Self-assemblying Classes of Shapes with a Minimum Number of Tiles, and in Optimal Time

Becker, Florent; Rapaport, Ivan; Rémila, Éric

doi:10.1007/11944836_7

Cited by 42 publications

(59 citation statements)

References 7 publications

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“…This result implies that we can simulate arbitrary tile systems that assemble a single line. From [6], we know that the insertion system can construct lines of arbitrary expected length with O(1) monomers. THEOREM 3.1.…”

Section: Expressive Powermentioning

confidence: 99%

Active Self-Assembly of Simple Units Using an Insertion Primitive

Dabby¹,

Chen²

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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While computer science has given us a framework for determining the complexity and difficulty of solving computational problems, we do not yet have a theoretical framework for knowing what actions, behaviors, and life-like qualities can emerge from a given set of simple modular units. There has been much interest in developing models for programming active self-assembly processes in both the reconfigurable robotics community and the nanotechnology community. With respect to materials science and nanotechnology, the models proposed to date are either not yet implementable with our current understanding of synthetic chemistry or those that are implementable are limited to a set of features that do not capture the power of active components. Prior implementable models of molecular assembly only considered the passive behaviors of attaching and detaching from a complex.Inspired by the algorithmic tile assembly model [Winfree, 1996] and the graph grammar assembly model [Klavins et al., 2004], we describe a formal model for studying the complexity of self-assembled structures with active molecular components. In particular, we add an insertion primitive and we show a direct mapping of our model to a molecular implementation using DNA. We show that the expressive power of this language is stronger than regular languages, but at most as strong as context free grammars. Here, we explore the trade-off between the complexity of the system (in terms of the number of unit types), and the behavior of the system and speed of its assembly. We find that we can grow a line of any given length n in expected time O(log 3 n) using O(log 2 n) monomers. If we grow a line with k insertion rules, either the expected final length is infinite or the expected length at time t is at most (2t + 2) k 2 , which is polynomial in t.

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Section: Expressive Powermentioning

confidence: 99%

Active Self-Assembly of Simple Units Using an Insertion Primitive

Dabby¹,

Chen²

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…In particular, we would like the assembly time model proposed in [3] to be derived as a special case of the model we propose, when only single-tile reactions with the seed-containing assembly are allowed. 9 With a model such as Gillespie's algorithm [27][28][29] using finite molecular counts, it is possible that no copy of the terminal assembly forms, so it is not clear how to sensibly ask how long it takes to form. 10 The mass-action model of kinetics [23] describes concentrations as a dynamical system that evolves continuously over time according to ordinary differential equations derived from reaction rates.…”

Section: Issues With Defining Hierarchical Time Complexitymentioning

confidence: 99%

“…However, rather than fixing transition rates at each time t ∈ R ≥0 as constant, 8 The fact that some directed systems may not require at least one of these attachments to happen in every terminal assembly tree is the reason we impose the partial order requirement when proving our time complexity lower bound. 9 As discussed in Section 1, the model of [3] is not exactly a special case of our model, since we assume tile concentrations deplete. However, the assumption of constant tile concentrations is itself a simplifying assumption of [3] that is approximated by a more realistic model in which tile concentrations deplete, but seed tile types have very low concentration compared to other tile types, implying that non-seed concentrations do not deplete too much.…”

Section: Issues With Defining Hierarchical Time Complexitymentioning

confidence: 99%

“…They showed that for most n, the problem of assembling an n × n square has tile complexity Ω( log n log log n ), and Adleman, Cheng, Goel, and Huang [3] exhibited a construction showing that this lower bound is asymptotically tight. Under natural generalizations of the model [1,6,9,13,16,18,19,33,34,37,50,51], tile complexity can be reduced for tasks such as square-building and assembly of more general shapes. See [20,39,58] for more background.…”

Section: Introductionmentioning

confidence: 99%

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Parallelism and Time in Hierarchical Self-Assembly

Chen¹,

Doty²

2017

SIAM J. Comput.

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We study the role that parallelism plays in time complexity of variants of Winfree's abstract Tile Assembly Model (aTAM), a model of molecular algorithmic self-assembly. In the "hierarchical" aTAM, two assemblies, both consisting of multiple tiles, are allowed to aggregate together, whereas in the "seeded" aTAM, tiles attach one at a time to a growing assembly. Adleman, Cheng, Goel, and Huang (Running Time and Program Size for Self-Assembled Squares, STOC 2001) showed how to assemble an n × n square in O(n) time in the seeded aTAM using O( log n log log n ) unique tile types, where both of these parameters are optimal. They asked whether the hierarchical aTAM could allow a tile system to use the ability to form large assemblies in parallel before they attach to break the Ω(n) lower bound for assembly time. We show that there is a tile system with the optimal O( log n log log n ) tile types that assembles an n× n square using O(log 2 n) parallel "stages", which is close to the optimal Ω(log n) stages, forming the final n × n square from four n/2 × n/2 squares, which are themselves recursively formed from n/4 × n/4 squares, etc. However, despite this nearly maximal parallelism, the system requires superlinear time to assemble the square. We extend the definition of partial order tile systems studied by Adleman et al. in a natural way to hierarchical assembly and show that no hierarchical partial order tile system can build any shape with diameter D in less than time Ω(D), demonstrating that in this case the hierarchical model affords no speedup whatsoever over the seeded model. We also strengthen the Ω(D) time lower bound for deterministic seeded systems of Adleman et al. to nondeterministic seeded systems. Finally, we show that for infinitely many n, a tile system can assemble an n × n ′ rectangle, with n > n ′ , in time O(n 4/5 log n), breaking the linear-time lower bound that applies to all seeded systems and partial order hierarchical systems.

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“…They showed that for most n, the problem of assembling an n × n square has tile complexity Ω( log n log log n ), and Adleman, Cheng, Goel, and Huang [3] exhibited a construction showing that this lower bound is asymptotically tight. Under natural generalizations of the model [1,6,8,[10][11][12][13]25,26,29,36,37], tile complexity can be reduced for the assembly of squares and more general shapes.…”

Section: Introductionmentioning

confidence: 99%

Program Size and Temperature in Self-Assembly

2014

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Winfree's abstract Tile Assembly Model (aTAM) is a model of molecular self-assembly of DNA complexes known as tiles, which float freely in solution and attach one at a time to a growing "seed" assembly based on specific binding sites on their four sides. We show that there is a polynomial-time algorithm that, given an n × n square, finds the minimal tile system (i.e., the system with the smallest number of distinct tile types) that uniquely self-assembles the square, answering an open question of Adleman, Cheng, Goel, Huang, Kempe, Moisset de Espanés, and Rothemund (Combinatorial Optimization Problems in Self-Assembly, STOC 2002). Our investigation leading to this algorithm reveals other positive and negative results about the relationship between the size of a tile system and its "temperature" (the binding strength threshold required for a tile to attach).

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Self-assemblying Classes of Shapes with a Minimum Number of Tiles, and in Optimal Time

Cited by 42 publications

References 7 publications

Active Self-Assembly of Simple Units Using an Insertion Primitive

Active Self-Assembly of Simple Units Using an Insertion Primitive

Parallelism and Time in Hierarchical Self-Assembly

Program Size and Temperature in Self-Assembly

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