1. Algorithms for Robust Self-Assembly1 1 1
Ho-Lin Chen
Ashish Goel
Qi Cheng 1 1 1 1 1 1 1
Counter made by self-assembly [Adleman, Cheng, Goel, Huang ’01]
(Chen, Goel, Cheng)

2. Self-Assembly
Self-Assembly is the process by which simple objects autonomously assemble into complexes.
Geometry, dynamics, combinatorics are all important
Inorganic: Crystals, supramolecules
Organic: Proteins, DNA, cells, organisms
Goals: Understand self-assembly, design self-assembling systems
A key problem in nano-technology, molecular robotics, molecular computation
(Chen, Goel, Cheng)

3. Applications of Self-Assembly
Self-assembly can be used to create small electrical devices such as FLASH memory. [Black et. Al. ’03]
Self-assembly can create nanostructures which “steer” light in the same way computer chips steer electrons. [Percec et. Al. ’03]
DNA strands can self-assemble into tiles and those tiles can further self-assemble into larger structures. This has many potential applications. [Winfree ’96]
DNA “rug” by Winfree
(Chen, Goel, Cheng)

4. DNA Tiles Glues = sticky ends Tiles = molecules [Winfree]
(Chen, Goel, Cheng)

5. Tile System: A tile can attach to an assembly
[Rothemund, Winfree, ’2000]
Temperature: A positive integer.
A set of tile types: Each tile is an oriented square with glues on its edges. Each glue has a non-negative strength.
An initial assembly (seed).
A tile can attach to an assembly
iff the combined strength of the
“matchings glues” is greater or
equal than the temperature.
(Chen, Goel, Cheng)

6. Example of a tile system
Temperature: 2
Set of tile types:
Seed:
(Cheng, Goel, Cheng)

7. Example of a SA process Temperature: 2 Set of tile types: Seed:
(Cheng, Goel, Cheng)

8. Example of a SA process Temperature: 2 Set of tile types: Seed:
(Cheng, Goel, Cheng)

9. Example of a SA process Temperature: 2 Set of tile types: Seed:
(Cheng, Goel, Cheng)

10. Theoretical and Algorithmic Issues
Efficiently assembling basic shapes with precisely controlled size and pattern
Constructing N X N squares with Ω(log n/log log n) tiles
[Adleman, Cheng, Goel, Huang, ’01]
Perform universal computation by simulating BCA
[Winfree ’99]
Library of primitives to use in designing nano-scale structures [Adleman, Cheng, Goel, Huang, ’01]
Automate the design process
[Adleman, Cheng, Goel, Huang, Kempe, Moisset de espanes, Rothemund ’01]
Robustness
(Cheng, Goel, Cheng)

11. Robustness
In practice, self-assembly is a thermodynamic process. When T=2, tiles with 0 or 1 matches also attach; tiles held by total strength 2 also fall off at a small rate.
Currently, there are 1-10% errors observed in experimental self-assembly.
[Winfree, Bekbolatov, ’03]
Possible schemes for error correction
Biochemistry tricks
Coding theory and error correction
(Cheng, Goel, Cheng)

12. Error-Reducing Designs
Biochemistry tricks
Strand Invasion mechanism
[Chen, Cheng, Goel, Huang, Moisset de espanes, ’04]
Coding theory and error correction
Proofreading tiles
[Winfree, Bekbolatov,’03]
Snake tiles
[Chen, Goel 05]
(Cheng, Goel, Cheng)

13. Strand Invasion
(Cheng, Goel, Cheng)

14. Strand Invasion
(Cheng, Goel, Cheng)

15. Strand Invasion
(Cheng, Goel, Cheng)

16. Strand Invasion
(Cheng, Goel, Cheng)

17. Strand Invasion
(Cheng, Goel, Cheng)

18. Strand Invasion
(Cheng, Goel, Cheng)

19. Strand Invasion
(Cheng, Goel, Cheng)

20. Strand Invasion
(Cheng, Goel, Cheng)

21. Strand Invasion
(Cheng, Goel, Cheng)

22. Strand Invasion
(Cheng, Goel, Cheng)

23. Strand Invasion
(Cheng, Goel, Cheng)

24. Strand Invasion
(Cheng, Goel, Cheng)

25. Strand Invasion
(Cheng, Goel, Cheng)

26. Strand Invasion
(Cheng, Goel, Cheng)

27. Strand Invasion
(Cheng, Goel, Cheng)

28. Strand Invasion
(Cheng, Goel, Cheng)

29. Strand Invasion
(Cheng, Goel, Cheng)

30. Strand Invasion
(Cheng, Goel, Cheng)

31. Strand Invasion
(Cheng, Goel, Cheng)

32. Strand Invasion
(Cheng, Goel, Cheng)

33. Strand Invasion (cont)
(Cheng, Goel, Cheng)

34. Strand Invasion
(Cheng, Goel, Cheng)

35. Example
T=2
(Cheng, Goel, Cheng)

36. Example
T=2
(Cheng, Goel, Cheng)

37. Example
T=2
(Cheng, Goel, Cheng)

38. What can go wrong?
T=2
(Cheng, Goel, Cheng)

39. What can go wrong?
T=2
(Cheng, Goel, Cheng)

40. Why it may not matter:
T=2
(Cheng, Goel, Cheng)

41. Why it may not matter:
T=2
(Cheng, Goel, Cheng)

42. What can go really wrong?
(Cheng, Goel, Cheng)

43. What can go really wrong?
(Cheng, Goel, Cheng)

44. What can go really wrong?
(Cheng, Goel, Cheng)

45. Safe attachments and invadable systems
Unsafe
Definition: A tile system is an invadable system iff for all assemblies that can be grown from the initial assembly, all possible attachments are safe.
(Cheng, Goel, Cheng)

46. Can we create efficient counters?
(Cheng, Goel, Cheng)

47. Can we create efficient counters?
Yes! Using “Chinese remaindering”
T=2
(Cheng, Goel, Cheng)

48. Can we create efficient counters?
Yes! Using “Chinese remaindering”
T=2
(Cheng, Goel, Cheng)

49. Can we create efficient counters?
Yes! Using “Chinese remaindering”
T=2
(Cheng, Goel, Cheng)

50. Can we create efficient counters?
Yes! Using “Chinese remaindering”
T=2
(Cheng, Goel, Cheng)

51. Can we create efficient counters?
Yes! Using “Chinese remaindering”
T=2
(Cheng, Goel, Cheng)

52. Can we create efficient counters?
Yes! Using “Chinese remaindering”
T=2
Generalizing:
Say
are distinct primes.
We can use tile
types to construct a
rectangle.
(Cheng, Goel, Cheng)

53. Invadable System
The restriction posed by invadable systems does not reduce the computational power of the model
For the problems we have considered, the loss in efficiency is limited to poly-logarithmic factors
This approach has not been tested in the lab
A complete solution is likely to include both coding/error correction and molecular biology tricks
(Cheng, Goel, Cheng)

54. Error Correcting Tile Sets
Do not change the tile model
Tile systems are designed to have error correction mechanisms.
Two most common errors
Crystallization error
Nucleation error
(Cheng, Goel, Cheng)

55. Example: Sierpinski Tile System1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

56. Example: Sierpinski Tile System1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

57. Example: Sierpinski Tile System1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

58. Example: Sierpinski Tile System1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

59. Crystallization Error1 1 1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

60. Crystallization Error1
mismatch 1 1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

61. Crystallization Error1 1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

62. Crystallization Error1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

63. Crystallization Error1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

64. [Winfree, Bekbolatov, ’03]
Proofreading Tiles
[Winfree, Bekbolatov, ’03] G2 G1 G3
G2a
G2b G4
G1b X3
G3b X4 X2
Each tile in the original system corresponds to
four tiles in the new system
The internal glues are unique to this block
G1a X1
G3a
G4a
G4b
(Cheng, Goel, Cheng)

65. How does this help? 1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

66. How does this help? 1
mismatch 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

67. How does this help? 1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

68. How does this help? No tile can attach at this location 1 1 1 1 1 1 11 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

69. How does this help? 1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

70. How does this help? 1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

71. How does this help? 1 1 1 1 1 1 1 1
(Cheng, Goel, Cheng)

72. Nucleation Error
(Cheng, Goel, Cheng)

73. Nucleation Error First tile attaches with a weak binding strength
(Cheng, Goel, Cheng)

74. Nucleation Error First tile attaches with a weak binding strength
Second tile attaches and secures the first tile
(Cheng, Goel, Cheng)

75. Nucleation Error First tile attaches with a weak binding strength
Second tile attaches and secures the first tile
Other tiles can attach and forms a layer of (possibly incorrect) tiles.
(Cheng, Goel, Cheng)

76. Snake Tiles G2 G1 G3 G2a G2b G4 G1b X2 G3b X1 X3
Each tile in the original system corresponds to four tiles in the new system
The internal glues are unique to this block
G1a
G3a
G4a
G4b
(Cheng, Goel, Cheng)

77. How does this help? First tile attaches with a weak binding strength
(Cheng, Goel, Cheng)

78. How does this help? First tile attaches with a weak binding strength
Second tile attaches and secures the first tile
(Cheng, Goel, Cheng)

79. How does this help? First tile attaches with a weak binding strength
Second tile attaches and secures the first tile
No Other tiles can attach without another nucleation error
(Cheng, Goel, Cheng)

80. Takes a row of tiles and complete into a triangle
Triangulation System
Takes a row of tiles and complete into a triangle 1 1 1 1
Seed row
(Cheng, Goel, Cheng)

81. Triangulation System 1 1 1 1
(Cheng, Goel, Cheng)

82. Triangulation System 1 1 1 1
(Cheng, Goel, Cheng)

83. Triangulation System 1 1 1 1
(Cheng, Goel, Cheng)

84. Triangulation System 1 1 1 1
(Cheng, Goel, Cheng)

85. Triangulation System 1 1 1 1
(Cheng, Goel, Cheng)

86. Four by Four Snake Tiles
G2a
G2b
G2c
G2d G2
G1d
G3d G1 G3
G1c
G3c G4
G1b
G3b
G1a
G3a
G4a
G4b
G4c
G4d
(Cheng, Goel, Cheng)

87. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

88. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

89. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

90. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

91. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

92. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

93. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

94. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

95. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

96. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

97. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

98. Four by Four Snake Tiles
(Cheng, Goel, Cheng)

99. Theoretical Analysis
The snake tile design can be extended to 2k by 2k blocks.
Prevents tile propagation even after k+1 nucleation error happen.
With 2k by 2k snake tile system, we can assemble an N by N square of blocks with time Õ(N1+4/k+1) and (with high probability) remain stable for time Õ(N1+4/k+1).
Assuming tiles held by strength 3 does not fall off
The error probability at each location changes from p to pk
(Cheng, Goel, Cheng)

100. Conclusion and Future Work
Robustness is one of the most important issues in DNA tile self-assembly
(Cheng, Goel, Cheng)

101. Conclusion and Future Work
Robustness is one of the most important issues in DNA tile self-assembly
Future work on robustness
Some other error correction designs
Experimental verification
(Cheng, Goel, Cheng)

102. Conclusion and Future Work
Robustness is one of the most important issues in DNA tile self-assembly
Future work on robustness
Some other error correction designs
Experimental verification
Open problems in self-assembly
Optimum design of tile system
Formal analysis of reversible tile systems
(Cheng, Goel, Cheng)