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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 (6 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
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 54 (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...
Toward Reliable Algorithmic SelfAssembly of DNA Tiles: A FixedWidth Cellular Automaton Pattern NANO LETTERS
, 2007
"... Bottomup fabrication of nanoscale structures relies on chemical processes to direct selfassembly. The complexity, precision, and yield achievable by a onepot reaction are limited by our ability to encode assembly instructions into the molecules themselves. Nucleic acids provide a platform for inv ..."
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Cited by 36 (3 self)
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Bottomup fabrication of nanoscale structures relies on chemical processes to direct selfassembly. The complexity, precision, and yield achievable by a onepot reaction are limited by our ability to encode assembly instructions into the molecules themselves. Nucleic acids provide a platform for investigating these issues, as molecular structure and intramolecular interactions can encode growth rules. Here, we use DNA tiles and DNA origami to grow crystals containing a cellular automaton pattern. In a onepot annealing reaction, 250 DNA strands first assemble into a set of 10 free tile types and a seed structure, then the free tiles grow algorithmically from the seed according to the automaton rules. In our experiments, crystals grew to ∼300 nm long, containing ∼300 tiles with an initial assembly error rate of ∼1.4 % per tile. This work provides evidence that programmable molecular selfassembly may be sufficient to create a wide range of complex objects in onepot reactions. The WatsonsCrick complementarity of DNA molecules allows one to design not only simple doublestranded helices but also complicated woven structures consisting of many DNA strands. 1 Welldesigned structures will selfassemble during annealing from a high initial temperature at which point all molecules are singlestranded to a lower final
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 28 (5 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
Parallelism and Time in Hierarchical SelfAssembly
, 2012
"... We study the role that parallelism plays in time complexity of variants of Winfree’s abstract Tile Assembly Model (aTAM), a model of molecular algorithmic selfassembly. In the “hierarchical ” aTAM, two assemblies, both consisting of multiple tiles, are allowed to aggregate together, whereas in the ..."
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Cited by 21 (8 self)
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We study the role that parallelism plays in time complexity of variants of Winfree’s abstract Tile Assembly Model (aTAM), a model of molecular algorithmic selfassembly. In the “hierarchical ” aTAM, two assemblies, both consisting of multiple tiles, are allowed to aggregate together, whereas in the “seeded” aTAM, tiles attach one at a time to a growing assembly. Adleman, Cheng, Goel, and Huang (Running Time and Program Size for SelfAssembled Squares, STOC 2001) showed how to assemble an n×n square in O(n) time in log n the seeded aTAM using O ( ) unique tile types, where log log n both of these parameters are optimal. They asked whether the hierarchical aTAM could allow a tile system to use the ability to form large assemblies in parallel before they attach to break the Ω(n) lower bound for assembly time. We show log n that there is a tile system with the optimal O ( ) tile log log n types that assembles an n×n square using O(log 2 n) parallel “stages”, which is close to the optimal Ω(log n) stages, forming the final n×n square from four n/2×n/2 squares, which are themselves recursively formed from n/4 × n/4 squares, etc. However, despite this nearly maximal parallelism, the system requires superlinear time to assemble the square. We extend the definition of partial order tile systems studied by Adleman et al. in a natural way to hierarchical assembly and show that no hierarchical partial order tile system can build any shape with diameter N in less than time Ω(N), demonstrating that in this case the hierarchical model affords no speedup whatsoever over the seeded model. We also strengthen the Ω(N) time lower bound for deterministic seeded systems of Adleman et al. to nondeterministic seeded systems. Finally, we show that for infinitely many n, a tile system can assemble an n × n ′ rectangle, with n> n ′, in time O(n 4/5 log n), breaking the lineartime lower bound that applies to all seeded systems and partial order hierarchical systems.
Selfhealing tile sets
 Foundations of Nanoscience: SelfAssembled Architectures and Devices, 2005
, 2005
"... Summary. Molecular selfassembly appears to be a promising route to bottomup fabrication of complex objects. Two major obstacles are how to create structures with more interesting organization than periodic or finite arrays, and how to reduce the fraction of side products and erroneous assemblies. ..."
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Cited by 17 (1 self)
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Summary. Molecular selfassembly appears to be a promising route to bottomup fabrication of complex objects. Two major obstacles are how to create structures with more interesting organization than periodic or finite arrays, and how to reduce the fraction of side products and erroneous assemblies. Algorithmic selfassembly provides a theoretical model for investigating these questions: the growth of arbitrarily complex objects can be programmed into a set of Wang tiles, and their robustness to a variety of possible errors can be studied. The ability to program the tiles presents an alternative to directly physical or chemical means for reducing error rates, since redundant information can be stored so that errors can be detected, corrected, and/or prevented during the selfassembly process. Here we study the ability of algorithmic selfassembly to heal damage to a selfassembled object. We present block transforms that convert an original errorprone tile set into a new tile set that performs the same construction task (at a slightly larger scale) and also is able to heal damaged areas where many tiles have been removed from the assembly. 1 Algorithmic Crystal Growth
Activatable Tiles: Compact, Robust Programmable Assembly and Other Applications
 in DNA Computing: DNA13 (edited by Max Garzon and Hao Yan), SpringerVerlag Lecture Notes for Computer Science (LNCS
, 2007
"... While algorithmic DNA selfassembly is, in theory, capable of forming complex patterns, its experimental demonstration has been limited by significant assembly errors. In this paper we describe a novel protection/deprotection strategy to strictly enforce the direction of tiling assembly growth to en ..."
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Cited by 11 (6 self)
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While algorithmic DNA selfassembly is, in theory, capable of forming complex patterns, its experimental demonstration has been limited by significant assembly errors. In this paper we describe a novel protection/deprotection strategy to strictly enforce the direction of tiling assembly growth to ensure the robustness of the assembly process. Tiles are initially inactive, meaning that each tile’s output pads are protected and cannot bind with other tiles. After other tiles bind to the tile’s input pads, the tile transitions to an active state and its output pads are exposed, allowing further growth. We describe abstract and kinetic models of activatable tile assembly and show that the error rate can be decreased significantly with respect to Winfree’s original kinetic tile assembly model without considerable decrease in assembly growth speed. We prove that an activatable tile set is an example of a compact, errorresilient and selfhealing tileset. We describe a DNA design of activatable tiles and a mechanism of deprotection using DNA polymerization and strand displacement. We conclude with a brief discussion on some applications of activatable tiles beyond computational tiling, both as a novel concentration system and a catalyst in chemical reactions. 1
One tile to rule them all: Simulating any Turing machine, tile assembly system, or tiling system with a single puzzle piece
, 2012
"... In this paper we explore the power of tile selfassembly models that extend the wellstudied abstract Tile Assembly Model (aTAM) by permitting tiles of shapes beyond unit squares. Our main result shows the surprising fact that any aTAM system, consisting of many different tile types, can be simulat ..."
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Cited by 9 (5 self)
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In this paper we explore the power of tile selfassembly models that extend the wellstudied abstract Tile Assembly Model (aTAM) by permitting tiles of shapes beyond unit squares. Our main result shows the surprising fact that any aTAM system, consisting of many different tile types, can be simulated by a single tile type of a general shape. As a consequence, we obtain a single universal tile type of a single (constantsize) shape that serves as a “universal tile machine”: the single universal tile type can simulate any desired aTAM system when given a single seed assembly that encodes the desired aTAM system. We also show how to adapt this result to convert any of a variety of plane tiling systems (such as Wang tiles) into a “nearly” plane tiling system with a single tile (but with small gaps between the tiles). All of these results rely on the ability to both rotate and translate tiles; by contrast, we show that a single nonrotatable tile, of arbitrary shape, can produce assemblies which either grow infinitely or cannot grow at all, implying drastically limited computational power. On the positive side, we
Capabilities and Limits of Compact Error Resilience Methods for Algorithmic SelfAssembly
 REIF@CS.DUKE.EDU
, 2007
"... Winfree’s pioneering work led the foundations in the area of errorreduction in algorithmic selfassembly (Winfree and Bekbolatov in DNA Based Computers 9, LNCS, vol. 2943, pp. 126–144, 2004), but the construction resulted in increase of the size of assembly. Reif et al. (Nanotechnol. Sci. Comput. 79 ..."
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Cited by 6 (4 self)
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Winfree’s pioneering work led the foundations in the area of errorreduction in algorithmic selfassembly (Winfree and Bekbolatov in DNA Based Computers 9, LNCS, vol. 2943, pp. 126–144, 2004), but the construction resulted in increase of the size of assembly. Reif et al. (Nanotechnol. Sci. Comput. 79–103, 2006) contributed further in this area with compact errorresilient schemes that maintained the original size of the assemblies, but required certain restrictions on the Boolean functions to be used in the algorithmic selfassembly. It is a critical challenge to improve these compact error resilient schemes to incorporate arbitrary Boolean functions, and to determine how far these prior results can be extended under different degrees of restrictions on the Boolean functions. In this work we present a considerably more complete theory of compact errorresilient schemes for algorithmic selfassembly in two and three dimensions. In our error model, ɛ is defined to be the probability that there is a mismatch between the neighboring sides of two juxtaposed tiles and they still stay together in the equilibrium. This probability is independent of any other match or mismatch and hence we term this probabilistic model as the