The Topology of Scaffold Routings on Non-Spherical Mesh Wireframes

The routing of a DNA-origami scaffold strand is often modelled as an Eulerian circuit of an Eulerian graph in combinatorial models of DNA origami design. The knot type of the scaffold strand dictates the feasibility of an Eulerian circuit to be used as the scaffold route in the design. Motivated by the topology of scaffold routings in 3D DNA origami, we investigate the knottedness of Eulerian circuits on surface-embedded graphs. We show that certain graph embeddings, checkerboard colorable, always admit unknotted Eulerian circuits. On the other hand, we prove that if a graph admits an embedding in a torus that is not checkerboard colorable, then it can be re-embedded so that all its non-intersecting Eulerian circuits are knotted. For surfaces of genus greater than one, we present an infinite family of checkerboard-colorable graph embeddings where there exist knotted Eulerian circuits

Combinatorial algorithms for the design of nanoscale systems

Over the past 30 years, DNA, with its exquisitely specific Watson-Crick base pairing rules, has found a novel use as a nanoscale construction material in DNA nanotechnology. DNA origami is a popular recent technique in DNA nanotechnology for the design and synthesis of DNA nanoscale shapes and patterns. DNA origami operates by the folding of a single long strand of DNA called a scaffold with the help of numerous shorter strands of DNA called staples. Recently, DNA origami design for polyhedral beam-frameworks has been proposed where a scaffold strand is conceptually routed over the beams of a polyhedron so that complementary strands potentially fold the scaffold to the framework in a solution. In this work, we modelled the problem of finding a scaffold routing path for polyhedral frameworks in graph-theoretic terms whereby the routing path was found to coincide with a specific type of Eulerian trail, called an A-trail, on the polyhedral skeleton. We studied the complexity of deciding whether an A-trail exists with an emphasis on rigid triangular frameworks or equivalently on plane triangulations. While the decision problem was found to be NP-complete in general, we learned that Eulerian triangulations always have A-trails if a long standing conjecture by Barnette on the Hamiltonicity of bipartite cubic polyhedral graphs holds. Given the general NP-completeness result, we developed a backtracking search algorithm for finding A-trails. To improve the backtrack search; we introduced an enumeration heuristic, tuned in particular to Eulerian triangulations, to schedule the nodes in the search tree. The algorithm, guided by the heuristic, efficiently found A-trails for a family of Eulerian triangulations as well as a family of braced grid graphs. Furthermore, we implemented a software package, BScOR (Beam Scaffolded-Origami Routing), which generates an A-trail, or equivalently a scaffold routing path, given a three-dimensional object description in a Polygon File Format

Algorithmic Design of Biomolecular Nanostructures

Biomolecular nanotechnology, a field where biomolecules such as DNA and RNA are used as programmable nanoscale construction materials, is emerging as a breakthrough technology with promising applications in nanomedicine, materials science and biophysical research. To accelerate the developments in nucleic acid nanotechnology, general and automated computer aided design tools which enable researchers from different fields to quickly design and synthesize nucleic acid nanostructures could play a significant role. Working in the framework of the robust DNA origami approach, this dissertation presents a novel, highly general and highly automated design approach for the design and synthesis of 2D and polyhedral DNA nanostructures suitable for e.g. biomedical applications. Grounded on graph-theoretic principles, the method introduces an Eulerian tour based approach for topologically routing DNA strands into nanoscale geometries exhibiting complex features. By employing an implementation of the design method, the impact of wireframe architecture on material efficiency and stiffness of DNA nanoscale assemblies was experimentally investigated. Motivated by the design of wireframe DNA nanostructures, we develop an algorithm for finding unknotted DNA strand routings on topologically more complex higher-genus mesh wireframes. Alternatively, cotranscriptionally folding RNA nanostructures have great potential for cell-based mass production of nucleic acid nanostructures. However, the presence of the cotranscriptional complex can present obstacles to folding a target shape. In this dissertation, we propose a graph-theoretic design framework which minimizes the risk of folding traps in a cotranscriptional setting

Computational modelling of the kinetic Tile Assembly Model using a rule-based approach

The (abstract) Tile Assembly Model (aTAM), is a mathematical paradigm for the study and algorithmic design of DNA self-assembly systems. It employs the use of so-called DNA-tiles, which are abstractions of experimentally achievable DNA nanostructure complexes with similar inter-matching behaviours. To this day, there are about half-dozen different experimental implementations of DNA tiles and their sub-sequent algorithmic assembly into larger complexes, see e.g. Reif et al. (2012) In order to provide further insight into the assembly process, the aTAM model has been extended to a kinetic counterpart (kTAM). Although there is a wide abundance of different variants of the abstract model, e.g., stage, step, hierarchical, temperature-k, signal-passing, etc. (see e.g. Patitz (2012) ), numerical simulations of the kinetic counterpart have been performed only for a few types of these systems. This might be due to the fact that the numerical models and simulations of kTAM were almost exclusively implemented using classical stochastic simulation algorithms frameworks, which are not designed for capturing models with theoretically un-bounded number of species. In this paper we introduce an agent- and rule-based modelling approach for kTAM, and its implementation on NFsim, one of the available platforms for such type of modelling. We show not only how the modelling of kTAM can be implemented, but we also explore the advantages of this modelling framework for kinetic simulations of kTAM and the easy way such models can be updated and modified. We present numerical comparisons both with classical numerical simulations of kTAM, as well as comparison in between four different kinetic variant of the TAM model, all implemented in NFsim as stand-alone rule-based models.Peer reviewe

Effects of design choices on the stiffness of wireframe DNA origami structures

DNA origami is a powerful method for the creation of 3D nanoscale objects, and in the past few years, interest in wireframe origami designs has increased due to their potential for biomedical applications. In DNA wireframe designs, the construction material is double-stranded DNA, which has a persistence length of around 50 nm. In this work, we study the effect of various design choices on the stiffness versus final size of nanoscale wireframe rods, given the constraints on origami designs set by the DNA origami scaffold size. An initial theoretical analysis predicts two competing mechanisms limiting rod stiffness, whose balancing results in an optimal edge length. For small edge lengths, the bending of the rod's overall frame geometry is the dominant factor, while the flexibility of individual DNA edges has a greater contribution at larger edge lengths. We evaluate our design choices through simulations and experiments and find that the stiffness of the structures increases with the number of sides in the cross-section polygon and that there are indications of an optimal member edge length. We also ascertain the effect of nicked DNA edges on the stiffness of the wireframe rods and demonstrate that ligation of the staple breakpoint nicks reduces the observed flexibility. Our simulations also indicate that the persistence length of wireframe DNA structures significantly decreases with increasing monovalent salt concentration.Peer reviewe

Algorithmic Design of 3D Wireframe RNA Polyhedra

Funding Information: The research of A.E., A.M., and P.O. was supported by Academy of Finland grant 311639. A.K. and A.K.N. have been supported by Academy of Finland grant 308992. The research of I.K. was supported by Japan Society for the Promotion of Science (JSPS) Early-Career Scientists 18K18144, Fund for the Promotion of Joint International Research (B) 19KK0261, and Young Researcher Dispatch Program (School of Engineering, Tohoku University). The research of A.M. was additionally supported by NSF-DMS (grant numbers 1800443/1764366) and Nokia Foundation (2017). The research of Y.S. has been supported by JSPS Grant-in-Aid for Scientific Research (KAKENHI; grant numbers 18K19831 and 19H04201). The research of L.O. and F.C.S. was supported by European Research Council grant agreement no. 694410, project AEDNA. Publisher Copyright: © 2022 The Authors. Published by American Chemical Society.We address the problem of de novo design and synthesis of nucleic acid nanostructures, a challenge that has been considered in the area of DNA nanotechnology since the 1980s and more recently in the area of RNA nanotechnology. Toward this goal, we introduce a general algorithmic design process and software pipeline for rendering 3D wireframe polyhedral nanostructures in single-stranded RNA. To initiate the pipeline, the user creates a model of the desired polyhedron using standard 3D graphic design software. As its output, the pipeline produces an RNA nucleotide sequence whose corresponding RNA primary structure can be transcribed from a DNA template and folded in the laboratory. As case examples, we design and characterize experimentally three 3D RNA nanostructures: a tetrahedron, a triangular bipyramid, and a triangular prism. The design software is openly available and also provides an export of the targeted 3D structure into the oxDNA molecular dynamics simulator for easy simulation and visualization.Peer reviewe

DNA rendering of polyhedral meshes at the nanoscale

VK: Orponen, P.; NC; TRITONIt was suggested1 more than thirty years ago that Watson–Crick base pairing might be used for the rational design of nanometre-scale structures from nucleic acids. Since then, and especially since the introduction of the origami technique2, DNA nanotechnology has enabled increasingly more complex structures3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18. But although general approaches for creating DNA origami polygonal meshes and design software are available14,16,17,19,20,21, there are still important constraints arising from DNA geometry and sense/antisense pairing, necessitating some manual adjustment during the design process. Here we present a general method of folding arbitrary polygonal digital meshes in DNA that readily produces structures that would be very difficult to realize using previous approaches. The design process is highly automated, using a routeing algorithm based on graph theory and a relaxation simulation that traces scaffold strands through the target structures. Moreover, unlike conventional origami designs built from close-packed helices, our structures have a more open conformation with one helix per edge and are therefore stable under the ionic conditions usually used in biological assays.Peer reviewe