Design principles for rapid folding of knotted DNA nanostructures

Kočar, Vid; Schreck, John S.; Čeru, Slavko; Gradišar, Helena; Bašić, Nino; Pisanski, Tomaž; Doye, Jonathan P. K.; Jerala, Roman

doi:10.1038/ncomms10803

Cited by 51 publications

(58 citation statements)

References 44 publications

(46 reference statements)

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“…Kočar et al have used a combination of simulation and experiment to devise design rules for the efficient self-assembly of highly knotted DNA structures. 52 These authors took as their target a hollow squarebased pyramid, in which each edge is a hybridised double helix of DNA. For a single DNA molecule to form such a structure, the strand must pass between the pyramid's five vertices on a route that travels along each edge exactly twice and in opposite directions (see Figure 22a).…”

Section: Self-assembly Of Macromolecular and Colloidal Knotsmentioning

confidence: 99%

Knot theory in modern chemistry

et al. 2016

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(2016) 'Knot theory in modern chemistry.', Chemical society reviews., 45 (23). pp. 6432-6448. Further information on publisher's website:https://doi.org/10.1039/C6CS00448BPublisher's copyright statement:Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Knot theory is a branch of pure mathematics, but it is increasingly being applied in a variety of sciences. Knots appear in chemistry, not only in synthetic molecular design, but also in an array of materials and media, including some not traditionally associated with knots. Mathematics and chemistry can now be used synergistically to identify, characterise and create knots, as well as to understand and predict their physical properties. This tutorial r eview provides a brief introduction to the mathematics of knots and related topological concepts in the context of the chemical sciences. We then survey the broad range of applications of the theory to contemporary research in the field. Key Learning Points Some fundamentals of knot theory.  Knot theory and closely related ideas in topology can be applied to modern chemistry.  Knots can be formed in single molecules as well as in materials and biological fibres using a mixture of selfassembly, metal templating and optical manipulation. The inclusion of knots in molecular structures can alter chemical and physical properties.  Knots are surprisingly ubiquitous in the chemical sciences.

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Section: Self-assembly Of Macromolecular and Colloidal Knotsmentioning

confidence: 99%

Knot theory in modern chemistry

et al. 2016

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“…Namely, a graph H ∈ Ξ admits retracting-free double circuits traversing every edge twice, but not necessarily in each direction, as is for graphs of Ω . The minimum graph from Ξ \ Ω is that of tetrahedron [9][10][11][12]. The following result is due to Sabidussi [13] and was later independently proven by Eggleton and Skilton ([3; Theorem 9]).…”

Section: Preliminariesmentioning

confidence: 91%

Traversing every edge in each direction once, but not at once: Cubic (polyhedral) graphs

Rosenfeld

2017

ejgta

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A retracting-free bidirectional circuit in a graph G is a closed walk which traverses every edge exactly once in each direction and such that no edge is succeeded by the same edge in the opposite direction. Such a circuit revisits each vertex only in a number of steps. Studying the class Ω of all graphs admitting at least one retracting-free bidirectional circuit was proposed by Ore (1951) and is by now of practical use to nanotechnology. The latter needs in various molecular polyhedra that are constructed from a single chain molecule in the retracting-free way. Some earlier results for simple graphs, obtained by Thomassen and, then, by other authors, are specially refined by us for a cubic graph Q. Most of such refinements depend only on the number n of vertices of Q.Keywords: cubic graph, spanning tree, cotree, retracting-free bidirectional circuit Mathematics Subject Classification : 05C10, 05C30, 05C45, 94C15 DOI:10.5614/ejgta.2017.5.1.13 PreliminariesLet Q = (V, E) be a simple cubic graph with the vertex set V and edge set E |V | = n, |E| = m = 3n/2 . A spanning tree T of Q is a subtree covering all the vertices of Q |V (T )| = n; |E(T )| = n − 1 . Its cotree Q − E(T ) V Q − E(T ) = n; E Q − E(T ) = (n + 2)/2 is a graph Q less all edges belonging to T .

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“…Furthermore, parallel edges (of double traces) represent pairs of two coiled-coil-forming segments aligned in the same direction while antiparallel edges represent pairs of two coiled-coilforming segments aligned in the opposite direction. With the recent demonstration by Kočar et al (2016) of DNA-based polyhedra assembled from a single chain and the fact that DNA segments are always aligned in opposite direction, antiparallel double traces became the focus of our research. At last, a polyhedron P composed from a single polymer chain corresponding to a stable trace with repetitions may fold to a polyhedron different from P , as a vertex may indeed split into a collection of independent vertices of smaller degree, see also Fig.…”

Section: Bionanostructuresmentioning

confidence: 99%

“…DNA nanotechnology (Seeman (2004) for example) has used this approach to design complex DNA assembles which are defined by the complementarity of antiparallel DNA strands. Typically several DNA strands are used to construct such structures although it has been recently demonstrated by Kočar et al (2016) that DNA-based polyhedra could also be assembled from a single chain. The straightforward complementarity of base pairs is crucial for the use of DNA for the design of complex structures.…”

Section: Introductionmentioning

confidence: 99%

Antiparallel d-stable traces and a stronger version of ore problem

Rus

2016

J. Math. Biol.

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In 2013 a novel self-assembly strategy for polypeptide nanostructure design which could lead to significant developments in biotechnology was presented in [Design of a single-chain polypeptide tetrahedron assembled from coiled-coil segments, Nature Chem. Bio. 9 (2013) 362-366]. It was since observed that a polyhedron P can be realized by interlocking pairs of polypeptide chains if its corresponding graph G(P ) admits a strong trace. It was since also demonstrated that a similar strategy can also be expanded to self-assembly of designed DNA [Design principles for rapid folding of knotted DNA nanostructures, Nature communications 7 (2016) 1-8.]. In this direction, in the present paper we characterize graphs which admit closed walk which traverses every edge exactly once in each direction and for every vertex v, there is no subset N of its neighbors, with 1 ≤ |N | ≤ d, such that every time the walk enters v from N , it also exits to a vertex in N . This extends C. Thomassen's characterization [Bidirectional retracting-free double tracings and upper embeddability of graphs, J. Combin. Theory Ser. B 50 (1990) 198-207] for the case d = 1.

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Design principles for rapid folding of knotted DNA nanostructures

Cited by 51 publications

References 44 publications

Knot theory in modern chemistry

Knot theory in modern chemistry

Traversing every edge in each direction once, but not at once: Cubic (polyhedral) graphs

Antiparallel d-stable traces and a stronger version of ore problem

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