AI 中文总结
该研究针对化学组织理论计算中的组合挑战，提出基本反应闭包（ERC）的层级结构，证明持续模块可由ERC组合生成，且协同与互补是最小生成的充分标准，还在438个生物网络中量化了相关结构。
AI 中文摘要
识别确保反应网络持续存在的结构特征，是理解生物实体进化与复杂化的基础。通过将组织定义为封闭且自我维持的物种子集，化学组织理论（Chemical Organization Theory, COT）表明，组织能够过滤掉吸引子可能存在的相空间区域。尽管已有多种理论探索，但当前计算组织的方法面临着不明确的组合挑战。本文中，我们首先对这类组合挑战进行系统研究，确定了具有生产性新颖性和不可约性的标准，以此区分相关组合与冗余组合。其次，我们确定了构建组织需结合的最小构建块，称为基本反应闭包（elementary reaction closures, ERC），并将其表征为一个层级结构。第三，我们证明所有持续模块仅可通过ERC的组合生成，且明确了ERC的两个属性——协同与互补，这两个属性定义了以最小方式构建此类生成器的充分标准。第四，我们证明每个相关的持续模块都可由此类ERC的最小序列构建。我们进一步表明，用于生成持续模块的协同与互补，其扩展速度远慢于常规组合方法。我们对来自BioModels和BiGG数据库的438个生物反应网络中的这些结构进行了量化。随着网络规模增长，所有ERC对中基础协同与互补的比例会缩小，而原始数量则增加，这证实了更大的生物网络通过更小型、更具选择性的集合实现持续存在。我们未指定计算组织的算法，但讨论了基于我们结果的算法如何能应用COT。
英文摘要
Identifying structural features ensuring the persistence of a reaction network is fundamental for understanding the evolution and complexification of biological entities. By defining organizations as subsets of species that are closed and self-maintaining, Chemical Organization Theory (COT) shows that organizations enable filtering out the regions of the phase space where attractors can exist. Despite various theoretical explorations current methods to compute the organizations face unclear combinatorial challenges. In this article, we first perform a systematic study of such combinatorial challenges and identify a criteria of productive novelty and irreducibility that separate relevant from redundant combinations. Second, we identify the minimal building blocks that shall be combined to build organizations, called elementary reaction closures (ERC), and characterize them as a hierarchy. Third, we show that all persistent modules can be generated as combination of ERCs only, and operationalize two properties among ERCs, synergy and complementarity, that define a sufficient criteria to build such generators in a minimal way. Fourth we show that every relevant persistent module can be built from such minimal sequences of ERCs. We next show that the synergies and complementarities we use to generate persistent modules not only scale radically slower than usual combinatorial methods. We quantify these structures across $438$ biological reaction networks from the BioModels and BiGG databases. The proportion of fundamental synergies and complementarities among all ERC pairs shrinks as networks grow even as raw counts increase, confirming that larger biological networks achieve persistence through a progressively smaller and more selective sets. We do not prescribe algorithms to compute organizations, discuss how algorithms based on our results could permit applying COT.
