Natural Computing

, Volume 14, Issue 3, pp 491–503 | Cite as

DNA origami and the complexity of Eulerian circuits with turning costs

  • Joanna A. Ellis-Monaghan
  • Andrew McDowell
  • Iain Moffatt
  • Greta Pangborn
Article

Abstract

Building a structure using self-assembly of DNA molecules by origami folding requires finding a route for the scaffolding strand through the desired structure. When the target structure is a 1-complex (or the geometric realization of a graph), an optimal route corresponds to an Eulerian circuit through the graph with minimum turning cost. By showing that it leads to a solution to the 3-SAT problem, we prove that the general problem of finding an optimal route for a scaffolding strand for such structures is NP-hard. We then show that the problem may readily be transformed into a traveling salesman problem (TSP), so that machinery that has been developed for the TSP may be applied to find optimal routes for the scaffolding strand in a DNA origami self-assembly process. We give results for a few special cases, showing for example that the problem remains intractable for graphs with maximum degree 8, but is polynomial time for 4-regular plane graphs if the circuit is restricted to following faces. We conclude with some implications of these results for related problems, such as biomolecular computing and mill routing problems.

Keywords

DNA origami DNA self-assembly Turning cost Eulerian circuit Hamiltonian cycle Threading strand Biomolecular computing Mill routing Computational complexity A-trails 

Mathematics Subject Classification

92E10 05C45 05C85 

Notes

Acknowledgments

We thank Ned Seeman for specific design problems leading to this research and for many related discussions. The work of the first and fourth authors was supported by the National Science Foundation (NSF) under grants DMS-1001408 and EFRI-1332411.

References

  1. Adelman L (1994) Molecular computation of solutions to combinatorial problems. Science 266:1021–1024CrossRefGoogle Scholar
  2. Andersen LD, Fleischner H (1995) The NP-completeness of finding A-trails in Eulerian graphs and of finding spanning trees in hypergraphs. Discret Appl Math 59(3):203–214MathSciNetCrossRefMATHGoogle Scholar
  3. Andersen LD, Bouchet A, Jackson B (1996) Orthogonal A-trails of 4-regular graphs embedded in surfaces of low genus. J Combin Theory Ser B 66(2):232–246MathSciNetCrossRefMATHGoogle Scholar
  4. Andersen LD, Fleischner H, Regner S (1998) Algorithms and outerplanar conditions for A-trails in plane Eulerian graphs. Discret Appl Math 85(2):99–112MathSciNetCrossRefMATHGoogle Scholar
  5. Arkin E, Bender M, Demaine E et al (2005) Optimal covering tours with turn costs. SIAM J Comput 35(3):531–566MathSciNetCrossRefMATHGoogle Scholar
  6. Bent SW, Manber U (1987) On non-intersecting Eulerian circuits. Discret Appl Math 18(1):87–94MathSciNetCrossRefMATHGoogle Scholar
  7. Chartrand G (1964) Graphs and their associated line-graphs. PhD thesis, Michigan State UniversityGoogle Scholar
  8. Chen J, Seeman N (1991) Synthesis from DNA of a molecule with the connectivity of a cube. Nature 350:631–633CrossRefGoogle Scholar
  9. Christofides N (1976) Worst-case analysis of a new heuristic for the travelling salesman problem. Report 388, Graduate School of Industrial Administration, CMUGoogle Scholar
  10. Dietz H, Douglas S, Shih W (2009) Folding DNA into twisted and curved nanoscale shapes. Science 325:725–730CrossRefGoogle Scholar
  11. Eiselt H, Gendreau M, Laporte G (1995) Arc routing problems, part I: the Chinese postman problem. Oper Res 43(2):231–242MathSciNetCrossRefMATHGoogle Scholar
  12. Ellis-Monaghan J, Moffatt I (2013) Graphs on surfaces: dualities, Polynomials, and Knots. Springer, BerlinCrossRefGoogle Scholar
  13. Ellis-Monaghan J, Pangborn G et al (2013) Minimal tile and bond-edge types for self-assembling DNA graphs. In: Jonoska N, Saito M (eds) Discrete and topological models in molecular biology. Springer, BerlinGoogle Scholar
  14. Fleischner H (1990) Eulerian graphs and related topics. Volume 45 annals of discrete mathematics part 1, vol 1. North-Holland Publishing Co., AmsterdamGoogle Scholar
  15. Fleischner H (1991) Eulerian graphs and related topics. Volume 50 annals of discrete mathematics Part 1, vol 2. North-Holland Publishing Co., AmsterdamGoogle Scholar
  16. Garey M, Johnson D (1979) Computers and intractability. A guide to the theory of NP-completeness. A series of books in the mathematical sciences. W. H. Freeman and Co., San FranciscoGoogle Scholar
  17. Harary F, Nash-Williams C (1965) On Eulerian and Hamiltonian graphs and line graphs. Canad Math Bull 8:701–709MathSciNetCrossRefMATHGoogle Scholar
  18. He Y, Ye T, Su M, Zhuang C, Ribbe A, Jiang W, Mao C (2008) Hierarchical self-assembly of DNA into symmetric supramolecular polyhedral. Nature 452:198–202CrossRefGoogle Scholar
  19. Held M, Karp R (1961) A dynamic programming approach to sequencing problems. In: Proceedings of the 1961 16th ACM national meeting, ACM, 71.201-71.204. ACM, New York, NYGoogle Scholar
  20. Hogberg B, Liedl T, Shih W (2009) Folding DNA origami from a double-stranded source of scaffold. J Am Chem Soc 131(XX):9154–9155CrossRefGoogle Scholar
  21. Jonoska N, Saito M (2002) Boundary components of thickened graphs. Lect Notes Comput Sci 2340:70–81MathSciNetCrossRefGoogle Scholar
  22. Jonoska N, Karl S, Saito M (1999) Three dimensional DNA structures in computing. BioSystems 52(XX):143–153CrossRefGoogle Scholar
  23. Jonoska N, Seeman NC, Wu G (2009) On existence of reporter strands in DNA-based graph structures. Theor Comput Sci 410(15):1448–1460MathSciNetCrossRefMATHGoogle Scholar
  24. Kleinberg J, Tardos E (2005) Algorithm design. Addison-Wesley Longman Publishing Co., Inc, BostonGoogle Scholar
  25. Kotzig A (1968) Eulerian lines in finite 4-valent graphs and their transformations. Theory Gr 1966:219–230MathSciNetGoogle Scholar
  26. Lawler EL, Lenstra JK, Rinnooy Kan AHG, Shmoys DB (eds) (1985) The traveling salesman problem: a guided tour of combinatorial optimization. Wiley, New YorkMATHGoogle Scholar
  27. Las Vergnas M (1981) Eulerian circuits of 4-valent graphs embedded in surfaces. Algebraic methods in graph theory, szeged, 1978, colloquia mathematics societatis Janos Bolyai, vol 25. North Holland, Amsterdam, pp 451–477Google Scholar
  28. Luo D (2003) The road from biology to materials. Mater Today 6(XX):38–43CrossRefGoogle Scholar
  29. Nangreave J, Han D, Liu Y, Yan H (2010) DNA origami: a history and current perspective. Curr Opin Chem Biol 14(5):608–615CrossRefGoogle Scholar
  30. New Graph Theory from and for Nanoconstruct Design Strategies (2012) https://sites.google.com/site/nanoselfassembly Cited 29 Aug 2013
  31. Pinheiro AV, Han D, Shih W, Yan H (2011) Challenges and opportunities for structural DNA nanotechnology. Nature Nanotechnology 6:763–72CrossRefGoogle Scholar
  32. Richter RB (1991) Spanning trees, Euler tours, medial graphs, left-right paths and cycle spaces. Discret Math 89(3):261–268CrossRefMATHGoogle Scholar
  33. Rothemund P (2006) Folding DNA to create nanoscale shapes and patterns. Nature 440:297–302CrossRefGoogle Scholar
  34. Sanderson K (2010) Bioengineering: what to make with DNA origami. Nature 464:158–159CrossRefGoogle Scholar
  35. Shih W, Quispe J, Joyce G (2004) A 1.7 kilobase single-stranded DNA that folds into a nanoscale octahedron. Nature 427:618–621CrossRefGoogle Scholar
  36. Žitnik A (2002) Plane graphs with Eulerian Petrie walks. Discret Math 244(1–3):539–549MATHGoogle Scholar
  37. Zheng J, Birktoft J, Chen Y, Wang T, Sha R, Constantinou P, Ginell S, Mao C, Seeman N (2009) From molecular to macroscopic via the rational design of a self-assembled 3D DNA crystal. Nature 461:74–77CrossRefGoogle Scholar
  38. Zhang Y, Seeman N (1994) Construction of a DNA-truncated octahedron. J Am Chem Soc 116(5):1661–1669CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Joanna A. Ellis-Monaghan
    • 1
  • Andrew McDowell
    • 2
  • Iain Moffatt
    • 2
  • Greta Pangborn
    • 3
  1. 1.Department of MathematicsSaint Michael’s CollegeColchesterUSA
  2. 2.Department of MathematicsRoyal Holloway, University of LondonEghamUK
  3. 3.Department of Computer ScienceSaint Michael’s CollegeColchesterUSA

Personalised recommendations