Limitations of self-assembly at temperature 1

Doty, David; Patitz, Matthew J.; Summers, Scott M.

doi:10.1016/j.tcs.2010.08.023

Cited by 67 publications

(52 citation statements)

References 12 publications

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“…exist in this model. Lower bounds on tile set complexity have also been shown for various shapes (see Rothemund and Winfree (2000); Aggarwal et al (2004); Doty et al (2011)). …”

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confidence: 93%

The Tile Complexity of Linear Assemblies

Chandran

Gopalkrishnan

Reif

2009

Automata, Languages and Programming

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Abstract. 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 produces 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 work, we extend TAM to take advantage of inherent probabilistic behavior in physically realized selfassembled 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 report. 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(log N ) using one pad per side of each tile. We prove a matching lower bound of Ω(log N ) 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 further demonstrate how linear assemblies can be modified to produce assemblies with sharp tail bounds on distribution of lengths by concatenating various assemblies together. In particular, we show that for infinitely many N we can get linear assemblies with exponentially dropping tail distributions using O(log 3 N ) tile types. 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 Θfor infinitely many N . We show that we cannot get smaller tile complexity by proving a lower bound of Ωfor 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...

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“…exist in this model. Lower bounds on tile set complexity have also been shown for various shapes (see Rothemund and Winfree (2000); Aggarwal et al (2004); Doty et al (2011)). …”

mentioning

confidence: 93%

The Tile Complexity of Linear Assemblies

Chandran

Gopalkrishnan

Reif

2009

Automata, Languages and Programming

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“…Meunier [12] was able to show the same lower bound for systems permitted to have mismatches under the assumption that the seed tile starts in the lower left of the assembly, and removing this restriction remains open. In a similar vein, Reif and Song [15] have shown that temperature-1 mismatch-free aTAM systems are not computationally universal, while the same problem for systems with mismatches permitted is a notoriously difficult problem that remains open, despite significant efforts [8,7,18,13].…”

Section: Introductionmentioning

confidence: 99%

Size-separable tile self-assembly: a tight bound for temperature-1 mismatch-free systems

Winslow

2015

Nat Comput

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Abstract. We introduce a new property of tile self-assembly systems that we call size-separability. A system is size-separable if every terminal assembly is a constant factor larger than any intermediate assembly. Sizeseparability is motivated by the practical problem of filtering completed assemblies from a variety of incomplete "garbage" assemblies using gel electrophoresis or other mass-based filtering techniques. Here we prove that any system without cooperative bonding assembling a unique mismatch-free terminal assembly can be used to construct a size-separable system uniquely assembling the same shape. The proof achieves optimal scale factor and temperature for the size-separable system. As part of the proof, we obtain two results of independent interest on mismatch-free temperature-1 two-handed systems.

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“…Such conjectures are appealing because the algorithmic design and verification of tile systems [44] as well as lower bounds and impossibility proofs [5,16,31] often rely on reasoning about directed tile systems, which are "better behaved" in many senses than arbitrary tile systems, even those that strictly self-assemble a shape. It would be helpful to begin such arguments with the phrase, "Assume without loss of generality that the tile system is directed.…”

Section: Introductionmentioning

confidence: 99%