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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

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

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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. Outline. - PowerPoint PPT Presentation

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Page 1: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

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

Presented By: Thomas Dickerson and Andrew ParentSaint Michael’s College

With Collaboration from:Jacob Girard, Brian Goodhue, Daniel Lewis, Andrew Gilbert, Mary Spuches, and Dan Koch

Page 2: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

OutlineIntroductory Information

Basic Definitions Labeling of TilesDefinitions II Labeling of RotationsOctahedral Group

Accomplishing the GoalsProblem Statement Computer Science ApproachApproach to Code Axis/Generator Correspondence

Exhaustive Case FinderCode Overview Program FlowIssues

ResultsTile Results Code Correctness

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Page 3: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Basic DefinitionsTile

◦A vertex with arms attached.

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Page 4: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Labeling of TilesArms 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:◦α1 α2 β1 - 3 arm tile.

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Page 5: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

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

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Page 6: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Definitions IILexicographically 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.

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Page 7: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Octahedral Group

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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

Page 8: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Labeling of RotationsEach 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

Page 9: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

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GraphingNaming Tiles

Schlegel diagramsIt is very helpful to be able to picture these molecules in two dimensions.

Page 10: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Examples of Rotation Labeling

α1β1γ1 @ 120°

α1 @ 180°

β2β3γ2γ3 @ 90°

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Page 11: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Problem StatementFinding 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.

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Page 12: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Approach to CodeGenerating 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 α1 α2 β1

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

Page 13: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Axis-Generator Correspondence

Single Rotation GeneratorsIdentity Noneα1β1γ1 @ 120° (α1β1γ1)α1β1γ1 @ 240° (α1β1γ1)2

α2β4γ1 @ 120° (α1β1γ1)2 (β2β3γ2γ3)3 α1α2β4γ1 @ 240° (α1β1γ1) (β2β3γ2γ3)2

α3β4γ4 @ 120° (α1β1γ1) (β2β3γ2γ3) α1α3β4γ4 @ 240° (α1β1γ1)2 (β2β3γ2γ3) α1α4β1γ4 @ 120° (α1β1γ1)2 (β2β3γ2γ3)2

α4β1γ4 @ 240° (α1β1γ1) (β2β3γ2γ3)3 α1α1 @ 180° α1α2 @ 180° (β2β3γ2γ3)2 α1β1 @ 180° (α1β1γ1)2 (β2β3γ2γ3)β2 @ 180° (α1β1γ1)2 (β2β3γ2γ3)2 α1γ1 @ 180° (α1β1γ1) (β2β3γ2γ3)3

γ2 @ 180° (α1β1γ1) (β2β3γ2γ3)2 α1α1α2γ1γ2 @ 90° (α1β1γ1)2 α1α1α2γ1γ2 @ 180° (β2β3γ2γ3) α1α1α2γ1γ2 @ 270° (α1β1γ1)2 (β2β3γ2γ3)3

α1α4β1β2 @ 90° (α1β1γ1) (β2β3γ2γ3)α1α4β1β2 @ 180° (β2β3γ2γ3)3 α1α1α4β1β2 @ 270° (α1β1γ1) α1β2β3γ2γ3 @ 90° (β2β3γ2γ3)β2β3γ2γ3 @ 180° (β2β3γ2γ3)2

β2β3γ2γ3 @ 270° (β2β3γ2γ3)3

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Page 14: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Exhaustive Case FinderComprised 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

Page 15: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Program Flow

Combo List removeCopies calcGroupings

CuboctahedronLUT

computeAngles

getResultantGroupings

CuboctahedronSymmetry

printRot

areSymmetric

AngleWrapper

Page 16: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

IssuesSome 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

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Page 17: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

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

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Page 18: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

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.

Page 19: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Code Correctness

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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 tile’s symmetry group

The sum of the order of the orbits for all n-armed tiles is 12Cn.

Page 20: Design Optimizations for DNA Nanostructures II:  Symmetries in the Octet Truss

Acknowledgements

03/10/10