DNA Tile Self-Assembly

Steffen Kopecki

Department of Computer Science

Natural Computing, Winter Term 2013/2014

S. Kopecki DNA Tile Self-Assembly 1 / 23

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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1

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0

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0

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0

0

S. Kopecki DNA Tile Self-Assembly 5 / 23

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

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1

1 1

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1

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1

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1 10

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01 0

1

00 0

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00 0

0

00 0

0

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0

0

S. Kopecki DNA Tile Self-Assembly 5 / 23

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

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1

0

q qqq qq

0

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01 0

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00 0

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00 0

0

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0

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0

0

S. Kopecki DNA Tile Self-Assembly 5 / 23

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

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1

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1

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0

S. Kopecki DNA Tile Self-Assembly 5 / 23

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

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1

0

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0

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1 10

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0

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01 0

1

00 0

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00 0

0

00 0

0

00 0

0

0

S. Kopecki DNA Tile Self-Assembly 5 / 23

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

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qqq

qqq

1

0

q qqq qq

qqq

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1

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1

0

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

�

�

+++

+++ F F

�

�

+++

+++

F

F

�

�

+++

+++ F F

�

�

F F

�

�

+++

+++

F

F

+++

+++

F

F

+++

+++

F

F

�

�

+++

+++

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+++

+++

�

�

+++

+++

F F

�

�

F F

�

�

F F

�

�

+++

+++

F

F

+++

+++

F

F �

�

+++

+++

S. Kopecki DNA Tile Self-Assembly 17 / 23

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

�

�

+++

+++ F F

�

�

+++

+++

F

F

�

�

+++

+++ F F

�

�

F F

�

�

+++

+++

F

F

+++

+++

F

F

+++

+++

F

F

�

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+++

+++

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+++

+++

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+++

+++

F F

�

�

F F

�

�

F F

�

�

+++

+++

F

F

+++

+++

F

F �

�

+++

+++

S. Kopecki DNA Tile Self-Assembly 17 / 23

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

�

�

+++

+++ F F

�

�

+++

+++

F

F

�

�

+++

+++ F F

�

�

F F

�

�

+++

+++

F

F

+++

+++

F

F

+++

+++

F

F

�

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+++

+++

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+++

+++

�

�

+++

+++

F F

�

�

F F

�

�

F F

�

�

+++

+++

F

F

+++

+++

F

F �

�

+++

+++

S. Kopecki DNA Tile Self-Assembly 17 / 23

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

�

�

+++

+++ F F

�

�

+++

+++

F

F

�

�

+++

+++ F F

�

�

F F

�

�

+++

+++

F

F

+++

+++

F

F

+++

+++

F

F

�

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+++

+++

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+++

+++

�

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+++

+++

F F

�

�

F F

�

�

F F

�

�

+++

+++

F

F

+++

+++

F

F �

�

+++

+++

S. Kopecki DNA Tile Self-Assembly 17 / 23

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

�

�

+++

+++ F F

�

�

+++

+++

F

F

�

�

+++

+++

F F

�

�

F F

�

�

+++

+++

F

F

+++

+++

F

F

+++

+++

F

F

�

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+++

+++

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+++

+++

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+++

+++

F F

�

�

F F

�

�

F F

�

�

+++

+++

F

F

+++

+++

F

F �

�

+++

+++

S. Kopecki DNA Tile Self-Assembly 17 / 23

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

�

�

+++

+++ F F

�

�

+++

+++

F

F

�

�

+++

+++ F F

�

�

F F

�

�

+++

+++

F

F

+++

+++

F

F

+++

+++

F

F

�

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+++

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+++

+++

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+++

+++

F F

�

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�

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F F

�

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+++

+++

F

F

+++

+++

F

F �

�

+++

+++

S. Kopecki DNA Tile Self-Assembly 17 / 23

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

�

�

+++

+++ F F

�

�

+++

+++

F

F

�

�

+++

+++ F F

�

�

F F

�

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+++

+++

F

F

+++

+++

F

F

+++

+++

F

F

�

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S. Kopecki DNA Tile Self-Assembly 17 / 23

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.

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S. Kopecki DNA Tile Self-Assembly 17 / 23

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

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

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

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

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

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

Table of Contents

S. Kopecki DNA Tile Self-Assembly 19 / 23

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Logic-Gated Nanorobot for Targeted TransportDouglas, Bachelet, Church (2012)

S. Kopecki DNA Tile Self-Assembly 23 / 23