American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On control synthesis for uncertain dynamical discrete-time systems through polyhedral techniques
  • PROC Home
  • This Issue
  • Next Article
    Non-holonomic constraints and their impact on discretizations of Klein-Gordon lattice dynamical models
2015, 2015(special): 705-722. doi: 10.3934/proc.2015.0705

Reduction of a kinetic model of active export of importins

Sarbaz H. A. Khoshnaw 1,

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.
Keywords: biochemical reaction networks, Quasi-steady state approximation, mathematical modelling, model reduction..
Mathematics Subject Classification: Primary: 92C42, 92C45; Secondary: 92B0.
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.   Google Scholar

[2]

O. Radulescu, A. N. Gorban, A. Zinovyev and A. Lilienbaum, Robust simplifications of multiscale biochemical networks,, BMC systems biology, 2 (2008), 86.   Google Scholar

[3]

A. N. Gorban, O. Radulescu and A. Y. Zinovyev, Asymptotology of chemical reaction networks,, Chemical Engineering Science, 65 (2010), 2310.   Google Scholar

[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.   Google Scholar

[5]

G. S. Yablonskii, V. I. Bykov, A. N. Gorban, V. I. Elohin, Kinetic Models of Catalytic Reactions,, Elsevier, (1991).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[9]

M. S. Okino and M. L. Mavrovouniotis, Simplification of Mathematical Models of Chemical Reaction Systems,, Chemical Reviews, 98 (1998), 391.   Google Scholar

[10]

L. Petzold, Model reduction for chemical kinetics: An optimization approach,, AIChE Journal, 45 (1999), 869.   Google Scholar

[11]

K. R. Schneider and T. Wilhelm, Model reduction by extended quasi-steady state approximation,, J. Math. Biol., 40 (2000), 443.   Google Scholar

[12]

N. Vora and P. Daoutidis, Non-linear model reduction of chemical reaction systems,, AIChE Journal, 47 (2001), 2320.   Google Scholar

[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.   Google Scholar

[14]

M. Bodenstein, Eine Theorie der Photochemischen Reaktionsgeschwindigkeiten,, Z. Phys. Chem., 85 (1913), 329.   Google Scholar

[15]

G. E. Briggs and J. B. Haldane, A Note on the Kinetics of Enzyme Action,, Biochemical Journal, 19 (1925), 338.   Google Scholar

[16]

L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation,, SIAM Rev., 31 (1989), 446.   Google Scholar

[17]

A. N. Gorban and M. Shahzad, The Michaelis-Menten Stueckelberg Theorem,, Entropy, 13 (2011), 966.   Google Scholar

[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.   Google Scholar

[19]

T. Turányi, Sensitivity analysis of complex kinetic systems, Tools and Applications,, Journal of mathematical chemistry, 5 (1990), 203.   Google Scholar

[20]

Z. Zi, Sensitivity analysis approaches applied to systems biology models,, IET systems biology, 5 (2011), 336.   Google Scholar

[21]

H. Rabitz, M. Kramer and D. Dacol, Sensitivity Analysis in Chemical Kinetics,, Annual Reviews Physics Chemistry, 34 (1983), 419.   Google Scholar

[22]

SimBiology., Available from: http://using-simBiology-for-pharmacokinetic-and-mechanistic-modeling &, http://www.mathworks.co.uk/products/simbiology/., ().   Google Scholar

[23]

BioSens, Available, from: http://www.chemengr.ucsb.edu/ceweb/faculty/doyle/biosens/ BioSens.htm., ().   Google Scholar

[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.   Google Scholar

[25]

H. Schmidt and M. Jirstrand, Systems biology toolbox for MATLAB: A computational platform for research in systems biology,, Bioinformatics, 22 (2006), 514.   Google Scholar

[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.   Google Scholar

[27]

D. Chandran, F. T. Bergmann and H. M. Sauro, TinkerCell: modular CAD tool for synthetic biology,, J. Biol. Eng., 3 (2009), 19.   Google Scholar

[28]

I. Segal, Non-linear semi-groups,, The Annals of Mathematics, 78 (1963), 339.   Google Scholar

[29]

T. Tao, Nonlinear dispersive equations: local and global analysis,, CBMS Regional Conference Series in Mathematics, 106 (2006).   Google Scholar

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.   Google Scholar

[2]

O. Radulescu, A. N. Gorban, A. Zinovyev and A. Lilienbaum, Robust simplifications of multiscale biochemical networks,, BMC systems biology, 2 (2008), 86.   Google Scholar

[3]

A. N. Gorban, O. Radulescu and A. Y. Zinovyev, Asymptotology of chemical reaction networks,, Chemical Engineering Science, 65 (2010), 2310.   Google Scholar

[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.   Google Scholar

[5]

G. S. Yablonskii, V. I. Bykov, A. N. Gorban, V. I. Elohin, Kinetic Models of Catalytic Reactions,, Elsevier, (1991).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[9]

M. S. Okino and M. L. Mavrovouniotis, Simplification of Mathematical Models of Chemical Reaction Systems,, Chemical Reviews, 98 (1998), 391.   Google Scholar

[10]

L. Petzold, Model reduction for chemical kinetics: An optimization approach,, AIChE Journal, 45 (1999), 869.   Google Scholar

[11]

K. R. Schneider and T. Wilhelm, Model reduction by extended quasi-steady state approximation,, J. Math. Biol., 40 (2000), 443.   Google Scholar

[12]

N. Vora and P. Daoutidis, Non-linear model reduction of chemical reaction systems,, AIChE Journal, 47 (2001), 2320.   Google Scholar

[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.   Google Scholar

[14]

M. Bodenstein, Eine Theorie der Photochemischen Reaktionsgeschwindigkeiten,, Z. Phys. Chem., 85 (1913), 329.   Google Scholar

[15]

G. E. Briggs and J. B. Haldane, A Note on the Kinetics of Enzyme Action,, Biochemical Journal, 19 (1925), 338.   Google Scholar

[16]

L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation,, SIAM Rev., 31 (1989), 446.   Google Scholar

[17]

A. N. Gorban and M. Shahzad, The Michaelis-Menten Stueckelberg Theorem,, Entropy, 13 (2011), 966.   Google Scholar

[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.   Google Scholar

[19]

T. Turányi, Sensitivity analysis of complex kinetic systems, Tools and Applications,, Journal of mathematical chemistry, 5 (1990), 203.   Google Scholar

[20]

Z. Zi, Sensitivity analysis approaches applied to systems biology models,, IET systems biology, 5 (2011), 336.   Google Scholar

[21]

H. Rabitz, M. Kramer and D. Dacol, Sensitivity Analysis in Chemical Kinetics,, Annual Reviews Physics Chemistry, 34 (1983), 419.   Google Scholar

[22]

SimBiology., Available from: http://using-simBiology-for-pharmacokinetic-and-mechanistic-modeling &, http://www.mathworks.co.uk/products/simbiology/., ().   Google Scholar

[23]

BioSens, Available, from: http://www.chemengr.ucsb.edu/ceweb/faculty/doyle/biosens/ BioSens.htm., ().   Google Scholar

[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.   Google Scholar

[25]

H. Schmidt and M. Jirstrand, Systems biology toolbox for MATLAB: A computational platform for research in systems biology,, Bioinformatics, 22 (2006), 514.   Google Scholar

[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.   Google Scholar

[27]

D. Chandran, F. T. Bergmann and H. M. Sauro, TinkerCell: modular CAD tool for synthetic biology,, J. Biol. Eng., 3 (2009), 19.   Google Scholar

[28]

I. Segal, Non-linear semi-groups,, The Annals of Mathematics, 78 (1963), 339.   Google Scholar

[29]

T. Tao, Nonlinear dispersive equations: local and global analysis,, CBMS Regional Conference Series in Mathematics, 106 (2006).   Google Scholar

[1]

Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945
[2]

Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi, Gregory I. Sivashinsky. A fully nonlinear equation for the flame front in a quasi-steady combustion model. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1415-1446. doi: 10.3934/dcds.2010.27.1415
[3]

Murat Arcak, Eduardo D. Sontag. A passivity-based stability criterion for a class of biochemical reaction networks. Mathematical Biosciences & Engineering, 2008, 5 (1) : 1-19. doi: 10.3934/mbe.2008.5.1
[4]

Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion. Networks & Heterogeneous Media, 2016, 11 (4) : 603-625. doi: 10.3934/nhm.2016011
[5]

Sze-Bi Hsu, Feng-Bin Wang. On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1479-1501. doi: 10.3934/cpaa.2011.10.1479
[6]

Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1373-1410. doi: 10.3934/dcdsb.2014.19.1373
[7]

Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105
[8]

Thomas Lepoutre, Salomé Martínez. Steady state analysis for a relaxed cross diffusion model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 613-633. doi: 10.3934/dcds.2014.34.613
[9]

Qi Wang. On the steady state of a shadow system to the SKT competition model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2941-2961. doi: 10.3934/dcdsb.2014.19.2941
[10]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39
[11]

Alessandro Bertuzzi, Alberto d'Onofrio, Antonio Fasano, Alberto Gandolfi. Modelling cell populations with spatial structure: Steady state and treatment-induced evolution. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 161-186. doi: 10.3934/dcdsb.2004.4.161
[12]

Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Fast algorithms for the approximation of a traffic flow model on networks. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 427-448. doi: 10.3934/dcdsb.2006.6.427
[13]

Martin Redmann, Peter Benner. Approximation and model order reduction for second order systems with Levy-noise. Conference Publications, 2015, 2015 (special) : 945-953. doi: 10.3934/proc.2015.0945
[14]

Alan D. Rendall. Multiple steady states in a mathematical model for interactions between T cells and macrophages. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 769-782. doi: 10.3934/dcdsb.2013.18.769
[15]

Antonio Fasano, Marco Gabrielli, Alberto Gandolfi. Investigating the steady state of multicellular spheroids by revisiting the two-fluid model. Mathematical Biosciences & Engineering, 2011, 8 (2) : 239-252. doi: 10.3934/mbe.2011.8.239
[16]

Lijuan Wang, Hongling Jiang, Ying Li. Positive steady state solutions of a plant-pollinator model with diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1805-1819. doi: 10.3934/dcdsb.2015.20.1805
[17]

Youcef Mammeri, Damien Sellier. A surface model of nonlinear, non-steady-state phloem transport. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1055-1069. doi: 10.3934/mbe.2017055
[18]

Mei-hua Wei, Jianhua Wu, Yinnian He. Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1147-1167. doi: 10.3934/cpaa.2015.14.1147
[19]

Hua Nie, Wenhao Xie, Jianhua Wu. Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1279-1297. doi: 10.3934/cpaa.2013.12.1279
[20]

Zhenzhen Zheng, Ching-Shan Chou, Tau-Mu Yi, Qing Nie. Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1135-1168. doi: 10.3934/mbe.2011.8.1135

 Impact Factor: 

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences