Origami fold as algebraic graph rewriting

AbstractWe formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewriting system (O,↬), where O is the set of abstract origamis and ↬ is a binary relation on O, that models fold. An abstract origami is a structure (Π,∽,≻), where Π is a set of faces constituting an origami, and ∽ and ≻ are binary relations on Π, each representing adjacency and superposition relations between the faces.We then address representation and transformation of abstract origamis and further reasoning about the construction for computational purposes. We present a labeled hypergraph of origami and define fold as algebraic graph transformation. The algebraic graph-theoretic formalism enables us to reason about origami in two separate domains of discourse, i.e. pure combinatorial domain where symbolic computation plays the main role and geometrical domain R×R. We detail the program language for the algebraic graph rewriting and graph rewriting algorithms for the fold, and show how fold is expressed by a set of graph rewrite rules

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