Staged Self-Assembly and Polyomino Context-Free Grammars

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Staged Self-Assembly and Polyomino Context-Free Grammars

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Staged Self-Assembly and Polyomino Context-Free Grammars

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Staged Self-Assembly andPolyomino Context-Free Grammars

Andrew Winslow

Defense Edition

Simple particles coalescing into

complex superstructures.

Simple particles coalescing into

complex superstructures.

Simple particles coalescing into

complex superstructures.

Simple particles coalescing into

complex superstructures.

Crystallization

Morphogenesis

Natural Self-Assembly

Synthetic Self-Assembly with DNA

Tile

Glues

Study an abstract self-assembly model

based on DNA tile systems.

Tile

Glues

Assembly

Mixing

Bin

Reagents

Products

Size

Staged self-assembly

Input

Output

(polyomino)

(small staged assembly system)

Input

n

Θ(n) system size

Θ(log(n)) system size

n

Θ(n)

Θ(log(n))

Input

Finding small systems is (NP-)hard.

Study a simplified model.

Rule

Grammar

Ο(1)

? ? ?

? ? ?

Ο(#glues2)

Ω(n1/2/log(n)), Ο(n2/3)

Staged System

Grammar

E. D. Demaine, S. Eisenstat, M. Ishaque, A. Winslow, One-dimensional staged self-assembly, DNA 17, 2011.

? ? ? ?

Polyomino Grammar

Ο(log(n))

w/scaling

Macrotile

…

…

Ω(n/log3n)

Grammar

Staged System

A. Winslow, Staged Self-Assembly and Polyomino Context-Free Grammars, DNA 19, 2013.

?

?

?

?

?

?

Values

Values are contiguous

Activation

Activation

? ? ?

? ? ?

? ? ?

[1..3] +4

Shifting

[4..6] +4

Shifting

[1..3] +4

Shifting

Activation

? ? ?

[2..5] +3

? ? ?

[1..4] +2

? ? ?

[1..4] +4

Product

Activated reagent 1

Activated & shifted reagent 2

Shifted reagent 2

Reagent 1

Reagent 2

?

?

?

?

?

?

A R1

A & S R2

S R2

A R1

A & S R2

S R2

A R1

A & S R2

S R2

A R1

A R1

A R1

A & S R2

A & S R2

A & S R2

S R2

S R2

S R2

Ο(log(n))

w/scaling

…

…

Ω(n/log3n)

…

…

Ω(n/loglog(n))

Ο(loglog(n))

…

…

Ω(n/log2n)

Ο(log(n))

…

…

Ω(n/log2n)

Ο(log(n))

Grammar

Staged System

Grammars are small,

staged systems are always smaller

Ο(log(n))

Ο(1)

Ω(n1/2/log(n)), Ο(n2/3)

Ω(n/log3n)

w/scaling

Committee:

Collaborators:

Lenore Cowen

Sarah Eisenstat

Erik Demaine

MashhoodIshaque

Ben Hescott

Matthew Patitz

Diane Souvaine

Robert Schweller

Hyunmin Yi

Damien Woods

NSF grants 0941538 and 0830734