Wu Wu Hsiung, Wang Feng Sheng, Chang Maw Shang
Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi 62102, Taiwan.
BMC Bioinformatics. 2008 Dec 12;9 Suppl 12(Suppl 12):S17. doi: 10.1186/1471-2105-9-S12-S17.
A mathematical model to understand, predict, control, or even design a real biological system is a central theme in systems biology. A dynamic biological system is always modeled as a nonlinear ordinary differential equation (ODE) system. How to simulate the dynamic behavior and dynamic parameter sensitivities of systems described by ODEs efficiently and accurately is a critical job. In many practical applications, e.g., the fed-batch fermentation systems, the system admissible input (corresponding to independent variables of the system) can be time-dependent. The main difficulty for investigating the dynamic log gains of these systems is the infinite dimension due to the time-dependent input. The classical dynamic sensitivity analysis does not take into account this case for the dynamic log gains.
We present an algorithm with an adaptive step size control that can be used for computing the solution and dynamic sensitivities of an autonomous ODE system simultaneously. Although our algorithm is one of the decouple direct methods in computing dynamic sensitivities of an ODE system, the step size determined by model equations can be used on the computations of the time profile and dynamic sensitivities with moderate accuracy even when sensitivity equations are more stiff than model equations. To show this algorithm can perform the dynamic sensitivity analysis on very stiff ODE systems with moderate accuracy, it is implemented and applied to two sets of chemical reactions: pyrolysis of ethane and oxidation of formaldehyde. The accuracy of this algorithm is demonstrated by comparing the dynamic parameter sensitivities obtained from this new algorithm and from the direct method with Rosenbrock stiff integrator based on the indirect method. The same dynamic sensitivity analysis was performed on an ethanol fed-batch fermentation system with a time-varying feed rate to evaluate the applicability of the algorithm to realistic models with time-dependent admissible input.
By combining the accuracy we show with the efficiency of being a decouple direct method, our algorithm is an excellent method for computing dynamic parameter sensitivities in stiff problems. We extend the scope of classical dynamic sensitivity analysis to the investigation of dynamic log gains of models with time-dependent admissible input.
理解、预测、控制甚至设计真实生物系统的数学模型是系统生物学的核心主题。动态生物系统通常被建模为非线性常微分方程（ODE）系统。如何高效且准确地模拟由ODE描述的系统的动态行为和动态参数敏感性是一项关键工作。在许多实际应用中，例如补料分批发酵系统，系统的允许输入（对应于系统的自变量）可能是时间相关的。研究这些系统动态对数增益的主要困难在于由于时间相关输入导致的无穷维。经典动态敏感性分析未考虑这种情况下的动态对数增益。
我们提出了一种具有自适应步长控制的算法，可用于同时计算自治ODE系统的解和动态敏感性。尽管我们的算法是计算ODE系统动态敏感性的解耦直接方法之一，但即使敏感性方程比模型方程更刚性，由模型方程确定的步长也可用于时间历程和动态敏感性的计算，且具有中等精度。为了表明该算法能够以中等精度对非常刚性的ODE系统进行动态敏感性分析，我们将其实现并应用于两组化学反应：乙烷热解和甲醛氧化。通过将基于间接方法的具有Rosenbrock刚性积分器的直接方法与该新算法获得的动态参数敏感性进行比较，证明了该算法的准确性。对具有时变进料速率的乙醇补料分批发酵系统进行了相同的动态敏感性分析，以评估该算法对具有时间相关允许输入的实际模型的适用性。
通过结合我们所展示的准确性和作为解耦直接方法的效率，我们的算法是计算刚性问题中动态参数敏感性的优秀方法。我们将经典动态敏感性分析的范围扩展到了对具有时间相关允许输入的模型的动态对数增益的研究。
