the heavier 2.12µm spheres (0.08µm). Because they are confined less tightly by gravity, the flexible DNA bonds allow the lighter spheres to fluctuate more in the z-direction and move over the surface of the bigger spheres. As a result, they sit slightly higher and exhibit noticeable changes in size and brightness as they move in and out of the focal plane. In contrast, the heavier 2.12µm spheres remain closer to the substrate and show minimal vertical motion.\nIn SI Movie 6, although the 1.35µm cubes have gravitational heights comparable to the smaller 1.25µm silica spheres, the cubes exhibit minimal out-of-plane motion because they arebondedtothesmallerspheresattheirverticalsurfaces, whichlimitstheirrelativevertical\nmotion (see Shelke et al. ACS Nano 2023).\nWe have integrated this information in the Materials and Methods section, in sample preparation and imaging.\nLine 185: “we experimentally coat colloidal beads with a fluid lipid membrane into which DNA linkers are embedded. [13, 15, 16]” There has been extensive work in this area and citations to coated colloidal particles with lipid membrane and DNA linkers should all be included here. References need to be updated to include the rest of the work accomplished utilizing this method.\nWe chose to refer to the works that have pioneered the method (Meulen, ref 13) and the work that established the protocol that we are using (Rinaldin ref 15, Chakraborty ref 16), but are happy to include additional references. We have now included other works where we made use of colloids functionalized with lipid membranes and DNA-linkers as requested.\nMovie 4: There is a large aggregate of green particles. If it’s complementary DNA, it should not bind to each other unless another noncovalent interaction is occurring.\nThe bright-green clusters observed in Movie 4 are due to cube–cube aggregation that occurred prior to functionalization with DNA and mixing with spheres. These clusters were foundtoformduringthewashingandsedimentationstepthatfollowslipidandDNAcoating.\nThis minor non-specific aggregation of a few particles is consistent with observations in all other samples, as seen in the initial frames of Movies 2, 5, and 8.\nImportantly, no further cube–cube or sphere–sphere non-specific aggregation was observed after mixing, even up to 18 hours of experimental observation. These pre-existing aggregates represent a small fraction of the total population and do not significantly affect the dynamics of the hetero-assembly pathway discussed in the manuscript. We have added this information to the Materials and Methods section.\nResponse to Reviewer 2\nIn the manuscript by Shelke et al., the authors investigated the crystal formation pathway in DNA coated colloids (DNACCs). They first found that in system of equally sized DNACCs, floppy crystals of square lattice first form and slowly transform to hexagonal lattices. They later replace one type of spherical DNACCs with cubes, and they found that the square lattice can be stabilised. Lastly, they tried with binary mixture of DNACCs of different sizes, in which they also found stable square lattices. I believe that the experiments have beenproperlydone, butIamnotsurewhetherthisworkisimportantenoughtobepublished\nin a high impact journal like Nature Communications. My questions are below.\nWe thank the reviewer for their time to read and evaluate our manuscript. We would like to take the opportunity to emphasize that our study goes beyond just trying three different mixtures of particles and finding floppy square lattices: it identifies how linker mobility and particle shape/size can be used to tune the kinetic assembly pathways towards floppy structures that are otherwise hard to achieve. As such, our results provide design rules for working with systems that possess flexible but irreversible bonds and strategies for building reconfigurable low-symmetry structures.\nFor systems of equally sized DNACCs, I think that the reason that they found square latticesfirstformedissimplyduetothefactthatsquarelatticeisametastablestate, and its interfacial tension with the dilute phase is small, as they found many particles on the boundary. Therefore, I disagree with the authors that “the formation of the square lattice stems from a combination of free energy minimization at the one hand and a competition between the timescales for adding another particle and reconfiguration of theexistingstructureontheother.”Ifnot, theauthorsmustoffermoredataorevidence\nto support their claims.\nWe thank the referee for pointing out that the sentence was not clear enough to explain our rationale. We agree that it can be interesting to think in terms of an interfacial tension, however, in the system studied here the bonds are irreversible making the assembly pathways moreimportanttounderstandtheobtainedstructuresthanthethermodynamicgroundstate.\nThe sentence was meant to allude to the following: The irreversibility of the binding implies that the transition to hexagonal order (Fig. 3c) relies on specific conditions, i.e. dangling particles, and has a characteristic timescale associated with it, as demonstrated in Fig. 3d (inset). It can be prevented by the addition of particles that re-stabilize the square lattice. Therefore the final state of the lattice depends on whichever happens first; either internal reconfiguration into a hexagonal lattice or addition of another particle that can prevent the transition. We utilize this understanding to demonstrate that we can control which lattice is more likely (see Fig. 5), and show that square lattices are stable at long timescales in both simulations and experiments (see Fig. 6). We have edited the sentence as well as the paragraph it was contained in to enhance its clarity.\nSimilarly, in the mixture of spheres and cubes and the mixture of binary DNACCs of different sizes, the shape simply stabilised the square lattice thermodynamically, which lead to the formation of long lasting square lattice in experiments. If the authors wish to claim that they did just manipulate the pathway to guide formation, they have to offer convincing evidence. In my opinion, it is trivial that cubes stabilise the square lattice in their system instead of purely guiding the pathway of formation.\nAs the referee has observed (and we point out in the manuscript) making a square lattice with cubic particles is somewhat trivial and also reduces the floppiness of the aggregates.\nWe would like to comment on the referee’s comment that the square lattice is simply “stabilized thermodynamically”. As we have stated, on the timescale of the experiments the bonds form irreversibly. In a system in which truly irreversible bonds are formed detailed balance is broken. Without detailed balance, there is no well defined “thermodynamic equilibrium” and things cannot “stabilize thermodynamically”. A consequence of this is that the pathways, such as those outlined in Fig. 2, are uni-directional (this is why they feature arrows). When we observe a specific cluster, there is only a finite number of pathways that it could have taken to arise. Thus changing the likelihood of a specific cluster is equivalent to changing the likelihood of the relevant pathways. We have amended the manuscript to make this point clearer.\nLast thing I am not very sure about is the general implication of this work. The authors were essentially playing with very simple toy model systems in 2D, and it is unclear to me whether their finding can be generalised for fabrication of any functional materials. Therefore, I think that this work is better suitable for more specialised journals, e.g., Soft Matter.\nWethankthereviewerforsharingtheirperspective. Webelievethatthisdifferentviewon the importance of our work stemmed from a misunderstanding of our system, which cannot reach the thermodynamic ground state but rather requires design of the kinetic pathways for reaching ordered floppy structures. Although our system is based on simplified 2D colloidal models, the core principles, such as how bond mobility and particle asymmetry control both the final structure and the assembly pathway, are broadly relevant to reconfigurable systems.\nAssuchtheycanbeusedtocreatelatticeswhichcanbeusedasmodelsfortestingpredictions about floppy modes in thermal systems (ref. 45), and assembling materials with unusual mechanical properties, i.e. Brownian metamaterials (ref. 11, 28), or microrobots (ref.15)\nThe ability to steer assembly away from kinetic traps using linker mobility and shape is a general strategy that extends to 3D and other programmable systems. We therefore believe these findings and our combined experimental-simulation approach offer insights of broad interest to the Nature Communications readership.\nResponse to Reviewer 3\nThe work by Shelke et al. studies, using combination of MC simulation and experiments, the assembly kinetics and distribution of different 2D DNA coated colloids. They look at ratio of different particles, number ratio, and particle shape on the target square lattice assembly.\nThe work is of interest to the general readership of the Nat Comm journal, and I recommend publication, provided the authors can address the following comments:\n1) Did the authors explore different temperature annealing protocols? Presumably, assembling at higher temperatures would also allow to escape some of low yields due to kinetic traps, and might lead to higher yield. Alternatively, it is also possible to shorten the DNA sequences to make binding reversible at room temperature. Have different annealing protocols been explored in MC simulations?\nResponse: We thank the reviewer for this suggestion. Indeed, in systems with surface-bound complementary DNA strands, annealing protocols or reversible binding are essential to allow structural rearrangements and improve crystallinity, especially when particles first form amorphous aggregates. In contrast, our study focuses on a different kind of system, which allows for structural rearrangements after binding due to the linker mobility, but without breaking bonds once formed. This allows us to single out the effect of rearranging bonds and how to use them to tailor the assembly pathways to avoid kinetic traps. While it is possible to design shorter DNA linkers and explore annealing protocols, it exceeds the focus of this study and hence we did not explore them. However, we believe that these can be powerful strategies to integrate in the future access other cluster types or increase yields.\nWe have added a sentence to the discussion section to integrate this suggestion.\n2) Have the authors explored different lengths of poly-T spacers in the DNA strands? This is a commmon technique used in 3D assembly for large crystal lattices, and might also lead to potentially higher yield.\nWe thank the referee for this suggestion. We have not explored this route since in our system, because the observed bond flexibility arises from the lateral mobility of DNA linkers embedded in the fluid lipid bilayer on the particle surface, rather than from the intrinsic flexibility of the DNA strands themselves. The 3D assemblies referenced by the reviewer typically involve surface-bound, immobile DNA linkers, where the poly-T spacer length can significantly influence bond flexibility and assembly yield. Since our DNA linkers are mobile themselves and not anchored rigidly to the particle surface, we did not explore variations in poly-T spacer length. Nonetheless, we agree that systematically investigating spacer length and sequence could be an interesting direction for future studies and have added a sentence to the introduction and discussion referring to this.\n3) It is not clear what is author’s criterion for ”large, flexible” self-assembled structures. There has been a lot of work dedicated to study of assembly of DNA coated particles (e.g. works by Oleg Gang, Chad Mirkin and Sharon Glotzer) where different lengths and the role of bond flexibility on the assembly has been discussed, eg. https://pubs.acs.org/doi/full/10.1021/jacs.3c03181 and https://doi.org/10.1126/science.adj1021. These strategies should be mentioned, and possibly dicsussed if insights from the role of shape and flexibility in these works is relevant to the findings in this work.\nTo clarify, by large, flexible structures we refer to lattices composed of tens to hundreds of particles that preserve lattice connectivity while allowing local angular reconfiguration such as are shown in the experimental overview images in the manuscript. In addition, we added a figure to the SI that measures the cluster size for self-assembled structures from equal-sized spheres after 2h. We also show it below in Fig. 3 for your convenience.\na b\nFigure 3: Cluster size distribution(a) Confocal microscopy image of equal-sized spheres after 2h. (b) Corresponding cluster map with different sizes represented by distinct colors. The inset shows the cluster size distribution.\n9"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The manuscript has strengthen significantly with the edits and new experiments. I have no problem to recommend publication at this stage.", "reviewer_2": "I have read the response letter and the revised manuscript, but I am still not convinced that this work can be published in Nature Communications. First, the authors are playing the word games. Even if the bond formation process is not reversible, this does not mean that the free energy/thermodynamics does not play any role in the process. In principle, the system is generally moving towards lowest free energy state, and it is not surprising that it gets stuck at some free energy local minima. Therefore, the evidence and data in this work cannot rule out the effect of metastability. Second, the most serious question in this work is that their system is too simple and over simplified. Although they mentioned potential implications in reconfigurable systems but I failed to see any direct usefulness of this work in any other systems. For 2D hexagonal and square colloidal lattices, I can hardly find any useful applications in whatever functional materials, and it is unclear whether this results from those oversimplified systems can be generalised into any other more complex lattices. At least the author did not offer convincing evidence to demonstrate the generalisation of their principles into more complex functional materials. Therefore, I still think this work is suitable for more specialised journals.", "reviewer_3": "The authors have addressed my concerns in sufficient level of detail."}, "author_response": "Response to referees\nReviewer #1\nThe manuscript has strengthened significantly with the edits and new experiments. I have no problem to recommend publication at this stage.\nWe thank the reviewer for their time and positive evaluation.\nReviewer #2\nI have read the response letter and the revised manuscript, but I am still not convinced that this work can be published in Nature Communications. First, the authors are playing the word games. Even if the bond formation process is not reversible, this does not mean that the free energy/thermodynamics does not play any role in the process. In principle, the system is generally moving towards lowest free energy state, and it is not surprising that it gets stuck at some free energy local minima. Therefore, the evidence and data in this work cannot rule out the effect of metastability. \nWe apologise if the referee feels we are being disingenuous in our arguments. This is not our intention. We do not argue that free energy/thermodynamics does not play a role – our system is generally moving towards the lowest free energy state by lowering the energy through bond formation and maximisation of the entropy, but cannot reach the ground state as it gets stuck in local minima due to the irreversible bond formation. As such, our whole experiment and simulation analysis is a study on the metastability of the various states. The surprise is that a despite hopping from local minimum to local minimum, a ordered structure still can be obtained. Understanding the metastability of the intermediate states, allows influencing the final structure and its general macroscopic properties. That is the central mechanism we explore in our manuscript.\nOur previous work on a very similar experimental system specifically investigated how entropy plays a role in the formation of clusters (Chakraborty et al, ACS nano (2022)). This previous work was performed on a single, specific aggregation pathway, which allowed us to calculate the entropy analytically. Such an approach is not possible here due to the increased complexity of the aggregates; thus, we estimate the effect numerically by determining the relative likelihoods of various aggregates. For example, the results presented in Fig. 2a are a measure of the relative importance of entropy and energy.\nThis was already mentioned in our manuscript, and we have further emphasized the above as follows (lines 343ff, additions in blue):\n“In our experiments, the probability for particles to unbind to optimize their position in the lattice and find the state of lowest energy is very low, i.e. the bonds are effectively irreversible. This implies that they cannot reach the thermodynamic ground state, as predicted by Monte Carlo simulations, but only local minima in the free energy landscape. However, while maintaining the bond they are still able to move with respect to each other allowing an increase of the number of bonds and lowering of their free energy. (…) We therefore hypothesize that the formation of the square lattice stems from a combination of free energy minimization through bond formation at the one hand, and a competition between the timescales for adding another particle and reconfiguration of the existing structure on the other.”\nLines 425ff (addition indicated in blue):\n“Since each specific cluster can arise from a finite number of pathways, the likelihood of a cluster is directly connected to the likelihood of the relevant pathways, which are all driven by a lowering of the free energy. “\nDiscussion, lines 952ff (additions in blue):\n“Our findings demonstrate that particle shape, size and number ratio, as well as reconfigurability all play crucial roles in the pathway towards and the time required for the formation of flexible lattices. Although our system is not able to reach the thermodynamic groundstate because of irreversible bonding, we showed that tuning the pathway along local minima allows achieving ordered flexible structures.”\nSecond, the most serious question in this work is that their system is too simple and over simplified. Although they mentioned potential implications in reconfigurable systems but I failed to see any direct usefulness of this work in any other systems. For 2D hexagonal and square colloidal lattices, I can hardly find any useful applications in whatever functional materials, and it is unclear whether this results from those oversimplified systems can be generalised into any other more complex lattices. At least the author did not offer convincing evidence to demonstrate the generalisation of their principles into more complex functional materials. Therefore, I still think this work is suitable for more specialised journals.\nWe respectfully disagree with this assessment of our work. The simplicity of our system is precisely what allows us to make such general statements about how the aggregation process relates to the final properties of the lattice. We demonstrate that even this simple option allows for control over macroscopic properties of the final material.\nWe also would like to point out that the possible applications we describe in the outlook paragraph, such as their use as model systems, microscopic metamaterials, and robots and machines, are all aims we are currently working on.\nReviewer #3\nThe authors have addressed my concerns in sufficient level of detail.\nWe thank the reviewer for their time and positive evaluation."}]

Decision letter