1. Complexities for Generalized Models of Self-Assembly Gagan Aggarwal Stanford University Michael H. Goldwasser St. Louis University Ming-Yang Kao Northwestern University Robert T. Schweller Northwestern University Some results were obtained independantly by Cheng, Espanes 2003

2. Outline Importance of DNA Self-Assembly –Synthesis of Nanostructures –DNA Computing Tile Self-Assembly DNA Word Design

4. TILE

5. G C A T C G C G T A G C

6. TILE G C A T C G C G T A G C

7. TILE

10. Super Small Circuits, Built Autonomously

11. Molecular-scale pattern for a RAM memory with demultiplexed addressing ( Winfree, 2003)

12. DNA Computers Computer Program + Input Output!

13. DNA Computers Computer Program + Input Output! Program

14. DNA Computers Computer Program + Input Output! Program Input +

15. DNA Computers Computer Program + Input Output! Program Input + Output!

16. Outline Importance of DNA Self-Assembly Tile Self-Assembly (Generalized Models) –Tile Complexity –Shape Verification –Error Resistance DNA Word Design

17. Tile Model of Self-Assembly (Rothemund, Winfree STOC 2000) Tile System: t : temperature, positive integer G: glue function T: tileset s: seed tile

18. How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

19. How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

20. How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

21. How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

22. How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

23. How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

24. How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

25. How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

26. How a tile system self assembles T = G(y,y) = 2 G(g,g) = 2 G(r, r) = 2 G(b,b) = 2 G(p,p) = 1 G(w,w) = 1 t = 2

27. New Models Multiple Temperature Model –temperature may go up and down Flexible Glue Model –Remove the restriction that G(x, y) = 0 for x!=y Multiple Tile Model –tiles may cluster together before being added Unique Shape Model –unique shape vs. unique supertile

28. New Models Multiple Temperature Model –temperature may go up and down Flexible Glue Model –Remove the restriction that G(x, y) = 0 for x!=y Multiple Tile Model –tiles may cluster together before being added Unique Shape Model –unique shape vs. unique supertile

29. New Models Multiple Temperature Model –temperature may go up and down Flexible Glue Model –Remove the restriction that G(x, y) = 0 for x!=y Multiple Tile Model –tiles may cluster together before being added Unique Shape Model –unique shape vs. unique supertile

30. New Models Multiple Temperature Model –temperature may go up and down Flexible Glue Model –Remove the restriction that G(x, y) = 0 for x!=y Multiple Tile Model –tiles may cluster together before being added Unique Shape Model –unique shape vs. unique supertile

31. Focus Multiple Temperature Model –Adjust temperature during assembly Flexible Glue Model –Remove the restriction that G(x, y) = 0 for x!=y Goal: Reduce Tile Complexity

32. Our Tile Complexity Results Multiple temperature model: k x N rectangles: beats standard model: Flexible Glue: N x N squares: beats standard model: ( Adleman, Cheng, Goel, Huang STOC 2001 ) (our paper)

33. Building k x N Rectangles k-digit, base N (1/k) counter: k N

34. Building k x N Rectangles k-digit, base N (1/k) counter: Tile Complexity: N k

35. S C1C1 C2C2 C3C3 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g S3S3 S2S2 0 0 S S1S1

36. S C1C1 C2C2 C3C3 0123 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g S3S3 S2S2 0 0 00 g g SC1C1 C2C2 C3C3 S1S1 S2S2 S3S3 S1S1 0 0 0 0 ggp

37. S C1C1 C2C2 C3C3 0123 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g gg S3S3 S2S2 0 0 1 00 g g pr 0 SC1C1 C2C2 C3C3 S1S1 00 S2S2 S3S3 00 00 01 S1S1 p

38. S C1C1 C2C2 C3C3 0123 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g gg S3S3 S2S2 0 0 1 00 g g pr 0 SC1C1 C2C2 C3C3 S1S1 0001 S2S2 S3S3 00 00 01 S1S1 gg

39. S C1C1 C2C2 C3C3 0123 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g gg S3S3 S2S2 0 0 1 00 g g pr 0 SC1C1 C2C2 C3C3 S1S1 0001 S2S2 S3S3 00 00 01 S1S1 C0C0 C1C1 C2C2 C3C3 00 00 p

40. S C1C1 C2C2 C3C3 0123 0 ggp Build a 4 x 256 rectangle: t = 2 C0C0 g gg S3S3 S2S2 0 0 1 00 g g pr 0 SC1C1 C2C2 C3C3 S1S1 0001 S2S2 S3S3 00 00 01 S1S1 C0C0 C1C1 C2C2 C3C3 00 00 p 11 00 00 12 23

41. S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 01 S1S1 p

42. S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 P C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 01 S1S1

43. S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 P 01 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 01 S1S1

44. S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 P 01 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 01 S1S1 R

45. S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 P 01 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 01 S1S1 R C0C0 C1C1 C2C2 …

46. S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 SC0C0 C1C1 C2C2 C3C3 S1S1 0001 C1C1 C2C2 C3C3 1122331223 S2S2 S3S3 00 00 00 00 00 00 00 00 00 00 00 00 00 00 P 01 R … 00 11 0000 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 01 S1S1

47. S C1C1 C2C2 C3C3 0123 0 ggp 12 P p 3 Build a 4 x 256 rectangle: t = 2 C0C0 g gg R 0 p r r S3S3 S2S2 0 0 1 23 00 g g pr 0 01 223P P P 33 33 33 33 33 212 33 33 33 33 1101 33 33 33 33 00 33 33 32 RP 333 2 3 3 2 3 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 C0C0 C1C1 C2C2 C3C3 S1S1

48. Building k x N Rectangles k-digit, base N (1/k) counter: Tile Complexity: N k

49. 2-temperature model 3 3 3 1 t = 4

50. 2-temperature model t = 4 6

51. 2-temperature model Kolmogorov Complexity (Rothemund, Winfree STOC 2000) Beats Standard Model (our paper)

52. Assembly of N x N Squares

53. k N - k k

54. Assembly of N x N Squares k N - k X Y k Complexity: ( Adleman, Cheng, Goel, Huang STOC 2001 )

55. N x N Squares --- Flexible Glue Model a b c d e f a 1 0 2 0 0 1 b 0 0 1 0 1 0 c 0 0 3 0 1 1 d 2 2 2 2 0 1 e 0 0 0 1 2 1 f 1 1 2 2 1 1 a b c d e f a 1 - - - - - b - 0 - - - - c - - 3 - - - d - - - 2 - - e - - - - 2 - f - - - - - 1 Standard Glue FunctionFlexible Glue Function Kolmogorov lower bounds: Standard Flexible (Rothemund, Winfree STOC 2000)

56. N x N Square --- Flexible Glue Model log N N – log N seed row

57. N x N Square --- Flexible Glue Model log N N – log N seed row Complexity:

58. N x N Square --- Flexible Glue Model goal: - seed binary counter to a given value - 2 log N 010000001111111111

59. ... 3334444445555 345012345012345 5 N x N Square --- Flexible Glue Model

60. ... 3334444445555 345012345012345 0 0 1 1 0 1 1 0 0 1 1 1 0 | | | | | | | | | | | | | 5 5 N x N Square --- Flexible Glue Model key idea:

61. 453 555 21 b4b4 5 5 w5w5 p5p5 G(b 4, p 5 ) = 1 G(b 4, w 5 ) = 0 N x N Square --- Flexible Glue Model

62. p0 p1 p2 p3 p4 p5 b0 0 1 1 0 1 1 b1 1 1 0 1 0 1 b2 0 1 0 1 1 1 b3 0 0 1 0 1 0 b4 0 0 0 0 0 1 b5 1 1 1 1 1 0 given B = 011011 110101 010111 … encode B into glue function B = 011011 110101 010111 … N x N Square --- Flexible Glue Model 4 b4b4 5 p5p5

63. Complexity: 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 build block N x N Square --- Flexible Glue Model

64. 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0

65. 2 x log N block N – log N log N

66. N – log N log N

67. N – log N X Y log N Complexity:

68. Our Tile Complexity Results Multiple temperature model: k x N rectangles: beats standard model: Flexible Glue: N x N squares: beats standard model: ( Adleman, Cheng, Goel, Huang STOC 2001 ) (our paper)

69. Molecular-scale pattern for a RAM memory with demultiplexed addressing ( Winfree, 2003)

70. Outline Importance of DNA Self-Assembly Tile Self-Assembly (Generalized Models) –Tile Complexity –Shape Verification –Error Resistance DNA Word Design

71. Shape Verification Unique Shape Problem Input: T, a tile system S, a shape Question: Does T uniquely assemble S. Standard:P (Adleman, Cheng, Goel, Huang, Kempe, Flexible Glue:P Espanes, Rothemund, STOC 2002) Unique Shape:Co-NPC (our paper) Multiple Temperature:NP-hard (our paper) Multiple Tile:Co-NPC (our paper)

72. x1x1 ** x2x2 x3x3 *TTTT ok c2c2 c1c1 c2c2 c2c2 c3c3 * * * * c1c1 0 1 1 SAT x1x1 ** x2x2 x3x3 *TTFF ok c2c2 c2c2 c1c1 c2c2 c2c2 c3c3 * * * * c1c1 0 0 1 Satisfied Not Satisfied (LaBean and Lagoudakis, 1999) Unique-Shape Model

73. ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * * * * ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * * * * Satisfied Not Satisfied ** Multiple Temperature Model

74. ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * 0 1 c1c1 ok c2c2 c2c2 1 T T T T * * * SAT * * * ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * 0 1 c1c1 ok c2c2 c2c2 0 T T c2c2 F F * * * NO * * * Satisfied Not Satisfied ** Multiple Temperature Model

75. Satisfied Not Satisfied Multiple Temperature Model ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * 0 1 c1c1 ok c2c2 c2c2 1 T T T T * * * SAT * * * * ** x1x1 x2x2 x3x3 c1c1 c2c2 c3c3 * * 0 1 c1c1 ok c2c2 c2c2 0 T T c2c2 F F * * * NO * * * *

76. Satisfied Not Satisfied Multiple Temperature Model * x1x1 x2x2 x3x3 * * * * * * x1x1 x2x2 x3x3 * * * * *

77. Unique Shape Problem Results StandardP Flexible GlueP Multiple TemperatureNP-hard Unique ShapeCo-NPC Multiple TileNP-hard (Adleman, Cheng, Goel, Huang, Kempe, Espanes, Rothemund, STOC 2002) (our paper)

78. Outline Importance of DNA Self-Assembly Tile Self-Assembly (Generalized Models) –Tile Complexity –Shape Verification –Error Resistance DNA Word Design

79. Further Research Error Resistance: Insufficient Bindings t = 2

80. Further Research Error Resistance: Insufficient Bindings t = 2

81. Further Research Error Resistance: Insufficient Bindings t = 2

82. Further Research Error Resistance: Insufficient Bindings t = 2

83. Further Research Error Resistance: Insufficient Bindings t = 2

84. Further Research Error Resistance: Insufficient Bindings t = 2

85. Further Research Error Resistance: Insufficient Bindings t = 2

86. Further Research Error Resistance: Insufficient Bindings a b temperature Standard Fluctuating

87. Outline Importance of DNA Self-Assembly Tile Self-Assembly (Generalized Models) DNA Word Design

88. 512346789 3 4 ACCT TGGA GCTA CGAT 5 DNA Word Design

89. 512346789 green: red: yellow: blue: purple: white: black: teal: ACCT GAAA GCTA CGTA CTCG CATG ACGA TTTA -Must be sufficiently different -Must have similar thermodynamic properties -Must be short

90. Hamming Constraint (k) ACCTGAGAGAGCTC GCGCAGCTGGCTCA TTAGCAGACTGACA GCTTCGTAGCATAG ATAGCTGCATCGAT TGCTAGCGTCAAGC AGCATTATAGATAC GCCCGTAGACTCGA TCGAGTAGATCGAT CGACGTAGGCTTTG CTGATGATTAGGCG TTCAGCTGCGGCTA TCGATGCGTAGCTA GAGTGCTGCTAGCT AGCTAGTCACTCGA TCGACTAGCTTCGA TTAGCCGCGTAGCT GACTAGTCGATCAG TCGCGCTTATATAT ATCGTAGTCTAGTC TACGATCGCTAGTC n strings X= GCTTCGTAGCATAG | | | Y= TTAGCCGCGTAGCT length L = 14 HAMM(X,Y) = 11 > k

91. Free Energy Constraint ACCTGAGAGAGCTC GCGCAGCTGGCTCA TTAGCAGACTGACA GCTTCGTAGCATAG ATAGCTGCATCGAT TGCTAGCGTCAAGC AGCATTATAGATAC GCCCGTAGACTCGA TCGAGTAGATCGAT CGACGTAGGCTTTG CTGATGATTAGGCG TTCAGCTGCGGCTA TCGATGCGTAGCTA GAGTGCTGCTAGCT AGCTAGTCACTCGA TCGACTAGCTTCGA TTAGCCGCGTAGCT GACTAGTCGATCAG TCGCGCTTATATAT ATCGTAGTCTAGTC TACGATCGCTAGTC n strings length L = 14 A C G T A 2 1 5 3 C 7 2 6 9 G 1 1 3 1 T 8 7 4 2 Pairwise free energies =

92. Free Energy Constraint ACCTGAGAGAGCTC GCGCAGCTGGCTCA TTAGCAGACTGACA GCTTCGTAGCATAG ATAGCTGCATCGAT TGCTAGCGTCAAGC AGCATTATAGATAC GCCCGTAGACTCGA TCGAGTAGATCGAT CGACGTAGGCTTTG CTGATGATTAGGCG TTCAGCTGCGGCTA TCGATGCGTAGCTA GAGTGCTGCTAGCT AGCTAGTCACTCGA TCGACTAGCTTCGA TTAGCCGCGTAGCT GACTAGTCGATCAG TCGCGCTTATATAT ATCGTAGTCTAGTC TACGATCGCTAGTC n strings length L = 14 A C G T A 2 1 5 3 C 7 2 6 9 G 1 1 3 1 T 8 7 4 2 Pairwise free energies = X= AGCATTATAGATAC FE(X) = 5+1+7+...

93. Free Energy Constraint ACCTGAGAGAGCTC GCGCAGCTGGCTCA TTAGCAGACTGACA GCTTCGTAGCATAG ATAGCTGCATCGAT TGCTAGCGTCAAGC AGCATTATAGATAC GCCCGTAGACTCGA TCGAGTAGATCGAT CGACGTAGGCTTTG CTGATGATTAGGCG TTCAGCTGCGGCTA TCGATGCGTAGCTA GAGTGCTGCTAGCT AGCTAGTCACTCGA TCGACTAGCTTCGA TTAGCCGCGTAGCT GACTAGTCGATCAG TCGCGCTTATATAT ATCGTAGTCTAGTC TACGATCGCTAGTC n strings length L = 14 A C G T A 2 1 5 3 C 7 2 6 9 G 1 1 3 1 T 8 7 4 2 Pairwise free energies = X= AGCATTATAGATAC FE(X) = 5+1+7+... For all strings X and Y: |FE(X) – FE(Y)| < C

94. DNA Word Design Word Design Problem Input: integers n and k Output: n strings of length L such that for all strings X and Y: 1) HAMM(X,Y) > k 2) |FE(X) – FE(Y)| < C Minimize L

95. DNA Word Design L > log n L > k L > ½(k + log n) Simple Lower Bound: Hamming Constraint: Set L = 5*(k + log n) Generate n strings of length L uniformly at random. -satisfies hamming constraint with high probability.