Abstract
BackgroundA 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.ResultsWe 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.ConclusionBy 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.
Highlights
A mathematical model to understand, predict, control, or even design a real biological system is a central theme in systems biology
Our algorithm is one of the decouple direct methods in computing dynamic sensitivities of an ordinary differential equation (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
The same dynamic sensitivity analysis is 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
Summary
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. The most often used mathematical models for dynamic biological systems are formulated as nonlinear ordinary differential equations (ODEs). To identify the structure and parameters of a dynamic model, the most important and essential job is to find the solution of ODEs efficiently and accurately. This job can be treated by analytical and numerical methods. There are several numerical methods can be used to solve ODEs [1,2,3], e.g., Taylor-series methods, modified Euler methods, and Runge-Kutta methods with variable step size control. We propose an adaptive step size control approach based on the Banach fixed point theorem for the modified collocation method in this paper
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