Program Size and Temperature in Self-Assembly

  • Ho-Lin Chen
  • David Doty
  • Shinnosuke Seki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)


Winfree’s abstract Tile Assembly Model (aTAM) 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 ×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, Cheng, Goel, Huang, Kempe, Moisset de Espanés, and Rothemund (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)


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  1. 1.
    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, Hersonissos, Greece, pp. 740–748. ACM (2001)Google Scholar
  2. 2.
    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: STOC 2002: Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, pp. 23–32 (2002)Google Scholar
  3. 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)CrossRefGoogle Scholar
  4. 4.
    Evans, C.: Personal communicationGoogle Scholar
  5. 5.
    Fujibayashi, K., Hariadi, R., Park, S.H., Winfree, E., Murata, S.: Toward reliable algorithmic self-assembly of DNA tiles: A fixed-width cellular automaton pattern. Nano Letters 8(7), 1791–1797 (2007)CrossRefGoogle Scholar
  6. 6.
    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, pp. 459–468 (2000)Google Scholar
  7. 7.
    Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology 2(12), 2041–2053 (2004)CrossRefGoogle Scholar
  8. 8.
    Seeman, N.C.: Nucleic-acid junctions and lattices. Journal of Theoretical Biology 99, 237–247 (1982)CrossRefGoogle Scholar
  9. 9.
    Wang, H.: Proving theorems by pattern recognition – II. The Bell System Technical Journal XL(1), 1–41 (1961)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Wang, H.: Dominoes and the AEA case of the decision problem. In: Proceedings of the Symposium on Mathematical Theory of Automata, New York, pp. 23–55 (1962); Polytechnic Press of Polytechnic Inst. of Brooklyn, Brooklyn, N.Y. (1963)Google Scholar
  11. 11.
    Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology (June 1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ho-Lin Chen
    • 1
  • David Doty
    • 1
  • Shinnosuke Seki
    • 2
  1. 1.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Department of System Sciences for Drug DiscoveryKyoto UniversityKyotoJapan

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