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Sensitivity analysis of dynamic biological systems with time-delays.

Wu WH, Wang FS, Chang MS - BMC Bioinformatics (2010)

Bottom Line: 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.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi 62102, Taiwan. wwh@cs.ccu.edu.tw

ABSTRACT

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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The semi-relative sensitivities of DISC. The semi-relative sensitivitiesof DISC (x21) with respect to the rate constants of theformation of survival complex (k9), the formation of deathcomplex (k15), the formation of DISC without TNFR1(k17 ), the caspase-8 activation(k20).
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Figure 10: The semi-relative sensitivities of DISC. The semi-relative sensitivitiesof DISC (x21) with respect to the rate constants of theformation of survival complex (k9), the formation of deathcomplex (k15), the formation of DISC without TNFR1(k17 ), the caspase-8 activation(k20).

Mentions: Following similar procedures mentioned above, we investigate the regulation of theapoptosis pathway. The DISC complex is essential for TNF-induced apoptosis and it isrequired for casepase-8 activation. To investigate the regulation of apoptosis, weidentify the important reactions that regulate the formation of DISC by sensitivityanalysis. The ranking of dynamic sensitivities of DISC based on the time-averagedsemi-relative sensitivities is shown in Table 5. The keyparameters -k9, k15, k17,and k20 - are identified and the dynamic sensitivities of DISCwith respect to these four parameters are shown in Figure 10.The reaction of dissociation of DISC from the death receptor TNFR1 is essential forthe following caspase-8 activation and its corresponding rate constantk17 is identify as an important parameter. This result is inagreement with the observation in an in vitro binding assay by Harper et al.[40].


Sensitivity analysis of dynamic biological systems with time-delays.

Wu WH, Wang FS, Chang MS - BMC Bioinformatics (2010)

The semi-relative sensitivities of DISC. The semi-relative sensitivitiesof DISC (x21) with respect to the rate constants of theformation of survival complex (k9), the formation of deathcomplex (k15), the formation of DISC without TNFR1(k17 ), the caspase-8 activation(k20).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2957680&req=5

Figure 10: The semi-relative sensitivities of DISC. The semi-relative sensitivitiesof DISC (x21) with respect to the rate constants of theformation of survival complex (k9), the formation of deathcomplex (k15), the formation of DISC without TNFR1(k17 ), the caspase-8 activation(k20).
Mentions: Following similar procedures mentioned above, we investigate the regulation of theapoptosis pathway. The DISC complex is essential for TNF-induced apoptosis and it isrequired for casepase-8 activation. To investigate the regulation of apoptosis, weidentify the important reactions that regulate the formation of DISC by sensitivityanalysis. The ranking of dynamic sensitivities of DISC based on the time-averagedsemi-relative sensitivities is shown in Table 5. The keyparameters -k9, k15, k17,and k20 - are identified and the dynamic sensitivities of DISCwith respect to these four parameters are shown in Figure 10.The reaction of dissociation of DISC from the death receptor TNFR1 is essential forthe following caspase-8 activation and its corresponding rate constantk17 is identify as an important parameter. This result is inagreement with the observation in an in vitro binding assay by Harper et al.[40].

Bottom Line: 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.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Computer Science and Information Engineering, National Chung Cheng University, Chiayi 62102, Taiwan. wwh@cs.ccu.edu.tw

ABSTRACT

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.

Show MeSH