1. 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

2. Self-Assembly
Biology

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

4. Self-Assembly
(Winfree, 1998)

5. Tile Self-Assembly
Type = green
Type = purple

6. Tile Self-Assembly
Strength = 2
Strength = 1

7. Tile Self-Assembly
Rotation
Translation

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

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

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

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

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

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

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

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

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

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

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

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

20. 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

21. 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)

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

23. Negative Glues
Tileset:
Temperature: 2 1

24. Negative Glues
Tileset:
Temperature: 2 1

25. Negative Glues
Tileset:
Temperature: 2 1

26. Negative Glues
Tileset:
Temperature: 2 1

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

28. 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?

29. 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)

30. 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

31. 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

32. 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

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

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

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

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

37. Shape Scaling
Target Shape S

38. Shape Scaling
Spanning Tree of S

39. Shape Scaling
Shape S at scale 2 x2

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

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

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

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

44. Shape Scaling
Shape S at scale 6 x3

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

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

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

48. 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

49. 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( )

50. Process Overview O( ) distinct tile types Higher-Base Representation1 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)

51. 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

52. 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

53. 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

54. Process Overview
Path-walking instructions

55. Process Overview
Inspired by (Schweller, Sherman, 2013)

56. Process Overview
Information Block

57. Process Overview

58. Process Overview

59. Process Overview

60. Process Overview

61. Process Overview
Outlining Path

62. Process Overview

63. Process Overview

64. Process Overview

65. Process Overview

66. Process Overview

67. Process Overview

68. Process Overview

69. Process Overview

70. Process Overview

71. Process Overview

72. Process Overview

73. Process Overview

74. Process Overview

75. Process Overview

76. Process Overview

77. Process Overview

78. Process Overview

79. Process Overview

80. Process Overview
Optimal description
of shape S:

81. 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

82. Thank You!