Journal of Mathematical Chemistry

, Volume 56, Issue 4, pp 1153–1183 | Cite as

Approximation of enzyme kinetics for high enzyme concentration by a first order perturbation approach

  • Sebastian Kram
  • Maximilian Schäfer
  • Rudolf RabensteinEmail author
Original Paper


This contribution presents an approximate solution of the enzyme kinetics problem for the case of excess of an enzyme over the substrate. A first order perturbation approach is adopted where the perturbation parameter is the relation of the substrate concentration to the total amount of enzyme. As a generalization over existing solutions for the same problem, the presented approximation allows for nonzero initial conditions for the substrate and the enzyme concentrations as well as for nonzero initial complex concentration. Nevertheless, the approximate solution is obtained in analytical form involving only elementary functions like exponentials and logarithms. The presentation discusses all steps of the procedure, starting from amplitude and time scaling for a non-dimensional representation and for the identification of the perturbation parameter. Suitable time constants lead to the short term and long term behaviour, also known as the inner and outer solution. Special attention is paid to the matching process by the definition of a suitable intermediate layer. The results are presented in concise form as a summary of the required calculations. An extended example compares the zero order and first order perturbation approximations for the short term and long term solution as well as the uniform solution. A comparison to the numerical solution of the initial set of nonlinear ordinary differential equations demonstrates the achievable accuracy.


Enzyme kinetics Perturbation approach Nonlinear ordinary differential equations 

Mathematics Subject Classification

34D10 80A30 92C45 93C15 


Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.


  1. 1.
    M.Y. Adamu, P. Ogenyi, Parameterized homotopy perturbation method. Nonlinear Sci. Lett. A 8(2), 240–243 (2017)Google Scholar
  2. 2.
    H. Awan, C.T. Chou, Improving the capacity of molecular communication using enzymatic reaction cycles. IEEE Trans. NanoBiosci. PP(99), 1 (2017). Google Scholar
  3. 3.
    A.M. Bersani, E. Bersani, G. Dell’Acqua, M.G. Pedersen, New trends and perspectives in nonlinear intracellular dynamics: one century from Michaelis–Menten paper. Contin. Mech. Thermodyn. 27(4), 659–684 (2015). CrossRefGoogle Scholar
  4. 4.
    A.M. Bersani, G. Dell’Acqua, Asymptotic expansions in enzyme reactions with high enzyme concentrations. Math. Methods Appl. Sci. 34(16), 1954–1960 (2011). CrossRefGoogle Scholar
  5. 5.
    J.A.M. Borghans, R.J.D. Boer, L.A. Siegel, Extending the quasi-steady state approximation by changing variables. Bull. Math. Biol. 58(1), 43–63 (1996)CrossRefGoogle Scholar
  6. 6.
    I. Bronshtein, K. Semendyayev, G. Musiol, H. Mühlig, Handbook of Mathematics, 6th edn. (Springer, New York, 2015)Google Scholar
  7. 7.
    Cho, Y.J., Yilmaz, H.B., Guo, W., Chae, C.B. (2017). Effective enzyme deployment for degradation of interference molecules in molecular communication, in 2017 IEEE Wireless Communications and Networking Conference (WCNC), pp. 1–6.
  8. 8.
    U.A.K. Chude-Okonkwo, R. Malekian, B.T. Maharaj, Diffusion-controlled interface kinetics-inclusive system-theoretic propagation models for molecular communication systems. EURASIP J. Adv. Signal Process. 2015(1), 89 (2015). CrossRefGoogle Scholar
  9. 9.
    G. Dell’Acqua, A.M. Bersani, A perturbation solution of Michaelis-Menten kinetics in a “total” framework. J. Math. Chem. 50(5), 1136–1148 (2012). CrossRefGoogle Scholar
  10. 10.
    J.W. Dingee, A.B. Anton, A new perturbation solution to the Michaelis–Menten problem. AIChE J. 54(5), 1344–1357 (2008). CrossRefGoogle Scholar
  11. 11.
    N. Farsad, H.B. Yilmaz, A. Eckford, C.B. Chae, W. Guo, A comprehensive survey of recent advancements in molecular communication. IEEE Commun. Surv. Tutor. 18(3), 1887–1919 (2016). CrossRefGoogle Scholar
  12. 12.
    J.H. He, Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 20(10), 1141–1199 (2006). CrossRefGoogle Scholar
  13. 13.
    V. Jamali, N. Farsad, R. Schober, A. Goldsmith, Diffusive molecular communications with reactive signaling. (2017). arXiv:1711.00131v1
  14. 14.
    J. Kevorkian, J.D. Cole, Perturbation Methods in Applied Mathematics, vol. 34, Applied Mathematical Sciences (Springer, Heidelberg, 1981)Google Scholar
  15. 15.
    S. Kram, M. Schäfer, R. Rabenstein, High enzyme concentration first order perturbation approximation (HiEC FOPA).
  16. 16.
    C.C. Lin, L.A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974)Google Scholar
  17. 17.
    M.U. Maheswari, L. Rajendran, Analytical solution of non-linear enzyme reaction equations arising in mathematical chemistry. J. Math. Chem. 49(8), 1713 (2011). CrossRefGoogle Scholar
  18. 18.
    D.M.C. Mary, T. Praveen, L. Rajendran, Mathematical modeling and analysis of nonlinear enzyme catalyzed reaction processes. J. Theor. Chem. (2013). Google Scholar
  19. 19.
    J.D. Murray, Mathematical Biology I: An Introduction, 3rd edn. (Springer, New York, 2008)Google Scholar
  20. 20.
    T. Nakano, A.W. Eckford, T. Haraguchi, Molecular Communication (Cambridge University Press, Cambridge, 2013)CrossRefGoogle Scholar
  21. 21.
    A. Noel, K. Cheung, R. Schober, Improving receiver performance of diffusive molecular communication with enzymes. IEEE Trans. NanoBiosci. 13(1), 31–43 (2014). CrossRefGoogle Scholar
  22. 22.
    M. Pedersen, A. Bersani, Introducing total substrates simplifies theoretical analysis at non-negligible enzyme concentrations: pseudo first-order kinetics and the loss of zero-order ultrasensitivity. J. Math. Biol. 60(2), 267–283 (2010). CrossRefGoogle Scholar
  23. 23.
    M.G. Pedersen, A.M. Bersani, E. Bersani, Quasi steady-state approximations in complex intracellular signal transduction networks—a word of caution. J. Math. Chem. 43(4), 1318–1344 (2008). CrossRefGoogle Scholar
  24. 24.
    M. Pierobon, I. Akyildiz, A physical end-to-end model for molecular communication in nanonetworks. IEEE J. Sel. Areas Commun. 28(4), 602–611 (2010). CrossRefGoogle Scholar
  25. 25.
    M. Pierobon, I. Akyildiz, A statistical-physical model of interference in diffusion-based molecular nanonetworks. IEEE Trans. Commun. 62(6), 2085–2095 (2014). CrossRefGoogle Scholar
  26. 26.
    R. Rabenstein, Design of a molecular communication channel by modelling enzyme kinetics. IFAC-PapersOnLine 48(1), 35–40 (2015). CrossRefGoogle Scholar
  27. 27.
    A. Tzafriri, Michaelis-Menten kinetics at high enzyme concentrations. Bull. Math. Biol. 65(6), 1111–1129 (2003). CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)ErlangenGermany

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