Design Optimizations for DNA Nanostructures II: Symmetries in the Octet Truss

1. Design Optimizations for DNA Nanostructures II: Symmetries in the Octet Truss Presented By: Thomas Dickerson and Andrew Parent Saint Michael’s College With Collaboration from: Jacob Girard, Brian Goodhue, Daniel Lewis, Andrew Gilbert, Mary Spuches, and Dan Koch

2. Outline Introductory Information Basic Definitions Labeling of Tiles Definitions II Labeling of Rotations Octahedral Group Accomplishing the Goals Problem Statement Computer Science Approach Approach to Code Axis/Generator Correspondence Exhaustive Case Finder Code Overview Program Flow Issues Results Tile Results Code Correctness 2

3. Basic Definitions • Tile • A vertex with arms attached. 3

4. Labeling of Tiles • Arms are labeled based on what plane they are in, in the cuboctahedron. • Labeled: ‘α’, ‘β’ and ‘γ’ • Each plane has four arms thus: • α1, α2, α3, α4, β1, β2, …, γ3, γ4. • A quick example: • α1α2β1 - 3 arm tile. 4

5. Labeling of Tiles (cont.) Cube Octahedron Arms 3 Armed Tile example 5

6. Definitions II • Lexicographically Minimal • A tile is lex-minimal when it has its arms in positions that are in the lowest orders, like a form of alphabetical order for tiles. • Unique • A unique tile type is the canonical representative for the group of tiles that are rotationally isomorphic to it. 6

7. Octahedral Group • The point group of symmetries for the cuboctahedron (and others). • Is of order 48 • Includes inversions • Inversions are bad • Has a pure rotational subgroup of order 24 7

8. Labeling of Rotations • Each rotation defined by a set of arms. • Arms define either a face or a vertex on the cuboctahedron’s surface. • Axis of rotation oriented through the cuboctahedron’s centroid. • Varies based on axis of rotation. • Three major types – divisible by: • 120° – 3 arms. – eg. α1β1γ1@ 120° • 180° – 1 arm. – eg. α1@ 180° • 90° – 4 arms. – eg. β2β3γ2γ3@ 90° • All rotations are counterclockwise. 8

9. Graphing Naming Tiles Schlegel diagrams It is very helpful to be able to picture these molecules in two dimensions.

10. Examples of Rotation Labeling α1 @ 180° α1β1γ1 @ 120° β2β3γ2γ3 @ 90° 10

11. Problem Statement • Finding all lex-minimal tiles. • Two approaches: • By hand • Pros: Easy to visualize, immediate satisfaction. • Cons: Slow, potentially inaccurate. • Computer Science • Pros: Fast, 100% correct (assuming the code is correct). • Cons: Difficult to visualize, results come all at once as a final product. 11

12. Approach to Code • Generating the combinations • Solve Hamming distance problem for n arms. • Tiles are created in sorted order. • Generates a bitstring, such as the following: • “110010000000” a 3 arm tile, namely α1α2β1 • Eliminating Duplicate Tiles • Use a combination of rotations and angular grouping to identify duplicates. • Angular grouping reduces the overhead of the rotation stage. • Generators • Represent all axes of rotation with 3 operations. 12

13. Axis-Generator Correspondence 13

14. Exhaustive Case Finder • Comprised of four classes • Combo • Generates every possible n-armed tile • CuboctahedronLUT • Lookup table used to group the cases generated by Combo based on the angles between arm pairs • CuboctahedronSymmetry • This performs the generator rotations in order to remove duplicates and distinguish between inversions • AngleWrapper • Wraps arrays representing angles between arm-pairs to override Java's built-in array comparison behavior (performs a shallow element comparison rather than a simple object-reference comparison) 14

15. Program Flow AngleWrapper CuboctahedronLUT computeAngles getResultantGroupings Combo List removeCopies calcGroupings CuboctahedronSymmetry printRot areSymmetric

16. Issues • Some tiles may not be useful in building actual structures. • Generates duplicates as “unique” tiles. • Certain tiles are rotationally isomorphic to their inversion • These cases can not be identified using only the rotations which are legal under the octahedral group • Four additional generator rotations are necessary • α1 and α2@ 60° and 120° • Only valid for tiles which lie entirely in a single hexagonal plane 16

17. Results • Tiles Found • 1 arm – 1 • 2 arm – 4 • 3 arm – 10 • 4 arm – 26 • 5 arm – 37 • 6 arm – 48 • 7 arm – 38 • 8 arm – 27 • 9 arm – 13 • 10 arm – 5 • 11 arm – 1 17

18. Results Continued. • Note that the number of tiles with n arms and tiles with (12 – n) arms are unequal • This is because of the extra rotations.

19. Code Correctness • We used the Orbits and Stabilizers theorem as a self check. • Number of rotations that leave a tile fixed (stabilizer subgroup) times number of new positions for the tile (orbit of the tile) is the order of the tile’s symmetry group • The sum of the order of the orbits for all n-armed tiles is 12Cn. 19

20. Acknowledgements