{"title":"Proving new directed tile complexity lower bounds at temperature 1 by folding between 2D and just-barely 3D self-assembly","authors":"David Furcy, Scott M. Summers, Hailey Vadnais","doi":"10.1007/s11047-024-09979-0","DOIUrl":null,"url":null,"abstract":"<p>We study the problem of determining the size of the smallest tile set that uniquely self-assembles into a given target shape in Winfree’s abstract Tile Assembly Model (aTAM), an elegant theoretical model of DNA tile self-assembly. This problem is also known as the “directed tile complexity” problem. We prove two main results related to the directed tile complexity problem within a variant of the aTAM in which the minimum binding strength threshold (temperature) is set to 1. For our first result, self-assembly happens in a “just-barely 3D” setting, where self-assembling unit cubes are allowed to be placed in the <span>\\(z=0\\)</span> and <span>\\(z=1\\)</span> planes. This is the same setting in which Furcy, Summers and Withers (DNA 2021) recently proved lower and upper bounds on the directed tile complexity of a just-barely 3D <span>\\(k \\times N\\)</span> rectangle at temperature 1 of <span>\\(\\Omega \\left( N^{\\frac{1}{k}}\\right) \\)</span> and <span>\\(O\\left( N^{\\frac{1}{k-1}}+\\log N\\right) \\)</span>, respectively, the latter of which does not hold for <span>\\(k=2\\)</span>. Our first result closes this gap for <span>\\(k=2\\)</span> by proving an asymptotically tight bound of <span>\\(\\Theta (N)\\)</span> on the directed tile complexity of a just-barely 3D <span>\\(2 \\times N\\)</span> rectangle at temperature 1. Our proof uses a novel process by which a just-barely 3D assembly sequence is “unfolded” to an equivalent 2D assembly sequence. For our second result, we use the aforementioned lower bound by Furcy, Summers and Withers and a novel process that is complementary-in-spirit to our 3D-to-2D unfolding process, by which we “fold” a 2D tile assembly to an equivalent just-barely 3D assembly to prove a new lower bound on the directed tile complexity of a 2D <span>\\(k \\times N\\)</span> rectangle at temperature 1 of <span>\\(\\Omega \\left( \\frac{N^{\\frac{2}{k + (k \\bmod 2)}}}{k} \\right) \\)</span>. For fixed <i>k</i>, our new bound gives a nearly quadratic improvement over, and matches for general even values of <span>\\(k < \\frac{\\log N}{\\log \\log N - \\log \\log \\log N}\\)</span> the state of the art lower bound on the directed tile complexity of a <span>\\(k \\times N\\)</span> rectangle at temperature 1 by Furcy, Summers and Wendlandt (DNA 2019) of <span>\\(\\Omega \\left( N^{\\frac{1}{k}}\\right) \\)</span>. While both of our results represent improvements over previous corresponding state of the art results, the proofs thereof are facilitated by novel examples of reasoning about tile self-assembly happening in 2D (just-barely 3D) as though it is happening in just-barely 3D (2D).</p>","PeriodicalId":49783,"journal":{"name":"Natural Computing","volume":"41 1","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Natural Computing","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1007/s11047-024-09979-0","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE","Score":null,"Total":0}
引用次数: 0
Abstract
We study the problem of determining the size of the smallest tile set that uniquely self-assembles into a given target shape in Winfree’s abstract Tile Assembly Model (aTAM), an elegant theoretical model of DNA tile self-assembly. This problem is also known as the “directed tile complexity” problem. We prove two main results related to the directed tile complexity problem within a variant of the aTAM in which the minimum binding strength threshold (temperature) is set to 1. For our first result, self-assembly happens in a “just-barely 3D” setting, where self-assembling unit cubes are allowed to be placed in the \(z=0\) and \(z=1\) planes. This is the same setting in which Furcy, Summers and Withers (DNA 2021) recently proved lower and upper bounds on the directed tile complexity of a just-barely 3D \(k \times N\) rectangle at temperature 1 of \(\Omega \left( N^{\frac{1}{k}}\right) \) and \(O\left( N^{\frac{1}{k-1}}+\log N\right) \), respectively, the latter of which does not hold for \(k=2\). Our first result closes this gap for \(k=2\) by proving an asymptotically tight bound of \(\Theta (N)\) on the directed tile complexity of a just-barely 3D \(2 \times N\) rectangle at temperature 1. Our proof uses a novel process by which a just-barely 3D assembly sequence is “unfolded” to an equivalent 2D assembly sequence. For our second result, we use the aforementioned lower bound by Furcy, Summers and Withers and a novel process that is complementary-in-spirit to our 3D-to-2D unfolding process, by which we “fold” a 2D tile assembly to an equivalent just-barely 3D assembly to prove a new lower bound on the directed tile complexity of a 2D \(k \times N\) rectangle at temperature 1 of \(\Omega \left( \frac{N^{\frac{2}{k + (k \bmod 2)}}}{k} \right) \). For fixed k, our new bound gives a nearly quadratic improvement over, and matches for general even values of \(k < \frac{\log N}{\log \log N - \log \log \log N}\) the state of the art lower bound on the directed tile complexity of a \(k \times N\) rectangle at temperature 1 by Furcy, Summers and Wendlandt (DNA 2019) of \(\Omega \left( N^{\frac{1}{k}}\right) \). While both of our results represent improvements over previous corresponding state of the art results, the proofs thereof are facilitated by novel examples of reasoning about tile self-assembly happening in 2D (just-barely 3D) as though it is happening in just-barely 3D (2D).
期刊介绍:
The journal is soliciting papers on all aspects of natural computing. Because of the interdisciplinary character of the journal a special effort will be made to solicit survey, review, and tutorial papers which would make research trends in a given subarea more accessible to the broad audience of the journal.