One Tile to Rule Them All: Simulating Any Tile Assembly System with a Single Universal Tile

In the classical model of tile self-assembly, unit square tiles translate in the plane and attach edgewise to form large crystalline structures. This model of self-assembly has been shown to be capable of asymptotically optimal assembly of arbitrary shapes and, via information-theoretic arguments, increasingly complex shapes necessarily require increasing numbers of distinct types of tiles. We explore the possibility of complex and efficient assembly using systems consisting of a single tile. Our main result shows that any system of square tiles can be simulated using a system with a single tile that is permitted to flip and rotate. We also show that systems of single tiles restricted to translation only can simulate cellular automata for a limited number of steps given an appropriate seed assembly, and that any longer-running simulation must induce infinite assembly

Self-replication and evolution of DNA crystals

Is it possible to create a simple physical system that is capable of replicating itself? Can such a system evolve interesting behaviors, thus allowing it to adapt to a wide range of environments? This paper presents a design for such a replicator constructed exclusively from synthetic DNA. The basis for the replicator is crystal growth: information is stored in the spatial arrangement of monomers and copied from layer to layer by templating. Replication is achieved by fragmentation of crystals, which produces new crystals that carry the same information. Crystal replication avoids intrinsic problems associated with template-directed mechanisms for replication of one-dimensional polymers. A key innovation of our work is that by using programmable DNA tiles as the crystal monomers, we can design crystal growth processes that apply interesting selective pressures to the evolving sequences. While evolution requires that copying occur with high accuracy, we show how to adapt error-correction techniques from algorithmic self-assembly to lower the replication error rate as much as is required

The Power of Duples (in Self-Assembly): It's Not So Hip To Be Square

In this paper we define the Dupled abstract Tile Assembly Model (DaTAM), which is a slight extension to the abstract Tile Assembly Model (aTAM) that allows for not only the standard square tiles, but also "duple" tiles which are rectangles pre-formed by the joining of two square tiles. We show that the addition of duples allows for powerful behaviors of self-assembling systems at temperature 1, meaning systems which exclude the requirement of cooperative binding by tiles (i.e., the requirement that a tile must be able to bind to at least 2 tiles in an existing assembly if it is to attach). Cooperative binding is conjectured to be required in the standard aTAM for Turing universal computation and the efficient self-assembly of shapes, but we show that in the DaTAM these behaviors can in fact be exhibited at temperature 1. We then show that the DaTAM doesn't provide asymptotic improvements over the aTAM in its ability to efficiently build thin rectangles. Finally, we present a series of results which prove that the temperature-2 aTAM and temperature-1 DaTAM have mutually exclusive powers. That is, each is able to self-assemble shapes that the other can't, and each has systems which cannot be simulated by the other. Beyond being of purely theoretical interest, these results have practical motivation as duples have already proven to be useful in laboratory implementations of DNA-based tiles

Program Size and Temperature in Self-Assembly

Winfree’s abstract Tile Assembly Model (aTAM) is a model of molecular self-assembly of DNA complexes known as tiles, which float freely in solution and attach one at a time to a growing “seed ” assembly based on specific binding sites on their four sides. We show that there is a polynomial-time algorithm that, given an n × n square, finds the minimal tile system (i.e., the system with the smallest number of distinct tile types) that uniquely self-assembles the square, answering an open question of Adleman, Cheng, Goel, Huang, Kempe, Moisset de Espanés, and Rothemund (Combinatorial Optimization Problems in Self-Assembly, STOC 2002). Our investigation leading to this algorithm reveals other positive and negative results about the relationship between the size of a tile system and its “temperature ” (the binding strength threshold required for a tile to attach)

Triangular and Hexagonal Tile Self-assembly Systems

Abstract. We discuss theoretical aspects of the self-assembly of triangular tiles, in particular, right triangular tiles and equilateral triangular tiles, and the self-assembly of hexagonal tiles. We show that triangular tile assembly systems and square tile assembly systems cannot be simulated by each other in a non-trivial way. More precisely, there exists a deterministic square (hexagonal) tile assembly system S such that no deterministic triangular tile assembly system that is a division of S produces an equivalent supertile (of the same shape and same border glues). There also exists a deterministic triangular tile assembly system T such that no deterministic square (hexagonal) tile assembly system produces the same final supertile while preserving border glues.

Programmable control of nucleation for algorithmic self-assembly

Abstract. Algorithmic self-assembly has been proposed as a mechanism for autonomous DNA computation and for bottom-up fabrication of complex nanodevices. Whereas much previous work has investigated self-assembly programs using an abstract model of irreversible, errorless assembly, experimental studies as well as more sophisticated reversible kinetic models indicate that algorithmic self-assembly is subject to several kinds of errors. Previously, it was shown that proofreading tile sets can reduce the occurrence of mismatch and facet errors. Here, we introduce the zig-zag tile set, which can reduce the occurrence of spurious nucleation errors. The zig-zag tile set takes advantage of the fact that assemblies must reach a critical size before their growth becomes favorable. By using a zig-zag tile set of greater width, we can increase the critical size of spurious assemblies without increasing the critical size of correctly seeded assemblies, exponentially reducing the spurious nucleation rate. In combination with proofreading results, this result indicates that algorithmic self-assembly can be performed with low error rates without a significant reduction in assembly speed. Furthermore, our zig-zag boundaries suggest methods for exquisite detection of DNA strands and for the replication of inheritable information without the use of enzymes.

Simple Evolution of Complex Crystal Species

Cairns-Smith has proposed that life began as structural patterns in clays that self-replicated during cycles of crystal growth and fragmentation. Complex, evolved crystal forms could then have catalyzed the formation of a more advanced genetic material. A crucial weakness of this theory is that it is unclear how complex crystals might arise through Darwinian selection. Here we investigate whether complex crystal patterns could evolve using a model system for crystal growth, DNA tile crystals, that is amenable to both theoretical and experimental inquiry. It was previously shown that in principle, the evolution of crystals assembled from a set of thousands of DNA tiles under very specific environmental conditions could produce arbitrarily complex patterns. Here we show that evolution driven only by the dearth of one monomer type could produce complex crystals from just 12 monomer types. The proposed mechanism of evolution is simple enough to test experimentally and is sufficiently general that it may apply to other DNA tile crystals or even to natural crystals, suggesting that complex crystals could evolve from simple starting materials because of relative differences in concentrations of the materials needed for growth

Complexity of compact proofreading for self-assembled patterns

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.