Self-Assembly with Geometric Tiles ICALP 2012 Bin Fu University of Texas – Pan American Matt Patitz University of Arkansas Robert Schweller (Speaker) University of Texas – Pan American Robert Sheline University of Texas – Pan American
Self-Assembly with Geometric Tiles ICALP 2012 Bin FuUniversity of Texas – Pan American Matt PatitzUniversity of Arkansas Robert Schweller (Speaker)University.
Dec 11, 2015
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Self-Assembly with Geometric TilesICALP 2012
Bin Fu University of Texas – Pan AmericanMatt Patitz University of ArkansasRobert Schweller (Speaker) University of Texas – Pan AmericanRobert Sheline University of Texas – Pan American
Outline
• Basic Tile Assembly Model• Geometric Tile Assembly Model
– Basic Model– Planar Model– More efficient n x n squares
• Future Directions
3
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:
x ed
cba
4
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
d
e
x ed
cba
Tile Assembly Model(Rothemund, Winfree, Adleman)
5
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
Tile Assembly Model(Rothemund, Winfree, Adleman)
6
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b c
Tile Assembly Model(Rothemund, Winfree, Adleman)
7
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b c
Tile Assembly Model(Rothemund, Winfree, Adleman)
8
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b c
Tile Assembly Model(Rothemund, Winfree, Adleman)
9
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b ca
Tile Assembly Model(Rothemund, Winfree, Adleman)
10
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b ca
Tile Assembly Model(Rothemund, Winfree, Adleman)
11
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b ca
Tile Assembly Model(Rothemund, Winfree, Adleman)
12
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2d
e
x ed
cba
b ca
Tile Assembly Model(Rothemund, Winfree, Adleman)
13
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
x ed
cba
a b c
d
e
Tile Assembly Model(Rothemund, Winfree, Adleman)
14
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
x ed
cba
x
a b c
d
e
Tile Assembly Model(Rothemund, Winfree, Adleman)
15
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
a b c
d
e
x
x ed
cba
Tile Assembly Model(Rothemund, Winfree, Adleman)
16
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
x ed
cba
a b c
d
e
x x
Tile Assembly Model(Rothemund, Winfree, Adleman)
17
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
x ed
cba
a b c
d
e
x x
x
Tile Assembly Model(Rothemund, Winfree, Adleman)
18
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
x ed
cba
a b c
d
e
x x
x x
Tile Assembly Model(Rothemund, Winfree, Adleman)
Geometric Tile Model
Geometric Tiles
Geometry Region
Geometric Tiles
Geometry Region
Geometric Tiles
Compatible Geometries
Geometric Tiles
Geometric Tiles
Incompatible Geometries
Geometric Tiles
Incompatible Geometries
n x n Results
Tile Complexity
)loglog
log(
n
nO
Geometric Tiles
Normal Tiles*
)log( nO
)loglog
log(
n
n
)log( n
Upper bound Lower bound
Planar Geometric Tiles
[*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]
n x n Squares, root(log n) tiles
log n0 1 0 1 1
Assembly of n x n Squares
n
log n
0 1 1 0 0
1 1 1 1 11 1 1 1 0
0 1 0 1 1
Assembly of n x n Squares
log n0 1 0 1 1
2
log n
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
Assembly of n x n Squares
-Build thicker 2 x log n seed row
)log()log(1
nOnkO k
3
3
2
3
1
3
0
3
3
2
2
2
1
2
0
2
3
1
2
1
1
1
0
1
3
0
2
0
1
002
log n
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
Assembly of n x n Squares
-Build thicker 2 x log n seed row
)log()log(1
nOnkO k
-But… can’t encode general binary strings:
0
-All the same
3
3
2
3
1
3
0
3
3
2
2
2
1
2
0
2
3
1
2
1
1
1
0
1
3
0
2
0
1
002
log n
Assembly of n x n Squares
0
B3 B2 B1 B0
A3 A2 A1 A0
Key Idea:Geometry Decoding Tiles
3
3
2
3
1
3
0
3
3
2
2
2
1
2
0
2
3
1
2
1
1
1
0
1
3
0
2
0
1
002
log n
Assembly of n x n Squares
0
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
B0
A0A1
B1
A2
B2
A3
B3
A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3
3
3
2
3
1
3
0
3
3
2
2
2
1
2
0
2
3
1
2
1
1
1
0
1
3
0
2
0
1
002
log n
Assembly of n x n Squares
0
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
B0
A0A1
B1
A2
B2
A3
B3
A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3
Assembly of n x n Squares
1
2
0
2
0
A2
B3
A3
3
3
2
3
1
3
0
3
3
2
2
2
1
2
0
2
3
1
2
1
1
1
0
1
3
0
2
0
1
002
log n
Assembly of n x n Squares
0
0 0 0 01 1 1 1
B0
A0A1
B1
A2
B2
A3
B3
A0B1B2A3B0A1A2
1
2
0
2
0
A2
B3
A3
3
3
2
3
1
3
0
3
3
2
2
2
1
2
0
2
3
1
2
1
1
1
0
1
3
0
2
0
1
002
log n
Assembly of n x n Squares
0
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
B0
A0A1
B1
A2
B2
A3
B3
A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3
3
3
2
3
1
3
0
3
3
2
2
2
1
2
0
2
3
1
2
1
1
1
0
1
3
0
2
0
1
002
log n
Assembly of n x n Squares
0
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
B0
A0A1
B1
A2
B2
A3
B3
A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3
• build 2 x log n block:• Decode geometry into log n bit string
)log( n
)loglog
log(
n
nO
)log( nO
)loglog
log(
n
n
)log( n
Upper bound Lower bound
n x n Results
Tile Complexity
Geometric Tiles
Normal Tiles*
[*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]
Planar Geometric Tiles
Planar Geometric Tile Assembly
Attachment requires a collision free path within the plane
Planar Geometric Tile Assembly
Attachment requires a collision free path within the plane
Attachment not permitted in the planar model
Planar Geometric Tile Assembly
Planar Geometric Tile Assembly
Planar Geometric Tile Assembly
Attachment not permitted in the planar model
n x n Results
Tile Complexity
)loglog
log(
n
nO
Geometric Tiles
Normal Tiles*
)log( nO
)loglog
log(
n
n
)log( n
Upper bound Lower bound
Planar Geometric Tiles ?
[*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]
n x n Results
Tile Complexity
)loglog
log(
n
nO
Geometric Tiles
Normal Tiles*
)log( nO
)loglog
log(
n
n
)log( n
Upper bound Lower bound
Planar Geometric Tiles O( loglog n )
[*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]
?
1 0 1 0 0 1 1 0
log n
Planar Geometric Tile Assembly
1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 loglog n
• Build log n columns with loglog n tile typesPlanar Geometric Tile Assembly
1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 loglog n
• Build log n columns with loglog n tile typesPlanar Geometric Tile Assembly
1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 loglog n
• Build log n columns with loglog n tile types• Columns must assemble in proper order
Planar Geometric Tile Assembly
1 0 1 0 0 1 1 0
log n
1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 loglog n
• Build log n columns with loglog n tile types• Columns must assemble in proper order• Somehow cap each column with specified ‘0’ or ‘1’ tile type.
Planar Geometric Tile Assembly
• Build log n columns with loglog n tile types• Columns must assemble in proper order• Somehow cap each column with specified ‘0’ or ‘1’ tile type.
10
0
0 1
1
• Build log n columns with loglog n tile types• Columns must assemble in proper order• Somehow cap each column with specified ‘0’ or ‘1’ tile type.
10
0
0 1
1
0
1
0
0
1
1
0
1
0 1
1
1
• Build log n columns with loglog n tile types• Columns must assemble in proper order• Somehow cap each column with specified ‘0’ or ‘1’ tile type.
10
0
0 1
1
0
1
0
0
1
1
0
1
0 1
1
1
1
0
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
1
0
0
0
1
1
Planar Geometric Tile Assembly
1
0
0
0
1
0
1
0
0
0
1
1
1 0 1 0 0 1 1 0
log n
1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 loglog n
• Build log n columns with loglog n tile types• Columns must assemble in proper order• Somehow cap each column with specified ‘0’ or ‘1’ tile type.
1 0 1 0 0 1 1 0
log n
1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 loglog n
• Build log n columns with loglog n tile types• Columns must assemble in proper order
• Somehow cap each column with specified ‘0’ or ‘1’ tile type.
1
0
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
0
1
0
Planar Geometric Tile Assembly
1
0
0
0
1
0
0
1
0
Planar Geometric Tile Assembly
1 0 1 0 0 1 1 0
log n
1
1
1
1
1
0101010
0
1
0
0
0
0
1
0
1
0 1
0 loglog n
• Build log n columns with loglog n tile types• Columns must assemble in proper order
• Somehow cap each column with specified ‘0’ or ‘1’ tile type.
• O( loglog n ) tile types
n – log n
n – log n
log n
X
Y
)log(log n
Complexity:
n x n Results
Tile Complexity
)loglog
log(
n
nO
Geometric Tiles
Normal Tiles*
)log( nO
)loglog
log(
n
n
)log( n
Upper bound Lower bound
Planar Geometric Tiles O( loglog n ) ?
[*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]
Outline
• Basic Tile Assembly Model– Rectangles– n x n squares
• Geometric Tile Assembly Model– More efficient n x n squares
• Planar Geometric Tile Assembly Model– Even MORE efficient n x n squares
(A strange game.. planarity restriction helps you…)• Future Directions and Other Results
Other Results
• Simulation of temperature-2 systems with temperature-1 geometric tile systems.
• Simulation of many glue systems with single glue geometric tile systems.
• Compact Geometry Design Problem– Algorithms, lower bounds
Future Directions
• Lower bound for the planar model?– Is O(1) tile complexity possible in the planar model?– If not, what about log*(n)?
• What can be done with just 1 tile type?– Stay tuned for:
• One Tile to Rule Them All: Simulating Any Turing Machine, Tile Assembly System, or Tiling System with One Rotatable Puzzle Piece by: Erik Demaine, Martin Demaine, Sandor Fekete, Matthew Patitz, Robert Schweller, Andrew Winslow, Damien Woods.
• What about no rotation, but relative translation placement:– Check out “One Tile...” -EXTENDED VERSION!
• SPOILER ALERT: There is totally 1 “universal” tile that can do anything that can be done.
PeopleBin Fu
Matt Patitz
Robbie Schweller
Bobby Sheline
79Barish, Shulman, Rothemund, Winfree, 2009
DNA Origami Tiles
[Masayuki Endo, Tsutomu Sugita, Yousuke Katsuda, Kumi Hidaka, and Hiroshi Sugiyama, 2010]
More DNA Origami Shapes
[Paul Rothemund, Nature 2006]
Alphabet of Shapes, Built with DNA Tiles
[Bryan Wei, Mingjie Dai, Peng Yin, Nature 2012]
83
n x n square’s with Geometric Tiles
Tile Complexity:
n - k
kk
n - k
)( /1 knk
x
Assembly of n x n Squares
n - k
k
)( /1 knkO
Complexity:
Assembly of n x n Squares
n – log n
log n)(log)(
2
log
/1
/1
nOnkO
n
nk
k
k
Complexity:
Assembly of n x n Squares
n – log n
log n)(log)(
2
log
/1
/1
nOnkO
n
nk
k
k
Complexity:
seed row
log n
0 1 0 0 0 0 01 1 1 1 1 1 1 1 1
Assembly of n x n Squares
-Build thicker 2 x log n seed row
n – log n
log n
n – log n
n – log n
log n
X
Y
)log( N
Complexity:
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