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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 93 (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.
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 57 (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 (") 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 56 (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...
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 49 (16 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
Computation with finite stochastic chemical reaction networks
 Natural Computing
, 2008
"... Abstract. A highly desired part of the synthetic biology toolbox is an embedded chemical microcontroller, capable of autonomously following a logic program specified by a set of instructions, and interacting with its cellular environment. Strategies for incorporating logic in aqueous chemistry have ..."
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Cited by 49 (14 self)
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Abstract. A highly desired part of the synthetic biology toolbox is an embedded chemical microcontroller, capable of autonomously following a logic program specified by a set of instructions, and interacting with its cellular environment. Strategies for incorporating logic in aqueous chemistry have focused primarily on implementing components, such as logic gates, that are composed into larger circuits, with each logic gate in the circuit corresponding to one or more molecular species. With this paradigm, designing and producing new molecular species is necessary to perform larger computations. An alternative approach begins by noticing that chemical systems on the small scale are fundamentally discrete and stochastic. In particular, the exact molecular counts of each molecular species present, is an intrinsically available form of information. This might appear to be a very weak form of information, perhaps quite difficult for computations to utilize. Indeed, it has been shown that errorfree Turing universal computation is impossible in this setting. Nevertheless, we show a design of a chemical computer that achieves fast and reliable Turinguniversal computation using molecular counts. Our scheme uses only a small number of different molecular species to do computation of arbitrary complexity. The total probability of error of the computation can be made arbitrarily small (but not zero) by adjusting the initial molecular counts of certain species. While physical implementations would be difficult, these results demonstrate that molecular counts can be a useful form of information for small molecular systems such as those operating within cellular environments. Key words. stochastic chemical kinetics; molecular counts; Turinguniversal computation; probabilistic computation 1. Introduction. Many
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 43 (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.
The Many Facets of Natural Computing
"... related. I am confident that at their interface great discoveries await those who seek them. ” (L.Adleman, [3]) 1. FOREWORD Natural computing is the field of research that investigates models and computational techniques inspired by nature and, dually, attempts to understand the world around us in t ..."
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Cited by 39 (2 self)
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related. I am confident that at their interface great discoveries await those who seek them. ” (L.Adleman, [3]) 1. FOREWORD Natural computing is the field of research that investigates models and computational techniques inspired by nature and, dually, attempts to understand the world around us in terms of information processing. It is a highly interdisciplinary field that connects the natural sciences with computing science, both at the level of information technology and at the level of fundamental research, [98]. As a matter of fact, natural computing areas and topics come in many flavours, including pure theoretical research, algorithms and software applications, as well as biology, chemistry and physics experimental laboratory research. In this review we describe computing paradigms abstracted
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 29 (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
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 26 (3 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
From molecular to macroscopic via the rational design of a selfassembled 3D DNA crystal. Nature 461
, 2009
"... We live in a macroscopic threedimensional world, but our best description of the structure of matter is at the atomic and molecular scale. Understanding the relationship between the two scales requires that we bridge from the molecular world to the macroscopic world. Connecting these two domains wi ..."
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Cited by 26 (1 self)
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We live in a macroscopic threedimensional world, but our best description of the structure of matter is at the atomic and molecular scale. Understanding the relationship between the two scales requires that we bridge from the molecular world to the macroscopic world. Connecting these two domains with atomic precision is a central goal of the natural sciences, but it requires high spatial control of the 3D structure of matter.1 The simplest practical route to producing precisely designed 3D macroscopic objects is to form a crystalline arrangement by selfassembly, because such a periodic array has only conceptually simple requirements: [1] A motif whose 3D structure is robust, [2] dominant affinity interactions between parts of the motif when it selfassociates, and [3] a predictable structures for these affinity interactions. Fulfilling all these criteria to produce a 3D periodic system is not easy, but it should readily be achieved by wellstructured branched DNA motifs tailed by sticky ends.2 Complementary sticky ends associate with each other preferentially and assume the wellknown BDNA structure when they do so;3 the helically repeating nature of DNA facilitates the construction of a periodic array. It is key that the directions of propagation associated with the sticky ends not share the same plane, but extend to form a 3D