Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

A Computational Model for Self-assembling Flexible Tiles

  • Conference paper
Unconventional Computation (UC 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3699))

Included in the following conference series:

Abstract

We present a theoretical model for self-assembling tiles with flexible branches motivated by DNA branched junction molecules. We encode an instance of a “problem” as a pot of such tiles, and a “solution” as an assembled complete complex without any free sticky ends (called ports), whose number of tiles is within predefined bounds. We develop an algebraic representation of this self-assembly process and use it to prove that this model of self-assembly precisely captures NP-computability when the number of tiles in the minimal complete complexes is bounded by a polynomial.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Adleman, L.: Molecular computation of solutions of combinatorial problems. Science 266, 1021–1024 (1994)

    Article  Google Scholar 

  2. Adleman, L.M., Cheng, Q., Goel, A., Huang, M.-D., Kempe, D., Moisset de Espanés, P., Rothemund, P.W.K.: Combinatorial optimization problems in self-assembly. In: STOC 2002 Proceedings, Montreal Quebec, Canada (2002)

    Google Scholar 

  3. Adleman, L.M., Kari, J., Kari, L., Reishus, D.: On the decidability of self-assembly of infinite ribbons. In: Proceedings of FOCS 2002, IEEE Symposium on Foundations of Computer Science, Washington, pp. 530–537 (2002)

    Google Scholar 

  4. Borosh, I., Treybig, L.B.: Bounds on the positive integral solutions of linear Diophantine equations. Proc. Amer. Math. Soc. 55, 299–304 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  5. Braich, R.S., Chelyapov, N., Johnson, C., Rothemund, P.W.K., Adleman, L.: Solution of a 20-variable 3-SAT problem on a DNA Computer. Science 296, 499–502 (2002)

    Article  Google Scholar 

  6. Chen, J.H., Seeman, N.C.: Synthesis from DNA of a molecule with the connectivity of a cube. Nature 350, 631–633 (1991)

    Article  Google Scholar 

  7. Cook, S.: The complexity of theorem proving procedures. In: Proc. 3rd. ACM Ann. Symp. Theory of Comput., pp. 151–158 (1971)

    Google Scholar 

  8. Ebbinghaus, H.-D., Flum, J.: Finite Model Theory, 2nd edn. Springer, Heidelberg (1999)

    MATH  Google Scholar 

  9. Fagin, R.: Contributions to the Model Theory of Finite Structures, UC Berkeley Ph.D. Thesis (1973)

    Google Scholar 

  10. Faulhammer, D., Curkas, A.R., Lipton, R.J., Landweber, L.F.: Molecular computation: RNA solution to chess problems. PNAS 97, 1385–1389 (2000)

    Article  Google Scholar 

  11. Fu, T.J., Seeman, N.C.: DNA double crossover structures. Biochemistry 32, 3211–3220 (1993)

    Article  Google Scholar 

  12. Garey, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman, New York (1979)

    MATH  Google Scholar 

  13. Head, T., et al.: Computing with DNA by operating on plasmids. BioSystems 57, 87–93 (2000)

    Article  Google Scholar 

  14. Immerman, N.: Descriptive Complexity Theory. Springer, Heidelberg (1999)

    Google Scholar 

  15. Jonoska, N., Sa-Ardyen, P., Seeman, N.C.: Computation by self-assembly of DNA graphs. Genetic Programming and Evolvable Machines 4, 123–137 (2003)

    Article  Google Scholar 

  16. Jonoska, N., Karl, S., Saito, M.: Three dimensional DNA structures in computing. BioSystems 52, 143–153 (1999)

    Article  Google Scholar 

  17. Jonoska, N., Karl, S., Saito, M.: Creating 3-dimensional graph structures with DNA. In: Rubin, H., Wood, D. (eds.) DNA based computers III. AMS DIMACS series, vol. 48, pp. 123–136 (1999)

    Google Scholar 

  18. Jonoska, N., McColm, G., Staninska, A.: Expectation and Variance of Self-Assembled Graph Structures. In: Proceedings of 11th International Meeting on DNA Computing, London, Ontario, Canada, June 6-9, pp. 49–58 (2005)

    Google Scholar 

  19. Jonoska, N., McColm, G.: On a Model of Computation by Self-Assembly (in preparation)

    Google Scholar 

  20. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, W.E. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

    Google Scholar 

  21. Kao, M.-Y., Ramachandran, V.: DNA self-assembly for constructing 3D boxes. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 429–440. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  22. LaBean, T.H., Yan, H., Kopatsch, J., Liu, F., Winfree, E., Reif, J.H., Seeman, N.C.: J. Am. Chem. Soc. 122, 1848–1860 (2000)

    Article  Google Scholar 

  23. Lehn, J.M.: Sopramolecular Chemistry. Science 260, 1762–1763 (1993)

    Article  Google Scholar 

  24. Lehn, J.M.: Toward complex matter: Supramolecular chemistry and self-organization. Proceedings of the National Academy of Science USA 99(8), 4763–4768 (2002)

    Article  Google Scholar 

  25. Mao, C., LaBean, T.H., Reif, J.H., Seeman, N.C.: Logical computation using algorithmic self-assembly of DNA triple-crossover molecules. Nature 407, 493–496 (2000)

    Article  Google Scholar 

  26. Mao, C., Sun, W., Shen, Z., Seeman, N.C.: A nanomechanical device based on the B-Z transition of DNA. Nature 397, 144–146 (1999)

    Article  Google Scholar 

  27. Reif, J.H., Sahu, S., Yin, P.: Complexity of Graph Self-Assembly in Accretive Systems and Self-Destructive Systems. In: Proceedings of 11th International Meeting on DNA Computing, London, Ontario, Canada, June 6-9, pp. 101–112 (2005)

    Google Scholar 

  28. Rothemund, P.W.K., Winfree, E.: The Program-Size Complexity of Self-Assembled Squares. In: Proceedings of 33rd ACM meeting STOC 2001, Portland, Oregon, May 21-23, pp. 459–468 (2001)

    Google Scholar 

  29. Rothemund, P., Papadakis, N., Winfree, E.: Algorithmic Self-assembly of DNA Sierpinski Triangles. PLoS Biology 2(12), e424 (2004) (13 pages)

    Google Scholar 

  30. Sa-Ardyen, P., Jonoska, N., Seeman, N.: Self-assembly of graphs represented by DNA helix axis topology. J. Am. Chem. Soc. 126(21), 6648–6657 (2004)

    Article  Google Scholar 

  31. Seeman, N.C.: DNA junctions and lattices. J. Theor. Biol. 99, 237–247 (1982)

    Article  Google Scholar 

  32. Seeman, N.C.: DNA nicks and nodes and nanotechnology. NanoLetters 1, 22–26 (2001)

    Google Scholar 

  33. Shihn, W.M., Quispe, J.D., Joyce, G.F.: A 1.7-kilobase single-stranded DNA folds into a nano-scale octahedron. Nature 427, 618–621 (2004)

    Article  Google Scholar 

  34. Soloveichik, D., Winfree, E.: Complexity of Self-Assembled Shapes, preprint at, http://arxiv.org/abs/cs.CC/0412096

  35. Wang, Y., Mueller, J.E., Kemper, B., Seeman, N.C.: The assembly and characterization of 5-arm and 6-arm DNA junctions. Biochemistry 30, 5667–5674 (1991)

    Article  Google Scholar 

  36. Winfree, E., Yang, X., Seeman, N.C.: Universal computation via self-assembly of DNA: some theory and experiments. In: Landweber, L., Baum, E. (eds.) DNA based computers II. AMS DIMACS series, vol. 44, pp. 191–214 (1998)

    Google Scholar 

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

    Article  Google Scholar 

  38. Yan, H., Zhang, X., Shen, Z., Seeman, N.C.: A robust DNA mechanical device controlled by hybridization topology. Nature 415, 62–65 (2002)

    Article  Google Scholar 

  39. Yurke, B., Turberfield, A.J., Mills, A.P., Simmel Jr, F.C.: A DNA fueled molecular machine made of DNA. Nature 406, 605–608 (2000)

    Article  Google Scholar 

  40. Zhang, Y., Seeman, N.C.: The construction of a DNA truncated octahedron. J. Am. Chem. Soc. 160, 1661–1669 (1994)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of South Florida Tampa, FL, 33620, USA

    Nataša Jonoska & Gregory L. McColm

Authors
  1. Nataša Jonoska
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gregory L. McColm
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Auckland, New Zealand

    Cristian S. Calude

  2. Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand

    Michael J. Dinneen

  3. Institute of Mathematics of the Romanian Academy, PO Box 1-764, 014700, Bucureşti, Romania

    Gheorghe Păun

  4. Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Seville, Avda. Reina Mercedes, 41012, Sevilla, Spain

    Mario J. Pérez-Jímenez

  5. Leiden Center of Advanced Computer Science (LIACS), Leiden University, Niels Bohrweg 1, 2333, Leiden, CA, The Netherlands

    Grzegorz Rozenberg

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Jonoska, N., McColm, G.L. (2005). A Computational Model for Self-assembling Flexible Tiles. In: Calude, C.S., Dinneen, M.J., Păun, G., Pérez-Jímenez, M.J., Rozenberg, G. (eds) Unconventional Computation. UC 2005. Lecture Notes in Computer Science, vol 3699. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11560319_14

Download citation

  • DOI: https://doi.org/10.1007/11560319_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29100-8

  • Online ISBN: 978-3-540-32022-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation