Skip to main content
SpringerLink
Log in

Complexities for High-Temperature Two-Handed Tile Self-assembly

  • Conference paper
  • First Online:
DNA Computing and Molecular Programming (DNA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10467))

Included in the following conference series:

Abstract

Tile self-assembly is a formal model of computation capturing DNA-based nanoscale systems. Here we consider the popular two-handed tile self-assembly model or 2HAM. Each 2HAM system includes a temperature parameter, which determines the threshold of bonding strength required for two assemblies to attach. Unlike most prior study, we consider general temperatures not limited to small, constant values. We obtain two results. First, we prove that the computational complexity of determining whether a given tile system uniquely assembles a given assembly is coNP-complete, confirming a conjecture of Cannon et al. (2013). Second, we prove that larger temperature values decrease the minimum number of tile types needed to assemble some shapes. In particular, for any temperature \(\tau \in \{3, \dots \}\), we give a class of shapes of size n such that the ratio of the minimum number of tiles needed to assemble these shapes at temperature \(\tau \) and any temperature less than \(\tau \) is \(\varOmega (n^{1/(2\tau +2)})\).

This research was supported in part by National Science Foundation Grant CCF-1555626.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Note that only matching glues have positive strength. The more general model of “flexible glues” where non-matching glue pairs may also have positive strength has been considered [7].

References

  1. Abel, Z., Benbernou, N., Damian, M., Demaine, E.D., Demaine, M.L., Flatland, R., Kominers, S.D., Schweller, R.T.: Shape replication through self-assembly and RNase enzymes. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1045–1064 (2010)

    Google Scholar 

  2. Adleman, L., Cheng, Q., Goel, A., Huang, M.-D.: Running time and program size for self-assembled squares. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pp. 740–748 (2001)

    Google Scholar 

  3. Adleman, L.M., Cheng, Q., Goel, A., Huang, M.-D.A., Kempe, D., de Espanés, P.M., Rothemund, P.W.K.: Combinatorial optimization problems in self-assembly. In: Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, pp. 23–32 (2002)

    Google Scholar 

  4. Cannon, S., Demaine, E.D., Demaine, E.D., Eisenstat, S., Patitz, M.J., Schweller, R., Summers, S.M., Winslow, A.: Two hands are better than one (up to constant factors): self-assembly in the 2HAM vs. aTAM. In: Proceedings of 30th International Symposium on Theoretical Aspects of Computer Science (STACS). LIPIcs, vol. 20, pp. 172–184. Schloss Dagstuhl (2013)

    Google Scholar 

  5. Chen, H.-L., Doty, D.: Parallelism and time in hierarchical self-assembly. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 1163–1182. SIAM (2012)

    Google Scholar 

  6. Chen, H.-L., Doty, D., Seki, S.: Program size and temperature in self-assembly. Algorithmica 72(3), 884–899 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, Q., Aggarwal, G., Goldwasser, M.H., Kao, M.-Y., Schweller, R.T., de Espanés, P.M.: Complexities for generalized models of self-assembly. SIAM J. Comput. 34, 1493–1515 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Demaine, E.D., Patitz, M.J., Rogers, T.A., Schweller, R.T., Summers, S.M., Woods, D.: The two-handed tile assembly model is not intrinsically universal. Algorithmica 74(2), 812–850 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Doty, D.: Producibility in hierarchical self-assembly. Nat. Comput. 15(1), 41–49 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Doty, D., Lutz, J.H., Patitz, M.J., Schweller, R., Summers, S.M., Woods, D.: The tile assembly model is intrinsically universal. In: Proceedings of the 53rd IEEE Conference on Foundations of Computer Science (FOCS), pp. 302–310 (2012)

    Google Scholar 

  11. Doty, D., Patitz, M.J., Reishus, D., Schweller, R.T., Summers, S.M.: Strong fault-tolerance for self-assembly with fuzzy temperature. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 417–426 (2010)

    Google Scholar 

  12. Doty, D., Patitz, M.J., Summers, S.M.: Limitations of self-assembly at temperature one. In: Deaton, R., Suyama, A. (eds.) DNA 2009. LNCS, vol. 5877, pp. 35–44. Springer, Heidelberg (2009). doi:10.1007/978-3-642-10604-0_4

    Chapter  Google Scholar 

  13. Itai, A., Papadimitriou, C.H., Szwarcfiter, J.L.: Hamilton paths in grid graphs. SIAM J. Comput. 11(4), 676–686 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kao, M.-Y., Schweller, R.T.: Reducing tile complexity for self-assembly through temperature programming. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, pp. 571–580 (2006)

    Google Scholar 

  15. Luhrs, C.: Polyomino-safe DNA self-assembly via block replacement. Nat. Comput. 9(1), 97–109 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Maňuch, J., Stacho, L., Stoll, C.: Two lower bounds for self-assemblies at temperature 1. J. Comput. Biol. 16(6), 841–852 (2010)

    MathSciNet  Google Scholar 

  17. Meunier, P.-E.: The self-assembly of paths and squares at temperature 1. Technical report, arXiv (2013). http://arxiv.org/abs/1312.1299

  18. Meunier, P.-E., Patitz, M.J., Summers, S.M., Theyssier, G., Winslow, A., Woods, D.: Intrinsic universality in tile self-assembly requires cooperation. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 752–771 (2014)

    Google Scholar 

  19. Patitz, M.J., Rogers, T.A., Schweller, R.T., Summers, S.M., Winslow, A.: Resiliency to multiple nucleation in temperature-1 self-assembly. In: Rondelez, Y., Woods, D. (eds.) DNA 2016. LNCS, vol. 9818, pp. 98–113. Springer, Cham (2016). doi:10.1007/978-3-319-43994-5_7

    Chapter  Google Scholar 

  20. Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: Proceedings of the 32nd ACM Symposium on Theory of Computing (STOC), pp. 459–468 (2000)

    Google Scholar 

  21. Schulman, R., Winfree, E.: Programmable control of nucleation for algorithmic self-assembly. SIAM J. Comput. 39(4), 1581–1616 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schweller, R., Winslow, A., Wylie, T.: Verification in staged tile self-assembly. In: Patitz, M.J., Stannett, M. (eds.) UCNC 2017. LNCS, vol. 10240, pp. 98–112. Springer, Cham (2017). doi:10.1007/978-3-319-58187-3_8

    Chapter  Google Scholar 

  23. Seeman, N.C.: Nucleic-acid junctions and lattices. J. Theor. Biol. 99, 237–247 (1982)

    Article  Google Scholar 

  24. Seki, S., Ukuno, Y.: On the behavior of tile assembly system at high temperatures. Computability 2(2), 107–124 (2013)

    MathSciNet  MATH  Google Scholar 

  25. Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM J. Comput. 36(6), 1544–1569 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Summers, S.M.: Reducing tile complexity for the self-assembly of scaled shapes through temperature programming. Algorithmica 63(1), 117–136 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology (1998)

    Google Scholar 

  28. Woods, D.: Intrinsic universality and the computational power of self-assembly. Philos. Trans. R. Soc. A 373, 2015 (2046)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Texas - Rio Grande Valley, Edinburg, TX, 78539, USA

    Robert Schweller, Andrew Winslow & Tim Wylie

Authors
  1. Robert Schweller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrew Winslow
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tim Wylie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrew Winslow .

Editor information

Editors and Affiliations

  1. Hasselt University, Diepenbeek, Belgium

    Robert Brijder

  2. California Institute of Technology, Pasadena, California, USA

    Lulu Qian

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Schweller, R., Winslow, A., Wylie, T. (2017). Complexities for High-Temperature Two-Handed Tile Self-assembly. In: Brijder, R., Qian, L. (eds) DNA Computing and Molecular Programming. DNA 2017. Lecture Notes in Computer Science(), vol 10467. Springer, Cham. https://doi.org/10.1007/978-3-319-66799-7_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-66799-7_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66798-0

  • Online ISBN: 978-3-319-66799-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation