Size limits of self-assembled colloidal structures made using specific interactions

Economical routes to size-specific assembly of self-closing structures

Programmable self-assembly has seen an explosion in the diversity of synthetic crystalline materials, but developing strategies that target "self-limiting" assemblies has remained a challenge. Among these, self-closing structures, in which the local curvature defines the finite global size, are prone to polymorphism due to thermal bending fluctuations, a problem that worsens with increasing target size. Here, we show that assembly complexity can be used to eliminate this source of polymorphism in the assembly of tubules. Using many distinct components, we prune the local density of off-target geometries, increasing the selectivity of the tubule width and helicity to nearly 100%. We further show that by reducing the design constraints to target either the pitch or the width alone, fewer components are needed to reach complete selectivity. Combining experiments with theory, we reveal an economical limit, which determines the minimum number of components required to create arbitrary assembly sizes with full selectivity.

Programming patchy particles for materials assembly design

Direct design of complex functional materials would revolutionize technologies ranging from printable organs to novel clean energy devices. However, even incremental steps toward designing functional materials have proven challenging. If the material is constructed from highly complex components, the design space of materials properties rapidly becomes too computationally expensive to search. On the other hand, very simple components such as uniform spherical particles are not powerful enough to capture rich functional behavior. Here, we introduce a differentiable materials design model with components that are simple enough to design yet powerful enough to capture complex materials properties: rigid bodies composed of spherical particles with directional interactions (patchy particles). We showcase the method with self-assembly designs ranging from open lattices to self-limiting clusters, all of which are notoriously challenging design goals to achieve using purely isotropic particles. By directly optimizing over the location and interaction of the patches on patchy particles using gradient descent, we dramatically reduce the computation time for finding the optimal building blocks.

Limits of economy and fidelity for programmable assembly of size-controlled triply periodic polyhedra

We propose and investigate an extension of the Caspar-Klug symmetry principles for viral capsid assembly to the programmable assembly of size-controlled triply periodic polyhedra, discrete variants of the Primitive, Diamond, and Gyroid cubic minimal surfaces. Inspired by a recent class of programmable DNA origami colloids, we demonstrate that the economy of design in these crystalline assemblies-in terms of the growth of the number of distinct particle species required with the increased size-scale (e.g., periodicity)-is comparable to viral shells. We further test the role of geometric specificity in these assemblies via dynamical assembly simulations, which show that conditions for simultaneously efficient and high-fidelity assembly require an intermediate degree of flexibility of local angles and lengths in programmed assembly. Off-target misassembly occurs via incorporation of a variant of disclination defects, generalized to the case of hyperbolic crystals. The possibility of these topological defects is a direct consequence of the very same symmetry principles that underlie the economical design, exposing a basic tradeoff between design economy and fidelity of programmable, size controlled assembly.

A computational toolbox for the assembly yield of complex and heterogeneous structures

The self-assembly of complex structures from a set of non-identical building blocks is a hallmark of soft matter and biological systems, including protein complexes, colloidal clusters, and DNA-based assemblies. Predicting the dependence of the equilibrium assembly yield on the concentrations and interaction energies of building blocks is highly challenging, owing to the difficulty of computing the entropic contributions to the free energy of the many structures that compete with the ground state configuration. While these calculations yield well known results for spherically symmetric building blocks, they do not hold when the building blocks have internal rotational degrees of freedom. Here we present an approach for solving this problem that works with arbitrary building blocks, including proteins with known structure and complex colloidal building blocks. Our algorithm combines classical statistical mechanics with recently developed computational tools for automatic differentiation. Automatic differentiation allows efficient evaluation of equilibrium averages over configurations that would otherwise be intractable. We demonstrate the validity of our framework by comparison to molecular dynamics simulations of simple examples, and apply it to calculate the yield curves for known protein complexes and for the assembly of colloidal shells.

Nonequilibrium design strategies for functional colloidal assemblies

We use a nonequilibrium variational principle to optimize the steady-state, shear-induced interconversion of self-assembled nanoclusters of DNA-coated colloids. Employing this principle within a stochastic optimization algorithm allows us to identify design strategies for functional materials. We find that far-from-equilibrium shear flow can significantly enhance the flux between specific colloidal states by decoupling trade-offs between stability and reactivity required by systems in equilibrium. For isolated nanoclusters, we find nonequilibrium strategies for amplifying transition rates by coupling a given reaction coordinate to the background shear flow. We also find that shear flow can be made to selectively break detailed balance and maximize probability currents by coupling orientational degrees of freedom to conformational transitions. For a microphase consisting of many nanoclusters, we study the flux of colloids hopping between clusters. We find that a shear flow can amplify the flux without a proportional compromise on the microphase structure. This approach provides a general means of uncovering design principles for nanoscale, autonomous, functional materials driven far from equilibrium.

Learning to learn by using nonequilibrium training protocols for adaptable materials

Evolution in time-varying environments naturally leads to adaptable biological systems that can easily switch functionalities. Advances in the synthesis of environmentally responsive materials therefore open up the possibility of creating a wide range of synthetic materials which can also be trained for adaptability. We consider high-dimensional inverse problems for materials where any particular functionality can be realized by numerous equivalent choices of design parameters. By periodically switching targets in a given design algorithm, we can teach a material to perform incompatible functionalities with minimal changes in design parameters. We exhibit this learning strategy for adaptability in two simulated settings: elastic networks that are designed to switch deformation modes with minimal bond changes and heteropolymers whose folding pathway selections are controlled by a minimal set of monomer affinities. The resulting designs can reveal physical principles, such as nucleation-controlled folding, that enable such adaptability.

Designing the Self-Assembly of Arbitrary Shapes Using Minimal Complexity Building Blocks

The design space for self-assembled multicomponent objects ranges from a solution in which every building block is unique to one with the minimum number of distinct building blocks that unambiguously define the target structure. We develop a pipeline to explore the design spaces for a set of structures of various sizes and complexities. To understand the implications of the different solutions, we analyze their assembly dynamics using patchy particle simulations and study the influence of the number of distinct building blocks, and the angular and spatial tolerances on their interactions, on the kinetics and yield of the target assembly. We show that the resource-saving solution with a minimum number of distinct blocks can often assemble just as well (or faster) than designs where each building block is unique. We further use our methods to design multifarious structures, where building blocks are shared between different target structures. Finally, we use coarse-grained DNA simulations to investigate the realization of multicomponent shapes using DNA nanostructures as building blocks.

Optimization of non-equilibrium self-assembly protocols using Markov state models

The promise of self-assembly to enable the bottom-up formation of materials with prescribed architectures and functions has driven intensive efforts to uncover rational design principles for maximizing the yield of a target structure. Yet, despite many successful examples of self-assembly, ensuring kinetic accessibility of the target structure remains an unsolved problem in many systems. In particular, long-lived kinetic traps can result in assembly times that vastly exceed experimentally accessible timescales. One proposed solution is to design non-equilibrium assembly protocols in which system parameters change over time to avoid such kinetic traps. Here, we develop a framework to combine Markov state model (MSM) analysis with optimal control theory to compute a time-dependent protocol that maximizes the yield of the target structure at a finite time. We present an adjoint-based gradient descent method that, in conjunction with MSMs for a system as a function of its control parameters, enables efficiently optimizing the assembly protocol. We also describe an interpolation approach to significantly reduce the number of simulations required to construct the MSMs. We demonstrate our approach with two examples; a simple semi-analytic model for the folding of a polymer of colloidal particles, and a more complex model for capsid assembly. Our results show that optimizing time-dependent protocols can achieve significant improvements in the yields of selected structures, including equilibrium free energy minima, long-lived metastable structures, and transient states.

Geometrically programmed self-limited assembly of tubules using DNA origami colloids

Nature is replete with self-assembled materials that have one or more self-limited dimensions, including shells, tubules, and fibers. Despite significant advances in making nanometer- and micrometer-scale subunits, the programmable assembly of similar self-limiting architectures from synthetic components has remained largely out of reach. In this article, we create geometrically programmed subunits using DNA origami and study their assembly into tubules with a self-limited width. We show that the average self-limited dimension can be tuned by changing the local curvature encoded in a single subunit. Exploiting the programmability of our system, we further test the tradeoffs between fidelity and complexity embodied by two paradigms for self-limited assembly: self-closure through programmed curvature and addressable assembly through programmed specific interactions.

Self-assembly is one of the most promising strategies for making functional materials at the nanoscale, yet new design principles for making self-limiting architectures, rather than spatially unlimited periodic lattice structures, are needed. To address this challenge, we explore the tradeoffs between addressable assembly and self-closing assembly of a specific class of self-limiting structures: cylindrical tubules. We make triangular subunits using DNA origami that have specific, valence-limited interactions and designed binding angles, and we study their assembly into tubules that have a self-limited width that is much larger than the size of an individual subunit. In the simplest case, the tubules are assembled from a single component by geometrically programming the dihedral angles between neighboring subunits. We show that the tubules can reach many micrometers in length and that their average width can be prescribed through the dihedral angles. We find that there is a distribution in the width and the chirality of the tubules, which we rationalize by developing a model that considers the finite bending rigidity of the assembled structure as well as the mechanism of self-closure. Finally, we demonstrate that the distributions of tubules can be further sculpted by increasing the number of subunit species, thereby increasing the assembly complexity, and demonstrate that using two subunit species successfully reduces the number of available end states by half. These results help to shed light on the roles of assembly complexity and geometry in self-limited assembly and could be extended to other self-limiting architectures, such as shells, toroids, or triply periodic frameworks.

Self-assembly of emulsion droplets through programmable folding

In the realm of particle self-assembly, it is possible to reliably construct nearly arbitrary structures if all the pieces are distinct^1-3, but systems with fewer flavours of building blocks have so far been limited to the assembly of exotic crystals^4-6. Here we introduce a minimal model system of colloidal droplet chains⁷, with programmable DNA interactions that guide their downhill folding into specific geometries. Droplets are observed in real space and time, unravelling the rules of folding. Combining experiments, simulations and theory, we show that controlling the order in which interactions are switched on directs folding into unique structures, which we call colloidal foldamers⁸. The simplest alternating sequences (ABAB...) of up to 13 droplets yield 11 foldamers in two dimensions and one in three dimensions. Optimizing the droplet sequence and adding an extra flavour uniquely encodes more than half of the 619 possible two-dimensional geometries. Foldamers consisting of at least 13 droplets exhibit open structures with holes, offering porous design. Numerical simulations show that foldamers can further interact to make complex supracolloidal architectures, such as dimers, ribbons and mosaics. Our results are independent of the dynamics and therefore apply to polymeric materials with hierarchical interactions on all length scales, from organic molecules all the way to Rubik's Snakes. This toolbox enables the encoding of large-scale design into sequences of short polymers, placing folding at the forefront of materials self-assembly.

Programming Directed Motion with DNA-Grafted Particles

Colloidal particles can be programmed to interact in complex ways by functionalizing them with DNA oligonucleotides. Adding DNA strand-displacement reactions to the system allows these interparticle interactions to respond to specific changes in temperature. We present the requirements for thermally driven directed motion of colloidal particles, and we explore how these conditions can be realized experimentally using strand-displacement reactions. To evaluate the concept, we build and test a colloidal "dancer": a single particle that can be driven to move through a programmed sequence of steps along a one-dimensional track composed of other particles. The results of these tests reveal the capabilities and limitations of using DNA-mediated interactions for applications in dynamic systems. Specifically, we discuss how to design the substrate to limit complexity while permitting full control of the motile component, how to ratchet the interactions to move over many substrate positions with a limited regime of control parameters, and how to use technological developments to reduce the probability of detachment without sacrificing speed.

Temperature protocols to guide selective self-assembly of competing structures

Multicomponent self-assembly mixtures offer the possibility of encoding multiple target structures with the same set of interacting components. Selective retrieval of one of the stored structures has been attempted by preparing an initial state that favors the assembly of the required target, through seeding, concentration patterning, or specific choices of interaction strengths. This may not be possible in an experiment where on-the-fly reconfiguration of the building blocks to switch functionality may be required. In this paper, we explore principles of inverse design of a multicomponent, self-assembly mixture capable of encoding two competing structures that can be selected through simple temperature protocols. We design the target structures to realize the generic situation in which one of the targets has the lower nucleation barrier, while the other is globally more stable. We observe that, to avoid the formation of spurious or chimeric aggregates, the number of neighboring component pairs that occur in both structures should be minimal. Our design also requires the inclusion of components that are part of only one of the target structures. We observe, however, that to maximize the selectivity of retrieval, the component library itself should be maximally shared by the two targets, within such a constraint. We demonstrate that temperature protocols can be designed that lead to the formation of either one of the target structures with high selectivity. We discuss the important role played by secondary aggregation products in improving selectivity, which we term "vestigial aggregates."

Peptide framework for screening the effects of amino acids on assembly

Discovery of peptide domains with unique intermolecular interactions is essential for engineering peptide-based materials. Rather than attempting a brute-force approach, we instead identify a previously unexplored strategy for discovery and study of intermolecular interactions: “co-assembly of oppositely charged peptide” (CoOP), a framework that “encourages” peptide assembly by mixing two oppositely charged hexapeptides. We used an integrated computational and experimental approach, probed the free energy of association and probability of amino acid contacts during co-assembly with atomic-resolution simulations, and correlated them to the physical properties of the aggregates. We introduce CoOP with three examples: dialanine, ditryptophan, and diisoleucine. Our results indicated that the opposite charges initiate the assembly, and the subsequent stability is enhanced by the presence of an undisturbed hydrophobic core. CoOP represents a unique, simple, and elegant framework that can be used to identify the structure-property relationships of self-assembling peptide-based materials.

Tiling a tubule: how increasing complexity improves the yield of self-limited assembly

The ability to design and synthesize ever more complicated colloidal particles opens the possibility of self-assembling a zoo of complex structures, including those with one or more self-limited length scales. An undesirable feature of systems with self-limited length scales is that thermal fluctuations can lead to the assembly of nearby, off-target states. We investigate strategies for limiting off-target assembly by using multiple types of subunits. Using simulations and energetics calculations, we explore this concept by considering the assembly of tubules built from triangular subunits that bind edge to edge. While in principle, a single type of triangle can assemble into tubules with a monodisperse width distribution, in practice, the finite bending rigidity of the binding sites leads to the formation of off-target structures. To increase the assembly specificity, we introduce tiling rules for assembling tubules from multiple species of triangles. We show that the selectivity of the target structure can be dramatically improved by using multiple species of subunits, and provide a prescription for choosing the minimum number of subunit species required for near-perfect yield. Our approach of increasing the system's complexity to reduce the accessibility of neighboring structures should be generalizable to other systems beyond the self-assembly of tubules.

Anatomy and Formation Mechanisms of Early Amyloid-β Oligomers with Lateral Branching: Graph Network Analysis on Large-Scale Simulations

Oligomeric amyloid-β aggregates (AβOs) effectively trigger Alzheimer's disease-related toxicity, generating great interest in understanding their structures and formation mechanisms. However, AβOs are heterogeneous and transient, making their structure and formation difficult to study. Here, we performed graph network analysis of tens of microsecond massive simulations of early amyloid-β (Aβ) aggregations at near-atomic resolution to characterize AβO structures with sizes up to 20-mers. We found that AβOs exhibit highly curvilinear, irregular shapes with occasional lateral branches, consistent with recent cryo-electron tomography experiments. We also found that Aβ40 oligomers were more likely to develop branches than Aβ42 oligomers, explaining an experimental observation that only Aβ40 was trapped in network-like aggregates and exhibited slower fibrillization kinetics. Moreover, AβO architecture dissection revealed that their curvilinear appearance is related to the local packing geometries of neighboring peptides and that Aβ40's greater branching ability originates from specific C-terminal interactions at branching interfaces. Finally, we demonstrate that whether Aβ oligomerization causes oligomers to elongate or to branch depends on the sizes and shapes of colliding aggregates. Collectively, this study provides bottom-up structural information for understanding early Aβ aggregation and AβO toxicity.

Graph network analysis on large-scale simulations uncovers the differential branching behaviours of large Aβ40 and Aβ42 oligomers.

Designing self-assembling kinetics with differentiable statistical physics models

The inverse problem of designing component interactions to target emergent structure is fundamental to numerous applications in biotechnology, materials science, and statistical physics. Equally important is the inverse problem of designing emergent kinetics, but this has received considerably less attention. Using recent advances in automatic differentiation, we show how kinetic pathways can be precisely designed by directly differentiating through statistical physics models, namely free energy calculations and molecular dynamics simulations. We consider two systems that are crucial to our understanding of structural self-assembly: bulk crystallization and small nanoclusters. In each case, we are able to assemble precise dynamical features. Using gradient information, we manipulate interactions among constituent particles to tune the rate at which these systems yield specific structures of interest. Moreover, we use this approach to learn nontrivial features about the high-dimensional design space, allowing us to accurately predict when multiple kinetic features can be simultaneously and independently controlled. These results provide a concrete and generalizable foundation for studying nonstructural self-assembly, including kinetic properties as well as other complex emergent properties, in a vast array of systems.

Thermodynamic stability versus kinetic accessibility: Pareto fronts for programmable self-assembly

A challenge in designing self-assembling building blocks is to ensure the target state is both thermodynamically stable and kinetically accessible. These two objectives are known to be typically in competition, but it is not known how to simultaneously optimize them. We consider this problem through the lens of multi-objective optimization theory: we develop a genetic algorithm to compute the Pareto fronts characterizing the tradeoff between equilibrium probability and folding rate, for a model system of small polymers of colloids with tunable short-ranged interaction energies. We use a coarse-grained model for the particles' dynamics that allows us to efficiently search over parameters, for systems small enough to be enumerated. For most target states there is a tradeoff when the number of types of particles is small, with medium-weak bonds favouring fast folding, and strong bonds favouring high equilibrium probability. The tradeoff disappears when the number of particle types reaches a value m*, that is usually much less than the total number of particles. This general approach of computing Pareto fronts allows one to identify the minimum number of design parameters to avoid a thermodynamic-kinetic tradeoff. However, we argue, by contrasting our coarse-grained model's predictions with those of Brownian dynamics simulations, that particles with short-ranged isotropic interactions should generically have a tradeoff, and avoiding it in larger systems will require orientation-dependent interactions.

Designer Nanomaterials through Programmable Assembly

Nanoparticles have long been recognized for their unique properties, leading to exciting potential applications across optics, electronics, magnetism, and catalysis. These specific functions often require a designed organization of particles, which includes the type of order as well as placement and relative orientation of particles of the same or different kinds. DNA nanotechnology offers the ability to introduce highly addressable bonds, tailor particle interactions, and control the geometry of bindings motifs. Here, we discuss how developments in structural DNA nanotechnology have enabled greater control over 1D, 2D, and 3D particle organizations through programmable assembly. This Review focuses on how the use of DNA binding between nanocomponents and DNA structural motifs has progressively allowed the rational formation of prescribed particle organizations. We offer insight into how DNA-based motifs and elements can be further developed to control particle organizations and how particles and DNA can be integrated into nanoscale building blocks, so-called "material voxels", to realize designer nanomaterials with desired functions.

Equilibrium mechanisms of self-limiting assembly

Self-assembly is a ubiquitous process in synthetic and biological systems, broadly defined as the spontaneous organization of multiple subunits (e.g. macromolecules, particles) into ordered multi-unit structures. The vast majority of equilibrium assembly processes give rise to two states: one consisting of dispersed disassociated subunits, and the other, a bulk-condensed state of unlimited size. This review focuses on the more specialized class of self-limiting assembly, which describes equilibrium assembly processes resulting in finite-size structures. These systems pose a generic and basic question, how do thermodynamic processes involving non-covalent interactions between identical subunits "measure" and select the size of assembled structures? In this review, we begin with an introduction to the basic statistical mechanical framework for assembly thermodynamics, and use this to highlight the key physical ingredients that ensure equilibrium assembly will terminate at finite dimensions. Then, we introduce examples of self-limiting assembly systems, and classify them within this framework based on two broad categories: self-closing assemblies and open-boundary assemblies. These include well-known cases in biology and synthetic soft matter - micellization of amphiphiles and shell/tubule formation of tapered subunits - as well as less widely known classes of assemblies, such as short-range attractive/long-range repulsive systems and geometrically-frustrated assemblies. For each of these self-limiting mechanisms, we describe the physical mechanisms that select equilibrium assembly size, as well as potential limitations of finite-size selection. Finally, we discuss alternative mechanisms for finite-size assemblies, and draw contrasts with the size-control that these can achieve relative to self-limitation in equilibrium, single-species assemblies.

Simulating sticky particles: A Monte Carlo method to sample a stratification

Many problems in materials science and biology involve particles interacting with strong, short-ranged bonds that can break and form on experimental timescales. Treating such bonds as constraints can significantly speed up sampling their equilibrium distribution, and there are several methods to sample probability distributions subject to fixed constraints. We introduce a Monte Carlo method to handle the case when constraints can break and form. More generally, the method samples a probability distribution on a stratification: a collection of manifolds of different dimensions, where the lower-dimensional manifolds lie on the boundaries of the higher-dimensional manifolds. We show several applications of the method in polymer physics, self-assembly of colloids, and volume calculation in high dimensions.

Self-assembly of finite-sized colloidal aggregates

One of the challenges of self-assembling finite-sized colloidal aggregates with a sought morphology is the necessity of precisely sorting the position of the colloids at the microscopic scale to avoid the formation of off-target structures. Microfluidic platforms address this problem by loading into single droplets the exact amount of colloids entering the targeted aggregate. Using theory and simulations, in this paper, we validate a more versatile design allowing us to fabricate different types of finite-sized aggregates, including colloidal molecules or core-shell clusters, starting from finite density suspensions of isotropic colloids in bulk. In our model, interactions between particles are mediated by DNA linkers with mobile tethering points, as found in experiments using DNA oligomers tagged with hydrophobic complexes immersed into supported bilayers. By fine-tuning the strength and number of the different types of linkers, we prove the possibility of controlling the morphology of the aggregates, in particular, the valency of the molecules and the size of the core-shell clusters. In general, our design shows how multivalent interactions can lead to microphase separation under equilibrium conditions.

Self-Assembly and Crystallization of DNA-Coated Colloids via Linker-Encoded Interactions

Coating colloidal particles with DNA is a promising strategy to make functional nanoscale materials because the particles can be programmed to spontaneously self-assemble into complex, ordered structures. In this Article, we explore the phase behavior and types of structures that can be formed when interactions between DNA-coated colloids are specified by linker DNA strands dispersed in solution. We show that linker-mediated interactions direct the self-assembly of colloids into equilibrium crystal structures. Furthermore, we demonstrate how different linker sequences and concentrations produce different crystal lattices, whose symmetry and compositional order are encoded exclusively by the linker-mediated interactions. These results illustrate how linkers can be used to separate the assembly instructions, encoded in the linkers, from the colloids themselves. We also examine the phase behavior of asymmetric linkers, which bind more strongly to one colloidal species than the other. We find that asymmetry strongly influences the concentration dependence of the colloidal interactions, which we explain using a mean-field model. We also find evidence that asymmetric linkers might help to reduce kinetic bottlenecks to colloidal crystallization. Together, our findings expand the design rules of linker-mediated self-assembly and make connections between the various schemes for programming assembly of DNA-coated colloids reported in the literature.

Photo-printing of faceted DNA patchy particles

Patchy particles with shape complementarity can serve as building blocks for assembling colloidal superstructures. Alternatively, encoding information on patches using DNA can direct assembly into a variety of crystalline or other preprogrammed structures. Here, we present a tool where DNA is used both to engineer shape and to encode information on colloidal particles. Two reactive oil emulsions with different but complementary DNA (cDNA) brushes are assembled into CsCl-like crystalline lattices. The DNA brushes are recruited to and ultimately localized at the junctions between neighboring droplets, which gives rise to DNA-encoded faceted patches. The emulsions are then solidified by ultraviolet (UV) polymerization, producing faceted patchy particles. The facet size and DNA distribution are determined by the balance between the DNA binding energy and the elastic deformation energy of droplets. This method leads to a variety of new patchy particles with directional interactions in scalable quantities.

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

Developing construction methods of materials tailored for given applications with absolute control over building block placement poses an immense challenge. DNA-coated colloids offer the possibility of realising programmable self-assembly, which, in principle, can assemble almost any structure in equilibrium, but remains challenging experimentally. Here, we propose an innovative system of linker-mediated mobile DNA-coated colloids (mDNACCs), in which mDNACCs are bridged by the free DNA linkers in solution, whose two single-stranded DNA tails can bind with specific single-stranded DNA receptors of complementary sequence coated on colloids. We formulate a mean-field theory efficiently calculating the effective interaction between mDNACCs, where the entropy of DNA linkers plays a nontrivial role. Particularly, when the binding between free DNA linkers in solution and the corresponding receptors on mDNACCs is strong, the linker-mediated colloidal interaction is determined by the linker entropy depending on the linker concentration.

High-Density DNA Coatings on Carboxylated Colloids by DMTMM- and Azide-Mediated Coupling Reactions

DNA-mediated colloidal interactions provide a powerful strategy for the self-assembly of ordered superstructures. We report a practical and efficient two-step chemical method to graft DNA brushes onto carboxylated particles, which resolves the previously reported issues such as irreversible aggregation, inhomogeneous coating, and relatively low DNA density that can hinder colloidal crystallization. First, carboxylated particles are functionalized with heterobifunctional poly(ethylene glycol) (NH₂-PEG_n-N₃) by 4-(4,6-dimethoxy-1,3,5-triazin-2-yl)-4-methylmorpholinium chloride (DMTMM)-activated esterification of carboxylic groups and amide coupling. Then, dibenzocyclooctyne (DBCO)-functionalized DNA strands are grafted onto the pegylated particles through strain-promoted alkyne-azide cycloaddition (SPAAC) on azide groups. The homogeneous PEG brushes provide dispersion stability to the particles and clickable functional groups, resulting in DNA coatings of 1 100 000 DNA per 1 μm particle or 1 DNA per 2.9 nm², about five times higher than previously reported. The DNA-coated particles exhibit a sharp association-dissociation transition and readily self-assemble into colloidal crystals upon annealing. In addition, fluorinated particles and lens-shaped particles with carboxylate groups are successfully grafted with DNA strands in this manner. Janus particles are also functionalized with DNA strands selectively on one of the two faces. Owing to the anisotropic attraction, the DNA-coated Janus particles self-assemble into self-limiting aggregates.

Recent Development in the Fabrication of Collagen Scaffolds for Tissue Engineering Applications: A Review

Tissue engineering focuses on developing biological substitutes to restore, maintain or improve tissue functions. The three main components of its application are scaffold, cell and growthstimulating signals. Scaffolds composed of biomaterials mainly function as the structural support for ex vivo cells to attach and proliferate. They also provide physical, mechanical and biochemical cues for the differentiation of cells before transferring to the in vivo site. Collagen has been long used in various clinical applications, including drug delivery. The wide usage of collagen in the clinical field can be attributed to its abundance in nature, biocompatibility, low antigenicity and biodegradability. In addition, the high tensile strength and fibril-forming ability of collagen enable its fabrication into various forms, such as sheet/membrane, sponge, hydrogel, beads, nanofibre and nanoparticle, and as a coating material. The wide option of fabrication technology together with the excellent biological and physicochemical characteristics of collagen has stimulated the use of collagen scaffolds in various tissue engineering applications. This review describes the fabrication methods used to produce various forms of scaffolds used in tissue engineering applications.

Collagen-Inspired Self-Assembly of Twisted Filaments

Collagen consists of three peptides twisted together through a periodic array of hydrogen bonds. Here we use this as inspiration to find design rules for programmed specific interactions for self-assembling synthetic collagenlike triple helices, starting from disordered configurations. The assembly generically nucleates defects in the triple helix, the characteristics of which can be manipulated by spatially varying the enthalpy of helix formation. Defect formation slows assembly, evoking kinetic pathologies that have been observed to mutations in the primary collagen amino acid sequence. The controlled formation and interaction between defects gives a route for hierarchical self-assembly of bundles of twisted filaments.

Surface-triggered cascade reactions between DNA linkers direct the self-assembly of colloidal crystals of controllable thickness

Functionalizing colloids with reactive DNA linkers is a versatile way of programming self-assembly. DNA selectivity provides direct control over colloid-colloid interactions allowing the engineering of structures such as complex crystals or gels. However, the self-assembly of localized and finite structures remains an open problem with many potential applications. In this work, we present a system in which functionalized surfaces initiate a cascade reaction between linkers leading to the self-assembly of crystals with a controllable number of layers. Specifically, we consider colloidal particles functionalized by two families of complementary DNA linkers with mobile anchoring points, as found in experiments using emulsions or lipid bilayers. In bulk, intra-particle linkages formed by pairs of complementary linkers prevent the formation of inter-particle bridges and therefore colloid-colloid aggregation. However, colloids interact strongly with the surface given that the latter can destabilize intra-particle linkages. When in direct contact with the surface, colloids are activated, meaning that they feature more unpaired DNA linkers ready to react. Activated colloids can then capture and activate other colloids from the bulk through the formation of inter-particle linkages. Using simulations and theory, validated by existing experiments, we clarify the thermodynamics of the activation and binding process and explain how particle-particle interactions, within the adsorbed phase, weaken as a function of the distance from the surface. The latter observation underlies the possibility of self-assembling finite aggregates with controllable thickness and flat solid-gas interfaces. Our design suggests a new avenue to fabricate heterogeneous and finite structures.

Structural hierarchy confers error tolerance in biological materials

Structural hierarchy, in which materials possess distinct features on multiple length scales, is ubiquitous in nature. Diverse biological materials, such as bone, cellulose, and muscle, have as many as 10 hierarchical levels. Structural hierarchy confers many mechanical advantages, including improved toughness and economy of material. However, it also presents a problem: Each hierarchical level adds a new source of assembly errors and substantially increases the information required for proper assembly. This seems to conflict with the prevalence of naturally occurring hierarchical structures, suggesting that a common mechanical source of hierarchical robustness may exist. However, our ability to identify such a unifying phenomenon is limited by the lack of a general mechanical framework for structures exhibiting organization on disparate length scales. Here, we use simulations to substantiate a generalized model for the tensile stiffness of hierarchical filamentous networks with a nested, dilute triangular lattice structure. Following seminal work by Maxwell and others on criteria for stiff frames, we extend the concept of connectivity in network mechanics and find a similar dependence of material stiffness upon each hierarchical level. Using this model, we find that stiffness becomes less sensitive to errors in assembly with additional levels of hierarchy; although surprising, we show that this result is analytically predictable from first principles and thus potentially model independent. More broadly, this work helps account for the success of hierarchical, filamentous materials in biology and materials design and offers a heuristic for ensuring that desired material properties are achieved within the required tolerance.

Pyrochlore lattice, self-assembly and photonic band gap optimizations

Non-spherical colloidal building blocks introduce new design principles for self-assembly, making it possible to realize optical structures that could not be assembled previously. With this added complexity, the phase space expands enormously so that computer simulation becomes a valuable tool to design and assemble structures with useful optical properties. We recently demonstrated that tetrahedral clusters and spheres, interacting through a DNA-mediated short-range attractive interaction, self-assemble into a superlattice of interpenetrating diamond and pyrochlore sublattices, but only if the clusters consist of partially overlapping spheres. Here we show how the domain of crystallization can be extended by implementing a longer range potential and consider how the resultant structures affect the photonic band gaps of the underlying pyrochlore sublattice. We show that with the proper design, using clusters of overlapping spheres lead to larger photonic band gaps that open up at lower optical contrast.

Freely Jointed Polymers Made of Droplets

An important goal of self-assembly is to achieve a preprogrammed structure with high fidelity. Here, we control the valence of DNA-functionalized emulsions to make linear and branched model polymers, or "colloidomers." The distribution of cluster sizes is consistent with a polymerization process in which the droplets achieve their prescribed valence. Conformational statistics reveal that the chains are freely jointed, so that the Kuhn length is close to one bead diameter. The end-to-end length scales with the number of bonds N as N^{ν}, where ν≈3/4, in agreement with the Flory theory in two dimensions. The chain diffusion coefficient D approximately scales as D∝N^{-ν}, as predicted by the Zimm model. Unlike molecular polymers, colloidomers can be repeatedly assembled and disassembled under temperature cycling, allowing for reconfigurable, responsive matter.

Radiation damage during in situ electron microscopy of DNA-mediated nanoparticle assemblies in solution

Oligonucleotide-nanoparticle conjugates, also called programmable atom equivalents, carry promise as building blocks for self-assembled colloidal crystals, reconfigurable or stimuli responsive functional materials, as well as bio-inspired hierarchical architectures in wet environments. In situ studies of the DNA-mediated self-assembly of nanoparticles have so far been limited to reciprocal space techniques. Liquid-cell electron microscopy could enable imaging such systems with high resolution in their native environment but to realize this potential, radiation damage to the oligonucleotide linkages needs to be understood and conditions for damage-free electron microscopy identified. Here, we analyze in situ observations of DNA-linked two-dimensional nanoparticle arrays, along with control experiments for different oligonucleotide configurations, to identify the mechanisms of radiation damage for ordered superlattices of DNA-nanoparticle conjugates. In a biological context, the results point to new avenues for studying direct and indirect radiation effects for small ensembles of DNA in solution by tracking conjugated nanoparticles. By establishing low-dose conditions suitable for extended in situ imaging of programmable atom equivalents, our work paves the way for real-space observations of DNA-mediated self-assembly processes.

Nonisotropic Self-Assembly of Nanoparticles: From Compact Packing to Functional Aggregates

Quantum strongly correlated systems that exhibit interesting features in condensed matter physics often need an unachievable temperature or pressure range in classical materials. One solution is to introduce a scaling factor, namely, the lattice parameter. Synthetic heterostructures named superlattices or supracrystals are synthesized by the assembling of colloidal atoms. These include semiconductors, metals, and insulators for the exploitation of their unique properties. Most of them are currently limited to dense packing. However, some of desired properties need to adjust the colloidal atoms neighboring number. Here, the current state of research in nondense packing is summarized, discussing the benefits, outlining possible scenarios and methodologies, describing examples reported in the literature, briefly discussing the challenges, and offering preliminary conclusions. Penetrating such new and intriguing research fields demands a multidisciplinary approach accounting for the coupling of statistic physics, solid state and quantum physics, chemistry, computational science, and mathematics. Standard interactions between colloidal atoms and emerging fields, such as the use of Casimir forces, are reported. In particular, the focus is on the novelty of patchy colloidal atoms to meet this challenge.

Nanoparticle self-assembly: from interactions in suspension to polymer nanocomposites

Recent experimental results using in particular small-angle scattering to characterize the self-assembly of mainly hard spherical nanoparticles into higher ordered structures ranging from fractal aggregates to ordered assemblies are reviewed. The crucial control of interparticle interactions is discussed, from chemical surface-modification, or the action of additives like depletion agents, to the generation of directional patches and the use of external fields. It is shown how the properties of interparticle interactions have been used to allow inducing and possibly controlling aggregation, opening the road to the generation of colloidal molecules or potentially metamaterials. In the last part, studies of the microstructure of polymer nanocomposites as an application of volume-spanning and stress-carrying aggregates are discussed.

Direct observation and rational design of nucleation behavior in addressable self-assembly

Current efforts aimed at constructing complex supramolecular structures often suffer from low yields or require long assembly protocols. We address these problems by demonstrating a facile strategy for optimizing the nucleation step of a multicomponent self-assembly reaction. By tracking the formation of multisubunit clusters in situ, our experiments show that modifying the critical nucleus required to initiate structure growth can broaden the range of conditions over which self-assembly occurs and, consequently, can dramatically improve the final yield of correctly formed structures. Since varying the design of only a small portion of the target structure optimizes its yield, this strategy provides a practical route to improve the speed and accuracy of self-assembly in biomolecular, colloidal, and nanoparticle systems.

To optimize a self-assembly reaction, it is essential to understand the factors that govern its pathway. Here, we examine the influence of nucleation pathways in a model system for addressable, multicomponent self-assembly based on a prototypical “DNA-brick” structure. By combining temperature-dependent dynamic light scattering and atomic force microscopy with coarse-grained simulations, we show how subtle changes in the nucleation pathway profoundly affect the yield of the correctly formed structures. In particular, we can increase the range of conditions over which self-assembly occurs by using stable multisubunit clusters that lower the nucleation barrier for assembling subunits in the interior of the structure. Consequently, modifying only a small portion of a structure is sufficient to optimize its assembly. Due to the generality of our coarse-grained model and the excellent agreement that we find with our experimental results, the design principles reported here are likely to apply generically to addressable, multicomponent self-assembly.

Self-assembled three-dimensional chiral colloidal architecture

Although stereochemistry has been a central focus of the molecular sciences since Pasteur, its province has previously been restricted to the nanometric scale. We have programmed the self-assembly of micron-sized colloidal clusters with structural information stemming from a nanometric arrangement. This was done by combining DNA nanotechnology with colloidal science. Using the functional flexibility of DNA origami in conjunction with the structural rigidity of colloidal particles, we demonstrate the parallel self-assembly of three-dimensional microconstructs, evincing highly specific geometry that includes control over position, dihedral angles, and cluster chirality.

Micrometer-sized TPM emulsion droplets with surface-mobile binding groups

Colloids coated with lipid membranes have been widely employed for fundamental studies of lipid membrane processes, biotechnological applications such as drug delivery and biosensing, and more recently, for self-assembly. The latter has been made possible by inserting DNA oligomers with covalently linked hydrophobic anchors into the membrane. The lateral mobility of the DNA linkers on micrometer-sized droplets and solid particles has opened the door to creating structures with unprecedented structural flexibility. Here, we investigate micro-emulsions of TPM (3-(trimethoxysilyl)propyl methacrylate) as a platform for lipid monolayers and further functionalization with proteins and DNA oligonucleotides. TPM droplets can be produced with a narrow size distribution and are polymerizable, thus providing supports for model lipid membranes with controlled size and curvature. With fluorescence recovery after photobleaching, we observed that droplet-attached lipids, NeutrAvidin proteins, as well as DNA oligonucleotides all show mobility on the surface. We explored the assembly of micron-sized particles on TPM-droplets by exploiting either avidin-biotin interactions or double-stranded DNA with complementary single-stranded end groups. While the single molecules are mobile, the particles that are attached to them are not. We propose that this is caused by the heterogeneous nature of emulsified TPM, which forms an oligomer network that limits the collective motion of linkers, but allows the surface mobility of individual molecules.

Dispersion enhancing effect of sonochemically functionalized graphene oxide for catalysing antioxidant efficacy of curcumin

The non-hazardous sonochemical approach has been developed for the functionalization of graphene oxide (GrO) with 5-Aminoindazole (5-AIND). The formation of f-(5-AIND) GrO is confirmed with ¹³C solid state NMR, HRXPS, XRD, Raman, TGA, DSC, FTIR etc. The >80% cell viabilities on MCF-7 and Vero cell lines have confirmed the high cytocompatibility of f-(5-AIND) GrO. Additionally, the morphological impact on Vero cell line has evidently confirmed the biocompatibility of f-(5-AIND) GrO. As compared to GrO, the f-(5-AIND) GrO has demonstrated an enhanced antioxidant efficacy of 69.4-72%, evaluated with 2, 2-diphenyl-1-picrylhydrazyl (DPPH) free radical assay. With a similar objective (0.01-0.05)mL peanut oil based curcumin micro and nanoemulsions have been prepared using ethanol and glycerol as co-solvent and co-surfactant respectively. The prepared emulsions are subsequently characterised with respect to morphological and physicochemical parameters via density, surface tension, viscosity, friccohesity measurement and DLS analysis. Henceforth, with an expectation to achieve higher dispersion, an ethanolic dispersion of f-(5-AIND) GrO has been mixed with curcumin carrying emulsions in 1:1. Notably, the radical scavenging activities (RSA) of the combined formulations are significantly enhanced to an extent of 26.6%.

Optimising minimal building blocks for addressable self-assembly

Addressable structures are characterised by the set of unique components from which they are built and by the specific location that each component occupies. For an addressable structure to self-assemble, its constituent building blocks must be encoded with sufficient information to define their positions with respect to each other and to enable them to navigate to those positions. DNA, with its vast scope for encoding specific interactions, has been successfully used to synthesise addressable systems of several hundred components. In this work we examine the complementary question of the minimal requirements for building blocks to undergo addressable self-assembly driven by a controlled temperature quench. Our testbed is an idealised model of cubic particles patterned with attractive interactions. We introduce a scheme for optimising the interactions using a variant of basin-hopping and a negative design principle. The designed building blocks are tested dynamically in simple target structures to establish how their complexity affects the limits of reliable self-assembly.

Spontaneous emergence of catalytic cycles with colloidal spheres

Colloidal particles endowed with specific time-dependent interactions are a promising route for realizing artificial materials that have the properties of living ones. Previous work has demonstrated how this system can give rise to self-replication. Here, we introduce the process of colloidal catalysis, in which clusters of particles catalyze the creation of other clusters through templating reactions. Surprisingly, we find that simple templating rules generically lead to the production of huge numbers of clusters. The templating reactions among this sea of clusters give rise to an exponentially growing catalytic cycle, a specific realization of Dyson's notion of an exponentially growing metabolism. We demonstrate this behavior with a fixed set of interactions between particles chosen to allow a catalysis of a specific six-particle cluster from a specific seven-particle cluster, yet giving rise to the catalytic production of a sea of clusters of sizes between 2 and 11 particles. The fact that an exponentially growing cycle emerges naturally from such a simple scheme demonstrates that the emergence of exponentially growing metabolisms could be simpler than previously imagined.

Free energy of singular sticky-sphere clusters

Networks of particles connected by springs model many condensed-matter systems, from colloids interacting with a short-range potential and complex fluids near jamming, to self-assembled lattices and various metamaterials. Under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero-frequency vibrational modes, yet such singular modes are at the epicenter of many interesting behaviors in the systems above. We consider a system of N spherical particles, and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged. Although the contribution to the partition function from singular clusters diverges in the limit, its asymptotic value can be calculated and depends on only two parameters, characterizing the depth and range of the potential. The result holds for systems that are second-order rigid, a geometric characterization that describes all known ground-state (rigid) sticky clusters. To illustrate the applications of our theory we address the question of emergence: how does crystalline order arise in large systems when it is strongly disfavored in small ones? We calculate the partition functions of all known rigid clusters up to N≤21 and show the cluster landscape is dominated by hyperstatic clusters (those with more than 3N-6 contacts); singular and isostatic clusters are far less frequent, despite their extra vibrational and configurational entropies. Since the most hyperstatic clusters are close to fragments of a close-packed lattice, this underlies the emergence of order in sticky-sphere systems, even those as small as N=10.

Using active colloids as machines to weave and braid on the micrometer scale

Controlling motion at the microscopic scale is a fundamental goal in the development of biologically inspired systems. We show that the motion of active, self-propelled colloids can be sufficiently controlled for use as a tool to assemble complex structures such as braids and weaves out of microscopic filaments. Unlike typical self-assembly paradigms, these structures are held together by geometric constraints rather than adhesive bonds. The out-of-equilibrium assembly that we propose involves precisely controlling the 2D motion of active colloids so that their path has a nontrivial topology. We demonstrate with proof-of-principle Brownian dynamics simulations that, when the colloids are attached to long semiflexible filaments, this motion causes the filaments to braid. The ability of the active particles to provide sufficient force necessary to bend the filaments into a braid depends on a number of factors, including the self-propulsion mechanism, the properties of the filament, and the maximum curvature in the braid. Our work demonstrates that nonequilibrium assembly pathways can be designed using active particles.

Hierarchical Self-Assembly of Cu7Te5 Nanorods into Superstructures with Enhanced SERS Performance

This paper reports a strategy to get self-assembly of Cu₇Te₅ nanorods into hierarchical superstructures: the side-by-side self-assembly of nanorods into microscale one-dimensional (1D) nanowires (primary structure), the side-by-side alignments of the 1D nanowires into two-dimensional (2D) nanowire bundles (secondary structure), and the further rolling up of the 2D bundles into three-dimensional (3D) microtubes (tertiary structure). It was found that the oleylamine (OLA)/n-dodecanethiol (DDT) mixture as a binary capping agent was key to produce Cu₇Te₅ nanorods in the quantum size regime with high monodispersity, and this was a prerequisite for their hierarchical self-assembly based on elaborate control of the solvent evaporation process. The obtained Cu₇Te₅ microtube superstructures were used as SERS substrate and showed much stronger SERS enhancement than the as-prepared Cu₇Te₅ nanorods before assembly. This was probably ascribed to the remarkably enhanced local electromagnetic field arising from the plasmon coupling of Cu₇Te₅ nanorods in the well-assembled superstructures.