Fuel Efficient Computation in Passive Self-Assembly

In this paper we show that passive self-assembly in the context of the tile self-assembly model is capable of performing fuel efficient, universal computation. The tile self-assembly model is a premiere model of self-assembly in which particles are modeled by four-sided squares with glue types assigned to each tile edge. The assembly process is driven by positive and negative force interactions between glue types, allowing for tile assemblies floating in the plane to combine and break apart over time. We refer to this type of assembly model as passive in that the constituent parts remain unchanged throughout the assembly process regardless of their interactions. A computationally universal system is said to be fuel efficient if the number of tiles used up per computation step is bounded by a constant. Work within this model has shown how fuel guzzling tile systems can perform universal computation with only positive strength glue interactions. Recent work has introduced space-efficient, fuel-guzzling universal computation with the addition of negative glue interactions and the use of a powerful non-diagonal class of glue interactions. Other recent work has shown how to achieve fuel efficient computation within active tile self-assembly. In this paper we utilize negative interactions in the tile self-assembly model to achieve the first computationally universal passive tile self-assembly system that is both space and fuel-efficient. In addition, we achieve this result using a limited diagonal class of glue interactions

Program Size and Temperature in Self-Assembly

Winfree’s abstract Tile Assembly Model is a model of molecular self-assembly of DNA complexes known as tiles, which float freely in solution and attach one at a time to a growing “seed” assembly based on specific binding sites on their four sides. We show that there is a polynomial-time algorithm that, given an n×n square, finds the minimal tile system (i.e., the system with the smallest number of distinct tile types) that uniquely self-assembles the square, answering an open question of Adleman et al. (Combinatorial optimization problems in self-assembly, STOC 2002). Our investigation leading to this algorithm reveals other positive and negative results about the relationship between the size of a tile system and its “temperature” (the binding strength threshold required for a tile to attach)

Nascent nanocomputers: DNA self-assembly in O(1) stages

DNA self-assembly offers a potential for nanoscale microcircuits and computers. To make that potential possible requires the development of reliable and efficient tile assembly models. Efficiency is often achieved by minimizing tile complexity, as well as by evaluating the cost and reliability of the specific elements of each tile assembly model. We consider a 2D tile assembly model at temperature 1. The standard 2D tile assembly model at temperature 1 has a tile complexity of O(n) for the construction of exact, complete n x n squares. However, previous research found a staged tile assembly model achieved a tile complexity of O(1) to construct n x n squares, with O(logn) stages. Our staged tile assembly model achieves a tile complexity of O(logn) using only O(1) stages to construct n x n squares

Signal Transmission Across Tile Assemblies: 3D Static Tiles Simulate Active Self-Assembly by 2D Signal-Passing Tiles

The 2-Handed Assembly Model (2HAM) is a tile-based self-assembly model in which, typically beginning from single tiles, arbitrarily large aggregations of static tiles combine in pairs to form structures. The Signal-passing Tile Assembly Model (STAM) is an extension of the 2HAM in which the tiles are dynamically changing components which are able to alter their binding domains as they bind together. For our first result, we demonstrate useful techniques and transformations for converting an arbitrarily complex STAM$^+$ tile set into an STAM$^+$ tile set where every tile has a constant, low amount of complexity, in terms of the number and types of ``signals'' they can send, with a trade off in scale factor. Using these simplifications, we prove that for each temperature $\tau>1$ there exists a 3D tile set in the 2HAM which is intrinsically universal for the class of all 2D STAM$^+$ systems at temperature $\tau$ (where the STAM$^+$ does not make use of the STAM's power of glue deactivation and assembly breaking, as the tile components of the 2HAM are static and unable to change or break bonds). This means that there is a single tile set $U$ in the 3D 2HAM which can, for an arbitrarily complex STAM$^+$ system $S$, be configured with a single input configuration which causes $U$ to exactly simulate $S$ at a scale factor dependent upon $S$. Furthermore, this simulation uses only two planes of the third dimension. This implies that there exists a 3D tile set at temperature $2$ in the 2HAM which is intrinsically universal for the class of all 2D STAM$^+$ systems at temperature $1$. Moreover, we show that for each temperature $\tau>1$ there exists an STAM$^+$ tile set which is intrinsically universal for the class of all 2D STAM$^+$ systems at temperature $\tau$, including the case where $\tau = 1$.Comment: A condensed version of this paper will appear in a special issue of Natural Computing for papers from DNA 19. This full version contains proofs not seen in the published versio

Randomness, information encoding, and shape replication in various models of DNA-inspired self-assembly

Self-assembly is the process by which simple, unorganized components autonomously combine to form larger, more complex structures. Researchers are turning to self-assembly technology for the design of ever smaller, more complex, and precise nanoscale devices, and as an emerging fundamental tool for nanotechnology. We introduce the robust random number generation problem, the problem of encoding a target string of bits in the form of a bit string pad, and the problem of shape replication in various models of tile-based self-assembly. Also included are preliminary results in each of these directions with discussion of possible future work directions

The Tile Complexity of Linear Assemblies

Self-assembly is fundamental to both biological processes and nanoscience. Key features of self-assembly are its probabilistic nature and local programmability. These features can be leveraged to design better self-assembled systems. The conventional Tile Assembly Model (TAM) developed by Winfree using Wang tiles is a powerful, Turing-universal theoretical framework which models varied self-assembly processes. A particular challenge in DNA nanoscience is to form linear assemblies or rulers of a specified length using the smallest possible tile set, where any tile type may appear more than once in the assembly. The tile complexity of a linear assembly is the cardinality of the tile set that produced it. These rulers can then be used as components for construction of other complex structures. While square assemblies have been extensively studied, many questions remain about fixed length linear assemblies, which are more basic constructs yet fundamental building blocks for molecular architectures. In this paper, we extend TAM to take advantage of inherent probabilistic behavior in physically realized self-assembled systems by introducing randomization. We describe a natural extension to TAM called the Probabilistic Tile Assembly Model (PTAM). A restriction of the model, which we call the standard PTAM is considered in this paper. Prior work in DNA self-assembly strongly suggests that standard PTAM can be realized in the laboratory. In TAM, a deterministic linear assembly of length N require

A Theoretical and Experimental Study of DNA Self-assembly

<p>The control of matter and phenomena at the nanoscale is fast becoming one of the most important challenges of the 21st century with wide-ranging applications from energy and health care to computing and material science. Conventional top-down approaches to nanotechnology, having served us well for long, are reaching their inherent limitations. Meanwhile, bottom-up methods such as self-assembly are emerging as viable alternatives for nanoscale fabrication and manipulation.</p><p>A particularly successful bottom up technique is DNA self-assembly where a set of carefully designed DNA strands form a nanoscale object as a consequence of specific, local interactions among the different components, without external direction. The final product of the self-assembly process might be a static nanostructure or a dynamic nanodevice that performs a specific function. Over the past two decades, DNA self-assembly has produced stunning nanoscale objects such as 2D and 3D lattices, polyhedra and addressable arbitrary shaped substrates, and a myriad of nanoscale devices such as molecular tweezers, computational circuits, biosensors and molecular assembly lines. In this dissertation we study multiple problems in the theory, simulations and experiments of DNA self-assembly. </p><p>We extend the Turing-universal mathematical framework of self-assembly known as the Tile Assembly Model by incorporating randomization during the assembly process. This allows us to reduce the tile complexity of linear assemblies. We develop multiple techniques to build linear assemblies of expected length N using far fewer tile types than previously possible.</p><p>We abstract the fundamental properties of DNA and develop a biochemical system, which we call meta-DNA, based entirely on strands of DNA as the only component molecule. We further develop various enzyme-free protocols to manipulate meta-DNA systems and provide strand level details along with abstract notations for these mechanisms. </p><p>We simulate DNA circuits by providing detailed designs for local molecular computations that involve spatially contiguous molecules arranged on addressable substrates via enzyme-free DNA hybridization reaction cascades. We use the Visual DSD simulation software in conjunction with localized reaction rates obtained from biophysical modeling to create chemical reaction networks of localized hybridization circuits that are then model checked using the PRISM model checking software.</p><p>We develop a DNA detection system employing the triggered self-assembly of a novel DNA dendritic nanostructure. Detection begins when a specific, single-stranded target DNA strand triggers a hybridization chain reaction between two distinct DNA hairpins. Each hairpin opens and hybridizes up to two copies of the other, and hence each layer of the growing dendritic nanostructure can in principle accommodate an exponentially increasing number of cognate molecules, generating a nanostructure with high molecular weight. </p><p>We build linear activatable assemblies employing a novel protection/deprotection strategy to strictly enforce the direction of tiling assembly growth to ensure the robustness of the assembly process. Our system consists of two tiles that can form a linear co-polymer. These tiles, which are initially protected such that they do not react with each other, can be activated to form linear co-polymers via the use of a strand displacing enzyme.</p>Dissertatio