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Exact Shapes and Turing Universality at Temperature 1 with a Single Negative Glue

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Book cover DNA Computing and Molecular Programming (DNA 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6937))

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Abstract

Is Winfree’s abstract Tile Assembly Model (aTAM) “powerful?” Well, if certain tiles are required to “cooperate” in order to be able to bind to a growing tile assembly (a.k.a., temperature 2 self-assembly), then Turing universal computation and the efficient self-assembly of N ×N squares is achievable in the aTAM (Rotemund and Winfree, STOC 2000). So yes, in a computational sense, the aTAM is quite powerful! However, if one completely removes this cooperativity condition (a.k.a., temperature 1 self-assembly), then the computational “power” of the aTAM (i.e., its ability to support Turing universal computation and the efficient self-assembly of N ×N squares) becomes unknown. On the plus side, the aTAM, at temperature 1, is not only Turing universal but also supports the efficient self-assembly N ×N squares if self-assembly is allowed to utilize three spatial dimensions (Fu, Schweller and Cook, SODA 2011). In this paper, we investigate the theoretical “power” of a seemingly simple, restrictive variant of Winfree’s aTAM in which (1) the absolute value of every glue strength is 1, (2) there is a single negative strength glue type and (3) unequal glues cannot interact (i.e., glue functions must be “diagonal”). We call this abstract model of self-assembly the restricted glue Tile Assembly Model (rgTAM). We achieve two positive results. First, we show that the tile complexity of uniquely producing an N ×N square in the rgTAM is O(logN). In our second result, we prove that the rgTAM is Turing universal.

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References

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

    Google Scholar 

  2. Adleman, L.M., Kari, J., Kari, L., Reishus, D., Sosík, P.: The undecidability of the infinite ribbon problem: Implications for computing by self-assembly. SIAM Journal on Computing 38(6), 2356–2381 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barish, R.D., Schulman, R., Rothemund, P.W., Winfree, E.: An information-bearing seed for nucleating algorithmic self-assembly. Proceedings of the National Academy of Sciences 106(15), 6054–6059 (2009)

    Article  Google Scholar 

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

    Article  Google Scholar 

  5. Cook, M., Fu, Y., Schweller, R.: Temperature 1 self-assembly: Deterministic assembly in 3d and probabilistic assembly in 2d. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (2011)

    Google Scholar 

  6. Doty, D.: Randomized self-assembly for exact shapes. SIAM Journal on Computing 39(8), 3521–3552 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Doty, D., Kari, L., Masson, B.: Negative interactions in irreversible self-assembly. In: Sakakibara, Y., Mi, Y. (eds.) DNA 16 2010. LNCS, vol. 6518, pp. 37–48. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  8. Doty, D., Lutz, J.H., Patitz, M.J., Summers, S.M., Woods, D.: Intrinsic universality in self-assembly. In: Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science, pp. 275–286 (2009)

    Google Scholar 

  9. 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 

  10. Doty, D., Patitz, M.J., Summers, S.M.: Limitations of self-assembly at temperature 1. Theoretical Computer Science 412, 145–158 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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), Miami, Florida, January 2006, pp. 571-580 (2007)

    Google Scholar 

  12. Kao, M.-Y., Schweller, R.T.: Randomized self-assembly for approximate shapes. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 370–384. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  13. Kinsella, J.M., Ivanisevic, A.: Enzymatic clipping of dna wires coated with magnetic nanoparticles. Journal of the American Chemical Society 127(10), 3276–3277 (2005)

    Article  Google Scholar 

  14. Lathrop, J.I., Lutz, J.H., Summers, S.M.: Strict self-assembly of discrete Sierpinski triangles. Theoretical Computer Science 410, 384–405 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Luhrs, C.: Polyomino-safe dna self-assembly via block replacement. DNA, 112–126 (2008)

    Google Scholar 

  16. Majumder, U., Reif, J.: A framework for designing novel magnetic tiles capable of complex self-assemblies. In: Calude, C.S., Costa, J.F., Freund, R., Oswald, M., Rozenberg, G. (eds.) UC 2008. LNCS, vol. 5204, pp. 129–145. Springer, Heidelberg (2008)

    Google Scholar 

  17. Mao, C., Sun, W., Seeman, N.C.: Designed two-dimensional DNA holliday junction arrays visualized by atomic force microscopy. Journal of the American Chemical Society 121(23), 5437–5443 (1999)

    Article  Google Scholar 

  18. Reif, J., Sahu, S., Yin, P.: Complexity of graph self-assembly in accretive systems and self-destructible systems. In: Carbone, A., Pierce, N.A. (eds.) DNA 2005. LNCS, vol. 3892, pp. 257–274. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  19. Rickwood, D., Lund, V.: Attachment of dna and oligonucleotides to magnetic particles: methods and applications. Fresenius’ Journal of Analytical Chemistry 330, 330–330 (1988), doi:10.1007/BF00469247

    Article  Google Scholar 

  20. Rothemund, P.W.K.: Theory and experiments in algorithmic self-assembly, Ph.D. thesis, University of Southern California (December 2001)

    Google Scholar 

  21. Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: STOC 2000: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, Portland, Oregon, United States, pp. 459–468. ACM, New York (2000)

    Chapter  Google Scholar 

  22. Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology 2(12), 2041–2053 (2004)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  25. Winfree, E., Liu, F., Wenzler, L.A., Seeman, N.C.: Design and self-assembly of two-dimensional DNA crystals. Nature 394(6693), 539–544 (1998)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Texas–Pan American, Edinburg, TX, 78539, USA

    Matthew J. Patitz & Robert T. Schweller

  2. Department of Computer Science and Software Engineering, University of Wisconsin, Platteville, WI, 53818, USA

    Scott M. Summers

Authors
  1. Matthew J. Patitz
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  2. Robert T. Schweller
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  3. Scott M. Summers
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Editor information

Editors and Affiliations

  1. Microsoft Research, 711 Thomson Avenue, CB3 0FB, Cambridge, UK

    Luca Cardelli

  2. Department of Biological Chemistry and Molecular Pharmacology, Dana-Farber Cancer Institute, 44 Binney Street, 02115, Boston, MA, USA

    William Shih

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Patitz, M.J., Schweller, R.T., Summers, S.M. (2011). Exact Shapes and Turing Universality at Temperature 1 with a Single Negative Glue. In: Cardelli, L., Shih, W. (eds) DNA Computing and Molecular Programming. DNA 2011. Lecture Notes in Computer Science, vol 6937. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23638-9_15

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  • DOI: https://doi.org/10.1007/978-3-642-23638-9_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23637-2

  • Online ISBN: 978-3-642-23638-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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