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COMPLEXITY OF SELFASSEMBLED SHAPES
, 2007
"... The connection between selfassembly and computation suggests that a shape can be considered the output of a selfassembly “program,” a set of tiles that fit together to create a shape. It seems plausible that the size of the smallest selfassembly program that builds a shape and the shape’s descrip ..."
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Cited by 86 (4 self)
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The connection between selfassembly and computation suggests that a shape can be considered the output of a selfassembly “program,” a set of tiles that fit together to create a shape. It seems plausible that the size of the smallest selfassembly program that builds a shape and the shape’s descriptional (Kolmogorov) complexity should be related. We show that when using a notion of a shape that is independent of scale, this is indeed so: in the tile assembly model, the minimal number of distinct tile types necessary to selfassemble a shape, at some scale, can be bounded both above and below in terms of the shape’s Kolmogorov complexity. As part of the proof, we develop a universal constructor for this model of selfassembly that can execute an arbitrary Turing machine program specifying how to grow a shape. Our result implies, somewhat counterintuitively, that selfassembly of a scaledup version of a shape often requires fewer tile types. Furthermore, the independence of scale in selfassembly theory appears to play the same crucial role as the independence of running time in the theory of computability. This leads to an elegant formulation of languages of shapes generated by selfassembly. Considering functions from bit strings to shapes, we show that the runningtime complexity, with respect to Turing machines, is polynomially equivalent to the scale complexity of the same function implemented via selfassembly by a finite set of tile types. Our results also hold for shapes defined by Wang tiling—where there is no sense of a selfassembly process—except that here time complexity must be measured with respect to nondeterministic Turing machines.
Two computational primitives for algorithmic selfassembly: Copying and counting
 Nano Letters
, 2005
"... Copying and counting are useful primitive operations for computation and construction. We have made DNA crystals that copy and crystals that count as they grow. For counting, 16 oligonucleotides assemble into four DNA Wang tiles that subsequently crystallize on a polymeric nucleating scaffold strand ..."
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Cited by 68 (5 self)
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Copying and counting are useful primitive operations for computation and construction. We have made DNA crystals that copy and crystals that count as they grow. For counting, 16 oligonucleotides assemble into four DNA Wang tiles that subsequently crystallize on a polymeric nucleating scaffold strand, arranging themselves in a binary counting pattern that could serve as a template for a molecular electronic demultiplexing circuit. Although the yield of counting crystals is low, and pertile error rates in such crystals is roughly 10%, this work demonstrates the potential of algorithmic selfassembly to create complex nanoscale patterns of technological interest. A subset of the tiles for counting form informationbearing DNA tubes that copy bit strings from layer to layer along their length. The challenge of engineering complex devices at the nanometer scale has been approached from two radically different directions. In topdown synthesis, information about the desired structure is imposed by an external apparatus, as in photolithography. In bottomup synthesis, structure arises spontaneously due to chemical and physical forces intrinsic to the molecular components themselves. A significant challenge for bottomup techniques is how to design
Proofreading tile sets: Error correction for algorithmic selfassembly
 DNA Computers
"... Abstract. For robust molecular implementation of tilebased algorithmic selfassembly, methods for reducing errors must be developed. Previous studies suggested that by control of physical conditions, such as temperature and the concentration of tiles, errors (") can be reduced to an arbitrar ..."
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Cited by 60 (11 self)
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Abstract. For robust molecular implementation of tilebased algorithmic selfassembly, methods for reducing errors must be developed. Previous studies suggested that by control of physical conditions, such as temperature and the concentration of tiles, errors (") can be reduced to an arbitrarily low rate { but at the cost of reduced speed (r) for the selfassembly process. For tile sets directly implementing blocked cellular automata, it was shown that r "2 was optimal. Here, we show that an improved construction, which we refer to as proofreading tile sets, can in principle exploit the cooperativity of tile assembly reactions to dramatically improve the scaling behavior to r " and better. This suggests that existing DNAbased molecular tile approaches may be improved to produce macroscopic algorithmic crystals with few errors. Generalizations and limitations of the proofreading tile set construction are discussed. 1
Compact ErrorResilient Computational DNA Tiling Assemblies
"... The selfassembly process for bottomup construction of nanostructures is of key importance to the emerging of the new scientific discipline of Nanoscience. For example, the selfassembly of DNA tile nanostructures into 2D and 3D lattices can be used to perform parallel universal computation and to ..."
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Cited by 53 (10 self)
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The selfassembly process for bottomup construction of nanostructures is of key importance to the emerging of the new scientific discipline of Nanoscience. For example, the selfassembly of DNA tile nanostructures into 2D and 3D lattices can be used to perform parallel universal computation and to manufacture patterned nanostructures from smaller unit components known as DNA tiles. However, selfassemblies at the molecular scale are prone to a quite high rate of error, and the key barrier to largescale experimental implementation of DNA tiling is the high error rate in the selfassembly process. One major challenge to nanostructure selfassembly is to eliminate/limit these errors. The goals of this paper are to develop theoretical methods for compact errorresilient selfassembly, to analyze these by stochastic analysis and computer simulation (at a future date we also intend to demonstrate these errorresilient selfassembly methods by a series of laboratory experiments). Prior work by Winfree provided a innovative approach to decrease tiling selfassembly errors without decreasing the intrinsic error rate # of assembling a single tile, however, his technique resulted in a final structure that is four times the size of the original one. This paper describes various compact errorresilient tiling methods that do not increase the size of the tiling assembly. These methods apply to assembly of boolean arrays which perform input sensitive computations (among other computations). We first describe an errorresilient tiling using 2way overlay redundancy such that a single pad mismatch between a tile and its immediate neighbor forces at least one further pad mismatch between a pair of adjacent tiles in the neighborhood of this tile. This drops the error rate from # to appr...
Selfassembled circuit patterns
 In DNA Computing 9
, 2004
"... Abstract. Selfassembly is a process in which basic units aggregate under attractive forces to form larger compound structures. Recent theoretical work has shown that pseudocrystalline selfassembly can be algorithmic, in the sense that complex logic can be programmed into the growth process [26]. ..."
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Cited by 44 (14 self)
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Abstract. Selfassembly is a process in which basic units aggregate under attractive forces to form larger compound structures. Recent theoretical work has shown that pseudocrystalline selfassembly can be algorithmic, in the sense that complex logic can be programmed into the growth process [26]. This theoretical work builds on the theory of twodimensional tilings [8], using rigid square tiles called Wang tiles [24] for the basic units of selfassembly, and leads to Turinguniversal models such as the Tile Assembly Model [28]. Using the Tile Assembly Model, we show how algorithmic selfassembly can be exploited for fabrication tasks such as constructing the patterns that define certain digital circuits, including demultiplexers, RAM arrays, pseudowavelet transforms, and Hadamard transforms. Since DNA selfassembly appears to be promising for implementing the arbitrary Wang tiles [30, 13] needed for programming in the Tile Assembly Model, algorithmic selfassembly methods such as those presented in this paper may eventually become a viable method of arranging molecular electronic components [18], such as carbon nanotubes [10, 1], into molecularscale circuits. 1
Programmable control of nucleation for algorithmic selfassembly
, 2009
"... Algorithmic selfassembly, a generalization of crystal growth processes, has been proposed as a mechanism for autonomous DNA computation and for bottomup fabrication of complex nanostructures. A “program” for growing a desired structure consists of a set of molecular “tiles” designed to have speci ..."
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Cited by 42 (11 self)
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Algorithmic selfassembly, a generalization of crystal growth processes, has been proposed as a mechanism for autonomous DNA computation and for bottomup fabrication of complex nanostructures. A “program” for growing a desired structure consists of a set of molecular “tiles” designed to have specific binding interactions. A key challenge to making algorithmic selfassembly practical is designing tile set programs that make assembly robust to errors that occur during initiation and growth. One method for the controlled initiation of assembly, often seen in biology, is the use of a seed or catalyst molecule that reduces an otherwise large kinetic barrier to nucleation. Here we show how to program algorithmic selfassembly similarly, such that seeded assembly proceeds quickly but there is an arbitrarily large kinetic barrier to unseeded growth. We demonstrate this technique by introducing a family of tile sets for which we rigorously prove that, under the right physical conditions, linearly increasing the size of the tile set exponentially reduces the rate of spurious nucleation. Simulations of these “zigzag” tile sets suggest that under plausible experimental conditions, it is possible to grow large seeded crystals in just a few hours such that less than 1 percent of crystals are spuriously nucleated. Simulation results also suggest that zigzag tile sets could be used for detection of single DNA strands. Together with prior work showing that tile sets can be made robust to errors during properly initiated growth, this work demonstrates that growth of objects via algorithmic selfassembly can proceed both efficiently and with an arbitrarily low error rate, even in a model where local growth rules are probabilistic.
DNABased Cryptography
 5th DIMACS Workshop on DNA Based Computers, MIT
, 1999
"... Recent research has considered DNA as a medium for ultrascale computation and for ultracompact information storage. One potential key application is DNAbased, molecular cryptography systems. We present some procedures for DNAbased cryptography based on onetimepads that are in principle unbre ..."
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Cited by 41 (6 self)
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Recent research has considered DNA as a medium for ultrascale computation and for ultracompact information storage. One potential key application is DNAbased, molecular cryptography systems. We present some procedures for DNAbased cryptography based on onetimepads that are in principle unbreakable. Practical applications of cryptographic systems based on onetime pads are limited in conventional electronic media by the size of the onetimepad; however DNA provides a much more compact storage medium, and an extremely small amount of DNA su#ces even for huge onetimepads. We detail procedures for two DNA onetimepad encryption schemes: (i) a substitution method using libraries of distinct pads, each of which defines a specific, randomly generated, pairwise mapping; and (ii) an XOR scheme utilizing molecular computation and indexed, random key strings. These methods can be applied either for the encryption of natural DNA or for artificial DNA encoding binary data. In the latter case, we also present a novel use of chipbased DNA microarray technology for 2D data input and output.
Complexity of compact proofreading for selfassembled patterns
 In Proc. 11th International Meeting on DNA Computing
, 2005
"... Abstract. Faulttolerance is a critical issue for biochemical computation. Recent theoretical work on algorithmic selfassembly has shown that error correcting tile sets are possible, and that they can achieve exponential decrease in error rates with a small increase in the number of tile types and ..."
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Cited by 25 (4 self)
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Abstract. Faulttolerance is a critical issue for biochemical computation. Recent theoretical work on algorithmic selfassembly has shown that error correcting tile sets are possible, and that they can achieve exponential decrease in error rates with a small increase in the number of tile types and the scale of the construction [24, 4]. Following [17], we consider the issue of applying similar schemes to achieve error correction without any increase in the scale of the assembled pattern. Using a new proofreading transformation, we show that compact proofreading can be performed for some patterns with a modest increase in the number of tile types. Other patterns appear to require an exponential number of tile types. A simple property of existing proofreading schemes – a strong kind of redundancy – is the culprit, suggesting that if general purpose compact proofreading schemes are to be found, this type of redundancy must be avoided. 1
The Tile Complexity of Linear Assemblies
"... Selfassembly is fundamental to both biological processes and nanoscience. Key features of selfassembly are its probabilistic nature and local programmability. These features can be leveraged to design better selfassembled systems. The conventional Tile Assembly Model (TAM) developed by Winfree us ..."
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Cited by 23 (5 self)
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Selfassembly is fundamental to both biological processes and nanoscience. Key features of selfassembly are its probabilistic nature and local programmability. These features can be leveraged to design better selfassembled systems. The conventional Tile Assembly Model (TAM) developed by Winfree using Wang tiles is a powerful, Turinguniversal theoretical framework which models varied selfassembly 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 selfassembled 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 selfassembly strongly suggests that standard PTAM can be realized in the laboratory. In TAM, a deterministic linear assembly of length N requires
Efficient Turinguniversal computation with DNA polymers (extended abstract)
"... Abstract. Bennett’s proposed chemical Turing machine is one of the most important thought experiments in the study of the thermodynamics of computation. Yet the sophistication of molecular engineering required to physically construct Bennett’s hypothetical polymer substrate and enzyme has deterred e ..."
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Cited by 20 (4 self)
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Abstract. Bennett’s proposed chemical Turing machine is one of the most important thought experiments in the study of the thermodynamics of computation. Yet the sophistication of molecular engineering required to physically construct Bennett’s hypothetical polymer substrate and enzyme has deterred experimental implementations. Here we propose a chemical implementation of stack machines — a Turinguniversal model of computation similar to Turing machines — using strand displacement cascades as the underlying chemical primitive. More specifically, the mechanism described herein is the addition and removal of monomers from the end of a polymer, controlled by strand displacement logic. We capture the motivating feature of Bennett’s scheme — that physical reversibility corresponds to logically reversible computation, and arbitrarily little energy per computation step is required. Further, as a method of embedding logic control into chemical and biological systems, polymerbased chemical computation is significantly more efficient than geometryfree chemical reaction networks. 1