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2015, 2015(special): 705-722. doi: 10.3934/proc.2015.0705

## Reduction of a kinetic model of active export of importins

 1 Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

Received  August 2014 Revised  February 2015 Published  November 2015

We study a kinetic model of active export of importins. The kinetic model is written as a system of ordinary differential equations. We developed some model reduction techniques to simplify the system. The techniques are: removal of very slow reactions, quasi-steady state approximation and simplification of kinetic equations based on stoichiometric conservation laws. Local sensitivity analysis is used for the identification of critical model parameters. After model reduction, the numbers of reactions and species are reduced from $28$ and $29$ to $20$ and $20$, respectively. The reduced model and original model are compared in numerical simulations using SBedit tools for Matlab, and the methods of further model reduction are discussed. Interestingly, we investigate an iterative algorithm based on the Duhamel iterates to calculate the analytical approximate solutions of the complex non--linear chemical kinetics. This technique plays as an explicit formula that can be studied in detail for wide regions of concentrations for optimization and parameter identification purposes. It seems that the third iterative solution of the suggested method is significantly close to the actual solution of the kinetic models in most cases.
Citation: Sarbaz H. A. Khoshnaw. Reduction of a kinetic model of active export of importins. Conference Publications, 2015, 2015 (special) : 705-722. doi: 10.3934/proc.2015.0705
##### References:
 [1] A. N. Kolodkin, H. V. Westerhoff, F. J. Bruggeman, N. Plant, M. J. Moné, B. M. Bakker, M. J. Campbell, V. Leeuwen, P. T. M. Johannes, C. Carlberg and J. L. Snoep, Design principles of nuclear receptor signaling: how complex networking improves signal transduction, Molecular Systems Biology, 6 (2010) 446-460. [2] O. Radulescu, A. N. Gorban, A. Zinovyev and A. Lilienbaum, Robust simplifications of multiscale biochemical networks, BMC systems biology, 2 (2008) 86-86. [3] A. N. Gorban, O. Radulescu and A. Y. Zinovyev, Asymptotology of chemical reaction networks, Chemical Engineering Science, 65 (2010) 2310-2324. [4] R. Hannemann-Tamás, A. Gábor, G. Szederkényi and K. M. Hangos, Model complexity reduction of chemical reaction networks using mixed-integer quadratic programming, Computers and Mathematics with Applications, 65 (2013) 1575-1595. [5] G. S. Yablonskii, V. I. Bykov, A. N. Gorban, V. I. Elohin, Kinetic Models of Catalytic Reactions, Elsevier, R. G. Compton (Ed.) Series omprehensive Chemical Kinetics, New York, 32( 1991). [6] O. Radulescu, A. N. Gorban, A. Zinovyev and V. Noel, Reduction of dynamical biochemical reactions networks in computational biology, Frontiers in genetics, 3 (2012) 131-148. [7] J. Choi, K. Yang, T. Lee and S. Y. Lee, New time-scale criteria for model simplification of bio-reaction systems, BMC Bioinformatics, 9 (2008) 338-346. [8] Z. Huang, Y. Chu and J. Hahn, Model simplification procedure for signal transduction pathway models: An application to IL-6 signaling, Chemical Engineering Science, 65 (2010) 1964-1975. [9] M. S. Okino and M. L. Mavrovouniotis, Simplification of Mathematical Models of Chemical Reaction Systems, Chemical Reviews, 98 (1998) 391-408. [10] L. Petzold, Model reduction for chemical kinetics: An optimization approach, AIChE Journal, 45 (1999) 869-886. [11] K. R. Schneider and T. Wilhelm, Model reduction by extended quasi-steady state approximation, J. Math. Biol., 40 (2000) 443-450. [12] N. Vora and P. Daoutidis, Non-linear model reduction of chemical reaction systems, AIChE Journal, 47 (2001) 2320-2332. [13] E. Kutumova, A. Zinovyev, R. Sharipov and F. Kolpakov, Model composition through model reduction: a combined model of CD95 and NF-kappaB signaling pathways, BMC Systems Biology,7 (2013) 13-34 & preprint, arXiv:1310.6314. [14] M. Bodenstein, Eine Theorie der Photochemischen Reaktionsgeschwindigkeiten, Z. Phys. Chem., 85 (1913) 329-397. [15] G. E. Briggs and J. B. Haldane, A Note on the Kinetics of Enzyme Action, Biochemical Journal, 19 (1925) 338-339. [16] L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, SIAM Rev., 31 (1989) 446-477. [17] A. N. Gorban and M. Shahzad, The Michaelis-Menten Stueckelberg Theorem, Entropy, 13 (2011) 966-1019. [18] A. S. Tomlin, M. J. Pilling, T. Turányi, J. H. Merkin and J. Brindley, Mechanism reduction for the oscillatory oxidation of hydrogen: sensitivity and quasi-steady state analyses, Combustion and Flame, 91 (1992) 107-130. [19] T. Turányi, Sensitivity analysis of complex kinetic systems, Tools and Applications, Journal of mathematical chemistry, 5 (1990) 203-248. [20] Z. Zi, Sensitivity analysis approaches applied to systems biology models, IET systems biology, 5 (2011) 336-346. [21] H. Rabitz, M. Kramer and D. Dacol, Sensitivity Analysis in Chemical Kinetics, Annual Reviews Physics Chemistry, 34 (1983) 419-461. [22] SimBiology., Available from: http://using-simBiology-for-pharmacokinetic-and-mechanistic-modeling & http://www.mathworks.co.uk/products/simbiology/. [23] BioSens, Available from: http://www.chemengr.ucsb.edu/ceweb/faculty/doyle/biosens/ BioSens.htm. [24] Z. Zi, Y. Zheng, A. E. Rundell and E. Klipp, SBML-SAT: A systems biology markup language (SBML) based sensitivity analysis tool, BMC Bioinf., 9 (2008) 342-356. [25] H. Schmidt and M. Jirstrand, Systems biology toolbox for MATLAB: A computational platform for research in systems biology, Bioinformatics, 22 (2006) 514-515. [26] M. Rodriguez-Fernandez and J. R. Banga, SensSB: A software toolbox for the development and sensitivity analysis of systems biology models, Bioinformatics, 26 (2010) 1675-1676. [27] D. Chandran, F. T. Bergmann and H. M. Sauro, TinkerCell: modular CAD tool for synthetic biology, J. Biol. Eng., 3 (2009) 19-36. [28] I. Segal, Non-linear semi-groups, The Annals of Mathematics, 78 (1963) 339-364. [29] T. Tao, Nonlinear dispersive equations: local and global analysis, CBMS Regional Conference Series in Mathematics, 106 (2006).

show all references

##### References:
 [1] A. N. Kolodkin, H. V. Westerhoff, F. J. Bruggeman, N. Plant, M. J. Moné, B. M. Bakker, M. J. Campbell, V. Leeuwen, P. T. M. Johannes, C. Carlberg and J. L. Snoep, Design principles of nuclear receptor signaling: how complex networking improves signal transduction, Molecular Systems Biology, 6 (2010) 446-460. [2] O. Radulescu, A. N. Gorban, A. Zinovyev and A. Lilienbaum, Robust simplifications of multiscale biochemical networks, BMC systems biology, 2 (2008) 86-86. [3] A. N. Gorban, O. Radulescu and A. Y. Zinovyev, Asymptotology of chemical reaction networks, Chemical Engineering Science, 65 (2010) 2310-2324. [4] R. Hannemann-Tamás, A. Gábor, G. Szederkényi and K. M. Hangos, Model complexity reduction of chemical reaction networks using mixed-integer quadratic programming, Computers and Mathematics with Applications, 65 (2013) 1575-1595. [5] G. S. Yablonskii, V. I. Bykov, A. N. Gorban, V. I. Elohin, Kinetic Models of Catalytic Reactions, Elsevier, R. G. Compton (Ed.) Series omprehensive Chemical Kinetics, New York, 32( 1991). [6] O. Radulescu, A. N. Gorban, A. Zinovyev and V. Noel, Reduction of dynamical biochemical reactions networks in computational biology, Frontiers in genetics, 3 (2012) 131-148. [7] J. Choi, K. Yang, T. Lee and S. Y. Lee, New time-scale criteria for model simplification of bio-reaction systems, BMC Bioinformatics, 9 (2008) 338-346. [8] Z. Huang, Y. Chu and J. Hahn, Model simplification procedure for signal transduction pathway models: An application to IL-6 signaling, Chemical Engineering Science, 65 (2010) 1964-1975. [9] M. S. Okino and M. L. Mavrovouniotis, Simplification of Mathematical Models of Chemical Reaction Systems, Chemical Reviews, 98 (1998) 391-408. [10] L. Petzold, Model reduction for chemical kinetics: An optimization approach, AIChE Journal, 45 (1999) 869-886. [11] K. R. Schneider and T. Wilhelm, Model reduction by extended quasi-steady state approximation, J. Math. Biol., 40 (2000) 443-450. [12] N. Vora and P. Daoutidis, Non-linear model reduction of chemical reaction systems, AIChE Journal, 47 (2001) 2320-2332. [13] E. Kutumova, A. Zinovyev, R. Sharipov and F. Kolpakov, Model composition through model reduction: a combined model of CD95 and NF-kappaB signaling pathways, BMC Systems Biology,7 (2013) 13-34 & preprint, arXiv:1310.6314. [14] M. Bodenstein, Eine Theorie der Photochemischen Reaktionsgeschwindigkeiten, Z. Phys. Chem., 85 (1913) 329-397. [15] G. E. Briggs and J. B. Haldane, A Note on the Kinetics of Enzyme Action, Biochemical Journal, 19 (1925) 338-339. [16] L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, SIAM Rev., 31 (1989) 446-477. [17] A. N. Gorban and M. Shahzad, The Michaelis-Menten Stueckelberg Theorem, Entropy, 13 (2011) 966-1019. [18] A. S. Tomlin, M. J. Pilling, T. Turányi, J. H. Merkin and J. Brindley, Mechanism reduction for the oscillatory oxidation of hydrogen: sensitivity and quasi-steady state analyses, Combustion and Flame, 91 (1992) 107-130. [19] T. Turányi, Sensitivity analysis of complex kinetic systems, Tools and Applications, Journal of mathematical chemistry, 5 (1990) 203-248. [20] Z. Zi, Sensitivity analysis approaches applied to systems biology models, IET systems biology, 5 (2011) 336-346. [21] H. Rabitz, M. Kramer and D. Dacol, Sensitivity Analysis in Chemical Kinetics, Annual Reviews Physics Chemistry, 34 (1983) 419-461. [22] SimBiology., Available from: http://using-simBiology-for-pharmacokinetic-and-mechanistic-modeling & http://www.mathworks.co.uk/products/simbiology/. [23] BioSens, Available from: http://www.chemengr.ucsb.edu/ceweb/faculty/doyle/biosens/ BioSens.htm. [24] Z. Zi, Y. Zheng, A. E. Rundell and E. Klipp, SBML-SAT: A systems biology markup language (SBML) based sensitivity analysis tool, BMC Bioinf., 9 (2008) 342-356. [25] H. Schmidt and M. Jirstrand, Systems biology toolbox for MATLAB: A computational platform for research in systems biology, Bioinformatics, 22 (2006) 514-515. [26] M. Rodriguez-Fernandez and J. R. Banga, SensSB: A software toolbox for the development and sensitivity analysis of systems biology models, Bioinformatics, 26 (2010) 1675-1676. [27] D. Chandran, F. T. Bergmann and H. M. Sauro, TinkerCell: modular CAD tool for synthetic biology, J. Biol. Eng., 3 (2009) 19-36. [28] I. Segal, Non-linear semi-groups, The Annals of Mathematics, 78 (1963) 339-364. [29] T. Tao, Nonlinear dispersive equations: local and global analysis, CBMS Regional Conference Series in Mathematics, 106 (2006).

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