Self-Assembly with Geometric Tiles

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

)logloglog(

nnO

Geometric Tiles

Normal Tiles*

)log( nO

)logloglog(

nn

)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

)logloglog(

nnO

)log( nO

)logloglog(

nn

)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

)logloglog(

nnO

Geometric Tiles

Normal Tiles*

)log( nO

)logloglog(

nn

)log( n

Upper bound Lower bound

Planar Geometric Tiles ?

[*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]

n x n Results

Tile Complexity

)logloglog(

nnO

Geometric Tiles

Normal Tiles*

)log( nO

)logloglog(

nn

)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

)logloglog(

nnO

Geometric Tiles

Normal Tiles*

)log( nO

)logloglog(

nn

)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: