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

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Outline Introductory Information Basic DefinitionsLabeling of Tiles Definitions IILabeling of Rotations Octahedral Group Accomplishing the Goals Problem StatementComputer Science Approach Approach to CodeAxis/Generator Correspondence Exhaustive Case Finder Code OverviewProgram Flow Issues Results Tile ResultsCode Correctness 2

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Basic Definitions Tile A vertex with arms attached. 3

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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: α 1, α 2, α 3, α 4, β 1, β 2, …, γ 3, γ 4. A quick example: α 1 α 2 β 1 - 3 arm tile. 4

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Labeling of Tiles (cont.) Cube Octahedron Arms3 Armed Tile example 5

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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

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Octahedral Group 7 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

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Labeling of Rotations Each rotation defined by a set of arms. Arms define either a face or a vertex on the cuboctahedrons surface. Axis of rotation oriented through the cuboctahedrons centroid. Varies based on axis of rotation. Three major types – divisible by: 120 ° – 3 arms. – eg. α 1 β 1 γ 1 @ 120° 180 ° – 1 arm. – eg. α 1 @ 180° 90 ° – 4 arms. – eg. β 2 β 3 γ 2 γ 3 @ 90° All rotations are counterclockwise. 8

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

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Examples of Rotation Labeling α 1 β 1 γ 1 @ 120° α 1 @ 180° β 2 β 3 γ 2 γ 3 @ 90° 10

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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

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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 α 1 α 2 β 1 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

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Axis-Generator Correspondence Single RotationGenerators IdentityNone α1β1γ1 @ 120°(α1β1γ1) α1β1γ1 @ 240°(α1β1γ1) 2 α2β4γ1 @ 120°(α1β1γ1) 2 (β2β3γ2γ3) 3 α1 α2β4γ1 @ 240°(α1β1γ1) (β2β3γ2γ3) 2 α3β4γ4 @ 120°(α1β1γ1) (β2β3γ2γ3) α1 α3β4γ4 @ 240°(α1β1γ1) 2 (β2β3γ2γ3) α1 α4β1γ4 @ 120°(α1β1γ1) 2 (β2β3γ2γ3) 2 α4β1γ4 @ 240°(α1β1γ1) (β2β3γ2γ3) 3 α1 α1 @ 180°α1 α2 @ 180°(β2β3γ2γ3) 2 α1 β1 @ 180°(α1β1γ1) 2 (β2β3γ2γ3) β2 @ 180°(α1β1γ1) 2 (β2β3γ2γ3) 2 α1 γ1 @ 180°(α1β1γ1) (β2β3γ2γ3) 3 γ2 @ 180°(α1β1γ1) (β2β3γ2γ3) 2 α1 α1α2γ1γ2 @ 90°(α1β1γ1) 2 α1 α1α2γ1γ2 @ 180°(β2β3γ2γ3) α1 α1α2γ1γ2 @ 270°(α1β1γ1) 2 (β2β3γ2γ3) 3 α1α4β1β2 @ 90°(α1β1γ1) (β2β3γ2γ3) α1α4β1β2 @ 180°(β2β3γ2γ3) 3 α1 α1α4β1β2 @ 270°(α1β1γ1) α1 β2β3γ2γ3 @ 90°(β2β3γ2γ3) β2β3γ2γ3 @ 180°(β2β3γ2γ3) 2 β2β3γ2γ3 @ 270°(β2β3γ2γ3) 3 13

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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

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

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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 α 1 and α 2 @ 60° and 120° Only valid for tiles which lie entirely in a single hexagonal plane 16

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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

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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.

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Code Correctness 19 We used the Orbits and Stabilizers theorem as a self check. o 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 tiles symmetry group The sum of the order of the orbits for all n- armed tiles is 12 C n.

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Acknowledgements 03/10/10

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