Know When to Fold ’Em: Self-assembly of Shapes by Folding in Oritatami

Demaine, Erik D.; Hendricks, Jacob; Olsen, Meagan; Patitz, Matthew J.; Rogers, Trent A.; Schabanel, Nicolas; Seki, Shinnosuke; Thomas, Hadley

doi:10.1007/978-3-030-00030-1_2

Cited by 16 publications

(9 citation statements)

References 24 publications

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“…Our binary counter is described for the inertial dynamics in Section 3. Many subsequent articles [38,39,40,41,42,43,44,45,46] chose the oblivious dynamics. Masuda et al.…”

Section: Model and Main Resultsmentioning

confidence: 99%

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Oritatami: A Computational Model for Molecular Co-Transcriptional Folding

Geary

Meunier

Schabanel

et al. 2019

IJMS

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We introduce and study the computational power of Oritatami, a theoretical model that explores greedy molecular folding, whereby a molecular strand begins to fold before its production is complete. This model is inspired by our recent experimental work demonstrating the construction of shapes at the nanoscale from RNA, where strands of RNA fold into programmable shapes during their transcription from an engineered sequence of synthetic DNA. In the model of Oritatami, we explore the process of folding a single-strand bit by bit in such a way that the final fold emerges as a space-time diagram of computation. One major requirement in order to compute within this model is the ability to program a single sequence to fold into different shapes dependent on the state of the surrounding inputs. Another challenge is to embed all of the computing components within a contiguous strand, and in such a way that different fold patterns of the same strand perform different functions of computation. Here, we introduce general design techniques to solve these challenges in the Oritatami model. Our main result in this direction is the demonstration of a periodic Oritatami system that folds upon itself algorithmically into a prescribed set of shapes, depending on its current local environment, and whose final folding displays the sequence of binary integers from 0 to N = 2 k − 1 with a seed of size O ( k ) . We prove that designing Oritatami is NP-hard in the number of possible local environments for the folding. Nevertheless, we provide an efficient algorithm, linear in the length of the sequence, that solves the Oritatami design problem when the number of local environments is a small fixed constant. This shows that this problem is in fact fixed parameter tractable (FPT) and can thus be solved in practice efficiently. We hope that the numerous structural strategies employed in Oritatami enabling computation will inspire new architectures for computing in RNA that take advantage of the rapid kinetic-folding of RNA.

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“…Our binary counter is described for the inertial dynamics in Section 3. Many subsequent articles [38,39,40,41,42,43,44,45,46] chose the oblivious dynamics. Masuda et al.…”

Section: Model and Main Resultsmentioning

confidence: 99%

“…Demaine et al. [39] and Han and Kim [40] independently conducted more comprehensive studies of the shape self-assembly by oritatami systems recently. In particular, Demaine et al.…”

Section: Introductionmentioning

confidence: 99%

Oritatami: A Computational Model for Molecular Co-Transcriptional Folding

Geary

Meunier

Schabanel

et al. 2019

IJMS

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“…There are a number related autonomous self-folding models, both 1D to 2D (Cheung et al 2011) and 2D to 3D (Connelly et al 2010), and reconfigurable robotic/programmable matter systems, e.g. (Aloupis et al 2009(Aloupis et al , 2008Demaine et al 2018;Geary et al 2016;Gmyr et al 2019;Michail et al 2019).…”

Section: Related and Future Workmentioning

confidence: 99%

Turning machines: a simple algorithmic model for molecular robotics

2022

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Molecular robotics is challenging, so it seems best to keep it simple. We consider an abstract molecular robotics model based on simple folding instructions that execute asynchronously. Turning Machines are a simple 1D to 2D folding model, also easily generalisable to 2D to 3D folding. A Turning Machine starts out as a line of connected monomers in the discrete plane, each with an associated turning number. A monomer turns relative to its neighbours, executing a unit-distance translation that drags other monomers along with it, and through collective motion the initial set of monomers eventually folds into a programmed shape. We provide a suite of tools for reasoning about Turning Machines by fully characterising their ability to execute line rotations: executing an almost-full line rotation of $$5\pi /3$$ 5 π / 3 radians is possible, yet a full $$2\pi$$ 2 π rotation is impossible. Furthermore, line rotations up to $$5\pi /3$$ 5 π / 3 are executed efficiently, in $$O(\log n)$$ O ( log n ) expected time in our continuous time Markov chain time model. We then show that such line-rotations represent a fundamental primitive in the model, by using them to efficiently and asynchronously fold shapes. In particular, arbitrarily large zig-zag-rastered squares and zig-zag paths are foldable, as are y-monotone shapes albeit with error (bounded by perimeter length). Finally, we give shapes that despite having paths that traverse all their points, are in fact impossible to fold, as well as techniques for folding certain classes of (scaled) shapes without error. Our approach relies on careful geometric-based analyses of the feats possible and impossible by a very simple robotic system, and pushes conceptional hardness towards mathematical analysis and away from molecular implementation.

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“…Its seed, colored in red, can be elongated by the first three beads w[1..3] = 579−580−581 in various ways, only three of which are shown in Figure 2 (left). The rule set R allows w [1] to be bound to 584, w [2] to 589, and w [3] to 588, but 584-bead is not around. In order for both w [2] and w [3] to be thus bound, the nascent fragment w[1..3] must be folded as bolded in Figure 2 (left).…”

Section: Preliminariesmentioning

confidence: 99%

“…Some of these modules do computation by folding into different shapes, which resembles somehow computation by cotranscriptional folding in nature [9]. Being thus motivated, the study of cotranscriptional folding of shapes in oritatami was initiated by Masuda, Seki, and Ubukata in [8] and extended independently by Domaine et al [2] as well as by Han and Kim [7] further. In [8], an arbitrary finite portion of the Heighway dragon fractal was folded by an oritatami system Ξ H .…”

Section: Introductionmentioning

confidence: 99%

A General Architecture of Oritatami Systems for Simulating Arbitrary Finite Automata

Han

Kim

Masuda

et al. 2019

Implementation and Application of Automata

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In this paper, we propose an architecture of oritatami systems with which one can simulate an arbitrary nondeterministic finite automaton (NFA) in a unified manner. The oritatami system is known to be Turinguniversal but the simulation available so far requires 542 bead types and O(t 4 log 2 t) steps in order to simulate t steps of a Turing machine. The architecture we propose employs only 329 bead types and requires just O(t|Q| 4 |Σ| 2 ) steps to simulate an NFA over an input alphabet Σ with a state set Q working on a word of length t.

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Know When to Fold ’Em: Self-assembly of Shapes by Folding in Oritatami

Cited by 16 publications

References 24 publications

Oritatami: A Computational Model for Molecular Co-Transcriptional Folding

Oritatami: A Computational Model for Molecular Co-Transcriptional Folding

Turning machines: a simple algorithmic model for molecular robotics

A General Architecture of Oritatami Systems for Simulating Arbitrary Finite Automata

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