The calculation of steady-state metabolite concentrations in metabolic reaction network models is the first step in the sensitivity analysis of a metabolic reaction system described by differential equations. However, this calculation becomes very difficult when the number of differential equations is more than 100. In the present study, therefore, we investigated a calculation procedure for obtaining true steady-state metabolite concentrations both efficiently and accurately even in large-scale network models. For convenience, a linear pathway model composed of a simple Michaelis-Menten rate law and two TCA cycle models were used as case studies. The calculation procedure is as follows: first solve the differential equations by a numerical method for solving initial-value problems until the upper several digits of the calculated values stabilize, and then use these values as initial guesses for a root-finding technique. An intensive investigation indicates that the S-system technique, finding roots in logarithmic space and providing a broader convergence region, is superior to the Newton-Raphson technique, and the algorithm using the S-system technique successfully provides true steady-state values with machine accuracy even with 1,500 differential equations. The complex-step method is also shown to contribute to shortening the calculation time and enhancing the accuracy. The program code has been deposited to https://github.com/BioprocessdesignLab/Steadystateconc.

中文翻译：

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一种在大规模代谢反应网络模型中计算真实稳态代谢物浓度的有前途的方法。

代谢反应网络模型中稳态代谢物浓度的计算是用微分方程描述的代谢反应系统敏感性分析的第一步。但是，当微分方程的数量大于100时，此计算将变得非常困难。因此，在本研究中，我们研究了即使在大规模网络模型中也能高效，准确地获得真实稳态代谢物浓度的计算程序。为方便起见，使用了一个简单的米利斯（Michaelis-Menten）速率定律和两个TCA周期模型组成的线性路径模型作为案例研究。计算过程如下：首先通过数值方法求解微分方程，以解决初值问题，直到计算值的高位数稳定为止；然后将这些值用作寻根技术的初始猜测。深入的研究表明，在对数空间中找到根并提供较宽收敛范围的S系统技术优于Newton-Raphson技术，并且使用S系统技术的算法可通过机器成功提供真实的稳态值即使有1,500个微分方程，也能保证准确性。还显示了复杂步骤方法，有助于缩短计算时间并提高准确性。该程序代码已保存到https://github.com/BioprocessdesignLab/Steadystateconc。优于Newton-Raphson技术，并且使用S系统技术的算法即使具有1,500个微分方程，也能够成功地提供具有机器精度的真实稳态值。还显示了复杂步骤方法，有助于缩短计算时间并提高准确性。该程序代码已保存到https://github.com/BioprocessdesignLab/Steadystateconc。优于Newton-Raphson技术，并且使用S系统技术的算法即使具有1,500个微分方程，也能够成功地提供具有机器精度的真实稳态值。还显示了复杂步骤方法，有助于缩短计算时间并提高准确性。该程序代码已保存到https://github.com/BioprocessdesignLab/Steadystateconc。