SLIDE 1
Limitations of Self-Assembly at Temperature 1 David Doty, Matt - - PowerPoint PPT Presentation
Limitations of Self-Assembly at Temperature 1 David Doty, Matt - - PowerPoint PPT Presentation
Limitations of Self-Assembly at Temperature 1 David Doty, Matt Patitz, Scott Summers Department of Computer Science Iowa State University, Ames, IA, USA DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998 DNA Tile Self-Assembly Winfree, Ph.D.
SLIDE 2
SLIDE 3
DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961 Refined in Paul Rothemund’s Ph.D. thesis, 2001
- Tile = unit square
SLIDE 4
DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961 Refined in Paul Rothemund’s Ph.D. thesis, 2001
X XY Z
Strength 0 Strength 1 Strength 2
- Tile = unit square
- Each side has glue
- f certain kind and
strength (0, 1, or 2).
SLIDE 5
DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961 Refined in Paul Rothemund’s Ph.D. thesis, 2001
- Tile = unit square
- Each side has glue
- f certain kind and
strength (0, 1, or 2).
- If tiles abut with
matching kinds of glue, then they bind with this glue’s strength.
X XY Z
Strength 0 Strength 1 Strength 2
SLIDE 6
DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961 Refined in Paul Rothemund’s Ph.D. thesis, 2001
- Tile = unit square
- Each side has glue
- f certain kind and
strength (0, 1, or 2).
- If tiles abut with
matching kinds of glue, then they bind with this glue’s strength.
- Tiles may have
labels.
Strength 0 Strength 1 Strength 2
X XY Z
R
SLIDE 7
DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961 Refined in Paul Rothemund’s Ph.D. thesis, 2001
- Tile = unit square
- Each side has glue
- f certain kind and
strength (0, 1, or 2).
- If tiles abut with
matching kinds of glue, then they bind with this glue’s strength.
- Tiles may have
labels.
- Tiles cannot be
rotated.
Strength 0 Strength 1 Strength 2
SLIDE 8
DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961 Refined in Paul Rothemund’s Ph.D. thesis, 2001
- Tile = unit square
- Each side has glue
- f certain kind and
strength (0, 1, or 2).
- If tiles abut with
matching kinds of glue, then they bind with this glue’s strength.
- Tiles may have
labels.
- Tiles cannot be
rotated.
Strength 0 Strength 1 Strength 2
X XY Z
R
- Finitely many tile types
SLIDE 9
DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961 Refined in Paul Rothemund’s Ph.D. thesis, 2001
- Tile = unit square
- Each side has glue
- f certain kind and
strength (0, 1, or 2).
- If tiles abut with
matching kinds of glue, then they bind with this glue’s strength.
- Tiles may have
labels.
- Tiles cannot be
rotated.
Strength 0 Strength 1 Strength 2
X XY Z
R
- Finitely many tile types
- Infinitely many of each
type available
SLIDE 10
DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961 Refined in Paul Rothemund’s Ph.D. thesis, 2001
- Tile = unit square
- Each side has glue
- f certain kind and
strength (0, 1, or 2).
- If tiles abut with
matching kinds of glue, then they bind with this glue’s strength.
- Tiles may have
labels.
- Tiles cannot be
rotated.
Strength 0 Strength 1 Strength 2
X XY Z
R
- Finitely many tile types
- Infinitely many of each
type available
- Assembly starts from a
seed tile (or seed assembly).
SLIDE 11
DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961 Refined in Paul Rothemund’s Ph.D. thesis, 2001
- Tile = unit square
- Each side has glue
- f certain kind and
strength (0, 1, or 2).
- If tiles abut with
matching kinds of glue, then they bind with this glue’s strength.
- Tiles may have
labels.
- Tiles cannot be
rotated.
Strength 0 Strength 1 Strength 2
X XY Z
R
- Finitely many tile types
- Infinitely many of each
type available
- Assembly starts from a
seed tile (or seed assembly).
- A tile can attach to the
existing assembly if it binds with total strength at least equal to the “temperature”.
SLIDE 12
Tile Assembly Example
S L
c R c R
L
c R 1 n c
1
L
c R 1 n c
1
S
1 c c 1 n c
1
1 1 n n
1
n n c R
L
Edge binding strengths: 1 2
Cooperation is key to computing with Tile Assembly Model.
SLIDE 13
Tile Assembly Example
S L
c R c R
L
c R 1 n c
1
L
c R 1 n c
1
S
1 c c 1 n c
1
1 1 n n
1
n n c R
L
Edge binding strengths: 1 2
Cooperation is key to computing with Tile Assembly Model.
Both of these have to be present...
SLIDE 14
Tile Assembly Example
S L
c R c R
L
c R 1 n c
1
L
c R 1 n c
1
S
1 c c 1 n c
1
1 1 n n
1
n n c R
L
Edge binding strengths: 1 2
Cooperation is key to computing with Tile Assembly Model.
Both of these have to be present... ...before this can attach
SLIDE 15
Tile Assembly Example
S L
c R c R
L
c R 1 n c
1
S
1 c c 1 n c
1
1 1 n n
1
n n c R
L
Edge binding strengths: 1 2
n n
L
c R 1 c c c R 1 n c
1 L
n n n n 1 n c
1
1 c c n n c R
L
1 1 n n
1
n n 1 n c
1
n n n n c R
L L
n n n n n n 1 c c n n n n n n 1 n c
1
n n n n 1 1 n n
1
n n 1 c c 1 n c
1
n n n n 1 1 n n
1
n n 1 n c
1
1 1 n n
1
n n 1 1 n n
1
n n
Cooperation is key to computing with Tile Assembly Model.
SLIDE 16
Tile Assembly Example
S L
c R c R
L
c R 1 n c
1
S
1 c c 1 n c
1
1 1 n n
1
n n c R
L
Edge binding strengths: 1 2
n n
L
c R 1 c c c R 1 n c
1 L
n n n n 1 n c
1
1 c c n n c R
L
1 1 n n
1
n n 1 n c
1
n n n n c R
L L
n n n n n n 1 c c n n n n n n 1 n c
1
n n n n 1 1 n n
1
n n 1 c c 1 n c
1
n n n n 1 1 n n
1
n n 1 n c
1
1 1 n n
1
n n 1 1 n n
1
n n
What if the tiles cannot cooperate?
SLIDE 17
- The binding graph of an assembly is the grid graph
induced by tiles binding with positive strength.
Binding Graph
SLIDE 18
- The binding graph of an assembly is the grid graph
induced by tiles binding with positive strength.
Binding Graph
SLIDE 19
- The binding graph of an assembly is the grid graph
induced by tiles binding with positive strength.
- “Paths” in this talk always refer to paths in the binding
graph of some assembly.
Binding Graph
SLIDE 20
Self-Assembly at Temperature 1 Tiling the Plane
If tiles abut with matching kinds of (positive strength) glue, then they bind.
Tile set:
SLIDE 21
Self-Assembly at Temperature 1 Tiling the Plane
If tiles abut with matching kinds of (positive strength) glue, then they bind.
Tile set:
SLIDE 22
Self-Assembly at Temperature 1 Tiling the Plane
If tiles abut with matching kinds of (positive strength) glue, then they bind.
Tile set:
SLIDE 23
Self-Assembly at Temperature 1 Tiling the Plane
If tiles abut with matching kinds of (positive strength) glue, then they bind.
Tile set:
SLIDE 24
Self-Assembly at Temperature 1 Tiling the Plane
If tiles abut with matching kinds of (positive strength) glue, then they bind.
Tile set:
SLIDE 25
Self-Assembly at Temperature 1 Tiling the Plane
If tiles abut with matching kinds of (positive strength) glue, then they bind.
Tile set:
SLIDE 26
Self-Assembly at Temperature 1 Tiling the Plane
SLIDE 27
Self-Assembly at Temperature 1 Building a Periodic Line
1
S
2 1
2
Tile set:
1 2
1
1
S
SLIDE 28
Self-Assembly at Temperature 1 Building a Periodic Line
1
S
2 1
2
Tile set:
1 2
1
1
S
1 2
1
SLIDE 29
Self-Assembly at Temperature 1 Building a Periodic Line
1
S
2 1
2
Tile set:
1 2
1
1
S
2 1
2
1 2
1
SLIDE 30
Self-Assembly at Temperature 1 Building a Periodic Line
1
S
2 1
2
Tile set:
1 2
1
1
S
2 1
2
1 2
1
1 2
1
SLIDE 31
Self-Assembly at Temperature 1 Building a Periodic Line
1
S
2 1
2
Tile set:
1 2
1
1
S
2 1
2
1 2
1
2 1
2
1 2
1
SLIDE 32
Self-Assembly at Temperature 1 Building a Periodic Line
1
S
2 1
2
Tile set:
1 2
1
1
S
2 1
2
1 2
1
2 1
2
1 2
1
1 2
1
SLIDE 33
Self-Assembly at Temperature 1 Building a Periodic Line
1
S
2 1
2
Tile set:
1 2
1
1 2
1
1
S
2 1
2
1 2
1
2 1
2
1 2
1
1 2
1
2 1
2
2 1
2
1 2
1
1 2
1
2 1
2
2 1
2
1 2
1
1 2
1
2 1
2
2 1
2
1 2
1
1 2
1
2 1
2
2 1
2
1 2
1
1 2
1
2
SLIDE 34
Self-Assembly at Temperature 1 Building a “Comb”
1
S
2 1
2
3 1 2
1
3 4
3
4 3
4
Tile set:
1
S
SLIDE 35
Self-Assembly at Temperature 1 Building a “Comb”
1
S
2 1
2
3 1 2
1
3 4
3
4 3
4
Tile set:
1
S
3 1 2
1
SLIDE 36
Self-Assembly at Temperature 1 Building a “Comb”
1
S
2 1
2
3 1 2
1
3 4
3
4 3
4
Tile set:
1
S
3 1 2
1
2 1
2
SLIDE 37
Self-Assembly at Temperature 1 Building a “Comb”
1
S
2 1
2
3 1 2
1
3 4
3
4 3
4
Tile set:
1
S
3 1 2
1
2 1
2
3 4
3
SLIDE 38
Self-Assembly at Temperature 1 Building a “Comb”
1
S
2 1
2
3 1 2
1
3 4
3
4 3
4
Tile set:
1
S
3 1 2
1
2 1
2
3 4
3
3 1 2
1
SLIDE 39
Self-Assembly at Temperature 1 Building a “Comb”
1
S
2 1
2
3 1 2
1
3 4
3
4 3
4
Tile set:
1
S
3 1 2
1
2 1
2
3 4
3
4 3
4
3 1 2
1
SLIDE 40
Self-Assembly at Temperature 1 Building a “Comb”
1
S
2 1
2
3 1 2
1
3 4
3
4 3
4
Tile set:
1
S
3 1 2
1
2 1
2
3 4
3
4 3
4
3 1 2
1
2 1
2
SLIDE 41
Self-Assembly at Temperature 1 Building a “Comb”
1
S
2 1
2
3 1 2
1
3 4
3
4 3
4
Tile set:
1
S
3 1 2
1
2 1
2
3 4
3
4 3
4
3 1 2
1
2 1
2
3 4
3
SLIDE 42
Self-Assembly at Temperature 1 Building a “Comb”
1
S
2 1
2
3 1 2
1
3 4
3
4 3
4
Tile set:
1
S
3 1 2
1
2 1
2
3 4
3
4 3
4
3 4
3
3 1 2
1
2 1
2
3 4
3
SLIDE 43
Self-Assembly at Temperature 1 Building a “Comb”
1
S
2 1
2
3 1 2
1
3 4
3
4 3
4
Tile set:
1
S
3 1 2
1
2 1
2
3 4
3
4 3
4
3 4
3
3 1 2
1
2 1
2
3 4
3
4 3
4
SLIDE 44
1
S
3 1 2
1
2 1
2
3 4
3
4 3
4
3 4
3
4 3
4
3 1 2
1
2 1
2
3 1 2
1
2 1
2
3 1 2
1
2 1
2
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 1 2
1
2 1
2
3 1 2
1
2 1
2
3 1 2
1
2 1
2
3 1 2
1
2 1
2
3 1 2
1
2 1
2
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
3 1 2
1
2 1
2
3 1 1 3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
3 4
3
4 3
4
3 4
3
4 3
4
3 4
3
4 4 4 4 4 4 4 4 4 4 4
Self-Assembly at Temperature 1 Building a “Comb”
SLIDE 45
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 46
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 47
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 48
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 49
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 50
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 51
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 52
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 53
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 54
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 55
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 56
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 57
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 58
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 59
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 60
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 61
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 62
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 63
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 64
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 65
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 66
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 67
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 68
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 69
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 70
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 71
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 72
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 73
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 74
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 75
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 76
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 77
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 78
Self-Assembly at Temperature 1 Building an “Eventual Comb”
SLIDE 79
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 80
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 81
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 82
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 83
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 84
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 85
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 86
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 87
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 88
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 89
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 90
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 91
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 92
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 93
Self-Assembly at Temperature 1 Building a “Plane-Filling” Grid
SLIDE 94
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 95
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 96
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 97
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 98
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 99
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 100
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 101
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 102
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 103
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 104
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 105
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 106
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 107
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 108
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 109
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 110
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 111
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 112
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 113
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 114
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 115
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 116
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 117
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 118
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 119
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 120
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 121
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 122
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 123
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 124
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 125
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 126
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 127
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 128
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 129
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 130
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 131
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 132
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 133
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 134
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 135
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 136
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 137
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 138
Self-Assembly at Temperature 1 Building A Complicated Structure
SLIDE 139
Our Result (Vaguely)
- The previous “complicated structure” intuitively
captures every set that can weakly self- assemble in a “nice” temperature 1 tile assembly system.
SLIDE 140
Our Result (Vaguely)
- The previous “complicated structure” intuitively
captures every set that can weakly self- assemble in a “nice” temperature 1 tile assembly system.
- Our proof relies on a notion of “pumpable
paths” in an assembly to define “nice”-ness.
SLIDE 141
Pumpability
Our result relies on a notion of pumpable paths in an assembly.
SLIDE 142
Pumpability
Our result relies on a notion of pumpable paths in an assembly. Partial assembly, with grey seed tile
SLIDE 143
Pumpability
“Sufficiently long” path Our result relies on a notion of pumpable paths in an assembly.
SLIDE 144
Pumpability
Our result relies on a notion of pumpable paths in an assembly. Look at just this partial path
SLIDE 145
Pumpability
Our result relies on a notion of pumpable paths in an assembly. Uniquely color each tile type
SLIDE 146
Pumpability
Our result relies on a notion of pumpable paths in an assembly. Note the repeating pattern, beginning with yellow tiles, which we call a pumpable segment
SLIDE 147
Pumpability
Our result relies on a notion of pumpable paths in an assembly. The pumpable segment can be infinitely repeated, or pumped, to create an infinite, periodic path
SLIDE 148
A “deterministic” (i.e., uniquely produces an assembly) temperature 1 tile assembly system T is c-pumpable if, given any two points p and q at least distance c apart in the terminal assembly of T, there is a path from p to q in the binding graph that contains a pumpable segment within the first c points on the path.
Pumpable Tile Assembly System
SLIDE 149
A “deterministic” (i.e., uniquely produces an assembly) temperature 1 tile assembly system T is c-pumpable if, given any two points p and q at least distance c apart in the terminal assembly of T, there is a path from p to q in the binding graph that contains a pumpable segment within the first c points on the path. In other words, every long path contains repetitions of a tile type (an obvious consequence of the pigeonhole principle) that can be pumped (repeated infinitely many times) to create a periodic path without colliding with the assembly up to that point.
Pumpable Tile Assembly System
SLIDE 150
Pumpability
Not all repeating patterns are pumpable!
SLIDE 151
Pumpability
Not all repeating patterns are pumpable!
SLIDE 152
Pumpability
Not all repeating patterns are pumpable!
SLIDE 153
Doubly Periodic Sets
The dark green tiles make up a doubly periodic set
A set of points is doubly periodic if every point in the set can be expressed as a nonnegative integer affine combination of two integer vectors:
SLIDE 154
Doubly Periodic Sets
An initial offset from the origin... base point b
The dark green tiles make up a doubly periodic set
A set of points is doubly periodic if every point in the set can be expressed as a nonnegative integer affine combination of two integer vectors:
SLIDE 155
Doubly Periodic Sets
An initial offset from the origin, plus a multiple of u... base point b
The dark green tiles make up a doubly periodic set
u
A set of points is doubly periodic if every point in the set can be expressed as a nonnegative integer affine combination of two integer vectors:
SLIDE 156
Doubly Periodic Sets
A set of points is doubly periodic if every point in the set can be expressed as a nonnegative integer affine combination of two integer vectors:
u v An initial offset from the origin, plus a multiple of u plus a multiple of v base point b
The dark green tiles make up a doubly periodic set
SLIDE 157
Doubly Periodic Sets
A set of points is doubly periodic if every point in the set can be expressed as a nonnegative integer affine combination of two integer vectors:
u An initial offset from the origin, plus a multiple of u plus a multiple of v The point b + 3u + 2v base point b
The dark green tiles make up a doubly periodic set
v
SLIDE 158
The Finite Closure of an Assembly
In our proof, we use the idea of a finite closure of a given (producible and non-terminal) assembly α.
SLIDE 159
The Finite Closure of an Assembly
In our proof, we use the idea of a finite closure of a given (producible and non-terminal) assembly α. Intuitively, if we extend α by only those “portions” that will eventually stop growing, the finite closure is the super- assembly that will be produced.
SLIDE 160
The Finite Closure of an Assembly
In our proof, we use the idea of a finite closure of a given (producible and non-terminal) assembly α. Intuitively, if we extend α by only those “portions” that will eventually stop growing, the finite closure is the super- assembly that will be produced. That is, any attempt to “leave” α through the finite closure and go infinitely far will eventually run into α again.
SLIDE 161
The Finite Closure of an Assembly
For example, consider the following (terminal) assembly.
SLIDE 162
The Finite Closure of an Assembly
For example, consider the following (terminal) assembly.
SLIDE 163
The Finite Closure of an Assembly
Consider the subassembly consisting of the dark red tiles...
SLIDE 164
The Finite Closure of an Assembly
Consider the subassembly consisting of the dark red tiles... The bottom row of the horizontal rectangle is incomplete, and all of the upward projections have yet to self-assemble.
SLIDE 165
The Finite Closure of an Assembly What is the finite closure this assembly?
SLIDE 166
The Finite Closure of an Assembly
It's the dark red tiles in this figure, because...
SLIDE 167
The Finite Closure of an Assembly
...a finite number of tiles can be added to each of these locations before they become terminal, but... It's the dark red tiles in this figure, because...
SLIDE 168
The Finite Closure of an Assembly
...a finite number of tiles can be added to each of these locations before they become terminal, but... ...an infinite number of tiles can be added to this location, so they are not on the finite closure. It's the dark red tiles in this figure, because...
SLIDE 169
The Finite Closure of an Assembly
The finite closure of an assembly is the "closure of the assembly under finite additions to a single frontier location." Formally, a point x is on the finite closure of an assembly α if every infinite path containing x in the terminal assembly intersects α.
Example x's in the finite closure
SLIDE 170
Main Result
- A subset X of the plane weakly self-assembles
in a finite tile assembly system if some of the tile types are colored “black,” and X consists of those points that eventually are tiled with a black tile.
SLIDE 171
Main Result
- A subset X of the plane weakly self-assembles
in a finite tile assembly system if some of the tile types are colored “black,” and X consists of those points that eventually are tiled with a black tile.
- Every set that weakly self-assembles in a
pumpable, deterministic, temperature 1 tile assembly system is a finite union of doubly periodic sets.
SLIDE 172
The Proof
- The finite closure of a periodic path or comb
defines a finite union of doubly periodic sets.
SLIDE 173
The Proof
- The finite closure of a periodic path or comb
defines a finite union of doubly periodic sets.
- There are only a finite number of periodic paths
and combs originating within a fixed radius around the origin.
SLIDE 174
The Proof
- The finite closure of a periodic path or comb
defines a finite union of doubly periodic sets.
- There are only a finite number of periodic paths
and combs originating within a fixed radius around the origin.
- We prove: If the tile system is c-pumpable, then
every black tile is on the finite closure of some periodic path, or comb, that originates within radius 2c of the origin.
SLIDE 175
The Proof (sketch)
seed seed
- 1. Pick a black tile
distance at least c from the origin.
SLIDE 176
The Proof (sketch)
seed seed
- 2. There is a pumpable
segment starting within radius c of the origin ...
- 1. Pick a black tile
distance at least c from the origin.
SLIDE 177
The Proof (sketch)
seed seed
- 2. There is a pumpable
segment starting within radius c of the origin ...
- 3. ...so this infinite periodic
path could happen instead.
- 1. Pick a black tile
distance at least c from the origin.
SLIDE 178
The Proof (sketch)
seed seed
- 3. ...so this infinite periodic
path could happen instead.
- 4. Maybe the black tile is on
the finite closure of the infinite periodic path. Done!
- 2. There is a pumpable
segment starting within radius c of the origin ...
- 1. Pick a black tile
distance at least c from the origin.
SLIDE 179
The Proof (sketch)
seed seed
- 4. Otherwise, another long
path that includes the first pumpable segment leads to the black tile.
- 2. There is a pumpable
segment starting within radius c of the origin ...
- 1. Pick a black tile
distance at least c from the origin.
SLIDE 180
The Proof (sketch)
seed seed
- 4. Otherwise, another long
path that includes the first pumpable segment leads to the black tile.
- 2. There is a pumpable
segment starting within radius c of the origin ...
- 1. Pick a black tile
distance at least c from the origin.
- 5. Another pumpable
segment...
SLIDE 181
The Proof (sketch)
seed seed
- 4. Otherwise, another long
path that includes the first pumpable segment leads to the black tile.
- 2. There is a pumpable
segment starting within radius c of the origin ...
- 1. Pick a black tile
distance at least c from the origin.
- 5. Another pumpable
segment... ...means more infinite periodic paths.
SLIDE 182
The Proof (sketch)
seed seed
- 4. Otherwise, another long
path that includes the first pumpable segment leads to the black tile.
- 2. There is a pumpable
segment starting within radius c of the origin ...
- 1. Pick a black tile
distance at least c from the origin.
- 5. Another pumpable
segment...
- 6. And we have a comb!
...means more infinite periodic paths.
SLIDE 183
The Proof (sketch)
The two vectors defining the comb form a basis for the region in which the black tile is contained, producing a regular array of parallelograms.
SLIDE 184
The Proof (sketch)
The two vectors defining the comb form a basis for the region in which the black tile is contained, producing a regular array of parallelograms. The “determinism” of the tile set enforces the same pattern in each parallelogram: a doubly periodic set.
SLIDE 185
The Proof (sketch)
There are a number of inaccuracies in the previous argument.
SLIDE 186
The Proof (sketch)
There are a number of inaccuracies in the previous argument.
But those would all result in this... A cycle intersecting two pumpable paths
SLIDE 187
The Proof (sketch)
There are a number of inaccuracies in the previous argument.
But those would all result in this... which we can prove results in this. A cycle intersecting two pumpable paths An infinite plane- covering grid
SLIDE 188
Conclusion
Informal thesis: if a computation can be performed with a tile assembly system, then the result of that computation can be represented as a set that weakly self-assembles in that system or a closely related system.
SLIDE 189
Conclusion
Informal thesis: If a computation can be performed with a tile assembly system, then the result of that computation can be represented as a set that weakly self-assembles in that system or a closely related system. Conclusion: Deterministic, pumpable, temperature 1 tile assembly systems are not capable of general-purpose computation.
SLIDE 190
Open Question
Conjecture: Every “deterministic” temperature 1 tile assembly system that produces an infinite assembly is pumpable.
SLIDE 191
Open Question
Conjecture: Every “deterministic” temperature 1 tile assembly system that produces an infinite assembly is pumpable. That is, no path can grow too far without growing
- ut of control (in a periodic manner) since the lack
- f cooperation inhibits controlled growth.
SLIDE 192
Open Question
Conjecture: Every “deterministic” temperature 1 tile assembly system that produces an infinite assembly is pumpable. That is, no path can grow too far without growing
- ut of control (in a periodic manner) since the lack
- f cooperation inhibits controlled growth.
We have no proof, so who are we to say this?
SLIDE 193
Non-pumpable Tile System?
Two tile types repeated on a path that get “blocked” eventually if they are pumped.
SLIDE 194
Non-pumpable Tile System?
Two tile types repeated on a path that get “blocked” eventually if they are pumped.
SLIDE 195
Non-pumpable Tile System?
Two tile types repeated on a path that get “blocked” eventually if they are pumped. But the blue tiles are NOT on an infinite path.
SLIDE 196