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Algorithmic selfassembly of DNA Sierpinski triangles
 PLoS Biology
"... Algorithms and information, fundamental to technological and biological organization, are also an essential aspect of many elementary physical phenomena, such as molecular selfassembly. Here we report the molecular realization, using twodimensional selfassembly of DNA tiles, of a cellular automat ..."
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Cited by 153 (13 self)
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Algorithms and information, fundamental to technological and biological organization, are also an essential aspect of many elementary physical phenomena, such as molecular selfassembly. Here we report the molecular realization, using twodimensional selfassembly of DNA tiles, of a cellular automaton whose update rule computes the binary function XOR and thus fabricates a fractal pattern—a Sierpinski triangle—as it grows. To achieve this, abstract tiles were translated into DNA tiles based on doublecrossover motifs. Serving as input for the computation, long singlestranded DNA molecules were used to nucleate growth of tiles into algorithmic crystals. For both of two independent molecular realizations, atomic force microscopy revealed recognizable Sierpinski triangles containing 100–200 correct tiles. Error rates during assembly appear to range from 1 % to 10%. Although imperfect, the growth of Sierpinski triangles demonstrates all the necessary mechanisms for the molecular implementation of arbitrary cellular automata. This shows that engineered DNA selfassembly can be treated as a Turinguniversal biomolecular system, capable of implementing any desired algorithm for computation or construction tasks.
Running Time and Program Size for Selfassembled Squares
, 2001
"... Recently Rothemund and Winfree [6] have considered the program size complexity of constructing squares by selfassembly. Here, we consider the time complexity of such constructions using a natural generalization of the Tile Assembly Model defined in [6]. In the generalized model, the RothemundWinf ..."
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Cited by 96 (8 self)
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Recently Rothemund and Winfree [6] have considered the program size complexity of constructing squares by selfassembly. Here, we consider the time complexity of such constructions using a natural generalization of the Tile Assembly Model defined in [6]. In the generalized model, the RothemundWinfree construction of n \Theta n squares requires time \Theta(n log n) and program size \Theta(log n). We present a new construction for assembling n \Theta n squares which uses optimal time \Theta(n) and program size \Theta( log n log log n ). This program size is also optimal since it matches the bound dictated by Kolmogorov complexity. Our improved time is achieved by demonstrating a set of tiles for parallel selfassembly of binary counters. Our improved program size is achieved by demonstrating that selfassembling systems can compute changes in the base representation of numbers. Selfassembly is emerging as a useful paradigm for computation. In addition the development of a computational theory of selfassembly promises to provide a new conduit by which results and methods of theoretical computer science might be applied to problems of interest in biology and the physical sciences. 1
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...
Complexities for Generalized Models of SelfAssembly
 IN SODA
, 2004
"... In this paper, we study the complexity of selfassembly under models that are natural generalizations of the tile selfassembly model. In particular, we extend Rothemund and Winfree’s log N study of the tile complexity of tile selfassembly [9]. They provided a lower bound of Ω ( log log N) on the ..."
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Cited by 52 (6 self)
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In this paper, we study the complexity of selfassembly under models that are natural generalizations of the tile selfassembly model. In particular, we extend Rothemund and Winfree’s log N study of the tile complexity of tile selfassembly [9]. They provided a lower bound of Ω ( log log N) on the tile complexity of assembling an N × N square for almost all N. Adleman et al. [1] gave a construction which achieves this bound. We consider whether the tile complexity for selfassembly can be reduced through several natural generalizations of the model. One of our results is a tile set of size O ( √ log N) which assembles an N × N square in a model which allows flexible glue strength between nonequal glues. This result is matched for almost all N by a lower bound dictated by log N Kolmogorov complexity. For three other generalizations, we show that the Ω ( ) lower bound log log N applies to N × N squares. At the same time, we demonstrate that there are some other shapes for which these generalizations allow reduced tile sets. Specifically, for thin rectangles with length N and width k, we provide a tighter lower bound of Ω ( N 1 k k log N construction which achieves O ( log log N) for the standard model, yet we also give a) complexity in a model in which the temperature of the tile system is adjusted during assembly. We also investigate the problem of verifying whether a given tile system uniquely assembles into a given shape; we show that this problem is NPhard for three of the generalized models.
Error Free SelfAssembly using Error Prone Tiles
"... DNA selfassembly is emerging as a key paradigm for nanotechnology, nanocomputation, and several related disciplines. In nature, DNA selfassembly is often equipped with explicit mechanisms for both error prevention and error correction. For artificial selfassembly, these problems are even more ..."
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Cited by 48 (5 self)
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DNA selfassembly is emerging as a key paradigm for nanotechnology, nanocomputation, and several related disciplines. In nature, DNA selfassembly is often equipped with explicit mechanisms for both error prevention and error correction. For artificial selfassembly, these problems are even more important since we are interested in assembling large systems with great precision. We present an
Combinatorial optimization problems in selfassembly
 In Proceedings of the thiryfourth annual ACM symposium on Theory of computing
, 2002
"... Selfassembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate selfassembly processes will ultimately be used in circuit fabrication, nanorobotics, DNA computation, and amorphous computing. In this paper, we stud ..."
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Cited by 43 (4 self)
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Selfassembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate selfassembly processes will ultimately be used in circuit fabrication, nanorobotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient selfassembly of shapes in the Tile Assembly Model of selfassembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum
Reducing tile complexity for selfassembly through temperature programming
 Proceedings of the 17th Annual ACMSIAM Symposium on Discrete Algorithms (SODA 2006
, 2006
"... We consider the tile selfassembly model and how tile complexity can be eliminated by permitting the temperature of the selfassembly system to be adjusted throughout the assembly process. To do this, we propose novel techniques for designing tile sets that permit an arbitrary length m binary number ..."
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Cited by 43 (6 self)
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We consider the tile selfassembly model and how tile complexity can be eliminated by permitting the temperature of the selfassembly system to be adjusted throughout the assembly process. To do this, we propose novel techniques for designing tile sets that permit an arbitrary length m binary number to be encoded into a sequence of O(m) temperature changes such that the tile set uniquely assembles a supertile that precisely encodes the corresponding binary number. As an application, we show how this provides a general tile set of size O(1) that is capable of uniquely assembling essentially any n × n square, where the assembled square is determined by a temperature sequence of length O(log n) that encodes a binary description of n. log n This yields an important decrease in tile complexity from the required Ω( log log n) for almost all n when the temperature of the system is fixed. We further show that for almost all n, no tile system log n log log n can simultaneously achieve both o(log n) temperature complexity and o ( ) tile complexity, showing that both versions of an optimal square building scheme have been discovered. This work suggests that temperature change can constitute a natural, dynamic method for providing input to selfassembly systems that is potentially superior to the current technique of designing large tile sets with specific inputs hardwired into the tileset. 1