# Elucidating the Mechanical Energy for Cyclization of a DNA Origami Tile
## 1 Introduction  
DNA self-assembly is a powerful bottom-up manufacturing approach based on spontaneous sequence complementarity [1, 2]. DNA nanotechnology can produce complex architectures with sub-nanometer accuracy and precision, using DNA tiles [3, 4, 5] and DNA bricks [6, 7, 8]. The tile-based assembly uses several short single-stranded DNA (ssDNA) oligonucleotides to assemble one or few motifs, which then form two-dimensional (2D) or three-dimensional (3D) configurations by using their sticky ends. In contrast, each DNA brick is a unique ssDNA strand, thus more intricate geometries may be constructed from a single step annealing reaction. Dynamic DNA nanostructures are also available, including tweezers [9, 10] and walkers [11, 12, 13, 14]. These mobile nanodevices rely on toehold-mediate strand displacement reaction [15, 16, 17], to detach from some of the strands in order to stay undefined or bind with new strands. The major difference between the two devices is the designed movement; DNA tweezers mostly switch between open and closed states, while DNA walkers can move along a track as long as fuel strands are available. DNA origami is a widely used method due to excellent programmability and structural predictability [18, 19, 20, 21, 22, 23]. In this approach, a long ssDNA'scaffold' strand (up to several thousand base-pairs or bp) is laid out in a folded pattern to approximate the target design. A number of short ssDNA'staple' strands are then added to bring selected parts of scaffold together by hybridization to form designed conformations. Various 2D and 3D DNA origami structures have been demonstrated, including shafts [24, 25, 26], tiles [27, 28, 29, 30], cubes [31, 32, 33],and wireframe polygons [34; 35; 36], to name a few. Dynamic, shape-changing DNA origami structures were also realized via two-step reactions: strand displacement and reannealing [37; 38; 39]. While the strand displacement reaction is possible at room-temperature due to the low energy barrier, the reannealing with new strands is often performed at elevated temperatures to provide necessary activation energies [40; 41; 42]. For both static and dynamic DNA origami architectures, their structural properties and mechanical behaviors are of great interest, including the twisting of a double-stranded DNA (dsDNA) bundle [43; 44], the interconnection between bundles by crossovers [45], the interaction between ds-regions in an origami structure [46], and the structural deformation by applied external forces [47]. A better understanding of such fundamental mechanics will significantly advance the design and development of DNA origami for a variety of applications. There are two typical lattices (honeycomb and square) for cross-section designs of DNA origami. These two design approaches may be distinguished by crossover arrangements as they are distinct and different based on the layouts of dsDNA bundles and crossovers. In a honeycomb lattice, a crossover is placed every 2/3 turn. Given the helicity of around 10.5 bp/turn for native B-form DNA, it is exactly 7 bp per crossover [48; 49; 50]. In contrast, a crossover is placed every 3/2 turns in a square lattice, yielding 15.75 bp per crossover. However, only an integer number of base pairs is possible in the design. Thus, 16 bp per crossover is used frequently, which corresponds to 10.67 bp/turn. The difference between 10.67 and 10.5 bp/turn in DNA helicity causes a right-handed twist in the entire structure [20; 24; 27]. The structural twist and curvature may be resolved by forcing the upper and lower boundaries together using linker staples [27] or by adding intercalators that directly modulate the helicity (e.g., from 10.5 to 10.67 bp/turn) [24; 51; 52; 53]. Regardless of the methods, external (mechanical or chemical) energy will be accounted for the structural correction, and this is the activation energy required for the deformation of the origami. In this work, we investigated a cyclization process of a single-layer DNA origami rectangle.