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 @ ACM-SIAM Symposium on Discrete Algorithms (SODA11), San Francisco, California, USA, January 23-25, 2011.

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)

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]

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

Abstract Tile-Assembly System A tile assembly system (TAS) is a triple (T, s, τ), where T is a finite set of tile types: A tile type is a tuple t ∈(Σ*×N)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 τ is the temperature N, 1 E, 3 S, 2 W, 1

Growing process of TASs Given T, an assembly is a partial function The shape of α is dom α Let’s consider the TAS shown right with 7 tile types working @ τ= 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 temperatureτ. example: binary counter @ τ= 2 strength 1 stre ngth 1 cooperation s

Determinism in TASs S 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. S

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

(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).

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

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. Simulation of M(1) Simulation of M(2) Non-determinism is necessary (f(n), 0)

Assembly of Finite Shapes Theorem 2. There is a finite shape S s.t. Ctc(S) < Cdtc(S). proof outline. We can construct a TAS (T, σ, 1) which strictly self-assembles the right shape with |T| = 2h+16 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 h+1 tiles which are neither green nor red tiles to assemble either the left or right pillars of L. Thus, Ctc(S) ≤ 2h+16 and 3h+17 ≤ Cdtc(S). 

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.

-completeness of MINTILESET Theorem 3. MINTILESET is -complete. MINTILESET being in : Observe that the following verification language is in P: Then

-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].

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.

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, 99:237-247, 1982. [Winfree 1998] E. Winfree. Algorithmic Self-Assembly of DNA. Ph.D. thesis, California Institute of Technology, June 1998.