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Toward minimum size self-assembled counters

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Abstract

DNA self-assembly is a promising paradigm for nanotechnology. In this paper we study the problem of finding tile systems of minimum size that assemble a given shape in the Tile Assembly Model, defined by Rothemund and Winfree (Proceedings of the thirty-second annual ACM symposium on theory of computing, 2000). We present a tile system that assembles an \(N\times\lceil\log_2 N\rceil\) rectangle in asymptotically optimal \(\Uptheta(N)\) time. This tile system has only 7 tiles. Earlier constructions need at least 8 tiles (Chen et al. Proceedings of symposium on discrete algorithms, 2004). We managed to reduce the number of tiles without increasing the assembly time. The new tile system works at temperature 3. The new construction was found by the combination of exhaustive computerized search of the design space and manual adjustment of the search output.

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Abbreviations

TAM:

Tile assembly model

GPC:

General purpose counter

FBC:

Full binary counter

SA:

Self-assembly

References

  • Adleman L, Cheng Q, Goel A, Huang M (2001) Running time and program size for self-assembled squares. In: Proceedings of the thirty-third annual ACM symposium on theory of computing. ACM Press, pp 740–748

  • Adleman L, Cheng Q, Goel A, Huang M, Kempe D, Moisset de Espanés P, Rothemund P (2002) Combinatorial optimization problems in self-assembly. In: Proceedings of the thirty-fourth annual ACM symposium on theory of computing. ACM Press, pp 23–32

  • Aggarwal G, Goldwasser M, Kao M, Schweller RT (2004) Complexities for generalized models of self-assembly. In: Proceedings of symposium on discrete algorithms. ACM Press

  • Chen H, Cheng Q, Goel A, Moisset de Espanés P (2004) Invadable self-assembly, combining robustness with efficiency. In: Proceedings of symposium on discrete algorithms. ACM Press

  • Cheng Q, Moisset P (2003) Resolving two open problems in the self-assembly of squares. Technical Report 03-793, University of Southern California

  • Cheng Q, Goel A, Moisset de Espanés P (2004) Optimal self-assembly of counters at temperature two. In: Foundation of Nanoscience

  • Cook M, Rothemund P, Winfree E (2003) Self assembled circuit patterns. In: Proceedings of DNA computing. Springer-Verlag

  • Goel A, Moisset de Espanés P (2007) Toward minimum size self-assembled counters. In: Proceedings of the 13th international meeting on DNA computing. Springer Verlag

  • Kao M, Schweller R (2006) Reducing tile complexity for self-assembly through temperature programming

  • Moisset de Espanés P (2005) Computerized exhaustive search for optimal self-assembly counters. In: FNANO05: Proccedings of the 2nd annual foundations of nanoscience conference, pp 24–25

  • Rothemund P (2001) Theory and experiments in algorithmic self-assembly. PhD thesis, University of Southern California

  • Rothemund PWK (2005) Design of DNA origami. In: ICCAD ’05: Proceedings of the 2005 IEEE/ACM international conference on computer-aided design. IEEE Computer Society, Washington, pp. 471–478

  • Rothemund P, Winfree E (2000) The program-size complexity of self-assembled squares (extended abstract). In: Proceedings of the thirty-second annual ACM symposium on theory of computing. ACM Press, pp 459–468

  • Rothemund PW, Papadakis N, Winfree E (2004) Algorithmic self-assembly of DNA sierpinski triangles. PLoS Biol 2(12):e424

    Google Scholar 

  • Wang H (1961) Proving theorems by pattern recognition ii. Bell Syst Tech J 40:1–42

    Google Scholar 

  • Winfree E (1998) Algorithmic self-assembly of DNA. PhD thesis, California Institute of Technology, Pasadena

  • Winfree E, Bekbolatov R (2003) Proofreading tile sets: error correction for algorithmic self-assembly. In DNA, pp 126–144

  • Winfree E, Yang X, Seeman N (1996) Universal computation via self-assembly of DNA: some theory and experiments. In: Proceedings of the second annual meeting on DNA based computers. Princeton University

  • Winfree E, Liu F, Wenzler L, Seeman N (1998) Design and self-assembly of two-dimensional DNA crystals. Nature 394:539–544

    Article  Google Scholar 

  • Yurke B, Turberfield A, Mills Jr A, Simmel F, Neumann J (2000) A DNA-fuelled molecular machine made of DNA. Nature 406:605–608

    Article  Google Scholar 

Download references

Acknowledgments

We would like to thank Len Adleman, Ming-Deh Huang, Yuri Brun and Manoj Gopalkrishnan for useful discussion, and especially Dustin Reishus for his comments on the first manuscript.

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Authors and Affiliations

  1. Centre for Biochemical Engineering and Biotechnology, Faculty of Physical and Mathematical Sciences, University of Chile, Beauchef 861, Santiago, Chile

    Pablo Moisset de Espanés

  2. Department of Management Science and Engineering and (by courtesy) Computer Science, Stanford University, Terman 311, Stanford, CA, 94305, USA

    Ashish Goel

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  1. Pablo Moisset de Espanés
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  2. Ashish Goel
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Correspondence to Pablo Moisset de Espanés.

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Moisset de Espanés, P., Goel, A. Toward minimum size self-assembled counters. Nat Comput 7, 317–334 (2008). https://doi.org/10.1007/s11047-008-9070-3

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  • DOI: https://doi.org/10.1007/s11047-008-9070-3

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