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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.
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 19 (4 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
Complexity of graph selfassembly in accretive systems and selfdestructible systems
 In Proc. 11th International Meeting on DNA Computing
, 2005
"... Abstract. Selfassembly is a process in which small objects autonomously associate with each other to form larger complexes. It is ubiquitous in biological constructions at the cellular and molecular scale and has also been identified by nanoscientists as a fundamental method for building nanoscale ..."
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Cited by 10 (2 self)
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Abstract. Selfassembly is a process in which small objects autonomously associate with each other to form larger complexes. It is ubiquitous in biological constructions at the cellular and molecular scale and has also been identified by nanoscientists as a fundamental method for building nanoscale structures. Recent years see convergent interest and efforts in studying selfassembly from mathematicians, computer scientists, physicists, chemists, and biologists. However most complexity theoretic studies of selfassembly utilize mathematical models with two limitations: 1) only attraction, while no repulsion, is studied; 2) only assembled structures of two dimensional square grids are studied. In this paper, we study the complexity of the assemblies resulting from the cooperative effect of repulsion and attraction in a more general setting of graphs. This allows for the study of a more general class of selfassembled structures than the previous tiling model. We define two novel assembly models, namely the accretive graph assembly model and the selfdestructible graph assembly model, and identify one fundamental problem in them: the sequential construction of a given graph, referred to as Accretive Graph Assembly Problem (AGAP) and SelfDestructible Graph Assembly Problem (DGAP), respectively. Our main results are: (i) AGAP is ¤¦ ¥complete even if the maximum degree of the graph is restricted to 4 or the graph is restricted to be planar with maximum degree 5; (ii) counting the number of sequential assembly orderings that result in a target graph (#AGAP) is §¨ ¥complete; and (iii) DGAP is ¥�©�¥����� �complete even if the maximum degree of the graph is restricted to 6 (this is the first ¥�©�¥����¨ �complete result in selfassembly). We also extend the accretive graph assembly model to a stochastic model, and prove that determining the probability of a given assembly in this model is §� ¥complete. 1
Selfhealing tile sets
 Foundations of Nanoscience: SelfAssembled Architectures and Devices, 2005
, 2005
"... Summary. Molecular selfassembly appears to be a promising route to bottomup fabrication of complex objects. Two major obstacles are how to create structures with more interesting organization than periodic or finite arrays, and how to reduce the fraction of side products and erroneous assemblies. ..."
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Cited by 10 (1 self)
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Summary. Molecular selfassembly appears to be a promising route to bottomup fabrication of complex objects. Two major obstacles are how to create structures with more interesting organization than periodic or finite arrays, and how to reduce the fraction of side products and erroneous assemblies. Algorithmic selfassembly provides a theoretical model for investigating these questions: the growth of arbitrarily complex objects can be programmed into a set of Wang tiles, and their robustness to a variety of possible errors can be studied. The ability to program the tiles presents an alternative to directly physical or chemical means for reducing error rates, since redundant information can be stored so that errors can be detected, corrected, and/or prevented during the selfassembly process. Here we study the ability of algorithmic selfassembly to heal damage to a selfassembled object. We present block transforms that convert an original errorprone tile set into a new tile set that performs the same construction task (at a slightly larger scale) and also is able to heal damaged areas where many tiles have been removed from the assembly. 1 Algorithmic Crystal Growth
Shape Replication through SelfAssembly and RNase Enzymes
"... We introduce the problem of shape replication in the Wang tile selfassembly model. Given an input shape, we consider the problem of designing a selfassembly system which will replicate that shape into either a specific number of copies, or an unbounded number of copies. Motivated by practical DNA ..."
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Cited by 8 (3 self)
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We introduce the problem of shape replication in the Wang tile selfassembly model. Given an input shape, we consider the problem of designing a selfassembly system which will replicate that shape into either a specific number of copies, or an unbounded number of copies. Motivated by practical DNA implementations of Wang tiles, we consider a model in which tiles consisting of DNA or RNA can be dynamically added in a sequence of stages. We further permit the addition of RNase enzymes capable of disintegrating RNA tiles. Under this model, we show that arbitrary genus0 shapes can be replicated infinitely many times using only O(1) distinct tile types and O(1) stages. Further, we show how to replicate precisely n copies of a shape using O(log n) stages and O(1) tile types. 1
How Crystals that Sense and Respond to Their Environments Could Evolve
"... Abstract. An enduring mystery in biology is how a physical entity simple enough to have arisen spontaneously could have evolved into the complex life seen on Earth today. CairnsSmith has proposed that life might have originated in clays which stored genomes consisting of an arrangement of crystal m ..."
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Cited by 6 (3 self)
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Abstract. An enduring mystery in biology is how a physical entity simple enough to have arisen spontaneously could have evolved into the complex life seen on Earth today. CairnsSmith has proposed that life might have originated in clays which stored genomes consisting of an arrangement of crystal monomers that was replicated during growth. While a clay genome is simple enough to have conceivably arisen spontaneously, it is not obvious how it might have produced more complex forms as a result of evolution. Here, we examine this possibility in the tile assembly model, a generalized model of crystal growth that has been used to study the selfassembly of DNA tiles. We describe hypothetical crystals for which evolution of complex crystal sequences is driven by the scarceness of resources required for growth. We show how, under certain circumstances, crystal growth that performs computation can predict which resources are abundant. In such cases, crystals executing programs that make these predictions most accurately will grow fastest. Since crystals can perform universal computation, the complexity of computation that can be used to optimize growth is unbounded. To the extent that lessons derived from the tile assembly model might be applicable to mineral crystals, our results suggest that resource scarcity could conceivably have provided the evolutionary pressures necessary to produce complex clay genomes that sense and respond to changes in their environment. 1
Temperature 1 SelfAssembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D
"... We investigate the power of the Wang tile selfassembly model at temperature 1, a threshold value that permits attachment between any two tiles that share even a single bond. When restricted to deterministic assembly in the plane, no temperature 1 assembly system has been shown to build a shape with ..."
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Cited by 5 (1 self)
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We investigate the power of the Wang tile selfassembly model at temperature 1, a threshold value that permits attachment between any two tiles that share even a single bond. When restricted to deterministic assembly in the plane, no temperature 1 assembly system has been shown to build a shape with a tile complexity smaller than the diameter of the shape. In contrast, we show that temperature 1 selfassembly in 3 dimensions, even when growth is restricted to at most 1 step into the third dimension, is capable of simulating a large class of temperature 2 systems, in turn permitting the simulation of arbitrary Turing machines and the assembly of n × n squares in near optimal O(log n) tile complexity. Further, we consider temperature 1 probabilistic assembly in 2D, and show that with a logarithmic scale up of tile complexity and shape scale, the same general class of temperature τ = 2 systems can be simulated with high probability, yielding Turing machine simulation and O(log 2 n) assembly of n × n squares with high probability. Our results show a sharp contrast in achievable tile complexity at temperature 1 if either growth into the third dimension or a small probability of error are permitted. Motivated by applications in nanotechnology and molecular computing, and the plausibility of implementing 3 dimensional selfassembly systems, our techniques may provide the needed power of temperature 2 systems, while at the same time avoiding the experimental challenges faced by those systems.
COMBINING SELFHEALING AND PROOFREADING IN SELFASSEMBLY
"... Abstract. Molecular selfassembly is a promising approach to bottomup fabrication of complex structures. A major impediment to the practical use of selfassembly to create complex structures is the high rate of error under existing experimental conditions. Recent theoretical work on algorithmic sel ..."
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Cited by 3 (0 self)
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Abstract. Molecular selfassembly is a promising approach to bottomup fabrication of complex structures. A major impediment to the practical use of selfassembly to create complex structures is the high rate of error under existing experimental conditions. Recent theoretical work on algorithmic selfassembly has shown that under a realistic model of tile addition and detachment, error correcting tile sets are possible that can recover from the attachment of incorrect tiles during the assembly process. An orthogonal type of error correction was recently considered as well: whether damage to a completed structure can be repaired. It was shown that such selfhealing tile sets are possible. However, these tile sets are not robust to the incorporation of incorrect tiles. It remained an open question whether it is possible to create tile sets that can simultaneously resist wholesale removal of tiles and the incorporation of incorrect ones. Here we present a method for converting a tile set producing a pattern on the quarter plane into a tile set that makes the same pattern (at a larger scale) but is able to withstand both of these types of errors.
A self assembly model of timedependent glue strength
 In Proc. 11th International Meeting on DNA Computing
, 2005
"... Abstract Selfassembly is a ubiquitous process in which small objects selforganize into larger and complex structures. In 2000, Rothemund and Winfree proposed a Tile Assembly Model as a mathematical model for theoretical studies of selfassembly. We propose a refined selfassembly model in which the ..."
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Cited by 3 (1 self)
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Abstract Selfassembly is a ubiquitous process in which small objects selforganize into larger and complex structures. In 2000, Rothemund and Winfree proposed a Tile Assembly Model as a mathematical model for theoretical studies of selfassembly. We propose a refined selfassembly model in which the glue strength between two juxtaposed tiles is a function of the time they have been in neighboring positions. We then present an implementation of our model using strand displacement reactions on DNA tiles. Under our model, we can demonstrate and study catalysis and selfreplication in the tile assembly. We then study the tile complexity for assembling shapes in our model and show that a thin rectangle of size k × N can be assembled using O((log(N)) / log log(N)) types of tiles, demonstrating the glue model has additional capabilities over the prior tiling assembly model. We also describe a method to implement with DNA tiles our model of timedependant glue strength.
Optimizing Tile Concentrations to Minimize Errors and Time for DNA Tile Selfassembly Systems
"... Abstract. DNA tile selfassembly has emerged as a rich and promising primitive for nanotechnology. This paper studies the problems of minimizing assembly time and error rate by changing the tile concentrations because changing the tile concentrations is easy to implement in actual lab experiments. ..."
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Cited by 2 (1 self)
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Abstract. DNA tile selfassembly has emerged as a rich and promising primitive for nanotechnology. This paper studies the problems of minimizing assembly time and error rate by changing the tile concentrations because changing the tile concentrations is easy to implement in actual lab experiments. We prove that setting the concentration of tile Ti proportional to the square root of Ni where Ni is the number of times Ti appears outside the seed structure in the final assembled shape minimizes the rate of growth errors for rectilinear tile systems. We also show that the same concentrations minimize the expected assembly time for a feasible class of tile systems. Moreover, for general tile systems, given tile concentrations, we can approximate the expected assembly time with high accuracy and probability by running only a polynomial number of simulations in the size of the target shape.