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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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Flux re-estimation initiated from 200 random starting points. The fluxes which were successfully re-estimated lie on the 1:1 line when plotted against true flux values (A). Correlation analysis by plotting the transaldolase (TA) [0, 1)-flux re-estimates (ϕ4r) versus those of transketolase 1 (TK1) (ϕ3r) (B) and the [0, 1)-flux re-estimate of TK1 (C) as well as of TA (D) versus ϕ2r of glucose 6-P isomerase. The dotted line indicates the solution of ϕ3r = 0.4337 and ϕ3r = 0.6341, respectively.
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Figure 5: Flux re-estimation initiated from 200 random starting points. The fluxes which were successfully re-estimated lie on the 1:1 line when plotted against true flux values (A). Correlation analysis by plotting the transaldolase (TA) [0, 1)-flux re-estimates (ϕ4r) versus those of transketolase 1 (TK1) (ϕ3r) (B) and the [0, 1)-flux re-estimate of TK1 (C) as well as of TA (D) versus ϕ2r of glucose 6-P isomerase. The dotted line indicates the solution of ϕ3r = 0.4337 and ϕ3r = 0.6341, respectively.

Mentions: Further to this, the flux re-estimates resulting from the 200 random simulations were compared with the true flux solutions calculated from Θdefault in advance. As shown in Figure 5-A, all fluxes except the flux pairs of transketolase 1 (TK1; ν3, ν3r) and transaldolase (TA; ν4, ν4r) could be re-estimated correctly. When plotting the flux re-estimates versus the true flux values given in Figure 5, the data points except ν3, ν3r, ν4, and ν4r lie exactly on a line with a slope of 1 (1:1 line). In comparison to this, only the net fluxes of the TK1 (ν3 - ν3r) and TA (ν4 - ν4r) reactions could be recalculated properly: the net fluxes could be calculated from stoichiometric relations with other flux re-estimates. The same was observed for the net flux of glucose 6-P isomerase, of which [0, 1)-flux ϕ2r had an arbitrary value at each simulation run.


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

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

Flux re-estimation initiated from 200 random starting points. The fluxes which were successfully re-estimated lie on the 1:1 line when plotted against true flux values (A). Correlation analysis by plotting the transaldolase (TA) [0, 1)-flux re-estimates (ϕ4r) versus those of transketolase 1 (TK1) (ϕ3r) (B) and the [0, 1)-flux re-estimate of TK1 (C) as well as of TA (D) versus ϕ2r of glucose 6-P isomerase. The dotted line indicates the solution of ϕ3r = 0.4337 and ϕ3r = 0.6341, respectively.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 5: Flux re-estimation initiated from 200 random starting points. The fluxes which were successfully re-estimated lie on the 1:1 line when plotted against true flux values (A). Correlation analysis by plotting the transaldolase (TA) [0, 1)-flux re-estimates (ϕ4r) versus those of transketolase 1 (TK1) (ϕ3r) (B) and the [0, 1)-flux re-estimate of TK1 (C) as well as of TA (D) versus ϕ2r of glucose 6-P isomerase. The dotted line indicates the solution of ϕ3r = 0.4337 and ϕ3r = 0.6341, respectively.
Mentions: Further to this, the flux re-estimates resulting from the 200 random simulations were compared with the true flux solutions calculated from Θdefault in advance. As shown in Figure 5-A, all fluxes except the flux pairs of transketolase 1 (TK1; ν3, ν3r) and transaldolase (TA; ν4, ν4r) could be re-estimated correctly. When plotting the flux re-estimates versus the true flux values given in Figure 5, the data points except ν3, ν3r, ν4, and ν4r lie exactly on a line with a slope of 1 (1:1 line). In comparison to this, only the net fluxes of the TK1 (ν3 - ν3r) and TA (ν4 - ν4r) reactions could be recalculated properly: the net fluxes could be calculated from stoichiometric relations with other flux re-estimates. The same was observed for the net flux of glucose 6-P isomerase, of which [0, 1)-flux ϕ2r had an arbitrary value at each simulation run.

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