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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 4, a process that em ..."
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Cited by 81 (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.
Local parallel biomolecular computing
 DNA Based Computers III, volume 48 of DIMACS
, 1999
"... Biomolecular Computation(BMC) is computation at the molecular scale, using biotechnology engineering techniques. Most proposed methods for BMC used distributed (molecular) parallelism (DP); where operations are executed in parallel on large numbers of distinct molecules. BMC done exclusively by DP r ..."
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Cited by 51 (15 self)
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Biomolecular Computation(BMC) is computation at the molecular scale, using biotechnology engineering techniques. Most proposed methods for BMC used distributed (molecular) parallelism (DP); where operations are executed in parallel on large numbers of distinct molecules. BMC done exclusively by DP requires that the computation execute sequentially within any given molecule (though done in parallel for multiple molecules). In contrast, local parallelism (LP) allows operations to be executed in parallel on each given molecule. Winfree, et al [W96, WYS96]) proposed an innovative method for LPBMC, that of computation by unmediated selfassembly of � arrays of DNA molecules, applying known domino tiling techniques (see Buchi [B62], Berger [B66], Robinson [R71], and Lewis and Papadimitriou [LP81]) in combination with the DNA selfassembly techniques of Seeman et al [SZC94]. The likelihood for successful unmediated selfassembly of computations has not been determined (we discuss a simple model of assembly where there may be blockages in selfassembly, but more sophisticated models may have a higher likelihood of success). We develop improved techniques to more fully exploit the potential power of LPBMC. To increase
Proofreading tile sets: Error correction for algorithmic selfassembly
 In DNA Based Computers 9, volume 2943 of LNCS
, 2004
"... 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 (ε) can be reduced to an arbitrarily low ..."
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Cited by 47 (10 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 (ε) 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 ≈ βε 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 ≈ βε 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 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...
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.
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 33 (3 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
Programmable control of nucleation for algorithmic selfassembly
 in DNA Computing 10, Lecture Notes in Comput. Sci. 3384
, 2005
"... Abstract. 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 ..."
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Cited by 28 (10 self)
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Abstract. 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.
Experimental Progress in Computation by SelfAssembly of DNA Tilings
, 1999
"... Approaches to DNAbased computing by selfassembly require the use of DNA nanostructures, called tiles, that have efficient chemistries, expressive computational power, and convenient input and output (I/O) mechanisms. We have designed two new classes of DNA tiles, TAO and TAE, both of which contain ..."
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Cited by 27 (14 self)
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Approaches to DNAbased computing by selfassembly require the use of DNA nanostructures, called tiles, that have efficient chemistries, expressive computational power, and convenient input and output (I/O) mechanisms. We have designed two new classes of DNA tiles, TAO and TAE, both of which contain three doublehelices linked by strand exchange. Structural analysis of a TAO molecule has shown that the molecule assembles efficiently from its four component strands. Here we demonstrate a novel method for I/O whereby multiple tiles assemble around a singlestranded (input) scaffold strand. Computation by tiling theoretically results in the formation of structures that contain singlestranded (output) reported strands, which can then be isolated for subsequent steps of computation if necessary. We illustrate the advantages of TAO and TAE designs by detailing two examples of massively parallel arithmetic: construction of complete XOR and addition tables by linear assemblies of DNA t...
Paradigms for computational nucleic acid design
 Nucleic Acids Res
"... The design of DNA and RNA sequences is critical for many endeavors, from DNA nanotechnology, to PCRbased applications, to DNA hybridization arrays. Results in the literature rely on a wide variety of design criteria adapted to the particular requirements of each application. Using an extensivelyst ..."
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Cited by 26 (3 self)
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The design of DNA and RNA sequences is critical for many endeavors, from DNA nanotechnology, to PCRbased applications, to DNA hybridization arrays. Results in the literature rely on a wide variety of design criteria adapted to the particular requirements of each application. Using an extensivelystudied thermodynamic model, we perform a detailed study of several criteria for designing sequences intended to adopt a target secondary structure. We conclude that superior design methods should explicitly implement both a positive design paradigm (optimize affinity for the target structure) and a negative design paradigm (optimize specificity for the target structure). The commonly used approaches of sequence symmetry minimization and minimum free energy satisfaction primarily implement negative design and can be strengthened by introducing a positive design component. Surprisingly, our findings hold for a wide range of secondary structures and are robust to modest perturbation of the thermodynamic parameters used for evaluating sequence quality, suggesting the feasibility and ongoing utility of a unified approach to nucleic acid design as parameter sets are further refined. Finally, we observe that designing for thermodynamic stability does not determine folding kinetics, emphasizing the opportunity for extending design criteria to target kinetic features of the energy landscape.