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154
Algorithmic SelfAssembly of DNA
, 1998
"... How can molecules compute? In his early studies of reversible computation, Bennett imagined an enzymatic Turing Machine which modified a heteropolymer (such as DNA) to perform computation with asymptotically low energy expenditures. Adleman's recent experimental demonstration of a DNA computation, ..."
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Cited by 104 (6 self)
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How can molecules compute? In his early studies of reversible computation, Bennett imagined an enzymatic Turing Machine which modified a heteropolymer (such as DNA) to perform computation with asymptotically low energy expenditures. Adleman's recent experimental demonstration of a DNA computation, using an entirely different approach, has led to a wealth of ideas for how to build DNAbased computers in the laboratory, whose energy efficiency, information density, and parallelism may have potential to surpass conventional electronic computers for some purposes. In this thesis, I examine one mechanism used in all designs for DNAbased computer  the selfassembly of DNA by hybridization and formation of the double helix  and show that this mechanism alone in theory can perform universal computation. To do so, I borrow an important result in the mathematical theory of tiling: Wang showed how jigsawshaped tiles can be designed to simulate the operation of any Turing Machine. I propose...
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 83 (19 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.
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 72 (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
Simulations of Computing by SelfAssembly
, 1998
"... Winfree (1996) proposed a Turinguniversal model of DNA selfassembly. In this abstract model, DNA doublecrossover molecules selfassemble to form an algorithmicallypatterned twodimensional lattice. Here, we develop a more realistic model based on the thermodynamics and kinetics of oligonucleo ..."
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Cited by 69 (15 self)
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Winfree (1996) proposed a Turinguniversal model of DNA selfassembly. In this abstract model, DNA doublecrossover molecules selfassemble to form an algorithmicallypatterned twodimensional lattice. Here, we develop a more realistic model based on the thermodynamics and kinetics of oligonucleotide hydridization. Using a computer simulation, we investigate what physical factors influence the error rates, i.e., when the more realistic model deviates from the ideal of the abstract model. We find, in agreement with rules of thumb for crystal growth, that the lowest error rates occur at the melting temperature when crystal growth is slowest, and that error rates can be made arbitrarily low by decreasing concentration and increasing binding strengths. 1 Introduction Early work in DNA computing (Adleman 1994; Lipton 1995; Boneh et al. 1996; Ouyang et al. 1997) showed how computations can be accomplished by first creating a combinatorial library of DNA and then, through successiv...
Toward a Mathematical Theory of SelfAssembly (Extended Abstract)
, 1999
"... October, 1999 Leonard M. Adleman University of Southern California Abstract Selfassembly is the ubiquitous process by which objects autonomously assemble into complexes. Nature provides many examples: Atoms react to form molecules. Molecules react to form crystals and supramolecules. Cells some ..."
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Cited by 67 (6 self)
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October, 1999 Leonard M. Adleman University of Southern California Abstract Selfassembly is the ubiquitous process by which objects autonomously assemble into complexes. Nature provides many examples: Atoms react to form molecules. Molecules react to form crystals and supramolecules. Cells sometimes coalesce to form organisms. Even heavenly bodies selfassemble into astronomical systems. It has been suggested that selfassembly will ultimately become an important technology, enabling the fabrication of great quantities of small complex objects such as computer circuits. Recent developments in DNA computing have highlighted the intimate connection between selfassembly and computation. Despite its importance, selfassembly is poorly understood. In this paper, an attempt is made to provide a basis for a mathematical theory of self assembly. A simple mathematical model of selfassembly with 'step counting' is presented and used to investigate the time complexity of polymerization. It...
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 61 (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
 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 48 (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 48 (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...
Three dimensional stochastic reconfigurations of modular robotics
 Proceedings of Robotics Science and Systems, MIT
"... Abstract — Here we introduce one simulated and two physical threedimensional stochastic modular robot systems, all capable of selfassembly and selfreconfiguration. We assume that individual units can only draw power when attached to the growing structure, and have no means of actuation. Instead t ..."
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Cited by 44 (7 self)
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Abstract — Here we introduce one simulated and two physical threedimensional stochastic modular robot systems, all capable of selfassembly and selfreconfiguration. We assume that individual units can only draw power when attached to the growing structure, and have no means of actuation. Instead they are subject to random motion induced by the surrounding medium when unattached. We present a simulation environment with a flexible scripting language that allows for parallel and serial selfassembly and selfreconfiguration processes. We explore factors that govern the rate of assembly and reconfiguration, and show that selfreconfiguration can be exploited to accelerate the assembly of a particular shape, as compared with static selfassembly. We then demonstrate the ability of two different physical threedimensional stochastic modular robot systems to selfreconfigure in a fluid. The second physical implementation is only composed of technologies that could be scaled down to achieve stochastic selfassembly and selfreconfiguration at the microscale. I.
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 43 (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