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Logical computation using algorithmic selfassembly of dna triplecrossover molecules
 Nature
, 2000
"... Recent work has demonstrated the selfassembly of designed periodic twodimensional arrays composed of DNA tiles, in which the intermolecular contacts are directed by 'sticky ' ends. In a mathematical context, aperiodic mosaics may be formed by the selfassembly of 'Wang ' tiles ..."
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Cited by 83 (18 self)
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Recent work has demonstrated the selfassembly of designed periodic twodimensional arrays composed of DNA tiles, in which the intermolecular contacts are directed by 'sticky ' ends. In a mathematical context, aperiodic mosaics may be formed by the selfassembly of 'Wang ' tiles 4, a process that emulates the operation of a Turing machine. Macroscopic selfassembly has been used to perform computations 5; there is also a logical equivalence between DNA sticky ends and Wang tile edges 6, 7. This suggests that the selfassembly of DNAbased tiles could be used to perform DNAbased computation 8. Algorithmic aperiodic selfassembly requires greater fidelity than periodic selfassembly, because correct tiles must compete with partially correct tiles. Here we report a onedimensional algorithmic selfassembly of DNA triplecrossover molecules 9 that can be used to execute four steps of a logical (cumulative XOR) operation on a string of binary bits. A variety of different DNA tile types have been used in previous assemblies, including doublecrossover molecules 1, triplecrossover molecules 9, and parallelograms produced from Holliday junction analogues 3.
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 60 (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.
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 46 (9 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 41 (13 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
Complexities for Generalized Models of SelfAssembly
 In SODA
, 2004
"... Abstract. 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 ..."
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Cited by 38 (4 self)
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Abstract. 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.
2D DNA SelfAssembly for Satisfiability
"... . DNA selfassembly has been proposed as a way to cope with huge combinatorial NPHARD problems, such as satisfiability. However, the algorithmic designs proposed so far either involve many biosteps or are highly dependent on the particular instance to be solved. This paper presents an algorithmic d ..."
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Cited by 36 (0 self)
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. DNA selfassembly has been proposed as a way to cope with huge combinatorial NPHARD problems, such as satisfiability. However, the algorithmic designs proposed so far either involve many biosteps or are highly dependent on the particular instance to be solved. This paper presents an algorithmic design for solving satisfiability problems using twodimensional DNA selfassembly (tiling). The main driving factor in this work was the design and encoding of the algorithm in a general way that separates the algorithm from the data and minimizes the dependency on particular instances. In effect, a large amount of work and preparation can be done in advance as a batch process. In practice, it is likely that the total time for computation will be decreased significantly and laboratory procedures will be simplified. 1. The Satisfiability (SAT) Problem The Boolean Satisfiability (SAT) problem is the most well known representative of the NPHARD and NPCOMPLETE classes of problems. These probl...
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 35 (4 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 30 (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
Strict selfassembly of discrete Sierpinski triangles
 Proceedings of The Third Conference on Computability in Europe
"... Winfree (1998) showed that discrete Sierpinski triangles can selfassemble in the Tile Assembly Model. A striking molecular realization of this selfassembly, using DNA tiles a few nanometers long and verifying the results by atomicforce microscopy, was achieved by Rothemund, Papadakis, and Winfree ..."
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Cited by 24 (11 self)
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Winfree (1998) showed that discrete Sierpinski triangles can selfassemble in the Tile Assembly Model. A striking molecular realization of this selfassembly, using DNA tiles a few nanometers long and verifying the results by atomicforce microscopy, was achieved by Rothemund, Papadakis, and Winfree (2004). Precisely speaking, the above selfassemblies tile completely filledin, twodimensional regions of the plane, with labeled subsets of these tiles representing discrete Sierpinski triangles. This paper addresses the more challenging problem of the strict selfassembly of discrete Sierpinski triangles, i.e., the task of tiling a discrete Sierpinski triangle and nothing else. We first prove that the standard discrete Sierpinski triangle cannot strictly selfassemble in the Tile Assembly Model. We then define the fibered Sierpinski triangle, a discrete Sierpinski triangle with the same fractal dimension as the standard one but with thin fibers that can carry data, and show that the fibered Sierpinski triangle strictly selfassembles in the Tile Assembly Model. In contrast with the simple XOR algorithm of the earlier, nonstrict selfassemblies, our strict selfassembly algorithm makes extensive, recursive use of optimal counters, coupled with measured delay and cornerturning operations. We verify our strict selfassembly using the local determinism method of Soloveichik and Winfree (2007). 1