The Power of Nondeterminism in Self-Assembly

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The Power of Nondeterminism in Self-Assembly

• Nathaniel Bryans, Ehsan Chiniforooshan, David
  Doty, Lila Kari, Shinnosuke Seki
• Department of Computer Science, University of
  Western Ontario, London, ON, N6A 5B7, Canada
• Department of Computer Science, California
  Institute of Technology, Pasadena, CA 91125, USA

Will be presented _at_ ACM-SIAM Symposium on
Discrete Algorithms (SODA11), San Francisco,
California, USA, January 23-25, 2011.
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Tile Self-Assembly

• In a well-mixed chemical soup, randomly-floating
  molecules tend to interact with each other so as
  to assemble information-processing biowares
  (programmable crystal growth).
• Tile self-assembly is an algorithmically rich
  model of this phenomena, in which tiles, which
  are one of the finite types, self-assemble
• a unique target shape
• due to only their local interactions (without any
  externally controlling device)

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Practical Implementation

• Self-assembling molecular tiles based on DNA
  complexes were experimentally implemented in 1982
  by Seeman Seeman 1982
• Experimental advances for reliable assembly with
  error rates
• 10 per tile Rothemund, Papadakis, Winfree 2004
• 1.4 per tile Fujibayashi et al. 2007
• 0.13 per tile Barish et al. 2009

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Abstract Tile-Assembly Model (aTAM)

• Introduced by E. Winfree Winfree 1998
• Wang tile the notion of time (growth)
• A simplified mathematical model of
  self-assembling DNA tiles
• Computationally universal
• Various resource bounds
• Tile complexity (min of tile types required to
  assemble a shape)

Discrete Sierpinski Triangle
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Abstract Tile-Assembly System
A tile assembly system (TAS) is a triple (T, s,
t), where

• T is a finite set of tile types
• A tile type is a tuple t ?(SN)4, i.e., a square
  tile each of whose 4 sides has a glue consisting
  of
• a finite string (label),
• a non-negative integer (strength).
• A tile cannot rotate.
• An infinite of copies of each tile type is
  available for computations, each copy referred to
  as a tile.
• s is a seed-assembly
• t is the temperature

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Growing process of TASs

• Given T, an assembly is a partial function
• The shape of a is dom a
• Lets consider the TAS shown right with 7 tile
  types working _at_ t 2.
• Starting from the seed placed on (0, 0).
• A tile can attach to an assembly via its edge(s)
  if the sum of binding strengths associated with
  these abutting edges is at least the temperaturet.

example binary counter _at_ t 2
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Determinism in TASs

• A TAS is directed (a.k.a. deterministic) if it
  has exactly one terminal assembly
• Any assembly process by the TAS reaches the same
  assembly.
• A TAS strictly self-assembles a shape S if all of
  its terminal assemblies are of the shape S.

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Main Problems

1. Is there any (infinite) shape which can be
   self-assembled by a non-deterministic TAS but
   cannot be by any deterministic one?
2. Does non-determinism decrease the tile complexity
   of some (finite) shape?
3. How difficult to exploit the non-determinism to
   design the smallest (and hence most economical)
   TAS for a given shape?

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(Directed) Tile Complexity of a Shape

• Tile complexity of a shape S is a measure of how
  complex the shape S is w.r.t. TAS defined as
• The directed tile complexity of S is
• Ctc(S) Cdtc(S).
• If S is finite, then Cdtc(S) dom S
  (hard-coding).

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Assembly of Infinite Shapes

• Theorem 1. There is a shape S such that some TAS
  strictly self-assembles S, but no directed one
  can.

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• Proof idea.
• Let and M be a TM for L.
• The simulation of M on the input n is carried out
  adjacent to the n-th ray.
• For a roughly quadratic function f, a special
  tile is placed at (f(n), 0) for all n (circled
  points in the figure).
• To the point (f(n), 0), a vertical pillar is
  extended from above iff n is in L.
• This vertical pillar gets longer as n increases.
  Thus, for any directed TAS, there exists a
  sufficiently-long vertical pillar on which this
  TAS has to put two tiles of the same type.

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Assembly of Finite Shapes

• Theorem 2. There is a finite shape S s.t. Ctc(S)
  lt Cdtc(S).

proof outline. We can construct a TAS (T, s, 1)
which strictly self-assembles the right shape
with T 2h16 as green tiles for A and red
tiles for B can be reused to assemble the loop L
(non-deterministic choice is made at the
top-middle point of L). Since any directed TAS
which assembles this shape has to avoid this
non-deterministic choice, it has to have h1
tiles which are neither green nor red tiles to
assemble either the left or right pillars of L.
Thus, Ctc(S) 2h16 and 3h17 Cdtc(S).
?
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Minimum Tile Set Problem

• It is of significant practical interest to find
  the smallest TAS which strictly self-assembles
  an objective shape.
• The minimum tile set problem is
• The minimum directed tile set problem is
• MINDIRECTEDTILESET was proved to be NP-complete
  Adleman et al. 2002.
• We will prove that MINTILESET is -complete.

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-completeness of MINTILESET

• Theorem 3. MINTILESET is -complete.

MINTILESET being in Observe that the
following verification language is in P Then
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-completeness of MINTILESET
2. Hardness of MINTILESET in We show that

, where This language is -complete.
Our strategy is similar to the reduction of
3SAT to MINDIRECTEDTILESET shown in Adleman et
al. 2002.
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Bibliography

• Adleman et al. 2002
• L. M. Adleman, Q. Cheng, A. Goel, M-D. A. Huang,
  D. Kempe, P. M. de Espanes, and P. W. K.
  Rothemund. Combinatorial optimization problems in
  self-assembly. In STOC 2002 Proceedings of the
  34th Annual ACM Symposium on Theory of Computing,
  pages 23-32, 2002.
• Barish et al. 2009
• R. D. Barish, R. Schulman, P. W. K. Rothemund,
  and E. Winfree. An information-bearing seed for
  nucleating algorithmic self-assembly. Proceedings
  of the National Academy of Sciences,
  106(15)6054-6059, 2009.
• Fujibayashi et al. 2007
• K. Fujibayashi, R. Hariadi, S. H. Park, E.
  Winfree, and S. Murata. Toward reliable
  algorithmic self-assembly of DNA tiles A
  fixed-width cellular automaton pattern. Nano
  Letters, 8(7)1791-1797, 2007.

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Bibliography

• Rothemund, Papadakis, Winfree 2004
• P. W. K. Rothemund, N. Papadakis, and E.
  Winfree. Algorithmic self-assembly of DNA
  Sierpinski triangles. PLoS Biology,
  2(12)2041-2053, 2004.
• Seeman 1982 N. Seeman. Nucleic-acid junctions
  and lattices. Journal of Theoretical Biology,
  99237-247, 1982.
• Winfree 1998 E. Winfree. Algorithmic
  Self-Assembly of DNA. Ph.D. thesis, California
  Institute of Technology, June 1998.