1. 1 Thirteenth International Meeting on DNA Computers June 5, 2007 Staged Self-Assembly: Nanomanufacture of Arbitrary Shapes with O(1) Glues Eric DemaineMassachusetts Institute of Technology Martin DemaineMassachusetts Institute of Technology Sandor FeketeTechnische Universität Braunschweig Mashood IshaqueTufts University Eynat RafalinGoogle Robert SchwellerUniversity of Texas Pan American Diane SouvaineTufts University

2. 2 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

3. 3 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)

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 bc 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 bca 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 x ed cba abc d e 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 x 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 abc d e x x ed cba 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 x ed cba abc d e xx 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 x 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 xx Tile Assembly Model (Rothemund, Winfree, Adleman)

18. 18 BEAKER Start with initial Tileset Non-Staged Assembly -Assembly occurs within 1 single container - Assembly occurs within 1 single stage

19. 19 BEAKER After some time... Start with initial Tileset Various Producible Supertiles exist in solution Non-Staged Assembly -Assembly occurs within 1 single container - Assembly occurs within 1 single stage

20. 20 BEAKER After some time... After enough time... Start with initial Tileset Various Producible Supertiles exist in solution Only Terminally Produced assemblies remain Non-Staged Assembly -Assembly occurs within 1 single container - Assembly occurs within 1 single stage

21. 21 Staged Assembly

22. 22 Staged Assembly -Pour multiple bins into a single bin

23. 23 Staged Assembly -Pour multiple bins into a single bin -Split contents of any given bin among multiple new bins

24. 24 Staged Assembly -Pour multiple bins into a single bin -Split contents of any given bin among multiple new bins

25. 25 Staged Assembly

26. 26 Staged Assembly Assembly occurs in a sequence of stages, and assemblies can be separated into separate bins Bin Complexity: 4 Stage Complexity: 3 Mix pattern:

27. 27 Staged Assembly Assembly occurs in a sequence of stages, and assemblies can be separated into separate bins Bins = Space Complexity Stages = Time Complexity Bin Complexity: 4 Stage Complexity: 3

28. 28 Staged Assembly Assembly occurs in a sequence of stages, and assemblies can be separated into separate bins Bin Complexity: 4 Stage Complexity: 3 Our Goal: Given a target shape, design mixing algorithms that: –Use only O(1) tiles/glues to build target shape. –Are efficient in terms of: Bin complexity Stage complexity.

29. 29 Simple Example: 1 x n line

30. 30 Simple Example: 1 x n line

31. 31 Simple Example: 1 x n line

32. 32 Simple Example: 1 x n line stage i stage i+3

33. 33 Simple Example: 1 x n line stage i stage i+3 tiles / gluesO(1) = 3 BinsO(1) StagesO(log n) Staged Assembly 1 x n line

34. 34 Simple Example: 1 x n line stage i stage i+3 tiles / gluesO(1) = 3 BinsO(1) StagesO(log n) Staged Assembly 1 x n line tiles / glues (n) Bins1 Stages1 Non-Staged Model 1 x n line

35. 35 n x n Square

36. 36 n x n Square Base Case 1 x n line: Use line algorithm tiles / gluesO(1) BinsO(1) StagesO(log n) Staged Assembly n x n square

37. 37 n x n Square: unstable?

38. 38 n x n Square: unstable?

39. 39 n x n Square: unstable?

40. 40 n x n Square: Full Connectivity Full Connectivity Constraint: All adjacent tiles in assembled shape must share a full strength bond [Rothemund, Winfree STOC 2000]

41. 41 n x n Square: Full Connectivity Full Connectivity Constraint: All adjacent tiles in assembled shape must share a full strength bond

42. 42 n x n Square: Full Connectivity Shifting Problem Full Connectivity Constraint: All adjacent tiles in assembled shape must share a full strength bond

43. 43 n x n Square: Full Connectivity Shifting Problem Jigsaw Technique: Use Geometry to enforce proper binding. Full Connectivity Constraint: All adjacent tiles in assembled shape must share a full strength bond

44. 44 n x n Square: Full Connectivity Jigsaw Technique: Use Geometry to enforce proper binding. Full Connectivity Constraint: All adjacent tiles in assembled shape must share a full strength bond

45. 45 n x n Square: Full Connectivity Jigsaw Technique: Use Geometry to enforce proper binding. Full Connectivity Constraint: All adjacent tiles in assembled shape must share a full strength bond

46. 46 n x n Square: Full Connectivity tiles / gluesO(1) BinsO(1) StagesO(log n) Temperature1 Staged Assembly Fully Connected n x n square tiles / glues (log n / log log n) Bins1 Stages1 Temperature2 Non-Staged Model Fully Connected n x n square [adleman, cheng, goel, huang STOC 2001]

47. 47 Arbitrary Shapes Spanning Tree Method Jigsaw Method for non-hole Shapes Simulation Method

48. 48 Simulate Large Tilesets

49. 49 Simulate Large Tilesets 0000 0001 0010 0011 0100 0101 0110

50. 50 Simulate Large Tilesets 0000 0001 0010 0011 0100 0101 0110 0 1

51. 51 Simulate Large Tilesets 0001 0000 0001 0011 0001 0011 0011 0000 0001 0010 0011 0100 0101 0110

52. 52 Simulate Large Tilesets 001 0011 0000 0001 0010 0011 0100 0101 0110 1

53. 53 Simulate Large Tilesets 00 0011 0000 0001 0010 0011 0100 0101 0110 10

54. 54 Simulate Large Tilesets

55. 55 c Simulate Large Tilesets b a...

56. 56 Simulate Large Tilesets c b a 001 0011 1 0 0 1 0 0 10 0 001 0011 1 0 0 1 0 0 10 0 001 0011 1 0 0 1 0 0 10 0... tiles / gluesO(1) BinsO(|T|) StagesO(log log |T|) Simulate temp=1 tileset T tiles / gluesO(1) BinsO(n) StagesO(log log n) ScaleO(log n) Arbitrary n tile Shape

57. 57 Arbitrary Shape Assembly Spanning Tree Method Jigsaw Method for non-hole Shapes Simulation Method tiles / gluesO(1) BinsO(n) StagesO(n) Connectivity FULL Scale2 GeneralityHole Free Jigsaw Method tiles / gluesO(1) BinsO(log n) StagesO(diameter) Connectivity Partial Scale1 GeneralityALL Spanning Tree Method tiles / gluesO(1) BinsO(n) StagesO(log log n) Connectivity FULL ScaleO(log n) GeneralityALL Simulation Method

58. 58 tiles / gluesO(1) BinsO(1) StagesO(log n) Staged Assembly n x n square First Result: What if we have B bins? Near Optimal Tradeoff: Bins versus Stages (Crazy Mixing)

59. 59 tiles / gluesO(1) BinsO(1) StagesO(log n) Staged Assembly n x n square First Result: What if we have B bins? B^2 edges, Can encode B^2 Bits of information Per stage. Near Optimal Tradeoff: Bins versus Stages (Crazy Mixing)

60. 60 Near Optimal Tradeoff: Bins versus Stages (Crazy Mixing) tiles / gluesO(1) BinsB Stages ( log n / B^2) Lower Bound for almost all n tiles / gluesO(1) BinsB Stages ( log n / B^2 + log B) Upper Bound Assembly of n x n squares with B bins: Upper bound technique: -Encode B^2 bits describing target square at each stage -Combine with Simulation macro tiles.

61. 61 Staged Assembly permits various techniques for the assembly of arbitrary shapes with O(1) tiles/glues. For some shapes (squares) we achieve near optimal tradeoffs in bin versus stage complexity. Staged assembly may shed light on natural assembly systems –Cells of body perhaps serve as bins –Staged assembly emphasizes importance of geometric shape for bonding, perhaps similar to protein shape determining function. Conclusions

62. 62 Problems with model? Applications in DNA code design using synthetic DNA words? Incorporating produced structures as well as terminally produced structures Experiments, simulations Apply more intense mixing patterns to general shapes Tradeoffs between tile complexity and bin/stage complexity. Simulation of t=2 systems Future Work 001 0011 1

63. 63 Thanks for listening. Questions?