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Producibility in hierarchical self-assembly

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Abstract

Three results are shown on producibility in the hierarchical model of tile self-assembly. It is shown that a simple greedy polynomial-time strategy decides whether an assembly α is producible. The algorithm can be optimized to use \(O(|\alpha | \log ^2 |\alpha |)\) time. Cannon et al. (STACS 2013: proceedings of the thirtieth international symposium on theoretical aspects of computer science. pp 172–184, 2013) showed that the problem of deciding if an assembly α is the unique producible terminal assembly of a tile system \({\mathcal {T}}\) can be solved in \(O(|\alpha |^2 |{\mathcal {T}}| + |\alpha | |{\mathcal {T}}|^2)\) time for the special case of noncooperative “temperature 1” systems. It is shown that this can be improved to \(O(|\alpha | |{\mathcal {T}}| \log |{\mathcal {T}}|)\) time. Finally, it is shown that if two assemblies are producible, and if they can be overlapped consistently—i.e., if the positions that they share have the same tile type in each assembly—then their union is also producible .

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Notes

  1. This assumption does not affect the results of this paper. It is irrelevant for Theorem 5.1 or the correctness of the algorithms in the other theorems. It also does not affect the running time results for algorithms taking a TAS as input, because we can preprocess T in linear time to find and set to null any functionally null glues. The number of glues is O(|T|), and we assume that each glue from glue set G is an integer in the set \(\{0,\ldots ,|G|-1\}\). We can use a Boolean array of size |G| to determine in time O(|T|) which glues appear on the north that do not appear on the south of some tile type. Repeat this for each of the remaining three directions. Then replace all functionally null glues in T with null glues, which takes time O(|T|). To do this replacement in an assembly \(\alpha \) takes time \(O(|\alpha |)\).

  2. For \(G^{\text{f}}_{{\text{dom}} \;\alpha }=(V_{{\text{dom}} \;\alpha },E_{{\text{dom}} \;\alpha })\) and \(G^{\text{b}}_\alpha =(V_\alpha ,E_\alpha )\), \(G^{\text{b}}_\alpha \) is a spanning subgraph of \(G^{\text{f}}_{{\text{dom}} \;\alpha }\): \(V_\alpha = V_{{\text{dom}} \;\alpha }\) and \(E_\alpha \subseteq E_{{\text{dom}} \;\alpha }\).

  3. Intuitively \(\alpha \rightarrow _1^{\mathcal {T}} \beta \) means that \(\alpha \) can grow into \(\beta \) by the addition of a single tile; the fact that we require both \(\alpha \) and \(\beta \) to be \(\tau \)-stable implies in particular that the new tile is able to bind to \(\alpha \) with strength at least \(\tau \). It is easy to check that had we instead required only \(\alpha \) to be \(\tau \)-stable, and required that the cut of \(\beta \) separating \(\alpha \) from the new tile has strength at least \(\tau \), then this implies that \(\beta \) is also \(\tau \)-stable.

  4. The following two convenient characterizations of “directed” are routine to verify. \({\mathcal {T}}\) is directed if and only if \(|\mathcal {A}_{\Box }[{\mathcal {T}}]| = 1\). \({\mathcal {T}}\) is not directed if and only if there exist \(\alpha ,\beta \in \mathcal {A}[{\mathcal {T}}]\) and \(p \in {\text{dom}} \;\alpha \cap {\text{dom}} \;\beta \) such that \(\alpha (p) \ne \beta (p)\).

  5. The restriction on overlap is a model of a chemical phenomenon known as steric hindrance (Wade 1991, Section 5.11) or, particularly when employed as a design tool for intentional prevention of unwanted binding in synthesized molecules, steric protection (Heller and Pugh 1954, 1960; Goto et al. 2000).

  6. We do not need to give the tile set T as input because the tiles in \(\alpha \) implicitly define a tile set, and the presence of extra tile types in T that do not appear in \(\alpha \) cannot affect its producibility.

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Acknowledgments

The author is very grateful to Ho-Lin Chen, David Soloveichik, Damien Woods, Matt Patitz, Scott Summers, Robbie Schweller, Ján Maňuch, Ladislav Stacho, Andrew Winslow for many insightful discussions, and to anonymous reviewers for their detailed and useful comments. The author was supported by NSF Grants CCF-1219274 and CCF-1162589 and the Molecular Programming Project under NSF Grants 0832824 and 1317694 and by a Computing Innovation Fellowship under NSF Grant 1019343.

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  1. California Institute of Technology, Pasadena, CA, USA

    David Doty

  2. University of California, Davis, CA, USA

    David Doty

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Doty, D. Producibility in hierarchical self-assembly. Nat Comput 15, 41–49 (2016). https://doi.org/10.1007/s11047-015-9517-2

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