Thirteenth International Meeting on DNA Computers

1
Staged Self-Assembly Nanomanufacture of
Arbitrary Shapes with O(1) Glues
Thirteenth International Meeting on DNA
Computers June 5, 2007
Eric Demaine Massachusetts Institute of
Technology Martin Demaine Massachusetts Institute
of Technology Sandor Fekete Technische
Universität Braunschweig Mashood Ishaque Tufts
University Eynat Rafalin Google Robert
Schweller University of Texas Pan American Diane
Souvaine Tufts University
2
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
Glue Function
Tile Set
Temperature
3
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
e
d
4
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
e
d
5
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
e
d
b
c
6
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
e
d
b
c
7
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
e
d
b
c
8
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
e
d
b
c
a
9
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
e
d
b
c
a
10
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
e
d
b
c
a
11
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
e
d
b
c
a
12
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
e
d
a
b
c
13
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
e
x
d
a
b
c
14
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
15
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
16
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
17
Tile Assembly Model (Rothemund, Winfree, Adleman)
G(y) 2 G(g) 2 G(r) 2 G(b) 2 G(p)
1 G(w) 1 t 2
T
18
Non-Staged Assembly
BEAKER
Start with initial Tileset

• Assembly occurs within 1 single container
• - Assembly occurs within 1 single stage

19
Non-Staged Assembly
BEAKER
BEAKER
After some time...
Start with initial Tileset
Various Producible Supertiles exist in solution

• Assembly occurs within 1 single container
• - Assembly occurs within 1 single stage

20
Non-Staged Assembly
BEAKER
BEAKER
BEAKER
After some time...
After enough time...
Start with initial Tileset
Various Producible Supertiles exist in solution
Only Terminally Produced assemblies remain

• Assembly occurs within 1 single container
• - Assembly occurs within 1 single stage

21
Staged Assembly
22
Staged Assembly

• Pour multiple bins into a single bin

23
Staged Assembly

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

24
Staged Assembly

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

25
Staged Assembly
26
Staged Assembly

• Assembly occurs in a sequence of stages, and
  assemblies can be separated into separate bins

Bin Complexity 4
Mix pattern
Stage Complexity 3
27
Staged Assembly

• Assembly occurs in a sequence of stages, and
  assemblies can be separated into separate bins

Bin Complexity 4
Bins Space Complexity Stages Time Complexity
Stage Complexity 3
28
Staged Assembly

• Assembly occurs in a sequence of stages, and
  assemblies can be separated into separate bins

• 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.

Bin Complexity 4
Stage Complexity 3
29
Simple Example 1 x n line
30
Simple Example 1 x n line
31
Simple Example 1 x n line
32
Simple Example 1 x n line
stage i
stage i3
33
Simple Example 1 x n line
Staged Assembly 1 x n line
tiles / glues O(1) 3
Bins O(1)
Stages O(log n)
34
Simple Example 1 x n line
Staged Assembly 1 x n line
Non-Staged Model 1 x n line
tiles / glues O(1) 3
Bins O(1)
Stages O(log n)
tiles / glues W(n)
Bins 1
Stages 1
35
n x n Square
36
n x n Square
Staged Assembly n x n square
Base Case 1 x n line Use line algorithm
tiles / glues O(1)
Bins O(1)
Stages O(log n)
37
n x n Square unstable?
38
n x n Square unstable?
39
n x n Square unstable?
40
n x n Square Full Connectivity
Rothemund, Winfree STOC 2000
Full Connectivity Constraint All adjacent
tiles in assembled shape must share a
full strength bond
41
n x n Square Full Connectivity
Full Connectivity Constraint All adjacent
tiles in assembled shape must share a
full strength bond
42
n x n Square Full Connectivity
Full Connectivity Constraint All adjacent
tiles in assembled shape must share a
full strength bond
Shifting Problem
43
n x n Square Full Connectivity
Full Connectivity Constraint All adjacent
tiles in assembled shape must share a
full strength bond
Jigsaw Technique Use Geometry to enforce
proper binding.
Shifting Problem
44
n x n Square Full Connectivity
Full Connectivity Constraint All adjacent
tiles in assembled shape must share a
full strength bond
Jigsaw Technique Use Geometry to enforce
proper binding.
45
n x n Square Full Connectivity
Full Connectivity Constraint All adjacent
tiles in assembled shape must share a
full strength bond
Jigsaw Technique Use Geometry to enforce
proper binding.
46
n x n Square Full Connectivity
Staged Assembly Fully Connected n x n square
Non-Staged Model Fully Connected n x n square
tiles / glues O(1)
Bins O(1)
Stages O(log n)
Temperature 1
tiles / glues Q(log n / log log n)
Bins 1
Stages 1
Temperature 2
adleman, cheng, goel, huang STOC 2001
47
Arbitrary Shapes

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

48
Simulate Large Tilesets
49
Simulate Large Tilesets
0000
0001
0010
0011
0100
0101
0110
50
Simulate Large Tilesets
0000
0001
0
0010
0011
1
0100
0101
0110
51
Simulate Large Tilesets
0
0
0
0
0000
0
0
0
1
0001
0
0
0
1
0010
0
0
1
1
0011
0
0
0
1
0100
0101
0
0
1
1
0110
0
0
1
1
52
Simulate Large Tilesets
0000
0001
0010
0
0
1
1
0011
0
0
1
1
0100
0101
0110
53
Simulate Large Tilesets
0000
0001
0010
0
0
1
0
0011
0100
0
0
1
1
0101
0110
54
Simulate Large Tilesets
55
Simulate Large Tilesets
a
b
c
. . .
56
Simulate Large Tilesets
Simulate temp1 tileset T
tiles / glues O(1)
Bins O(T)
Stages O(log log T)
Arbitrary n tile Shape
tiles / glues O(1)
Bins O(n)
Stages O(log log n)
Scale O(log n)
57
Arbitrary Shape Assembly

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

Jigsaw Method
Spanning Tree Method
Simulation Method
tiles / glues O(1)
Bins O(n)
Stages O(n)
Connectivity FULL
Scale 2
Generality Hole Free
tiles / glues O(1)
Bins O(log n)
Stages O(diameter)
Connectivity Partial
Scale 1
Generality ALL
tiles / glues O(1)
Bins O(n)
Stages O(log log n)
Connectivity FULL
Scale O(log n)
Generality ALL
58
Near Optimal Tradeoff Bins versus Stages(Crazy
Mixing)
First Result
What if we have B bins?
Staged Assembly n x n square
tiles / glues O(1)
Bins O(1)
Stages O(log n)
59
Near Optimal Tradeoff Bins versus Stages(Crazy
Mixing)
First Result
What if we have B bins?
Staged Assembly n x n square
tiles / glues O(1)
Bins O(1)
Stages O(log n)
B2 edges, Can encode B2 Bits of
information Per stage.
60
Near Optimal Tradeoff Bins versus Stages(Crazy
Mixing)
Assembly of n x n squares with B bins
Lower Bound for almost all n
Upper Bound
tiles / glues O(1)
Bins B
Stages W( log n / B2)
tiles / glues O(1)
Bins B
Stages O( log n / B2 log B)

• Upper bound technique
• Encode B2 bits describing
• target square at each stage
• Combine with Simulation
• macro tiles.

61
Conclusions

• 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.

62
Future Work

• 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 t2 systems

63
Thanks for listening. Questions?