Regulation of 2D DNA Nanostructures by the Coupling of Tile Curvatures and Arm Twists

DNA overwinding and
underwinding between adjacent Holliday junctions have been applied in DNA
origami constructs to design both left-handed and right-handed nanostructures. For
a variety of DNA tubes assembled from small tiles, only a theoretical
approach of the intrinsic tile curvature was previously used to explain their
formation. Details regarding the quantitative and structural descriptions of
the intrinsic tile curvature and its evolution in DNA tubes by coupling with
arm twists were missing. In this work, we designed three types of tile cores
from a circular 128 nucleotide scaffold by longitudinal weaving (LW), bridging
longitudinal weaving (bLW), and transverse weaving (TW) and assembled their 2D
planar or tubular nanostructures via inter-tile arms with a distance of an odd
or even number of DNA half-turns. The biotin/streptavidin (SA) labeling
technique was applied to define the tube configuration with addressable inside
and outside surfaces and thus their component tile conformation with
addressable concave and convex curvatures. Both chiral tubes possessing
left-handed and right-handed curvatures could be generated by finely tuning p
and q in bLW-Ep/q designs (bLW tile cores joined together by inter-tile
arms of an even number of half-turns with the arm length of p base
pairs (bp) and the sticky end length of q nucleotides (nt)). We were able to
assign the chiral indices (n,m) to each specific tube from the high-resolution
AFM images, and thus estimated the tile curvature angle with a regular polygon
model that approximates each tube’s transverse section. We attribute the
curvature evolution of bLW-Ep/q tubes composed of the same tile core
to the coupling of the intrinsic tile curvature and different arm twists. A
better understanding of the integrated actions of different types of twisting
forces on DNA tubes will be much more helpful in engineering DNA nanostructures
in the future.

(2) cutting out of the corresponding gel bands by a razor blade under UV light; (3) chopping and crushing of the gel bands into fine pieces and transfer into a 2.0 mL Eppendorf tube; (4) addition of deionized water at least twice the size of the gel volume into the tube and elution at room temperature overnight; (5) filtering to collect the supernatants, recovery of any residual DNAs by rinsing with small volume of deionized water and filtering again to combine the supernatant fractions; (6)    with each helper strand carrying two protruding 8Ts (gray shadowed) on both ends out of the tile core. The yields of tile cores, labeled under their corresponding bands, were processed using ImageJ. S1.5 AFM imaging. AFM images were obtained in both "ScanAsyst mode in air" and "ScanAsyst mode in fluid" (Dimension FastScan, Bruker) with FastScan-C or Scanasyst-Fluid+ tips (Bruker). The sample preparation and imaging protocol was as follows: 2 μL sample were deposited onto a freshly cleaved mica (Ted Pella) and incubated between 2 and 4 minutes, then the specimen was washed twice with 50 μL of deionized water, and finally the specimen was imaged in either air or in fluid mode.
The fluid imaging was carried out by adding 80 μL TAE-Mg buffer onto the specimen and 40 μL TAE-Mg buffer onto the AFM tip.

S1.6
In-situ binding of streptavidin to biotin-labeled lattices. Streptavidin (SA) was ordered from Sangon Biotech, Shanghai. The SA stock solution was prepared as follows: 1 mL of ultrapure water was added to a tube containing 1 mg of lyophilized SA protein, then incubated for 10 minutes at room temperature and gently mixed to obtain a 1 mg/mL stock solution (According to the molecular weight of 55,000 Da of streptavidin, the streptavidin concentration of stock solution was calculated to be 18.2 µM), 20 µL of the stock solution were taken out and diluted to 5 µM with ultrapure water, and the ready-for-use SA solution was stored in a fridge. The remaining solution was divided into aliquots and stored frozen at -20°C (to avoid freeze-thaw cycles). during which period the density of physically adsorbed streptavidin on bare mica as background was very low so that it didn't interfere with the assignment of specifically bound SAs on DNA tubes. In-situ biotin/SA binding yield over 70% was high enough to define the geometric relationship of a SA to its host tile. AFM images were obtained in the ScanAsyst mode in fluid (Dimension FastScan, Bruker) with FastScan-C or ScanAsyst-Fluid+ tips (Bruker). Similarly, the unit cell parameters (a, b, c, and ϕ) for each lattice of other assemblies, E-tiling assemblies of LW-E20/4, bLW-E26/6, LW-E31/5, LW-E31/7, TW-E32/6, and three Otiling assemblies of LW-O26/4, bLW-O26/4, TW-O26/4, could be estimated in the same way.

S2 Theoretical estimation of unit cell parameters
All experimental and theoretical unit cell parameters are listed in Table S1.

S3 Tube parameters calculated from the chiral indices (n,m)
The chiral indices (n,m) of a specific tube were used to calculate the tube perimeter, helical angle α, and the tube axial periodicity (|T|) as follows: Figure S4. The chiral indices (n,m) of a specific DNA nanotube and its unit cell parameters.
The grid pattern represents a 2D lattice background, which is defined with the basis vectors a and b, the primitive vectors c1 and c2, and the inter-angle φ = 120° at the bottom left corner. The shadowed green rectangle is the radial projection of a DNA nanotube (3,3) unit cell, which can be illustrated with the perimeter vector Ch(n,m), the tubule axis vector T, and the helical angle α (defined as the inter-angle between Ch(n,m) and c2). The unit cell parameters can be theoretically estimated from the chiral indices (n,m), c, and the inter-angle φ.
For a DNA nanotube of given chiral indices (n,m), its perimeter vector is which has a magnitude (perimeter of the tube) When ϕ = 120°, Here, to keep both n and m of the chiral indices (n,m) in positive integers, we use ϕ = 120°, different from the ϕ = 60° for carbon nanotubes.
The diameter, D, of the DNA nanotube is: Similar to carbon nanotubes' formula, through trigonometry, the helicity α, was obtained as follows: d, the highest common divisor of (n,m) dR, the highest common divisor of (2n+m,2m+n) According to reference 1 , by defining φ = 120° (in practice, φ could vary a little around 120° due to the DNA spring-like structure), all lattice parameters (the helicity α and the axial periodicity T) can easily be obtained from the chiral indices (n,m) and c, as shown in equations (1) to (5). Except for the tube perimeters, other parameters are not discussed further here because they are beyond the scope of this work.

S4 Assignment of the chiral indices (n,m) to a specific tube based on its highresolution AFM image
Assignment of the chiral indices (n,m) to a specific tube was made possible by directly counting the number of c1-and c2-frames to form a perimeter vector according to its high-resolution AFM images. For example, the bLW-E31/7 assemblies were uniform tubes with the same perimeter, which can be defined only to the chiral indices (3,3) by counting the number of c1-and c2-frames from the well-resolved uniform tubes (block 1 of Figure S18). However, for most of the infinite tubes built by periodic packing of the same tile such as bLW-E21/5, bLW-E32/6, and TW-E21/5, each design generated the same type of infinite tubes with very similar structures but perimeter variations occurred in a narrow window due to small changes of the curvature during the assembly.
A cluster of chiral indices (n,m) were assigned to the same type of tubes with small perimeter variations. In Figures S5 to S9, we illustrate the assignments of (6,3), (8,3), (8,4), (10,4), and (10, 5) by directly counting the number of c1 and c2-frames to form a perimeter vector in their corresponding high-resolution AFM images of the bLW-E21/5 tubes with both open and closed structures.   (1) in Table S2.  Table S2.

S5 Assignments of a cluster of chiral indices by the numerical approximation method to both bLW-E21/5 and TW-E21/5 tubes
As shown in the above Section S4, to assign the chiral indices (n,m) to every individual tube by means of its high-resolution AFM images was possible but very time- consuming. An uncertainty also exists for the correct counting of n, the number of c2frames, because we may count 1 fewer than the real number at the top layer of a tube.
For the TW-E21/5 tubes, we judged the chiral indices (n,m) with a limitation of n = m from their high-resolution images ( Figures 3B(c) and S21). Therefore, it was straightforward to calculate the perimeters from equation (1) with (n,m) from (4,4) to (9,9), which are listed in Table S3. The tube abundances falling in 6 perimeter windows are presented in Figure 3B(g) and also listed in Table S3.

S6 Saddle-like tile oligomer model to simultaneously grow "large" and "giant" tubes
The paired group of "orthogonal" tubes in bLW-E32/6 assemblies have been seldom observed in other DNA assemblies. Growth of "giant" tubes with their axes along b is thermodynamically disfavored. To simultaneously grow such "orthogonal" tubes, we suggested the following saddle-like model of tile oligomers at the initial growth stage as their growth mechanism. Figure S11. Saddle-like tile oligomer model to grow "large" and "giant" tubes simultaneously.
The mechanism of simultaneous growth of "large" and "giant" tubes is suggested: 1) A saddle-like oligomer forms at the initial assembly stage, 2) the competing closure of ring seeds leads to epitaxial growth of "large" and "giant" tubes.
Tiling, a term originated from the construction architecture, refers here to the assembly of DNA tiles into either finite or infinite nanostructures by joining tiles with inter-tile arms via sticky end cohesion. In our current work, we term the joining with the arm length of an even number of half-turns as E-tiling and that of an odd number of half-turns as O-tiling. In tiles, the intrinsic tile curvature is originated from weaving, whereas the arm twist is generated from tiling. Coupling of both will finely tune the 2D DNA nanostructures into different morphologies and chirality. From 1D to 3D periodic DNA nanostructures, Bravais lattices can be abstracted, with tile cores as lattice points, joining arms as bonds, and the smallest repeating structures as unit cells. Thus, the 2D planar and tubular lattices in this work will be described via crystal terms and analyzed using crystal theory.

Design strategies
A small DNA tile, such as a double crossover (DX) 9 tile or a multi-arm junction tile [18][19]

Definition and estimation of tile curvature
Because of the spring-like structure of DNA, a well-known phenomenon in DNA nanotechnology is that DNA tubes and planar ribbons are often observed when E-tiling and O-tiling are applied, respectively; this has been attributed to the intrinsic tile curvature. 18 Following the canonical winding phase of B-DNA, E-tiling requires that all tile faces to be aligned identically, and O-tiling requires that adjacently joined tile faces to be aligned alternately. By assuming that the arms in both E-tiling and O-tiling are straight and do not generate any torque, the intrinsic tile curvatures will be accumulated by E-tiling to generate tubes, whereas they will be cancelled surfaces, as well as the tube perimeters and morphologies. [35][36] Many questions regarding the intrinsic tile curvature and its evolution with changing arm and sticky end lengths still remain unanswered.
In DNA tubular structures of DAE-E tubes, 35 six helix bundle tubes, [37][38] and single-stranded tile tubes, 16, 39 the universal polygon model has been applied to explain the curving of tile arrays to tubes. Such polygon models are based on the B-DNA winding criteria including the major/minor grooving effect at crossovers by presupposing that all helixes are rigid, tangent between adjacent helixes, and parallel to the tube axis. However, a precise physical description of the tile curvature and its cooperation with arm-twisting forces have not yet been illustrated clearly in the polygon models mentioned above.
In this work, in order to quantify the tile curvature, we first relied on high-resolution AFM images of each type of 2D lattices with a lateral resolution at about 2.0 nm to abstract its 2D Bravais lattice of centered rectangle and precisely measured the unit cell parameters (Section S2, S3 and S4). It is obvious that only the E-tiling tubes are adequate for the curvature analysis. Secondly, we described each specific tube with the chiral indices (n,m) 33, 40 based on its high-resolution AFM image. The correspondence of the tube perimeters from theoretical calculations based on (n,m) and lattice linear and angular constants to experimental measurements confirmed the correct assignment of (n,m). Thirdly, we projected the tube unit cell possessing the chiral indices (n,m) (in this work, we limited our analysis to two types of stable tubes, n = 2m and n = m) onto its transverse section and approximate the projection to a regular polygon model. The exterior angle of the regular polygon is assigned as the tile curvature angle. We define this semi-quantified curvature as the global tile curvature, which integrates the intrinsic tile curvature and the arm twist together. We define the intrinsic tile curvature ideally as the curvature of a free tile, which depends on its weaving architecture. However, its measurement is beyond our current capabilities. In practice, we could assign the global tile curvature defined from an E-tiling tube with an arm length of 21 bp and a sticky end length of 5 nt (specified as E21/5) as the intrinsic tile curvature, which is based on the postulation that the E21/5 arm is straight and does not yield any additional torque. 22 Under such a postulation, we also considered the intrinsic tile curvature as the intrinsic curvature of the tile core.
We relied on the combination of both biotin/SA labeling and high-resolution AFM imaging techniques to define the configurations of the E-tiling tubes and thus the conformations of the DNA tiles, primarily for the stable and minimally distorted tubes wound according to the canonical B-DNA helixes. We represent the stereo-conformation (i.e. chirality) of a tile with the right-hand or lefthand grip rule. To demonstrate, with the thumb points towards the inside surface of the tube or the concave face of the tile, the rotation direction of c128nt from 5' to 3' can be followed by either right-hand or left-hand grip. For concise notation, we use l-or r-as the prefix of the chiral indices (n,m) of a specific tube as l-(n,m) or r-(n,m) to represent its left-handed or right-handed curvature. Similarly, we separated and identified the two opposite faces of a tile using either the l-face or the r-face. Moreover, the prefix of l-or r-in front of (n,m) indicates a clear geometry of the tube, whereas l-or r-in front of "face"only indicates one of the two opposite faces of a tile exclusively.

Synthesis of c128nt and tile stability
The circular c128nt was synthesized using two linear, phosphorylated 64 nt oligonucleotides and their two corresponding splints by T4 ligation. The c128nt was purified through denaturing polyacrylamide gel electrophoresis (PAGE) with a yield of 40~60% (Section S1.2 and Figure S1). We used the native PAGE to test the stability and yield of three tile cores of LW, bLW, and TW (Section S1.4 and Figure S2). For each core, a single band confirms its stability. The yield was 80% for each core based on an analysis with the software "imageJ" of the gel band intensity in each lane.

A unified 2D Bravais lattice of centered rectangle
From both theoretical designs and experimental results of AFM images, we applied a unified 2D Bravais lattice of centered rectangle to describe all the 2D DNA lattices originated from c128nt-derived tiles. For example, in the brick assembly model of Figure 1A(a), a compound unit cell of centered rectangle containing two tiles is defined by the mutually orthogonal basis vectors a and b with unit lengths a and b, respectively, and a primitive cell of rhombus containing one tile is superimposed by the primitive vectors c1 and c2 with unit length c and inter-angle ϕ. By achieving a lateral resolution of about 2.0 nm in AFM images, we were able to measure the unit cell constants of each 2D lattice. The measured lattice constants were in line with the simple theoretical estimations of our designs well (theoretical descriptions refer to Section S2 and S3, data refer to Table 1 and Table S1).

2D arrays of longitudinally woven (LW) tiles
In Figure 1  will cohere together to form arms whose length is defined as the inter-tile distance between adjacent HJs. In both the brick assembly models of Figures 1A(a)  Because we did not use any AFM imaging markers on the LW-derived tiles, their densely woven 2D arrays (as well as TW-E21/5 and TW-O26/4 assemblies which are described later) were primarily imaged in fluid as brighter patches of planar ribbons, handkerchieflike structures, or blurring tubes without any texture, shown in Figures 1A(c), 1B(c), S12, S13 (as well as in Figures 3A(c), 3B(b), S21, and S22). To obtain high-resolution images with nanoscale textures, we employed two imaging skills: 1) On a wide tube, the top monolayer sits flatways on a bottom monolayer and both layers are electrostatically repulsive. The nanoscale rhombus texture of the top layer was easily resolved using between 1 to 3 scans, shown in Figure 1A(d) and magnified in the left bottom insert. In comparison with the brick assembly model of Figure 1A(a), each bright intersection point represents a LW core, each edge a 21 bp joining arm, and each dim pit a rhombus center torn a little bit apart by scanning. 2) For the LW-O26/4 monolayer array which was strongly adsorbed on the hard substrate of mica, its high-resolution image in Figure 1B(d) was achieved through at least 4 scans.
However, repeated scans removed some tiles from the lattice patch and generated scattered empty pores. The hierarchical and distinctive imaging features in Figure 1B Table S1). The high-resolution imaging details also confirm the correct Bravais lattice assignment of the centered rectangle.

Chiral indices and curvature evolution of the bLW-Ep/q tubes
The center-bridging strategies to build 2D bLW-derived nanostructures are demonstrated in Figure 2. The insertion of a flexible 10 bp helix instead of the central HJ releases more space and provides more flexibility for the bLW-derived tiles so that they can regulate themselves during the assembly.
We show the O-tiling bLW-O26/4 assemblies first. Figure 2A two tiny faint pores can be seen separated by the 10 bp bridge. Thus, the geometry of the bLW core is defined as an X-shape with two pairs of opposite angles of around 60 0 and 120 0 . The X-shaped geometry can be explained by the fact that the 2 three-branch junctions at both sides could be kinked more easily than the two continuous helixes without any breaks at the top and bottom, as shown in the helix tile model of Figure 2A(b). The bLW-O26/4 assemblies provide the dominant tubes with scattered perimeters ( Figure   S14).
In Figure 2B, we indicate three sets of chiral indices (n,m) to specify the DNA tubes. The chiral indices have been widely used in defining the carbon nanotube structures, 41 but they have rarely been used in describing the DNA tube structures. 33 On the hexagonal lattice background (ϕ = 120 0 ) of the bLW-Ep/q assemblies with l-faces facing up, we drew the Bravais unit cells at the bottom left, and three sets of chiral indices: (8,4) in purple shadow representing a typical bLW-E21/5 tube unit cell, (3,3) in green shadow representing the unique bLW-E31/7 tube unit cell, and (23,-23) in blue shadow representing a typical "giant" bLW-E32/6 tube unit cell. The chiral indices (n,m) are abstracted from nc1 + mc2, which is the chiral (or perimeter) vector Ch(n,m) originated from the origin O.
For example, (8,4) and (3,3) are abstracted from Ch(8,4) = 8c1 + 4c2 and Ch(3,3) = 3c1 + 3c2, respectively. The tube unit cell of (23,-23) is represented with repeating units at the top right, in which vectors of 23c1 and -23c2 are omitted for illustration purposes. The unit cell parameters of c and ϕ were measured directly from the AFM images, thus, the chiral indices (n,m) could precisely define many tube unit cell parameters, including the perimeter (diameter and radius), the helical angle of the inter-angle between c2 and Ch(n,m), and the axial periodicity (T) (Section S3 and Figure S4), and also the global tile curvature being investigated in our study. 40 Combining both the biotin/SA labeling and the AFM imaging techniques, we analyzed in detail the E-tiling tubes of bLW-E21/5 in Figures 2C(a-f), bLW-E31/7 in Figures 2D(a-e), and bLW-E32/6 in Figures 2E(a-e). The evolution of the perimeters of the bLW-Ep/q tubes, their configurations, their tile conformations, and their global tile curvatures by manipulating both the arm and sticky end lengths enabled us to gain a much deeper understanding of the tube formation mechanism.
For the bLW-E21/5 tubes, Figure 2C(a) shows the schematic brick model of the typical tube (8,4) (top) and the helix model of its component tile (bottom) with a biotin label at overhang 2. Figure 2C(b) provides a zoomed-out AFM image containing many tubes (top) and a zoomed-in image of only one typical tube (bottom). Densely distributed tubes in the top panel suggest a high yield for the products. Figure 2C(c) shows an abundance of 73 tubes falling in 4 perimeter windows, in which tube perimeters were measured from more than 5 batches of assemblies (Section S5, Table S2, and block 1 of Figure S15). The tubes were clearly seen as mostly homogeneous, falling within the perimeter window of 110.0 to 230.0 nm. An obvious feature of the bLW-E21/5 tubes in all the zoomedin images ( Figures 2C(b,d), Figures S4-9 and S15) was that the two groups of c1 and c2-frames were not symmetric. One group of frames were aligned nearly parallel to the tube axis, and the other group curled around the tube axis with left-handed rolling. specimen with water and blew the water drop away with air streams (Section S1.6 and Figure S3). The top layers of the tubes were either partly or completely removed with water due to the surface tension interactions, whereas the bottom layers were strongly adsorbed and remained intact on the mica. Then, the AFM scanning in either air or buffer mode was carried out. The completely broken and partly broken bLW-E21/5 tubes were imaged mostly as monolayer strips, which were either locally straight or kinked at some sites. The monolayer strips exposing the inside surfaces of the tubes outside are a requirement to define the direction of the curvature. For conciseness purposes, we only show the results of biotin/SA labeling at the 5'-end of the outer helper strand (green strand) of overhang 2 for bLW-E21/5, bLW-E31/7, and bLW-E32/6 in Figure 2 (additional AFM images of each design are shown in Figures S15, S19, and S20). The other biotin/SA labeling results at overhang 1 and 1* are also shown in Figures S15-20. The in-situ binding of the SAs on biotin-labeled lattices was carried out under a dilute SA solution according to previously reported protocols. 18,39 Although the biotin-labeled nucleotide is located at the buried r-face in Figure 2C(d), high yield of binding of the SAs (70~90 %) was confirmed by AFM imaging because the empty space between the arms allowed biotins to be bent and stand up from the mica, exposed their functional groups and bound the SAs easily. From the massive SA dots overlying on bLW-E21/5, bLW-E31/7, and bLW-E32/6 monolayer lattice strips, we could in most cases precisely define the position of an SA dot relative to its corresponding tile. For example, in the bottom panel of Figure 3C(d), the SA dots always sit on the longitudinal c2-frames along the tube axis. With the center of each X-shaped tile as a lattice point, via the acute angle from each SA dot at an overhang 2 of a c2-frame to its two adjacent adjacent c1-frames would be counterclockwise. Furthermore, we assigned the c2-frames via the connection of 2*-2 aligned parallel to the tube axis, and the c1-frames via the connection of 1*-1 curled around the tube axis with a left-handed rolling. In summary, we assayed over 60 individual bLW-E21/5 tubes at high-resolution from more than 5 batches with biotin/SA labels on overhangs 1, 1*, and 2, all imaging results supported the left-handed curvature (more AFM images are in Figure S15).
Quantitative analysis of the curvature angle of a specific tube relies on assigning the chiral indices (n,m) to the tube. In practice, we could assign (n,m) to a perforated bLW-Ep/q tube based on its high-resolution AFM images illustrating the tube lattices in both the open and closed forms, such as in Figures S5-9. The high-resolution AFM images indicated that around 90% of the bLW-E21/5 tubes had 4, 5, or 6 tube-axis-parallel c2-frames from the bottom layer stuck on the mica, and a few other irregular tubes (around 10%), had c2-frames either clearly tilted against the tube axis or had wider perimeters. By directly counting the chiral indices (n,m) from a highly resolved tube in both the open and closed forms ( Figures S4-9), we were able to assign tubes with 4 bottom c2-frames stuck on the mica mostly to l- (6,2), l- (6,3), and l-(7,3) ( Figures S5, S10 and Table S2), 5 bottom c2-frames mostly to l- (8,3), l- (8,4), and l- (9,4) ( Figures S6, S7, S10, and Table S2), and 6 bottom c2-frames to l- (10,4), l- (10,5), and l- (11,5) (Figures S8-10, and Table S2). We also applied the numerical approximation method (Section S3-5, Table S2, and Figure S10) to match the chiral indices to the tube perimeter windows of Figure 2C(c). The numerical calculations led to the conclusion that the bLW-E21/5 tubes in Figure 2C(c) could be represented with a paraxial cluster of the chiral indices (n,m) along the linear segment n = 2m+1, where m is 3, 4 and 5 in the 2D Cartesian coordinates (m,n) ( Figure S10). The range of the curvature angle was estimated from these clustered tubes, which was sufficient to support our claim of semi-quantitation of the tile curvature. We took the most abundant bLW-E21/5 tube of l-(8,4) ( Figure 2C(c), Table   S2, and Figure S10), as an example to describe the quantitation approach. First, we define the curvature of a tile as the bending of two halves of a tile along an axis passing through the tile center to a dihedral angle, which could clearly be compared to a butt hinge with a pair of leaves bending along a shaft. It is reasonable to assume that the arms along c1 in the bLW-E21/5 tube l- (8,4) bend at their centers of sticky end cohesion sites with the same curvature as the tile core. Thus, we approximated the transverse section of the perforated tube l- (8,4) by projecting its tube unit cell in Figure 2B to a 16-gon, as shown in Figure 2C(e). The physical appearance for the 16-gon was recognized as follows: 8 tile centers (lattice points) and 8 arm centers (lattice bond centers) along c1 are projected to 16 vertices, and 16 straight half-arms (half lattice bonds) along c1 to 16 edges. Therefore, the curvature angle θ of l- (8,4) was estimated as the exterior angle 360 0 /16 = 22.5 0 , schematically shown in Figure 2B(f). Similarly, a 12-gon was approximated from the narrowest tube l-(6,3) and a 24-gon from the widest tube l- (12,6), and their curvature angles were estimated at 30 0 and 15 0 , separately.
From the above approximation approaches, the intrinsic tile curvature angle of the bLW-E21/5 tubes was estimated to be limited in the window of 22.5 0 ± 8 0 .
Furthermore, we investigated the evolution of the global tile curvature in both the bLW-E20/4 and bLW-E22/6 tubes, which have 1 bp deletion and 1 bp insertion from bLW-E21/5 in both the arms and sticky ends, respectively. The biotin/SA labeling results ( Figure S16) showed that the bLW-E20/4 tubes possessed a left-handed curvature (6 individual tubes from 2 batches were measured), whereas the bLW-E22/6 tubes ( Figure S17) had a right-handed curvature (6 individual tubes from 2 batches were measured). The unique bLW-E20/4 tube was assigned to the chiral indices l- (4,2), and the typical bLW-E22/6 tube to r- (6,6). The opposite curvatures can be explained as follows: 1) The arm in bLW-E20/4 with a helical twist density of 10.0 bp/turn (less than 10.5 bp/turn of canonical B-DNA) and 4 nt sticky end cohesion would generate a left-handed arm twist, whereas the arm in bLW-E22/6 with a helical twist density of 11.0 bp/turn (larger than 10.5 bp/turn) and 6 nt sticky end cohesion would yield a right-handed arm twist. 2) Depending on the coupling of the arm twist and the intrinsic tile curvature, bLW-E20/4 would enhance the left-handed curling to the unique tube l-(4,2) ( Figure S16), while bLW-E22/6 would flip over the intrinsic tile curvature from left-handed to right-handed, thus resulting in tubes r- (5,5), r- (6,6), and r- (7,7), among which, r- (6,6) was the most abundant one ( Figure S17). Both tubes coexisted with many other residual fragments of tile oligomers, which indicated that the deviation of ±1 bp of arms from the canonical B-DNA would disturb the tube formation process to some degree. In the bLW-E20/4 tube of l- (4,2), c1 and c2-frames played their respective framing roles as in bLW-E21/5 (i.e., c1-frames curled around the tube axis with left-handed rolling and c2-frames aligned nearly parallel to the tube axis). For the bLW-E22/6 tubes of r- (5,5) to r- (7,7), both the c1-and c2-frames were symmetric and played equal framing roles.
We also extended the bLW-Ep/q assemblies to the arm length to three full turns and the sticky end length from 5 to 7 nt. We tested the assemblies of bLW-E31/5, bLW-E31/7, and bLW-E32/6. For the bLW-E31/5 assemblies, the helical twist density of 31/3 = 10.3 bp/turn (less than10.5 bp/turn) in the arm and 5 nt sticky end cohesion would generate a left-handed arm twist. Similar to bLW-E20/4, the coupling of both the left-handed twisting forces from the arms and the tile cores generated the typical tube with the chiral indices l- (10,5), where the left-handed curvature was defined from the biotin/SA labeling results (8 individual tubes from 2 batches were measured) and the yield for tube products was moderate ( Figure S18).
The bLW-E31/7 assemblies gave high yield products of uniform tubes with the chiral indices (3,3), as shown in Figure 2D(b) and in block 1 of Figure S19. tile conformation assignment. We tried to image more than a hundred of individual tubes, but were only able to clearly define 15 tubes' configuration with the right-handed curvature from more than 5 batches, as shown in Figure 2D(c) and in blocks 2 and 3 of Figure S19. With the regular polygon approximation method, projection of the r- (3,3) tube unit cell elements to its transverse section could be approximated with a 12-gon, and the exterior angle (i.e. the global tile curvature angle) was 30 0 . The global tile curvature of the bLW-E31/7 tube of r- (3,3) is schematically shown in Figure 2D(d), with the dihedral axis along a.
Schematic and AFM images of the bLW-E32/6 tubes and their curvatures are shown in Figures 2E(a-e). Much different from previously assembled tubes with similar configurations, two types of tubes were achieved with high yield and high quality (a very few lattice fragments were imaged on mica), the "large" tube (bottom of Figure 2E(a) and its AFM images at the bottom of Figure 2E(b) and in block 1 of Figure S20) with its axis along a having an average perimeter of 407.1 nm, represented with the chiral indices (16,16), and the "giant" tube (top of Figure 2E(a) and its AFM images at the top of Figure 2E(b) and in block 1 of Figure  and block 3 of Figure S20), whereas "large" tubes possessed the right-handed curvature ( Figure 2E(d) and block 2 of Figure S20) (over 20 individual tubes of each type from 3 more batches were imaged). The global tile curvature angle of the "large" tube r- (16,16) was estimated at 9 0 from a regular 64-gon (bottom of Figure 2E(e)), and that of the "giant" tube l- (23,-23)  3) There should be a delicate balance between the two opposite curving forces at the initial assembly stage, which drives the tile oligomers to a saddle-like shape (Section S6 and Figure S11). In the saddle-like oligomers, the deeper curvature surrounding a is driven by the right-handed curving force and the other shallower curvature surrounding b is driven by the left-handed curving force. 4) With further growth, depending on which curved surface closes first to form a ring as a tube seed, which ends up growing epitaxially to a full tube. The saddle-like model of the tile oligomers is strongly supported by two evidences: 1) The directions of growth of the tube axes ("large" tubes along a and "giant" ones along b) are perpendicular to each other.
2) The global tile curvatures ("large" tubes with the right-handed curvature and "giant" ones with the left-handed one) are opposite to each other. We named the specifically paired group of "large" and "giant" tubes as "orthogonal" tubes. The phenomenon of tube widening with b as the tube axis had also been observed in the DAE-E tubes. However, the authors addressed that the mechanism of formation and configuration of the tube were not clearly described. 35 In our case, the saddle-like oligomer model reasonably explains how "giant" tubes overcome the higher energy barrier to form a ring seed and finally grow up epitaxially with their tube axes along b. representing typical E-tiling tubes: a typical bLW-E21/5 tube unit cell of (8,4) shadowed in gold with its tube axis along c2, the unique bLW-E31/7 tube unit cell of (3,3) shadowed in green with its tube axis along a, and a "giant" bLW-E32/6 tube unit cell of (23,-23) (3,3) represented with a dihedral angle along the a axis following the right-hand grip rule. E, bLW-E32/6 tubes: a) a brick model of a "giant" tube l- (23,-23) (top) and that of a "large" tube r- (16,16) (bottom), and the helix tile model of bLW-E32/6 with a biotin/SA label at overhang 2 is similar to bLW-E31/7 and thus omitted, b) AFM images of both "large" and "giant" tubes and their corresponding zoomed-in images in insets illustrating their different tube axes, c) a zoomed-in AFM image of the inside surface lattice of a "giant" tube with biotin/SA dots and its corresponding l-faced brick model (inset), d) a zoomed-in AFM image of an inside surface lattice of a "large" tube with biotin/SA dots and its corresponding r-faced brick model (inset), e) the global tile curvature of the "giant" tube (top) represented with a dihedral along the axis b following the left-hand grip rule and that of the "large" tube (bottom) with a dihedral angle along the a axis following the right-hand grip rule.
In Table 1, we summarize the linear and angular constants of 2D lattice (a, b, c, and φ) of the bLW-Ep/q tubes, followed by their chiral indices (n,m), perimeters (C), curvature angles (θ), and length ranges of the most abundant tube for each design. # For all six designs, bLW-E20/4 and bLW-E31/7 have their corresponding unique tubes with the chiral indices Ch(n,m) of l-(4,2) and r- (3,3), respectively; bLW-E21/5, bLW-E22/6, and bLW-E31/5 are represented with their most abundant tubes l- (8,4), r- (6,6), and l-(10,5), respectively; and the "giant" and "large" tubes in bLW-E32/6 are represented with l- (23,-23) and r- (16,16) at their average perimeters, respectively. The tube perimeter C was calculated as C = � 2 + 2 + 2 cos , where c and φ measured experimentally are listed in the same row (refer to Section S3 of SI). The curvature angle θ correlated with the tile dihedral axis was estimated according to the following approximation rule: For each bLW-Ep/q tube represented with (n,m), when n = 2m, θ was calculated as the exterior angle of a regular 2n-gon; when n = |m|, θ was calculated as the exterior angle of a regular 4n-gon.  Figures 3A(c) and S21. To achieve high-resolution images with individual tile textures, as shown in Figure 3A(d) and in Figure   S21, more than 4 scans were needed. However, repeated scans moved some tiles away and left dim pores clearly showing shadows in the lattice. In contrast with the LW-E21/5 and LW-O26/4 ribbon-like lattices having ragged edges (stepped faces) along their longitudinal directions, TW-O26/4 lattices presented sharp vertices and straight edges (flat faces) along the c1 and c2 directions, due to the strong bonds with three helixes integrated together as an arm for sticky end cohesion. 46 As shown in Figure 3(d), with assignments of c1 and c2 along the sharp edges, a and b were easily defined.

2D arrays of transversely woven (TW) tiles
The E-tiling assembly of TW-E21/5 shows relatively homogenous tubes ( Figure 3B(b) and block 1 of Figure S22). Although the TW core has the highest HJ density, formation of tubes instead of planar arrays was an indication that the TW core has an intrinsic curvature, probably due to the lack of a central HJ and therefore the bending of both half-parts along a passing through the TW core.
Similarly, we achieved high-resolution AFM images with individual tile textures by repeated scans in the same region more than 4 times. As shown in Figure 3B(c) and block 1 of Figure S22, parts of the top layers were torn open along the flat face of c1 or c2. The TW-E21/5 tube axis is along the direction of the arm helix extension of a because it is thermodynamically favored. The number of counted tiles along c1 and c2 in Figure 3B(c) were 10, thus, this tube was assigned to the chiral indices (5,5).
The tile conformation was defined with the in-situ biotin/SA labeling technique too. In contrast with the perforated bLW-Ep/q tubes, TW-E21/5 assemblies were sealed tubes without any space between the tiles. In this case, the biotin orientation on mica-supported lattices was key for the in-situ binding of the SAs. When biotins were exposed on the top of the DNA lattice, the SAs bound to biotins efficiently, whereas when biotins were buried in a monolayered lattice, no binding occurred. We applied three biotin-labeling strategies: 1) an iBiodT label at overhang 2 pointing toward the l-face of every TW core, 2) a biotin label at the 5'-end of a helper strand at overhang 1* pointing toward the r-face of every TW core, 3) both types of biotin labels at the r-and l-faces of every TW core, as shown in the TW-E21/5 tile helix model of Figure 3B(a) (bottom). After in-situ binding of SAs, the biotin labels exposed on the l-faces bind SAs only at the top of the double-layers (i.e. sealed tubes), but not on the monolayers (i.e. open tubes), as shown in Figure 3B(d) and block 2 of Figure S22; the biotin labels exposed on the r-faces bind SAs only on the monolayers but not at the top of the doublelayers, as shown in Figure 3B(e) and block 3 of Figure S22; the biotin labels exposed on the r-and l-faces bind SAs both on monolayers and at the top of the double-layers, as shown in block 4 of Figure S22. For each labeling design, we have imaged more than 30 individual tubes from 3 separate batches. All imaging results support that the intrinsic tile curvature of TW-E21/5 followed the righthand grip rule.
We measured the tube perimeters of 67 tubes in more than 5 different batches by doubling the width. The abundances of different perimeter windows are plotted in Figure 3B(g), indicating a nearly normal distribution from 45 to 135 nm. Combining both the direct counting of the chiral indices (n,m) from high-resolution AFM images and the numerical approximation method in Section S3, we assigned the most abundant tube (21/67 = 31.3%) at the perimeter window of 75.0~90.0 nm to r- (6,6), the second most abundant tubes (14/67 = 21%) at 60.0~75.0 nm to r- (5,5) and at 90.0~105.0 nm to r- (7,7), and the less abundant tubes at 45.0~60.0 nm to r-(4,4) (6/67 = 9.0%), at 105.0~120.0 nm to r- (8,8) (9/67 = 13.4%), and at 120.0~135.0 nm to r- (9,9) (3/67 = 4.5%) (Section S5 and Table S3). Because the TW-E21/5 tubes have their helixes closely juxtaposed without any space between them, and the arm helixes are parallel to the tube axis, there will be no bending occurring in the arm regions. Therefore, we approximated the transverse section of the most abundant r-(6,6) to a regular 12-gon, that of the narrowest r- (4,4) to a regular 8-gon, and that of the widest tube r- (9,9) to a 18-gon. The curvature angle window was estimated to be around 30 0 ± 15 0 , which is schematically represented in Figure 3B(f).

COMPRESHENSIVE DISCUSSION
In this work, the most interesting phenomenon was the modification of the global tile curvatures of bLW-Ep/q tubes with different arm lengths at 4 or 6 DNA half-turns and sticky end lengths in the range of 4 to 7 nt. As shown, many parameters influence the global tile curvature such as the intrinsic tile curvature, the arm length, the sticky end length, the tile orientation vs the tube axis. We simplified the multiple factors into a model in which the tube configuration would be tuned with two twisting forces. One of them results from the intrinsic tile curvature and the other emanates from the arm twist, which was affected by both the arm and the sticky end length.
The intrinsic tile curvature of a tile core depended on its weaving architecture. For example, the bLW cores had an intrinsic lefthanded curvature, while the TW cores possessed the intrinsic right-handed curvature, both of which were inferred from the bLW-E21/5 and TW-E21/5 tubes without any extra twist in the arm. The extra arm twist nearly follows the overwinding/underwinding rule 22 by regulating the inter-tile arm lengths between two adjacent HJs at the exception of the assembly of the bLW-E31/7 tubes. In briefly describing the overwinding/underwinding rule, when the helical twist density in an arm is at 10.5 bp/turn (10.44 bp/turn to be precise) 24 of the canonical B-DNA, the arm would be straight; when it is less than 10.5 bp/turn, the arm would gain a left-handed twisting torque; and when it is larger than 10.5 bp/turn, the arm would gain a right-handed twisting torque. 22 In our case, according to the experimental results, the rule needed small modifications: When a sticky end cohesion exists in the arm and the sticky end length is at 4 or 5 nt, the extra twisting torque generated from the sticky end cohesion could be ignored, thus, the above overwinding/underwinding rule was observed; however when the sticky end length was at either 6 or 7 nt, the extra right-handed twisting torque generated from the sticky end cohesion needed to be counted. [47][48] Taking the bLW-E31/7 tubes as the example, although its helical twist density of the arm of 31/3 = 10.3 bp/turn was less than 10.5 bp/turn, its much stronger 7 nt sticky end cohesion could generate an extra right-handed twisting torque. As a consequence the coupling of the right-handed arm twist and the intrinsic left-handed tile curvature generated the unique bLW-E31/7 tube r- (3,3) in high yield with the right-handed curvature.
We grew tubes using a very slow annealing process over 70 hours, allowing for the assembly of all the lattice structures to be thermodynamically controlled. From the AFM imaging results, we found out that the bLW-Ep/q tubes always presented either one of two clusters of the chiral indices (n,m) with 1) n ≈ 2m and 2) n ≈ m when 0 ≤ m ≤ n. When n ≈ 2m, tubes possessing the left-handed curvature were governed by the intrinsic tile curvature, while when n ≈ m, the tubes followed the right-handed curvature due to the control by the extra right-handed arm twist. The flip-over of the tile curvature could be compared to that of a contact lens. We suggest that the bLW tile core is always curved due to its rigidity, and its intrinsic left-handed curvature can be forced by stronger opposite arm twists to switch to the right-handed curvature during the assembly process. The two types of chiral indices (n,m) of n ≈ 2m and n ≈ m should represent the most stable structures with global energy minima for tubes possessing the left-handed and right-handed curvatures, respectively. Other intermediate tubes with n and m (0 ≤ m ≤ n) deviating from n ≈ 2m and n ≈ m should possess higher deformation energies and would not be thermodynamically favored. In fact, we imaged a few of these tubes in the bLW-E21/5 assemblies, but they represented less than 5 % of the total. In all the tested bLW-Ep/q tubes, we only achieved three tube assemblies in high yield, bLW-E21/5, bLW-E31/7, and bLW-E32/6. Thus, the tube assembly performance depends not only on a delicate balance between the intrinsic tile curvature and the arm twists, but also on the helix winding phase and geometry match between the tile cores and the arms.
Formation of nonhomogeneous tubes assembled from small tiles by O-tiling is often attributed to the decrease of the global free energy of the system. We observed dominant tubular structures and minor planar lattices in the bLW-O26/4 assemblies, which might be caused by the rigidity of bLW tile cores which always possess a curvature. The curvature flip-over effect observed in the E-tiling tubes might also occur in the O-tiling assemblies. Once a ring seed spontaneously forms, epitaxial extension of the tube lattices could force the oppositely curled tiles in the bLW-O26/4 assemblies to switch to the same curled ones. As for the less rigid TW and DX tiles, the intrinsic tile curvature is more flexible, thus leading O-tiling of TW and DAE tiles to generate dominant planar ribbons and minor nonhomogeneous tubes with random stoichiometric ratios, suggesting that the global curling bias is much weaker on the TW-O26/4 and the DAE-O26/4 assemblies 2 than on the bLW-O26/4 ones. Overall, coupling of shallower intrinsic tile curvatures with weaker arm twisting torques would generate either larger lattices or wider tubes easily in higher probability. On the other hand, the coupling of the deeper intrinsic curvatures with stronger arm twisting torques to generate large arrays would be difficult to achieve, and would most probably yield ill-behaved fragments of tile oligomers, narrower tubes, and even fibers rather than well-behaved DNA lattices.
Other couplings between the two extremes would generate moderate structures in a complicated way.
In summary, we not only realized three weaving architectures to construct three new tile cores, LW, bLW, TW, and their 2D lattices, but we also deciphered each tube configuration and their component tile's conformation with the biotin/SA labeling technique. By abstracting the 2D Bravais lattice of centered rectangle for each assembly, we introduced the chiral indices to quantify the global tile curvature. We used a simplified model to analyze in detail the curving forces acting on the two types of DNA tubes (the intrinsic tile curving force and the extra arm curving force), The coupling of these two forces allowed us to understand the mechanism of formation of the DNA tubes into a plethora of different shapes, diameters, and configurations. Such detailed investigation of the DNA tube curvatures including structure and quantitation will on one hand be helpful in the future for engineering DNA nanostructures with high yield and high quality, and for the investigation of the different physicochemical properties and biological functionalities of the DNA chiral assemblies on the other hand.