Atom tracing is essential for understanding the fate of labeled atoms in biochemical reaction networks, yet existing computational methods either simplify label correlations or suffer from combinatorial explosion. We introduce a saturation-based framework for enumerating labeling patterns that directly operates on atom-atom maps without requiring flux data or experimental measurements. The approach models reaction semantics using Kleisli morphisms in the powerset monad, allowing for compositional propagation of atom provenance through reaction networks. By iteratively saturating all possible educt combinations of reaction rules, the method exhaustively enumerates labeled molecular configurations, including multiplicities and reuse. Allowing arbitrary initial labeling patterns - including identical or distinct labels - the method expands only isotopomers reachable from these inputs, keeping the configuration space as small as necessary and avoids the full combinatorial growth characteristic of previous approaches. In principle, even every atom could carry a distinct identifier (e.g., tracing all carbon atoms individually), illustrating the generality of the framework beyond practical experimental limitations. The resulting template instance hypergraph captures the complete flow of atoms between compounds and supports projections tailored to experimental targets. Customizable labeling sets significantly reduce generated network sizes, providing efficient and exact atom traces focused on specific compounds or available isotopes. Applications to the tricarboxylic acid cycle, and glycolytic pathways demonstrate that the method fully automatically reproduces known labeling patterns and discovers steady-state labeling behavior. The framework offers a scalable, mechanistically transparent, and generalizable foundation for isotopomer modeling and experiment design.

翻译：原子追踪对于理解标记原子在生化反应网络中的归宿至关重要，然而现有的计算方法要么简化标记关联性，要么面临组合爆炸问题。我们提出了一种基于饱和度的框架，用于枚举标记模式，该框架直接在原子-原子映射上操作，无需通量数据或实验测量。该方法利用幂集单子中的Kleisli态射对反应语义进行建模，实现了原子来源在反应网络中的组合式传播。通过对反应规则的所有可能反应物组合进行迭代饱和，该方法能够穷举标记分子构型，包括多重性和重复利用。该方法允许任意初始标记模式——包括相同或不同的标记——仅扩展从这些输入可达的同位素异构体，将构型空间保持在必要的最小范围，避免了先前方法中典型的完全组合增长。原则上，每个原子甚至可以携带不同的标识符（例如，单独追踪所有碳原子），这体现了该框架超越实际实验限制的普适性。生成的模板实例超图捕获了化合物间原子的完整流动，并支持针对实验目标定制的投影。可定制的标记集显著减少了生成网络的规模，为特定化合物或可用同位素提供了高效且精确的原子追踪。在三羧酸循环和糖酵解途径中的应用表明，该方法能全自动复现已知标记模式并发现稳态标记行为。该框架为同位素异构体建模和实验设计提供了一个可扩展、机理透明且可推广的基础。