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New time-scale criteria for model simplification of bio-reaction systems.

Choi J, Yang KW, Lee TY, Lee SY - BMC Bioinformatics (2008)

Bottom Line: Therefore, an iterative procedure was also developed to find the possible multiple boundary layers and to derive an appropriate reduced model.By successive calculation of the newly derived time-scale criteria, it was possible to detect multiple boundary layers of full ordinary differential equation (ODE) models.Besides, the iterative procedure could derive the appropriate reduced differential algebraic equation (DAE) model with consistent initial values, which was tested with simple examples and a practical example.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Chemical and Biomolecular Engineering (BK21 Program), KAIST, 335 Gwahangro, Yuseong-gu, Daejeon, 305-701, Republic of Korea. martial@kaist.ac.kr

ABSTRACT

Background: Quasi-steady state approximation (QSSA) based on time-scale analysis is known to be an effective method for simplifying metabolic reaction system, but the conventional analysis becomes time-consuming and tedious when the system is large. Although there are automatic methods, they are based on eigenvalue calculations of the Jacobian matrix and on linear transformations, which have a high computation cost. A more efficient estimation approach is necessary for complex systems.

Results: This work derived new time-scale factor by focusing on the problem structure. By mathematically reasoning the balancing behavior of fast species, new time-scale criteria were derived with a simple expression that uses the Jacobian matrix directly. The algorithm requires no linear transformation or decomposition of the Jacobian matrix, which has been an essential part for previous automatic time-scaling methods. Furthermore, the proposed scale factor is estimated locally. Therefore, an iterative procedure was also developed to find the possible multiple boundary layers and to derive an appropriate reduced model.

Conclusion: By successive calculation of the newly derived time-scale criteria, it was possible to detect multiple boundary layers of full ordinary differential equation (ODE) models. Besides, the iterative procedure could derive the appropriate reduced differential algebraic equation (DAE) model with consistent initial values, which was tested with simple examples and a practical example.

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Existence of multiple boundary layers. (a) Semi-log plot of s and (b) that of e, c1, and c2 for the Michaelis-Menten system. The existence of two boundary layers at the initial region are observed.
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Figure 2: Existence of multiple boundary layers. (a) Semi-log plot of s and (b) that of e, c1, and c2 for the Michaelis-Menten system. The existence of two boundary layers at the initial region are observed.

Mentions: are considered in this study. The parameters are (k1, k2, k3, k4, k5) = (500000, 5, 1000, 100, 0.16) and the initial values are (e0, s0, c10, c20, p0) = (1, 100, 0, 0, 0) and (e0, s0, c10, c20, p0, i0, ei0) = (1, 100, 0, 0, 0, 1000, 0) [24]. As can be seen, there are two boundary layers at the initial region and near t = 900 (see Figure 1). Since p is only produced, its dynamics are not considered when searching for the fast balancing species. In the initial region, the estimated value of δt ≈ 3.96 × 10-8 and species e, s, and c1 were selected as fast variables, as expected. The subsystem composed of the chosen species was opened by the second reaction, hence the second reaction was removed from the subsystem. Based on the solution from the first iteration, species c1 and c2 were selected. Since the δt of e and s remains small, there indices were maintained as fast variables. The updated subsystem was opened by the third reaction, and consequently, the reaction was excluded. After the second iteration, the solution converged and was stored, then the open system was solved in a small interval period. Since a very large δt of s was identified in this refining step, s was excluded from the fast variable set. Finally, e, c1, and c2 were selected as QSSA variables before the second boundary layer. The values of δti at each iteration after three iterations of the iterative process are listed in Table 2. At the second boundary layer, the predefined criteria gave another iterative process and relocated the solution toward the outer area in the same manner as described above. For comparison with the conventional manual QSSA approach, the time scales of each species were also derived by mathematical balancing [25]. The meaning of the time-scales of the fast variables from the conventional derivation is the time to exit the boundary layer. Therefore, the summation of δti for every iteration until the species enters the outer region is the direct comparative value of the conventional scales (see Table 3). A similar tendency is observed between the sums of δti and the mathematical scales. As in Table 3, the sums of δti and the mathematically scaled values indicate that there are two boundary layers near t ≈ 10-8 and t ≈ 10-3 before exiting the initial regions. A semi-log plot of the full model simulation (Figure 2) supports this expectation.


New time-scale criteria for model simplification of bio-reaction systems.

Choi J, Yang KW, Lee TY, Lee SY - BMC Bioinformatics (2008)

Existence of multiple boundary layers. (a) Semi-log plot of s and (b) that of e, c1, and c2 for the Michaelis-Menten system. The existence of two boundary layers at the initial region are observed.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2553091&req=5

Figure 2: Existence of multiple boundary layers. (a) Semi-log plot of s and (b) that of e, c1, and c2 for the Michaelis-Menten system. The existence of two boundary layers at the initial region are observed.
Mentions: are considered in this study. The parameters are (k1, k2, k3, k4, k5) = (500000, 5, 1000, 100, 0.16) and the initial values are (e0, s0, c10, c20, p0) = (1, 100, 0, 0, 0) and (e0, s0, c10, c20, p0, i0, ei0) = (1, 100, 0, 0, 0, 1000, 0) [24]. As can be seen, there are two boundary layers at the initial region and near t = 900 (see Figure 1). Since p is only produced, its dynamics are not considered when searching for the fast balancing species. In the initial region, the estimated value of δt ≈ 3.96 × 10-8 and species e, s, and c1 were selected as fast variables, as expected. The subsystem composed of the chosen species was opened by the second reaction, hence the second reaction was removed from the subsystem. Based on the solution from the first iteration, species c1 and c2 were selected. Since the δt of e and s remains small, there indices were maintained as fast variables. The updated subsystem was opened by the third reaction, and consequently, the reaction was excluded. After the second iteration, the solution converged and was stored, then the open system was solved in a small interval period. Since a very large δt of s was identified in this refining step, s was excluded from the fast variable set. Finally, e, c1, and c2 were selected as QSSA variables before the second boundary layer. The values of δti at each iteration after three iterations of the iterative process are listed in Table 2. At the second boundary layer, the predefined criteria gave another iterative process and relocated the solution toward the outer area in the same manner as described above. For comparison with the conventional manual QSSA approach, the time scales of each species were also derived by mathematical balancing [25]. The meaning of the time-scales of the fast variables from the conventional derivation is the time to exit the boundary layer. Therefore, the summation of δti for every iteration until the species enters the outer region is the direct comparative value of the conventional scales (see Table 3). A similar tendency is observed between the sums of δti and the mathematical scales. As in Table 3, the sums of δti and the mathematically scaled values indicate that there are two boundary layers near t ≈ 10-8 and t ≈ 10-3 before exiting the initial regions. A semi-log plot of the full model simulation (Figure 2) supports this expectation.

Bottom Line: Therefore, an iterative procedure was also developed to find the possible multiple boundary layers and to derive an appropriate reduced model.By successive calculation of the newly derived time-scale criteria, it was possible to detect multiple boundary layers of full ordinary differential equation (ODE) models.Besides, the iterative procedure could derive the appropriate reduced differential algebraic equation (DAE) model with consistent initial values, which was tested with simple examples and a practical example.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Chemical and Biomolecular Engineering (BK21 Program), KAIST, 335 Gwahangro, Yuseong-gu, Daejeon, 305-701, Republic of Korea. martial@kaist.ac.kr

ABSTRACT

Background: Quasi-steady state approximation (QSSA) based on time-scale analysis is known to be an effective method for simplifying metabolic reaction system, but the conventional analysis becomes time-consuming and tedious when the system is large. Although there are automatic methods, they are based on eigenvalue calculations of the Jacobian matrix and on linear transformations, which have a high computation cost. A more efficient estimation approach is necessary for complex systems.

Results: This work derived new time-scale factor by focusing on the problem structure. By mathematically reasoning the balancing behavior of fast species, new time-scale criteria were derived with a simple expression that uses the Jacobian matrix directly. The algorithm requires no linear transformation or decomposition of the Jacobian matrix, which has been an essential part for previous automatic time-scaling methods. Furthermore, the proposed scale factor is estimated locally. Therefore, an iterative procedure was also developed to find the possible multiple boundary layers and to derive an appropriate reduced model.

Conclusion: By successive calculation of the newly derived time-scale criteria, it was possible to detect multiple boundary layers of full ordinary differential equation (ODE) models. Besides, the iterative procedure could derive the appropriate reduced differential algebraic equation (DAE) model with consistent initial values, which was tested with simple examples and a practical example.

Show MeSH
Related in: MedlinePlus