Staged self assembly and polyomino context free grammars
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Staged Self-Assembly and Polyomino Context-Free Grammars. Andrew Winslow. Defense Edition. Self-Assembly. S imple particles coalescing into complex superstructures. Self-Assembly. S imple particles coalescing into complex superstructures. Self-Assembly.

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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


Self-Assembly

Simple particles coalescing into

complex superstructures.


Self-Assembly

Simple particles coalescing into

complex superstructures.


Self-Assembly

Simple particles coalescing into

complex superstructures.


Self-Assembly

Simple particles coalescing into

complex superstructures.


Crystallization


Morphogenesis


Natural Self-Assembly


Synthetic Self-Assembly with DNA


Tile


Glues


Our work:

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


One dimension

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


Two dimensions

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 distinct

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


Thank you

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


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