Staged Self-Assembly and Polyomino Context-Free Grammars Andrew Winslow Defense Edition.

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Presentation transcript:

Staged Self-Assembly and Polyomino Context-Free Grammars Andrew Winslow Defense Edition

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

InputOutput (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 (#glues 2 ) Ω (n 1/2 /log(n)), Ο (n 2/3 ) Staged System ? ? ? One dimension Ο (1) E. D. Demaine, S. Eisenstat, M. Ishaque, A. Winslow, One-dimensional staged self-assembly, DNA 17, Grammar ? ? ?

? ?

Polyomino Grammar

w/scaling Two dimensions GrammarStaged System … … A. Winslow, Staged Self-Assembly and Polyomino Context-Free Grammars, DNA 19, Ω (n/log 3 n) Ο (log(n)) Macrotile

? ? ? ? ? ?

Values

Values are distinct Values are contiguous

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/scaling Ο (log(n))

… … Ω (n/log 3 n)

Ο (loglog(n)) … Ω (n/loglog(n)) …

… … Ο (log(n)) Ω (n/log 2 n)

Ο (log(n)) … Ω (n/log 2 n) …

GrammarStaged System Ο (1) Ω (n 1/2 /log(n)), Ο (n 2/3 ) w/scaling Ο (log(n)) Ω (n/log 3 n) Grammars are small, staged systems are always smaller

Thank you Committee: Diane Souvaine Erik Demaine Ben Hescott Lenore Cowen Hyunmin Yi Collaborators: Matthew Patitz Sarah Eisenstat Robert Schweller Mashhood Ishaque Damien Woods NSF grants and