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In this study, we provide constructive proof that Transformers can recognize and generate hierarchical language efficiently with respect to model size, even without the need for a specific positional encoding. Specifically, we show that causal masking and a starting token enable Transformers to compute positional information and depth within hierarchical structures. We demonstrate that Transformers without positional encoding can generate hierarchical languages. Furthermore, we suggest that explicit positional encoding might have a detrimental effect on generalization with respect to sequence length., Comment: 55 pages, 11 figures

Attaching enzymes to nanostructures has proven useful to the study of enzyme functionality under controlled conditions and has led to new technologies. Often, the utility and interest of enzyme-tethered nanostructures lie in how the enzymatic activity is affected by how the enzymes are arranged in space. Therefore, being able to conjugate enzymes to nanostructures while preserving the enzymatic activity is essential. In this paper, we present a method to conjugate single-stranded DNA to the enzyme urease while maintaining enzymatic activity. We show evidence of successful conjugation and quantify the variables that affect the conjugation yield. We also show that the enzymatic activity is unchanged after conjugation compared to the enzyme in its native state. Finally, we demonstrate the tethering of urease to nanostructures made using DNA origami with high site-specificity. Decorating nanostructures with enzymatically-active urease may prove to be useful in studying, or even utilizing, the functionality of urease in disciplines ranging from biotechnology to soft-matter physics. The techniques we present in this paper will enable researchers across these fields to modify enzymes without disrupting their functionality, thus allowing for more insightful studies into their behavior and utility., Comment: 17 pages, 5 figures, SI is 2 pages

Recent advances enable the creation of nanoscale building blocks with complex geometries and interaction specificities for self-assembly. This nearly boundless design space necessitates design principles for defining the mutual interactions between multiple particle species to target a user-specified complex structure or pattern. In this article, we develop a symmetry-based method to generate the interaction matrices that specify the assembly of two-dimensional tilings which we illustrate using equilateral triangles. By exploiting the allowed 2D symmetries, we develop an algorithmic approach by which any periodic 2D tiling can be generated from an arbitrarily large number of subunit species, notably addressing an unmet challenge of engineering 2D crystals with periodicities that can be arbitrarily larger than subunit size. To demonstrate the utility of our design approach, we encode specific interactions between triangular subunits synthesized by DNA origami and show that we can guide their self-assembly into tilings with a wide variety of symmetries, using up to 12 unique species of triangles. By conjugating specific triangles with gold nanoparticles, we fabricate gold-nanoparticle supercrystals whose lattice parameter spans up to 300 nm. Finally, to generate economical design rules, we compare the design economy of various tilings. In particular, we show that (1) higher symmetries allow assembly of larger unit cells with fewer subunits and (2) linear supercrystals can be designed more economically using linear primitive unit cells. This work provides a simple algorithmic approach to designing periodic assemblies, which may open new doors to the multiscale assembly of superlattices of nanostructured "metatoms" with engineered plasmonic functions., Comment: Main text 10 pages, 4 figures, SI 27 pages, 19 figures, and 3 tables

Self-assembly is one of the prevalent strategies used by living systems to fabricate ensembles of precision nanometer-scale structures and devices. The push for analogous approaches to create synthetic nanomaterials has led to the development of a large class of programmable crystalline structures. However, many applications require `self-limiting' assemblies, which autonomously terminate growth at a well-defined size and geometry. For example, curved architectures such as tubules, vesicles, or capsids can be designed to self-close at a particular size, symmetry, and topology. But developing synthetic strategies for self-closing assembly has been challenging, in part because such structures are prone to polymorphism that arises from thermal fluctuations of their local curvature, a problem that worsens with increased target size. Here we demonstrate a strategy to eliminate this source of polymorphism in self-closing assembly of tubules by increasing the assembly complexity. In the limit of single-component assembly, we find that thermal fluctuations allow the system to assemble nearby, off-target structures with varying widths, helicities, and chirality. By increasing the number of distinct components, we reduce the density of off-target states, thereby increasing the selectivity of a user-specified target structure to nearly 100%. We further show that by reducing the design constraints by targeting either the pitch or the width of tubules, fewer components are needed to reach complete selectivity. Combining experiments with theory, our results reveal an economical limit, which determines the minimum number of components that are required to create arbitrary assembly sizes with full selectivity. In the future, this approach could be extended to more complex self-limited structures, such as shells or triply periodic surfaces., Comment: Main text 8 pages, 5 figures, SI 22 pages, 12 figures, 5 tables. Slight changes to the text from previous version

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 trade-offs 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 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., Comment: Main text 11 pages and 4 figures; SI 44 pages, 19 figures, 9 tables

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., Comment: Main text 10 pages and 6 figures, SI 12 pages and 10 figures