, Volume 66, Issue 1, pp 153–172 | Cite as

Negative Interactions in Irreversible Self-assembly



This paper explores the use of negative (i.e., repulsive) interactions in the abstract Tile Assembly Model defined by Winfree. Winfree in his Ph.D. thesis postulated negative interactions to be physically plausible, and Reif, Sahu, and Yin studied them in the context of reversible attachment operations. We investigate the power of negative interactions with irreversible attachments, and we achieve two main results. Our first result is an impossibility theorem: after t steps of assembly, Ω(t) tiles will be forever bound to an assembly, unable to detach. Thus negative glue strengths do not afford unlimited power to reuse tiles. Our second result is a positive one: we construct a set of tiles that can simulate an s-space-bounded, t-time-bounded Turing machine, while ensuring that no intermediate assembly grows larger than O(s), rather than O(st) as required by the standard Turing machine simulation with tiles. In addition to the space-bounded Turing machine simulation, we show another example application of negative glues: reducing the number of tile types required to assemble “thin” (n×o(logn/loglogn)) rectangles.


Self-assembly Self-destructible Negative bond strength Molecular computing DNA computing Space-bounded computation 



The authors are grateful to Shinnosuke Seki for insightful discussions and anonymous referees for their suggestions.


  1. 1.
    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 2004 MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001) MATHGoogle Scholar
  3. 3.
    Doty, D., Kari, L., Masson, B.: Negative interactions in irreversible self-assembly. In: DNA 16: Proceedings of the Sixteenth International Meeting on DNA Computing and Molecular Programming. Lecture Notes in Computer Science, vol. 6518, pp. 37–48. Springer, Berlin (2010) CrossRefGoogle Scholar
  4. 4.
    Doty, D., Lutz, J.H., Patitz, M.J., Summers, S.M., Woods, D.: Random number selection in self-assembly. In: UC 2009: Proceedings of the Eighth International Conference on Unconventional Computation. Lecture Notes in Computer Science, vol. 5715, pp. 143–157. Springer, Berlin (2009) Google Scholar
  5. 5.
    Doty, D., Lutz, J.H., Patitz, M.J., Summers, S.M., Woods, D.: Intrinsic universality in self-assembly. In: STACS 2010: Proceedings of the Twenty-Seventh International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), vol. 5, pp. 275–286 (2010) Google Scholar
  6. 6.
    Patitz, M.J., Schweller, R.T., Summers, S.M.: Exact shapes and Turing universality at temperature 1 with a single negative glue. In: DNA 17: Proceedings of the 17th International Conference on DNA Computing and Molecular Programming, DNA’11, pp. 175–189. Springer, Berlin (2011) CrossRefGoogle Scholar
  7. 7.
    Reif, J.H., Sahu, S., Yin, P.: Complexity of graph self-assembly in accretive systems and self-destructible systems. Theor. Comput. Sci. 412, 1592–1605 (2011) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    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) CrossRefGoogle Scholar
  9. 9.
    Schoen, R., Yau, S.-T.: On the positive mass conjecture in general relativity. Commun. Math. Phys. 65(45), 45–76 (1979) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Seeman, N.C.: Nucleic-acid junctions and lattices. J. Theor. Biol. 99, 237–247 (1982) CrossRefGoogle Scholar
  11. 11.
    Soloveichik, D., Cook, M., Winfree, E., Bruck, J.: Computation with finite stochastic chemical reaction networks. Nat. Comput. 7(4), 615–633 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Wang, H.: Proving theorems by pattern recognition—II. Bell Syst. Tech. J. XL(1), 1–41 (1961) Google Scholar
  13. 13.
    Winfree, E.: Algorithmic Self-assembly of DNA. Ph.D. thesis, California Institute of Technology (June 1998) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Department of Computer ScienceUniversity of Western OntarioLondonCanada
  3. 3.IRISA (INRIA)RennesFrance

Personalised recommendations