Sensitivity analysis of dynamic biological systems with time-delays.
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These systems are represented as delay differential equation (DDE) models.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.
Affiliation: Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi 62102, Taiwan. wwh@cs.ccu.edu.tw
ABSTRACT
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Background: Mathematical 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. Results: We 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. Conclusions: By 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. |
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Mentions: To verify the result obtained by the EAMCM algorithm, it is compared with the finitedifference method using the dde23 as the DDE solver. The dde23 solver is available inMATLAB 6.5 and later. Forward difference is considered in the finite differencemethod. The dynamic sensitivities of these two systems mentioned above are solved bythe finite difference method with spacing ratio 0.1 and 0.01, respectively. Therelative parameter sensitivities of heart rate and blood pressure with respect toβ obtained by the finite difference method with spacing ratio 0.1and 0.01, respectively, and the EAMCM method are shown in Figure 11 as an illustration (another data is similar and not shown here).According to the definition of relative sensitivity, the theoretical value ofrelative sensitivity is obtained when the spacing ratio is approaching to zero. FromFigure 11, the relative sensitivities obtained by the EAMCMare close to the theoretical values. We analyze the performance of the finitedifference method and the EAMCM method for computing the dynamic sensitivities bymeasuring the number of evaluations of model equations. The results are shown inTable 6. The CPU time in second running by a 1.86 GHz IntelXeon CPU with 4 GMB RAM is shown in the parenthesis for reference. Based on thecomparison, the EAMCM program surely outperforms the finite difference method usingdde23 solver. The EAMCM program can be accessed fromhttp://www.che.ccu.edu.tw/~bioproc/index_english.files/page00064.htmand a brief manual can be found in the Additional file 2. |
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Affiliation: Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi 62102, Taiwan. wwh@cs.ccu.edu.tw
Background: Mathematical 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.
Results: We 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.
Conclusions: By 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.