Background: The first objective for realizing and handling biological systems is to choose a suitable model prototype and then perform structure and parameter identification. Afterwards, a theoretical analysis is needed to understand the characteristics, abilities, and limitations of the underlying systems. Generalized Michaelis–Menten kinetics (MM) and S-systems are two well-known biochemical system theory-based models. Research on steady-state estimation of generalized MM systems is difficult because of their complex structure. Further, theoretical analysis of S-systems is still difficult because of the power-law structure, and even the estimation of steady states can be easily achieved via algebraic equations. Aim: We focus on how to flexibly use control technologies to perform deeper biological system analysis. Methods: For generalized MM systems, the root locus method (proposed by Walter R. Evans) is used to predict the direction and rate (flux) limitations of the reaction and to estimate the steady states and stability margins (relative stability). Mode analysis is additionally introduced to discuss the transient behavior and the setting time. For S-systems, the concept of root locus, mode analysis, and the converse theorem are used to predict the dynamic behavior, to estimate the setting time and to analyze the relative stability of systems. Theoretical results were examined via simulation in a Simulink/MATLAB environment. Results: Four kinds of small functional modules (a system with reversible MM kinetics, a system with a singular or nearly singular system matrix and systems with cascade or branch pathways) are used to describe the proposed strategies clearly. For the reversible MM kinetics system, we successfully predict the direction and the rate (flux) limitations of reactions and obtain the values of steady state and net flux. We observe that theoretically derived results are consistent with simulation results. Good prediction is observed (>99.7% accuracy). For the system with a (nearly) singular matrix, we demonstrate that the system is neither globally exponentially stable nor globally asymptotically stable but globally semistable. The system possesses an infinite gain margin (GM denoting how much the gain can increase before the system becomes unstable) regardless of how large or how small the values of independent variables are, but the setting time decreases and then increases or always decreases as the values of independent variables increase. For S-systems, we first demonstrate that the stability of S-systems can be determined by linearized systems via root loci, mode analysis, and block diagram-based simulation. The relevant S-systems possess infinite GM for the values of independent variables varying from zero to infinity, and the setting time increases as the values of independent variables increase. Furthermore, the branch pathway maintains oscillation until a steady state is reached, but the oscillation phenomenon does not exist in the cascade pathway because in this system, all of the root loci are located on real lines. The theoretical predictions of dynamic behavior for these two systems are consistent with the simulation results. This study provides a guideline describing how to choose suitable independent variables such that systems possess satisfactory performance for stability margins, setting time and dynamic behavior. Conclusion: The proposed root locus-based analysis can be applied to any kind of differential equation-based biological system. This research initiates a method to examine system dynamic behavior and to discuss operating principles.

中文翻译：

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基于根位点的生物系统稳定性分析

背景：实现和处理生物系统的首要目标是选择合适的模型原型，然后进行结构和参数识别。之后，需要进行理论分析以了解底层系统的特征、能力和局限性。广义 Michaelis-Menten 动力学 (MM) 和 S-系统是两个著名的基于生化系统理论的模型。广义MM系统的稳态估计研究由于其复杂的结构而变得困难。此外，由于幂律结构，S系统的理论分析仍然很困难，甚至可以通过代数方程轻松实现稳态估计。目的：我们专注于如何灵活地使用控制技术进行更深层次的生物系统分析。方法：对于广义的 MM 系统，根轨迹法（由 Walter R. Evans 提出）用于预测反应的方向和速率（通量）限制，并估计稳态和稳定裕度（相对稳定性）。另外引入了模式分析来讨论瞬态行为和设置时间。对于 S 系统，使用根轨迹、模态分析和逆定理的概念来预测动态行为，估计设置时间并分析系统的相对稳定性。通过在 Simulink/MATLAB 环境中的仿真检查了理论结果。结果：四种小功能模块（具有可逆 MM 动力学的系统、具有奇异或接近奇异系统矩阵的系统以及具有级联或分支路径的系统）用于清楚地描述所提出的策略。对于可逆 MM 动力学系统，我们成功地预测了反应的方向和速率（通量）限制，并获得了稳态和净通量的值。我们观察到理论上得出的结果与模拟结果一致。观察到良好的预测（>99.7％ 准确性）。对于具有（接近）奇异矩阵的系统，我们证明了该系统既不是全局指数稳定的，也不是全局渐近稳定的，而是全局半稳定的。无论自变量的值有多大或多小，系统都具有无限的增益裕度（GM 表示在系统变得不稳定之前可以增加多少增益），但设置时间会随着值的增加而减少然后增加或始终减少自变量的增加。对于 S 系统，我们首先证明 S 系统的稳定性可以通过线性化系统通过根轨迹、模式分析和基于框图的仿真来确定。对于从零到无穷大的自变量值，相关的 S 系统具有无限的 GM，并且设定时间随着自变量值的增加而增加。此外，分支通路保持振荡直到达到稳定状态，但在级联通路中不存在振荡现象，因为在该系统中，所有根基因座都位于实线上。这两个系统的动态行为的理论预测与模拟结果一致。本研究提供了一个指南，描述了如何选择合适的自变量，以使系统在稳定性裕度、设置时间和动态行为方面具有令人满意的性能。所有的根基因座都位于实线上。这两个系统的动态行为的理论预测与模拟结果一致。本研究提供了一个指南，描述了如何选择合适的自变量，以使系统在稳定性裕度、设置时间和动态行为方面具有令人满意的性能。所有的根基因座都位于实线上。这两个系统的动态行为的理论预测与模拟结果一致。本研究提供了一个指南，描述了如何选择合适的自变量，以使系统在稳定性裕度、设置时间和动态行为方面具有令人满意的性能。结论：所提出的基于根轨迹的分析可以应用于任何类型的基于微分方程的生物系统。这项研究启动了一种检查系统动态行为和讨论操作原理的方法。