On the Computational Power of DNA Annealing and Ligation

In [20] it was shown that the DNA primitives of Separate, Merge, and Amplify were not sufficiently powerful to invert functions defined by circuits in linear time. Dan Boneh et al [4] show that the addition of a ligation primitive, Append, provides the missing power. The question becomes, "How powerful is ligation? Are Separate, Merge, and Amplify necessary at all?" This paper proposes to informally explore the power of annealing and ligation for DNA computation. We conclude, in fact, that annealing and ligation alone are theoretically capable of universal computation

Experimental Progress in Computation by Self-Assembly of DNA Tilings

Approaches to DNA-based computing by self-assembly require the use of D. T A nanostructures, called tiles, that have efficient chemistries, expressive computational power: and convenient input and output (I/O) mechanisms. We have designed two new classes of DNA tiles: TAO and TAE, both of which contain three double-helices linked by strand exchange. Structural analysis of a TAO molecule has shown that the molecule assembles efficiently from its four component strands. Here we demonstrate a novel method for I/O whereby multiple tiles assemble around a single-stranded (input) scaffold strand. Computation by tiling theoretically results in the formation of structures that contain single-stranded (output) reported strands, which can then be isolated for subsequent steps of computation if necessary. We illustrate the advantages of TAO and TAE designs by detailing two examples of massively parallel arithmetic: construction of complete XOR and addition tables by linear assemblies of DNA tiles. The three helix structures provide flexibility for topological routing of strands in the computation: allowing the implementation of string tile models

Algorithmic Self-Assembly of DNA: Theoretical Motivations and 2D Assembly Experiments

Biology makes things far smaller and more complex than anything produced by human engineering. The biotechnology revolution has for the first time given us the tools necessary to consider engineering on the molecular level. Research in DNA computation, launched by Len Adleman, has opened the door for experimental study of programmable biochemical reactions. Here we focus on a single biochemical mechanism, the self-assembly of DNA structures, that is theoretically sufficient for Turing-universal computation. The theory combines Hao Wang?s purely mathematical Tiling Problem with the branched DNA constructions of Ned Seeman. In the context of mathematical logic, Wang showed how jigsaw-shaped tiles can be designed to simulate the operation of any Turing Machine. For a biochemical implementation, we will need molecular Wang tiles. DNA molecular structures and intermolecular interactions are particularly amenable to design and are sufficient for the creation of complex molecular objects. The structure of individual molecules can be designed by maximizing desired and minimizing undesired Watson-Crick complementarity. Intermolecular interactions are programmed by the design of sticky ends that determine which molecules associate, and how. The theory has been demonstrated experimentally using a system of synthetic DNA double-crossover molecules that self-assemble into two-dimensional crystals that have been visualized by atomic force microscopy. This experimental system provides an excellent platform for exploring the relationship between computation and molecular self-assembly, and thus represents a first step toward the ability to program molecular reactions and molecular structures

Sticker systems over monoids

Molecular computing has gained many interests among researchers since Head introduced the first theoretical model for DNA based computation using the splicing operation in 1987. Another model for DNA computing was proposed by using the sticker operation which Adlemanused in his successful experiment for the computation of Hamiltonian paths in a graph: a double stranded DNA sequence is composed by prolonging to the left and to the right a sequence of (single or double) symbols by using given single stranded strings or even more complex dominoes with sticky ends, gluing these ends together with the sticky ends of the current sequence according to a complementarity relation. According to this sticker operation, a language generative mechanism, called a sticker system, can be defined: a set of (incomplete) double-stranded sequences (axioms) and a set of pairs of single or double-stranded complementary sequences are given. The initial sequences are prolonged to the left and to the right by using sequences from the latter set, respectively. The iterations of these prolongations produce “computations” of possibly arbitrary length. These processes stop when a complete double stranded sequence is obtained. Sticker systems will generate only regular languages without restrictions. Additional restrictions can be imposed on the matching pairs of strands to obtain more powerful languages. Several types of sticker systems are shown to have the same power as regular grammars; one type is found to represent all linear languages whereas another one is proved to be able to represent any recursively enumerable language. The main aim of this research is to introduce and study sticker systems over monoids in which with each sticker operation, an element of a monoid is associated and a complete double stranded sequence is considered to be valid if the computation of the associated elements of the monoid produces the neutral element. Moreover, the sticker system over monoids is defined in this study

An information-bearing seed for nucleating algorithmic self-assembly

Self-assembly creates natural mineral, chemical, and biological structures of great complexity. Often, the same starting materials have the potential to form an infinite variety of distinct structures; information in a seed molecule can determine which form is grown as well as where and when. These phenomena can be exploited to program the growth of complex supramolecular structures, as demonstrated by the algorithmic self-assembly of DNA tiles. However, the lack of effective seeds has limited the reliability and yield of algorithmic crystals. Here, we present a programmable DNA origami seed that can display up to 32 distinct binding sites and demonstrate the use of seeds to nucleate three types of algorithmic crystals. In the simplest case, the starting materials are a set of tiles that can form crystalline ribbons of any width; the seed directs assembly of a chosen width with >90% yield. Increased structural diversity is obtained by using tiles that copy a binary string from layer to layer; the seed specifies the initial string and triggers growth under near-optimal conditions where the bit copying error rate is 17 kb of sequence information. In sum, this work demonstrates how DNA origami seeds enable the easy, high-yield, low-error-rate growth of algorithmic crystals as a route toward programmable bottom-up fabrication

Two computational primitives for algorithmic self-assembly: Copying and counting

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

Toward reliable algorithmic self-assembly of DNA tiles: A fixed-width cellular automaton pattern

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