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Self-Assembly of Any Shape with Constant Tile Types using High Temperature

Authors Cameron Chalk, Austin Luchsinger, Robert Schweller, Tim Wylie

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Author Details

Cameron Chalk
  • Department of Electrical and Computer Engineering, University of Texas - Austin, Austin, TX, USA
Austin Luchsinger
  • Department of Computer Science, University of Texas - Rio Grande Valley, Edinburg, TX, USA
Robert Schweller
  • Department of Computer Science, University of Texas - Rio Grande Valley, Edinburg, TX, USA
Tim Wylie
  • Department of Computer Science, University of Texas - Rio Grande Valley, Edinburg, TX, USA

Funding

  • Chalk, Cameron: This author’s research was supported in part by National Science Foundation Grants CCF-1618895 and CCF-1652824.
  • Luchsinger, Austin: This author’s research was supported in part by National Science Foundation Grant CCF-1817602.
  • Schweller, Robert: This author’s research was supported in part by National Science Foundation Grant CCF-1817602.
  • Wylie, Tim: This author’s research was supported in part by National Science Foundation Grant CCF-1817602.

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Cameron Chalk, Austin Luchsinger, Robert Schweller, and Tim Wylie. Self-Assembly of Any Shape with Constant Tile Types using High Temperature. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ESA.2018.14

Abstract

Inspired by nature and motivated by a lack of top-down tools for precise nanoscale manufacture, self-assembly is a bottom-up process where simple, unorganized components autonomously combine to form larger more complex structures. Such systems hide rich algorithmic properties - notably, Turing universality - and a self-assembly system can be seen as both the object to be manufactured as well as the machine controlling the manufacturing process. Thus, a benchmark problem in self-assembly is the unique assembly of shapes: to design a set of simple agents which, based on aggregation rules and random movement, self-assemble into a particular shape and nothing else. We use a popular model of self-assembly, the 2-handed or hierarchical tile assembly model, and allow the existence of repulsive forces, which is a well-studied variant. The technique utilizes a finely-tuned temperature (the minimum required affinity required for aggregation of separate complexes).
We show that calibrating the temperature and the strength of the aggregation between the tiles, one can encode the shape to be assembled without increasing the number of distinct tile types. Precisely, we show one tile set for which the following holds: for any finite connected shape S, there exists a setting of binding strengths between tiles and a temperature under which the system uniquely assembles S at some scale factor. Our tile system only uses one repulsive glue type and the system is growth-only (it produces no unstable assemblies). The best previous unique shape assembly results in tile assembly models use O(K(S)/(log K(S))) distinct tile types, where K(S) is the Kolmogorov (descriptional) complexity of the shape S.

Subject Classification

ACM Subject Classification
  • Theory of computation → Self-organization
Keywords
  • self-assembly
  • molecular computing
  • tiling
  • tile
  • shapes

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References

  1. Sarah Cannon, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Matthew J. Patitz, Robert Schweller, Scott M. Summers, and Andrew Winslow. Two hands are better than one (up to constant factors): Self-assembly in the 2ham vs. atam. In Natacha Portier and Thomas Wilke, editors, STACS, volume 20 of LIPIcs, pages 172-184. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2013.172.
  2. Cameron Chalk, Erik D. Demiane, Martin L. Demaine, Eric Martinez, Robert Schweller, Luis Vega, and Tim Wylie. Universal shape replicators via self-assembly with attractive and repulsive forces. In Proc. of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'17), 2017. Google Scholar
  3. Cameron Chalk, Eric Martinez, Robert Schweller, Luis Vega, Andrew Winslow, and Tim Wylie. Optimal staged self-assembly of general shapes. Algorithmica, May 2017. URL: http://dx.doi.org/10.1007/s00453-017-0318-0.
  4. H.-L. Chen, D. Doty, and S. Seki. Program size and temperature in self-assembly. Algorithmica, 72(3):884-899, 2015. Google Scholar
  5. Qi Cheng, Gagan Aggarwal, Michael H. Goldwasser, Ming-Yang Kao, Robert T. Schweller, and Pablo Moisset de Espanés. Complexities for generalized models of self-assembly. SIAM Journal on Computing, 34:1493-1515, 2005. Google Scholar
  6. Erik D. Demaine, Matthew J. Patitz, Trent A. Rogers, Robert T. Schweller, Scott M. Summers, and Damien Woods. The two-handed tile assembly model is not intrinsically universal. Algorithmica, 74(2):812-850, 2016. Google Scholar
  7. Erik D. Demaine, Matthew J. Patitz, Robert T. Schweller, and Scott M. Summers. Self-assembly of arbitrary shapes using rnase enzymes: Meeting the kolmogorov bound with small scale factor (extended abstract). In Proc. of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS'11), 2011. Google Scholar
  8. David Doty, Lila Kari, and Benoît Masson. Negative interactions in irreversible self-assembly. Algorithmica, 66(1):153-172, 2013. URL: http://dx.doi.org/10.1007/s00453-012-9631-9.
  9. Constantine Evans. Crystals that Count! Physical Principles and Experimental Investigations of DNA Tile Self-Assembly. PhD thesis, California Inst. of Tech., 2014. Google Scholar
  10. Ming-Yang Kao and Robert T. Schweller. Reducing tile complexity for self-assembly through temperature programming. In SODA 2006: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 571-580, 2006. Google Scholar
  11. M. Li and P. Vitanyi. An Introduction to Kolmogorov and Its Applications (Second Edition). Springer Verlag, New York, 1997. Google Scholar
  12. Austin Luchsinger, Robert Schweller, and Tim Wylie. Self-assembly of shapes at constant scale using repulsive forces. In Matthew J. Patitz and Mike Stannett, editors, Unconventional Computation and Natural Computation, pages 82-97, Cham, 2017. Springer International Publishing. Google Scholar
  13. Matthew J. Patitz, Trent A. Rogers, Robert Schweller, Scott M. Summers, and Andrew Winslow. Resiliency to multiple nucleation in temperature-1 self-assembly. In DNA Computing and Molecular Programming. Springer International Publishing, 2016. Google Scholar
  14. Matthew J. Patitz, Robert T. Schweller, and Scott M. Summers. Exact shapes and turing universality at temperature 1 with a single negative glue. In DNA Computing and Molecular Programming, volume 6937 of LNCS, pages 175-189. Springer, 2011. Google Scholar
  15. John H. Reif, Sudheer Sahu, and Peng Yin. Complexity of graph self-assembly in accretive systems and self-destructible systems. Theoretical Comp. Sci., 412(17):1592-1605, 2011. URL: http://dx.doi.org/10.1016/j.tcs.2010.10.034.
  16. Paul W. K. Rothemund. Using lateral capillary forces to compute by self-assembly. Proceedings of the National Academy of Sciences, 97(3):984-989, 2000. URL: http://dx.doi.org/10.1073/pnas.97.3.984.
  17. Paul W. K. Rothemund. Folding DNA to create nanoscale shapes and patterns. Nature, 440(7082):297-302, March 2006. URL: http://dx.doi.org/10.1038/nature04586.
  18. Paul W. K. Rothemund and Erik Winfree. The program-size complexity of self-assembled squares (extended abstract). In Proc. of the 32nd ACM Sym. on Theory of Computing, STOC'00, pages 459-468, 2000. Google Scholar
  19. Nicholas Schiefer and Erik Winfree. Universal Computation and Optimal Construction in the Chemical Reaction Network-Controlled Tile Assembly Model, pages 34-54. Springer International Publishing, Cham, 2015. URL: http://dx.doi.org/10.1007/978-3-319-21999-8_3.
  20. Robert Schweller and Michael Sherman. Fuel efficient computation in passive self-assembly. In SODA 2013: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1513-1525. SIAM, 2013. Google Scholar
  21. Robert Schweller, Andrew Winslow, and Tim Wylie. Complexities for high-temperature two-handed tile self-assembly. In Robert Brijder and Lulu Qian, editors, DNA Computing and Molecular Programming, pages 98-109, Cham, 2017. Springer International Publishing. Google Scholar
  22. Sinnosuke Seki and Yasushi Ukuno. On the behavior of tile assembly system at high temperatures. Computability, 2(2):107-124, 2013. Google Scholar
  23. David Soloveichik and Erik Winfree. Complexity of self-assembled shapes. SIAM Journal on Computing, 36(6):1544-1569, 2007. URL: http://dx.doi.org/10.1137/S0097539704446712.
  24. Scott M. Summers. Reducing tile complexity for the self-assembly of scaled shapes through temperature programming. Algorithmica, 63(1):117-136, 2012. Google Scholar
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