标题： 当代数拓扑遇见系统生物学：关于复杂化学反应网络中Gröbner基结构的猜想 显示英文标题

标题： When algebra twinks system biology: a conjecture on the structure of Gröbner bases in complex chemical reaction networks

摘要： 我们解决了在由质量作用动力学控制的化学反应网络（CRNs）中识别所有实正稳态的挑战。 传统的数值方法通常需要特定的初始猜测，并且在表现出多稳态的系统中可能无法找到所有解。 Gröbner基提供了一个代数框架，可以系统地将多项式方程转换为更简单的形式，从而促进全面的解枚举。 在本工作中，我们提出一个猜想，即在适当的假设下，最多成对相互作用的CRNs会产生具有近似“三角”结构的Gröbner基。 我们使用基因调控网络和Wnt信号通路中的例子来说明这一现象，其中Gröbner基方法可靠地捕捉到了所有实正解。 我们的计算实验揭示了Gröbner基克服局部数值方法在寻找复杂生物系统稳态方面的局限性的潜力，使其成为理解跨多种生化模型的动力学过程的强大工具。 显示英文摘要

摘要： We address the challenge of identifying all real positive steady states in chemical reaction networks (CRNs) governed by mass-action kinetics. Traditional numerical methods often require specific initial guesses and may fail to find all the solutions in systems exhibiting multistability. Gr\"obner bases offer an algebraic framework that systematically transforms polynomial equations into simpler forms, facilitating comprehensive solution enumeration. In this work, we propose a conjecture that CRNs with at most pairwise interactions yield Gr\"obner bases possessing a near-"triangular" structure, under appropriate assumptions. We illustrate this phenomenon using examples from a gene regulatory network and the Wnt signaling pathway, where the Gr\"obner basis approach reliably captures all real positive solutions. Our computational experiments reveal the potential of Gr\"obner bases to overcome limitations of local numerical methods for finding the steady states of complex biological systems, making them a powerful tool for understanding dynamical processes across diverse biochemical models.