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The Power of Nondeterminism in Self-Assembly

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Title: The Power of Nondeterminism in Self-Assembly


1
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.
2
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)

3
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

4
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
5
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

6
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
7
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.

8
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?

9
(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).

10
Assembly of Infinite Shapes
  • Theorem 1. There is a shape S such that some TAS
    strictly self-assembles S, but no directed one
    can.

11
  • 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.

12
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).
?
13
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.

14
-completeness of MINTILESET
  • Theorem 3. MINTILESET is -complete.

MINTILESET being in Observe that the
following verification language is in P Then
15
-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.
16
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.

17
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.
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