Reaction networks (RNs) comprise a set X of species and a set $$\mathscr {R}$$ of reactions $$Y\rightarrow Y'$$ , each converting a multiset of educts $$Y\subseteq X$$ into a multiset $$Y'\subseteq X$$ of products. RNs are equivalent to directed hypergraphs. However, not all RNs necessarily admit a chemical interpretation. Instead, they might contradict fundamental principles of physics such as the conservation of energy and mass or the reversibility of chemical reactions. The consequences of these necessary conditions for the stoichiometric matrix $$\mathbf {S}\in \mathbb {R}^{X\times \mathscr {R}}$$ have been discussed extensively in the chemical literature. Here, we provide sufficient conditions for $$\mathbf {S}$$ that guarantee the interpretation of RNs in terms of balanced sum formulas and structural formulas, respectively. Chemically plausible RNs allow neither a perpetuum mobile, i.e., a “futile cycle” of reactions with non-vanishing energy production, nor the creation or annihilation of mass. Such RNs are said to be thermodynamically sound and conservative. For finite RNs, both conditions can be expressed equivalently as properties of the stoichiometric matrix $$\mathbf {S}$$ . The first condition is vacuous for reversible networks, but it excludes irreversible futile cycles and—in a stricter sense—futile cycles that even contain an irreversible reaction. The second condition is equivalent to the existence of a strictly positive reaction invariant. It is also sufficient for the existence of a realization in terms of sum formulas, obeying conservation of “atoms”. In particular, these realizations can be chosen such that any two species have distinct sum formulas, unless $$\mathbf {S}$$ implies that they are “obligatory isomers”. In terms of structural formulas, every compound is a labeled multigraph, in essence a Lewis formula, and reactions comprise only a rearrangement of bonds such that the total bond order is preserved. In particular, for every conservative RN, there exists a Lewis realization, in which any two compounds are realized by pairwisely distinct multigraphs. Finally, we show that, in general, there are infinitely many realizations for a given conservative RN. “Chemical” RNs are directed hypergraphs with a stoichiometric matrix $$\mathbf {S}$$ whose left kernel contains a strictly positive vector and whose right kernel does not contain a futile cycle involving an irreversible reaction. This simple characterization also provides a concise specification of random models for chemical RNs that additionally constrain $$\mathbf {S}$$ by rank, sparsity, or distribution of the non-zero entries. Furthermore, it suggests several interesting avenues for future research, in particular, concerning alternative representations of reaction networks and infinite chemical universes.

中文翻译：

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是什么让反应网络变得“化学”？


反应网络 (RN) 包含一组 X 种物质和一组 $$\mathscr {R}$$ 反应 $$Y\rightarrow Y'$$ ，每个反应网络将多组离析物 $$Y\subseteq X$$ 转换为产品的多重集 $$Y'\subseteq X$$。 RN 相当于有向超图。然而，并非所有 RN 都一定承认化学解释。相反，它们可能与物理学的基本原理相矛盾，例如能量和质量守恒或化学反应的可逆性。化学计量矩阵 $$\mathbf {S}\in \mathbb {R}^{X\times \mathscr {R}}$$ 的这些必要条件的结果已在化学文献中进行了广泛讨论。在这里，我们为 $$\mathbf {S}$$ 提供了充分的条件，分别保证了 RN 的平衡和公式和结构公式的解释。化学上合理的RNs既不允许永动机，即产生不为零的能量的反应的“无效循环”，也不允许质量的产生或湮灭。据说这样的 RN 在热力学上是合理且保守的。对于有限 RN，这两个条件都可以等效地表示为化学计量矩阵 $$\mathbf {S}$$ 的属性。第一个条件对于可逆网络来说是空洞的，但它排除了不可逆的无效循环，以及更严格意义上的甚至包含不可逆反应的无效循环。第二个条件相当于存在严格正反应不变量。满足“原子”守恒定律的求和公式的实现的存在也是足够的。特别是，可以选择这些实现，使得任何两个物种都具有不同的求和公式，除非 $$\mathbf {S}$$ 暗示它们是“强制异构体”。 就结构式而言，每个化合物都是一个标记的多重图，本质上是一个路易斯公式，并且反应仅包含键的重排，从而保留了总键序。特别是，对于每个保守的 RN，都存在一个 Lewis 实现，其中任何两个化合物都通过成对不同的多重图来实现。最后，我们表明，一般来说，对于给定的保守 RN，有无限多种实现。 “化学”RN 是具有化学计量矩阵 $$\mathbf {S}$$ 的有向超图，其左核包含严格的正向量，其右核不包含涉及不可逆反应的无效循环。这个简单的表征还提供了化学 RN 随机模型的简明规范，该随机模型还通过非零条目的等级、稀疏性或分布来约束 $$\mathbf {S}$$。此外，它还为未来的研究提出了一些有趣的途径，特别是关于反应网络和无限化学宇宙的替代表示。