1. Ashish Goel, ashishg@stanford.edu 1 The Source of Errors: Thermodynamics Rate of correct growth ¼ exp(-G A ) Probability of incorrect growth ¼ exp(-G A + G B ) Constraint: 2 G B > G A (system goes forward) ) Error probability ¸ exp(-G A /2) ) Rate has quadratic dependence on error probability ) Time to reliably assemble an n £ n square ¼ n 5 G A = Activation energy G B = Bond energy GAGA GBGB GAGA 2G B + Correct Growth Incorrect Growth

2. Ashish Goel, ashishg@stanford.edu 2 Error-Reducing Designs  Error correction via redundancy: do not change the model  Tile systems are designed to have error correction mechanisms  The Electrical Engineering approach -- error correcting codes But can not use existing coding/decoding techniques  Proofreading tiles [Winfree, Bekbolatov,’03]  Snake tiles [Chen, Goel ‘04]  Biochemistry techniques  Strand Invasion mechanism [Chen, Cheng, Goel, Huang, Moisset de espanes, ’04]

3. Ashish Goel, ashishg@stanford.edu 3 Example: Sierpinski Tile System 0 0 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0

4. Ashish Goel, ashishg@stanford.edu 4 Example: Sierpinski Tile System 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0

5. Ashish Goel, ashishg@stanford.edu 5 Example: Sierpinski Tile System 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0

6. Ashish Goel, ashishg@stanford.edu 6 Example: Sierpinski Tile System 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0

7. Ashish Goel, ashishg@stanford.edu 7 Growth Error 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 0

8. Ashish Goel, ashishg@stanford.edu 8 Growth Error 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 mismatch 0 1 1 0

9. Ashish Goel, ashishg@stanford.edu 9 Growth Error 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 1 0

10. Ashish Goel, ashishg@stanford.edu 10 Growth Error 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0

11. Ashish Goel, ashishg@stanford.edu 11 Growth Error 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0

12. Ashish Goel, ashishg@stanford.edu 12 Proofreading Tiles Each tile in the original system corresponds to four tiles in the new system The internal glues are unique to this block G1G1 G4G4 G3G3 G2G2 G 1b X4X4 X3X3 G 2a X2X2 G 3b G 2b G 1a G 4a X1X1 G 4b G 3a [Winfree, Bekbolatov, ’03]

13. Ashish Goel, ashishg@stanford.edu 13 How does this help? 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0

14. Ashish Goel, ashishg@stanford.edu 14 How does this help? 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 mismatch

15. Ashish Goel, ashishg@stanford.edu 15 How does this help? 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0

16. Ashish Goel, ashishg@stanford.edu 16 How does this help? 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0 No tile can attach at this location

17. Ashish Goel, ashishg@stanford.edu 17 How does this help? 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0

18. Ashish Goel, ashishg@stanford.edu 18 How does this help? 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0

19. Ashish Goel, ashishg@stanford.edu 19 How does this help? 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 0

20. Ashish Goel, ashishg@stanford.edu 20 Nucleation Error

21. Ashish Goel, ashishg@stanford.edu 21 Nucleation Error First tile attaches with a weak binding strength

22. Ashish Goel, ashishg@stanford.edu 22 Nucleation Error First tile attaches with a weak binding strength Second tile attaches and secures the first tile

23. Ashish Goel, ashishg@stanford.edu 23 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.

24. Ashish Goel, ashishg@stanford.edu 24 Snake Tiles Each tile in the original system corresponds to four tiles in the new system The internal glues are unique to this block G1G1 G4G4 G3G3 G2G2 G 1b X1X1 X2X2 G 2a X3X3 G 3b G 2b G 1a G 4a G 4b G 3a

25. Ashish Goel, ashishg@stanford.edu 25 How does this help? First tile attaches with a weak binding strength

26. Ashish Goel, ashishg@stanford.edu 26 How does this help? First tile attaches with a weak binding strength Second tile attaches and secures the first tile

27. Ashish Goel, ashishg@stanford.edu 27 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

28. Ashish Goel, ashishg@stanford.edu 28 Preliminary Experimental Results (Obtained by Chen, Goel, Schulman, Winfree)

29. Ashish Goel, ashishg@stanford.edu 29

30. Ashish Goel, ashishg@stanford.edu 30

31. Ashish Goel, ashishg@stanford.edu 31

32. Ashish Goel, ashishg@stanford.edu 32

33. Ashish Goel, ashishg@stanford.edu 33 Four by Four Snake Tiles

34. Ashish Goel, ashishg@stanford.edu 34 Four by Four Snake Tiles

35. Ashish Goel, ashishg@stanford.edu 35 Four by Four Snake Tiles

36. Ashish Goel, ashishg@stanford.edu 36 Four by Four Snake Tiles

37. Ashish Goel, ashishg@stanford.edu 37 Four by Four Snake Tiles

38. Ashish Goel, ashishg@stanford.edu 38 Four by Four Snake Tiles

39. Ashish Goel, ashishg@stanford.edu 39 Four by Four Snake Tiles

40. Ashish Goel, ashishg@stanford.edu 40 Four by Four Snake Tiles

41. Ashish Goel, ashishg@stanford.edu 41 Four by Four Snake Tiles

42. Ashish Goel, ashishg@stanford.edu 42 Four by Four Snake Tiles

43. Ashish Goel, ashishg@stanford.edu 43 Four by Four Snake Tiles

44. Ashish Goel, ashishg@stanford.edu 44 Four by Four Snake Tiles

45. Ashish Goel, ashishg@stanford.edu 45 Analysis  Snake tile design extends to 2k £ 2k blocks.  Prevents tile propagation even after k+1 nucleation/growth errors  The error probability changes from p to roughly p k  We can assemble an N £ N square in time O(N polylog N) and it remains stable for time  (N) (with high probability).  Resolution loss of O(log N)  Assuming tiles held by strength 3 do not fall off  Matches the time for ideal, irreversible assembly  Compare to N 3 for basic proof-reading and N 5 with no error-correction in the thermodynamic model [Chen, Goel; DNA ‘04]  Extensions, variations by Reif’s group, Winfree’s group, our group, and others  Recent result: Simple combinatorial criteria; Can avoid resolution loss by using third dimension [Chen, Goel, Luhrs; SODA ‘08]

46. Ashish Goel, ashishg@stanford.edu 46 Interesting Open Problems - I  General theorems for analyzing reversible self- assembly?  Example: Imagine you are given an “L”, with each arm being length N From each “convex corner”, a tile can fall off at rate r At each “concave” corner, a tile can attach at rate f > r What is the first time that the (N,N) location is occupied? We believe that the right answer is O(N), can prove O(N log N)  General theorems which relate the combinatorial structure of an error-correction scheme to the error probability?  We have combinatorial criteria for error correction, but they are not all encompassing

47. Ashish Goel, ashishg@stanford.edu 47 Interesting Open Problems – II Robust, efficient counting  We replace a tile by a k £ k block, where k ! 1 as N ! 1  Or, by a k £ 1 block if we use the third dimension  Codes (eg. Reed-Solomon) can do much better  Can we use codes to design more efficient counters?  Specifically: Do there exist one-to-one functions (code-words) W: {1,..N} ! {1..N 2 } such that 1. Given a row of 2 log N tiles encoding W(k), there is some simple “tiling subroutine” to assemble W(k+1) on top 2. Even if there are p log N errors in the tiling process for each row, this process stops after “counting” from 1 to N  Motivation: Correctly assembling large shapes up-to molecular precision will be a new engineering paradigm – so an exciting opportunity for theoreticians

48. Ashish Goel, ashishg@stanford.edu 48 (1,1)(1,0)(0,1)(0,0)(1,1) (0,1)(1,1)(0,1)(1,1)(0,1) (1,0) (0,0) (1,1) (1,0) (0,0) (0,1)(1,1)(0,0) (1,1) (1,0) (1,1) Another Mode of Error -- Damage 1W 1S (1,1) (1,0)(0,1) (1,1) (0,0) (1,0) S S 1W 1S (0,0) (0,1) (1,1) (1,0)

49. Ashish Goel, ashishg@stanford.edu 49 What went wrong?  When tiles attach from unexpected directions the “correct” tile is not guaranteed.  Potential fix: Design systems more carefully so that the system can reassemble from small pieces all over.  Previous work: [Winfree ’06] Rectilinear Systems that will grow back correctly as long as the seed remains in place by forcing growth only from the seed direction.  Single point of failure: Lose the seed and the structure cannot regrow  Akin to a lizard regenerating a limb  Our goal: Tile systems that heal from small fragments anywhere  Akin to two parts of a starfish growing into complete separate starfish  Almost a “reproductive property”

50. Ashish Goel, ashishg@stanford.edu 50 Two pieces of self-healing: Immutability and Progressiveness Immutability: Only correct tiles may attach. (As opposed to the Sierpinski example.) Progressiveness: Eventually, all tiles attach. (Provided one of a set of pieces containing enough information remains) Example: The Chinese remainder counter is almost self-healing from any row