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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

Decrease of the objective function at each termination of SQP sub-optimization using tolerance adjustment (A) and its comparison with SQP carried out at a constant tolerance during optimization (B).
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Figure 3: Decrease of the objective function at each termination of SQP sub-optimization using tolerance adjustment (A) and its comparison with SQP carried out at a constant tolerance during optimization (B).

Mentions: Prior to testing the hybrid optimization with tolerance adjustment (HATA), we examined whether the tolerance adjustment is beneficial for optimization. This was checked by performing the SQP optimization by providing the gradients for the objective function and for the flux inequality constraints (∇c = (-∂νdepend/∂Θ)T) analytically. As shown in Figure 3-A, the objective function value f(Θ*) of each optimization trial decreased with respect to tolerance adjusted at each optimization restart. It was observed that f(Θ*k) <f(Θ*k-1) always holds when starting the kth trial from the (k - 1)th local minimizer Θ*k-1. Restarting the failed (k - 1)th trial from the feasible iterate Θ° recorded for the smallest function value up to the current trial and increasing the tolerance was observed to give the same result, i.e., f(Θ*k) <f(Θ°). The efficiency of tolerance adjustment was further compared to the SQP optimization carried out at a constant tolerance of 1 × 10-20 (Figure 3-B). The SQP using the tolerance adjustment was observed to be more efficient in accuracy but more time-consuming than the case without adjusting. The SQP without adjustment reached a local minimum of around 10-7 much more rapidly but did not result in any further improvement, whereas the SQP with adjustment made slower but continuous progression. This gives an idea that the tolerance adjustment strategy might be useful to escape from possible local stationary regions and to achieve a lower minimum.


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

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

Decrease of the objective function at each termination of SQP sub-optimization using tolerance adjustment (A) and its comparison with SQP carried out at a constant tolerance during optimization (B).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 3: Decrease of the objective function at each termination of SQP sub-optimization using tolerance adjustment (A) and its comparison with SQP carried out at a constant tolerance during optimization (B).
Mentions: Prior to testing the hybrid optimization with tolerance adjustment (HATA), we examined whether the tolerance adjustment is beneficial for optimization. This was checked by performing the SQP optimization by providing the gradients for the objective function and for the flux inequality constraints (∇c = (-∂νdepend/∂Θ)T) analytically. As shown in Figure 3-A, the objective function value f(Θ*) of each optimization trial decreased with respect to tolerance adjusted at each optimization restart. It was observed that f(Θ*k) <f(Θ*k-1) always holds when starting the kth trial from the (k - 1)th local minimizer Θ*k-1. Restarting the failed (k - 1)th trial from the feasible iterate Θ° recorded for the smallest function value up to the current trial and increasing the tolerance was observed to give the same result, i.e., f(Θ*k) <f(Θ°). The efficiency of tolerance adjustment was further compared to the SQP optimization carried out at a constant tolerance of 1 × 10-20 (Figure 3-B). The SQP using the tolerance adjustment was observed to be more efficient in accuracy but more time-consuming than the case without adjusting. The SQP without adjustment reached a local minimum of around 10-7 much more rapidly but did not result in any further improvement, whereas the SQP with adjustment made slower but continuous progression. This gives an idea that the tolerance adjustment strategy might be useful to escape from possible local stationary regions and to achieve a lower minimum.

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