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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

Comparison of the hybrid optimization with tolerance adjustment (HATA) with its parent algorithms of the SQP with user-supplied analytical gradient (SQP ∇user) and the STRiN with user-supplied analytical gradient and Hessian (STRiN ∇user Huser) as well as with other algorithms, such as SQP with numerical gradient by finite differentiation (SQP ∇finite), genetic algorithm (GA), and simulated annealing (SA) (A). All algorithms were initiated from an identical starting point. The objective function value at the ith iterate is registered only if f(Θi) ≤ f(Θi-1). Time efficiency of the HATA represented by histogram plot of time taken for termination of 200 runs of optimization using different starting points (B).
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Figure 4: Comparison of the hybrid optimization with tolerance adjustment (HATA) with its parent algorithms of the SQP with user-supplied analytical gradient (SQP ∇user) and the STRiN with user-supplied analytical gradient and Hessian (STRiN ∇user Huser) as well as with other algorithms, such as SQP with numerical gradient by finite differentiation (SQP ∇finite), genetic algorithm (GA), and simulated annealing (SA) (A). All algorithms were initiated from an identical starting point. The objective function value at the ith iterate is registered only if f(Θi) ≤ f(Θi-1). Time efficiency of the HATA represented by histogram plot of time taken for termination of 200 runs of optimization using different starting points (B).

Mentions: Using the tolerance adjustment, the hybrid algorithm consisting of SQP and STRiN (HATA, Figure 1) was compared with its parent algorithms and two global optimization methods. All optimizations except the genetic algorithm (GA) were initiated from an identical starting point for the numerical flux re-estimation. The GA applied does not need an external initial value set. At each initiation of the HATA trials, α was updated by choosing an integer between 1 and 10 that yields the best-conditioned Jacobian matrix of the model as mentioned previously. As shown in Figure 4-A, HATA accomplished the re-estimation with the best efficiency regarding its accuracy and speed. It took about 300 seconds until the objective function became 10-16. The SQP optimization with analytical gradient (SQP ∇user) yielded the next satisfactory result. In comparison, the SQP optimization using the numerical gradient obtained by the finite difference (SQP ∇finite) resulted in a poorer progress and was much slower and less accurate than HATA or SQP ∇user. During the optimization using SQP ∇finite, we observed a discrepancy between the gradient obtained by the finite difference and the analytical approach (A.1). The inaccuracy of the finite difference seemed to cause the poorer result of SQP ∇finite.


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

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

Comparison of the hybrid optimization with tolerance adjustment (HATA) with its parent algorithms of the SQP with user-supplied analytical gradient (SQP ∇user) and the STRiN with user-supplied analytical gradient and Hessian (STRiN ∇user Huser) as well as with other algorithms, such as SQP with numerical gradient by finite differentiation (SQP ∇finite), genetic algorithm (GA), and simulated annealing (SA) (A). All algorithms were initiated from an identical starting point. The objective function value at the ith iterate is registered only if f(Θi) ≤ f(Θi-1). Time efficiency of the HATA represented by histogram plot of time taken for termination of 200 runs of optimization using different starting points (B).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 4: Comparison of the hybrid optimization with tolerance adjustment (HATA) with its parent algorithms of the SQP with user-supplied analytical gradient (SQP ∇user) and the STRiN with user-supplied analytical gradient and Hessian (STRiN ∇user Huser) as well as with other algorithms, such as SQP with numerical gradient by finite differentiation (SQP ∇finite), genetic algorithm (GA), and simulated annealing (SA) (A). All algorithms were initiated from an identical starting point. The objective function value at the ith iterate is registered only if f(Θi) ≤ f(Θi-1). Time efficiency of the HATA represented by histogram plot of time taken for termination of 200 runs of optimization using different starting points (B).
Mentions: Using the tolerance adjustment, the hybrid algorithm consisting of SQP and STRiN (HATA, Figure 1) was compared with its parent algorithms and two global optimization methods. All optimizations except the genetic algorithm (GA) were initiated from an identical starting point for the numerical flux re-estimation. The GA applied does not need an external initial value set. At each initiation of the HATA trials, α was updated by choosing an integer between 1 and 10 that yields the best-conditioned Jacobian matrix of the model as mentioned previously. As shown in Figure 4-A, HATA accomplished the re-estimation with the best efficiency regarding its accuracy and speed. It took about 300 seconds until the objective function became 10-16. The SQP optimization with analytical gradient (SQP ∇user) yielded the next satisfactory result. In comparison, the SQP optimization using the numerical gradient obtained by the finite difference (SQP ∇finite) resulted in a poorer progress and was much slower and less accurate than HATA or SQP ∇user. During the optimization using SQP ∇finite, we observed a discrepancy between the gradient obtained by the finite difference and the analytical approach (A.1). The inaccuracy of the finite difference seemed to cause the poorer result of SQP ∇finite.

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