Numerical Analysis of Multiple Steady States, Limit Cycles, Period-Doubling, and Chaos in Enzymatic Reactions Involving Oxidation of L-tyrosine to Produce L-DOPA


We analyze the nonlinear dynamics of an isothermal system involving complex enzymatic reactions for L-DOPA (L-3,4-dihydroxyphenylalanine) production by numerical simulation. The mass action kinetics of the system forms a family of 9ordinary differential equations with 22 parameters. The multiple steady states are calculated by the chemical reaction network toolbox. Starting from one of the steady states, a limit point is guaranteed to be detected due to a change in the system parameters using the numerical continuation software MatCont. Other bifurcations are also obtained via the bifurcation continuations of the limit point, such as cusp bifurcations, Hopf bifurcations, limit cycles, zero Hopf bifurcations, generalized Hopf bifurcations, period-doubling, and so on. A transition of a period-doubling bifurcation to chaos occurs by numerical simulations. Positive values, 0.25~0.71, of Lyapunov exponents are obtained for the chaotic dynamics. Poincare maps and power spectrum densities are also plot. The Feigenbaum constant is computed to be 4.681~4.705.

This is a preview of subscription content, access via your institution.

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.


  1. 1

    Scheeline, A., Olson, D.L., Williksen, E.P., Horras, G.A., Klein, M.L., and Larter, R., The peroxidase−oxidase oscillator and its constituent chemistries, Chem. Rev., 1997, vol. 97, no. 3, p. 739.

    CAS  Article  Google Scholar 

  2. 2

    Degn, H., Olsen, L.F., and Perram, J.W., Bistability, oscillation, and chaos in an enzyme reaction, Ann. N. Y. Acad. Sci., 1979, vol. 316, no. 1, p. 623.

    CAS  Article  Google Scholar 

  3. 3

    Hauck, T. and Schneider, F.W., Mixed-mode and quasiperiodic oscillations in the peroxidase-oxidase reaction, J. Phys. Chem., 1993, vol. 97, no. 2, p. 391.

    CAS  Article  Google Scholar 

  4. 4

    Samples, M., Hung, Y.-F., and Ross, J., Further experimental studies on the horseradish peroxidase-oxidase reaction, J. Phys. Chem., 1992, vol. 96, no. 18, p. 7338.

    CAS  Article  Google Scholar 

  5. 5

    Samples, M. and Ross, J., Theoretical studies and comparison with experiments on the horseradish peroxidase-oxidase reaction, J. Phys. Chem., 1992, vol. 96, no. 18, p. 7342.

    CAS  Article  Google Scholar 

  6. 6

    Geest, T., Steinmetz, C.G., Larter, R., and Olsen, L.F., Period-doubling bifurcations and chaos in an enzyme reaction, J. Phys. Chem., 1992, vol. 96, no. 14, p. 5678.

    CAS  Article  Google Scholar 

  7. 7

    Hauser, M.J.B. and Olsen, L.F., Mixed-mode oscillations and homoclinic chaos in an enzyme reaction, J. Chem. Soc., Faraday Trans., 1996, vol. 92, no. 16, p. 2857.

    CAS  Article  Google Scholar 

  8. 8

    Olsen, L.F. and Degn, H., Chaos in an enzyme reaction, Nature, 1977, vol. 267, p. 177.

    CAS  Article  Google Scholar 

  9. 9

    Steinmetz, C.G., Geest, T., and Larter, R., Universality in the peroxidase-oxidase reaction: Period doublings, chaos, period three, and unstable limit cycles, J. Phys. Chem., 1993, vol. 97, no. 21, p. 5649.

    CAS  Article  Google Scholar 

  10. 10

    Hauck, T. and Schneider, F.W., Chaos in a Farey sequence through period doubling in the peroxidase-oxidase reaction, J. Phys. Chem., 1994, vol. 98, no. 8, p. 2072.

    CAS  Article  Google Scholar 

  11. 11

    Chattopadhyay, S., Datta, S.K., and Mahato, S.B., Production of L-DOPA from cell suspension culture of Mucuna pruriens f. pruriens, Plant Cell Rep., 1994, vol. 13, no. 9, p. 519.

    CAS  Article  PubMed  Google Scholar 

  12. 12

    Foor, F., Morin, N., and Bostian, K.A., Production of L-dihydroxyphenylalanine in Escherichia coli with the tyrosine phenol-lyase gene cloned from Erwinia herbicola, Appl. Environ. Microbiol., 1993, vol. 59, no. 9, p. 3070.

    CAS  Article  Google Scholar 

  13. 13

    Vilanova, E., Manjon, A., and Iborra, J.L., Tyrosine hydroxylase activity of immobilized tyrosinase on enzacryl-AA and CPG-AA supports: Stabilization and properties, Biotechnol. Bioeng., 1984, vol. 26, no. 11, p. 1306.

    CAS  Article  Google Scholar 

  14. 14

    Freire, D.M.G., Carvalho, G.M.J., and Alves, T.L.M., L-DOPA production by immobilized tyrosinase, Appl. Biochem. Biotechnol., 2000, vols. 84–86, p. 791.

    PubMed  Google Scholar 

  15. 15

    Saville, B.A. and Seetharam, G., L-DOPA production from tyrosinase immobilized on zeolite, Enzyme Microb. Technol., 2002, vol. 31, no. 6, p. 747.

    Article  Google Scholar 

  16. 16

    Horn, F., Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Ration. Mech. Anal., 1972, vol. 49, no. 3, p. 172.

    Article  Google Scholar 

  17. 17

    Horn, F. and Jackson, R., General mass action kinetics, Arch. Ration. Mech. Anal., 1972, vol. 47, no. 2, p. 81.

    Article  Google Scholar 

  18. 18

    Feinberg, M., Complex balancing in general kinetic systems, Arch. Ration. Mech. Anal., 1972, vol. 49, no. 3, p. 187.

    Article  Google Scholar 

  19. 19

    Ji, H., Ellison, P.R., Knight, D., and Feinberg, M., The Chemical Reaction Network Toolbox, Version 2.3. Accessed March 15, 2017.

  20. 20

    Feinberg, M., Chemical reaction network structure and the stability of complex isothermal reactors—II. Multiple steady states for networks of deficiency one, Chem. Eng. Sci., 1988, vol. 43, no. 1, p. 1.

    CAS  Article  Google Scholar 

  21. 21

    Ellison, P. and Feinberg, M., How catalytic mechanisms reveal themselves in multiple steady-state data: I. Basic principles, J. Mol. Catal. A: Chem., 2000, vol. 154, nos. 1–2, p. 155.

    CAS  Article  Google Scholar 

  22. 22

    Ajbar, A. and Alhumaizi, K., Dynamics of the Chemostat: A Bifurcation Theory Approach, Boca Raton, Fla.: Chapman and Hall/CRC, 2012.

    Google Scholar 

  23. 23

    Blanchard, P., Devaney, R., and Hall, G., Differential Equations, London: Thompson, 2006.

    Google Scholar 

  24. 24

    Dhooge, A., Govaerts, W., Kuznetsov, Yu.A., Meijer, H.G.E., and Sautois, B., New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Modell. Dyn. Syst., 2008, vol. 14, no. 2, pp. 147–175.

    Article  Google Scholar 

  25. 25

    Meijer, H. and Govaerts, W., MatCont: Numerical Bifurcation Analysis Toolbox in Matlab. Accessed May 25, 2016.

  26. 26

    Makeev, A.G. and Nieuwenhuys, B.E., Mathematical modeling of the NO+H2/Pt(100) reaction: “Surface explosion,” kinetic oscillations and chaos, J. Chem. Phys., 1988, vol. 108, no. 9, p. 3740.

    Article  Google Scholar 

  27. 27

    Sensse, A., Hauser, M.J.B., and Eiswirth, M., Feedback loops for Shil’nikov chaos: The peroxidase-oxidase reaction, J. Chem. Phys., 2006, vol. 125, no. 1, p. 014901.

    Article  Google Scholar 

  28. 28

    Ho, P.Y., Chuang, G.S., and Li, H.Y., Computational multiple steady states for enzymatic production of L-dopa in an isothermal CSTR, Process Biochem., 2005, vol. 40, no. 1, p. 469.

    CAS  Article  Google Scholar 

  29. 29

    Luo, Y.H., Chien, Y.S., Chiou, M.S., Lin, Y.I., and Li, H.Y., Numerical study of isothermal heterogeneous catalysis exhibiting multiple steady states, limit cycles, and chaos in a complex reaction network, Asia-Pac. J. Chem. Eng., 2018, vol. 13, no. 5, p. e2244.

    CAS  Article  Google Scholar 

  30. 30

    Sánchez-Ferrer, A., Rodr’ıguez-López, J.N., Garc’ıa-Cánovas, F., and Garc’ıa-Carmona, F., Tyrosinase: A comprehensive review of its mechanism, Biochim. Biophys. Acta, 1995, vol. 1247, no. 1, p. 1.

    Article  Google Scholar 

  31. 31

    Ho, P.Y., Chiou, M.S., and Chao, A.C., Production of L-DOPA by tyrosinase immobilized on modified polystyrene, Appl. Biochem. Biotechnol., 2003, vol. 111, no. 3, p. 139.

    CAS  Article  Google Scholar 

  32. 32

    Hearing, V.J., Mammalian monophenol monooxygenase (tyrosinase): Purification properties and reactions catalyzed, Methods Enzymol., 1987, vol. 142, p. 154.

    CAS  Article  Google Scholar 

  33. 33

    Hamann, M.C.J. and Saville, B.A., Enhancement of tyrosinase stability by immobilization on nylon 66, Food Bioprod. Process., 1996, vol. 74, p. 47.

    Google Scholar 

  34. 34

    Pialis, P. and Saville, B.A., Production of L-DOPA from tyrosinase immobilizedon nylon 6,6: Enzyme stability and scale up, Enzyme Microb. Technol., 1998, vol. 22, no. 4, p. 261.

    CAS  Article  Google Scholar 

  35. 35

    Craciun, G., Tang, Y., and Feinberg, M., Understanding bistability in complex enzyme-driven reaction networks, Proc. Natl. Acad. Sci. U. S. A., 2006, vol. 103, no. 23, p. 8697.

    CAS  Article  PubMed  PubMed Central  Google Scholar 

  36. 36

    Kuznetsov, Y., Elements of Applied Bifurcation Theory, New York: Springer-Verlag, 2004.

    Google Scholar 

  37. 37

    Ermentrout, B., XPPAUT. ~bard/xpp/xpp.html. Accessed October 23, 2017.

  38. 38

    Özer, A.B. and Akin, E., Tools for detecting chaos, SAÜ Fen Bilimleri Enst. Derg., 2005, vol. 9, p. 60.

    Google Scholar 

  39. 39

    Siu, S., Lyapunov Exponents Toolbox, https://cn. focused=5039116&tab=function. Accessed October 23, 2017.

  40. 40

    Feigenbaum, M.J., Quantitative universality for a class of non-linear transformations, J. Stat. Phys., 1978, vol. 19, no. 1, p. 25.

    Article  Google Scholar 

  41. 41

    De Kepper, P. and Boissonade, J., Theoretical and experimental analysis of phase diagrams and related dynamical properties in the Belousov–Zhabotinskii system, J. Chem. Phys., 1998, vol. 75, no. 1, p. 189.

    Article  Google Scholar 

Download references


We are grateful to the National Center for High-Performance Computing (NCHC) of Taiwan for computer time and facilities. We also would like to thank Dr. Hil Meijer for kind help with the MatCont in bifurcation analysis.

Author information



Corresponding authors

Correspondence to Yuan-Hong Luo or Hsing-Ya Li.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yuan-Hong Luo, Hsing-Ya Li Numerical Analysis of Multiple Steady States, Limit Cycles, Period-Doubling, and Chaos in Enzymatic Reactions Involving Oxidation of L-tyrosine to Produce L-DOPA. Theor Found Chem Eng 54, 1340–1352 (2020).

Download citation


  • multiple steady states
  • Hopf bifurcation
  • limit cycle
  • period-doubling bifurcation
  • chaos