Staged Self-Assembly andPolyomino Context-Free

GrammarsAndrew Winslow

Tile

Glues

Assembly

×∞ ×∞ ×∞ ×∞Tile set

Mixing

Bin

Reagents

Products

Size

Staged self-assembly

Three glue types for any shape!

How to assemble n-tile stick?

n

n≈n size for n-tile stick

n ≈log(n) size for n-tile stick

n

≈n ≈log(n)

Study a simplified model.

Finding small systems is difficult.

Rule

Grammar

#glues2×

Staged System

? ? ?

One dimension

6×

n0.5×

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

Grammar

? ? ?

? ? ? ?

Grammar

Two dimensions

Grammar Staged System

≈n×

log(n)×

? ? ?? ? ?w/scaling

at most ≈log(n)

…

at least ≈n

…

at most ≈log(n)

…

at least ≈n

…

at most ≈log(n)

…

at least ≈n

…

Grammar Staged System

log(n)×

n×

6×

n0.5×

w/scalingThis work

DNA 17

Grammar Staged System

log(n)×

≈n×

6×

n0.5×

w/scaling

Grammars are small,staged systems are always

smaller

Thank you