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

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Simple particles coalescing into complex superstructures. Self-Assembly

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Simple particles coalescing into complex superstructures.

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Self-Assembly Simple particles coalescing into complex superstructures.

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Self-Assembly Simple particles coalescing into complex superstructures.

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Crystallization

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Morphogenesis

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Natural Self-Assembly

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Synthetic Self-Assembly with DNA

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Tile

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Glues

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Our work: Study an abstract self-assembly model based on DNA tile systems.

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Tile

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Glues

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Assembly

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Mixing

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Bin

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Reagents

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Products

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Size

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Staged self-assembly

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InputOutput (polyomino) (small staged assembly system)

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n Input

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Θ( n) system size

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Θ (log(n)) system size

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n Θ (n) Θ (log(n))

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Input

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Study a simplified model. Finding small systems is (NP-)hard.

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Rule

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Grammar

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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, 2011. Grammar ? ? ?

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? ?

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Polyomino Grammar

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w/scaling Two dimensions GrammarStaged System … … A. Winslow, Staged Self-Assembly and Polyomino Context-Free Grammars, DNA 19, 2013. Ω (n/log 3 n) Ο (log(n)) Macrotile

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? ? ? ? ? ?

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Values

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Values are distinct Values are contiguous

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Activation

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? ? ?

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Shifting [1..3] +4

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Shifting [4..6] +4

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Activation Shifting [1..3] +4

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? ? ?

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[2..5] +3

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? ? ?

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[1..4] +2

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? ? ?

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[1..4] +4

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Reagent 1 Reagent 2 Product Activated reagent 1 Shifted reagent 2 Activated & shifted reagent 2

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? ? ? ? ? ?

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A R1 S R2 A & S R2 A R1 S R2 A & S R2 A R1 S R2 A & S R2

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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))

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… … Ω (n/log 3 n)

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Ο (loglog(n)) … Ω (n/loglog(n)) …

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… … Ο (log(n)) Ω (n/log 2 n)

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Ο (log(n)) … Ω (n/log 2 n) …

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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

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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 0941538 and 0830734

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An Introduction to Algorithmic Tile Self-Assembly.

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