january 18, 2010
DESCRIPTION
Shape Replication through Self-Assembly and Rnase Enzymes . January 18, 2010. Read:. Replicate:. Zachary Abel Harvard University Nadia Benbernou Massachusetts Institute of Technology Mirela Damian Villanova University Erik D. Demaine Massachusetts Institute of Technology - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/1.jpg)
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January 18, 2010
Shape Replication through Self-Assembly and Rnase Enzymes
Zachary Abel Harvard UniversityNadia Benbernou Massachusetts Institute of TechnologyMirela Damian Villanova UniversityErik D. Demaine Massachusetts Institute of TechnologyMartin Demaine Massachusetts Institute of TechnologyRobin Flatland Siena CollegeSkott D. Kominers Harvard UniversityRobert Schweller University of Texas Pan American
Read: Replicate:
![Page 2: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/2.jpg)
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Outline
• Basic Model• RNA enzyme model• Shape replication
• Precise yield shape replication• Infinite yield shape replication
![Page 3: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/3.jpg)
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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
![Page 4: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/4.jpg)
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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)
![Page 5: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/5.jpg)
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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)
![Page 6: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/6.jpg)
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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)
![Page 7: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/7.jpg)
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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)
![Page 8: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/8.jpg)
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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)
![Page 9: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/9.jpg)
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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)
![Page 10: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/10.jpg)
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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)
![Page 11: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/11.jpg)
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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)
![Page 12: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/12.jpg)
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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)
![Page 13: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/13.jpg)
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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
Tile Assembly Model(Rothemund, Winfree, Adleman)
![Page 14: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/14.jpg)
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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)
![Page 15: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/15.jpg)
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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)
![Page 16: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/16.jpg)
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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
Tile Assembly Model(Rothemund, Winfree, Adleman)
![Page 17: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/17.jpg)
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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
Tile Assembly Model(Rothemund, Winfree, Adleman)
![Page 18: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/18.jpg)
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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
(Basic)Tile Assembly Model(Rothemund, Winfree, Adleman)
![Page 19: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/19.jpg)
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Outline
• Basic Model• RNA enzyme model• Shape replication
• Precise yield shape replication• Infinite yield shape replication
![Page 20: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/20.jpg)
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RNA enzyme Self-Assembly(suggested by Rothemund, Winfree 2000)
RNA tile types DNA tile types
RNA assembly model: • Assembly occurs over a number of stages.
• At each stage you may:1) Add a new collection of tile types
- Allow for further growth- All added types have infinite count
2) Add an Rnase enzyme- Dissolve all RNA tile types- May break apart assemblies
All tile types are of either DNA or RNA makeup:
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RNA enzyme Self-Assembly
Stage 1:
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RNA enzyme Self-Assembly
Stage 1:
![Page 23: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/23.jpg)
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RNA enzyme Self-Assembly
Stage 1:
Stage 2:
![Page 24: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/24.jpg)
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RNA enzyme Self-Assembly
Stage 1:
Stage 2:
![Page 25: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/25.jpg)
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RNA enzyme Self-Assembly
Stage 1:
Stage 2:
Stage 3: Enzyme
![Page 26: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/26.jpg)
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RNA enzyme Self-Assembly
Stage 1:
Stage 2:
Stage 3: Enzyme
![Page 27: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/27.jpg)
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RNA enzyme Self-Assembly
Stage 1:
Stage 2:
Stage 3: Enzyme
![Page 28: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/28.jpg)
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RNA enzyme Self-Assembly
Stage 1:
Stage 2:
Stage 3: Enzyme
Stage 4:
![Page 29: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/29.jpg)
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RNA enzyme Self-Assembly
Stage 1:
Stage 2:
Stage 3: Enzyme
Stage 4:
![Page 30: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/30.jpg)
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RNA enzyme Self-Assembly
Stage 1:
Stage 2:
Stage 3: Enzyme
Stage 4:
![Page 31: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/31.jpg)
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RNA enzyme Self-Assembly
Stage 1:
Stage 2:
Stage 3: Enzyme
Stage 4:
![Page 32: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/32.jpg)
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RNA enzyme Self-AssemblyMetrics for efficiency:
• Tile complexity: total number of distinct tile types used in the system.
• Stage complexity: total number of distinct stages used.
Stage 1:
Stage 2:
Stage 3: Enzyme
Stage 4:
![Page 33: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/33.jpg)
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Outline
• Basic Model• RNA enzyme model• Shape replication
• Precise yield shape replication• Infinite yield shape replication
![Page 34: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/34.jpg)
Shape Replication Problem
Design an assembly system (algorithm) that will replicate a large number of copies given a single copy of a pre-assembled input shape.
Precise Yield: Replicate exactly n copies for a given n
Infinite Yield: Replicate infinite copies-in practice, the number of copies should only be limited by the volume of particles available.
![Page 35: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/35.jpg)
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Outline
• Basic Model• RNA enzyme model• Shape replication
• Precise yield shape replication• Infinite yield shape replication
![Page 36: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/36.jpg)
Precise Yield: rectangles
![Page 37: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/37.jpg)
Precise Yield: rectangles
a a a aaaaaaa
a a a a
aaaaaa
![Page 38: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/38.jpg)
Precise Yield: rectangles
n n n neeeeee
wwwwww
s s ss
![Page 39: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/39.jpg)
Precise Yield: rectangles
n n n neeeeee
wwwwww
s s ss
n
w
x xyy
![Page 40: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/40.jpg)
Precise Yield: rectangles
n n n neeeeee
wwwwww
s s ss
nw
![Page 41: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/41.jpg)
Precise Yield: rectangles
n n n neeeeee
wwwwww
s s ss
nw n
e
sw s
e
![Page 42: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/42.jpg)
Precise Yield: rectangles
n n n neeeeee
wwwwww
s s ss
nw
ne
sw s
e
![Page 43: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/43.jpg)
Precise Yield: rectangles
n n n neeeeee
ww
w
w
ww
s s ss
a
wa
a
a
![Page 44: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/44.jpg)
Precise Yield: rectangles
n ne
e
w
wss
Step 1: Coat shape with layer of RNA
![Page 45: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/45.jpg)
Precise Yield: rectangles
n ne
e
w
wss
Step 2: Coat shape with layer of DNAStep 1: Coat shape with layer of RNA
![Page 46: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/46.jpg)
Precise Yield: rectanglesStep 2: Coat shape with layer of DNA.Step 3: Add enzyme.
Step 1: Coat shape with layer of RNA.
![Page 47: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/47.jpg)
Precise Yield: rectanglesStep 2: Coat shape with layer of DNA.Step 3: Add enzyme.
Step 1: Coat shape with layer of RNA.
![Page 48: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/48.jpg)
Precise Yield: rectanglesStep 2: Coat shape with layer of DNA.Step 3: Add enzyme.Step 4: Coat frame with layer of RNA.
Step 1: Coat shape with layer of RNA.
![Page 49: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/49.jpg)
Precise Yield: rectanglesStep 2: Coat shape with layer of DNA.Step 3: Add enzyme.Step 4: Coat frame with layer of RNA.
Step 1: Coat shape with layer of RNA.
![Page 50: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/50.jpg)
Precise Yield: rectanglesStep 2: Coat shape with layer of DNA.Step 3: Add enzyme.Step 4: Coat frame with layer of RNA.Step 5: Fill frame with DNA.
Step 1: Coat shape with layer of RNA.
![Page 51: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/51.jpg)
Precise Yield: rectanglesStep 2: Coat shape with layer of DNA.Step 3: Add enzyme.Step 4: Coat frame with layer of RNA.Step 5: Fill frame with DNA.
Step 1: Coat shape with layer of RNA.
![Page 52: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/52.jpg)
Precise Yield: rectanglesStep 2: Coat shape with layer of DNA.Step 3: Add enzyme.Step 4: Coat frame with layer of RNA.Step 5: Fill frame with DNA.Step 6: Add enzyme.
Step 1: Coat shape with layer of RNA.
![Page 53: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/53.jpg)
Precise Yield: rectanglesStep 2: Coat shape with layer of DNA.Step 3: Add enzyme.Step 4: Coat frame with layer of RNA.Step 5: Fill frame with DNA.Step 6: Add enzyme.
Step 1: Coat shape with layer of RNA.
![Page 54: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/54.jpg)
Precise Yield: General Shapes
![Page 55: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/55.jpg)
Precise Yield: General Shapes
![Page 56: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/56.jpg)
Precise Yield: General Shapes
![Page 57: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/57.jpg)
Precise Yield: rectangles
![Page 58: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/58.jpg)
Tile types O(1)
Stages O(log n)
Precise Yield:n copies
Precise Yield: rectangles
Tile types O(log n)
Stages O(1)
Precise Yield:n copies
![Page 59: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/59.jpg)
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Outline
• Basic Model• RNA enzyme model• Shape replication
• Precise yield shape replication• Infinite yield shape replication
![Page 60: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/60.jpg)
Infinite Yield: Rectangles
n n n neeeeee
wwwwww
s s ss
![Page 61: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/61.jpg)
Infinite Yield: Rectangles
n n n neeeeee
wwwwww
s s ss
sw
a
![Page 62: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/62.jpg)
Infinite Yield: Rectangles
as
bs
bs
a
b
s sa
s
Stair step tiles:
xx
![Page 63: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/63.jpg)
Infinite Yield: Rectangles
as
bs
bs
a
b
s s
Stair step tiles:
xx
bx
![Page 64: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/64.jpg)
Infinite Yield: Rectangles
as
bs
bs
a
b
s s
Stair step tiles:
xx
a
b
![Page 65: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/65.jpg)
Infinite Yield: Rectangles
as
bs
bs
a
b
s
s
Stair step tiles:
xx
ab
![Page 66: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/66.jpg)
Infinite Yield: Rectangles
as
bs
bs
a
b
s
Stair step tiles:
xx
b
a
b
![Page 67: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/67.jpg)
Infinite Yield: Rectangles
as
bs
bs
a
b
s
Stair step tiles:
xx
b
a
b
![Page 68: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/68.jpg)
Infinite Yield: Rectangles
as
bs
bs
a
b
s
Stair step tiles:
xx
b
a
b
![Page 69: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/69.jpg)
Infinite Yield: Rectangles
as
bs
bs
a
b
Stair step tiles:
xx
b
bb
a
![Page 70: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/70.jpg)
Infinite Yield: Rectangles
as
bs
bs
a
b
Stair step tiles:
xx
![Page 71: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/71.jpg)
Infinite Yield: Rectangles
as
bs
bs
a
b
Stair step tiles:
xx
![Page 72: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/72.jpg)
Infinite Yield: Rectangles
as
bs
bs
a
b
Stair step tiles:
xx
…
![Page 73: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/73.jpg)
Infinite Yield: Rectangles
…
Tile types O(1)
Stages O(1)
Infinite Yield:Rectangles
![Page 74: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/74.jpg)
Infinite Yield: General Shapes
![Page 75: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/75.jpg)
Infinite Yield: General ShapesStep 1: Coat with RNA
![Page 76: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/76.jpg)
Infinite Yield: General ShapesStep 2: Create rectangular DNA encasing
![Page 77: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/77.jpg)
Infinite Yield: Binary counter tool
c c c cc c c c c c c c c c c c c c c cc c cc
![Page 78: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/78.jpg)
Infinite Yield: Binary counter tool
1 c c c cc c c c c c c c c c c c c c c cc c cm
0
0 10 0 1 1nn
nn
0 11 0 0 1cc
nc
1 m
m
xx
Binary counter tiles:
c01 mm
n
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1
Infinite Yield: Binary counter tool
1 c c c cc c c c c c c c c c c c c c cc c c0 0
m
0
0 10 0 1 1nn
nn
0 11 0 0 1cc
nc
1 m
m
xx
Binary counter tiles:
c01 mm
n
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1
Infinite Yield: Binary counter tool
1 c c c cc c c c c c c c c c c c c cc c c0
m
1 1n
0
0 10 0 1 1nn
nn
0 11 0 0 1cc
nc
1 m
m
xx
Binary counter tiles:
c01 mm
n
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1
Infinite Yield: Binary counter tool
1 c c c cc c c c c c c c c c c c c cc c c
0
0 10 0 1 1nn
nn
0 11 0 0 1cc
nc
1 m
m
xx
Binary counter tiles:
c
0
0
m
1 1
1 mmn
1
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1
Infinite Yield: Binary counter tool
1 c c c cc c c c c c c c c c c c cc c c
0
0 10 0 1 1nn
nn
0 11 0 0 1cc
nc
1 m
m
xx
Binary counter tiles:
c
0
0
1
1 mmn
10 0cm
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1
Infinite Yield: Binary counter tool
1 c c c cc c c c c c c c c c c c cc c c
0
0 10 0 1 1nn
nn
0 11 0 0 1cc
nc
1 m
m
xx
Binary counter tiles:
c
0
0
m
1
1 mmn
10 0
10 0
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1
Infinite Yield: Binary counter tool
1 c c c c c c c c c c c c c c c cc c c
0
0 10 0 1 1nn
nn
0 11 0 0 1cc
nc
1 m
m
xx
Binary counter tiles:
c
0
0
m
1
1 mmn
10
10 0
1 1n
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1
Infinite Yield: Binary counter tool
1 c c c c c c c c c c c c c c c cc c c
0
0 10 0 1 1nn
nn
0 11 0 0 1cc
nc
1 m
m
xx
Binary counter tiles:
c
0
0
m
1
1 mmn
10
10
1 1
n0 0
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1
Infinite Yield: Binary counter tool
1 c c c c c c c c c c c c c c c cc c c
0
0 10 0 1 1nn
nn
0 11 0 0 1cc
nc
1 m
m
xx
Binary counter tiles:
c
0
0
m
1
1 mmn
10
10
1 1
0 01
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1
Infinite Yield: Binary counter tool
1
0
0 10 0 1 1nn
nn
0 11 0 0 1cc
nc
1 m
m
xx
Binary counter tiles:
c
0
0
1
1 mmn
10
10
101
011
111
000
100
010
101
1 1 1 1
011
111
1 1
0001
0 100
010
101
011
111
0 0 0 01 1 1 1
01
11
000
100
010
101
011
11
11
11
11
![Page 88: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/88.jpg)
Infinite Yield: General Shapes
…
…
![Page 89: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/89.jpg)
Infinite Yield: General Shapes
1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
1 1 1 1 1
1 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1 …
…
![Page 90: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/90.jpg)
Infinite Yield: General Shapes
1 1 1 1 0 0 0 0 0 01 1 1 1 1 1
1 1 1 1 11 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 00 0 0 01 1 1 1
1 1 1 11 1 1 1 11 1 1 1 1
0 0 0 0 0 1 1 1 1 11 1 1 1 1
1 1 1 1 11 1 1 1 1
11
1 1 1 1 1 1 1 1 1 1 1 0 0 0 01 1 1 1
0 0 0 00 0 0 0
![Page 91: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/91.jpg)
Infinite Yield: General Shapes
1 1 1 1 0 0 0 0 0 01 1 1 1 1 1
1 1 1 1 11 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 00 0 0 01 1 1 1
1 1 1 11 1 1 1 11 1 1 1 1
0 0 0 0 0 1 1 1 1 11 1 1 1 1
1 1 1 1 11 1 1 1 1
11
1 1 1 1 1 1 1 1 1 1 1 0 0 0 01 1 1 1
0 0 0 00 0 0 0
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 000 0 0 000 0 0 0
00
0 0 000 0 000 0 0 00 0 0 00 00 0 0 0 0 0 0 0 0 0
1 1 1 1 0 0 0 0 01 1 1 1 1
1 1 1 1 11 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 00 0 0 01 1 1 1
1 1 1 1
0 0 0 00 0 0 000 0 0 00 0 0 0
00
0 0 000 0 000 0 0 0 0 0 00 00 0 0 0 0 0
01
1
0
0 0 0 0 0 01 1 1 1 1 1
0 0 0 0 0 0
0 0 0 00 0
0 01 1
0 0
0 0
1 1 1 100 0 000 0 000 0 0
1000
Step 3: Label each face with unique binary code
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Infinite Yield: General Shapes
1 1 1 1 0 0 0 0 0 01 1 1 1 1 1
1 1 1 1 11 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 00 0 0 01 1 1 1
1 1 1 11 1 1 1 11 1 1 1 1
0 0 0 0 0 1 1 1 1 11 1 1 1 1
1 1 1 1 11 1 1 1 1
11
1 1 1 1 1 1 1 1 1 1 1 0 0 0 01 1 1 1
0 0 0 00 0 0 0
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 000 0 0 000 0 0 0
00
0 0 000 0 000 0 0 00 0 0 00 00 0 0 0 0 0 0 0 0 0
1 1 1 1 0 0 0 0 01 1 1 1 1
1 1 1 1 11 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 00 0 0 01 1 1 1
1 1 1 1
0 0 0 00 0 0 000 0 0 00 0 0 0
00
0 0 000 0 000 0 0 0 0 0 00 00 0 0 0 0 0
01
1
0
0 0 0 0 0 01 1 1 1 1 1
0 0 0 0 0 0
0 0 0 00 0
0 01 1
0 0
0 0
1 1 1 100 0 000 0 000 0 0
1000
Step 4: Enzyme.
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Infinite Yield: General Shapes
0 0 0 01 1 1 1
1 1 1 1
0 0 0 0
…
0 0 0 01 1 1 1
1 1 1 1
0 0 0 0
0 0 0 01 1 1 1
1 1 1 1
0 0 0 0
0 0 0 01 1 1 1
1 1 1 1
0 0 0 0
0 0 0 01 1 1 1
1 1 1 1
0 0 0 0
Step 5: Infinitely replicate all labeled rectangles
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Infinite Yield: General Shapes
1 1 1 1 0 0 0 0 0 01 1 1 1 1 1
1 1 1 1 11 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 00 0 0 01 1 1 1
1 1 1 11 1 1 1 11 1 1 1 1
0 0 0 0 0 1 1 1 1 11 1 1 1 1
1 1 1 1 11 1 1 1 1
11
1 1 1 1 1 1 1 1 1 1 1 0 0 0 01 1 1 1
0 0 0 00 0 0 0
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 000 0 0 000 0 0 0
00
0 0 000 0 000 0 0 00 0 0 00 00 0 0 0 0 0 0 0 0 0
1 1 1 1 0 0 0 0 01 1 1 1 1
1 1 1 1 11 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 00 0 0 01 1 1 1
1 1 1 1
0 0 0 00 0 0 000 0 0 00 0 0 0
00
0 0 000 0 000 0 0 0 0 0 00 00 0 0 0 0 0
01
1
0
0 0 0 0 0 01 1 1 1 1 1
0 0 0 0 0 0
0 0 0 00 0
0 01 1
0 0
0 0
1 1 1 100 0 000 0 000 0 0
1000
![Page 95: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/95.jpg)
Infinite Yield: General ShapesReassembly?
0 0 0 01 1 1 1
1 1 1 1
0 0 0 0
0 0 0 0 0 01 1 1 1 1 1
0 0 0 0 0 0
0 0 0 00 0
![Page 96: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/96.jpg)
Infinite Yield: General ShapesReassembly?
0 0 0 11 1 1 1
1 1 1 0
0 0 0 0
0 0 0 0 0 01 1 1 1 1 1
0 0 0 0 0 1
0 0 0 00 0
![Page 97: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/97.jpg)
Infinite Yield: General ShapesReassembly?
0 0 0 11 1 1 1
1 1 1 0
0 0 0 0
0 0 0 0 0 01 1 1 1 1 1
0 0 0 0 0 1
0 0 0 00 0
![Page 98: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/98.jpg)
Infinite Yield: General ShapesReassembly?
0 0 0 11 1 1 1
1 1 1 0
0 0 0 0
0 0 0 0 0 01 1 1 1 1 1
0 0 0 0 0 1
0 0 0 00 00101
0 11 00 01 1
10
1
1
![Page 99: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/99.jpg)
Infinite Yield: General Shapes
1 1 1 1 0 0 0 0 0 01 1 1 1 1 1
1 1 1 1 11 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 00 0 0 01 1 1 1
1 1 1 11 1 1 1 11 1 1 1 1
0 0 0 0 0 1 1 1 1 11 1 1 1 1
1 1 1 1 11 1 1 1 1
11
1 1 1 1 1 1 1 1 1 1 1 0 0 0 01 1 1 1
0 0 0 00 0 0 0
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 000 0 0 000 0 0 0
00
0 0 000 0 000 0 0 00 0 0 00 00 0 0 0 0 0 0 0 0 0
1 1 1 1 0 0 0 0 01 1 1 1 1
1 1 1 1 11 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 00 0 0 01 1 1 1
1 1 1 1
0 0 0 00 0 0 000 0 0 00 0 0 0
00
0 0 000 0 000 0 0 0 0 0 00 00 0 0 0 0 0
01
1
0
0 0 0 0 0 01 1 1 1 1 1
0 0 0 0 0 0
0 0 0 00 0
0 01 1
0 0
0 0
1 1 1 100 0 000 0 000 0 0
1000
Step 6: Reassemble, fill in frame, break out copies with enzyme.
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Infinite Yield: General Shapes
1 1 1 1 0 0 0 0 0 01 1 1 1 1 1
1 1 1 1 11 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 00 0 0 01 1 1 1
1 1 1 11 1 1 1 11 1 1 1 1
0 0 0 0 0 1 1 1 1 11 1 1 1 1
1 1 1 1 11 1 1 1 1
11
1 1 1 1 1 1 1 1 1 1 1 0 0 0 01 1 1 1
0 0 0 00 0 0 0
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 000 0 0 000 0 0 0
00
0 0 000 0 000 0 0 00 0 0 00 00 0 0 0 0 0 0 0 0 0
1 1 1 1 0 0 0 0 01 1 1 1 1
1 1 1 1 11 1 1 1 1 0 0 0 0 0 0
1 1 1 1 1 1
0 0 0 0 0 00 0 0 01 1 1 1
1 1 1 1
0 0 0 00 0 0 000 0 0 00 0 0 0
00
0 0 000 0 000 0 0 0 0 0 00 00 0 0 0 0 0
01
1
0
0 0 0 0 0 01 1 1 1 1 1
0 0 0 0 0 0
0 0 0 00 0
0 01 1
0 0
0 0
1 1 1 100 0 000 0 000 0 0
1000
Tile types O(1)
Stages O(1)
Infinite Yield:Vertically convex
![Page 101: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/101.jpg)
Infinite Yield: Non-vertically convex shapes
• Grow counter along surface of shape
![Page 102: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/102.jpg)
000100100011
0100
0101
0110
0111
1000
1001
1010
• Grow counter along surface of shape
Start end
Infinite Yield: Non-vertically convex shapes
![Page 103: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/103.jpg)
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
• Grow counter along surface of shape
• Break apart with enzyme
Infinite Yield: Non-vertically convex shapes
![Page 104: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/104.jpg)
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
• Grow counter along surface of shape
• Break apart
with enzyme
• Replicate
Infinite Yield: Non-vertically convex shapes
![Page 105: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/105.jpg)
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
• Grow counter along surface of shape
• Break apart
with enzyme
• Replicate
• Reassemble
Infinite Yield: Non-vertically convex shapes
![Page 106: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/106.jpg)
0001
00100011
0100
0101
0110
0111
1000
1001
1010
• Grow counter along surface of shape
• Break apart
with enzyme
• Replicate
• Reassemble
Infinite Yield: Non-vertically convex shapes
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107
Tile types O(1)
Stages O(1)
Infinite Yield:
Infinite Yield: General Shapes
![Page 108: January 18, 2010](https://reader035.vdocuments.us/reader035/viewer/2022070420/56815f55550346895dce33ef/html5/thumbnails/108.jpg)
Future Work
• Replicate and improve-Hybrid algorithms for replication and modification
• Extension to 3D-Planarity/spacial constraint
• Replication of internal pattern
• Staged enzyme model for assembly from scratch- Seems to be very powerful for this
• Temperature changes to perform replication
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109
January 18, 2010
Thank you. Questions?
Zachary Abel Harvard UniversityNadia Benbernou Massachusetts Institute of TechnologyMirela Damian Villanova UniversityErik D. Demaine Massachusetts Institute of TechnologyMartin Demaine Massachusetts Institute of TechnologyRobin Flatland Siena CollegeSkott D. Kominers Harvard UniversityRobert Schweller University of Texas Pan American
Read: Replicate: