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Strict Self-Assembly of Discrete Sierpinski Triangles

Strict Self-Assembly of Discrete Sierpinski Triangles. Scott M. Summers Iowa State University. DNA tile, oversimplified:. Four single DNA strands bound by Watson-Crick pairing (A-T, C-G). DNA Tile Self-Assembly: Ned Seeman, starting in 1980s. DNA Tile Self-Assembly:

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Strict Self-Assembly of Discrete Sierpinski Triangles

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  1. Strict Self-Assembly of Discrete Sierpinski Triangles Scott M. Summers Iowa State University

  2. DNA tile, oversimplified: Four single DNA strandsbound by Watson-Crickpairing (A-T, C-G). DNA Tile Self-Assembly: Ned Seeman, starting in 1980s

  3. DNA Tile Self-Assembly: Ned Seeman, starting in 1980s DNA tile, oversimplified:

  4. DNA Tile Self-Assembly: Ned Seeman, starting in 1980s DNA tile, oversimplified: “Sticky ends” bind with their Watson-Crick complements,so that a regular array self-assembles.

  5. DNA Tile Self-Assembly: Ned Seeman, starting in 1980s DNA tile, oversimplified: “Sticky ends” bind with their Watson-Crick complements,so that a regular array self-assembles.

  6. DNA Tile Self-Assembly: Ned Seeman, starting in 1980s DNA tile, oversimplified: Choice of sticky endsallows one to programthe pattern of the array. “Sticky ends” bind with their Watson-Crick complements,so that a regular array self-assembles.

  7. Theoretical Tile Self-Assembly: Erik Winfree, 1998

  8. Tile = unit square Theoretical Tile Self-Assembly: Erik Winfree, 1998

  9. Tile = unit square • Each side has a gluelabel and strength (0, 1, or 2 notches) Theoretical Tile Self-Assembly: Erik Winfree, 1998 Y X Z

  10. Tile = unit square • Each side has a gluelabel and strength (0, 1, or 2 notches) • If tiles abut with matching glue label and strength, then they bind with this glue’s strength Theoretical Tile Self-Assembly: Erik Winfree, 1998 Y X Z

  11. Tile = unit square • Each side has a gluelabel and strength (0, 1, or 2 notches) • If tiles abut with matching glue label and strength, then they bind with this glue’s strength • Tiles may have labels Theoretical Tile Self-Assembly: Erik Winfree, 1998 R Y X Z

  12. Tile = unit square • Each side has a gluelabel and strength (0, 1, or 2 notches) • If tiles abut with matching glue label and strength, then they bind with this glue’s strength • Tiles may have labels • Tiles cannot be rotated Theoretical Tile Self-Assembly: Erik Winfree, 1998 Z R Y X

  13. Tile = unit square • Each side has a gluelabel and strength (0, 1, or 2 notches) • If tiles abut with matching glue label and strength, then they bind with this glue’s strength • Tiles may have labels • Tiles cannot be rotated • Finitely many tile types Theoretical Tile Self-Assembly: Erik Winfree, 1998 R Y X Z

  14. Tile = unit square • Each side has a gluelabel and strength (0, 1, or 2 notches) • If tiles abut with matching glue label and strength, then they bind with this glue’s strength • Tiles may have labels • Tiles cannot be rotated • Finitely many tile types • Infinitely many of each type available Theoretical Tile Self-Assembly: Erik Winfree, 1998 R Y X Z

  15. Tile = unit square • Each side has a gluelabel and strength (0, 1, or 2 notches) • If tiles abut with matching glue label and strength, then they bind with this glue’s strength • Tiles may have labels • Tiles cannot be rotated • Finitely many tile types • Infinitely many of each type available • Assembly starts from a seed tile Theoretical Tile Self-Assembly: Erik Winfree, 1998 R Y X Z

  16. Tile = unit square • Each side has a gluelabel and strength (0, 1, or 2 notches) • If tiles abut with matching glue label and strength, then they bind with this glue’s strength • Tiles may have labels • Tiles cannot be rotated • Finitely many tile types • Infinitely many of each type available • Assembly starts from a seed tile • Self-assembly proceeds in a random fashion with tiles attaching one at a time Theoretical Tile Self-Assembly: Erik Winfree, 1998 R Y X Z

  17. Tile = unit square • Each side has a gluelabel and strength (0, 1, or 2 notches) • If tiles abut with matching glue label and strength, then they bind with this glue’s strength • Tiles may have labels • Tiles cannot be rotated • Finitely many tile types • Infinitely many of each type available • Assembly starts from a seed tile • Self-assembly proceeds in a random fashion with tiles attaching one at a time • A tile can attach to the existing assembly if it binds with total strength at least the “temperature” Theoretical Tile Self-Assembly: Erik Winfree, 1998 R Y X Z

  18. Y R c R n n 0 0 1 0 n 0 1 1 c 1 c c n n S R L L 0 X L 1 0 0 1 Tile Assembly Example

  19. 1 1 0 0 n n n n 1 0 Y Y Y 1 1 1 1 0 0 Y R c c c n n c c c c c 0 0 1 R X X X 0 0 0 S S R R R R R R R L L L L L Tile Assembly Example R L L L Temperature = 2

  20. 1 1 0 0 n n n n 1 0 Y Y Y 1 1 1 1 0 0 Y R c c c n n c c c c c 0 0 1 R X X X 0 0 0 S S R c Y R R 1 1 n c R X 0 L L R 0 R R L L L L L Tile Assembly Example R L L L Temperature = 2 R R Cooperation is the key to computing with the Tile Assembly Model.

  21. R 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n R 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 R Y Y Y Y Y Y Y Y 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Y R R c c c c c c c c n n n n n n n n n c c c c c c c c c c c c c c c c c c c c c c c c c c 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 R R R X X X X X X X X X 0 0 0 0 0 0 0 0 0 S S R R L L L L L L L L L L L L Tile Assembly Example R L L L R Temperature = 2 R R R R R R R R L L L L L

  22. 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Y Y Y Y Y Y Y Y 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Y R c c c c c c c c n n n n n n n n n c c c c c c c c c c c c c c c c c c c c c c c c c c 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 R X X X X X X X X X 0 0 0 0 0 0 0 0 0 S S Tile Assembly Example R R R R R R R R L L L R R Temperature = 2 R R R R R R R R L L L L L L L L L L L L L L L L L

  23. 1 1 1 0 0 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 Another Tile Assembly Example

  24. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 The “discrete Sierpinski triangle” 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Another Tile Assembly Example Temperature = 2 1 1 1 1 1 1 1 1

  25. From: Algorithmic Self-Assembly of DNA Sierpinski Triangles Rothemund PWK, Papadakis N, Winfree E PLoS Biology Vol. 2, No. 12, e424 doi:10.1371/journal.pbio.0020424 Experimental Self-Assembly: Rothemund, Papadakis and Winfree, 2004

  26. Study the self-assembly of discrete fractal structures in the Tile Assembly Model. Typical test bed for new research on fractals: Sierpinski triangles Objective

  27. Self-Assembly of Sierpinski Triangles

  28. Self-Assembly of Sierpinski Triangles • We have already seen theoretical and molecular self-assemblies of Sierpinski triangles. • But these are really just self-assemblies of entire two-dimensional surfaces onto which a picture of the Sierpinski triangle is “painted.” • But I want to study the more difficult problem of the self-assembly of shapes and nothing else, i.e., strict self-assembly.

  29. Some Formal Definitions

  30. Some Formal Definitions • Let X be a set of grid points. • The set Xweakly self-assembles if there exists a finite set of tile types T that places “black” tiles on—and only on—every point that belongs to X. • The set Xstrictly self-assembles if there exists a finite set of tile types T that places tiles on—and only on—every point that belongs to X.

  31. Study the strict self-assembly of discrete Sierpinski triangles in the Tile Assembly Model. Today’s Objective

  32. THEOREM (Summers, Lathrop and Lutz, 2007). The discrete Sierpinski triangle does NOT strictly self-assemble in the Tile Assembly Model. Why? Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle

  33. Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle

  34. Assume (for the sake of contradiction) that S strictly self-assembles in the tile set denoted as T Denote the discrete Sierpinski triangle as S Now look at the points rk = (2k + 1,2k) for all natural numbers k = 0, 1, 2, … Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle

  35. Assume (for the sake of contradiction) that S strictly self-assembles in the tile set denoted as T Now look at the points rk = (2k + 1,2k) for all natural numbers k = 0, 1, 2, … Since T is finite and S is infinite, there must be two numbers i and j such that T places the same tile type at ri and rj Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle

  36. Assume (for the sake of contradiction) that S strictly self-assembles in the tile set denoted as T Now look at the points rk = (2k + 1,2k) for all natural numbers k = 0, 1, 2, … Since T is finite and S is infinite, there must be two numbers i and j such that T places the same tile type at ri and rj But then some structure other thanS could just as easily strictly self-assemble in T—a contradiction! Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle

  37. Assume (for the sake of contradiction) that S strictly self-assembles in the tile set denoted as T Now look at the points rk = (2k + 1,2k) for all natural numbers k = 0, 1, 2, … Since T is finite and S is infinite, there must be two numbers i and j such that T places the same tile type at ri and rj But then some structure other thanS could just as easily strictly self-assemble in T—a contradiction! Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle

  38. Perhaps we could “approximately” strictly self-assemble the discrete Sierpinski triangle Now What?

  39. The Fibered Sierpinski Triangle

  40. The First Stage The Fibered Sierpinski Triangle

  41. The Fibered Sierpinski Triangle

  42. The Second Stage The Fibered Sierpinski Triangle

  43. The Fibered Sierpinski Triangle

  44. The Third Stage The Fibered Sierpinski Triangle

  45. The Fibered Sierpinski Triangle

  46. The Fourth Stage The Fibered Sierpinski Triangle

  47. Both fractals even share the same discrete fractal dimension (i.e., log23 ≈ 1.585) Similarity Between Fibered and Standard Sierpinski Triangle

  48. THEOREM (Summers, Lathrop and Lutz, 2007). The fibered Sierpinski triangle strictly self-assembles in the Tile Assembly Model. In fact, our tile set contains only 51 unique tile types. Strict Self-Assembly of the Fibered Sierpinski Triangle

  49. The fibered Sierpinski triangle is made up of a bunch of squares and rectangles. The Key Observation

  50. Standard fixed-width counter Modified fixed-width counter 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 1 Strict Self-Assembly of the Fibered Sierpinski Triangle (sketch)

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