The Tile Complexity of Linear Assemblies

Self-assembly is fundamental to both biological processes and nanoscience. Key features of self-assembly are its probabilistic nature and local programmability. These features can be leveraged to design better self-assembled systems. 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. A particular challenge in DNA nanoscience is to form linear assemblies or rulers of a specified length using the smallest possible tile set, where any tile type may appear more than once in the assembly. The tile complexity of a linear assembly is the cardinality of the tile set that produced it. These rulers can then be used as components for construction of other complex structures. While square assemblies have been extensively studied, many questions remain about fixed length linear assemblies, which are more basic constructs yet fundamental building blocks for molecular architectures. In this paper, we extend TAM to take advantage of inherent probabilistic behavior in physically realized self-assembled systems by introducing randomization. We describe a natural extension to TAM called the Probabilistic Tile Assembly Model (PTAM). A restriction of the model, which we call the standard PTAM is considered in this paper. Prior work in DNA self-assembly strongly suggests that standard PTAM can be realized in the laboratory. In TAM, a deterministic linear assembly of length N requires a tile set of cardinality at least N . In contrast, we show various non-trivial probabilistic constructions for forming linear assemblies in PTAM with tile sets of sub-linear cardinality, using techniques that differ considerably from existing assembly techniques. In particular, for any given N , we demonstrate linear assemblies of expected length N with a tile set of cardinality Θ(logN) using one pad per side of each tile. We prove a matching lower bound of Ω(logN) on the tile complexity of linear assemblies of any given expected length N in standard PTAM systems using one pad per side of each tile. 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. This gives linear assemblies of expected length N with a 2-pad (two pads per side of each tile) tile set of cardinality Θ( log N log log N ) for infinitely many N . We show that we cannot get smaller tile complexity by proving a lower bound of Ω( log N log log N ) for each N on the cardinality of the κ-pad (κ-pads per side of each tile) tile set required to form linear assemblies of expected length N in standard κ-pad PTAM systems for any constant κ. The techniques that we use for deriving these tile complexity lower bounds are notable as they differ from traditional Kolmogorov complexity based information theoretic methods used for lower bounds on tile complexity. Also, Kolmogorov complexity based lower bounds do not preclude the possibility of achieving assemblies of very small tile multiset cardinality for infinitely many N . In contrast, our lower bounds are stronger as they hold for every N , rather than for almost all N . All our probabilistic constructions 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.