DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23....
Transcript of DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23....
![Page 1: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/1.jpg)
DNA Tile Self-Assembly
Steffen Kopecki
Department of Computer Science
Natural Computing, Winter Term 2013/2014
S. Kopecki DNA Tile Self-Assembly 1 / 23
![Page 2: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/2.jpg)
Table of Contents
(I) Self-Assembly Systems with a Temperature
(II) Directed vs. Undirected Self-Assembly Systems
(III) Staged Self-Assembly
(IV) Assembly of Patterns
(V) Assembly of “Smart Tiles” and “Smart Structures”
S. Kopecki DNA Tile Self-Assembly 2 / 23
![Page 3: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/3.jpg)
Abstract Tile Self-Assembly Model (aTAM)Winfree (1998)
The abstract tile self-assembly model was defined in order to capturethe the process of DNA self-assembly in a simplified formal model. AnaTAM consists of
I finite set of tile types T with glues from Γ ,I temperature τ ∈Z+,I glue strength function g : Γ →N, andI seed tile (or structure) σ .
A tile can attach to the growing structure if its binding strength is atleast the temperature τ .Let τ = 2.
σppp
q qqp pp
qqq
q qqq qq
p pp
ppp
p pp
ppp
qqq
qqqq qq
qqq
ppp
q qq
qqq
ppp
q qq
qqq
ppp
qqq
ppp
p ppppp
p pp
ppp
qqq
ppp
qqq
ppp
An assembly is terminal if no further tiles can be attached.
S. Kopecki DNA Tile Self-Assembly 3 / 23
![Page 4: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/4.jpg)
Abstract Tile Self-Assembly Model (aTAM)Winfree (1998)
The abstract tile self-assembly model was defined in order to capturethe the process of DNA self-assembly in a simplified formal model. AnaTAM consists of
I finite set of tile types T with glues from Γ ,I temperature τ ∈Z+,I glue strength function g : Γ →N, andI seed tile (or structure) σ .
A tile can attach to the growing structure if its binding strength is atleast the temperature τ .
Let τ = 2.
σppp
q qqp pp
qqq
q qqq qq
p pp
ppp
p pp
ppp
qqq
qqqq qq
qqq
ppp
q qq
qqq
ppp
q qq
qqq
ppp
qqq
ppp
p ppppp
p pp
ppp
qqq
ppp
qqq
ppp
An assembly is terminal if no further tiles can be attached.
S. Kopecki DNA Tile Self-Assembly 3 / 23
![Page 5: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/5.jpg)
Abstract Tile Self-Assembly Model (aTAM)Winfree (1998)
The abstract tile self-assembly model was defined in order to capturethe the process of DNA self-assembly in a simplified formal model. AnaTAM consists of
I finite set of tile types T with glues from Γ ,I temperature τ ∈Z+,I glue strength function g : Γ →N, andI seed tile (or structure) σ .
A tile can attach to the growing structure if its binding strength is atleast the temperature τ .Let τ = 2.
σppp
q qqp pp
qqq
q qqq qq
p pp
ppp
p pp
ppp
qqq
qqqq qq
qqq
ppp
q qq
qqq
ppp
q qq
qqq
ppp
qqq
ppp
p ppppp
p pp
ppp
qqq
ppp
qqq
ppp
An assembly is terminal if no further tiles can be attached.
S. Kopecki DNA Tile Self-Assembly 3 / 23
![Page 6: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/6.jpg)
Abstract Tile Self-Assembly Model (aTAM)Winfree (1998)
The abstract tile self-assembly model was defined in order to capturethe the process of DNA self-assembly in a simplified formal model. AnaTAM consists of
I finite set of tile types T with glues from Γ ,I temperature τ ∈Z+,I glue strength function g : Γ →N, andI seed tile (or structure) σ .
A tile can attach to the growing structure if its binding strength is atleast the temperature τ .Let τ = 2.
σppp
q qqp pp
qqq
q qqq qq
p pp
ppp
p pp
ppp
qqq
qqqq qq
qqq
ppp
q qq
qqq
ppp
q qq
qqq
ppp
qqq
ppp
p ppppp
p pp
ppp
qqq
ppp
qqq
ppp
An assembly is terminal if no further tiles can be attached.
S. Kopecki DNA Tile Self-Assembly 3 / 23
![Page 7: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/7.jpg)
Abstract Tile Self-Assembly Model (aTAM)Winfree (1998)
The abstract tile self-assembly model was defined in order to capturethe the process of DNA self-assembly in a simplified formal model. AnaTAM consists of
I finite set of tile types T with glues from Γ ,I temperature τ ∈Z+,I glue strength function g : Γ →N, andI seed tile (or structure) σ .
A tile can attach to the growing structure if its binding strength is atleast the temperature τ .Let τ = 2.
σppp
q qqp pp
qqq
q qqq qq
p pp
ppp
p pp
ppp
qqq
qqqq qq
qqq
ppp
q qq
qqq
ppp
q qq
qqq
ppp
qqq
ppp
p ppppp
p pp
ppp
qqq
ppp
qqq
ppp
An assembly is terminal if no further tiles can be attached.
S. Kopecki DNA Tile Self-Assembly 3 / 23
![Page 8: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/8.jpg)
Abstract Tile Self-Assembly Model (aTAM)Winfree (1998)
The abstract tile self-assembly model was defined in order to capturethe the process of DNA self-assembly in a simplified formal model. AnaTAM consists of
I finite set of tile types T with glues from Γ ,I temperature τ ∈Z+,I glue strength function g : Γ →N, andI seed tile (or structure) σ .
A tile can attach to the growing structure if its binding strength is atleast the temperature τ .Let τ = 2.
σppp
q qqp pp
qqq
q qqq qq
p pp
ppp
p pp
ppp
qqq
qqqq qq
qqq
ppp
q qq
qqq
ppp
q qq
qqq
ppp
qqq
ppp
p ppppp
p pp
ppp
qqq
ppp
qqq
ppp
An assembly is terminal if no further tiles can be attached.
S. Kopecki DNA Tile Self-Assembly 3 / 23
![Page 9: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/9.jpg)
Abstract Tile Self-Assembly Model (aTAM)Winfree (1998)
The abstract tile self-assembly model was defined in order to capturethe the process of DNA self-assembly in a simplified formal model. AnaTAM consists of
I finite set of tile types T with glues from Γ ,I temperature τ ∈Z+,I glue strength function g : Γ →N, andI seed tile (or structure) σ .
A tile can attach to the growing structure if its binding strength is atleast the temperature τ .Let τ = 2.
σppp
q qqp pp
qqq
q qqq qq
p pp
ppp
p pp
ppp
qqq
qqqq qq
qqq
ppp
q qq
qqq
ppp
q qq
qqq
ppp
qqq
ppp
p ppppp
p pp
ppp
qqq
ppp
qqq
ppp
An assembly is terminal if no further tiles can be attached.
S. Kopecki DNA Tile Self-Assembly 3 / 23
![Page 10: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/10.jpg)
Abstract Tile Self-Assembly Model (aTAM)Winfree (1998)
The abstract tile self-assembly model was defined in order to capturethe the process of DNA self-assembly in a simplified formal model. AnaTAM consists of
I finite set of tile types T with glues from Γ ,I temperature τ ∈Z+,I glue strength function g : Γ →N, andI seed tile (or structure) σ .
A tile can attach to the growing structure if its binding strength is atleast the temperature τ .Let τ = 2.
σppp
q qqp pp
qqq
q qqq qq
p pp
ppp
p pp
ppp
qqq
qqqq qq
qqq
ppp
q qq
qqq
ppp
q qq
qqq
ppp
qqq
ppp
p ppppp
p pp
ppp
qqq
ppp
qqq
ppp
An assembly is terminal if no further tiles can be attached.
S. Kopecki DNA Tile Self-Assembly 3 / 23
![Page 11: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/11.jpg)
Abstract Tile Self-Assembly Model (aTAM)Winfree (1998)
The abstract tile self-assembly model was defined in order to capturethe the process of DNA self-assembly in a simplified formal model. AnaTAM consists of
I finite set of tile types T with glues from Γ ,I temperature τ ∈Z+,I glue strength function g : Γ →N, andI seed tile (or structure) σ .
A tile can attach to the growing structure if its binding strength is atleast the temperature τ .Let τ = 2.
σppp
q qqp pp
qqq
q qqq qq
p pp
ppp
p pp
ppp
qqq
qqqq qq
qqq
ppp
q qq
qqq
ppp
q qq
qqq
ppp
qqq
ppp
p ppppp
p pp
ppp
qqq
ppp
qqq
ppp
An assembly is terminal if no further tiles can be attached.
S. Kopecki DNA Tile Self-Assembly 3 / 23
![Page 12: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/12.jpg)
Modeling of Chemical Properties
Glues are implemented by complementary DNA sticky ends u and u∗.The glue strength is the energy needed to break the hydrogen bondsbetween the sticky ends.
I the length of the sticky ends,I G,C-content (G−C pairs 3 hydrogen
bonds whereas A−T pairs have 2),I possible mismatches in u and u∗.
Depending on the temperature of the solution “weak bonds” willfrequently assemble and disassemble, but will not be stable.
Other factors can influence the glue strength, like solvents (oftensalts) in the solution.
S. Kopecki DNA Tile Self-Assembly 4 / 23
![Page 13: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/13.jpg)
Modeling of Chemical Properties
Glues are implemented by complementary DNA sticky ends u and u∗.The glue strength is the energy needed to break the hydrogen bondsbetween the sticky ends.
I the length of the sticky ends,I G,C-content (G−C pairs 3 hydrogen
bonds whereas A−T pairs have 2),I possible mismatches in u and u∗.
Depending on the temperature of the solution “weak bonds” willfrequently assemble and disassemble, but will not be stable.
Other factors can influence the glue strength, like solvents (oftensalts) in the solution.
S. Kopecki DNA Tile Self-Assembly 4 / 23
![Page 14: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/14.jpg)
Modeling of Chemical Properties
Glues are implemented by complementary DNA sticky ends u and u∗.The glue strength is the energy needed to break the hydrogen bondsbetween the sticky ends.
I the length of the sticky ends,I G,C-content (G−C pairs 3 hydrogen
bonds whereas A−T pairs have 2),I possible mismatches in u and u∗.
Depending on the temperature of the solution “weak bonds” willfrequently assemble and disassemble, but will not be stable.
Other factors can influence the glue strength, like solvents (oftensalts) in the solution.
S. Kopecki DNA Tile Self-Assembly 4 / 23
![Page 15: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/15.jpg)
Self-Assembly of a Counter at Temperature τ = 2
Seed
σqqq
qqq
Frameqqq
qqq
1 0
q qqq qq
Half-adder0 1
1
1 0 0
0
0
1 1
0
0 1 0
1
0
sum a⊕ binput ainput bcarry a∧ b
g(qqq) = 2
g(0) = g(1) = 1
σqqq
qqq
qqq
qqq
1
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
0
q qqq qq
0
q qqq qq
1 1
0
0
0
q qqq qq
qqq
qqq
1
1 1
0
0
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
1 1
0
0
1 1
0
0
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
0 0
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
00
1 1
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
000 0
0
0
1 1
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
000 0
0
00 0
0
0
0 1
1
10 1
1
11 1
0
00 0
0
00 0
000 0
0
00 0
0
00 0
0
0
1 1
0
00 0
0
01 0
1
00 0
000 0
0
00 0
0
00 0
0
00 0
0
0
0 1
1
11 1
0
01 0
100 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 1
0
01 0
101 0
1
00 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
0 1
110 1
1
10 1
1
11 1
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 10
00 0
0
00 0
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
0
0
S. Kopecki DNA Tile Self-Assembly 5 / 23
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Self-Assembly of a Counter at Temperature τ = 2
Seed
σqqq
qqq
Frameqqq
qqq
1 0
q qqq qq
Half-adder0 1
1
1 0 0
0
0
1 1
0
0 1 0
1
0
sum a⊕ binput ainput bcarry a∧ b
g(qqq) = 2
g(0) = g(1) = 1
σqqq
qqq
qqq
qqq
1
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
0
q qqq qq
0
q qqq qq
1 1
0
0
0
q qqq qq
qqq
qqq
1
1 1
0
0
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
1 1
0
0
1 1
0
0
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
0 0
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
00
1 1
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
000 0
0
0
1 1
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
000 0
0
00 0
0
0
0 1
1
10 1
1
11 1
0
00 0
0
00 0
000 0
0
00 0
0
00 0
0
0
1 1
0
00 0
0
01 0
1
00 0
000 0
0
00 0
0
00 0
0
00 0
0
0
0 1
1
11 1
0
01 0
100 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 1
0
01 0
101 0
1
00 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
0 1
110 1
1
10 1
1
11 1
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 10
00 0
0
00 0
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
0
0
S. Kopecki DNA Tile Self-Assembly 5 / 23
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Self-Assembly of a Counter at Temperature τ = 2
Seed
σqqq
qqq
Frameqqq
qqq
1 0
q qqq qq
Half-adder0 1
1
1 0 0
0
0
1 1
0
0 1 0
1
0
sum a⊕ binput ainput bcarry a∧ b
g(qqq) = 2
g(0) = g(1) = 1
σqqq
qqq
qqq
qqq
1
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
0
q qqq qq
0
q qqq qq
1 1
0
0
0
q qqq qq
qqq
qqq
1
1 1
0
0
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
1 1
0
0
1 1
0
0
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
0 0
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
00
1 1
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
000 0
0
0
1 1
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
000 0
0
00 0
0
0
0 1
1
10 1
1
11 1
0
00 0
0
00 0
000 0
0
00 0
0
00 0
0
0
1 1
0
00 0
0
01 0
1
00 0
000 0
0
00 0
0
00 0
0
00 0
0
0
0 1
1
11 1
0
01 0
100 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 1
0
01 0
101 0
1
00 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
0 1
110 1
1
10 1
1
11 1
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 10
00 0
0
00 0
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
0
0
S. Kopecki DNA Tile Self-Assembly 5 / 23
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Self-Assembly of a Counter at Temperature τ = 2
Seed
σqqq
qqq
Frameqqq
qqq
1 0
q qqq qq
Half-adder0 1
1
1 0 0
0
0
1 1
0
0 1 0
1
0
sum a⊕ binput ainput bcarry a∧ b
g(qqq) = 2
g(0) = g(1) = 1
σqqq
qqq
qqq
qqq
1
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
0
q qqq qq
0
q qqq qq
1 1
0
0
0
q qqq qq
qqq
qqq
1
1 1
0
0
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
1 1
0
0
1 1
0
0
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
0 0
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
00
1 1
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
000 0
0
0
1 1
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
000 0
0
00 0
0
0
0 1
1
10 1
1
11 1
0
00 0
0
00 0
000 0
0
00 0
0
00 0
0
0
1 1
0
00 0
0
01 0
1
00 0
000 0
0
00 0
0
00 0
0
00 0
0
0
0 1
1
11 1
0
01 0
100 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 1
0
01 0
101 0
1
00 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
0 1
110 1
1
10 1
1
11 1
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 10
00 0
0
00 0
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
0
0
S. Kopecki DNA Tile Self-Assembly 5 / 23
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Self-Assembly of a Counter at Temperature τ = 2
Seed
σqqq
qqq
Frameqqq
qqq
1 0
q qqq qq
Half-adder0 1
1
1 0 0
0
0
1 1
0
0 1 0
1
0
sum a⊕ binput ainput bcarry a∧ b
g(qqq) = 2
g(0) = g(1) = 1
σqqq
qqq
qqq
qqq
1
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
0
q qqq qq
0
q qqq qq
1 1
0
0
0
q qqq qq
qqq
qqq
1
1 1
0
0
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
1 1
0
0
1 1
0
0
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
0 0
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
00
1 1
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
000 0
0
0
1 1
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
000 0
0
00 0
0
0
0 1
1
10 1
1
11 1
0
00 0
0
00 0
000 0
0
00 0
0
00 0
0
0
1 1
0
00 0
0
01 0
1
00 0
000 0
0
00 0
0
00 0
0
00 0
0
0
0 1
1
11 1
0
01 0
100 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 1
0
01 0
101 0
1
00 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
0 1
110 1
1
10 1
1
11 1
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 10
00 0
0
00 0
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
0
0
S. Kopecki DNA Tile Self-Assembly 5 / 23
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Self-Assembly of a Counter at Temperature τ = 2
Seed
σqqq
qqq
Frameqqq
qqq
1 0
q qqq qq
Half-adder0 1
1
1 0 0
0
0
1 1
0
0 1 0
1
0
sum a⊕ binput ainput bcarry a∧ b
g(qqq) = 2
g(0) = g(1) = 1
σqqq
qqq
qqq
qqq
1
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
0
q qqq qq
0
q qqq qq
1 1
0
0
0
q qqq qq
qqq
qqq
1
1 1
0
0
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
1 1
0
0
1 1
0
0
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
0 0
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
00
1 1
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
000 0
0
0
1 1
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
000 0
0
00 0
0
0
0 1
1
10 1
1
11 1
0
00 0
0
00 0
000 0
0
00 0
0
00 0
0
0
1 1
0
00 0
0
01 0
1
00 0
000 0
0
00 0
0
00 0
0
00 0
0
0
0 1
1
11 1
0
01 0
100 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 1
0
01 0
101 0
1
00 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
0 1
110 1
1
10 1
1
11 1
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 10
00 0
0
00 0
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
0
0
S. Kopecki DNA Tile Self-Assembly 5 / 23
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Self-Assembly of a Counter at Temperature τ = 2
Seed
σqqq
qqq
Frameqqq
qqq
1 0
q qqq qq
Half-adder0 1
1
1 0 0
0
0
1 1
0
0 1 0
1
0
sum a⊕ binput ainput bcarry a∧ b
g(qqq) = 2
g(0) = g(1) = 1
σqqq
qqq
qqq
qqq
1
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
0
q qqq qq
0
q qqq qq
1 1
0
0
0
q qqq qq
qqq
qqq
1
1 1
0
0
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
0 1
1
1
0
q qqq qq
qqq
qqq
1
0 0
0
0
1 1
0
0
1 1
0
0
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
0
q qqq qq
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
qqq
qqq
1
0 0
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
00
1 1
0
00 0
0
00 0
0
00 0
0
00 0
0
00 0
000 0
0
0
1 1
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
000 0
0
00 0
0
0
0 1
1
10 1
1
11 1
0
00 0
0
00 0
000 0
0
00 0
0
00 0
0
0
1 1
0
00 0
0
01 0
1
00 0
000 0
0
00 0
0
00 0
0
00 0
0
0
0 1
1
11 1
0
01 0
100 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 1
0
01 0
101 0
1
00 0
0
00 0
0
00 0
0
00 0
0
00 0
0
0
0 1
110 1
1
10 1
1
11 1
0
00 0
0
00 0
0
00 0
0
00 0
0
0
1 10
00 0
0
00 0
0
01 0
1
00 0
0
00 0
0
00 0
0
00 0
0
0
S. Kopecki DNA Tile Self-Assembly 5 / 23
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Self-Assembly of DNA Sierpinski TrianglesRothemund, Papadakis, Winfree (2004)
The fractal structure of the Sierpinski triangle can also be generatedby the the xor logic gate from a string · · ·000010000 · · ·
S. Kopecki DNA Tile Self-Assembly 6 / 23
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Self-Assembly of DNA Sierpinski TrianglesRothemund, Papadakis, Winfree (2004)
The fractal structure of the Sierpinski triangle can also be generatedby the the xor logic gate from a string · · ·000010000 · · ·
S. Kopecki DNA Tile Self-Assembly 6 / 23
![Page 24: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/24.jpg)
Self-Assembly of DNA Sierpinski TrianglesRothemund, Papadakis, Winfree (2004)
The fractal structure of the Sierpinski triangle can also be generatedby the the xor logic gate from a string · · ·000010000 · · ·
S. Kopecki DNA Tile Self-Assembly 6 / 23
![Page 25: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/25.jpg)
Self-Assembly of Sierpinski Triangles
000
0 110
1 101
1 011
0
output a⊕ boutput a⊕ binput binput a
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
0qqq
qqq
10qqq
qqq
0 0qqq
qqq
00qqq
qqq
0 0qqq
qqq
00qqq
qqq
0 0qqq
qqq
00qqq
qqq
0 0qqq
qqq
00qqq
qqq
0 0qqq
qqq
00qqq
qqq
0
110
1000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
011
0000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
101
1 110
1000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0011
0 011
0 011
0
101
1 110
1000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
011
0 101
1 110
1 011
0000
0 000
0 000
0 000
0 000
0 000
0 000
0
101
1 110
1 101
1 110
1 101
1 110
1000
0 000
0 000
0 000
0
000
0 000
0 000
0 000
0011
0 011
0 011
0 011
0 011
0 011
0 011
0
101
1 110
1000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
011
0 101
1 110
1 011
0000
0 000
0 000
0 000
0 000
0 000
0 000
0
101
1 110
1 101
1 110
1 110
1 101
1000
0 000
0 000
0 000
0
101
1 110
1011
0 011
0 011
0 011
0 011
0 011
0000
0 000
0 000
0
S. Kopecki DNA Tile Self-Assembly 7 / 23
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Self-Assembly of Sierpinski Triangles
000
0 110
1 101
1 011
0
output a⊕ boutput a⊕ binput binput a
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
qqqqqq
0qqq
qqq
10qqq
qqq
0 0qqq
qqq
00qqq
qqq
0 0qqq
qqq
00qqq
qqq
0 0qqq
qqq
00qqq
qqq
0 0qqq
qqq
00qqq
qqq
0 0qqq
qqq
00qqq
qqq
0
110
1000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
110
1 101
1
000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
011
0000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
101
1 110
1000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0011
0 011
0 011
0
101
1 110
1000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
011
0 101
1 110
1 011
0000
0 000
0 000
0 000
0 000
0 000
0 000
0
101
1 110
1 101
1 110
1 101
1 110
1000
0 000
0 000
0 000
0
000
0 000
0 000
0 000
0011
0 011
0 011
0 011
0 011
0 011
0 011
0
101
1 110
1000
0 000
0 000
0 000
0 000
0 000
0 000
0 000
0
011
0 101
1 110
1 011
0000
0 000
0 000
0 000
0 000
0 000
0 000
0
101
1 110
1 101
1 110
1 110
1 101
1000
0 000
0 000
0 000
0
101
1 110
1011
0 011
0 011
0 011
0 011
0 011
0000
0 000
0 000
0
S. Kopecki DNA Tile Self-Assembly 7 / 23
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Table of Contents
(I) Self-Assembly Systems with a Temperature
(II) Directed vs. Undirected Self-Assembly Systems
(III) Staged Self-Assembly
(IV) Assembly of Patterns
(V) Assembly of “Smart Tiles” and “Smart Structures”
S. Kopecki DNA Tile Self-Assembly 8 / 23
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Directed Self-assembly Systems
An aTAM is directed (a. k. a. deterministic) if it forms one uniqueterminal assembly, where an assembly is defined by which tile type isplaced at each position.
An aTAM strictly self-assembles a shape if all of its terminalassemblies are guaranteed to have that shape, although some of theassemblies may have different tile types at the same position.
S. Kopecki DNA Tile Self-Assembly 9 / 23
![Page 29: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/29.jpg)
Directed Self-assembly Systems
An aTAM is directed (a. k. a. deterministic) if it forms one uniqueterminal assembly, where an assembly is defined by which tile type isplaced at each position.
An aTAM strictly self-assembles a shape if all of its terminalassemblies are guaranteed to have that shape, although some of theassemblies may have different tile types at the same position.
S. Kopecki DNA Tile Self-Assembly 9 / 23
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The Power of Undirected Systems
Theorem
For n ∈N, there is a finite shape S that is strictly self-assembled by anaTAM with c tile types, but every directed aTAM that (strictly)self-assembles S requires at least c+n tile types.
n
A1
A2
A3
...
An-1
An
B1
B2
B3
...
Bn-1
Bn
A1
A2
A3
...
An-1
An
B1
B2
B3
...
Bn-1
Bn
C1
C2
C3
...
Cn-1
Cn
Theorem
There is an infinite shape S such that some aTAM strictlyself-assembles S, but no directed aTAM (strictly) self-assembles S.
S. Kopecki DNA Tile Self-Assembly 10 / 23
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The Power of Undirected Systems
Theorem
For n ∈N, there is a finite shape S that is strictly self-assembled by anaTAM with c tile types, but every directed aTAM that (strictly)self-assembles S requires at least c+n tile types.
n
A1
A2
A3
...
An-1
An
B1
B2
B3
...
Bn-1
Bn
A1
A2
A3
...
An-1
An
B1
B2
B3
...
Bn-1
Bn
C1
C2
C3
...
Cn-1
Cn
Theorem
There is an infinite shape S such that some aTAM strictlyself-assembles S, but no directed aTAM (strictly) self-assembles S.
S. Kopecki DNA Tile Self-Assembly 10 / 23
![Page 32: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/32.jpg)
The Power of Undirected Systems
Theorem
For n ∈N, there is a finite shape S that is strictly self-assembled by anaTAM with c tile types, but every directed aTAM that (strictly)self-assembles S requires at least c+n tile types.
n
A1
A2
A3
...
An-1
An
B1
B2
B3
...
Bn-1
Bn
A1
A2
A3
...
An-1
An
B1
B2
B3
...
Bn-1
Bn
C1
C2
C3
...
Cn-1
Cn
Theorem
There is an infinite shape S such that some aTAM strictlyself-assembles S, but no directed aTAM (strictly) self-assembles S.
S. Kopecki DNA Tile Self-Assembly 10 / 23
![Page 33: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/33.jpg)
The Power of Undirected Systems
Theorem
For n ∈N, there is a finite shape S that is strictly self-assembled by anaTAM with c tile types, but every directed aTAM that (strictly)self-assembles S requires at least c+n tile types.
n
A1
A2
A3
...
An-1
An
B1
B2
B3
...
Bn-1
Bn
A1
A2
A3
...
An-1
An
B1
B2
B3
...
Bn-1
Bn
C1
C2
C3
...
Cn-1
Cn
Theorem
There is an infinite shape S such that some aTAM strictlyself-assembles S, but no directed aTAM (strictly) self-assembles S.
S. Kopecki DNA Tile Self-Assembly 10 / 23
![Page 34: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/34.jpg)
The Power of Undirected Systems
Theorem
For n ∈N, there is a finite shape S that is strictly self-assembled by anaTAM with c tile types, but every directed aTAM that (strictly)self-assembles S requires at least c+n tile types.
n
A1
A2
A3
...
An-1
An
B1
B2
B3
...
Bn-1
Bn
A1
A2
A3
...
An-1
An
B1
B2
B3
...
Bn-1
Bn
C1
C2
C3
...
Cn-1
Cn
Theorem
There is an infinite shape S such that some aTAM strictlyself-assembles S, but no directed aTAM (strictly) self-assembles S.
S. Kopecki DNA Tile Self-Assembly 10 / 23
![Page 35: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/35.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→
pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 36: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/36.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→
pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 37: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/37.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→
pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 38: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/38.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→
pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 39: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/39.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→
pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 40: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/40.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→
pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 41: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/41.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→
pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 42: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/42.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→
pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 43: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/43.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→
pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 44: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/44.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→
pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 45: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/45.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 46: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/46.jpg)
Universality of Directed aTAM with τ = 2
Theorem
The directed (zig-zag) aTAM at temperature τ = 2 is Turing-universal.
· · ·��001010111�� · · ·
s
· · ·��101010111�� · · ·
δ(s,0) = (p,1,R)
p
· · ·��101010111�� · · ·
δ(p,0) = (q,0,L)
q
· · ·��0011101�� · · ·
∗
f
I s0 0 1 0 1 0 1 1 1 H
H p1 0 1 0 1 0 1 1 1 J
I 1 p0 1 0 1 0 1 1 1 H
H q1 0 1 0 1 0 1 1 1 J
I q1 1 1 0 1 0 1 1 1 H
I 0 0 1 1 f 1 0 1 H
I 0 0 1 1 1 0 1 J
qqq
qq q qq q qq q qq q qq q qq q qq q qq q qq q qq q
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
←
qqq
pp p pp p pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp ppp ppp
→
qqq
qqq
pp p pp p pp p pp p pp p pp p pp p pp p
ppp ppp ppp ppp ppp ppp ppp ppp
ppp ppp ppp ppp ppp ppp ppp ppp
→pp p pp p pp p pp p pp p pp p pp p pp p ←
Open Problem
Is the directed aTAM Turing-universal at temperature τ = 1?
S. Kopecki DNA Tile Self-Assembly 11 / 23
![Page 47: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/47.jpg)
Table of Contents
(I) Self-Assembly Systems with a Temperature
(II) Directed vs. Undirected Self-Assembly Systems
(III) Staged Self-Assembly
(IV) Assembly of Patterns
(V) Assembly of “Smart Tiles” and “Smart Structures”
S. Kopecki DNA Tile Self-Assembly 12 / 23
![Page 48: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/48.jpg)
Staged Self-assembly
More external control is added to the assembly process by usingdifferent sets of tile types in each of several stages.
Start with a seed structure σ and sets of tile types T1, . . . ,Tn. For eachstage i = 1, . . . ,n
1. add the tile types Ti to the solution,
2. wait for a terminal structure to assemble,
3. then, “wash away” all unbound tile types.
After the n-th stage start over with the 1-st stage again.
In biochemistry wet-labs the repeated process of mixing DNAstructures (in our case tile types) into a solution and then purifying thesolution to obtain certain structures is a commonly used technique.
S. Kopecki DNA Tile Self-Assembly 13 / 23
![Page 49: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/49.jpg)
Staged Self-assembly
More external control is added to the assembly process by usingdifferent sets of tile types in each of several stages.
Start with a seed structure σ and sets of tile types T1, . . . ,Tn. For eachstage i = 1, . . . ,n
1. add the tile types Ti to the solution,
2. wait for a terminal structure to assemble,
3. then, “wash away” all unbound tile types.
After the n-th stage start over with the 1-st stage again.
In biochemistry wet-labs the repeated process of mixing DNAstructures (in our case tile types) into a solution and then purifying thesolution to obtain certain structures is a commonly used technique.
S. Kopecki DNA Tile Self-Assembly 13 / 23
![Page 50: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/50.jpg)
Universality of Staged aTAM with τ = 1
Theorem
The directed, staged aTAM at temperature τ = 1 is Turing-universal.
Staged aTAM with τ = 2 can simulate zig-zag aTAM with τ = 2.
East-west glues are actual glues while north-south glues aresimulated by the geometry of the tile.
0x
a
b
x
c
a
c d ed efg
f
0x
g
b h
1x
ab
xb h ih ji k lj
m
k
l
1xm
S. Kopecki DNA Tile Self-Assembly 14 / 23
![Page 51: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/51.jpg)
Universality of Staged aTAM with τ = 1
Theorem
The directed, staged aTAM at temperature τ = 1 is Turing-universal.
Staged aTAM with τ = 2 can simulate zig-zag aTAM with τ = 2.
East-west glues are actual glues while north-south glues aresimulated by the geometry of the tile.
0x
a
b
x
c
a
c d ed efg
f
0x
g
b h
1x
ab
xb h ih ji k lj
m
k
l
1xm
S. Kopecki DNA Tile Self-Assembly 14 / 23
![Page 52: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/52.jpg)
Universality of Staged aTAM with τ = 1
Theorem
The directed, staged aTAM at temperature τ = 1 is Turing-universal.
Staged aTAM with τ = 2 can simulate zig-zag aTAM with τ = 2.
East-west glues are actual glues while north-south glues aresimulated by the geometry of the tile.
0x a
b
x
c
a
c d ed efg
f
0x
g
b h
1x ab
x
b h ih ji k lj
m
k
l
1xm
S. Kopecki DNA Tile Self-Assembly 14 / 23
![Page 53: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/53.jpg)
Universality of Staged aTAM with τ = 1
Theorem
The directed, staged aTAM at temperature τ = 1 is Turing-universal.
Staged aTAM with τ = 2 can simulate zig-zag aTAM with τ = 2.
East-west glues are actual glues while north-south glues aresimulated by the geometry of the tile.
0x a
b
x
c
a
c d ed efg
f
0x
g
b h1x a
bx
b h ih ji k lj
m
k
l
1xm
S. Kopecki DNA Tile Self-Assembly 14 / 23
![Page 54: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/54.jpg)
Table of Contents
(I) Self-Assembly Systems with a Temperature
(II) Directed vs. Undirected Self-Assembly Systems
(III) Staged Self-Assembly
(IV) Assembly of Patterns
(V) Assembly of “Smart Tiles” and “Smart Structures”
S. Kopecki DNA Tile Self-Assembly 15 / 23
![Page 55: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/55.jpg)
What are Nanoscopic Patterns?
“Molecular pegboards” (addressable nanoarrays), which are cheap toproduce, for
a.) arranging nanoparticles (gold, silver, . . . ),
b.) molecular and logic circuits (in vitro and in vivo),
c.) enzyme interaction or enzyme detection,
d.) nano-factories like “artificial leafs”,
e.) quantum dot assembly.
a.) d.)
S. Kopecki DNA Tile Self-Assembly 16 / 23
![Page 56: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/56.jpg)
Pattern Assembly
grid where everynode has a property
pattern where everypixel has a color
Pattern assembly is an aTAM whereI the temperature is τ = 2,I every tile type has a color,I every glue has strength 1, andI we start from an L-shaped seed.
+++ � F +++ �
+ ++�
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F �
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+++
+++
S. Kopecki DNA Tile Self-Assembly 17 / 23
![Page 57: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/57.jpg)
Pattern Assembly
grid where everynode has a property
pattern where everypixel has a color
Pattern assembly is an aTAM whereI the temperature is τ = 2,I every tile type has a color,I every glue has strength 1, andI we start from an L-shaped seed.
+++ � F +++ �
+ ++�
F
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+++ F F
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S. Kopecki DNA Tile Self-Assembly 17 / 23
![Page 58: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/58.jpg)
Pattern Assembly
grid where everynode has a property
pattern where everypixel has a color
Pattern assembly is an aTAM whereI the temperature is τ = 2,I every tile type has a color,I every glue has strength 1, andI we start from an L-shaped seed.
+++ � F +++ �
+ ++�
F
�
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F �
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+++
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S. Kopecki DNA Tile Self-Assembly 17 / 23
![Page 59: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/59.jpg)
Pattern Assembly
grid where everynode has a property
pattern where everypixel has a color
Pattern assembly is an aTAM whereI the temperature is τ = 2,I every tile type has a color,I every glue has strength 1, andI we start from an L-shaped seed.
+++ � F +++ �
+ ++�
F
�
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+++
+++ F F
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+++
+++
F
F
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+++
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S. Kopecki DNA Tile Self-Assembly 17 / 23
![Page 60: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/60.jpg)
Pattern Assembly
grid where everynode has a property
pattern where everypixel has a color
Pattern assembly is an aTAM whereI the temperature is τ = 2,I every tile type has a color,I every glue has strength 1, andI we start from an L-shaped seed.
+++ � F +++ �
+ ++�
F
�
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+++
+++ F F
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+++
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F
F
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S. Kopecki DNA Tile Self-Assembly 17 / 23
![Page 61: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/61.jpg)
Pattern Assembly
grid where everynode has a property
pattern where everypixel has a color
Pattern assembly is an aTAM whereI the temperature is τ = 2,I every tile type has a color,I every glue has strength 1, andI we start from an L-shaped seed.
+++ � F +++ �
+ ++�
F
�
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+++
+++ F F
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+++
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F
F
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S. Kopecki DNA Tile Self-Assembly 17 / 23
![Page 62: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/62.jpg)
Pattern Assembly
grid where everynode has a property
pattern where everypixel has a color
Pattern assembly is an aTAM whereI the temperature is τ = 2,I every tile type has a color,I every glue has strength 1, andI we start from an L-shaped seed.
+++ � F +++ �
+ ++�
F
�
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+++
+++ F F
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F
F
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S. Kopecki DNA Tile Self-Assembly 17 / 23
![Page 63: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/63.jpg)
Pattern Assembly
grid where everynode has a property
pattern where everypixel has a color
Pattern assembly is an aTAM whereI the temperature is τ = 2,I every tile type has a color,I every glue has strength 1, andI we start from an L-shaped seed.
+++ � F +++ �
+ ++�
F
�
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+++
+++ F F
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F
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S. Kopecki DNA Tile Self-Assembly 17 / 23
![Page 64: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/64.jpg)
Pattern Assembly
grid where everynode has a property
pattern where everypixel has a color
Pattern assembly is an aTAM whereI the temperature is τ = 2,I every tile type has a color,I every glue has strength 1, andI we start from an L-shaped seed.
+++ � F +++ �
+ ++�
F
�
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S. Kopecki DNA Tile Self-Assembly 17 / 23
![Page 65: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/65.jpg)
Minimal Tile Sets for Patterns
For a given pattern P , among all aTAMs which strictly self-assemble P ,find an aTAM with the minimal number of tile types.Obvious bounds: #colors ≤ #tile types ≤ pattern size
77167
62216
15461
27153
16272
21531
16722
Theorem
A minimal tile set which strictly self-assembles a pattern P is directed.
Theorem
It is NP-hard to find a minimal tile set that strictly self-assembles agiven binary pattern P .
NP-hard: no algorithm is known which solves the problem efficiently.
S. Kopecki DNA Tile Self-Assembly 18 / 23
![Page 66: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/66.jpg)
Minimal Tile Sets for Patterns
For a given pattern P , among all aTAMs which strictly self-assemble P ,find an aTAM with the minimal number of tile types.Obvious bounds: #colors ≤ #tile types ≤ pattern size
77167
62216
15461
27153
16272
21531
16722
Theorem
A minimal tile set which strictly self-assembles a pattern P is directed.
Theorem
It is NP-hard to find a minimal tile set that strictly self-assembles agiven binary pattern P .
NP-hard: no algorithm is known which solves the problem efficiently.
S. Kopecki DNA Tile Self-Assembly 18 / 23
![Page 67: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/67.jpg)
Minimal Tile Sets for Patterns
For a given pattern P , among all aTAMs which strictly self-assemble P ,find an aTAM with the minimal number of tile types.Obvious bounds: #colors ≤ #tile types ≤ pattern size
77167
62216
15461
27153
16272
21531
16722
Theorem
A minimal tile set which strictly self-assembles a pattern P is directed.
Theorem
It is NP-hard to find a minimal tile set that strictly self-assembles agiven binary pattern P .
NP-hard: no algorithm is known which solves the problem efficiently.
S. Kopecki DNA Tile Self-Assembly 18 / 23
![Page 68: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/68.jpg)
Minimal Tile Sets for Patterns
For a given pattern P , among all aTAMs which strictly self-assemble P ,find an aTAM with the minimal number of tile types.Obvious bounds: #colors ≤ #tile types ≤ pattern size
77167
62216
15461
27153
16272
21531
16722
Theorem
A minimal tile set which strictly self-assembles a pattern P is directed.
Theorem
It is NP-hard to find a minimal tile set that strictly self-assembles agiven binary pattern P .
NP-hard: no algorithm is known which solves the problem efficiently.
S. Kopecki DNA Tile Self-Assembly 18 / 23
![Page 69: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/69.jpg)
Minimal Tile Sets for Patterns
For a given pattern P , among all aTAMs which strictly self-assemble P ,find an aTAM with the minimal number of tile types.Obvious bounds: #colors ≤ #tile types ≤ pattern size
77167
62216
15461
27153
16272
21531
16722
Theorem
A minimal tile set which strictly self-assembles a pattern P is directed.
Theorem
It is NP-hard to find a minimal tile set that strictly self-assembles agiven binary pattern P .
NP-hard: no algorithm is known which solves the problem efficiently.
S. Kopecki DNA Tile Self-Assembly 18 / 23
![Page 70: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/70.jpg)
Table of Contents
S. Kopecki DNA Tile Self-Assembly 19 / 23
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Signals and Logic Gates on Tiles
Signal Passing
Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.
Logic Gates
Several signals on one tile can be combinedvia logic gates.
Signals and logic gates can be implementedusing strand displacement.
S. Kopecki DNA Tile Self-Assembly 20 / 23
![Page 72: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/72.jpg)
Signals and Logic Gates on Tiles
Signal Passing
Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.
Logic Gates
Several signals on one tile can be combinedvia logic gates.
Signals and logic gates can be implementedusing strand displacement.
S. Kopecki DNA Tile Self-Assembly 20 / 23
![Page 73: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/73.jpg)
Signals and Logic Gates on Tiles
Signal Passing
Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.
Logic Gates
Several signals on one tile can be combinedvia logic gates.
Signals and logic gates can be implementedusing strand displacement.
S. Kopecki DNA Tile Self-Assembly 20 / 23
![Page 74: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/74.jpg)
Signals and Logic Gates on Tiles
Signal Passing
Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.
Logic Gates
Several signals on one tile can be combinedvia logic gates.
Signals and logic gates can be implementedusing strand displacement.
S. Kopecki DNA Tile Self-Assembly 20 / 23
![Page 75: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/75.jpg)
Signals and Logic Gates on Tiles
Signal Passing
Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.
Logic Gates
Several signals on one tile can be combinedvia logic gates.
Signals and logic gates can be implementedusing strand displacement.
S. Kopecki DNA Tile Self-Assembly 20 / 23
![Page 76: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/76.jpg)
Signals and Logic Gates on Tiles
Signal Passing
Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.
Logic Gates
Several signals on one tile can be combinedvia logic gates.
Signals and logic gates can be implementedusing strand displacement.
S. Kopecki DNA Tile Self-Assembly 20 / 23
![Page 77: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/77.jpg)
Signals and Logic Gates on Tiles
Signal Passing
Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.
Logic Gates
Several signals on one tile can be combinedvia logic gates.
Signals and logic gates can be implementedusing strand displacement.
S. Kopecki DNA Tile Self-Assembly 20 / 23
![Page 78: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/78.jpg)
Signals and Logic Gates on Tiles
Signal Passing
Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.
Logic Gates
Several signals on one tile can be combinedvia logic gates.
Signals and logic gates can be implementedusing strand displacement.
S. Kopecki DNA Tile Self-Assembly 20 / 23
![Page 79: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/79.jpg)
Signals and Logic Gates on Tiles
Signal Passing
Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.
Logic Gates
Several signals on one tile can be combinedvia logic gates.
Signals and logic gates can be implementedusing strand displacement.
S. Kopecki DNA Tile Self-Assembly 20 / 23
![Page 80: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/80.jpg)
Signals and Logic Gates on Tiles
Signal Passing
Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.
Logic Gates
Several signals on one tile can be combinedvia logic gates.
Signals and logic gates can be implementedusing strand displacement.
S. Kopecki DNA Tile Self-Assembly 20 / 23
![Page 81: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/81.jpg)
Signals and Logic Gates on Tiles
Signal Passing
Attach signals on top of tiles which aretriggered when the tile assembles. Signalscan activate glues, deactivate glues, ortrigger a signal on a neighbouring tile.
Logic Gates
Several signals on one tile can be combinedvia logic gates.
Signals and logic gates can be implementedusing strand displacement.
S. Kopecki DNA Tile Self-Assembly 20 / 23
![Page 82: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/82.jpg)
Signal Passing for Tile Self-AssemblyPadilla, Liu, Seeman (2011)
Smart tiles which can interac-tively self-assemble larger struc-tures are in turn self-assembledfrom smaller “DNA structures”.
S. Kopecki DNA Tile Self-Assembly 21 / 23
![Page 83: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/83.jpg)
Signal Passing for Tile Self-AssemblyPadilla, Liu, Seeman (2011)
Smart tiles which can interac-tively self-assemble larger struc-tures are in turn self-assembledfrom smaller “DNA structures”.
S. Kopecki DNA Tile Self-Assembly 21 / 23
![Page 84: DNA Tile Self-Assemblylkari/NatCompLecture_full.pdf · S. KopeckiDNA Tile Self-Assembly2 / 23. Abstract Tile Self-Assembly Model (aTAM) Winfree (1998) The abstract tile self-assembly](https://reader035.fdocuments.net/reader035/viewer/2022070821/5f1f63516635f508620c5ae0/html5/thumbnails/84.jpg)
Robot Pebbles a. k. a. “Smart Sand”Gilpin, Rus et al. (2009–2012)
http://www.youtube.com/watch?v=swxTTlHjN5Q
S. Kopecki DNA Tile Self-Assembly 22 / 23
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Logic-Gated Nanorobot for Targeted TransportDouglas, Bachelet, Church (2012)
S. Kopecki DNA Tile Self-Assembly 23 / 23