Self-Assembly Model with Time Dependent Glue Strength

1. Self-Assembly Model with Time Dependent Glue Strength Sudheer Sahu Peng Yin John Reif Department of Computer Science Duke University

2. Roadmap • Background • Time-dependent glue model • Implementation • Catalysis and self-replication • Tile-complexity

3. Background • Self-assembly: • Fundamental process in nature • Recent uses in constructions and computations • Tile self-assembly • 1960s (Wang tiling etc.) • Recent times (Winfree, Adleman etc.)

4. Winfree’s Model GlueΣ TileΣ4 • Strength function : g: Σ x Σ → R • g(σ,σ’) > 0 if σ = σ’ • = 0 otherwise • Temperature  • A tile can be added to the aggregate only if the glue-strength of new bonds formed is more than . Tile-system (T,S,g,)

5. Toy example Assembly of Square Temperature = 2 Given Tile Set

6. Our Model • Time Dependent Glue Model: • Glue strength depends on interaction time of tiles • Results: • Catalysis • Self-replication • Reduction in tile-complexity

7. Time Dependent Glue Model • Glue strength increases monotonically before becoming constant • Glue strength function • Time for maximum strength • Minimum interaction time

8. Strand Displacement as Random Walk

9. Implementation A s1 s2 s3 B

10. Implementation A s1 s2 s3 B

11. Implementation A s1 s2 s3 B

12. Implementation A s2 s3 B

13. Implementation A s2 s3 B

14. Implementation A s3 B

15. Implementation A s3 B

16. Implementation A B

17. How Glue Strength Varies with Time

18. Catalysis

19. Self-Replication • A-B acts as a catalyst for formation of C-D which in turn acts as a catalyst for the formation of A-B • Conditions:

20. Self-Replication A-B • Two states: • Dormant state • Replicating state • Exponential growth • Low probability of going from dormant to replicating state C-D as catalyst for A-B A-B as catalyst for C-D A,B, C,D C-D dormant replicating

21. Tile Complexity • Tile Complexity: • Minimum number of distinct tile types required to construct a shape uniquely.

22. Generalized Models • Multi-temperature Model • Thin rectangle [Aggarwal04] • Flexible Glue Model • Square [Aggarwal04]

23. Tile Complexity Results k N N N

24. Rectangle • Construct a k x mk rectangle using O(k+m) type of tiles. • Base m counter of k-digits

26. Construction of thin rectangles • Thin Rectangle: k x n for k < • Construct a j x n rectangle using O(j+n1/j ) type of tiles, where j > k. • The glue of bottom k rows become strong after mit, and the glue of top j-k rows (volatile rows) do not.

35. Decreasing it further

36. Shapes with holes nxn square with a hole of k x k in center • Lower Bound in standard model: • Upper Bound in our model: N N

37. Lower Bound • Proof by contradiction: • Assume fewer than k2/(n-k) tile-types required • Divide into regions s.t. seed tile is in longer rectangle • Number of different possible rows=2k • Two or more rows that are identical rows

38. Upper Bound • Grow four different rectangles

39. Discussion and Future Work • Kinetic analysis of catalysis and self-replication • Theoretical analysis is hard • the rate-constant changes assuming rate proportional to exp(-bond strength) • Computer program • Experiments or simulation • Tile-complexities for more shapes

40. THANKS!!

41. Concept of Minimum Interaction Time