1 soda january 23, 2011 temperature 1 self-assembly: deterministic assembly in 3d and probabilistic...
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![Page 1: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/1.jpg)
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SODA January 23, 2011
Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D
Matthew Cook University of Zurich and ETH ZurichYunhui Fu Clemson UniversityRobert Schweller University of Texas Pan American
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Outline
• Background Information• Model• Results
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A C
G C
T G C G
Molecular Building Blocks
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Molecular Building Blocks
A T A G CT A T C G
T G A T C G G AA C T A G C C T
A C T A G C C TA C T A G C C T
C T A G C C G TG A T C G G C A
G C T T G A C CC G A A C T G G
A G A
T C G
A C
T C T
A G C
T G
T A C
C G
C A
TA T
G G
C G
T A
T G A
A T A
G C
A C T
T A T
C G
A C T
A G C
C T
A C T
A G C
C T
A T A G CT A T C G
A T A G CT A T C G
G T A C AC A T G T
A T A
G C
T A T
C G
A T A
G C
T A T
C G
A T A
G C
T A T
C G
A T A
G C
T A T
C G
C G G T C
T T C C A
G A C
A G
T T A
G T
[Reif’s Group, Duke University]
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DNA Scaffolding
[Sung Ha Park, Constantin Pistol, Sang Jung Ahn, John H. Reif, Alvin R. Lebeck, Chris Dwyer, and Thomas H. LaBean, 2006]
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Paul Rothemund, Nick Papadakis, Erik Winfree, PLoS Biology 2: e424 (2004)
340nm
Simulation of Cellular Automata
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Example of 3D Self-Assembly[Shaw, University of Southern California]
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3D DNA Cube
[Seeman, New York University]
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3D DNA Truncated Octahedron
[Seeman, New York University]
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Outline
• Background Information• Model• Results
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Tile Assembly Model(Rothemund, Winfree, Adleman)
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
Tile Set:
Glue Function:
Temperature:
S
Seed Tile:
x dc
baS
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How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
S
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S a
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a
c
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a
c
d
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a b
c
d
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a b
c
d
x
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a b
c
d
x x
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a b
c
d
x x
x
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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S a b
c
d
x x
x x
How a tile system self assembles
x dc
baST = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
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How efficiently can you build an n x n square?
n
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How efficiently can you build an n x n square?
x
Tile Complexity:2n
n
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How efficiently can you build an n x n square?
0 0 00
log n
-Use log n tile types to seedcounter:
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How efficiently can you build an n x n square?
0 0 00
log n
-Use 8 additional tile types capable of binary counting:
-Use log n tile types capable ofBinary counting:
![Page 25: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/25.jpg)
How efficiently can you build an n x n square?
0 0 00
log n
000
00
0 00
1 0 10
1 1 00
1 1 10
0 0 0
0 1
1 0
1
1
11
1
000 1
-Use 8 additional tile types capable of binary counting:
-Use log n tile types capable ofBinary counting:
![Page 26: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/26.jpg)
How efficiently can you build an n x n square?
0 0 00
000
000
00
0 00
1 0 10
1 1 00
1 1 10
0 0 01
0 0 11
0 1 01
0 1 11
1 0 01
1 0 11
1 1 11
1 1 01
1
1
11
1
-Use 8 additional tile types capable of binary counting:
-Use log n tile types capable ofBinary counting:
log n
![Page 27: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/27.jpg)
How efficiently can you build an n x n square?
0 0 00
000
000
00
0 00
1 0 10
1 1 00
1 1 10
0 0 01
0 0 11
0 1 01
0 1 11
1 0 01
1 0 11
1 1 11
1 1 01
1
1
11
1
0
0
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
1
1
0
1
1
1
0
0
0
0
1
1
0
0
1
0
1
0
1
1
1
0
1
0
0
1
1
1
0
1
1
0
1
1
1
1
1
1
1
-Use 8 additional tile types capable of binary counting:
-Use log n tile types capable ofBinary counting:
![Page 28: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/28.jpg)
How efficiently can you build an n x n square?
0 0 00
000
000
00
0 00
1 0 10
1 1 00
1 1 10
0 0 01
0 0 11
0 1 01
0 1 11
1 0 01
1 0 11
1 1 11
1 1 01
1
1
11
1
0
0
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
1
1
0
1
1
1
0
0
0
0
1
1
0
0
1
0
1
0
1
1
1
0
1
0
0
1
1
1
0
1
1
0
1
1
1
1
1
1
1
n – log n
log n
x
y
Tile Complexity:O(log n)
(Rothemund, Winfree 2000)
![Page 29: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/29.jpg)
How efficiently can you build an n x n square?
0 0 00
000
000
00
0 00
1 0 10
1 1 00
1 1 10
0 0 01
0 0 11
0 1 01
0 1 11
1 0 01
1 0 11
1 1 11
1 1 01
1
1
11
1
0
0
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
1
1
0
1
1
1
0
0
0
0
1
1
0
0
1
0
1
0
1
1
1
0
1
0
0
1
1
1
0
1
1
0
1
1
1
1
1
1
1
n – log n
log n
x
y
Tile Complexity:O(log n)
With optimalcounter:Tile Complexity:O(log n / loglog n)
Meets lower bound:W(log n / loglog n)
(Rothemund, Winfree 2000)
(Adleman, Cheng, Goel, Huang 2001)
(Rothemund, Winfree 2000)
![Page 30: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/30.jpg)
30Barish, Shulman, Rothemund, Winfree, 2009
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Why is Temperature 1 Theory Important?
• Temperature 2 self-assembly is powerful
• Efficient assembly of squares and more general shapes
• Universal Computation
• But….• Precise laboratory settings required
• High error rates
xy
![Page 32: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/32.jpg)
Why is Temperature 1 Theory Important?
• Temperature 2 self-assembly is powerful
• Efficient assembly of squares and more general shapes
• Universal Computation
• But….• Precise laboratory settings required
• High error rates
xy
![Page 33: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/33.jpg)
Why is Temperature 1 Theory Important?
• Temperature 2 self-assembly is powerful
• Efficient assembly of squares and more general shapes
• Universal Computation
• But….• Precise laboratory settings required
• High error rates
Error locked in place
![Page 34: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/34.jpg)
Why is Temperature 1 Theory Important?
• Temperature 2 self-assembly is powerful
• Efficient assembly of squares and more general shapes
• Universal Computation
• But….• Precise laboratory settings required
• High error rates Error locked in place
Question:• Is temperature 1 substantially less
powerful than temperature 2?• Is temperature 1 powerful enough
to warrant consideration considering it’s potential experimental advantages?
![Page 35: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/35.jpg)
Build an n x n square at Temperature 1
sa1
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Build an n x n square at Temperature 1
s A1 A2 A3 A4 A5a1 a2 a3 a4 a5
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Build an n x n square at Temperature 1
s A1 A2 A3 A4 A5a1 a2 a3 a4 a5
B1
B2
B3
B4
B5
b1
b1
b1
b1
b1
b1 b1 b1 b1 b1 b1
![Page 38: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/38.jpg)
Build an n x n square at Temperature 1
s A1 A2 A3 A4 A5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
![Page 39: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/39.jpg)
Build an n x n square at Temperature 1
s A1 A2 A3 A4 A5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
• Distinct tile types: 2n-1
• Probably optimal, but no substantial lower boundproof has been given.
![Page 40: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/40.jpg)
Build an n x n square at Temperature 1
s A1 A2 A3 A4 A5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
B1
B2
B3
B4
B5
• Distinct tile types: 2n-1
• Probably optimal, but no substantial lower boundproof has been given.
Two directions to consider
• Can we do better if consider 3D assembly? (3D deterministic assembly)
• Can we do better if we permit a small chance of error? (2D probabilistic assembly)
![Page 41: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/41.jpg)
Our Temperature 1 Results
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Outline
• Background Information• Model• Results
– Temperature 1 in 3D– Temperature 1 in 2D, probabilistic
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Simulating Temp 2 Systems at Temp 1
0 0 00
000
000
00
0
0
0
1
1
11
1
*
*
*
*
c
0
1
x
0
100*
c
1
c
c
0
1x x
10
0
x x
1
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Simulating Temp 2 Systems at Temp 1
0 0 00
000
000
00
0
0
0
1
1
11
1
*
*
*
*
c
0
1
x
0
100*
c
1
c
c
0
1x x
10
0
x x
1
0 c
1
c
0
![Page 45: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/45.jpg)
Simulating Temp 2 Systems at Temp 1
0 0 00
000
000
00
0
0
0
1
1
11
1
*
*
*
*
c
0
1
x
0
00*
c
1
c
0
1x x
10
0
x x
1
0
0c
0
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Simulating Temp 2 Systems at Temp 1
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Simulating Temp 2 Systems at Temp 1
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Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 49: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/49.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting of a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 50: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/50.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 51: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/51.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 52: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/52.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 53: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/53.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 54: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/54.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 55: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/55.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 56: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/56.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 57: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/57.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 58: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/58.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 59: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/59.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 60: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/60.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 61: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/61.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 62: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/62.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 63: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/63.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 64: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/64.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 65: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/65.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 66: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/66.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 67: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/67.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 68: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/68.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 69: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/69.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 70: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/70.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 71: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/71.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)
![Page 72: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/72.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
2 inputs
2 outputs
Key idea: • Map each single temperature 2 tile to a collection
of tiles constituting a “macro” tile• Use 3D geometry to encode north outputs.
(X,Y)A
B“0” “1”
![Page 73: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/73.jpg)
Simulating Temp 2 Systems at Temp 1
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(X,Y)A
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YX
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Simulating Temp 2 Systems at Temp 1
Y
X
A
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W
VYX
V
W
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Simulating Temperature 2 Systems at Temperature 1
Y
Geometry decoding tiles:Y
X
A
B
W
VYX
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Simulating Temp 2 Systems at Temp 1
Y
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Simulating Temp 2 Systems at Temp 1
Y
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Simulating Temp 2 Systems at Temp 1
Y
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Simulating Temp 2 Systems at Temp 1
Y
X
A
B
W
VYX
(X,Y)
![Page 80: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/80.jpg)
Simulating Temp 2 Systems at Temp 1
Y
X
A
B
W
VYX
A
B“0” “1”
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Outline
• Background Information• Model• Results
– Temperature 1 in 3D– Temperature 1 in 2D, probabilistic
![Page 84: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/84.jpg)
Simulating Temperature 2 Systems at Temperature 1:2D with high probability
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Summary• 3D temperature 1 and 2D probabilistic
temperature 1 offer much of the power of temperature 2.
• Temperature 1 self-assembly may have important experimental motivation.
• The use of steric hindrance and steric protection seems inline with nature:– Steric hindrance is a common mechansim
in nature.– The physical shape of proteins in biology is
closely related to function.
![Page 86: 1 SODA January 23, 2011 Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D Matthew CookUniversity of Zurich and](https://reader030.vdocuments.us/reader030/viewer/2022032516/56649c755503460f9492900a/html5/thumbnails/86.jpg)
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Future Work– Lower bound for nxn squares for
temperature 1, 2D, deterministic.– Multiple nucleation.– Can the nxn 3D result be improve to O(log
n / loglog n)?– Combine ideas from this work with other
techniques for robustness and error correction.
– Improve sturdiness or connectivity of constructions.