Skip to main content
SpringerLink
Log in
Book cover

International Conference on DNA-Based Computers

DNA 2016: DNA Computing and Molecular Programming pp 98–113Cite as

Resiliency to Multiple Nucleation in Temperature-1 Self-Assembly

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9818))

Abstract

We consider problems in variations of the two-handed abstract Tile Assembly Model (2HAM), a generalization of Erik Winfree’s abstract Tile Assembly Model (aTAM). In the latter, tiles attach one-at-a-time to a seed-containing assembly. In the former, tiles aggregate into supertiles that then further combine to form larger supertiles; hence, constructions must be robust to the choice of seed (nucleation) tiles. We obtain three distinct results in two 2HAM variants whose aTAM siblings are well-studied.

In the first variant, called the restricted glue 2HAM (rg2HAM), glue strengths are restricted to \(-1\), 0, or 1. We prove this model is Turing universal, overcoming undesired growth by breaking apart undesired computation assembly via repulsive forces.

In the second 2HAM variant, the 3D 2HAM (3D2HAM), tiles are (three-dimensional) cubes. We prove that assembling a (roughly two-layer) \(n \times n\) square in this model is possible with \(O(\log ^2{n})\) tile types. The construction uses “cyclic, colliding” binary counters, and assembles the shape non-deterministically. Finally, we prove that there exist 3D2HAM systems that only assemble infinite aperiodic shapes.

M.J. Patitz—This author’s research was supported in part by National Science Foundation Grant CCF-1422152.

T.A. Rogers—This author’s research was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1450079, and National Science Foundation Grant CCF-1422152.

R.T. Schweller—This author’s research was supported in part by National Science Foundation Grants CCF-1117672 and CCF-1555626.

This is a preview of subscription content, 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

Learn about institutional subscriptions

Notes

  1. 1.

    Such a distinction is only needed in two-handed models, where the seed cannot be used as a “reference point”.

References

  1. 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 

  2. Barish, R.D., Schulman, R., Rothemund, P.W., Winfree, E.: An information-bearing seed for nucleating algorithmic self-assembly. Proc. Natl. Acad. Sci. 106(15), 6054–6059 (2009)

    Article  Google Scholar 

  3. Berger, R.: The undecidability of the domino problem. Mem. Am. Math. Soc. 66, 1–72 (1966)

    MathSciNet  MATH  Google Scholar 

  4. Cannon, S., Demaine, E.D., Demaine, M.L., 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., Manuch, J., Rafiey, A., Stacho, L.: Pattern overlap implies runaway growth in hierarchical tile systems. In: Arge, L., Pach, J. (eds.) 31st International Symposium on Computational Geometry (SoCG). LIPIcs, vol. 34, pp. 360–373. Schloss Dagstuhl (2015)

    Google Scholar 

  6. Chen, H.-L., Schulman, R., Goel, A., Winfree, E.: Reducing facet nucleation during algorithmic self-assembly. Nano Lett. 7(9), 2913–2919 (2007)

    Article  Google Scholar 

  7. Cook, M., Fu, Y., Schweller, R.T.: Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D. In: Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pp. 570–589 (2011)

    Google Scholar 

  8. Demaine, E.D., Demaine, M.L., Fekete, S.P., Ishaque, M., Rafalin, E., Schweller, R.T., Souvaine, D.L.: Staged self-assembly: nanomanufacture of arbitrary shapes with \({O}(1)\) glues. Nat. Comput. 7(3), 347–370 (2008)

    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., Patitz, M.J., Summers, S.M.: Limitations of self-assembly at temperature 1. Theor. Comput. Sci. 412, 145–158 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fekete, S.P., Hendricks, J., Patitz, M.J., Rogers, T.A., Schweller, R.T.: Universal computation with arbitrary polyomino tiles in non-cooperative self-assembly. In: Proceedings of the 25th ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, pp. 148–167. SIAM (2015)

    Google Scholar 

  12. Goodman-Strauss, C.: Open questions in tiling (2000). http://comp.uark.edu/strauss/papers/survey.pdf

  13. Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W.H. Freeman and Company, New York (1987)

    MATH  Google Scholar 

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

    Google Scholar 

  15. Padilla, J.E., et al.: Asynchronous signal passing for tile self-assembly: fuel efficient computation and efficient assembly of shapes. In: Mauri, G., Dennunzio, A., Manzoni, L., Porreca, A.E. (eds.) UCNC 2013. LNCS, vol. 7956, pp. 174–185. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  16. Patitz, M.J., Schweller, R.T., Summers, S.M.: Exact shapes and turing universality at temperature 1 with a single negative glue. In: Cardelli, L., Shih, W. (eds.) DNA 17 2011. LNCS, vol. 6937, pp. 175–189. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  17. 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 

  18. Schulman, R.: The self-replication and evolution of DNA crystals. PhD thesis (2007)

    Google Scholar 

  19. Schulman, R., Winfree, E.: Synthesis of crystals with a programmable kinetic barrier to nucleation. Proc. Natl. Acad. Sci. 104(39), 15236–15241 (2007)

    Article  Google Scholar 

  20. 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 

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

    Article  Google Scholar 

  22. Socolar, J.E.S., Taylor, J.M.: An aperiodic hexagonal tile. J. Comb. Theor. Series A 118(8), 2207–2231 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Winfree, E.: Algorithmic self-assembly of DNA. PhD thesis, Caltech (1998)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, USA

    Matthew J. Patitz & Trent A. Rogers

  2. Department of Computer Science, University of Texas–Rio Grande Valley, Edinburg, USA

    Robert T. Schweller

  3. Computer Science Department, University of Wisconsin–Oshkosh, Oshkosh, USA

    Scott M. Summers

  4. Département d’Informatique, Université libre de Bruxelles, Brussels, Belgium

    Andrew Winslow

Authors
  1. Matthew J. Patitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Trent A. Rogers
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Robert T. Schweller
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Scott M. Summers
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Andrew Winslow
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew J. Patitz .

Editor information

Editors and Affiliations

  1. ESPCI, Laboratoire Gulliver UMR 7083, Paris, France

    Yannick Rondelez

  2. Centre de Recherche Inria de Paris, Paris, France

    Damien Woods

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Patitz, M.J., Rogers, T.A., Schweller, R.T., Summers, S.M., Winslow, A. (2016). Resiliency to Multiple Nucleation in Temperature-1 Self-Assembly. In: Rondelez, Y., Woods, D. (eds) DNA Computing and Molecular Programming. DNA 2016. Lecture Notes in Computer Science(), vol 9818. Springer, Cham. https://doi.org/10.1007/978-3-319-43994-5_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-43994-5_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-43993-8

  • Online ISBN: 978-3-319-43994-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation