Less Haste, Less Waste: On Recycling and Its Limits in Strand Displacement Systems

  • Anne Condon
  • Alan Hu
  • Ján Maňuch
  • Chris Thachuk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6937)


We study the potential for molecule recycling in chemical reaction systems and their DNA strand displacement realizations. Recycling happens when a product of one reaction is a reactant in a later reaction. Recycling has the benefits of reducing consumption, or waste, of molecules and of avoiding fuel depletion. We present a binary counter that recycles molecules efficiently while incurring just a moderate slowdown compared to alternative counters that do not recycle strands. This counter is an n-bit binary reflecting Gray code counter that advances through 2 n states. In the strand displacement realization of this counter, the waste—total number of nucleotides of the DNA strands consumed—is O(n 3), while alternative counters have Ω(2 n ) waste. We also show that our n-bit counter fails to work correctly when Θ(n) copies of the species that represent the state (bits) of the counter are present initially. The proof applies more generally to show that a class of chemical reaction systems, in which all but one reactant of each reaction are catalysts, are not capable of computations longer than \(\tfrac{1}{2}n^2\) steps when there are at least n copies.


Signal Molecule Gray Code Strand Displacement Template Strand Initial Species 
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.


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  1. 1.
    Cardelli, L.: Strand algebras for DNA computing. Natural Computing 10(1), 407–428 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Doty, D., Kari, L., Masson, B.: Negative interactions in irreversible self-assembly. In: Sakakibara, Y., Mi, Y. (eds.) DNA 16 2010. LNCS, vol. 6518, pp. 37–48. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1. Wiley, Chichester (1971)zbMATHGoogle Scholar
  4. 4.
    Hongzhou, G., Chao, J., Xiao, S.-J., Seeman, N.C.: A proximity-based programmable DNA nanoscale assembly line. Nature 465, 202–205 (2010)CrossRefGoogle Scholar
  5. 5.
    Kharam, A., Jiang, H., Riedel, M., Parhi, K.: Binary counting with chemical reactions. In: Proceedings of the 2011 Pacific Symposium on Biocomputing, pp. 302–313. World Scientific Publishing, Singapore (2011)Google Scholar
  6. 6.
    Cardelli, L.: Two-domain DNA strand displacement. In: Proc. of Developments in Computational Models (DCM 2010). Electronic Proceedings in Theoretical Computer Science, vol. 26, pp. 47–61 (2010)Google Scholar
  7. 7.
    Lund, K., Manzo, A.T., Dabby, N., Michelotti, N., Johnson-Buck, A., Nangreave, J., Taylor, N., Pei, R., Stojanovic, M.N., Walter, N.G., Winfree, E., Yan, H.: Molecular robots guided by prescriptive landscapes. Nature 465, 206–210 (2010)CrossRefGoogle Scholar
  8. 8.
    Omabegho, T., Sha, R., Seeman, N.C.: A bipedal DNA brownian motor with coordinated legs. Science 324(5923), 67–71 (2009)CrossRefGoogle Scholar
  9. 9.
    Qian, L., Soloveichik, D., Winfree, E.: Efficient turing-universal computation with DNA polymers. In: Sakakibara, Y., Mi, Y. (eds.) DNA 16 2010. LNCS, vol. 6518, pp. 123–140. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Qian, L., Winfree, E.: A simple DNA gate motif for synthesizing large-scale circuits. In: J. R. Soc. Interface (2011)Google Scholar
  11. 11.
    Reif, J.H., Sahu, S., Yin, P.: Complexity of graph self-assembly in accretive systems and self-destructible systems. In: Carbone, A., Pierce, N.A. (eds.) DNA 2005. LNCS, vol. 3892, pp. 257–274. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pp. 459–468 (2000)Google Scholar
  13. 13.
    Savage, C.: A survey of combinatorial Gray codes. SIAM Review 39(4), 605–629 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Schulman, L.J., Zuckerman, D.: Asymptotically good codes correcting insertions, deletions, and transpositions. IEEE Transactions on Information Theory 45, 2552–2557 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Seelig, G., Soloveichik, D., Zhang, D.Y., Winfree, E.: Enzyme-free nucleic acid logic circuits. Science 314(5805), 1585–1588 (2006)CrossRefGoogle Scholar
  16. 16.
    Shin, J.-S., Pierce, N.A.: A synthetic DNA walker for molecular transport. J. Am. Chem. Soc. 126, 10834–10835 (2004)CrossRefGoogle Scholar
  17. 17.
    Soloveichik, D., Seelig, G., Winfree, E.: DNA as a universal substrate for chemical kinetics. Proc. Nat. Acad. Sci. USA 107(12), 5393–5398 (2010)CrossRefGoogle Scholar
  18. 18.
    Venkataraman, S., Dirks, R.M., Rothemund, P.W.K., Winfree, E., Pierce, N.A.: An autonomous polymerization motor powered by DNA hybridization. Nature Nanotech 2(8), 490–494 (2007)CrossRefGoogle Scholar
  19. 19.
    Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, Caltech (1998)Google Scholar
  20. 20.
    Zhang, D.Y., Seelig, G.: Dynamic DNA nanotechnology using strand displacement reactions. Nature Chemistry 3, 103–113 (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Anne Condon
    • 1
  • Alan Hu
    • 1
  • Ján Maňuch
    • 1
  • Chris Thachuk
    • 1
  1. 1.The Department of Computer ScienceUniversity of British ColumbiaVancouverCanada

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