Staged Self-Assembly andPolyomino Context-Free

GrammarsAndrew Winslow

Defense Edition

Simple particles coalescing into complex superstructures.

Self-Assembly

Self-AssemblySimple particles coalescing into

complex superstructures.

Self-AssemblySimple particles coalescing into

complex superstructures.

Self-AssemblySimple 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)

n

Input

Θ(n) system size

Θ(log(n)) system size

n

Θ(n) Θ(log(n))

Input

Study a simplified model.

Finding small systems is (NP-)hard.

Rule

Grammar

Ο(#glues2)Ω(n1/2/log(n)), Ο(n2/3)

Staged System

? ? ?

One dimension

Ο(1)

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

Grammar

? ? ?

? ? ? ?

Polyomino Grammar

w/scaling

Two dimensions

Grammar Staged System

……

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

Ω(n/log3n)

Ο(log(n))

Macrotile

? ?

??

? ?

Values

Values are distinct

Values are contiguous

Activation

Activation

? ? ?

? ? ?

? ? ?

Shifting

[1..3] +4

Shifting

[4..6] +4

Activation

Shifting

[1..3] +4

? ? ?

[2..5] +3

? ? ?

[1..4] +2

? ? ?

[1..4] +4

Reagent 1

Reagent 2

Product

Activated reagent 1

Shifted reagent 2

Activated & shifted reagent 2

? ?

??

? ?

A R1

S R2

A & S R2

A R1

S R2

A & S R2

A R1

S R2

A & S R2

A R1

S R2

A & S R2

A R1

S R2

A & S R2

A R1

S R2

A & S R2

w/scalingΟ(log(n))

…… Ω(n/log3n)

Ο(loglog(n))

…

Ω(n/loglog(n))

…

……

Ο(log(n))Ω(n/log2n)

Ο(log(n))

…

Ω(n/log2n)

…

Grammar Staged System Ο(1)

Ω(n1/2/log(n)), Ο(n2/3)

w/scaling

Ο(log(n))

Ω(n/log3n)

Grammars are small,staged systems are always

smaller

Thank youCommittee:

Diane Souvaine

Erik DemaineBen Hescott

Lenore Cowen

Hyunmin Yi

Collaborators:

Matthew Patitz

Sarah Eisenstat

Robert Schweller

Mashhood Ishaque

Damien Woods

NSF grants 0941538 and 0830734