AI 中文总结
本文确定了极小二次零一反应网络的多稳态性，发现（3,6,3）族网络最多有3个正稳态，识别出373个双稳态网络，为细胞信号双稳态研究提供了测试案例。
AI 中文摘要
具有单位化学计量比的生化网络自然出现在受体-配体结合和多位点磷酸化系统中。核心问题是确定哪些网络具有多稳态性。由于该性质可从小子网络遗传到更大的网络，因此寻找具有该性质的最小网络是自然的。本文中，我们完全确定了所有表现出多稳态性的极小二次零一网络。基于近期进展已将搜索范围缩小到具有3个物种、6个反应和3维的零一网络，即（3,6,3）族，我们开发了计算流水线以完整表征该类网络中的多稳态性。主要理论贡献是一个结构简化，表明对于（3,m,3）二次零一网络，连续正稳态处的雅可比行列式符号相反，且恰好一半的这些状态是稳定的。这将多稳态性检测和稳定性验证简化为局部符号检查问题，无需边界或渐近分析。应用该结果，我们识别出373个表现出双稳态（两个稳定和一个不稳定正稳态）的二次零一网络。值得注意的是，没有（3,6,3）二次零一网络具有超过3个正稳态——这一严格界限远低于5的理论BKK界和8的贝祖界。在恰好具有3个正稳态的375个网络中，373个是多稳态的，仅2个不是。这些极小网络为理解细胞信号中双稳态如何出现提供了必要的测试案例，而无需更高阶的分子碰撞。
英文摘要
Biochemical networks with unit stoichiometry arise naturally in receptor-ligand binding and multisite phosphorylation systems. A central question is to identify which networks admit multistability. Because this property can be inherited from smaller subnetworks to larger ones, it is natural to seek the smallest networks with it. In this paper, we completely determine all minimal quadratic zero-one networks that exhibit multistability. Building on recent progress that has narrowed the search to the zero-one networks with 3 species, 6 reactions, and dimension 3, say (3, 6, 3) family. We develop a computational pipeline to provide a complete characterization of multistability within this class. The primary theoretical contribution is a structural simplification showing that for (3, m, 3) quadratic zero-one networks, the Jacobian determinants at consecutive positive steady states have opposite signs, and exactly half of these states are stable. This reduces multistationarity detection and stability verification to a local sign-checking problem, eliminating the need for boundary or asymptotic analysis. Applying this result, we identify 373 quadratic zero-one networks that exhibit bistability (two stable and one unstable positive steady states). Strikingly, no (3, 6, 3) quadratic zero-one network admits more than 3 positive steady states--a sharp bound that falls well below the theoretical BKK bound of 5 and the Bezout bound of 8. Among the 375 networks that admit exactly 3 positive steady states, 373 are multistable and only 2 are not. These minimal networks provide essential test cases for understanding how bistability emerges in cell signaling without requiring higher-order molecular collisions.
