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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...
Selfreplication and evolution of DNA crystals
 Advances in Artificial Life: 8th European Conference (ECAL), volume LNCS 3630
, 2005
"... I came to Caltech a scatterbrained but enthusiastic young scientist. Without the constant nurturing and tutelage of my PhD advisor, Erik Winfree, I can’t imagine what would have happened. Erik’s gifts are many – a generous spirit, stratospheric intellectual standards, a razorsharp intuition for the ..."
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Cited by 15 (7 self)
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I came to Caltech a scatterbrained but enthusiastic young scientist. Without the constant nurturing and tutelage of my PhD advisor, Erik Winfree, I can’t imagine what would have happened. Erik’s gifts are many – a generous spirit, stratospheric intellectual standards, a razorsharp intuition for the truth, and a boundless imagination. It has been a pleasure and a privilege to work with him, to hear his constant feedback on my own imperfect thoughts. I hope in the future I can honor a tiny portion of his gifts to me by teaching others. As a PhD student I have been privileged to stand on the shoulders of other both brilliant and kind intellectual giants, without whom this work would never have been. First and foremost, my thesis work owes an unpayable intellectual debt to the work of Graham CairnsSmith. His unconventional thoughts about the first life on earth were the catalyst for this work on selfreplication. I am flattered and grateful for his continued support in the form of visits, talks, and letters during his retirement. No one was more honest about the rigors of the PhD process and a life in science than Paul Rothemund. As human and as good a friend as Paul has been, he also been someone to aspire to be like. Simply, Paul is a whiz, and a big friendly intellectual giant. I am excited about everything
Toward Reliable Algorithmic SelfAssembly of DNA Tiles: A FixedWidth Cellular Automaton Pattern NANO LETTERS
, 2007
"... Bottomup fabrication of nanoscale structures relies on chemical processes to direct selfassembly. The complexity, precision, and yield achievable by a onepot reaction are limited by our ability to encode assembly instructions into the molecules themselves. Nucleic acids provide a platform for inv ..."
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Cited by 13 (1 self)
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Bottomup fabrication of nanoscale structures relies on chemical processes to direct selfassembly. The complexity, precision, and yield achievable by a onepot reaction are limited by our ability to encode assembly instructions into the molecules themselves. Nucleic acids provide a platform for investigating these issues, as molecular structure and intramolecular interactions can encode growth rules. Here, we use DNA tiles and DNA origami to grow crystals containing a cellular automaton pattern. In a onepot annealing reaction, 250 DNA strands first assemble into a set of 10 free tile types and a seed structure, then the free tiles grow algorithmically from the seed according to the automaton rules. In our experiments, crystals grew to ∼300 nm long, containing ∼300 tiles with an initial assembly error rate of ∼1.4 % per tile. This work provides evidence that programmable molecular selfassembly may be sufficient to create a wide range of complex objects in onepot reactions. The WatsonsCrick complementarity of DNA molecules allows one to design not only simple doublestranded helices but also complicated woven structures consisting of many DNA strands. 1 Welldesigned structures will selfassemble during annealing from a high initial temperature at which point all molecules are singlestranded to a lower final
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 11 (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
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
Analysis of selfcorrecting selfassembly growth models
 IN PROC. OF THE ELEVENTH INTERNATIONAL MEETING ON DNA COMPUTING
, 2005
"... In many respects, the current state of DNAbased computing resembles the state of standard, electronic computing a half century ago: a fascinating prospect is slow to develop owing to inflexible interfaces and unacceptably low reliability of the computational processes. We concentrate in this paper ..."
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Cited by 10 (1 self)
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In many respects, the current state of DNAbased computing resembles the state of standard, electronic computing a half century ago: a fascinating prospect is slow to develop owing to inflexible interfaces and unacceptably low reliability of the computational processes. We concentrate in this paper on the latter aspect,
Reducing Facet Nucleation During Algorithmic SelfAssembly
"... Abstract: Algorithmic selfassembly, a generalization of crystal growth, has been proposed as a mechanism for bottomup fabrication of complex nanostructures and autonomous DNA computation. In principle, growth can be programmed by designing a set of molecular tiles with binding interactions that en ..."
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Cited by 6 (1 self)
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Abstract: Algorithmic selfassembly, a generalization of crystal growth, has been proposed as a mechanism for bottomup fabrication of complex nanostructures and autonomous DNA computation. In principle, growth can be programmed by designing a set of molecular tiles with binding interactions that enforce assembly rules. In practice, however, errors during assembly cause undesired products, drastically reducing yields. Here we provide experimental evidence that assembly can be made more robust to errors by adding redundant tiles that “proofread ” assembly. We construct DNA tile sets for two methods, uniform and snaked proofreading. While both tile sets are predicted to reduce errors during growth, the snaked proofreading tile set is also designed to reduce nucleation errors on crystal facets. Using atomic force microscopy to image growth of proofreading tiles on ribbonlike crystals presenting long facets, we show that under the physical conditions we studied, the rate of facet nucleation is fourfold smaller for snaked proofreading tile sets than for uniform proofreading tile sets.
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 5 (2 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.
Error Correction for DNA SelfAssembly: Preventing Facet Nucleation
"... Abstract. Algorithmic selfassembly has been proposed as a mechanism for bottomup construction of nanostructures and autonomous DNA computation. For these applications, we are often interested in assembling large systems with great precision. However, several effects present in real systems result ..."
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Cited by 2 (2 self)
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Abstract. Algorithmic selfassembly has been proposed as a mechanism for bottomup construction of nanostructures and autonomous DNA computation. For these applications, we are often interested in assembling large systems with great precision. However, several effects present in real systems result in errors with respect the the abstract Tile Assembly Model used for most theoretical studies. Hence the high error rate is becoming a key issue for algorithmic selfassembly. Several error correction mechanisms have been proposed for the selfassembly of DNA tiles [7, 4, 3, 1]. The “snaked proofreading” scheme of Chen and Goel [1], which builds on the simpler proofreading scheme of Winfree and Bekbolatov [7], provides a means to prevent undesired nucleation on facets of a growing crystal. This allowed for the first provable results that arbitrarily low error rates can be achieved with little cost (under some mild assumptions). Prior to this work, none of these schemes have been experimentally demonstrated. Here, we have implemented a twobytwo snaked proofreading system, and, for comparison, a twobytwo original proofreading system. As shown in Figure 1, the snaked system is a twobytwo block composed of a double tile and two single tiles which have an inert edge between them. To create long facets (the worst case situation for error control), we use the zigzag tile set [6] implemented as DNA tiles [5] that selfassemble into long ribbons. One facet of the ribbon has sticky ends matching with side CD, and the other facet has sticky ends matching with side AB. Each snaked proofreading tile has at most one sticky end that can bind to either facet. Hence, according to the abstract Tile Assembly Model with attachment threshold τ = 2, no tiles are supposed to be
On the Complexity of Graph Selfassembly in Accretive Systems
"... Abstract. We study the complexity of the Accretive Graph Assembly Problem (AGAP). An instance of AGAP consists of an edgeweighted graph G, a seed vertex in G, and a temperature τ. The goal is to determine if there is a sequence of vertex additions which constructs G starting from the seed. The edge ..."
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Cited by 2 (1 self)
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Abstract. We study the complexity of the Accretive Graph Assembly Problem (AGAP). An instance of AGAP consists of an edgeweighted graph G, a seed vertex in G, and a temperature τ. The goal is to determine if there is a sequence of vertex additions which constructs G starting from the seed. The edge weights model the forces of attraction and repulsion, and determine which vertices can be added to a partially assembled graph at the given temperature. Our first result is that AGAP is NPcomplete even on degree 3 planar graphs when edges have only two different types of weights. This resolves the complexity of AGAP in the sense that the problem is polytime solvable when either the degree is bounded by 2 or the number of distinct edge weights is one, and is NPcomplete otherwise. Our second result is a dichotomy theorem that completely characterizes the complexity of AGAP on degree 3 bounded graphs with two distinct weights: wp,wn. Wegive a simple system of linear constraints on wp,wn, and τ that determines whether the problem is NPcomplete or is polytime solvable. In the process of establishing this dichotomy, we give the first polytime algorithm to solve a nontrivial class of AGAP. Finally, we consider the optimization version of AGAP where the goal is to realize a largestpossible subgraph of the given input graph. We show that even on constructible graphs of degree at most 3, it is NPhard to realize a (1/n 1−ɛ)fraction of the input graph for any ɛ>0; here n denotes the number of vertices in G. 1