1. 1 January 18, 2010 Shape Replication through Self-Assembly and Rnase Enzymes Zachary AbelHarvard University Nadia BenbernouMassachusetts Institute of Technology Mirela DamianVillanova University Erik D. DemaineMassachusetts Institute of Technology Martin DemaineMassachusetts Institute of Technology Robin FlatlandSiena College Skott D. KominersHarvard University Robert SchwellerUniversity of Texas Pan American Read: Replicate:

2. 2 Outline Basic Model RNA enzyme model Shape replication Precise yield shape replication Infinite yield shape replication

3. 3 Tile Assembly Model (Rothemund, Winfree, Adleman) T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 Tile Set: Glue Function: Temperature: x ed cba

4. 4 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

5. 5 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

6. 6 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

7. 7 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

8. 8 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

9. 9 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

10. 10 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

11. 11 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

12. 12 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

13. 13 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e Tile Assembly Model (Rothemund, Winfree, Adleman)

14. 14 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba x abc d e Tile Assembly Model (Rothemund, Winfree, Adleman)

15. 15 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 abc d e x x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

16. 16 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx Tile Assembly Model (Rothemund, Winfree, Adleman)

17. 17 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx x Tile Assembly Model (Rothemund, Winfree, Adleman)

18. 18 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx xx (Basic)Tile Assembly Model (Rothemund, Winfree, Adleman)

19. 19 Outline Basic Model RNA enzyme model Shape replication Precise yield shape replication Infinite yield shape replication

20. 20 RNA enzyme Self-Assembly (suggested by Rothemund, Winfree 2000) RNA tile types DNA tile types RNA assembly model: Assembly occurs over a number of stages. At each stage you may: 1) Add a new collection of tile types - Allow for further growth - All added types have infinite count 2) Add an Rnase enzyme - Dissolve all RNA tile types - May break apart assemblies All tile types are of either DNA or RNA makeup:

21. 21 RNA enzyme Self-Assembly Stage 1:

22. 22 RNA enzyme Self-Assembly Stage 1:

23. 23 RNA enzyme Self-Assembly Stage 1: Stage 2:

24. 24 RNA enzyme Self-Assembly Stage 1: Stage 2:

25. 25 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme

26. 26 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme

27. 27 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme

28. 28 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme Stage 4:

29. 29 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme Stage 4:

30. 30 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme Stage 4:

31. 31 RNA enzyme Self-Assembly Stage 1: Stage 2: Stage 3: Enzyme Stage 4:

32. 32 RNA enzyme Self-Assembly Metrics for efficiency: Tile complexity: total number of distinct tile types used in the system. Stage complexity: total number of distinct stages used.

33. 33 Outline Basic Model RNA enzyme model Shape replication Precise yield shape replication Infinite yield shape replication

34. Shape Replication Problem Design an assembly system (algorithm) that will replicate a large number of copies given a single copy of a pre-assembled input shape. Precise Yield: Replicate exactly n copies for a given n Infinite Yield: Replicate infinite copies -in practice, the number of copies should only be limited by the volume of particles available.

35. 35 Outline Basic Model RNA enzyme model Shape replication Precise yield shape replication Infinite yield shape replication

36. Precise Yield: rectangles

37. aaaa a a a a a a aaaa a a a a a a

38. nnnn e e e e e e w w w w w w ssss

39. nnnn e e e e e e w w w w w w ssss n w xx y y

40. nnnn e e e e e e w w w w w w ssss n w

41. nnnn e e e e e e w w w w w w ssss n w n e s ws e

42. nnnn e e e e e e w w w w w w ssss n w n e s w s e

43. n nn n e e e e e e w w w w w w ssss a w a a a

44. nn e e w w ss Step 1: Coat shape with layer of RNA

45. Precise Yield: rectangles nn e e w w ss Step 2: Coat shape with layer of DNA Step 1: Coat shape with layer of RNA

46. Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 1: Coat shape with layer of RNA.

47. Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 1: Coat shape with layer of RNA.

48. Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 1: Coat shape with layer of RNA.

49. Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 1: Coat shape with layer of RNA.

50. Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 5: Fill frame with DNA. Step 1: Coat shape with layer of RNA.

51. Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 5: Fill frame with DNA. Step 1: Coat shape with layer of RNA.

52. Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 5: Fill frame with DNA. Step 6: Add enzyme. Step 1: Coat shape with layer of RNA.

53. Precise Yield: rectangles Step 2: Coat shape with layer of DNA. Step 3: Add enzyme. Step 4: Coat frame with layer of RNA. Step 5: Fill frame with DNA. Step 6: Add enzyme. Step 1: Coat shape with layer of RNA.

54. Precise Yield: General Shapes

57. Precise Yield: rectangles

58. Tile typesO(1) StagesO(log n) Precise Yield: n copies Precise Yield: rectangles Tile typesO(log n) StagesO(1) Precise Yield: n copies

59. 59 Outline Basic Model RNA enzyme model Shape replication Precise yield shape replication Infinite yield shape replication

60. Infinite Yield: Rectangles nnnn e e e e e e w w w w w w ssss

61. nnnn e e e e e e w w w w w w ssss s w a

62. a s b s b s a b ss a s Stair step tiles: x x

63. Infinite Yield: Rectangles a s b s b s a b ss Stair step tiles: x x b x

64. Infinite Yield: Rectangles a s b s b s a b ss Stair step tiles: x x a b

65. Infinite Yield: Rectangles a s b s b s a b s s Stair step tiles: x x a b

66. Infinite Yield: Rectangles a s b s b s a b s Stair step tiles: x x b a b

67. Infinite Yield: Rectangles a s b s b s a b s Stair step tiles: x x b a b

68. Infinite Yield: Rectangles a s b s b s a b s Stair step tiles: x x b a b

69. Infinite Yield: Rectangles a s b s b s a b Stair step tiles: x x b b b a

70. Infinite Yield: Rectangles a s b s b s a b Stair step tiles: x x

71. Infinite Yield: Rectangles a s b s b s a b Stair step tiles: x x

72. Infinite Yield: Rectangles a s b s b s a b Stair step tiles: x x …

73. Infinite Yield: Rectangles Tile typesO(1) StagesO(1) Infinite Yield: Rectangles

74. Infinite Yield: General Shapes

75. Step 1: Coat with RNA

76. Infinite Yield: General Shapes Step 2: Create rectangular DNA encasing

77. Infinite Yield: Binary counter tool cccccccccccccccccccccccc

78. 1 ccccccccccccccccccccccc m 0 01 00 11 nn nn 01 10 01 cc nc 1 m m x x Binary counter tiles: c 0 1 mm n

79. 1 Infinite Yield: Binary counter tool 1 cccccccccccccccccccccc 0 0 m 0 01 00 11 nn nn 01 10 01 cc nc 1 m m x x Binary counter tiles: c 0 1 mm n

80. 1 Infinite Yield: Binary counter tool 1 ccccccccccccccccccccc 0 m 1 1 n 0 01 00 11 nn nn 01 10 01 cc nc 1 m m x x Binary counter tiles: c 0 1 mm n

81. 1 Infinite Yield: Binary counter tool 1 ccccccccccccccccccccc 0 01 00 11 nn nn 01 10 01 cc nc 1 m m x x Binary counter tiles: c 0 0 m 1 1 1 mm n 1

82. 1 Infinite Yield: Binary counter tool 1 cccccccccccccccccccc 0 01 00 11 nn nn 01 10 01 cc nc 1 m m x x Binary counter tiles: c 0 0 1 1 mm n 1 0 0 c m

83. 1 Infinite Yield: Binary counter tool 1 cccccccccccccccccccc 0 01 00 11 nn nn 01 10 01 cc nc 1 m m x x Binary counter tiles: c 0 0 m 1 1 mm n 1 0 0 1 0 0

84. 1 Infinite Yield: Binary counter tool 1 ccccccccccccccccccc 0 01 00 11 nn nn 01 10 01 cc nc 1 m m x x Binary counter tiles: c 0 0 m 1 1 mm n 1 0 1 0 0 1 1 n

85. 1 Infinite Yield: Binary counter tool 1 ccccccccccccccccccc 0 01 00 11 nn nn 01 10 01 cc nc 1 m m x x Binary counter tiles: c 0 0 m 1 1 mm n 1 0 1 0 1 1 n 0 0

86. 1 Infinite Yield: Binary counter tool 1 ccccccccccccccccccc 0 01 00 11 nn nn 01 10 01 cc nc 1 m m x x Binary counter tiles: c 0 0 m 1 1 mm n 1 0 1 0 1 1 0 0 1

87. 1 Infinite Yield: Binary counter tool 1 0 01 00 11 nn nn 01 10 01 cc nc 1 m m x x Binary counter tiles: c 0 0 1 1 mm n 1 0 1 0 1 0 1 0 1 1 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 1111 0 1 1 1 1 1 11 0 0 0 1 01 0 0 0 1 0 1 0 1 0 1 1 1 1 1 0000 1111 0 1 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1

88. Infinite Yield: General Shapes … …

89. 1111000000 111111 11111 11111000000 111111 000000 0000 1111 1111 … …

90. 1111000000 111111 11111 11111000000 111111 000000 0000 1111 1111 11111 11111 0000011111 11111 11111 11111 1 1 111111111110000 1111 0000 0000

91. 1111000000 111111 11111 11111000000 111111 000000 0000 1111 1111 11111 11111 0000011111 11111 11111 11111 1 1 111111111110000 1111 0000 0000 00000000000000000000 0000000000 0 0 000 0000 000000000000000000000 111100000 11111 11111 11111000000 111111 000000 0000 1111 1111 00000000 000000000 0 0 000 0000 0000000000000000 0 1 1 0 000000 111111 000000 000000 00 11 00 00 1111 0000 0000 0000 1 0 0 0 Step 3: Label each face with unique binary code

92. Infinite Yield: General Shapes 1111000000 111111 11111 11111000000 111111 000000 0000 1111 1111 11111 11111 0000011111 11111 11111 11111 1 1 111111111110000 1111 0000 0000 00000000000000000000 0000000000 0 0 000 0000 000000000000000000000 111100000 11111 11111 11111000000 111111 000000 0000 1111 1111 00000000 000000000 0 0 000 0000 0000000000000000 0 1 1 0 000000 111111 000000 000000 00 11 00 00 1111 0000 0000 0000 1 0 0 0 Step 4: Enzyme.

93. Infinite Yield: General Shapes 0000 1111 1111 0000 … 0000 1111 1111 0000 0000 1111 1111 0000 0000 1111 1111 0000 0000 1111 1111 0000 Step 5: Infinitely replicate all labeled rectangles

94. Infinite Yield: General Shapes 1111000000 111111 11111 11111000000 111111 000000 0000 1111 1111 11111 11111 0000011111 11111 11111 11111 1 1 111111111110000 1111 0000 0000 00000000000000000000 0000000000 0 0 000 0000 000000000000000000000 111100000 11111 11111 11111000000 111111 000000 0000 1111 1111 00000000 000000000 0 0 000 0000 0000000000000000 0 1 1 0 000000 111111 000000 000000 00 11 00 00 1111 0000 0000 0000 1 0 0 0

95. Reassembly? 0000 1111 1111 0000 000000 111111 000000 000000

96. Infinite Yield: General Shapes Reassembly? 000 1 111 1 111 0 000 0 00000 0 11111 1 00000 1 000 0 00

97. Infinite Yield: General Shapes Reassembly? 0001 1111 1110 0000 000000 111111 000001 000000

98. Infinite Yield: General Shapes Reassembly? 0001 1111 1110 0000 000000 111111 000001 000000 0 1 0 1 01 10 00 11 1 0 1 1

99. Infinite Yield: General Shapes 1111000000 111111 11111 11111000000 111111 000000 0000 1111 1111 11111 11111 0000011111 11111 11111 11111 1 1 111111111110000 1111 0000 0000 00000000000000000000 0000000000 0 0 000 0000 000000000000000000000 111100000 11111 11111 11111000000 111111 000000 0000 1111 1111 00000000 000000000 0 0 000 0000 0000000000000000 0 1 1 0 000000 111111 000000 000000 00 11 00 00 1111 0000 0000 0000 1 0 0 0 Step 6: Reassemble, fill in frame, break out copies with enzyme.

100. Infinite Yield: General Shapes Tile typesO(1) StagesO(1) Infinite Yield: Vertically convex

101. Infinite Yield: Non-vertically convex shapes Grow counter along surface of shape

102. 0001 00100010 0011 01000100 0101 01100110 0111 10001000 1001 10101010 Grow counter along surface of shape Start end Infinite Yield: Non-vertically convex shapes

103. 0001 00100010 0011 01000100 0101 01100110 0111 10001000 1001 10101010 Grow counter along surface of shape Break apart with enzyme Infinite Yield: Non-vertically convex shapes

104. 0001 00100010 0011 01000100 0101 01100110 0111 10001000 1001 10101010 Grow counter along surface of shape Break apart with enzyme Replicate Infinite Yield: Non-vertically convex shapes

105. 0001 00100010 0011 01000100 0101 01100110 0111 10001000 1001 10101010 Grow counter along surface of shape Break apart with enzyme Replicate Reassemble Infinite Yield: Non-vertically convex shapes

106. 0001 00100010 0011 01000100 0101 01100110 0111 10001000 1001 10101010 Grow counter along surface of shape Break apart with enzyme Replicate Reassemble Infinite Yield: Non-vertically convex shapes

107. 107 Tile typesO(1) StagesO(1) Infinite Yield: Infinite Yield: General Shapes

108. Future Work Replicate and improve -Hybrid algorithms for replication and modification Extension to 3D -Planarity/spacial constraint Replication of internal pattern Staged enzyme model for assembly from scratch - Seems to be very powerful for this Temperature changes to perform replication

109. 109 January 18, 2010 Thank you. Questions? Zachary AbelHarvard University Nadia BenbernouMassachusetts Institute of Technology Mirela DamianVillanova University Erik D. DemaineMassachusetts Institute of Technology Martin DemaineMassachusetts Institute of Technology Robin FlatlandSiena College Skott D. KominersHarvard University Robert SchwellerUniversity of Texas Pan American Read: Replicate: