Thirteenth International Meeting on DNA Computers June 5, 2007

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Thirteenth International Meeting on DNA Computers

June 5, 2007

Staged Self-Assembly: Nanomanufacture of Arbitrary Shapes with O(1) Glues

Eric Demaine Massachusetts Institute of TechnologyMartin Demaine Massachusetts Institute of TechnologySandor Fekete Technische Universität BraunschweigMashood Ishaque Tufts UniversityEynat Rafalin GoogleRobert Schweller University of Texas Pan AmericanDiane Souvaine Tufts University

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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:

x ed

cba

3

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

d

e

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


5

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

d

e

x ed

cba

b c


6

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

d

e

x ed

cba

b c


7

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

d

e

x ed

cba

b c


8

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

d

e

x ed

cba

b ca


9

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

d

e

x ed

cba

b ca


10

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

d

e

x ed

cba

b ca


11

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

d

e

x ed

cba

b ca


12

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

x ed

cba

a b c

d

e


13

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


14

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


15

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


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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


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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


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BEAKER

Start with initial Tileset

Non-Staged Assembly

-Assembly occurs within 1 single container

- Assembly occurs within 1 single stage

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BEAKERBEAKER

Aftersome time...

Start with initial Tileset Various Producible Supertilesexist in solution

Non-Staged Assembly



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BEAKERBEAKER BEAKER

Aftersome time...

After enough time...

Start with initial Tileset Various Producible Supertilesexist in solution

Only Terminally Producedassemblies remain

Non-Staged Assembly



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Staged Assembly

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Staged Assembly

-Pour multiple bins into a single bin

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Staged Assembly

-Pour multiple bins into a single bin-Split contents of any given bin among multiple new bins

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Staged Assembly

-Pour multiple bins into a single bin-Split contents of any given bin among multiple new bins

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Staged Assembly

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Staged Assembly• Assembly occurs in a sequence of stages, and

assemblies can be separated into separate bins

Bin Complexity: 4

Stage Complexity: 3

Mix pattern:

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Bins = Space ComplexityStages = Time Complexity

Bin Complexity: 4

Stage Complexity: 3

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Bin Complexity: 4

Stage Complexity: 3

• Our Goal:

Given a target shape, design mixing algorithms that: – Use only O(1) tiles/glues to build target

shape.– Are efficient in terms of:

• Bin complexity• Stage complexity.

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Simple Example: 1 x n line

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stage i

stage i+3

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stage i

stage i+3

tiles / glues O(1) = 3

Bins O(1)

Stages O(log n)

Staged Assembly1 x n line

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stage i

stage i+3

tiles / glues O(1) = 3

Bins O(1)

Stages O(log n)

Staged Assembly1 x n line

tiles / glues (n)

Bins 1

Stages 1

Non-Staged Model1 x n line

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n x n Square

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n x n Square

Base Case1 x n line:Use linealgorithm

tiles / glues O(1)

Bins O(1)

Stages O(log n)

Staged Assemblyn x n square

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n x n Square: unstable?

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n x n Square: Full Connectivity

Full Connectivity Constraint: All adjacent tiles inassembled shape mustshare a full strength bond

[Rothemund, Winfree STOC 2000]

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Shifting Problem


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Shifting Problem

Jigsaw Technique:Use Geometryto enforce properbinding.


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tiles / glues O(1)

Bins O(1)

Stages O(log n)

Temperature 1

Staged AssemblyFully Connected

n x n square

tiles / glues (log n / log log n)

Bins 1

Stages 1

Temperature 2

Non-Staged ModelFully Connected

n x n square

[adleman, cheng, goel, huang STOC 2001]

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Arbitrary Shapes• Spanning Tree Method• Jigsaw Method for non-hole Shapes• Simulation Method

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Simulate Large Tilesets

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0000

0001

0010

0011

0100

0101

0110

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0000

0001

0010

0011

0100

0101

0110

0

1

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0 0 0 1

0 0 0 0

0 0 01

0 0 1 1

0 0 01

0 01 1

0 01 1

0000

0001

0010

0011

0100

0101

0110

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0 01

0 01 1

0000

0001

0010

0011

0100

0101

0110

1

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0 0

0 01 1

0000

0001

0010

0011

0100

0101

0110

10

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0 01

0 01 1

1

00

1

00

1 0

0

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c


b

a

0 01

0 01 1

1

00

1

00

1 0

0

0 01

0 01 1

1

00

1

00

1 0

0

0 01

0 01 1

1

00

1

00

1 0

0

. . .

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c

b

a

0 01

0 01 1

1

00

1

00

1 0

0

0 01

0 01 1

1

00

1

00

1 0

0

0 01

0 01 1

1

00

1

00

1 0

0

. . .

tiles / glues O(1)

Bins O(|T|)

Stages O(log log |T|)

Simulate temp=1 tileset T

tiles / glues O(1)

Bins O(n)

Stages O(log log n)

Scale O(log n)

Arbitrary n tile Shape

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Arbitrary Shape Assembly

• Spanning Tree Method• Jigsaw Method for non-hole Shapes• Simulation Method

tiles / glues O(1)

Bins O(n)

Stages O(n)

Connectivity FULL

Scale 2

Generality Hole Free

Jigsaw Method

tiles / glues O(1)

Bins O(log n)

Stages O(diameter)

Connectivity Partial

Scale 1

Generality ALL

Spanning Tree Method

tiles / glues O(1)

Bins O(n)

Stages O(log log n)

Connectivity FULL

Scale O(log n)

Generality ALL

Simulation Method

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tiles / glues O(1)

Bins O(1)

Stages O(log n)


First Result:

What if we have B bins?

Near Optimal Tradeoff: Bins versus Stages(Crazy Mixing)

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tiles / glues O(1)

Bins O(1)

Stages O(log n)


First Result:

What if we have B bins?

B^2 edges, Can encode B^2Bits of informationPer stage.


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tiles / glues O(1)

Bins B

Stages ( log n / B^2)

Lower Bound for almost all n

tiles / glues O(1)

Bins B

Stages ( log n / B^2 + log B)

Upper Bound

Assembly of n x n squares with B bins:

Upper bound technique:

-Encode B^2 bits describing target square at each stage

-Combine with Simulation macro tiles.

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• Staged Assembly permits various techniques for the assembly of arbitrary shapes with O(1) tiles/glues.

• For some shapes (squares) we achieve near optimal tradeoffs in bin versus stage complexity.

• Staged assembly may shed light on natural assembly systems– Cells of body perhaps serve as bins

– Staged assembly emphasizes importance of geometric shape for bonding, perhaps similar to protein shape determining function.

Conclusions

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• Problems with model?• Applications in DNA code design using synthetic DNA words?

• Incorporating produced structures as well as terminally produced structures

• Experiments, simulations• Apply more intense mixing patterns to general shapes• Tradeoffs between tile complexity and bin/stage complexity.• Simulation of t=2 systems

Future Work

0 01

0 01 1

1

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Thanks for listening. Questions?

Thirteenth International Meeting on DNA Computers June 5, 2007

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