Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model

We consider the self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-handed Tile Assembly Model (2HAM). In particular, we focus our attention on a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpi\'nski carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpi\'nski carpet with scale factor 1. Moreover, the 2HAM system that we give lends itself to being generalized and we describe how this system can be modified to obtain a 2HAM system that finitely self-assembles one of any fractal from an infinite set of fractals which we call 4-sided fractals. The 2HAM systems we give in this paper are the first examples of systems that finitely self-assemble discrete self-similar fractals at scale factor 1 in a purely growth model of self-assembly. Finally, we show that there exists a 3-sided fractal (which is not a tree fractal) that cannot be finitely self-assembled by any 2HAM system

Binary pattern tile set synthesis is NP-hard

In the field of algorithmic self-assembly, a long-standing unproven conjecture has been that of the NP-hardness of binary pattern tile set synthesis (2-PATS). The $k$-PATS problem is that of designing a tile assembly system with the smallest number of tile types which will self-assemble an input pattern of $k$ colors. Of both theoretical and practical significance, $k$-PATS has been studied in a series of papers which have shown $k$-PATS to be NP-hard for $k = 60$, $k = 29$, and then $k = 11$. In this paper, we close the fundamental conjecture that 2-PATS is NP-hard, concluding this line of study. While most of our proof relies on standard mathematical proof techniques, one crucial lemma makes use of a computer-assisted proof, which is a relatively novel but increasingly utilized paradigm for deriving proofs for complex mathematical problems. This tool is especially powerful for attacking combinatorial problems, as exemplified by the proof of the four color theorem by Appel and Haken (simplified later by Robertson, Sanders, Seymour, and Thomas) or the recent important advance on the Erd\H{o}s discrepancy problem by Konev and Lisitsa using computer programs. We utilize a massively parallel algorithm and thus turn an otherwise intractable portion of our proof into a program which requires approximately a year of computation time, bringing the use of computer-assisted proofs to a new scale. We fully detail the algorithm employed by our code, and make the code freely available online

Fractal assembly of micrometre-scale DNA origami arrays with arbitrary patterns

Self-assembled DNA nanostructures enable nanometre-precise patterning that can be used to create programmable molecular machines and arrays of functional materials. DNA origami is particularly versatile in this context because each DNA strand in the origami nanostructure occupies a unique position and can serve as a uniquely addressable pixel. However, the scale of such structures has been limited to about 0.05 square micrometres, hindering applications that demand a larger layout and integration with more conventional patterning methods. Hierarchical multistage assembly of simple sets of tiles can in principle overcome this limitation, but so far has not been sufficiently robust to enable successful implementation of larger structures using DNA origami tiles. Here we show that by using simple local assembly rules that are modified and applied recursively throughout a hierarchical, multistage assembly process, a small and constant set of unique DNA strands can be used to create DNA origami arrays of increasing size and with arbitrary patterns. We illustrate this method, which we term ‘fractal assembly’, by producing DNA origami arrays with sizes of up to 0.5 square micrometres and with up to 8,704 pixels, allowing us to render images such as the Mona Lisa and a rooster. We find that self-assembly of the tiles into arrays is unaffected by changes in surface patterns on the tiles, and that the yield of the fractal assembly process corresponds to about 0.95^(m − 1) for arrays containing m tiles. When used in conjunction with a software tool that we developed that converts an arbitrary pattern into DNA sequences and experimental protocols, our assembly method is readily accessible and will facilitate the construction of sophisticated materials and devices with sizes similar to that of a bacterium using DNA nanostructures

Self-assembly of highly symmetrical, ultrasmall inorganic cages directed by surfactant micelles

Nanometre-sized objects with highly symmetrical, cage-like polyhedral shapes, often with icosahedral symmetry, have recently been assembled from DNA(1-3), RNA(4) or proteins(5,6) for applications in biology and medicine. These achievements relied on advances in the development of programmable self-assembling biological materials(7-10), and on rapidly developing techniques for generating three-dimensional (3D) reconstructions from cryo-electron microscopy images of single particles, which provide high-resolution structural characterization of biological complexes(11-13). Such single-particle 3D reconstruction approaches have not yet been successfully applied to the identification of synthetic inorganic nanomaterials with highly symmetrical cage-like shapes. Here, however, using a combination of cryo-electron microscopy and single-particle 3D reconstruction, we suggest the existence of isolated ultrasmall (less than 10 nm) silica cages ('silicages') with dodecahedral structure. We propose that such highly symmetrical, self-assembled cages form through the arrangement of primary silica clusters in aqueous solutions on the surface of oppositely charged surfactant micelles. This discovery paves the way for nanoscale cages made from silica and other inorganic materials to be used as building blocks for a wide range of advanced functional-materials applications

Pathways to cellular supremacy in biocomputing

Synthetic biology uses living cells as the substrate for performing human-defined computations. Many current implementations of cellular computing are based on the “genetic circuit” metaphor, an approximation of the operation of silicon-based computers. Although this conceptual mapping has been relatively successful, we argue that it fundamentally limits the types of computation that may be engineered inside the cell, and fails to exploit the rich and diverse functionality available in natural living systems. We propose the notion of “cellular supremacy” to focus attention on domains in which biocomputing might offer superior performance over traditional computers. We consider potential pathways toward cellular supremacy, and suggest application areas in which it may be found.A.G.-M. was supported by the SynBio3D project of the UK Engineering and Physical Sciences Research Council (EP/R019002/1) and the European CSA on biological standardization BIOROBOOST (EU grant number 820699). T.E.G. was supported by a Royal Society University Research Fellowship (grant UF160357) and BrisSynBio, a BBSRC/ EPSRC Synthetic Biology Research Centre (grant BB/L01386X/1). P.Z. was supported by the EPSRC Portabolomics project (grant EP/N031962/1). P.C. was supported by SynBioChem, a BBSRC/EPSRC Centre for Synthetic Biology of Fine and Specialty Chemicals (grant BB/M017702/1) and the ShikiFactory100 project of the European Union’s Horizon 2020 research and innovation programme under grant agreement 814408

Pushing Lines Helps: Efficient Universal Centralised Transformations for Programmable Matter

In this paper, we study a discrete system of entities residing on a two-dimensional square grid. Each entity is modelled as a node occupying a distinct cell of the grid. The set of all $n$ nodes forms initially a connected shape $A$. Entities are equipped with a linear-strength pushing mechanism that can push a whole line of entities, from 1 to $n$, in parallel in a single time-step. A target connected shape $B$ is also provided and the goal is to \emph{transform} $A$ into $B$ via a sequence of line movements. Existing models based on local movement of individual nodes, such as rotating or sliding a single node, can be shown to be special cases of the present model, therefore their (inefficient, $\Theta(n^2)$) \emph{universal transformations} carry over. Our main goal is to investigate whether the parallelism inherent in this new type of movement can be exploited for efficient, i.e., sub-quadratic worst-case, transformations. As a first step towards this, we restrict attention solely to centralised transformations and leave the distributed case as a direction for future research. Our results are positive. By focusing on the apparently hard instance of transforming a diagonal $A$ into a straight line $B$, we first obtain transformations of time $O(n\sqrt{n})$ without and with preserving the connectivity of the shape throughout the transformation. Then, we further improve by providing two $O(n\log n)$-time transformations for this problem. By building upon these ideas, we first manage to develop an $O(n\sqrt{n})$-time universal transformation. Our main result is then an $O(n \log n)$-time universal transformation. We leave as an interesting open problem a suspected $\Omega(n\log n)$-time lower bound.Comment: 40 pages, 27 figure

Computing with bacterial constituents, cells and populations: from bioputing to bactoputing

The relevance of biological materials and processes to computing—aliasbioputing—has been explored for decades. These materials include DNA, RNA and proteins, while the processes include transcription, translation, signal transduction and regulation. Recently, the use of bacteria themselves as living computers has been explored but this use generally falls within the classical paradigm of computing. Computer scientists, however, have a variety of problems to which they seek solutions, while microbiologists are having new insights into the problems bacteria are solving and how they are solving them. Here, we envisage that bacteria might be used for new sorts of computing. These could be based on the capacity of bacteria to grow, move and adapt to a myriad different fickle environments both as individuals and as populations of bacteria plus bacteriophage. New principles might be based on the way that bacteria explore phenotype space via hyperstructure dynamics and the fundamental nature of the cell cycle. This computing might even extend to developing a high level language appropriate to using populations of bacteria and bacteriophage. Here, we offer a speculative tour of what we term bactoputing, namely the use of the natural behaviour of bacteria for calculating

From Cleanroom to Desktop: Emerging Micro-Nanofabrication Technology for Biomedical Applications

This review is motivated by the growing demand for low-cost, easy-to-use, compact-size yet powerful micro-nanofabrication technology to address emerging challenges of fundamental biology and translational medicine in regular laboratory settings. Recent advancements in the field benefit considerably from rapidly expanding material selections, ranging from inorganics to organics and from nanoparticles to self-assembled molecules. Meanwhile a great number of novel methodologies, employing off-the-shelf consumer electronics, intriguing interfacial phenomena, bottom-up self-assembly principles, etc., have been implemented to transit micro-nanofabrication from a cleanroom environment to a desktop setup. Furthermore, the latest application of micro-nanofabrication to emerging biomedical research will be presented in detail, which includes point-of-care diagnostics, on-chip cell culture as well as bio-manipulation. While significant progresses have been made in the rapidly growing field, both apparent and unrevealed roadblocks will need to be addressed in the future. We conclude this review by offering our perspectives on the current technical challenges and future research opportunities