Optimal Tile-Based DNA Self-Assembly Designs for Lattice Graphs and Platonic Solids (original) (raw)
2021
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Abstract
AI
This repository presents optimal tile-based DNA self-assembly designs for lattice graphs and Platonic solids, focusing on the minimum number of tile and bond-edge types required for constructing the octahedron and icosahedron. Additionally, it explores partial results for the dodecahedron and defined scenarios in square lattice graphs, highlighting ongoing challenges in determining optimal parameters for larger dimensions.
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We consider the problem of finding, for a given 2D pattern of coloured tiles, a minimal set of tile types self-assembling to this pattern in the abstract Tile Assembly Model of . This Patterned self-Assembly Tile set Synthesis (PATS) problem was first introduced by Ma and Lombardi (2008), and subsequently studied by Göös and Orponen (2011), who presented an exhaustive partition-search branch-and-bound algorithm (briefly PS-BB) for it. However, finding the true minimal tile sets is very time consuming, and the algorithm PS-BB is not well-suited for finding small but not necessarily minimal solutions. In this paper, we modify the basic partitionsearch framework by using a heuristic to optimize the order in which the algorithm traverses its search space. We find that by running several parallel executions of the modified algorithm PS-H, the search time for small tile sets can be shortened considerably. Additionally, we suggest a new approach, answer set programmin (ASP), to solving the PATS problem. We also introduce a method for computing the reliability of a given tile set, i.e. the probability of its error-free self-assembly to the desired target tiling, based on Winfree's analysis of the kinetic Tile Assembly Model (1998). We present empirical data on the reliability of tile sets found by the PS-BB and PS-H algorithms and find that also here the PS-H algorithm constitutes a significant improvement over the earlier PS-BB algorithm.
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References (5)
- Leyda Almodóvar, Jo Ellis-Monaghan, Amanda Harsy, Cory Johnson, and Jessica Sorrells. Computational complexity and pragmatic solutions for flexible tile-based dna self-assembly. 2021. Submitted. https://arxiv.org/abs/2108.00035.
- Leyda Almodóvar, Samantha Mauro, Sydney Martin, and Heiko Todt. Minimal tile and bond- edge types for self-assembling DNA graphs of triangular lattice graphs. Congressus Numer- atium, 232:241-263, 2019.
- Joanna Ellis-Monaghan, Greta Pangborn, Laura Beaudin, David Miller, Nick Bruno, and Akie Hashimoto. Minimal tile and bond-edge types for self-assembling DNA graphs. In Discrete and Topological Models in Molecular Biology, pages 241-270. Springer, 2014.
- Chris Godsil and Gordon Royle. Algebraic graph theory, volume 207 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2001.
- Gabriel Lopez. Self-assembling dna complexes with a wheel graph structure. 2021. In prepa- ration.
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