Unknotted Strand Routings of Triangulated Meshes

In molecular self-assembly such as DNA origami, a circular strand’s topological routing determines the feasibility of a design to assemble to a target. In this regard, the Chinese-postman DNA scaffold routings of Benson et al. (2015) only ensure the unknottedness of the scaffold strand for triangulated topological spheres. In this paper, we present a cubic-time \(\frac{5}{3}-\)approximation algorithm to compute unknotted Chinese-postman scaffold routings on triangulated orientable surfaces of higher genus. Our algorithm guarantees every edge is routed at most twice, hence permitting low-packed designs suitable for physiological conditions.

[1]  Tamal K. Dey,et al.  A new technique to compute polygonal schema for 2-manifolds with application to null-homotopy detection , 1995, Discret. Comput. Geom..

[2]  Dieter Jungnickel,et al.  Graphs, Networks, and Algorithms , 1980 .

[3]  P. Rothemund.  Folding DNA to create nanoscale shapes and patterns , 2006, Nature.

[4]  William M. Shih,et al.  A 1.7-kilobase single-stranded DNA that folds into a nanoscale octahedron , 2004, Nature.

[5]  Greta Pangborn,et al.  DNA origami and unknotted A-trails in torus graphs , 2020, Journal of Knot Theory and Its Ramifications.

[6]  Michael S. Floater,et al.  Parametrization and smooth approximation of surface triangulations , 1997, Comput. Aided Geom. Des..

[7]  Gang Wu,et al.  On existence of reporter strands in DNA-based graph structures , 2009, Theor. Comput. Sci..

[8]  John M. Lee.  Introduction to Topological Manifolds , 2000 .

[9]  C. Hierholzer,et al.  Ueber die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren , 1873 .

[10]  C. Mao,et al.  Hierarchical self-assembly of DNA into symmetric supramolecular polyhedra , 2008, Nature.

[11]  Ben M. Webb,et al.  Design of a single-chain polypeptide tetrahedron assembled from coiled-coil segments , 2013, Nature chemical biology.

[12]  W. Chiu,et al.  Designer nanoscale DNA assemblies programmed from the top down , 2016, Science.

[13]  Sandi Klavzar,et al.  Stable Traces as a Model for Self-Assembly of Polypeptide Nanoscale Polyhedrons , 2013 .

[14]  Helena Gradišar,et al.  Design principles for rapid folding of knotted DNA nanostructures , 2016, Nature Communications.

[15]  Douglas B. West,et al.  A new proof of 3-colorability of Eulerian triangulations , 2011, Ars Math. Contemp..

[16]  Gang Wu,et al.  Construction of a DNA nano-object directly demonstrates computation , 2009, Biosyst..

[17]  H. Fleischner.  Eulerian graphs and related topics , 1990 .

[18]  W. Lickorish,et al.  An Introduction to Knot Theory , 1997 .

[19]  E. Winfree,et al.  Algorithmic Self-Assembly of DNA Sierpinski Triangles , 2004, PLoS biology.

[20]  Pekka Orponen,et al.  DNA rendering of polyhedral meshes at the nanoscale , 2015, Nature.

[21]  Jack Edmonds,et al.  Matching, Euler tours and the Chinese postman , 1973, Math. Program..

[22]  Pekka Orponen,et al.  Computer‐Aided Production of Scaffolded DNA Nanostructures from Flat Sheet Meshes , 2016, Angewandte Chemie.

[23]  Russell P. Goodman,et al.  Rapid Chiral Assembly of Rigid DNA Building Blocks for Molecular Nanofabrication , 2005, Science.

[24]  Greta Pangborn,et al.  Design tools for reporter strands and DNA origami scaffold strands , 2017, Theor. Comput. Sci..

[25]  Pamela E. Constantinou,et al.  From Molecular to Macroscopic via the Rational Design of a Self-Assembled 3D DNA Crystal , 2009, Nature.

[26]  Jonathan L. Gross,et al.  Handbook of graph theory , 2007, Discrete mathematics and its applications.

[27]  Shawn M. Douglas,et al.  Self-assembly of DNA into nanoscale three-dimensional shapes , 2009, Nature.

[28]  D. Rolfsen.  Knots and Links , 2003 .

[29]  Jeff Erickson,et al.  Optimally Cutting a Surface into a Disk , 2002, SCG '02.

[30]  N. Seeman,et al.  Synthesis from DNA of a molecule with the connectivity of a cube , 1991, Nature.

[31]  N. Seeman.  Nucleic acid junctions and lattices. , 1982, Journal of theoretical biology.

[32]  P. Yin,et al.  Complex shapes self-assembled from single-stranded DNA tiles , 2012, Nature.

[33]  Luvena L. Ong,et al.  Three-Dimensional Structures Self-Assembled from DNA Bricks , 2012, Science.