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To achieve curvature control, we investigated how two variables—the length of the biotin linker and the distance between biotin molecules—affect curvature.
We evaluated the curvature angle using the radius of curvature.
It is assumed that two biotin-modified DNA molecules bind to one avidin.
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Avidin has four binding sites, each with an edge length of approximately 2.4 nm.
The four binding sites approximate a regular tetrahedral structure. Treating the center of the tetrahedron as the center of a sphere and approximating avidin as a rigid sphere yields a radius of approximately 2.95 nm.
Based on the distance between the binding sites, we considered that at least one turn was necessary to sufficiently bend the DNA.
The length of the linker refers to the distance from the end of the biotin to the end of the spacer, as shown in the figure on the right.
The following simulation represents the curved structure formed by biotin-modified DNA via the avidin-biotin interaction as a rigid body approximation model using Pymunk.
The DNA zigzag approximates a structure composed of numerous connected cylindrical rigid body segments.
Each cylinder has a radius of 1.0 nm and a thickness of 3.24 Å, corresponding to one base pair.
Using Pymunk's collision handling, we set up physical repulsion between DNA segments and the avidin circle.
１, Effect of Biotin Linker Length on Curvature
Using Pymunk's rigid body approximation model, we examined how the curvature of the bent structure differs when the biotin linker length is 1 nm, 2 nm, or 3 nm.
To ensure sufficient curvature for comparison, the distance between biotin groups was fixed at 6.8 nm (2 turn).
The curvature radius is measured at the point when the DNA's curved structure has finally stabilized. However, if the curved structure collapses, the value is taken immediately before the collapse occurs.
Simulation images of other patterns are included in the appendix.
linker length：1.onm
linker length：2.0nm
linker length：3.0nm
This simulation was created using source code １ (see Appendix).
Findings
When the linker length is 1 nm, the distance between DNA and avidin is too close, preventing the formation of a curvature structure due to steric hindrance.. The radius of curvature is approximately 3.2 nm.
When the linker length is 2 nm, a curved structure can form. However, due to the close proximity between the DNA and avidin, the curve becomes distorted and unstable. The radius of curvature is approximately 3.5 nm.
When the linker length is 3 nm, the curved structure forms stably because the distance between DNA and avidin is appropriate. The radius of curvature is approximately 3.8 nm.
If the length of the linker is too long, the avidin-biotin interaction becomes less easily transmitted to the DNA. Therefore, it can be inferred that a shorter linker length is more appropriate. However, if the linker length is too short, the curved structure cannot form due to steric hindrance issues with the avidin. Setting an appropriate linker length is necessary.
Furthermore, it can be seen that as the length of the linker increases, the radius of curvature becomes larger, meaning the bending angle becomes smaller. This is due to the avidin-biotin interaction becoming less easily transmitted to the DNA.
２, The correlation between linker length and radius of curvature
As the length of the linker increases, the binding force becomes less easily transmitted to the DNA, making it difficult to maintain the curved structure.
The correlation between linker length and radius of curvature is shown for each biotin-biotin distance within the linker length range of 0 nm to 5 nm.
Biotin spacer were determined to be 3.4 nm(1 turn), 4.6nm(1 turn + major groove), 5.6nm(1 turn + minor groove)
, 6.8 nm(2turn)
Vertical axis: curvature radius
Horizontal axis: Linker length
A sharp decrease in the radius of curvature indicates that a curved surface structure is not formed.
Simulation images of other patterns are included in the appendix.
Biotin spacing：3.4 nm(1 turn)
Biotin spacing：4.6nm(1 turn + major groove)
Biotin spacing：5.6nm(1 turn + minor groove)
Biotin spacing：6.8 nm(2turn)
This simulation was created using source code ２ (see Appendix).
Findings
As the biotin spacing increases, the maximum length of the linker capable of forming a curved structure also increases.
３, Effect of Biotin Spacing on Curvature
Using Pymunk's rigid body approximation model, we investigated how the curvature of the bent structure differs for biotin inter-distance values of 3.4 nm(1 turn), 6.8 nm(2turn), and 10.2 nm(3turn).
The length of the biotin linker was fixed at 3.05 nm, the value actually used in the experiment.
The curvature radius is measured at the point when the DNA's curved structure has finally stabilized. However, if the curved structure collapses, the value is taken immediately before the collapse occurs.
Simulation images of other patterns are included in the appendix.
Biotin spacing：3.4 nm(1 turn)
Biotin spacing：6.8 nm(2 turn)
Biotin spacing：10.2 nm(3 turn)
Findings
When the biotin spacing is 1 turn, a curved structure can be formed, but its curvature is gentle. The curvature radius is approximately 3.6 nm.
When the biotin spacing is 2 turn , a curved structure can be formed, and this curved structure appears stable. The curvature radius is approximately 3.8 nm.
When the biotin interval is 3 turn, it can form a curved structure, but because the DNA is long, that curved structure sags and appears unstable. The curvature radius is approximately 3.9 nm.
As the biotin spacing increases, the curvature radius becomes larger. However, it was suggested that if the DNA length becomes excessively long, a stable curved structure cannot be formed.
Furthermore, while the curvature radius increases as the biotin spacing , the rate of change is smaller than when altering the length of the biotin linker.
４, Visualization of Curvature in Planar Lattice-Like DNA Origami Tiles
We conducted a three-dimensional plane simulation using Pybullet. Origami DNA models are constructed by arranging small particles in a lattice pattern and connecting adjacent particles with springs.
The biotin spacing was set to 10.2 nm (3 turns), and the biotin linker length was set to 3.05 nm.
Yellow indicates convex surfaces, while blue indicates concave surfaces.
This simulation was created using source code ３ (see Appendix).
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