Skip to main content

Automatic Complexity Analysis and Model Reduction of Nonlinear Biochemical Systems

  • Conference paper
Computational Methods in Systems Biology (CMSB 2008)

Part of the book series: Lecture Notes in Computer Science ((LNBI,volume 5307))

Included in the following conference series:

Abstract

Kinetic models for biochemical systems often comprise a large amount of coupled differential equations with species concentrations varying on different time scales. In this paper we present and apply two novel methods aimed at automatic complexity and model reduction by numerical algorithms. The first method combines dynamic sensitivity analysis with singular value decomposition. The aim is to determine the minimal dimension of the kinetic model necessary to describe the active dynamics of the system accurately enough within a user-defined error tolerance for particular species concentrations and to determine each species’ contribution to the active dynamics. The second method treats the explicit numerical reduction of the model to a lower dimension according to the results of the first method and allows any species combination to be chosen as a parameterization of the reduced model which may either be tabulated in the form of look-up tables or computed in situ during numerical simulations. A reduced representation of a multiple time scale system is particularly beneficial in the context of spatiotemporal simulations which require high computational efforts. Both the complexity analysis and model reduction method operate in a fully automatic and numerically highly efficient way and have been implemented in a software package. The methods are applied to a biochemical example model describing the ERK signaling pathway. With this example, we demonstrate the value of the methods for various applications in systems biology.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Bauer, I., Finocchi, F., Duschl, W.J., Gail, H.-P., Schlöder, J.P.: Simulation of chemical reactions and dust destruction in protoplanetary accretion discs. Astron. Astrophys. 317, 273–289 (1997)

    CAS  Google Scholar 

  2. Bock, H.G.: Numerical treatment of inverse problems in chemical reaction kinetics. In: Ebert, K.H., Deuflhard, P., Jäger, W. (eds.) Modeling of Chemical Reaction Systems. Springer Series in Chemical Physics, vol. 18, pp. 102–125. Springer, Heidelberg (1981)

    Chapter  Google Scholar 

  3. Bock, H.G.: Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichlinearer Differentialgleichungen. Bonner Mathematische Schriften, vol. 183. University of Bonn, Bonn (1987)

    Google Scholar 

  4. Bock, H.G., Plitt, K.J.: A multiple shooting algorithm for direct solution of optimal control problems. In: Proc. 9th IFAC World Congress Budapest. Pergamon Press, Oxford (1984)

    Google Scholar 

  5. Bodenstein, M.: Eine Theorie der photochemischen Reaktionsgeschwindigkeiten. Z. Phys. Chem. 85, 329–397 (1913)

    Google Scholar 

  6. Chapman, D., Underhill, L.: The interaction of chlorine and hydrogen. The influence of mass. J. Chem. Soc. Trans. 103, 496–508 (1913)

    Article  CAS  Google Scholar 

  7. Cho, K.-H., Shin, S.-Y., Kim, H.-W., Wolkenhauer, O., McFerran, B., Kolch, W.: Mathematical modeling of the influence of RKIP on the ERK signaling pathway. In: Priami, C. (ed.) CMSB 2003. LNCS, vol. 2602, pp. 127–141. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  8. de Donder, T., van Rysselberghe, P.: Thermodynamic Theory of Affinity: A Book of Principles. Stanford University, Menlo Park (1936)

    Google Scholar 

  9. Gorban, A., Karlin, I.: Invariant Manifolds for Physical and Chemical Kinetics. Springer - Lecture Notes in Physics, vol. 660. Springer, Heidelberg (2005)

    Google Scholar 

  10. Gorban, A., Karlin, I., Zinovyev, A.: Constructive methods of invariant manifolds for kinetic problems. Phys. Rep. 396, 197–403 (2004)

    Article  CAS  Google Scholar 

  11. Hadjinicolaou, M., Goussis, D.A.: Asymptotic solutions of stiff PDEs with the CSP method: the reaction diffusion equation. SIAM J. Sci. Comput. 20, 781–810 (1999)

    Article  Google Scholar 

  12. Lam, S.H., Goussis, D.A.: The CSP method for simplifying kinetics. J. Chem. Kinet. 26, 461–486 (1994)

    Article  CAS  Google Scholar 

  13. Lebiedz, D.: Computing minimal entropy production trajectories: An approach to model reduction in chemical kinetics. J. Chem. Phys. 120, 6890–6897 (2004)

    Article  CAS  PubMed  Google Scholar 

  14. Lebiedz, D., Kammerer, J., Brandt-Pollmann, U.: Automatic network coupling analysis for dynamical systems based on detailed kinetic models. Phys. Rev. E 72(041911) (2005)

    Google Scholar 

  15. Leineweber, D.B.: Efficient reduced SQP methods for the optimization of chemical processes described by large sparse DAE models. Fortschritt-Berichte VDI Reihe 3, Verfahrenstechnik, vol. 613. VDI-Verlag GmbH, Düsseldorf (1999)

    Google Scholar 

  16. Leineweber, D.B., Schäfer, A., Bock, H.G., Schlöder, J.P.: An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization – part I: Theoretical aspects. Comput. Chem. Engng. 27, 157–166 (2003)

    Article  CAS  Google Scholar 

  17. Leineweber, D.B., Schäfer, A., Bock, H.G., Schlöder, J.P.: An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization – part II: Software aspects and applications. Comput. Chem. Engng. 27, 167–174 (2003)

    Article  CAS  Google Scholar 

  18. Li, G., Pope, S.B., Rabitz, H.: New approaches to determination of constrained lumping schemes for a reaction system in the whole composition space. Chem. Eng. Sci. 46, 95–111 (1991)

    Article  CAS  Google Scholar 

  19. Maas, U.: Coupling of chemical reaction with flow and molecular transport. Appl. Math. 40, 249–266 (1995)

    Google Scholar 

  20. Maas, U.: Efficient calculation of intrinsic low-dimensional manifolds for the simplification of chemical kinetics. Springer – Computing and Visualization in Science 1, 69–81 (1998)

    Article  Google Scholar 

  21. Maas, U., Pope, S.B.: Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space. Combust. Flame 88, 239–264 (1992)

    Article  CAS  Google Scholar 

  22. Michaelis, L., Menten, M.L.: Die Kinetik der Invertinwirkung. Biochem. Z. 49, 333–369 (1913)

    CAS  Google Scholar 

  23. Okino, M.S., Mavrovouniotis, M.L.: Simplification of mathematical models of chemical systems. Chem. Rev. 98, 391–406 (1998)

    Article  CAS  PubMed  Google Scholar 

  24. Petrov, V., Nikolova, E., Wolkenhauer, O.: Reduction of nonlinear dynamic systems with an application to signal transduction pathways. IET Syst. Biol. 1(1), 2–9 (2007)

    Article  CAS  PubMed  Google Scholar 

  25. Rabitz, H., Kramer, M., Dacol, D.: Sensitivity analysis in chemical kinetics. Annu. Rev. Phys. Chem. 34, 419–461 (1983)

    Article  CAS  Google Scholar 

  26. Reinhardt, V., Winckler, M., Lebiedz, D.: Approximation of slow attracting manifolds by trajectory-based optimization approaches. J. Phys. Chem. A 112, 1712–1718 (2008)

    Article  CAS  PubMed  Google Scholar 

  27. Shaik, O.S., Kammerer, J., Górecki, J., Lebiedz, D.: Derivation of a quantitative minimal model for the photosensitive Belousov-Zhabotinsky reaction from a detailed elementary-step mechanism. J. Chem. Phys. 123(234103) (2005)

    Google Scholar 

  28. Tomlin, A.S., Pilling, M.J., Turányi, T., Merkin, J.H., Brindley, J.: Mechanism reduction for the oscillatory oxidation of hydrogen sensitivity and quasi-steady state analyses. Combust. Flame 91, 107–130 (1992)

    Article  CAS  Google Scholar 

  29. Trefethen, L.N., Bau, D.: Numerical Linear Algebra. SIAM, Philadelphia (1997)

    Book  Google Scholar 

  30. Turányi, T.: Sensitivity analysis of complex kinetic systems. Tools and applications. J. Math. Chem. 5, 203–248 (1990)

    Article  Google Scholar 

  31. Warnatz, J., Maas, U., Dibble, R.W.: Combustion. Physical and Chemical Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation, 3rd edn. Springer, Heidelberg (2001)

    Google Scholar 

  32. Zobeley, J., Lebiedz, D., Kammerer, J., Ishmurzin, A., Kummer, U.: A new time-dependent complexity reduction method for biological systems. Trans. Comput. Syst. Biol. 1, 90–110 (2005)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lebiedz, D., Skanda, D., Fein, M. (2008). Automatic Complexity Analysis and Model Reduction of Nonlinear Biochemical Systems. In: Heiner, M., Uhrmacher, A.M. (eds) Computational Methods in Systems Biology. CMSB 2008. Lecture Notes in Computer Science(), vol 5307. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88562-7_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-88562-7_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-88561-0

  • Online ISBN: 978-3-540-88562-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics