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arXiv · 1703.03799

DNA Origami and Unknotted A-trails in Torus Graphs

Ada Morse·William Adkisson·Jessica Greene·David Perry·Brenna Smith·Jo Ellis-Monaghan·Greta Pangborn
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Abstract

Motivated by the problem of determining unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine existence and knottedness of A-trails in graphs embedded on the torus. We show that any A-trail in a checkerboard-colorable torus graph is unknotted and characterize the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated embeddings. Surface meshes are frequent targets for DNA nanostructure self-assembly, and so we study both triangular and rectangular torus grids. We show that, aside from one exceptional family, a triangular torus grid contains an A-trail if and only if it has an odd number of vertices, and that such an A-trail is necessarily unknotted. On the other hand, while every rectangular torus grid contains an unknotted A-trail, we also show that any torus knot can be realized as an A-trail in some rectangular grid. Lastly, we use a gluing operation to construct infinite families of triangular and rectangular grids containing unknotted A-trails on surfaces of arbitrary genus. We also give infinite families of triangular grids containing no unknotted A-trail on surfaces of arbitrary nonzero genus.

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Ada Morse, William Adkisson, Jessica Greene, David Perry, Brenna Smith, Jo Ellis-Monaghan, Greta Pangborn. 2017-03-10. DNA Origami and Unknotted A-trails in Torus Graphs. https://arxiv.org/abs/1703.03799

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