Abstract
BackgroundMathematical modeling has been applied to the study and analysis of complex biological systems for a long time. Some processes in biological systems, such as the gene expression and feedback control in signal transduction networks, involve a time delay. These systems are represented as delay differential equation (DDE) models. Numerical sensitivity analysis of a DDE model by the direct method requires the solutions of model and sensitivity equations with time-delays. The major effort is the computation of Jacobian matrix when computing the solution of sensitivity equations. The computation of partial derivatives of complex equations either by the analytic method or by symbolic manipulation is time consuming, inconvenient, and prone to introduce human errors. To address this problem, an automatic approach to obtain the derivatives of complex functions efficiently and accurately is necessary.ResultsWe have proposed an efficient algorithm with an adaptive step size control to compute the solution and dynamic sensitivities of biological systems described by ordinal differential equations (ODEs). The adaptive direct-decoupled algorithm is extended to solve the solution and dynamic sensitivities of time-delay systems describing by DDEs. To save the human effort and avoid the human errors in the computation of partial derivatives, an automatic differentiation technique is embedded in the extended algorithm to evaluate the Jacobian matrix. The extended algorithm is implemented and applied to two realistic models with time-delays: the cardiovascular control system and the TNF-α signal transduction network. The results show that the extended algorithm is a good tool for dynamic sensitivity analysis on DDE models with less user intervention.ConclusionsBy comparing with direct-coupled methods in theory, the extended algorithm is efficient, accurate, and easy to use for end users without programming background to do dynamic sensitivity analysis on complex biological systems with time-delays.
Highlights
Mathematical modeling has been applied to the study and analysis of complex biological systems for a long time
We have proposed an efficient algorithm with an adaptive step size control, called adaptive modified collocation method (AMCM), to compute the solution and dynamic sensitivities of biological systems described by ordinary differential equations (ODEs) [21]
By combining the adaptive direct-decoupled AMCM algorithm with automatic differentiation technique and extending its usage to delay differential equation (DDE) models, the extended algorithm, extended adaptive modified collocation method (EAMCM), is efficient, accurate, and easy to use for end users without programming background to do dynamic sensitivity analysis on complex biological systems with time-delays
Summary
Mathematical modeling has been applied to the study and analysis of complex biological systems for a long time. The computation of partial derivatives of complex equations either by the analytic method or by symbolic manipulation is time consuming, inconvenient, and prone to introduce human errors To address this problem, an automatic approach to obtain the derivatives of complex functions efficiently and accurately is necessary. Some processes in biological systems, such as the gene expression and feedback control in signal transduction networks, involve a time delay These dynamic sensitivity analysis on DDE models, an efficient and accurate approach to compute the solution of DDEs is the basic requirement. The computation of partial derivatives of complicated equations either by the analytic method or by symbolic manipulation is time consuming, inconvenient, and prone to introduce human errors To surmount this problem, an automatic approach to obtain the derivatives of complex functions efficiently and accurately is necessary
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