Self-Assembly of Shapes at Constant Scale Using Repulsive Forces June 9, 2017 Austin Luchsinger1, Robert Schweller1, Tim Wylie1 1University of Texas – Rio Grande Valley

Self-Assembly Biology

Self-Assembly - Yan, H., Park, S.H., Ginkelstein, G., Reif, J.H. & LaBean, T.H.

Self-Assembly (Winfree, 1998)

Tile Self-Assembly Type = green Type = purple

Tile Self-Assembly Strength = 2 Strength = 1

Tile Self-Assembly Rotation Translation

Tile Self-Assembly (2HAM) (Rothemund, Winfree, Adleman) Tileset: Temperature: 2 1 ∞ counts of each tile type

Tile Self-Assembly (2HAM) Tileset: Temperature: 2 1

Tile Self-Assembly (2HAM) Tileset: Temperature: 2 1

Tile Self-Assembly (2HAM) Tileset: Temperature: 2 1

Tile Self-Assembly (2HAM) Tileset: Temperature: 2 1

Tile Self-Assembly (2HAM) Tileset: Temperature: 2 1

Tile Self-Assembly (2HAM) Tileset: Temperature: 2 1

Tile Self-Assembly (2HAM) Tileset: Temperature: 2 1

Tile Self-Assembly (2HAM) Tileset: Temperature: 2 1

Tile Self-Assembly (2HAM) Tileset: Temperature: 2 1

Tile Self-Assembly (2HAM) Tileset: Temperature: Terminal 2 1

Tile Self-Assembly (2HAM) This system produces a 3x3 square: Tileset: Temperature: 2 1 Tile Complexity = 6

Tile Self-Assembly (2HAM) This system produces a 3x3 square: Tileset: Temperature: 2 1 Goal: What is the minimum tile complexity for assembling an n x n square? Tile Complexity = 6

Tile Self-Assembly (2HAM) This system produces a 3x3 square: Tileset: Temperature: 2 1 Ω( ) log n log log n Tile Complexity = 6 n x n squares: (Rothemund, Winfree, 2000)

(Patitz, Schweller, Summers, 2011) Negative Glues (Patitz, Schweller, Summers, 2011) Tileset: Temperature: 2 1

Negative Glues Tileset: Temperature: 2 1

Negative Glues Tileset: Temperature: 2 1

Negative Glues Tileset: Temperature: 2 1

Negative Glues Tileset: Temperature: 2 1

Self-Assembly of Shapes What if you wanted to assemble an arbitrary shape (S)? S =

Self-Assembly of Shapes What if you wanted to assemble an arbitrary shape (S)? S = Goal: What is the minimum tile complexity for assembling arbitrary shapes?

Ω( ) Self-Assembly of Shapes K(S) log(K(S)) What if you wanted to assemble an arbitrary shape (S)? S = log(K(S)) K(S) Ω( ) (Soloveichik, Winfree, 2007)

O( ) Self-Assembly of Shapes arbitrary aTAM Staged RNAse K(S) Model Tile Complexity Scale Factor aTAM (Soloveichik, Winfree, 2007) Staged RNAse (Demaine, Patitz, Schweller, and Summers, 2010) Negative Glue 2HAM log(K(S)) K(S) O( ) arbitrary

O( ) O( ) Self-Assembly of Shapes arbitrary logarithmic aTAM Model Tile Complexity Scale Factor aTAM (Soloveichik, Winfree, 2007) Staged RNAse (Demaine, Patitz, Schweller, and Summers, 2010) Negative Glue 2HAM log(K(S)) K(S) O( ) arbitrary log(K(S)) K(S) O( ) logarithmic

O( ) O( ) O( ) Self-Assembly of Shapes arbitrary logarithmic constant Model Tile Complexity Scale Factor aTAM (Soloveichik, Winfree, 2007) Staged RNAse (Demaine, Patitz, Schweller, and Summers, 2010) Negative Glue 2HAM log(K(S)) K(S) O( ) arbitrary log(K(S)) K(S) O( ) logarithmic log(K(S)) K(S) O( ) constant

Self-Assembly of Shapes Input Optimal description of shape S:

Self-Assembly of Shapes Input Optimal description of shape S: Tileset Temperature 10

Self-Assembly of Shapes Input Optimal description of shape S: Tileset Temperature 10

Self-Assembly of Shapes Input Output Optimal description of shape S: Tileset Temperature 10 An assembly of shape S at constant scale

Shape Scaling Target Shape S

Shape Scaling Spanning Tree of S

Shape Scaling Shape S at scale 2 x2

Path around spanning tree of S Shape Scaling Path around spanning tree of S

Path around spanning tree of S Shape Scaling Path around spanning tree of S

Path around spanning tree of S Shape Scaling Path around spanning tree of S

Path around spanning tree of S Shape Scaling Path around spanning tree of S

Shape Scaling Shape S at scale 6 x3

Path around spanning tree of S with “buffer” Shape Scaling Path around spanning tree of S with “buffer”

Path around spanning tree of S with “buffer” Shape Scaling Path around spanning tree of S with “buffer”

Process Overview log(K(S)) K(S) O( ) distinct tile types

Process Overview O( ) distinct tile types Higher-Base Representation log(K(S)) K(S) O( ) distinct tile types Higher-Base Representation 4 1 7 3 2 8 Self Assembles

Process Overview O( ) O( ) distinct tile types log(K(S)) K(S) O( ) distinct tile types Higher-Base Representation 4 1 7 3 2 8 Binary Representation of length K(S) Self Assembles 1 TM using tile types log(K(S)) K(S) O( )

Process Overview O( ) distinct tile types Higher-Base Representation 1 4 7 3 2 8 distinct tile types Higher-Base Representation Binary Representation of length K(S) Self Assembles log(K(S)) K(S) O( ) TM using tile types Method introduced by (Soloveichik, Winfree)

Process Overview O( ) O( ) distinct tile types log(K(S)) K(S) O( ) distinct tile types Higher-Base Representation 4 1 7 3 2 8 Binary Representation of length K(S) Self Assembles 1 Explicit Encoding TM using tile types log(K(S)) K(S) O( ) F L R TM using O(1) tile types

Process Overview O( ) O( ) distinct tile types log(K(S)) K(S) O( ) distinct tile types Higher-Base Representation 4 1 7 3 2 8 Binary Representation of length K(S) Self Assembles 1 Explicit Encoding TM using tile types log(K(S)) K(S) O( ) F L R Scaled Shape TM using O(1) tile types O(1) tile types

O( ) Process Overview O( ) O( ) K(S) log(K(S)) distinct tile types Higher-Base Representation 4 1 7 3 2 8 Binary Representation of length K(S) Self Assembles 1 Explicit Encoding TM using tile types log(K(S)) K(S) O( ) F L R Scaled Shape TM using O(1) tile types log(K(S)) K(S) O( ) O(1) tile types

Process Overview Path-walking instructions

Process Overview Inspired by (Schweller, Sherman, 2013)

Process Overview Information Block

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview Outlining Path

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview

Process Overview Optimal description of shape S:

O( ) O( ) O( ) Summary arbitrary logarithmic constant Future Work aTAM Model Tile Complexity Scale Factor aTAM (Soloveichik, Winfree, 2007) Staged RNAse (Demaine, Patitz, Schweller, and Summers, 2010) Negative Glue 2HAM log(K(S)) K(S) O( ) arbitrary log(K(S)) K(S) O( ) logarithmic log(K(S)) K(S) O( ) constant Can we lower the temperature? - currently a temperature 10 system - high temperatures allow for varying degrees of interaction Future Work

Thank You!