Optimizing Tile Concentrations to Minimize Errors and Time for DNA Tile Self-assembly Systems

  • Ho-Lin Chen
  • Ming-Yang Kao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6518)

Abstract

DNA tile self-assembly has emerged as a rich and promising primitive for nano-technology. This paper studies the problems of minimizing assembly time and error rate by changing the tile concentrations because changing the tile concentrations is easy to implement in actual lab experiments. We prove that setting the concentration of tile Ti proportional to the square root of Ni where Ni is the number of times Ti appears outside the seed structure in the final assembled shape minimizes the rate of growth errors for rectilinear tile systems. We also show that the same concentrations minimize the expected assembly time for a feasible class of tile systems. Moreover, for general tile systems, given tile concentrations, we can approximate the expected assembly time with high accuracy and probability by running only a polynomial number of simulations in the size of the target shape.

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References

  1. 1.
    Adleman, L., Cheng, Q., Goel, A., Huang, M.-D.: Running time and program size for self-assembled squares. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 740–748 (2001)Google Scholar
  2. 2.
    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 34th Annual ACM Symposium on Theory of Computing, pp. 23–32 (2002)Google Scholar
  3. 3.
    Barish, R.D., Rothemund, P.W.K., Winfree, E.: Two computational primitives for algorithmic self-assembly: Copying and counting. Nano Letters 5(12), 2586–2592 (2005)CrossRefGoogle Scholar
  4. 4.
    Barish, R.D., Schulman, R., Rothemund, P.W.K., Winfree, E.: An information-bearing seed for nucleating algorithmic self-assembly. Proceedings of the National Academy of Sciences 106, 6054–6059 (2009)CrossRefGoogle Scholar
  5. 5.
    Bishop, J., Klavins, E.: An improved autonomous DNA nanomotor. Nano Letters 7(9), 2574–2577 (2007)CrossRefGoogle Scholar
  6. 6.
    Chen, H.-L., Goel, A.: Error free self-assembly using error prone tiles. In: Proceedings of the 10th International Meeting on DNA Based Computers, pp. 62–75 (2004)Google Scholar
  7. 7.
    Chen, H.-L., Luhrs, C., Goel, A.: Dimension augmentation and combinatorial criteria for efficient error-resistant DNA self-assembly. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 409–418 (2008)Google Scholar
  8. 8.
    Cheng, Q., Goel, A., Moisset, P.: Optimal self-assembly of counters at temperature two. In: Proceedings of the 1st Conference on Foundations of Nanoscience: Self-Assembled Architectures and Devices, pp. 62–75 (2004)Google Scholar
  9. 9.
    Dietz, H., Douglas, S., Shih, W.: Folding DNA into twisted and curved nanoscale shapes. Science 325, 725–730 (2009)CrossRefGoogle Scholar
  10. 10.
    Ding, B., Seeman, N.: Operation of a DNA robot arm inserted into a 2D DNA crystalline substrate. Science 384, 1583–1585 (2006)CrossRefGoogle Scholar
  11. 11.
    Doty, D.: Randomized self-assembly for exact shapes. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 85–94 (2009)Google Scholar
  12. 12.
    Douglas, S., Dietz, H., Liedl, T., Hogberg, B., Graf, F., Shih, W.: Self-assembly of DNA into nanoscale three-dimensional shapes. Nature (459), 414–418 (2009)Google Scholar
  13. 13.
    Green, S., Bath, J., Turberfield, A.: Coordinated chemomechanical cycles: a mechanism for autonomous molecular motion. Physical Review Letters (101), 238101 (2008)Google Scholar
  14. 14.
    Sahu, S., Reif, J., Yin, P.: Compact error-resilient computational DNA tiling assemblies. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA 2004. LNCS, vol. 3384, pp. 293–307. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Kao, M.-Y., Schweller, R.: Reducing tile complexity for self-assembly through temperature programming. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 571–580 (2006)Google Scholar
  16. 16.
    Lagoudakis, M., LaBean, T.: 2D DNA self-assembly for satisfiability. In: Proceedings of the 5th DIMACS Workshop on DNA Based Computers. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 54, pp. 141–154 (1999)Google Scholar
  17. 17.
    Rothemund, P., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pp. 459–468 (2000)Google Scholar
  18. 18.
    Rothemund, P.W.K.: Folding DNA to create nanoscale shapes and patterns. Nature (440), 297–302 (March 2006)Google Scholar
  19. 19.
    Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLOS Biology 2, 424–436 (2004)CrossRefGoogle Scholar
  20. 20.
    Schulman, R., Winfree, E.: Programmable control of nucleation for algorithmic self-assembly. In: Proceedings of the 10th International Meeting on DNA Based Computers, pp. 319–328 (2004)Google Scholar
  21. 21.
    Schulman, R., Winfree, E.: Self-replication and evolution of DNA crystals. In: Proceedings of the 5th European Conference on Artificial Life, pp. 734–743 (2005)Google Scholar
  22. 22.
    Seelig, G., Soloveichik, D., Zhang, D., Winfree, E.: Enzyme-free nucleic acid logic circuits. Science 314, 1585–1588 (2006)CrossRefGoogle Scholar
  23. 23.
    Sherman, W.B., Seeman, N.C.: A precisely controlled DNA bipedal walking device. Nano Letters 4, 1203–1207 (2004)CrossRefGoogle Scholar
  24. 24.
    Shih, W.M., Quispe, J.D., Joyce, G.F.A.: A 1.7-kilobase single-stranded DNA that folds into a nanoscale octahedron. Nature (427), 618–621 (2004)Google Scholar
  25. 25.
    Shin, J.-S., Pierce, N.A.: A synthetic DNA walker for molecular transport. Journal of American Chemistry Society 126, 10834–10835 (2004)CrossRefGoogle Scholar
  26. 26.
    Soloveichik, D., Cook, M., Winfree, E.: Combining self-healing and proofreading in self-assembly. Natural Computing (7), 203–218 (2008)Google Scholar
  27. 27.
    Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM Journal on Computing 36, 1544–1569 (2007)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wang, H.: Proving theorems by pattern recognition ii. Bell Systems Technical Journal 40, 1–42 (1961)CrossRefGoogle Scholar
  29. 29.
    Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology, Pasadena (1998)Google Scholar
  30. 30.
    Winfree, E., Bekbolatov, R.: Proofreading tile sets: Error correction for algorithmic self-assembly. In: Proceedings of the 9th International Meeting on DNA Based Computers, pp. 126–144 (2003)Google Scholar
  31. 31.
    Winfree, E., Liu, F., Wenzler, L., Seeman, N.: Design and self-assembly of two-dimensional DNA crystals, 6 pages. Nature (394), 539–544 (August 1998)Google Scholar
  32. 32.
    Yurke, B., Turberfield, A., Mills Jr., A., Simmel, F., Neumann, J.: A DNA-fuelled molecular machine made of DNA. Nature (406), 605–608 (August 2000)Google Scholar
  33. 33.
    Zhang, Y., Seeman, N.: Construction of a DNA-truncated octahedron. Journal of American Chemical Society 116(5), 1661 (1994)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ho-Lin Chen
    • 1
    • 2
  • Ming-Yang Kao
    • 1
    • 2
  1. 1.Center for Mathematics of InformationCalifornia Institute of TechnologyPasadenaUSA
  2. 2.Department of Electrical Engineering and Computer ScienceNorthwestern UniversityEvanstonUSA

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