Hill型动力学系统的正平衡
Positive Equilibria of Hill-Type Kinetic Systems
摘要：
本文介绍了一种新的方法来研究具有Hill型动力学的化学反应网络$\mathscr{N}$的正平衡态性质，称为Hill型动力学（HTK）系统$\left(\mathscr{N},K\right)$，包括它们在物种中的多重性和浓度鲁棒性。我们关联一个独特的正线性组合的幂律动力学系统称为聚PL动力学（PYK）系统$\left( {\mathscr{N},{K_\text{PY}}} \right)$给定的HTK系统。关联系统的关键性质是其平衡集与Hill型系统的平衡集一致，即，${E_ + }\left( {\mathscr{N},K} \right) = {E_ + }\left( {\mathscr{N},{K_\text{PY}}} \right)$和${Z_ + }\left( {\mathscr{N},K} \right) = {Z_ + }\left( {\mathscr{N},{K_\text{PY}}} \right)$。这使我们能够确定两个新的子集的山型动力学，PL平衡和PL复杂的平衡动力学，最近的结果绝对浓度鲁棒性（ACR）的物种和复杂的平衡在正平衡的幂律（PL）动力学系统可以应用。我们的主要结果还包括PL平衡HT-RDK系统的Shinar-Feinberg ACR定理（即，复可因子分解HTK系统的一个子集），为HTK系统的ACR分析奠定了基础，并将Müller和Regensburger关于广义质量作用系统的结果推广到PL-复平衡HT-RDK系统.此外，我们推导出平衡浓度鲁棒性（BCR）的理论，以类似的方式ACR PL平衡系统。最后，我们提供了进一步的扩展，我们的结果更一般的一类动力学，其中包括多PL功能的concernents。
英文摘要：
This work introduces a novel approach to study properties of positive equilibria of a chemical reaction network $\mathscr{N}$ endowed with Hill-type kinetics $K$, called a Hill-type kinetic (HTK) system $\left(\mathscr{N},K\right)$, including their multiplicity and concentration robustness in a species. We associate a unique positive linear combination of power-law kinetic systems called poly-PL kinetic (PYK) system $\left( {\mathscr{N},{K_\text{PY}}} \right)$ to the given HTK system. The associated system has the key property that its equilibria sets coincide with those of the Hill-type system, i.e., ${E_ + }\left( {\mathscr{N},K} \right) = {E_ + }\left( {\mathscr{N},{K_\text{PY}}} \right)$ and ${Z_ + }\left( {\mathscr{N},K} \right) = {Z_ + }\left( {\mathscr{N},{K_\text{PY}}} \right)$. This allows us to identify two novel subsets of the Hill-type kinetics, called PL-equilibrated and PL-complex balanced kinetics, to which recent results on absolute concentration robustness (ACR) of species and complex balancing at positive equilibria of power-law (PL) kinetic systems can be applied. Our main results also include the Shinar-Feinberg ACR Theorem for PL-equilibrated HT-RDK systems (i.e., subset of complex factorizable HTK systems), which establishes a foundation for the analysis of ACR in HTK systems, and the extension of the results of Müller and Regensburger on generalized mass action systems to PL-complex balanced HT-RDK systems. In addition, we derive the theory of balanced concentration robustness (BCR) in an analogous manner to ACR for PL-equilibrated systems. Finally, we provide further extensions of our results to a more general class of kinetics, which includes quotients of poly-PL functions.
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