Staged Self-Assembly and
Polyomino Context-Free Grammars
Andrew Winslow
Tile
Glues
Assembly
×∞
×∞ ×∞
Tile set
×∞
Mixing
Bin
Reagents
Products
Size
Staged self-assembly
Three glue types for any shape!
n
How to assemble n-tile stick?
n
≈n size for n-tile stick
n
≈log(n) size for n-tile stick
n
≈n
≈log(n)
Finding small systems is difficult.
Study a simplified model.
Rule
Grammar
One dimension
? ? ?
Grammar
6×
n0.5×2×
#glues
? ? ?
Staged System
E. D. Demaine, S. Eisenstat, M. Ishaque, A. Winslow, One-dimensional staged self-assembly, DNA 17, 2011.
?
?
?
?
Grammar
Two dimensions
log(n)×
w/scaling
? ? ?
Grammar
≈n×
? ? ?
Staged System
…
…
at least ≈n
at most ≈log(n)
…
…
at least ≈n
at most ≈log(n)
…
…
at least ≈n
at most ≈log(n)
Grammar
Staged System
6×
DNA
17
0.5
n ×
log(n)×
w/scaling
This work
n×
Grammar
Staged System
6×
n0.5×
Grammars are small,
staged systems are always smaller
log(n)×
w/scaling
≈n×
Thank you