The Tile Complexity of Linear Assemblies

  • Harish Chandran
  • Nikhil Gopalkrishnan
  • John Reif
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5555)

Abstract

The conventional Tile Assembly Model (TAM) developed by Winfree using Wang tiles is a powerful, Turing-universal theoretical framework which models varied self-assembly processes. We describe a natural extension to TAM called the Probabilistic Tile Assembly Model (PTAM) to model the inherent probabilistic behavior in physically realized self-assembled systems. A particular challenge in DNA nanoscience is to form linear assemblies or rulers of a specified length using the smallest possible tile set. These rulers can then be used as components for construction of other complex structures. In TAM, a deterministic linear assembly of length N requires a tile set of cardinality at least N. In contrast, for any given N, we demonstrate linear assemblies of expected length N with a tile set of cardinality Θ(logN) and prove a matching lower bound of Ω(logN). We also propose a simple extension to PTAM called κ-pad systems in which we associate κ pads with each side of a tile, allowing abutting tiles to bind when at least one pair of corresponding pads match and prove analogous results. All our probabilistic constructions are free from co-operative tile binding errors and can be modified to produce assemblies whose probability distribution of lengths has arbitrarily small tail bounds dropping exponentially with a given multiplicative factor increase in number of tile types. Thus, for linear assembly systems, we have shown that randomization can be exploited to get large improvements in tile complexity at a small expense of precision in length.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Harish Chandran
    • 1
  • Nikhil Gopalkrishnan
    • 1
  • John Reif
    • 1
  1. 1.Department of Computer ScienceDuke UniversityDurhamUSA

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