A New Time-Dependent Complexity Reduction Method for Biochemical Systems

  • Jürgen Zobeley
  • Dirk Lebiedz
  • Julia Kammerer
  • Anton Ishmurzin
  • Ursula Kummer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3380)


Systems biology aims at an understanding of increasingly large and complex cellular systems making use of computational approaches, e.g. numerical simulations. The size and complexity of the underlying biochemical reaction networks call for methods to speed up simulations and/or dissect the biochemical network into smaller subsystems which can be studied independently. Both goals can be achieved by so-called complexity reduction algorithms. However, existing complexity reduction approaches for biochemical reaction networks are mostly based on studying the steady state behavior of a system and/or are based on heuristics. Given the fact that many complex biochemical systems display highly nonlinear dynamics and that this dynamics plays a crucial role in the functioning of the organism, a new methodology has to be developed. Therefore, we present a new complexity reduction method which is time-dependent and suited not only for steady states, but for all possible dynamics of a biochemical system. It makes use of the evolution of the different time–scales in the system, allowing to reduce the number of equations necessary to describe the system which is speeding up the computation time. In addition, it is possible to study the way different variables/metabolites contribute to the reduced equation system which indicates how strongly they interact and couple. In the extreme case of variables decoupling in a specific state, the method allows the complete dissection of the system resulting in subsystems that can be studied in isolation. The whole method provides a systematic tool for an automated complexity reduction of arbitrary biochemical reaction networks. With the aid of a specific example, the oscillatory peroxidase-oxidase system, we show that coupling of time–scales depends heavily on the specific dynamics of the system. Therefore, neither computational improvement nor systematic understanding can be achieved by studying these aspects solely under steady state conditions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kitano, H.: Computational systems biology. Nature 420, 206–210 (2002), References thereinGoogle Scholar
  2. 2.
    Kauffman, K.J., Pajerowski, J.D., Jamshidi, N., Palsson, B.O., Edwards, J.E.: Description and analysis of metabolic connectivity and dynamics in the human red blood cell. Biophys. J. 83, 646–662 (2002)CrossRefGoogle Scholar
  3. 3.
    Price, N.D., Reed, J.L., Papin, J.A., Famili, I., Palsson, B.O.: Analysis of metabolic capabilities using singular value decomposition of extreme pathway matrices. Biophys. J. 84, 794–804 (2003)CrossRefGoogle Scholar
  4. 4.
    Schuster, S., Pfeiffer, T., Moldenhauer, F., Koch, I., Dandekar, T.: Exploring the pathway structure of metabolism: decomposition into subnetworks and application to Mycoplasma pneumoniae. Bioinformatics 18, 351–361 (2002)CrossRefGoogle Scholar
  5. 5.
    Holme, P., Huss, M., Jeong, H.: Subnetwork hierarchies of biochemical pathways. Bioinformatics 19, 532–538 (2003)CrossRefGoogle Scholar
  6. 6.
    Berridge, M.J., Bootman, M.D., Lipp, P.: Calcium - a life and death signal. Nature 395, 645–648 (1998)CrossRefGoogle Scholar
  7. 7.
    Petty, H.R., Worth, R.G., Kindzelskii, A.L.: Imaging sustained dissipative patterns in the metabolism of individual cells. Phys. Rev. Lett. 84, 2754–2757 (2000)CrossRefGoogle Scholar
  8. 8.
    Duysens, L.N.M., Amesz, J.: Fluorescence sprectrophotometry of reduced phosphopyridine nucleotide in intact cells in the near-ultraviolet and visible region. Biochim. Biophys. Acta 24, 19–26 (1957)CrossRefGoogle Scholar
  9. 9.
    Frenkel, R.: Control of reduced diphosphopyridine nucleotide oscillations in beef heart extracts.I. Effect of modifiers of phosphofructokinase activity. Arch. Biochem. Biophys. 125, 151–156 (1968)CrossRefGoogle Scholar
  10. 10.
    Scheeline, A., Olson, D.L., Williksen, E.P., Horras, G.A., Klein, M.L., Larter, R.: The peroxidase-oxidase oscillator and its constituent chemistries. Chem. Rev. 97, 739–756 (1997)CrossRefGoogle Scholar
  11. 11.
    Bronnikova, T.V., Fed’kina, V.R., Schaffer, W.M., Olsen, L.F.: Period-doubling bifurcations and chaos in a detailed model of the peroxidase-oxidase reaction. J. Phys. Chem. 99, 9309–9312 (1995)CrossRefGoogle Scholar
  12. 12.
    Okino, M.S., Mavrovouniotis, M.L.: Simplification of mathematical models of chemical reaction systems. Chem. Rev. 98, 391–408 (1998)CrossRefGoogle Scholar
  13. 13.
    Tomlin, A.S., Turanyi, T., Pilling, M.J.: Mathematical tools for the construction, investigation and reduction of combustion mechanisms. In: Pilling, M.J. (ed.) Low Temperature Combustion and Autoignition, pp. 293–437. Elsevier, Amsterdam (1997)CrossRefGoogle Scholar
  14. 14.
    Lam, S.H., Goussis, D.A.: The CSP method for simplifying kinetics. Int. J. Chem. Kinet. 26, 461–486 (1994), References thereinGoogle Scholar
  15. 15.
    Maas, U., Pope, S.B.: Simplifying chemical reaction kinetics: Intrinsic low-dimensional manifolds in composition space. Combustion and Flame 88, 239–264 (1992)CrossRefGoogle Scholar
  16. 16.
    Davis, M.J., Skodje, R.T.: Geometric investigation of low-dimensional manifolds in systems approaching equilibrium. J. Chem. Phys. 111, 847–859 (1999)CrossRefGoogle Scholar
  17. 17.
    Skodje, R.T., Davis, M.J.: Geometrical simplification of complex kinetic systems. J. Phys. Chem. A 105, 10356–10365 (2001)CrossRefGoogle Scholar
  18. 18.
    Roussel, M.R., Fraser, S.J.: Invariant manifold methods for metabolic model reduction. Chaos 11, 196–206 (2001)MATHCrossRefGoogle Scholar
  19. 19.
    Valorani, M., Goussis, D.A.: Explicit time-scale splitting algorithm for stiff problems: auto-ignition of gaseous mixtures behind a steady shock. J. Comput. Phys. 169, 44–79 (2001)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Schmidt, D., Blasenbrey, T., Maas, U.: Intrinsic low-dimensional manifolds of strained and unstrained flames. Combustion Theory and Modelling 2, 135–152 (1998)MATHCrossRefGoogle Scholar
  21. 21.
    Correa, C., Niemann, H., Schramm, B., Warnatz, J.: Reaction mechanisms reduction for higher hydrocarbons by the ILDM method. Proc. Comb. Inst. 28, 1607–1614 (2001)CrossRefGoogle Scholar
  22. 22.
    Voet, D., Voet, J.G.: Biochemistry. Wiley, New York (1990)Google Scholar
  23. 23.
    Agarwal, P.K., Billeter, S.R., Ravi Rajagopalan, P.T., Benkovic, S.J., Hammes-Schiffer, S.: Network of coupled promoting motions in enzyme catalysis. Proc. Natl. Acad. Sci. 99, 2794–2799 (2002)CrossRefGoogle Scholar
  24. 24.
    Segel, L.A., Slemrod, M.: The Quasi-steady state assumption: a case study in perturbation. SIAM Review 31, 446–477 (1989)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Heinrich, R., Schuster, S.: The regulation of cellular systems. Chapman and Hall, New York (1996)MATHGoogle Scholar
  26. 26.
    Reder, C.: Metabolic control theory: a structural approach. J. Theor. Biol. 135, 175–201 (1988)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Deuflhard, P., Heroth, J.: Dynamic dimension reduction in ODE models. In: Scientific Computing in Chemical Engineering, pp. 29–43. Springer, Berlin (1996)Google Scholar
  28. 28.
    Golub, G.H., van Loan, C.F.: Matrix computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)MATHGoogle Scholar
  29. 29.
    Golub, G.H., Wilkinson, J.H.: Ill-conditioned eigensystems and computation of the Jordan canonical form. SIAM review 18, 578–619 (1976)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Deuflhard, P., Nowak, U.: Extrapolation Integrators for Quasilinear Implicit ODEs. In: Large Scale Scientific Computing. Progress in Scientific Computing, vol. 7, pp. 37–50. Birkhäuser, Boston (1987)Google Scholar
  31. 31.
    Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LA-PACK Users’ Guide, 3rd edn. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  32. 32.
    Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical recipes in FORTRAN 77, 2nd edn. Cambridge University Press, Cambridge (1993)Google Scholar
  33. 33.
    Thompson, D.R., Larter, R.: Multiple time-scale analysis of two models for the peroxidase-oxidase reaction. Chaos 5, 448–457 (1995)CrossRefGoogle Scholar
  34. 34.
    Hauser, M.J.B., Kummer, U., Larsen, A.Z., Olsen, L.F.: Oscillatory dynamics protect enzymes and possibly cells against toxic intermediates. Faraday Discuss. 120, 215–227 (2001)CrossRefGoogle Scholar
  35. 35.
    Amit, A., Kindzelskii, A.L., Zanoni, J., Jarvis, J.N., Petty, H.R.: Complement deposition on immune complexes reduces the frequencies of metabolic, proteolytic, and superoxide oscillations of migrating neutrophils. Cell. Immunol. 194, 47–53 (1999)CrossRefGoogle Scholar
  36. 36.
    Klann, E., Robertson, E.D., Knapp, L.T., Sweat, J.D.: A role for superoxide in protein kinase C activation and long-term potentiation. J. Biol. Chem. 273, 4516–4522 (1998)CrossRefGoogle Scholar
  37. 37.
    Carafoli, E., Santella, L., Brance, D., Brini, M.: Generation, control, and processing of cellular calcium signals. Crit. Rev. Biochem. Mol. Biol. 36, 107–260 (2001)CrossRefGoogle Scholar
  38. 38.
    Olson, D.L., Williksen, E.P., Scheeline, A.: An experimentally based model of the Peroxidase-NADH biochemical oscillator: an enzyme-mediated chemical switch. J. Am. Chem. Soc. 117, 2–15 (1995); Biol. 36, 107–260 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jürgen Zobeley
    • 1
  • Dirk Lebiedz
    • 2
  • Julia Kammerer
    • 2
  • Anton Ishmurzin
    • 2
  • Ursula Kummer
    • 1
  1. 1.EML Research gGmbHHeidelbergGermany
  2. 2.IWRUniversity of HeidelbergHeidelbergGermany

Personalised recommendations