Abstract
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 \(n\geqslant 3\), 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 \(\delta '<\delta \).
These results serve as a foundation for the study of shape-building in this new model of self-assembly, and have the potential to provide better understanding of cotranscriptional folding in biology, as well as improved abilities of experimentalists to design artificial systems that self-assemble via this complex dynamical process.
M. J. Patitz and T. A. Rogers—Supported in part by NSF Grant CCF-1422152 and CAREER-1553166.
N. Schabanel—Supported by Moprexprogmol CNRS MI grant.
S. Seki—Supported in part by JST Program to Disseminate Tenure Tracking System, MEXT, Japan, No. 6F36, JSPS Grant-in-Aid for Young Scientists (A) No. 16H05854, and JSPS Bilateral Program No. YB29004.
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Notes
- 1.
The triangular lattice is defined as \(\mathbb {T} = (\mathbb {Z}^2, \sim )\), where \((x, y) \sim (u, v)\) if and only if \((u, v) \in \cup _{\epsilon = \pm 1}\{{(x+\epsilon , y)}, {(x, y+\epsilon )}, {(x+\epsilon , y+\epsilon )}\}\). Every position (x, y) in \({\mathbb {T}}\) is mapped in the euclidean plane to \(x\cdot X + y \cdot Y\) using the vector basis \(X = (1,0)\) and \(Y = {\text {RotateClockwise}}\left( X,120^\circ \right) = (-\frac{1}{2}, -\frac{\sqrt{3}}{2})\).
- 2.
Our app Scary Pacman can be freely downloaded from the app store at https://apple.co/2qP9aCX and its source code can be downloaded and compiled from the public Darcs repository at https://bit.ly/2qQjzy6.
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Demaine, E.D. et al. (2018). Know When to Fold ’Em: Self-assembly of Shapes by Folding in Oritatami. In: Doty, D., Dietz, H. (eds) DNA Computing and Molecular Programming. DNA 2018. Lecture Notes in Computer Science(), vol 11145. Springer, Cham. https://doi.org/10.1007/978-3-030-00030-1_2
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