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Hybrid optimization for 13C metabolic flux analysis using systems parametrized by compactification.

Yang TH, Frick O, Heinzle E - BMC Syst Biol (2008)

Bottom Line: To solve the problem more efficiently, improved numerical optimization techniques are necessary.In the metabolic network studied, some fluxes were found to be either non-identifiable or nonlinearly correlated.In this way, it contributes to future quantitative studies of central metabolic networks in the framework of systems biology.

View Article: PubMed Central - HTML - PubMed

Affiliation: James Graham Brown Cancer Center & Department of Surgery, 2210 S, Brook St, Rm 342, Belknap Research Building, University of Louisville, Louisville, KY 40208, USA. th.yang@louisville.edu

ABSTRACT

Background: The importance and power of isotope-based metabolic flux analysis and its contribution to understanding the metabolic network is increasingly recognized. Its application is, however, still limited partly due to computational inefficiency. 13C metabolic flux analysis aims to compute in vivo metabolic fluxes in terms of metabolite balancing extended by carbon isotopomer balances and involves a nonlinear least-squares problem. To solve the problem more efficiently, improved numerical optimization techniques are necessary.

Results: For flux computation, we developed a gradient-based hybrid optimization algorithm. Here, independent flux variables were compactified into [0, 1)-ranged variables using a single transformation rule. The compactified parameters could be discriminated between non-identifiable and identifiable variables after model linearization. The developed hybrid algorithm was applied to the central metabolism of Bacillus subtilis with only succinate and glutamate as carbon sources. This creates difficulties caused by symmetry of succinate leading to limited introduction of 13C labeling information into the system. The algorithm was found to be superior to its parent algorithms and to global optimization methods both in accuracy and speed. The hybrid optimization with tolerance adjustment quickly converged to the minimum with close to zero deviation and exactly re-estimated flux variables. In the metabolic network studied, some fluxes were found to be either non-identifiable or nonlinearly correlated. The non-identifiable fluxes could correctly be predicted a priori using the model identification method applied, whereas the nonlinear flux correlation was revealed only by identification runs using different starting values a posteriori.

Conclusion: This fast, robust and accurate optimization method is useful for high-throughput metabolic flux analysis, a posteriori identification of possible parameter correlations, and also for Monte Carlo simulations to obtain statistical qualities for flux estimates. In this way, it contributes to future quantitative studies of central metabolic networks in the framework of systems biology.

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Related in: MedlinePlus

Behaviors of the objective function in the parameter space of non-correlated ϕ5r and ϕ6r (A) as well as correlated ϕ3r and ϕ4r (B). Two parameters were varied from 0.05 to 0.95 with a step of 0.005 while other parameters were fixed at the default solution Θdefault.
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Figure 6: Behaviors of the objective function in the parameter space of non-correlated ϕ5r and ϕ6r (A) as well as correlated ϕ3r and ϕ4r (B). Two parameters were varied from 0.05 to 0.95 with a step of 0.005 while other parameters were fixed at the default solution Θdefault.

Mentions: We could find unique flux solutions except the fluxes recognized a priori as non-identifiable or those that were a posteriori found to have a significant nonlinear correlation. Such nonlinear parameter correlation can be identified only by the tedious process of executing flux estimation using different sets of starting values a posteriori [36]. The non-correlated [0, 1)-fluxes of which true values could be re-estimated, e.g., ϕ5r and ϕ6r, give a unique minimum of f(Θ), which is obviously achieved disregarding starting points (Figure 6-A). In comparison to this, the nonlinearly correlated [0, 1)-fluxes ϕ3r and ϕ4r result in an infinite number of minima that meet the correlation shown in Figure 5-B depending on starting points (Figure 6-B). Because the objective function value is nearly zero at each minimum (f() < 1 × 10-10 with a weighting factor of 108), the system can be considered to have an infinite number of global minima when parameters are nonlinearly correlated. In this case, a global optimization method can be extremely expensive for identifying such unknown parameter correlations due to its time-inefficiency. As a consequence, a fast and reliable numerical method of flux estimation is highly desirable to repeat optimization runs at different starting points.


Hybrid optimization for 13C metabolic flux analysis using systems parametrized by compactification.

Yang TH, Frick O, Heinzle E - BMC Syst Biol (2008)

Behaviors of the objective function in the parameter space of non-correlated ϕ5r and ϕ6r (A) as well as correlated ϕ3r and ϕ4r (B). Two parameters were varied from 0.05 to 0.95 with a step of 0.005 while other parameters were fixed at the default solution Θdefault.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2333969&req=5

Figure 6: Behaviors of the objective function in the parameter space of non-correlated ϕ5r and ϕ6r (A) as well as correlated ϕ3r and ϕ4r (B). Two parameters were varied from 0.05 to 0.95 with a step of 0.005 while other parameters were fixed at the default solution Θdefault.
Mentions: We could find unique flux solutions except the fluxes recognized a priori as non-identifiable or those that were a posteriori found to have a significant nonlinear correlation. Such nonlinear parameter correlation can be identified only by the tedious process of executing flux estimation using different sets of starting values a posteriori [36]. The non-correlated [0, 1)-fluxes of which true values could be re-estimated, e.g., ϕ5r and ϕ6r, give a unique minimum of f(Θ), which is obviously achieved disregarding starting points (Figure 6-A). In comparison to this, the nonlinearly correlated [0, 1)-fluxes ϕ3r and ϕ4r result in an infinite number of minima that meet the correlation shown in Figure 5-B depending on starting points (Figure 6-B). Because the objective function value is nearly zero at each minimum (f() < 1 × 10-10 with a weighting factor of 108), the system can be considered to have an infinite number of global minima when parameters are nonlinearly correlated. In this case, a global optimization method can be extremely expensive for identifying such unknown parameter correlations due to its time-inefficiency. As a consequence, a fast and reliable numerical method of flux estimation is highly desirable to repeat optimization runs at different starting points.

Bottom Line: To solve the problem more efficiently, improved numerical optimization techniques are necessary.In the metabolic network studied, some fluxes were found to be either non-identifiable or nonlinearly correlated.In this way, it contributes to future quantitative studies of central metabolic networks in the framework of systems biology.

View Article: PubMed Central - HTML - PubMed

Affiliation: James Graham Brown Cancer Center & Department of Surgery, 2210 S, Brook St, Rm 342, Belknap Research Building, University of Louisville, Louisville, KY 40208, USA. th.yang@louisville.edu

ABSTRACT

Background: The importance and power of isotope-based metabolic flux analysis and its contribution to understanding the metabolic network is increasingly recognized. Its application is, however, still limited partly due to computational inefficiency. 13C metabolic flux analysis aims to compute in vivo metabolic fluxes in terms of metabolite balancing extended by carbon isotopomer balances and involves a nonlinear least-squares problem. To solve the problem more efficiently, improved numerical optimization techniques are necessary.

Results: For flux computation, we developed a gradient-based hybrid optimization algorithm. Here, independent flux variables were compactified into [0, 1)-ranged variables using a single transformation rule. The compactified parameters could be discriminated between non-identifiable and identifiable variables after model linearization. The developed hybrid algorithm was applied to the central metabolism of Bacillus subtilis with only succinate and glutamate as carbon sources. This creates difficulties caused by symmetry of succinate leading to limited introduction of 13C labeling information into the system. The algorithm was found to be superior to its parent algorithms and to global optimization methods both in accuracy and speed. The hybrid optimization with tolerance adjustment quickly converged to the minimum with close to zero deviation and exactly re-estimated flux variables. In the metabolic network studied, some fluxes were found to be either non-identifiable or nonlinearly correlated. The non-identifiable fluxes could correctly be predicted a priori using the model identification method applied, whereas the nonlinear flux correlation was revealed only by identification runs using different starting values a posteriori.

Conclusion: This fast, robust and accurate optimization method is useful for high-throughput metabolic flux analysis, a posteriori identification of possible parameter correlations, and also for Monte Carlo simulations to obtain statistical qualities for flux estimates. In this way, it contributes to future quantitative studies of central metabolic networks in the framework of systems biology.

Show MeSH
Related in: MedlinePlus