Communication: theoretical prediction of free-energy landscapes for complex self-assembly

Hydrogen-Bonded Fibrous Nanotubes Assembled from Trigonal Prismatic Building Blocks

In reticular chemistry, molecular building blocks are designed to create crystalline open frameworks. A key principle of reticular chemistry is that the most symmetrical networks are the likely outcomes of reactions, particularly when highly symmetrical building blocks are involved. The strategy of synthesizing low-dimensional networks aims to reduce explicitly the symmetry of the molecular building blocks. Here we report the spontaneous formation of hydrogen-bonded fibrous structures from trigonal prismatic building blocks, which were designed to form three-dimensional crystalline networks on account of their highly symmetrical structures. Utilizing different microscopic and spectroscopic techniques, we identify the structures at the early stages of the assembly process in order to and understand the growth mechanism. The symmetrical molecular building blocks are incorporated preferentially in the longitudinal direction, giving rise to anisotropic hydrogen-bonded porous organic nanotubes. Entropy-driven anisotropic growth provides micrometer-scale unidirectional nanotubes with high porosity. By combining experimental evidence and theoretical modeling, we have obtained a deep understanding of the nucleation and growth processes. Our findings offer fundamental insight into the molecular design of tubular structures. The nanotubes evolve further in the transverse directions to provide extended higher-order fibrous structures [nano- and microfibers], ultimately leading to large-scale interconnected hydrogen-bonded fiber-like structures with twists and turns. Our work provides fundamental understanding and paves the way for innovative molecular designs in low-dimensional networks.

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."

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.

Self-assembly of millimeter-scale magnetic particles in suspension

Self-assembly is ubiquitous at all scales in nature. Most studies have focused on the self-assembly of micron-scale and nano-scale components. In this study, we explore the self-assembly of millimeter-scale magnetic particles in a bubble-column reactor to form 9 different structures. Two component systems (N-N and S-S particles) assemble faster than one-component systems (all particles have N-S poles) because they have more numerous bonding pathways. In addition, two-components add control to process initiation and evolution, and enable the formation of complex structures such as squares, tetrahedra and cubes. Self-assembly is collision-limited, thus, the formation time increases with the total number of bonds required to form the structure and the injected power. The dimensionless Mason number captures the interplay between hydrodynamic forces and magnetic interactions: self-assembly is most efficient at intermediate Mason numbers (the system is quasi-static at low Mason numbers with limited chances for particle interaction; on the other hand, hydrodynamic forces prevail over dipole-dipole interactions and hinder bonding at high Mason numbers). Two strategies to improve yield involve (1) the inclusion of pre-assembled nucleation templates to prevent the formation of incorrect initial structures that lead to kinetic traps, and (2) the presence of boundaries to geometrically filter unwanted configurations and to overcome kinetic traps through particle-wall collisions. Yield maximization involves system operation at an optimal Mason number, the inclusion of nucleation templates and the use of engineered boundaries (size and shape).

The Art of Designing DNA Nanostructures with CAD Software

Since the arrival of DNA nanotechnology nearly 40 years ago, the field has progressed from its beginnings of envisioning rather simple DNA structures having a branched, multi-strand architecture into creating beautifully complex structures comprising hundreds or even thousands of unique strands, with the possibility to exactly control the positions down to the molecular level. While the earliest construction methodologies, such as simple Holliday junctions or tiles, could reasonably be designed on pen and paper in a short amount of time, the advent of complex techniques, such as DNA origami or DNA bricks, require software to reduce the time required and propensity for human error within the design process. Where available, readily accessible design software catalyzes our ability to bring techniques to researchers in diverse fields and it has helped to speed the penetration of methods, such as DNA origami, into a wide range of applications from biomedicine to photonics. Here, we review the historical and current state of CAD software to enable a variety of methods that are fundamental to using structural DNA technology. Beginning with the first tools for predicting sequence-based secondary structure of nucleotides, we trace the development and significance of different software packages to the current state-of-the-art, with a particular focus on programs that are open source.

Numerical calculation of free-energy barriers for entangled polymer nucleation

The crystallization of entangled polymers from their melt is investigated using computer simulation with a coarse-grained model. Using hybrid Monte Carlo simulations enables us to probe the behavior of long polymer chains. We identify solid-like beads with a centrosymmetry local order parameter and compute the nucleation free-energy barrier at relatively high supercooling with adaptive-bias windowed umbrella sampling. Our results demonstrate that the critical nucleus sizes and the heights of free-energy barriers do not significantly depend on the molecular weight of the polymer; however, the nucleation rate decreases with the increase in molecular weight. Moreover, an analysis of the composition of the critical nucleus suggests that intra-molecular growth of the nucleated cluster does not contribute significantly to crystallization for this system.

Programming the Nucleation of DNA Brick Self-Assembly with a Seeding Strand

Recently, the DNA brick strategy has provided a highly modular and scalable approach for the construction of complex structures, which can be used as nanoscale pegboards for the precise organization of molecules and nanoparticles for many applications. Despite the dramatic increase of structural complexity provided by the DNA brick method, the assembly pathways are still poorly understood. Herein, we introduce a "seed" strand to control the crucial nucleation and assembly pathway in DNA brick assembly. Through experimental studies and computer simulations, we successfully demonstrate that the regulation of the assembly pathways through seeded growth can accelerate the assembly kinetics and increase the optimal temperature by circa 4-7 °C for isothermal assembly. By improving our understanding of the assembly pathways, we provide new guidelines for the design of programmable pathways to improve the self-assembly of DNA nanostructures.

Controlling Fragment Competition on Pathways to Addressable Self-Assembly

Addressable self-assembly is the formation of a target structure from a set of unique molecular or colloidal building blocks, each of which occupies a defined location in the target. The requirement that each type of building block appears exactly once in each copy of the target introduces severe restrictions on the combinations of particles and on the pathways that lead to successful self-assembly. These restrictions can limit the efficiency of self-assembly and the final yield of the product. In particular, partially formed fragments may compete with each other if their compositions overlap, since they cannot be combined. Here, we introduce a "completability" algorithm to quantify competition between self-assembling fragments and use it to deduce general principles for suppressing the effects of fragment incompatibility in the self-assembly of small addressable clusters. Competition originates from loops in the bonding network of the target structure, but loops may be needed to provide structural rigidity and thermodynamic stability. An optimal compromise can be achieved by careful choice of bonding networks and by promoting semihierarchical pathways that rule out competition between early fragments. These concepts are illustrated in simulations of self-assembly in two contrasting addressable targets of 20 unique components each.

Self-Assembly of Nanocubic Molecular Capsules via Solvent-Guided Formation of Rectangular Blocks

We investigate the mechanism underlying the self-assembly of gear-shaped amphiphilic molecules into a highly ordered nanocubic capsule ("nanocube") in aqueous methanol. Simulation results show that the solvent molecules play a significant role in the assembly process by directing the primitive intermediates to orthogonal/rectangular shapes, thus creating appropriate building blocks for cubic assembly while avoiding off-pathway stacked aggregates. Free-energy analyses reveal that the interplay of the direct intermonomer interaction and the solvent-mediated repulsion between large aromatic cores (via preferential solvation of methanol on hydrophobic surfaces) leads to the strong trend for perpendicular binding of monomers and hence the solvent-guided formation of rectangular blocks. Furthermore, we report the self-assembly simulation of the nanocube using replica exchange with solute tempering and demonstrate that the simulation can predict a highly ordered nanocapsule structure, assembly intermediates, and encapsulated molecules, which helps promote computer-aided design of functional molecular self-assemblies in explicit solvent.

Multi-scale coarse-graining for the study of assembly pathways in DNA-brick self-assembly

Inspired by recent successes using single-stranded DNA tiles to produce complex structures, we develop a two-step coarse-graining approach that uses detailed thermodynamic calculations with oxDNA, a nucleotide-based model of DNA, to parametrize a coarser kinetic model that can reach the time and length scales needed to study the assembly mechanisms of these structures. We test the model by performing a detailed study of the assembly pathways for a two-dimensional target structure made up of 334 unique strands each of which are 42 nucleotides long. Without adjustable parameters, the model reproduces a critical temperature for the formation of the assembly that is close to the temperature at which assembly first occurs in experiments. Furthermore, the model allows us to investigate in detail the nucleation barriers and the distribution of critical nucleus shapes for the assembly of a single target structure. The assembly intermediates are compact and highly connected (although not maximally so), and classical nucleation theory provides a good fit to the height and shape of the nucleation barrier at temperatures close to where assembly first occurs.

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.

Molecular control over colloidal assembly

Contemporary chemical and material engineering often takes inspiration from nature, targeting for example strong yet light materials and materials composed of highly ordered domains at multiple different lengthscales for fundamental science and applications in e.g. sensing, catalysis, coating technology, and delivery. The preparation of such hierarchically structured functional materials through guided bottom-up assembly of synthetic building blocks requires a high level of control over their synthesis, interactions and assembly pathways. In this perspective we showcase recent work demonstrating how molecular control can be exploited to direct colloidal assembly into responsive materials with mechanical, optical or electrical properties that can be adjusted post-synthesis with external cues.

Eye patches: Protein assembly of index-gradient squid lenses

A parabolic relationship between lens radius and refractive index allows spherical lenses to avoid spherical aberration. We show that in squid, patchy colloidal physics resulted from an evolutionary radiation of globular S-crystallin proteins. Small-angle x-ray scattering experiments on lens tissue show colloidal gels of S-crystallins at all radial positions. Sparse lens materials form via low-valence linkages between disordered loops protruding from the protein surface. The loops are polydisperse and bind via a set of hydrogen bonds between disordered side chains. Peripheral lens regions with low particle valence form stable, volume-spanning gels at low density, whereas central regions with higher average valence gel at higher densities. The proteins demonstrate an evolved set of linkers for self-assembly of nanoparticles into volumetric materials.

Structure-Based Prediction of Protein-Folding Transition Paths

We propose a general theory to describe the distribution of protein-folding transition paths. We show that transition paths follow a predictable sequence of high-free-energy transient states that are separated by free-energy barriers. Each transient state corresponds to the assembly of one or more discrete, cooperative units, which are determined directly from the native structure. We show that the transition state on a folding pathway is reached when a small number of critical contacts are formed between a specific set of substructures, after which folding proceeds downhill in free energy. This approach suggests a natural resolution for distinguishing parallel folding pathways and provides a simple means to predict the rate-limiting step in a folding reaction. Our theory identifies a common folding mechanism for proteins with diverse native structures and establishes general principles for the self-assembly of polymers with specific interactions.

Free-energy predictions and absorption spectra calculations for supramolecular nanocarriers and their photoactive cargo

We reconstructed the free-energy landscape for supramolecular nanoparticles of amphiphilic methacrylated-based co-polymers. Their self-assembly in aqueous solution and encapsulation of borondipyrromethene (BODIPY) derivatives were enforced through atomistic free-energy simulations. The BODIPY binding modes detected in each of the free-energy basins were validated through a comparison of theoretical absorption spectra, calculated at the TD-DFT level, to their experimental counterparts. The nanoparticle distribution is controlled within a thermodynamic regime, with free-energy barriers approaching 8 kcal mol^-1, enabling the existence of different-sized nanoparticles in aqueous solution at room temperature. Two types of supramolecular morphologies were identified. One is compact and spherical in shape and the other is large and donut-like, with the former more stable than the latter by 4 kcal mol^-1. The morphology of the supramolecular host affects the binding mode of the BODIPY guests. Stacked BODIPY aggregates are encapsulated in the spherical nanocarriers, whereas isolated chromophores associate with the donut-shaped assemblies.

Investigating the role of boundary bricks in DNA brick self-assembly

In the standard DNA brick set-up, distinct 32-nucleotide strands of single-stranded DNA are each designed to bind specifically to four other such molecules. Experimentally, it has been demonstrated that the overall yield is increased if certain bricks which occur on the outer faces of target structures are merged with adjacent bricks. However, it is not well understood by what mechanism such 'boundary bricks' increase the yield, as they likely influence both the nucleation process and the final stability of the target structure. Here, we use Monte Carlo simulations with a patchy particle model of DNA bricks to investigate the role of boundary bricks in the self-assembly of complex multicomponent target structures. We demonstrate that boundary bricks lower the free-energy barrier to nucleation and that boundary bricks on edges stabilize the final structure. However, boundary bricks are also more prone to aggregation, as they can stabilize partially assembled intermediates. We explore some design strategies that permit us to benefit from the stabilizing role of boundary bricks whilst minimizing their ability to hinder assembly; in particular, we show that maximizing the total number of boundary bricks is not an optimal strategy.

DNA brick self-assembly with an off-lattice potential

We report Monte Carlo simulations of a simple off-lattice patchy-particle model for DNA 'bricks'. We relate the parameters that characterise this model with the binding free energy of pairs of single-stranded DNA molecules. We verify that an off-lattice potential parameterised in this way reproduces much of the behaviour seen with a simpler lattice model we introduced previously, although the relaxation of the geometric constraints leads to a more error-prone self-assembly pathway. We investigate the self-assembly process as a function of the strength of the non-specific interactions. We show that our off-lattice model for DNA bricks results in robust self-assembly into a variety of target structures.

State-space reduction and equivalence class sampling for a molecular self-assembly model

Direct simulation of a model with a large state space will generate enormous volumes of data, much of which is not relevant to the questions under study. In this paper, we consider a molecular self-assembly model as a typical example of a large state-space model, and present a method for selectively retrieving 'target information' from this model. This method partitions the state space into equivalence classes, as identified by an appropriate equivalence relation. The set of equivalence classes H, which serves as a reduced state space, contains none of the superfluous information of the original model. After construction and characterization of a Markov chain with state space H, the target information is efficiently retrieved via Markov chain Monte Carlo sampling. This approach represents a new breed of simulation techniques which are highly optimized for studying molecular self-assembly and, moreover, serves as a valuable guideline for analysis of other large state-space models.

Information capacity of specific interactions

Specific interactions are a hallmark feature of self-assembly and signal-processing systems in both synthetic and biological settings. Specificity between components may arise from a wide variety of physical and chemical mechanisms in diverse contexts, from DNA hybridization to shape-sensitive depletion interactions. Despite this diversity, all systems that rely on interaction specificity operate under the constraint that increasing the number of distinct components inevitably increases off-target binding. Here we introduce "capacity," the maximal information encodable using specific interactions, to compare specificity across diverse experimental systems and to compute how specificity changes with physical parameters. Using this framework, we find that "shape" coding of interactions has higher capacity than chemical ("color") coding because the strength of off-target binding is strongly sublinear in binding-site size for shapes while being linear for colors. We also find that different specificity mechanisms, such as shape and color, can be combined in a synergistic manner, giving a capacity greater than the sum of the parts.

Effects of co-ordination number on the nucleation behaviour in many-component self-assembly

We report canonical and grand-canonical lattice Monte Carlo simulations of the self-assembly of addressable structures comprising hundreds of distinct component types. The nucleation behaviour, in the form of free-energy barriers to nucleation, changes significantly as the co-ordination number of the building blocks is changed from 4 to 8 to 12. Unlike tetrahedral structures – which roughly correspond to DNA bricks that have been studied in experiments – the shapes of the free-energy barriers of higher co-ordination structures depend strongly on the supersaturation, and such structures require a very significant driving force for structure growth before nucleation becomes thermally accessible. Although growth at high supersaturation results in more defects during self-assembly, we show that high co-ordination number structures can still be assembled successfully in computer simulations and that they exhibit self-assembly behaviour analogous to DNA bricks. In particular, the self-assembly remains modular, enabling in principle a wide variety of nanostructures to be assembled, with a greater spatial resolution than is possible in low co-ordination structures.

Self-assembly protocol design for periodic multicomponent structures

Assembling molecular superstructures with many distinct components will allow unprecedented control over morphology at the nanoscale. Recently, this approach has been used to assemble periodic structures with precisely defined features, such as repeating arrays of pores and channels, using a large number of building blocks. Here we propose a predictive tool that allows us to optimize the nucleation and growth of unbounded, ordered structures. In what follows, we call these structures 'crystals', even though they may only be periodic in one or two dimensions. We find that the nucleation barriers and growth pathways for crystals consisting of many components exhibit generic features that are very different from those of simple crystals. To illustrate the very non-classical nature of the nucleation and growth of such structures, we study the formation of one and two-dimensional crystals with multicomponent unit cells. We find that, whilst the boundaries in the non-periodic dimensions significantly affect the stabilities of these crystals, the nucleation barriers are largely determined by the local connectivity of the associated bulk crystal and are independent of the number of distinct components in the unit cell. We predict that the self-assembly of crystals with complex morphologies can be made to follow specific pathways toward the target structure that successively incorporate key features of the three-dimensional target structure. In contrast with simple crystals, it is possible to tune the kinetics of nucleation and growth separately, thus minimizing defect formation. We show how control over self-assembly pathways can be used to optimize the kinetics of formation of extended structures with arbitrary nanoscale patterns.

Hierarchical assembly may be a way to make large information-rich structures

Self-assembly in the laboratory can now yield 'information-rich' nanostructures in which each component is of a distinct type and has a defined spatial position. Ensuring the thermodynamic stability of such structures requires inter-component interaction energies to increase logarithmically with structure size, in order to counter the entropy gained upon mixing component types in solution. However, self-assembly in the presence of strong interactions results in general in kinetic trapping, so suggesting a limit to the size of an (equilibrium) structure that can be self-assembled from distinguishable components. Here we study numerically a two-dimensional hierarchical assembly scheme already considered in experiment. We show that this scheme is immune to the kinetic traps associated with strong 'native' interactions (interactions designed to stabilize the intended structure), and so, in principle, offers a way to make large information-rich structures. In this scheme the size of an assembled structure scales exponentially with the stage of assembly, and assembly can continue as long as random motion is able to bring structures into contact. The resulting superstructure could provide a template for building in the third dimension. The chief drawback of this scheme is that it is particularly susceptible to kinetic traps that result from 'non-native' interactions (interactions not required to stabilize the intended structure); the scale on which such a scheme can be realized therefore depends upon how effectively this latter kind of interaction can be suppressed.

Design strategies for self-assembly of discrete targets

Both biological and artificial self-assembly processes can take place by a range of different schemes, from the successive addition of identical building blocks to hierarchical sequences of intermediates, all the way to the fully addressable limit in which each component is unique. In this paper, we introduce an idealized model of cubic particles with patterned faces that allows self-assembly strategies to be compared and tested. We consider a simple octameric target, starting with the minimal requirements for successful self-assembly and comparing the benefits and limitations of more sophisticated hierarchical and addressable schemes. Simulations are performed using a hybrid dynamical Monte Carlo protocol that allows self-assembling clusters to rearrange internally while still providing Stokes-Einstein-like diffusion of aggregates of different sizes. Our simulations explicitly capture the thermodynamic, dynamic, and steric challenges typically faced by self-assembly processes, including competition between multiple partially completed structures. Self-assembly pathways are extracted from the simulation trajectories by a fully extendable scheme for identifying structural fragments, which are then assembled into history diagrams for successfully completed target structures. For the simple target, a one-component assembly scheme is most efficient and robust overall, but hierarchical and addressable strategies can have an advantage under some conditions if high yield is a priority.

Rational design of self-assembly pathways for complex multicomponent structures

The field of complex self-assembly is moving toward the design of multiparticle structures consisting of thousands of distinct building blocks. To exploit the potential benefits of structures with such "addressable complexity," we need to understand the factors that optimize the yield and the kinetics of self-assembly. Here we use a simple theoretical method to explain the key features responsible for the unexpected success of DNA-brick experiments, which are currently the only demonstration of reliable self-assembly with such a large number of components. Simulations confirm that our theory accurately predicts the narrow temperature window in which error-free assembly can occur. Even more strikingly, our theory predicts that correct assembly of the complete structure may require a time-dependent experimental protocol. Furthermore, we predict that low coordination numbers result in nonclassical nucleation behavior, which we find to be essential for achieving optimal nucleation kinetics under mild growth conditions. We also show that, rather surprisingly, the use of heterogeneous bond energies improves the nucleation kinetics and in fact appears to be necessary for assembling certain intricate 3D structures. This observation makes it possible to sculpt nucleation pathways by tuning the distribution of interaction strengths. These insights not only suggest how to improve the design of structures based on DNA bricks, but also point the way toward the creation of a much wider class of chemical or colloidal structures with addressable complexity.