Complexity of Graph Self-assembly in Accretive Systems and Self-destructible Systems

  • John H. Reif
  • Sudheer Sahu
  • Peng Yin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3892)


Self-assembly is a process in which small objects autonomously associate with each other to form larger complexes. It is ubiquitous in biological constructions at the cellular and molecular scale and has also been identified by nanoscientists as a fundamental method for building nano-scale structures. Recent years see convergent interest and efforts in studying self-assembly from mathematicians, computer scientists, physicists, chemists, and biologists. However most complexity theoretic studies of self-assembly utilize mathematical models with two limitations: 1) only attraction, while no repulsion, is studied; 2) only assembled structures of two dimensional square grids are studied. In this paper, we study the complexity of the assemblies resulting from the cooperative effect of repulsion and attraction in a more general setting of graphs. This allows for the study of a more general class of self-assembled structures than the previous tiling model. We define two novel assembly models, namely the accretive graph assembly model and the self-destructible graph assembly model, and identify one fundamental problem in them: the sequential construction of a given graph, referred to as Accretive Graph Assembly Problem (\(\textsc{AGAP}\)) and Self-Destructible Graph Assembly Problem (\(\textsc{DGAP}\)), respectively. Our main results are: (i) \(\textsc{AGAP}\)  is NP-complete even if the maximum degree of the graph is restricted to 4 or the graph is restricted to be planar with maximum degree 5; (ii) counting the number of sequential assembly orderings that result in a target graph (\(\textsc{\#AGAP}\)) is #P-complete; and (iii) \(\textsc{DGAP}\)  is PSPACE-complete even if the maximum degree of the graph is restricted to 6 (this is the first PSPACE-complete result in self-assembly). We also extend the accretive graph assembly model to a stochastic model, and prove that determining the probability of a given assembly in this model is #P-complete.


Black Vertex Negative Edge Connector Vertex Target Graph Graph Assembly 
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  1. 1.
  2. 2.
    Adleman, L.: Towards a mathematical theory of self-assembly. Technical Report 00-722, University of Southern California (2000)Google Scholar
  3. 3.
    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)Google Scholar
  4. 4.
    Adleman, L., Cheng, Q., Goel, A., Huang, M.D., Kempe, D., de Espans, P.M., Rothemund, P.W.K.: Combinatorial optimization problems in self-assembly. In: Proceedings of the thirty-fourth annual ACM symposium on Theory of computing, pp. 23–32. ACM Press, New York (2002)Google Scholar
  5. 5.
    Adleman, L., Cheng, Q., Goel, A., Huang, M.D., Wasserman, H.: Linear self-assemblies: Equilibria, entropy, and convergence rate. In: Sixth International Conference on Difference Equations and Applications (2001)Google Scholar
  6. 6.
    Aggarwal, G., Goldwasser, M.H., Kao, M.Y., Schweller, R.T.: Complexities for generalized models of self-assembly. In: Proceedings of 15th annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 880–889. ACM Press, New York (2004)Google Scholar
  7. 7.
    Bowden, N., Terfort, A., Carbeck, J., Whitesides, G.M.: Self-assembly of mesoscale objects into ordered two-dimensional arrays. Science 276(11), 233–235 (1997)CrossRefGoogle Scholar
  8. 8.
    Bruinsma, R.F., Gelbart, W.M., Reguera, D., Rudnick, J., Zandi, R.: Viral self-assembly as a thermodynamic process. Phys. Rev. Lett. 90(24), 248101 (2003)CrossRefMATHGoogle Scholar
  9. 9.
    Chelyapov, N., Brun, Y., Gopalkrishnan, M., Reishus, D., Shaw, B., Adleman, L.: DNA triangles and self-assembled hexagonal tilings. J. Am. Chem. Soc. 126, 13924–13925 (2004)CrossRefGoogle Scholar
  10. 10.
    Chen, H.L., Cheng, Q., Goel, A., Huang, M.D., de Espanes, P.M.: Invadable self-assembly: Combining robustness with efficiency. In: Proceedings of the 15th annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 890–899 (2004)Google Scholar
  11. 11.
    Chen, H.L., Goel, A.: Error free self-assembly using error prone tiles. In: DNA Based Computers 10, pp. 274–283 (2004)Google Scholar
  12. 12.
    Cheng, Q., Goel, A., Moisset, P.: Optimal self-assembly of counters at temperature two. In: Proceedings of the first conference on Foundations of nanoscience: self-assembled architectures and devices (2004)Google Scholar
  13. 13.
    Cook, M., Rothemund, P.W.K., Winfree, E.: Self-assembled circuit patterns. In: Chen, J., Reif, J.H. (eds.) DAN 2003. LNCS, vol. 2943, pp. 91–107. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Fujibayashi, K., Murata, S.: A method for error suppression for self-assembling DNA tiles. In: DNA Based Computing 10, pp. 284–293 (2004)Google Scholar
  15. 15.
    He, Y., Chen, Y., Liu, H., Ribbe, A.E., Mao, C.: Self-assembly of hexagonal DNA two-dimensional (2D) arrays. J. Am. Chem. Soc. 127, 12202–12203 (2005)CrossRefGoogle Scholar
  16. 16.
    Jonoska, N., Karl, S.A., Saito, M.: Three dimensional DNA structures in computing. BioSystems 52, 143–153 (1999)CrossRefGoogle Scholar
  17. 17.
    Jonoska, N., McColm, G.L.: A computational model for self-assembling flexible tiles. Unconventional Computing (to appear, 2005)Google Scholar
  18. 18.
    Jonoska, N., Sa-Ardyen, P., Seeman, N.C.: Genetic programming and evolvable machines. Computation by Self-assembly of DNA Graphs 4Google Scholar
  19. 19.
    Kao, M., Schweller, R.: Reduce complexity for tile self-assembly through temperature programming. In: Proceedings of 17th annual ACM-SIAM Symposium on Discrete Algorithms (SODA). ACM Press, New York (to appear, 2006)Google Scholar
  20. 20.
    Klavins, E.: Toward the control of self-assembling systems. In: Control Problems in Robotics, vol. 4, pp. 153–168. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  21. 21.
    Klavins, E.: Directed self-assembly using graph grammars. In: Foundations of Nanoscience: Self Assembled Architectures and Devices, Snowbird, UT (2004)Google Scholar
  22. 22.
    Klavins, E., Ghrist, R., Lipsky, D.: Graph grammars for self-assembling robotic systems. In: Proceedings of the International Conference on Robotics and Automation (2004)Google Scholar
  23. 23.
    LaBean, T.H., Yan, H., Kopatsch, J., Liu, F., Winfree, E., Reif, J.H., Seeman, N.C.: The construction, analysis, ligation and self-assembly of DNA triple crossover complexes. J. Am. Chem. Soc. 122, 1848–1860 (2000)CrossRefGoogle Scholar
  24. 24.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Malo, J., Mitchell, J.C., Venien-Bryan, C., Harris, J.R., Wille, H., Sherratt, D.J., Turberfield, A.J.: Engineering a 2D protein-DNA crystal. Angew. Chem. Intl. Ed. 44, 3057–3061 (2005)CrossRefGoogle Scholar
  26. 26.
    Papadimitriou, C.M.: Computational complexity, 1st edn. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  27. 27.
    Reif, J.H., Sahu, S., Yin, P.: Compact error-resilient computational DNA tiling assemblies. In: Ferretti, C., Mauri, G., Zandron, C. (eds.) DNA 2004. LNCS, vol. 3384, pp. 248–260. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  28. 28.
    Robinson, R.M.: Undecidability and non periodicity of tilings of the plane. Inventiones Math. 12, 177–209 (1971)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Rothemund, P.W.K.: Using lateral capillary forces to compute by self-assembly. Proc. Natl. Acad. Sci. USA 97(3), 984–989 (2000)CrossRefGoogle Scholar
  30. 30.
    Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA sierpinski triangles. PLoS Biology 2(12), 2:e424 (2004)Google Scholar
  31. 31.
    Rothemund, P.W.K., 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)Google Scholar
  32. 32.
    Sa-Ardyen, P., Jonoska, N., Seeman, N.C.: Self-assembling DNA graphs. In: Hagiya, M., Ohuchi, A. (eds.) DNA 2002. LNCS, vol. 2568, pp. 1–9. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  33. 33.
    Sahu, S., Yin, P., Reif, J.H.: A self assembly model of time-dependent glue strength. In: Carbone, A., Pierce, N.A. (eds.) DNA 2005. LNCS, vol. 3892, pp. 113–124. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  34. 34.
    Schulman, R., Lee, S., Papadakis, N., Winfree, E.: One dimensional boundaries for DNA tile self-assembly. In: Chen, J., Reif, J.H. (eds.) DAN 2003. LNCS, vol. 2943, pp. 108–125. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  35. 35.
    Schulman, R., Winfree, E.: Programmable control of nucleation for algorithmic self-assembly. In: DNA Based Computers 10. LNCS (2005)Google Scholar
  36. 36.
    Schulman, R., Winfree, E.: Self-replication and evolution of DNA crystals. In: Capcarrère, M.S., Freitas, A.A., Bentley, P.J., Johnson, C.G., Timmis, J. (eds.) ECAL 2005. LNCS, vol. 3630, pp. 734–743. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  37. 37.
    Soloveichik, D., Winfree, E.: Complexity of compact proofreading for self-assembled patterns. In: Carbone, A., Pierce, N.A. (eds.) DNA 2005. LNCS, vol. 3892, pp. 125–135. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  38. 38.
    Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. In: DNA Based Computers 10. LNCS. Springer, Heidelberg (2005)Google Scholar
  39. 39.
    Strasser, A., O’Connor, L., Dixit, V.M.: Apoptosis signaling. Annu. Rev. Biochem. 69, 217–245 (2000)CrossRefGoogle Scholar
  40. 40.
    Wang, H.: Proving theorems by pattern recognition ii. Bell Systems Technical Journal 40, 1–41 (1961)CrossRefGoogle Scholar
  41. 41.
    Winfree, E.: Self-healing tile sets. Draft (2005)Google Scholar
  42. 42.
    Winfree, E., Bekbolatov, R.: Proofreading tile sets: Error correction for algorithmic self-assembly. In: Chen, J., Reif, J.H. (eds.) DAN 2003. LNCS, vol. 2943, pp. 126–144. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  43. 43.
    Winfree, E., Liu, F., Wenzler, L.A., Seeman, N.C.: Design and self-assembly of two-dimensional DNA crystals. Nature 394(6693), 539–544 (1998)CrossRefGoogle Scholar
  44. 44.
    Yan, H., LaBean, T.H., Feng, L., Reif, J.H.: Directed nucleation assembly of DNA tile complexes for barcode patterned DNA lattices. Proc. Natl. Acad. Sci. USA 100(14), 8103–8108 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • John H. Reif
    • 1
  • Sudheer Sahu
    • 1
  • Peng Yin
    • 1
  1. 1.Department of Computer ScienceDuke UniversityDurhamUSA

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