DNA 2005: DNA Computing pp 257-274

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

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

## Abstract

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.

## Keywords

Black Vertex Negative Edge Connector Vertex Target Graph Graph Assembly
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.

## References

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)
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)
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)
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)
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)
16. 16.
Jonoska, N., Karl, S.A., Saito, M.: Three dimensional DNA structures in computing. BioSystems 52, 143–153 (1999)
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)
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)
24. 24.
Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)
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)
26. 26.
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)
28. 28.
Robinson, R.M.: Undecidability and non periodicity of tilings of the plane. Inventiones Math. 12, 177–209 (1971)
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)
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)
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)
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)
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)
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)
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)
40. 40.
Wang, H.: Proving theorems by pattern recognition ii. Bell Systems Technical Journal 40, 1–41 (1961)
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)
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)
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)

© 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