Algorithmica

, Volume 72, Issue 3, pp 884–899 | Cite as

Program Size and Temperature in Self-Assembly

Article

Abstract

Winfree’s abstract Tile Assembly Model is a model of molecular self-assembly of DNA complexes known as tiles, which float freely in solution and attach one at a time to a growing “seed” assembly based on specific binding sites on their four sides. We show that there is a polynomial-time algorithm that, given an \(n \times n\) square, finds the minimal tile system (i.e., the system with the smallest number of distinct tile types) that uniquely self-assembles the square, answering an open question of Adleman et al. (Combinatorial optimization problems in self-assembly, STOC 2002). Our investigation leading to this algorithm reveals other positive and negative results about the relationship between the size of a tile system and its “temperature” (the binding strength threshold required for a tile to attach).

Keywords

Self-assembly Tile complexity Temperature Optimization 

Notes

Acknowledgments

The authors thank Ehsan Chiniforooshan especially, for many fruitful and illuminating discussions that led to the results on temperature, and also Adam Marblestone and the members of Erik Winfree’s group, particularly David Soloveichik, Joe Schaeffer, Damien Woods, and Erik Winfree, for insightful discussion and comments. The second author is grateful to Aaron Meyerowitz (via the website http://mathoverflow.net) for pointing out the Dedekind numbers as a way to count the number of collections of subsets of a given set that are closed under the superset operation. Ho-Lin Chen was supported by the Molecular Programming Project under NSF Grant 0832824. David Doty was supported by an Computing Innovation Fellowship under NSF Grant 1019343 and NSF Grants CCF-1219274 and CCF-1162589 and the Molecular Programming Project under NSF Grant 0832824. Shinnosuke Seki was supported by NSERC Discovery Grant R2824A01 and the Canada Research Chair Award in Biocomputing to Lila Kari, by Kyoto University Start-up Grant-in-Aid for Young Scientists (No. 021530), by HIIT Pump Priming Project 902184/T30606, and by Academy of Finland, Postdoctoral Researcher Grant No. 13266670/T30606.

References

  1. 1.
    Abel, Z., Benbernou, N., Damian, M., Demaine, E.D., Demaine, M.L., Flatland, R., Kominers, S., Schweller, R.: Shape replication through self-assembly and RNase enzymes. In: SODA 2010 Proceedings of the Twenty-first Annual ACM-SIAM Symposium on Discrete Algorithms, Austin, Texas. Society for Industrial and Applied Mathematics (2010)Google Scholar
  2. 2.
    Adleman, L.M., Cheng, Q., Goel, A., Huang, M.-D.: Running time and program size for self-assembled squares. In: STOC 2001 Proceedings of the thirty-third annual ACM Symposium on Theory of Computing, ACM, Hersonissos, Greece, 740–748 (2001)Google Scholar
  3. 3.
    Adleman, L.M., Cheng, Q., Goel, A., Huang, M.-D.A., Kempe, D., Moisset de Espanés, P., Rothemund, P. W. K.: Combinatorial optimization problems in self-assembly. In: STOC 2002 Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of, Computing, 23–32, (2002)Google Scholar
  4. 4.
    Aggarwal, G., Cheng, Q., Goldwasser, M.H., Kao, M.-Y., Moisset de Espanés, P., Schweller, R.T.: Complexities for generalized models of self-assembly. SIAM J. Comput. 34, 1493–1515 (2005). Preliminary version appeared in SODA 2004MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Barish, R.D., Schulman, R., Rothemund, P.W.K., Winfree, E.: An information-bearing seed for nucleating algorithmic self-assembly. Proc. Natl. Acad. Sci. 106(15), 6054–6059 (2009)CrossRefGoogle Scholar
  6. 6.
    Becker, F., Rapaport, I., Eric Rémila.: Self-assembling classes of shapes with a minimum number of tiles, and in optimal time. In: FSTTCS 2006 Foundations of Software Technology and Theoretical Computer Science, 45–56, (2006)Google Scholar
  7. 7.
    Chandran, H., Gopalkrishnan, N., Reif, J.H.: Tile complexity of linear assemblies. SIAM J. Comput. 41(4), 1051–1073 (2012). Preliminary version appeared in ICALP 2009MATHMathSciNetCrossRefGoogle Scholar
  8. 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). Preliminary version appeared in, DNA 2007MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Demaine, E.D., Eisenstat, S., Ishaque, M., Winslow, A.: One-dimensional staged self-assembly. In: Cardelli, Luca, Shih, William (eds.) DNA Computing and Molecular Programming. Lecture Notes in Computer Science, pp. 100–114. Springer, Berlin / Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Demaine, E.D., Patitz, M.J., Schweller, R.T., Summers, S.M.: Self-assembly of arbitrary shapes using RNase enzymes: meeting the Kolmogorov bound with small scale factor. In: STACS 2011 Proceedings of the 28th International Symposium on Theoretical Aspects of Computer, Science, (2011)Google Scholar
  11. 11.
    Doty, David: Randomized self-assembly for exact shapes. SIAM J. Comput. 39(8), 3521–3552 (2010). Preliminary version appeared in FOCS 2009MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Doty, David: Theory of algorithmic self-assembly. Commun. ACM 55(12), 78–88 (2012)CrossRefGoogle Scholar
  13. 13.
    Fu, B., Patitz, M.J., Schweller, R.T., Sheline,R.: Self-assembly with geometric tiles. In: ICALP 2012 Proceedings of the 39th International Colloquium on Automata, Languages and Programming, 714–725, July 2012Google Scholar
  14. 14.
    Fujibayashi, Kenichi, Hariadi, Rizal, Park, Sung Ha, Winfree, Erik, Murata, Satoshi: Toward reliable algorithmic self-assembly of DNA tiles: a fixed-width cellular automaton pattern. Nano Lett. 8(7), 1791–1797 (2007)CrossRefGoogle Scholar
  15. 15.
    Kao,M.-Y., Schweller,R.T.: Reducing tile complexity for self-assembly through temperature programming. In: SODA 2006 Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, 571–580 (2006)Google Scholar
  16. 16.
    Kao,M.-Y., Schweller, R.T.: Randomized self-assembly for approximate shapes. In: ICALP 2008: International Colloqium on Automata, Languages, and Programming, Lecture Notes in Computer Science, vol. 5125 pp. 370–384. Springer, (2008)Google Scholar
  17. 17.
    Lathrop, James I., Lutz, Jack H., Summers, Scott M.: Strict self-assembly of discrete Sierpinski triangles. Theor. Comput. Sci. 410, 384–405 (2009). Preliminary version appeared in CiE 2007MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Manuch, J., Stacho, L., Stoll, C.: Step-assembly with a constant number of tile types. In: ISAAC 2009 Proceedings of the 20th International Symposium on Algorithms and Computation, pp. 954–963. Springer-Verlag, Berlin, Heidelberg (2009)Google Scholar
  19. 19.
    Rothemund, P.W.K.: Theory and experiments in algorithmic self-assembly. PhD thesis, University of Southern California (2001)Google Scholar
  20. 20.
    Rothemund, P.W.K., Papadakis, Nick, Winfree, Erik: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol. 2(12), 2041–2053 (2004)CrossRefGoogle Scholar
  21. 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, 459–468 (2000)Google Scholar
  22. 22.
    Seeman, Nadrian C.: Nucleic-acid junctions and lattices. J. Theor. Biol. 99, 237–247 (1982)CrossRefGoogle Scholar
  23. 23.
    Seki, Shinnouke, Okuno, Y.: On the behavior of tile assembly model at high temperatures. Computability to appear. SIAM J. Comput. 41(4), 1051–1073 (2012). Preliminary version appeared in CiE 2012MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sloane, N.J.: The on-line encyclopedia of integer sequences. Dedekind numbers: number of monotone Boolean functions of n variables or number of antichains of subsets of an n-set. http://oeis.org/wiki/A000372. Accessed 1 Nov 2010
  25. 25.
    Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM J. Comput. 36(6), 1544–1569 (2007). Preliminary version appeared in, DNA 2004MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Summers, Scott M.: Reducing tile complexity for the self-assembly of scaled shapes through temperature programming. Algorithmica 63(1–2), 117–136 (2012)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, H.: Proving theorems by pattern recognition—II. Bell Syst. Tech. J. XL(1), 1–41 (1963)Google Scholar
  28. 28.
    Wang, H.,. Dominoes and the AEA case of the decision problem. In: Proceedings of the Symposium on Mathematical Theory of Automata (New York, 1962) pp 23–55. Polytechnic Press of Polytechnic Institute of Brooklyn, Brooklyn (1963)Google Scholar
  29. 29.
    Winfree, E.: Algorithmic self-assembly of DNA. PhD thesis, California Institute of Technology (1998)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.National Taiwan UniversityTaipeiTaiwan
  2. 2.California Institute of TechnologyPasadenaUSA
  3. 3.Aalto UniversityHelsinkiFinland

Personalised recommendations