DNA origami and the complexity of Eulerian circuits with turning costs
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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.
KeywordsDNA origami DNA self-assembly Turning cost Eulerian circuit Hamiltonian cycle Threading strand Biomolecular computing Mill routing Computational complexity A-trails
Mathematics Subject Classification92E10 05C45 05C85
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.
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