Some of the most common mathematical models in biology, chemistry, physics, and engineering, are polynomial dynamical systems, i.e., systems of differential equations with polynomial right-hand sides. Inspired by notions and results that have been developed for the analysis of reaction networks in biochemistry and chemical engineering, we show that any polynomial dynamical system on the positive orthant $\mathbb R^n_{> 0}$ can be regarded as being generated by an oriented graph embedded in $\mathbb R^n$, called $\mathit{Euclidean \ embedded \ graph}$. This allows us to recast key conjectures about reaction network models (such as the Global Attractor Conjecture, or the Persistence Conjecture) into more general versions about some important classes of polynomial dynamical systems. Then, we introduce $\mathit{toric \ differential \ inclusions}$, which are piecewise constant autonomous dynamical systems with a remarkable geometric structure. We show that if a Euclidean embedded graph $G$ has some reversibility properties, then any polynomial dynamical system generated by $G$ can be embedded into a toric differential inclusion. We discuss how this embedding suggests an approach for the proof of the Global Attractor Conjecture and Persistence Conjecture.
A flux-based approach for analyzing the disguised toric locus of reaction networks
Bolyai Institute, University of Szeged（博莱研究所，塞格德大学）；National Laboratory for Health Security, University of Szeged（国家健康安全实验室，塞格德大学）；Department of Mathematics, University of Wisconsin-Madison（威斯康星大学麦迪逊分校数学系）；Department of Biomolecular Chemistry, University of Wisconsin-Madison（威斯康星大学麦迪逊分校生物分子化学系）；Max Planck Institute of Molecular Cell Biology and Genetics, Dresden（德累斯顿马克斯·普朗克分子细胞生物学与遗传学研究所）；Center for Systems Biology Dresden（德累斯顿系统生物学中心）；Department of Mathematics, University of Louisiana at Lafayette（路易斯安那大学拉斐特分校数学系）
