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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

CT 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

CA C

T T A T

C G

A C T A

G C C

TA 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

CT A

T C G

A T A G

CT A

T C G

A T A G

CT A

T C G

A T A G

CT 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

13

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

How efficiently can you build an n x n square?

0 0 00

log n

-Use log n tile types to seedcounter:

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:

How efficiently can you build an n x n square?

0 0 00

log n

00000

0 001 0 101 1 001 1 100 0 0

0 11 0

1

111

1

000 1

-Use 8 additional tile types capable of binary counting:

-Use log n tile types capable ofBinary counting:

How efficiently can you build an n x n square?

0 0 00000

00000

0 001 0 101 1 001 1 100 0 010 0 110 1 010 1 111 0 011 0 11

1 1 111 1 01

11

111

-Use 8 additional tile types capable of binary counting:

-Use log n tile types capable ofBinary counting:

log n

How efficiently can you build an n x n square?

0 0 00000

00000

0 001 0 101 1 001 1 100 0 010 0 110 1 010 1 111 0 011 0 11

1 1 111 1 01

11

111

0000

1000

0100

1100

0010

1010

0110

1110

0001

1001

0101

1101

0011

1011

0111

1111

-Use 8 additional tile types capable of binary counting:

-Use log n tile types capable ofBinary counting:

How efficiently can you build an n x n square?

0 0 00000

00000

0 001 0 101 1 001 1 100 0 010 0 110 1 010 1 111 0 011 0 11

1 1 111 1 01

11

111

0000

1000

0100

1100

0010

1010

0110

1110

0001

1001

0101

1101

0011

1011

0111

1111

n – log n

log n

x

y

Tile Complexity:O(log n)

(Rothemund, Winfree 2000)

How efficiently can you build an n x n square?

0 0 00000

00000

0 001 0 101 1 001 1 100 0 010 0 110 1 010 1 111 0 011 0 11

1 1 111 1 01

11

111

0000

1000

0100

1100

0010

1010

0110

1110

0001

1001

0101

1101

0011

1011

0111

1111

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)

30Barish, Shulman, Rothemund, Winfree, 2009

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

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

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

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?

Build an n x n square at Temperature 1

sa1

Build an n x n square at Temperature 1

s A1 A2 A3 A4 A5a1 a2 a3 a4 a5

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

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

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.

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)

Our Temperature 1 Results

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Outline

• Background Information• Model• Results

– Temperature 1 in 3D– Temperature 1 in 2D, probabilistic

Simulating Temp 2 Systems at Temp 1

0 0 00

000

000

00

0

0

0

1

1

11

1

***

*

c

0

1x

0

100*

c

1

c

c

0

1x x

10

0x x

1

Simulating Temp 2 Systems at Temp 1

0 0 00000

000

00

0

0

0

1

1

11

1

***

*

c

0

1x

0

100*

c

1

c

c

0

1x x

10

0x x

1

0 c

1

c

0

Simulating Temp 2 Systems at Temp 1

0 0 00

000

000

00

0

0

0

1

1

11

1

***

*

c

0

1x

0

00*

c

1

c

0

1x x

10

0x x

1

0

0c0

Simulating Temp 2 Systems at Temp 1

0 0 00

000

000

00

0

0

0

1

1

11

1

***

*

c

0

1x

0

00*

c

1

c

0

1x x

10

0x x

1

0

0c0

1 c

0

1x

Simulating Temp 2 Systems at Temp 1

0 0 00

000

000

00

0

0

0

1

1

11

1

***

*

c

0

1x

0

0*

c

1

c

0

1x x

10

0x x

1

0

001

x 1

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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)

Simulating Temp 2 Systems at Temp 1

YX

AB

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”

Simulating Temp 2 Systems at Temp 1

YX

AB

(X,Y)A

B“0” “1”

YX

Simulating Temp 2 Systems at Temp 1

YX

AB

WVY

X

V

W

Simulating Temperature 2 Systems at Temperature 1

Y

Geometry decoding tiles:Y

XA

B

WVY

X

Simulating Temp 2 Systems at Temp 1

Y

Simulating Temp 2 Systems at Temp 1

Y

Simulating Temp 2 Systems at Temp 1

Y

Simulating Temp 2 Systems at Temp 1

YX

AB

WVY

X

(X,Y)

Simulating Temp 2 Systems at Temp 1

YX

AB

WVY

X

A

B“0” “1”

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82

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Outline

• Background Information• Model• Results

– Temperature 1 in 3D– Temperature 1 in 2D, probabilistic

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.

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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.