Advertisement

Applying Symmetric Enumeration Method to One-Dimensional Assembly of Rotatable Tiles

  • Satoshi KobayashiEmail author
Chapter
Part of the Natural Computing Series book series (NCS)

Abstract

Motivated by the increasing importance of the analysis of a complicated reaction system where molecules are interacting in various ways to produce a huge number of compounds of molecules, the author’s previous work proposed a new approach, called Symmetric Enumeration Method (SEM), to the efficient analysis of such reaction systems. The proposed theory provided a general method for the efficient computation of equilibrium states. In this paper, we will review the results of the theory of SEM, and apply it to the equilibria analysis of a one-dimensional assembly system of tiles which can be rotated around three axes, i.e., the x-, y-, and z-axes. Although the growth of a tile assembly is restricted to only one-dimensional direction, the exhaustive method of generating all tile assemblies and obtaining equilibria is intractable since the number of assemblies could be exponential with respect to the number of tiles given to the system. We will show that this equilibria analysis can be transformed into a convex programming problem with a set of variables of size polynomial with respect to the number of input tiles.

Keywords

Assembly System Enumeration Scheme Free Energy Function Graph Component Convex Programming Problem 
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 (1994) Molecular computation of solutions to combinatorial problems. Science 266:1021–1024 CrossRefGoogle Scholar
  2. 2.
    Adleman L, Cheng Q, Goel A, Huang M, Wasserman H (2000) Linear self-assemblies: equilibria, entropy, and convergence rates. Unpublished manuscript Google Scholar
  3. 3.
    Benneson A, Gil B, Ben-Dor U, Adar R, Shapiro E (2004) An autonomous molecular computer for logical control of gene expression. Nature 429:423–429 CrossRefGoogle Scholar
  4. 4.
    Condon AE (2003) Problems on rna secondary structure prediction and design. In: Proceedings of ICALP’2003. Lecture notes in computer science, vol 2719. Springer, Berlin, pp 22–32 Google Scholar
  5. 5.
    Dirks R, Bois J, Schaeffer J, Winfree E, Pierce N (2007) Thermodynamic analysis of interacting nucleic acid strands. SIAM Rev 49:65–88 zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gallo G, Longo G, Nguyen S, Pallottino S (1993) Directed hypergraphs and applications. Discrete Appl Math 40:177–201 CrossRefMathSciNetGoogle Scholar
  7. 7.
    Gallo G, Scutella MG (1999) Directed hypergraphs as a modelling paradigm. Technical report TR-99-02, Dipartmento di Informatica, Universita di Pisa Google Scholar
  8. 8.
    Kobayashi S (2006) A new approach to computing equilibrium state of combinatorial chemical reaction systems. Technical report CS 06-01, Department of Computer Science, University of Electro-Communications Google Scholar
  9. 9.
    Kobayashi S (2007) A new approach to computing equilibrium state of combinatorial hybridization reaction systems. In: Proceedings of workshop on computing and communications from biological systems: theory and applications. CD-ROM, paper 2376 Google Scholar
  10. 10.
    Kobayashi S (2008) Symmetric enumeration method: a new approach to computing equilibria. Technical report CS 08-01, Department of Computer Science, University of Electro-Communications Google Scholar
  11. 11.
    Lipton RJ (1995) DNA solution of hard computational problems. Science 268:542–545 CrossRefGoogle Scholar
  12. 12.
    Nesterov Y, Nemirovskii A (1993) Interior-point polynomial algorithms in convex programming. SIAM, Philadelphia Google Scholar
  13. 13.
    Păun G, Rozenberg G, Salomaa A (1998) DNA computing—new computing paradigms. Texts in theoretical computer science—an EATCS series. Springer, Berlin zbMATHGoogle Scholar
  14. 14.
    Rothemund P, Papadakis N, Winfree E (2004) Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol 2:e424 CrossRefGoogle Scholar
  15. 15.
    Winfree E, Liu F, Wenzler L, Seeman NC (1998) Design and self-assembly of two-dimensional DNA crystals. Nature 394:539–544 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Electro-CommunicationsTokyoJapan

Personalised recommendations