Self-Assembly Model with Time
Dependent Glue Strength
Sudheer Sahu
Peng Yin
John Reif
Department of Computer Science
Duke University
Roadmap
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Background
Time-dependent glue model
Implementation
Catalysis and self-replication
Tile-complexity
Background
• Self-assembly:
–Fundamental process in nature
–Recent uses in constructions and computations
–Tile self-assembly
•1960s (Wang tiling etc.)
•Recent times (Winfree, Adleman etc.)
Winfree’s Model
Glue
Tile
Σ
Σ4
Strength function : g: Σ x Σ → R
g(σ,σ’) > 0 if σ = σ’
= 0 otherwise
Temperature 
A tile can be added to the
aggregate only if the gluestrength of new bonds formed
is more than .
Tile-system (T,S,g,)
Toy example
Given Tile Set
Assembly of Square
Temperature = 2
Our Model
• Time Dependent Glue Model:
– Glue strength depends on interaction time of
tiles
• Results:
– Catalysis
– Self-replication
– Reduction in tile-complexity
Time Dependent Glue Model
• Glue strength increases monotonically before
becoming constant
• Glue strength function
g : R  R
g ( ,  ' , t )
• Time for maximum strength
tms :     R
• Minimum interaction time
mit :     R
Strand Displacement as Random Walk
Implementation
A
B
s1
s2
s3
Implementation
A
s1
s2
s3
B
Implementation
A
s1
B
s2
s3
Implementation
A
s2
s3
B
Implementation
A
B
s2
s3
Implementation
B
A
s3
Implementation
B
A
s3
Implementation
A
B
How Glue Strength Varies with Time
Catalysis
Self-Replication
• A-B acts as a catalyst for formation of C-D which
in turn acts as a catalyst for the formation of A-B
• Conditions:
Self-Replication
A-B
• Two states:
– Dormant state
– Replicating state
• Exponential growth
A,B,
C,D
dormant
C-D as
catalyst
for A-B
A-B as
catalyst
for C-D
C-D
replicating
• Low probability of going from dormant to
replicating state
Tile Complexity
• Tile Complexity:
– Minimum number of distinct tile types required
to construct a shape uniquely.
• In standard model, tile complexity of an n  n
log n
square is (
)
log log n
[Adleman01 ]
Generalized Models
• Multi-temperature Model
– Thin rectangle
log n
O(
) [Aggarwal04]
log log n
• Flexible Glue Model
– Square
O( log n )
[Aggarwal04]
Tile Complexity Results
Thin Rectangle( k  N rectangle)
N 1/ k
(
) [Standard M odel, Aggarwal04 ]
k
log N
O(
) [Time Dependent Glue Strength]
log log N
k
N
N  N Square with square hole of size k  k in center
( k
2
N k
) [Standard M odel]
log N
O(
) [Time Dependent Glue M odel]
log log N
N
N
Rectangle
• Construct a k x mk rectangle using
O(k+m) type of tiles.
– Base m counter
of k-digits
Construction of thin rectangles
• Thin Rectangle: k x n for k <
log n
log log n  log log log n
• Construct a j x n rectangle using O(j+n1/j )
type of tiles, where j > k.
• The glue of bottom k rows become strong
after mit, and the glue of top j-k rows
(volatile rows) do not.
Decreasing it further
We constructe d a k  n rectangle using O( j  n )
1/ j
type of tiles.
For j 
log n
log log n  log log log n
log n
1/ j
O( j  n )  O(
)
log log n
,
Shapes with holes
nxn square with a hole of k x k in center
• Lower Bound in standard model:
(k
2 /( n  k )
)
• Upper Bound in our model:
log k
O(
)
log log k
N
N
Lower Bound
• Proof by contradiction:
– Assume fewer than k2/(n-k) tile-types required
– Divide into regions s.t. seed tile is in longer rectangle
– Number of different possible rows=2k
–Two or more rows that are identical rows
Upper Bound
• Grow four different rectangles
Size (
O(
nk 2
)  ( k  2)
2
log k
)
log log k
Connector Tiles
Filler Tiles
Discussion and Future Work
• Kinetic analysis of catalysis and selfreplication
– Theoretical analysis is hard
• the rate-constant changes assuming rate
proportional to exp(-bond strength)
– Computer program
• Experiments or simulation
• Tile-complexities for more shapes
THANKS!!
Concept of Minimum Interaction Time
f (t )  probabilit y that no dissociati on took place in time t
g (t )  glue strength function w rt t
p (b)  probabilit y of dissociati on when glue strength is b
f (t  t )  f (t )  (1  p( g (t  t )))  t
f (t  t )
 (1  p( g (t  t )))  t
f (t )
E[mit ]  lim
t 0

 (t  t )  f (t )  p( g (t  t ))  t
t 0