Know When to Fold 'Em: Self-Assembly of Shapes by Folding in Oritatami

• Published in: arXiv.org 2018-07
• Main Authors: Demaine, Erik D, Hendricks, Jacob, Olsen, Meagan, Patitz, Matthew J, Rogers, Trent A, Schabanel, Nicolas, Seki, Shinnosuke, Hadley, Thomas
• Format: Article
• Language: English
• Subjects: Beads; Delay; Folding; Self-assembly
• ISSN: 2331-8422

An oritatami system (OS) is a theoretical model of self-assembly via co-transcriptional folding. It consists of a growing chain of beads which can form bonds with each other as they are transcribed. During the transcription process, the \(\delta\) most recently produced beads dynamically fold so as to maximize the number of bonds formed, self-assemblying into a shape incrementally. The parameter \(\delta\) is called the delay and is related to the transcription rate in nature. This article initiates the study of shape self-assembly using oritatami. A shape is a connected set of points in the triangular lattice. We first show that oritatami systems differ fundamentally from tile-assembly systems by exhibiting a family of infinite shapes that can be tile-assembled but cannot be folded by any OS. As it is NP-hard in general to determine whether there is an OS that folds into (self-assembles) a given finite shape, we explore the folding of upscaled versions of finite shapes. We show that any shape can be folded from a constant size seed, at any scale n >= 3, by an OS with delay 1. We also show that any shape can be folded at the smaller scale 2 by an OS with unbounded delay. This leads us to investigate the influence of delay and to prove that, for all {\delta} > 2, there are shapes that can be folded (at scale 1) with delay {\delta} but not with delay {\delta}'