The Tile Complexity of Linear Assemblies
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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