知道何时折叠：Oritatami中折叠的自组装
Know When to Fold 'Em: Self-Assembly of Shapes by Folding in Oritatami
摘要：
oritatami系统（OS）是一种通过共转录折叠进行自组装的理论模型。它由不断增长的珠子链组成，这些珠子在转录时可以相互结合。在转录过程中，最近产生的$δ$珠粒动态折叠，以使形成的键的数量最大化，逐渐自组装成形状。参数$δ$被称为延迟，并且本质上与转录速率相关。本文首次利用织纹进行形状自组装的研究。形状是三角形晶格中的一组相连的点。我们首先表明，oritatami系统从根本上不同于瓷砖组装系统，表现出一个家庭的无限形状，可以瓷砖组装，但不能被任何操作系统折叠。因为它是NP-难的一般，以确定是否有一个操作系统折叠成（自组装）一个给定的有限形状，我们探讨折叠的放大版本的有限形状。我们表明，任何形状都可以折叠从一个恒定大小的种子，在任何规模n >= 3，延迟1的OS。我们还表明，任何形状都可以折叠在较小的规模2的操作系统具有无限的延迟。这导致我们研究延迟的影响，并证明，对于所有δ > 2，存在可以折叠的形状（在尺度1），延迟δ，但不能延迟δ '<δ。这些结果作为在这种新的自组装模型中研究形状构建的基础，并有可能提供更好地理解生物学中的共转录折叠，以及提高实验人员设计通过这种复杂的动力学过程自组装的人工系统的能力。
英文摘要：
An oritatami system (OS) is a theoretical model of self-assembly via co-transcriptional folding. It consists of a growing chain of beads which can form bonds with each other as they are transcribed. During the transcription process, the $δ$ most recently produced beads dynamically fold so as to maximize the number of bonds formed, self-assemblying into a shape incrementally. The parameter $δ$ is called the delay and is related to the transcription rate in nature. This article initiates the study of shape self-assembly using oritatami. A shape is a connected set of points in the triangular lattice. We first show that oritatami systems differ fundamentally from tile-assembly systems by exhibiting a family of infinite shapes that can be tile-assembled but cannot be folded by any OS. As it is NP-hard in general to determine whether there is an OS that folds into (self-assembles) a given finite shape, we explore the folding of upscaled versions of finite shapes. We show that any shape can be folded from a constant size seed, at any scale n >= 3, by an OS with delay 1. We also show that any shape can be folded at the smaller scale 2 by an OS with unbounded delay. This leads us to investigate the influence of delay and to prove that, for all δ > 2, there are shapes that can be folded (at scale 1) with delay δ but not with delay δ'<δ. These results serve as a foundation for the study of shape-building in this new model of self-assembly, and have the potential to provide better understanding of cotranscriptional folding in biology, as well as improved abilities of experimentalists to design artificial systems that self-assemble via this complex dynamical process.
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