Abstract
RNA cotranscriptional folding is the phenomenon in which an RNA transcript folds upon itself while being synthesized out of a gene. The oritatami system is a computation model of this phenomenon, which lets its sequence (transcript) of beads (abstract molecules) fold cotranscriptionally by the interactions between beads according to the binding ruleset. In such models based on self-assembly, one of the key questions is the ability to construct fractal structures. We focus on the problem of generating an infinite fractal curves using a cyclic oritatami system, which has an infinite periodic transcript. We first establish a formal definition of drawing a curve using an oritatami system, proposing conditions and restrictions with reference to prior oritatami designs for possibly infinite conformations. Under such definition, we prove that it is impossible to draw a Koch curve or a Minkowski curve infinitely. We then establish sufficient conditions of infinite aperiodic curves that a cyclic oritatami system cannot fold.
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References
Demaine ED, Hendricks J, Olsen M, Patitz MJ, Rogers TA, Schabanel N, Seki S, Thomas H (2018) Know when to fold ’em: Self-assembly of shapes by folding in oritatami. In: Proceedings of the 24th international conference on DNA computing and molecular programming, pp. 19–36
Geary C, Meunier P, Schabanel N, Seki S (2016) Programming biomolecules that fold greedily during transcription. In: Proceedings of the 41st international symposium on mathematical foundations of computer science, pp. 43:1–43:14
Geary C, Rothemund PWK, Andersen ES (2014) A single-stranded architecture for cotranscriptional folding of RNA nanostructures. Science 345:799–804
Geary CW, Meunier P, Schabanel N, Seki S (2018) Proving the turing universality of oritatami co-transcriptional folding. In: Proceedings of the 29th international symposium on algorithms and computation, pp. 23:1–23:13
Han Y, Kim H (2018) Construction of geometric structure by oritatami system. In: Proceedings of the 24th International Conference on DNA Computing and Molecular Programming, pp. 173–188
Han Y, Kim H, Ota M, Seki S (2018) Nondeterministic seedless oritatami systems and hardness of testing their equivalence. Nat Comput 17(1):67–79
Han Y, Kim H, Seki S (2020) Transcript design problem of oritatami systems. Nat Comput 19(2):323–335
Hendricks J, Olsen M, Patitz MJ, Rogers TA, Thomas H (2018) Hierarchical self-assembly of fractals with signal-passing tiles. Nat Comput 17(1):47–65
Hendricks J, Opseth J (2017) Self-assembly of 4-sided fractals in the two-handed tile assembly model. In: Proceedings of the 16th international conference on unconventional computation and natural computation, pp. 113–128
Lathrop JI, Lutz JH, Summers SM (2009) Strict self-assembly of discrete Sierpinski triangles. Theor Comput Sci 410:384–405
Maruyama K, Seki S (2020) Counting infinitely by oritatami co-transcriptional folding. In: Proceedings of the 46th international conference on current trends in theory and practice of informatics, pp. 566–575
Masuda Y, Seki S, Ubukata Y (2018) Towards the algorithmic molecular self-assembly of fractals by cotranscriptional folding. In: Proceedings of the 23rd international conference on implementation and applications of automata, pp. 261–273 (2018)
Pchelina D, Schabanel N, Seki S, Ubukata Y (2020) Simple intrinsic simulation of cellular automata in oritatami molecular folding model. In: Proceedings of the 14th Latin American theoretical informatics symposium, pp. 425–436
Rogers TA, Seki S (2017) Oritatami system; a survey and the impossibility of simple simulation at small delays. Fundamenta Informaticae 154(1–4):359–372
Rothemund PWK, Papadakis N, Winfree E (2004) Algorithmic self-assembly of DNA sierpinski triangles. PLoS Biology 2:e424
Rozenburg G, Salomaa A (1980) The mathematical theory of L systems. Academic Press, Cambridge
Tesoro S, Ahnert SE (2016) Nondeterministic self-assembly of two tile types on a lattice. Phys Rev E 93:042412
Tikhomirov G, Petersen P, Qian L (2017) Fractal assembly of micrometre-scale DNA origami arrays with arbitrary patterns. Nature 552(7683):67–71
Whitesides GM, Boncheva M (2002) Beyond molecules: self-assembly of mesoscopic and macroscopic components. Proc Natl Acad Sci USA 99(8):4769–4774
Winfree E (1998) Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology
Wolfram S (2002) A New Kind of Science. Wolfram Media
Acknowledgements
Han was supported by the Basic Science Research Program through NRF (2018R1D1A1A09084107). Kim was supported in part by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIT) (No. 2020R1F1A1072738).
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Han, YS., Kim, H. Impossibility of strict assembly of infinite fractals by oritatami. Nat Comput 20, 691–701 (2021). https://doi.org/10.1007/s11047-021-09868-w
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DOI: https://doi.org/10.1007/s11047-021-09868-w