1 thirteenth international meeting on dna computers june 5, 2007 staged self-assembly:...
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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
Tile Assembly Model(Rothemund, Winfree, Adleman)
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 = 2
d
e
x ed
cba
b c
Tile Assembly Model(Rothemund, Winfree, Adleman)
6
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
d
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 = 2
d
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 = 2
d
e
x ed
cba
b ca
Tile Assembly Model(Rothemund, Winfree, Adleman)
9
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1
t = 2
d
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 = 2
d
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 = 2
d
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 = 2
x ed
cba
a b c
d
e
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
x
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
a b c
d
e
x
x ed
cba
Tile Assembly Model(Rothemund, Winfree, Adleman)
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
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
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 x
Tile Assembly Model(Rothemund, Winfree, Adleman)
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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
-Assembly occurs within 1 single container
- Assembly occurs within 1 single stage
20
BEAKERBEAKER BEAKER
Aftersome time...
After enough time...
Start with initial Tileset Various Producible Supertilesexist in solution
Only Terminally Producedassemblies remain
Non-Staged Assembly
-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
Stage Complexity: 3
Mix pattern:
27
Staged Assembly• Assembly occurs in a sequence of stages, and
assemblies can be separated into separate bins
Bins = Space ComplexityStages = Time Complexity
Bin Complexity: 4
Stage Complexity: 3
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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
• 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.
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 i+3
33
Simple Example: 1 x n line
stage i
stage i+3
tiles / glues O(1) = 3
Bins O(1)
Stages O(log n)
Staged Assembly1 x n line
34
Simple Example: 1 x n line
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
35
n x n Square
36
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
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
Full Connectivity Constraint: All adjacent tiles inassembled shape mustshare a full strength bond
[Rothemund, Winfree STOC 2000]
41
n x n Square: Full Connectivity
Full Connectivity Constraint: All adjacent tiles inassembled shape mustshare a full strength bond
42
n x n Square: Full Connectivity
Shifting Problem
Full Connectivity Constraint: All adjacent tiles inassembled shape mustshare a full strength bond
43
n x n Square: Full Connectivity
Shifting Problem
Jigsaw Technique:Use Geometryto enforce properbinding.
Full Connectivity Constraint: All adjacent tiles inassembled shape mustshare a full strength bond
44
n x n Square: Full Connectivity
Jigsaw Technique:Use Geometryto enforce properbinding.
Full Connectivity Constraint: All adjacent tiles inassembled shape mustshare a full strength bond
45
n x n Square: Full Connectivity
Jigsaw Technique:Use Geometryto enforce properbinding.
Full Connectivity Constraint: All adjacent tiles inassembled shape mustshare a full strength bond
46
n x n Square: Full Connectivity
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]
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
0010
0011
0100
0101
0110
0
1
51
Simulate Large Tilesets
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
52
Simulate Large Tilesets
0 01
0 01 1
0000
0001
0010
0011
0100
0101
0110
1
53
Simulate Large Tilesets
0 0
0 01 1
0000
0001
0010
0011
0100
0101
0110
10
54
Simulate Large Tilesets
0 01
0 01 1
1
00
1
00
1 0
0
55
c
Simulate Large Tilesets
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
. . .
56
Simulate Large Tilesets
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
57
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
58
tiles / glues O(1)
Bins O(1)
Stages O(log n)
Staged Assemblyn x n square
First Result:
What if we have B bins?
Near Optimal Tradeoff: Bins versus Stages(Crazy Mixing)
59
tiles / glues O(1)
Bins O(1)
Stages O(log n)
Staged Assemblyn x n square
First Result:
What if we have B bins?
B^2 edges, Can encode B^2Bits of informationPer stage.
Near Optimal Tradeoff: Bins versus Stages(Crazy Mixing)
60
Near Optimal Tradeoff: Bins versus Stages(Crazy Mixing)
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.
61
• 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
62
• 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
63
Thanks for listening. Questions?
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