Self-assembly with Geometric Tiles

  • Bin Fu
  • Matthew J. Patitz
  • Robert T. Schweller
  • Robert Sheline
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

In this work we propose a generalization of Winfree’s abstract Tile Assembly Model (aTAM) in which tile types are assigned rigid shapes, or geometries, along each tile face. We examine the number of distinct tile types needed to assemble shapes within this model, the temperature required for efficient assembly, and the problem of designing compact geometric faces to meet given compatibility specifications. We pose the following question: can complex geometric tile faces arbitrarily reduce the number of distinct tile types to assemble shapes? Within the most basic generalization of the aTAM, we show that the answer is no. For almost all n at least \(\Omega(\sqrt{\log n})\) tile types are required to uniquely assemble an n×n square, regardless of how much complexity is pumped into the face of each tile type. However, we show for all n we can achieve a matching \(O(\sqrt{\log n})\) tile types, beating the known lower bound of Θ(logn / loglogn) that holds for almost all n within the aTAM. Further, our result holds at temperature τ = 1. Our next result considers a geometric tile model that is a generalization of the 2-handed abstract tile assembly model in which tile aggregates must move together through obstacle free paths within the plane. Within this model we present a novel construction that harnesses the collision free path requirement to allow for the unique assembly of any n×n square with a sleek O(loglogn) distinct tile types. This construction is of interest in that it is the first tile self-assembly result to harness collision free planar translation to increase efficiency, whereas previous work has simply used the planarity restriction as a desireable quality that could be achieved at reduced efficiency. This surprisingly low tile type result further emphasizes a fundamental open question: Is it possible to assemble n×n squares with O(1) distinct tile types? Essentially, how far can the trade off between the number of distinct tile types required for an assembly and the complexity of each tile type itself be taken?

Keywords

Tile Type Tile Assembly Model Tile Assembly System Tile Complexity Adjacent Tile 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)CrossRefGoogle Scholar
  2. 2.
    Adleman, L., Cheng, Q., Goel, A., Huang, M.-D., Wasserman, H.: Linear self-assemblies: Equilibria, entropy and convergence rates. In: Sixth International Conference on Difference Equations and Applications. Taylor and Francis (2001)Google Scholar
  3. 3.
    Aggarwal, G., Goldwasser, M.H., Kao, M.-Y., Schweller, R.T.: Complexities for generalized models of self-assembly. In: Proceedings of ACM-SIAM Symposium on Discrete Algorithms (2004)Google Scholar
  4. 4.
    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
  5. 5.
    Chandran, H., Gopalkrishnan, N., Reif, J.: The Tile Complexity of Linear Assemblies. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 235–253. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Chen, H.-L., Schulman, R., Goel, A., Winfree, E.: Reducing facet nucleation during algorithmic self-assembly. Nano Letters 7(9), 2913–2919 (2007)CrossRefGoogle Scholar
  7. 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 Journal on Computing 34, 1493–1515 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Cook, M., Fu, Y., Schweller, R.T.: Temperature 1 self-assembly: Deterministic assembly in 3d and probabilistic assembly in 2d. In: Randall, D. (ed.) Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, pp. 570–589. SIAM (2011)Google Scholar
  9. 9.
    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. Natural Computing 7(3), 347–370 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    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
  11. 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. 12.
    Endo, M., Sugita, T., Katsuda, Y., Hidaka, K., Sugiyama, H.: Programmed-assembly system using DNA jigsaw pieces. Chemistry: A European Journal, 5362–5368 (2010)Google Scholar
  13. 13.
    Fu, B., Patitz, M.J., Schweller, R., Sheline, R.: Self-assembly with geometric tiles. Arxiv preprint arXiv:1104.2809 (2012)Google Scholar
  14. 14.
    LaBean, T.H., Winfree, E., Reif, J.H.: Experimental progress in computation by self-assembly of DNA tilings. DNA Based Computers 5, 123–140 (1999)Google Scholar
  15. 15.
    Luhrs, C.: Polyomino-Safe DNA Self-assembly via Block Replacement. In: Goel, A., Simmel, F.C., Sosík, P. (eds.) DNA. LNCS, vol. 5347, pp. 112–126. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Mao, C., LaBean, T.H., Relf, J.H., Seeman, N.C.: Logical computation using algorithmic self-assembly of DNA triple-crossover molecules. Nature 407(6803), 493–496 (2000)CrossRefGoogle Scholar
  17. 17.
    Reif, J.H., Sahu, S., Yin, P.: Compact Error-Resilient Computational DNA Tiling Assemblies. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA 2004. LNCS, vol. 3384, pp. 293–307. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  18. 18.
    Rothemund, P.W.K.: Folding DNA to create nanoscale shapes and patterns. Nature 440(7082), 297–302 (2006)CrossRefGoogle Scholar
  19. 19.
    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 Press (2000)Google Scholar
  20. 20.
    Schulman, R., Winfree, E.: Synthesis of crystals with a programmable kinetic barrier to nucleation. Proceedings of the National Academy of Sciences 104(39), 15236–15241 (2007)CrossRefGoogle Scholar
  21. 21.
    Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM Journal on Computing 36(6), 1544–1569 (2007)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology (June 1998)Google Scholar
  23. 23.
    Winfree, E.: Self-healing tile sets. In: Chen, J., Jonoska, N., Rozenberg, G. (eds.) Nanotechnology: Science and Computation. Natural Computing Series, pp. 55–78. Springer (2006)Google Scholar
  24. 24.
    Woo, S., Rothemund, P.W.K.: Stacking bonds: Programming molecular recognition based on the geometry of dna nanostructures. Nature Chemistry 3, 620–627 (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Bin Fu
    • 1
  • Matthew J. Patitz
    • 1
  • Robert T. Schweller
    • 1
  • Robert Sheline
    • 1
  1. 1.Department of Computer ScienceUniversity of Texas - Pan AmericanUSA

Personalised recommendations