Copying and counting are useful primitive operations for computation and construction. We have made DNA crystals that copy and crystals that count as they grow. For counting, 16 oligonucleotides assemble into four DNA Wang tiles that subsequently crystallize on a polymeric nucleating scaffold strand, arranging themselves in a binary counting pattern that could serve as a template for a molecular electronic demultiplexing circuit. Although the yield of counting crystals is low, and per-tile error rates in such crystals is roughly 10%, this work demonstrates the potential of algorithmic self-assembly to create complex nanoscale patterns of technological interest. A subset of the tiles for counting form information-bearing DNA tubes that copy bit strings from layer to layer along their length. The challenge of engineering complex devices at the nanometer scale has been approached from two radically different directions. In top-down synthesis, information about the desired structure is imposed by an external apparatus, as in photolithography. In bottom-up synthesis, structure arises spontaneously due to chemical and physical forces intrinsic to the molecular components themselves. A significant challenge for bottom-up techniques is how to design

Abstract. Fault-tolerance is a critical issue for biochemical computation. Recent theoretical work on algorithmic self-assembly 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

Summary. Molecular self-assembly appears to be a promising route to bottom-up 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 self-assembly 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 self-assembly process. Here we study the ability of algorithmic self-assembly to heal damage to a self-assembled object. We present block transforms that convert an original error-prone 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

Abstract. Self-assembly 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 nano-scale structures. Recent years see convergent interest and efforts in studying self-assembly from mathematicians, computer scientists, physicists, chemists, and biologists. However most complexity theoretic studies of self-assembly 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 self-assembled structures than the previous tiling model. We define two novel assembly models, namely the accretive graph assembly model and the self-destructible 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 Self-Destructible 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 self-assembly). 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

Bottom-up fabrication of nanoscale structures relies on chemical processes to direct self-assembly. The complexity, precision, and yield achievable by a one-pot 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 one-pot 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 self-assembly may be sufficient to create a wide range of complex objects in one-pot reactions. The WatsonsCrick complementarity of DNA molecules allows one to design not only simple double-stranded helices but also complicated woven structures consisting of many DNA strands. 1 Well-designed structures will self-assemble during annealing from a high initial temperature at which point all molecules are single-stranded to a lower final

Winfree’s pioneering work led the foundations in the area of errorreduction in algorithmic self-assembly (Winfree and Bekbolatov in DNA Based Computers 9, LNCS, vol. 2943, pp. 126–144, 2004), but the construction resulted in increase of the size of assembly. Reif et al. (Nanotechnol. Sci. Comput. 79–103, 2006) contributed further in this area with compact error-resilient schemes that maintained the original size of the assemblies, but required certain restrictions on the Boolean functions to be used in the algorithmic self-assembly. It is a critical challenge to improve these compact error resilient schemes to incorporate arbitrary Boolean functions, and to determine how far these prior results can be extended under different degrees of restrictions on the Boolean functions. In this work we present a considerably more complete theory of compact error-resilient schemes for algorithmic selfassembly in two and three dimensions. In our error model, ɛ is defined to be the probability that there is a mismatch between the neighboring sides of two juxtaposed tiles and they still stay together in the equilibrium. This probability is independent of any other match or mismatch and hence we term this probabilistic model as the

Abstract. Molecular self-assembly is a promising approach to bottom-up fabrication of complex structures. A major impediment to the practical use of self-assembly to create complex structures is the high rate of error under existing experimental conditions. Recent theoretical work on algorithmic self-assembly 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 self-healing 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.

Tile systems have proved to be a useful model for understanding self-assembly at the nano scale. Self-healing tile systems, introduced by Winfree, have the property that the self-assembled shape can recover from the loss of arbitrarily many tiles, provided the seed tile of the assembly remains intact and the assembly remains connected. In this paper, we present improved self-healing tile systems for the self-assembly of several interesting classes of shapes, including counters, squares, and Turing-computable shapes. Our tile systems can recover from the loss of arbitrary many tiles, including the seed, provided that a large enough fragment (logarithmic in the size of the desired assembly for the case of counters and squares) is left intact. Molecular self-assembly has emerged as an important tool for nano-scale fabrication, molecular computation, crystallography, and nano-machines. On the experimental side, the combinatorial nature of DNA molecules, their relatively simple geometry (compared to smaller molecules such as the protein), and the existence of well developed laboratory techniques for creating and manipulating DNA molecules have made DNA an

Abstract. Algorithmic self-assembly has been proposed as a mechanism for bottom-up 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 self-assembly. Several error correction mechanisms have been proposed for the self-assembly 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 twoby-two snaked proofreading system, and, for comparison, a two-by-two original proofreading system. As shown in Figure 1, the snaked system is a two-by-two 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 zig-zag tile set [6] implemented as DNA tiles [5] that self-assemble 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

Abstract. We propose a self-assembly 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 self-replication in the tile assembly. We then study the tile complexity for assembling shapes in our model and show that a thin rectangle of ¤¦¥¨ § size can be assembled ©¦�� � ����� using types of tiles.