Resiliency to multiple nucleation in temperature-1 self-assembly

  • Matthew J. Patitz
  • Robert Schweller
  • Trent A. Rogers
  • Scott M. Summers
  • Andrew Winslow
Article
  • 337 Downloads

Abstract

We consider problems in variations of the two-handed abstract Tile Assembly Model (2HAM), a generalization of Erik Winfree’s abstract Tile Assembly Model (aTAM). In the latter, tiles attach one-at-a-time to a seed-containing assembly. In the former, tiles aggregate into supertiles that then further combine to form larger supertiles; hence, constructions must be robust to the choice of seed (nucleation) tiles. We obtain three distinct temperature-1 results in two 2HAM variants whose aTAM siblings are well-studied. In the first variant, called the restricted glue 2HAM (rg2HAM), glue strengths are restricted to \(-\,1\), 0, or 1. We prove this model is Turing universal, overcoming undesired growth by breaking apart undesired computation assembly via repulsive forces. In the second 2HAM variant, the 3D 2HAM (3D2HAM), tiles are (three-dimensional) cubes. We prove that assembling a (roughly two-layer) \(n \times n\) square in this model at temperature 1 is possible with \(O(\log ^2{n})\) tile types. The construction uses “cyclic, colliding” binary counters, and assembles the shape non-deterministically. Finally, we prove that there exist 3D2HAM systems that only assemble infinite aperiodic shapes.

Keywords

DNA computing Biocomputing 2HAM 

Notes

Acknowledgements

Matthew J. Patitz: This author’s research was supported in part by National Science Foundation Grants CCF-1117672, CCF-1422152, and CAREER-1553166. Robert Schweller: This author’s research was supported in part by National Science Foundation Grants CCF-1117672 and CCF-1555626. Trent A. Rogers: This author’s research was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1450079, and National Science Foundation Grant CCF-1422152

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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  • Matthew J. Patitz
    • 1
  • Robert Schweller
    • 2
  • Trent A. Rogers
    • 1
  • Scott M. Summers
    • 3
  • Andrew Winslow
    • 2
  1. 1.Department of Computer Science and Computer EngineeringUniversity of ArkansasFayettevilleUSA
  2. 2.Department of Computer ScienceUniversity of Texas–Rio Grande ValleyEdinburgUSA
  3. 3.Computer Science DepartmentUniversity of Wisconsin–OshkoshOshkoshUSA

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