标题： 分层自组装的结构和组成复杂性：超图方法 显示英文标题

标题： Structural and Compositional Complexities of Hierarchical Self-Assembly: a Hypergraph Approach

摘要： 可编程自组装使从精心设计的构建模块中构建复杂的分子、超分子和晶体结构成为可能。 在这里，我们引入了一种基于超图的框架，称为块与键（B&B），它扩展了经典的化学图论，以编码定向多色相互作用、构建模块的内部对称性以及层次化组织。 在该框架内，我们开发了一种通用脚本，结构代码，用于编码复杂的超图结构。 本着柯尔莫哥洛夫的方法，我们将结构复杂性定义为编码自组装结构所需的最小信息量。 它由组成复杂性补充，捕捉构建模块的多样性。 这两个度量通过复杂性不等式相关联，该不等式指出，在可编程组装中，结构复杂性不能超过组成复杂性，并且可以识别出违反情况，这些情况表明出现复杂性。 对分子系统（乙二醇、葡萄糖）和可编程DNA折纸晶格的应用展示了B&B超图和结构代码如何自然地捕捉模块化、立体化学和晶体学秩序，同时实现结构信息的显著压缩。 这种方法提供了一种统一且可扩展的语言，用于在不同尺度上分类复杂性，将信息理论与可编程物质的设计联系起来。 显示英文摘要

摘要： Programmable self-assembly enables the construction of complex molecular, supramolecular, and crystalline architectures from well-designed building blocks. Here we introduce a hypergraph-based framework, termed Blocks \& Bonds (B\&B), that extends classical chemical graph theory to encode directed multi-colored interactions, internal symmetries of building blocs, and hierarchical organization. Within this framework, we develop a universal script, Structure Code, for encoding complex hypergraph organization. In the spirit of Kolmogorov's approach, we define Structural Complexity as the minimal information required to encode a self-assembled structure. It is complemented by Compositional Complexity, capturing the diversity of building blocks. The two measures are related through Complexity Inequality, stating that structural complexity cannot exceed compositional complexity for programmable assembly, and identify cases where violations signal emergent complexity. Applications to molecular systems (ethylene glycol, glucose) and programmable DNA-origami lattices demonstrate how B\&B hypergraphs and the structure code naturally capture modularity, stereochemistry, and crystallographic order while enabling significant compression of structural information. This approach provides a unified and scalable language for classifying complexity across scales, bridging information theory with the design of programmable matter.