Advertisement

Error Free Self-assembly Using Error Prone Tiles

  • Ho-Lin Chen
  • Ashish Goel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3384)

Abstract

DNA self-assembly is emerging as a key paradigm for nano-technology, nano-computation, and several related disciplines. In nature, DNA self-assembly is often equipped with explicit mechanisms for both error prevention and error correction. For artificial self-assembly, these problems are even more important since we are interested in assembling large systems with great precision.

We present an error-correction scheme, called snaked proof-reading, which can correct both growth and nucleation errors in a self-assembling system. This builds upon an earlier construction of Winfree and Bekbolatov [11], which could correct a limited class of growth errors. Like their construction, our system also replaces each tile in the system by a k × k block of tiles, and does not require changing the basic tile assembly model proposed by Rothemund and Winfree [8].

We perform a theoretical analysis of our system under fairly general assumptions: tiles can both attach and fall off depending on the thermodynamic rate parameters which also govern the error rate. We prove that with appropriate values of the block size, a seed row of n tiles can be extended into an n × n square of tiles without errors in expected time \(\widetilde{O}(n)\), and further, this square remains stable for an expected time of \(\widetilde{\Omega}(n)\). This is the first error-correction system for DNA self-assembly that has provably good assembly time (close to linear) and provable error-correction. The assembly time is thesame, up to logarithmic factors, as the time for an irreversible, error-free assembly.

We also did a preliminary simulation study of our scheme. In simulations, our scheme performs much better (in terms of error-correction) than the earlier scheme of Winfree and Bekbolatov, and also much better than the unaltered tile system.

Our basic construction (and analysis) applies to all rectilinear tile systems (where growth happens from south to north and west to east). These systems include the Sierpinski tile system, the square-completion tile system, and the block cellular automata for simulating Turing machines. It also applies to counters, a basic primitive in many self-assembly constructions and computations.

Keywords

Tile System Growth Error Block Error Single Tile Tile Assembly Model 
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.: Towards a mathematical theory of self-assembly. Technical Report 00-722, Department of Computer Science, University of Southern California (2000)Google Scholar
  2. 2.
    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 Press, New York (2001)CrossRefGoogle Scholar
  3. 3.
    Adleman, L., Cheng, Q., Goel, A., Huang, M.-D., Kempe, D., Moisset de Espans, P., Rothemund, P.: Combinatorial optimization problems in self-assembly. In: Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pp. 23–32. ACM Press, New York (2002)CrossRefGoogle Scholar
  4. 4.
    Chen, H., Cheng, Q., Goel, A., Huang, M.-D., Moisset de Espans, P.: Invadable self-assembly: Combining robustness with efficiency. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 883–892 (2004)Google Scholar
  5. 5.
    Lagoudakis, M., LaBean, T.: 2D DNA self-assembly for satisfiability. In: Proceedings of the 5th DIMACS Workshop on DNA Based Computers in DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 54, MIT, Cambridge (1999)Google Scholar
  6. 6.
    Reif, J.: Local parallel biomolecular computation. In: Rubin, H. (ed.) Third Annual DIMACS Workshop on DNA Based Computers. DIMACS Series in Discrete Mathematics and Theoretical Computer Science (1998)Google Scholar
  7. 7.
    Rothemund, P.: Theory and Experiments in Algorithmic Self-Assembly. PhD thesis, University of Southern California (2001)Google Scholar
  8. 8.
    Rothemund, P., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: Proceedings of the thirty-second annual ACM symposium on Theory of computing, pp. 459–468. ACM Press, New York (2000)CrossRefGoogle Scholar
  9. 9.
    Wang, H.: Proving theorems by pattern recognition II. Bell Systems Technical Journal 40, 1–42 (1961)Google Scholar
  10. 10.
    Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology, Pasadena (1998)Google Scholar
  11. 11.
    Winfree, E., Bekbolatov, R.: Proofreading tile sets: Error correction for algorithmic self-assembly. In: Chen, J., Reif, J.H. (eds.) DNA 2003. LNCS, vol. 2943. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Winfree, E., Liu, F., Wenzler, L., Seeman, N.: Design and self-assembly of two-dimensional DNA crystals. Nature 394, 539–544 (1998)CrossRefGoogle Scholar
  13. 13.
    Winfree, E., Yang, X., Seeman, N.: Universal computation via self-assembly of DNA: Some theory and experiments. In: Proceedings of the Second Annual Meeting on DNA Based Computers, Princeton University, Princeton (1996)Google Scholar
  14. 14.
    Winfree, E., et al.: The xgrow simulator, http://www.dna.caltech.edu/Xgrow/xgrow_www.html
  15. 15.
    Yurke, B., Turberfield, A., Mills Jr, A., Simmel, F., Neumann, J.: A DNA-fuelled molecular machine made of DNA. Nature 406, 605–608 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ho-Lin Chen
    • 1
  • Ashish Goel
    • 2
  1. 1.Department of Computer ScienceStanford University 
  2. 2.Department of Management Science and Engineering and (by courtesy) Computer ScienceStanford UniversityStanfordUSA

Personalised recommendations