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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Algorithms play a primary role in programming an orchestrated self-assembly of shapes into molecules. In this paper, we study the algorithmic self-assembly of squares by RNA co-transcriptional folding in its oritatami model. We formalize the square self-assembly problem in oritatami and propose a universal oritatami transcript made of 939 types of abstract molecules (beads) and of period 1294 that folds deterministically and co-transcriptionally at delay 3 and maximum arity into the n × n square modulo horizontal and vertical scaling factors for all sufficiently large n’s after building a Θ(log n) width "ruler" that measures n upon the seed of size Θ(log n) on which n is encoded in binary.

Szilárd Zsolt Fazekas, Hwee Kim, Ryuichi Matsuoka, Shinnosuke Seki, and Hinano Takeuchi. On Algorithmic Self-Assembly of Squares by Co-Transcriptional Folding. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fazekas_et_al:LIPIcs.ISAAC.2022.37, author = {Fazekas, Szil\'{a}rd Zsolt and Kim, Hwee and Matsuoka, Ryuichi and Seki, Shinnosuke and Takeuchi, Hinano}, title = {{On Algorithmic Self-Assembly of Squares by Co-Transcriptional Folding}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {37:1--37:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.37}, URN = {urn:nbn:de:0030-drops-173228}, doi = {10.4230/LIPIcs.ISAAC.2022.37}, annote = {Keywords: Algorithmic molecular self-assembly, Co-transcriptional folding, Oritatami system, Self-assembly of squares} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Different models have been proposed to understand natural phenomena at the molecular scale from a computational point of view. Oritatami systems are a model of molecular co-transcriptional folding: the transcript (the "molecule") folds as it is synthesized according to a local energy optimisation process, in a similar way to how actual biomolecules such as RNA fold into complex shapes and functions. We introduce a new model, called turedo, which is a self-avoiding Turing machine on the plane that evolves by marking visited positions and that can only move to unmarked positions. Any oritatami can be seen as a particular turedo. We show that any turedo with lookup radius 1 can conversely be simulated by an oritatami, using a universal bead type set. Our notion of simulation is strong enough to preserve the geometrical and dynamical features of these models up to a constant spatio-temporal rescaling (as in intrinsic simulation). As a consequence, turedo can be used as a readable oritatami "higher-level" programming language to build readily oritatami "smart robots", using our explicit simulation result as a compiler.
As an application of our simulation result, we prove two new complexity results on the (infinite) limit configurations of oritatami systems (and radius-1 turedos), assembled from a finite seed configuration. First, we show that such limit configurations can embed any recursively enumerable set, and are thus exactly as complex as aTAM limit configurations. Second, we characterize the possible densities of occupied positions in such limit configurations: they are exactly the Π₂-computable numbers between 0 and 1. We also show that all such limit densities can be produced by one single oritatami system, just by changing the finite seed configuration.
None of these results is implied by previous constructions of oritatami embedding tag systems or 1D cellular automata, which produce only computable limit configurations with constrained density.

Daria Pchelina, Nicolas Schabanel, Shinnosuke Seki, and Guillaume Theyssier. Oritatami Systems Assemble Shapes No Less Complex Than Tile Assembly Model (ATAM). In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 51:1-51:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{pchelina_et_al:LIPIcs.STACS.2022.51, author = {Pchelina, Daria and Schabanel, Nicolas and Seki, Shinnosuke and Theyssier, Guillaume}, title = {{Oritatami Systems Assemble Shapes No Less Complex Than Tile Assembly Model (ATAM)}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {51:1--51:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.51}, URN = {urn:nbn:de:0030-drops-158618}, doi = {10.4230/LIPIcs.STACS.2022.51}, annote = {Keywords: Molecular Self-assembly, Co-transcriptional folding, Intrinsic simulation, Arithmetical hierarchy of real numbers, 2D Turing machines, Computability} }

Document

**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We study the oritatami model for molecular co-transcriptional folding. In oritatami systems, the transcript (the "molecule") folds as it is synthesized (transcribed), according to a local energy optimisation process, which is similar to how actual biomolecules such as RNA fold into complex shapes and functions as they are transcribed. We prove that there is an oritatami system embedding universal computation in the folding process itself.
Our result relies on the development of a generic toolbox, which is easily reusable for future work to design complex functions in oritatami systems. We develop "low-level" tools that allow to easily spread apart the encoding of different "functions" in the transcript, even if they are required to be applied at the same geometrical location in the folding. We build upon these low-level tools, a programming framework with increasing levels of abstraction, from encoding of instructions into the transcript to logical analysis. This framework is similar to the hardware-to-algorithm levels of abstractions in standard algorithm theory. These various levels of abstractions allow to separate the proof of correctness of the global behavior of our system, from the proof of correctness of its implementation. Thanks to this framework, we were able to computerise the proof of correctness of its implementation and produce certificates, in the form of a relatively small number of proof trees, compact and easily readable/checkable by human, while encapsulating huge case enumerations. We believe this particular type of certificates can be generalised to other discrete dynamical systems, where proofs involve large case enumerations as well.

Cody Geary, Pierre-Étienne Meunier, Nicolas Schabanel, and Shinnosuke Seki. Proving the Turing Universality of Oritatami Co-Transcriptional Folding. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 23:1-23:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{geary_et_al:LIPIcs.ISAAC.2018.23, author = {Geary, Cody and Meunier, Pierre-\'{E}tienne and Schabanel, Nicolas and Seki, Shinnosuke}, title = {{Proving the Turing Universality of Oritatami Co-Transcriptional Folding}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {23:1--23:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.23}, URN = {urn:nbn:de:0030-drops-99715}, doi = {10.4230/LIPIcs.ISAAC.2018.23}, annote = {Keywords: Molecular computing, Turing universality, co-transcriptional folding} }

Document

**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

We introduce and study the computational power of Oritatami, a theoretical model to explore greedy molecular folding, by which a molecule begins to fold before awaiting the end of its production. This model is inspired by a recent experimental work demonstrating the construction of shapes at the nanoscale by folding an RNA molecule during its transcription from an engineered sequence of synthetic DNA.
An important challenge of this model, also encountered in experiments, is to get a single sequence to fold into different shapes, depending on the surrounding molecules. Another big challenge is that not all parts of the sequence are meaningful for all possible inputs. Hence, to prevent them from interfering with subsequent operations in the Oritatami folding pathway we must structure the unused portions of the sequence depending on the context in which it folds.
Next, we introduce general design techniques to solve these challenges and program molecules. Our main result in this direction is an algorithm that is time linear in the sequence length that finds a rule for folding the sequence deterministically into a prescribed set of shapes, dependent on its local environment. This shows that the corresponding problem is fixed-parameter tractable, although we also prove it NP-complete in the number of possible environments.

Cody Geary, Pierre-Etienne Meunier, Nicolas Schabanel, and Shinnosuke Seki. Programming Biomolecules That Fold Greedily During Transcription. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 43:1-43:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{geary_et_al:LIPIcs.MFCS.2016.43, author = {Geary, Cody and Meunier, Pierre-Etienne and Schabanel, Nicolas and Seki, Shinnosuke}, title = {{Programming Biomolecules That Fold Greedily During Transcription}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {43:1--43:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.43}, URN = {urn:nbn:de:0030-drops-64567}, doi = {10.4230/LIPIcs.MFCS.2016.43}, annote = {Keywords: natural computing, self-assembly, molecular folding} }

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