A simple and highly accurate numerical differentiation method for sensitivity analysis of large-scale metabolic reaction systems. (English) Zbl 1119.92035
Summary: Numerical differentiation is known to be one of the most difficult numerical calculation methods to obtain reliable calculated values at all times. A simple numerical differentiation method using a combination of finite-difference formulas, derived by approximation of Taylor-series equations, is investigated in order to efficiently perform the sensitivity analysis of large-scale metabolic reaction systems. A result of the application to four basic mathematical functions reveals that the use of the eight-point differentiation formula with a non-dimensionalized stepsize close to 0.01 mostly provides more than 14 digits of accuracy in double precision for the numerical derivatives.
Moreover, a result of the application to the modified TCA cycle model indicates that the numerical differentiation method gives the calculated values of steady-state metabolite concentrations within a range of round-off errors and also makes it possible to transform the Michaelis-Menten equations into the S-system equations having the kinetic orders whose accuracies are mostly more than 14 significant digits. Because of the simple structure of the numerical differentiation formula and its promising high accuracy, it is evident that the present numerical differentiation method is useful for the analysis of large-scale metabolic reaction systems according to the systematic procedure of Biological Systems Theory (BST).
Moreover, a result of the application to the modified TCA cycle model indicates that the numerical differentiation method gives the calculated values of steady-state metabolite concentrations within a range of round-off errors and also makes it possible to transform the Michaelis-Menten equations into the S-system equations having the kinetic orders whose accuracies are mostly more than 14 significant digits. Because of the simple structure of the numerical differentiation formula and its promising high accuracy, it is evident that the present numerical differentiation method is useful for the analysis of large-scale metabolic reaction systems according to the systematic procedure of Biological Systems Theory (BST).
MSC:
| 92C45 | Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) |
| 65D25 | Numerical differentiation |
| 92E20 | Classical flows, reactions, etc. in chemistry |
Keywords:
Biochemical systems theory; Finite-difference formula; High accuracy; Sensitivity analysis; tricarboxylic acid cycleReferences:
| [1] | Varma, A.; Morbidelli, M.; Wu, H., Parametric Sensitivity in Chemical Systems (1999), Cambridge University: Cambridge University Cambridge |
| [2] | Savageau, M. A., Biochemical systems analysis I. Some mathematical properties of the rate law for the component enzymatic reactions, J. Theor. Biol., 25, 365 (1969) |
| [3] | Savageau, M. A., Biochemical Systems Analysis: A Study of Function and Design in Molecular Biology (1976), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0398.92013 |
| [4] | Sorribas, A.; Savageau, M. A., A comparison of variant theories of intact biochemical systems. I. Enzyme-enzyme interactions and biochemical systems theory, Math. Biosci., 94, 161 (1989) · Zbl 0696.92012 |
| [5] | Atkinson, K., Elementary Numerical Analysis (1993), John Wiley & Sons: John Wiley & Sons New York · Zbl 0782.65002 |
| [6] | Shiraishi, F.; Savageau, M. A., The tricarboxylic acid cycle in Dictyostelium discoideum: systemic effects of including protein turnover in the current model, J. Biol. Chem., 268, 16917 (1993) |
| [7] | Savageau, M. A.; Sorribas, A., Recasting nonlinear differential equations as S-systems: a canonical nonlinear form, J. Theor. Biol., 141, 93 (1989) |
| [8] | Shiraishi, F.; Savageau, M. A., The tricarboxylic acid cycle in Dictyostelium discoideum: I. Formulation of alternative kinetic representations, J. Biol. Chem., 267, 22912 (1992) |
| [9] | Shiraishi, F.; Savageau, M. A., The tricarboxylic acid cycle in Dictyostelium discoideum: II. Evaluation of model consistency and robustness, J. Biol. Chem., 267, 22919 (1992) |
| [10] | Shiraishi, F.; Savageau, M. A., The tricarboxylic acid cycle in Dictyostelium discoideum: III. Analysis of steady-state and dynamic behavior, J. Biol. Chem., 267, 22926 (1992) |
| [11] | Shiraishi, F.; Savageau, M. A., The tricarboxylic acid cycle in Dictyostelium discoideum: IV. Resolution of discrepancies between alternative methods of analysis, J. Biol. Chem., 267, 22934 (1992) |
| [12] | Albe, K. R.; Wright, B. E., Systems analysis of the tricarboxylic acid cycle in Dictyostelium discoideum. II. Control analysis, J. Biol. Chem., 267, 3106 (1992) |
| [13] | Wright, B. E.; Butler, M. H.; Albe, K. R., Systems analysis of the tricarboxylic acid cycle in Dictyostelium discoideum. I. The basis for model construction, J. Biol. Chem., 267, 3101 (1992) |
| [14] | McCormick, J. M.; Salvadori, M. G., Numerical Methods in Fortran (1964), Prentice-Hall: Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0136.38604 |
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