Mathematical Modelling for Complex Biochemical Networks and Identification of Fast and Slow Reactions

  • Sarbaz H. A. KhoshnawEmail author
  • Hemn M. Rasool
Conference paper
Part of the Lecture Notes in Networks and Systems book series (LNNS, volume 123)


This paper reviews some mathematical models of biochemical reaction networks. The models contain a large number species and reactions. This is a difficult task and it requires some effective computational tools. Techniques of model reduction are important approaches for minimizing the number of elements. One of the classical and common techniques of model reduction is quasi equilibrium approximation (QEA). According to this approach, the fast reactions simply reach equilibrium very fast. It allows one to classify the model reaction rates into slow and fast terms. This study suggest QEA technique to simplify and calculate analytical approximate solutions for non–competitive inhibition enzymatic reactions in different cases. On the other hand, the suggested method may not work very well analytically with higher dimensional biochemical networks. As a result, we propose an algorithm to identify slow and fast reactions in complex systems. The proposed algorithm provides a great step further in developing QEA technique. This is applied to dihydrofolate reductase (DHFR) cell signaling pathways. The algorithm would be easily applied by biologists and chemists for various purposes such as identifying slow–fast reactions and critical model elements. Finally, computational simulations show that many cell signalling pathways can reach equilibrium in a short interval of time. Interestingly, all reactions mainly become slow for a long range of time.


Mathematical modelling Quasi–equilibrium approximation Slow and fast reactions Computational simulations Cell signalling pathways 


  1. 1.
    Jones, C.K.R.T.: Geometric singular perturbation theory. In: Dynamical Systems. Lecture Notes in Mathematics, vol. 1609, pp. 44–118 (1995)Google Scholar
  2. 2.
    Briggs, G.E., Haldane, J.B.: A Note on the Kinetics of Enzyme Action. Biochem. J. 19, 338–339 (1925). Scholar
  3. 3.
    Vasiliev, V.M., Volpert, A.I., Hudiaev, S.I.: A method of quasi stationary concentrations for chemical kinetics equations. Zhurnal vychislitel noimatematiki matematicheskoi fiziki 13, 683–697 (1973).
  4. 4.
    Schnell, S., Maini, P.K.: Enzyme kinetics far from the standard quasi steady state and equilibrium approximations. Math. Comput. Model. 35, 137–144 (2002). Scholar
  5. 5.
    Gorban, A.N., Radulescu, O., Zinovyev, A.Y.: Asymptotology of chemical reaction networks. Chem. Eng. 65, 2310–2324 (2010). Scholar
  6. 6.
    Prescott, T.P., Papachristodoulou, A.: Layered decomposition for the model order reduction of timescale separated biochemical reaction networks. J. Theor. Biol. 356, 113–122 (2014). Scholar
  7. 7.
    Khoshnaw, S.H.A.: Model Reductions in Biochemical Reaction Networks. Thesis. University of Leicester, UK (2015)Google Scholar
  8. 8.
    Huang, Y.J., Yong, W.A.: Partial equilibrium approximations in apoptosis I. The intracellular-signaling subsystem. Math. Biosci. 246, 27–37 (2013). Scholar
  9. 9.
    Kijima, H., Kijima, S.K.: Steady/equilibrium approximation in relaxation and fluctuation: II. Mathematical theory of approximations in first–order reaction. Biophys. Chem. 17, 261–283 (1983). Scholar
  10. 10.
    Volk, L., Richardson, W., Lau, K., Hall, M., Lin, S.: Steady state and equilibrium approximations in reaction kinetics. J. Chem. Educ. 54, 95 (1977). Scholar
  11. 11.
    Khoshnaw, S. H. A.: Reduction of a kinetic model of active export of importins. In: AIMS Conference on Dynamical Systems, Differential Equations and Applications, Madrid, pp. 7–11 (2015).
  12. 12.
    Khoshnaw, S.H., Mohammad, N.A., Salih, R.H.: Identifying critical parameters in SIR model for spread of disease. Open J. Model. Simul. 5, 32 (2016). Scholar
  13. 13.
    Gorban, A.N., Karlin, I.V.: Method of invariant manifold for chemical kinetics. Chem. Eng. Sci. 58, 4751–4768 (2003). Scholar
  14. 14.
    Khoshnaw, S.H.A.: Iterative approximate solutions of kinetic equations for reversible enzyme reactions. Nat. Sci. 5, 740–755 (2013). Scholar
  15. 15.
    Ciliberto, A., Capuani, F., Tyson, J.: Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation. PLoS Comput. Biol. 3, e45 (2007). Scholar
  16. 16.
    Goeke, A., Schilli, C., Walcher, S., Zerz, E.: Computing quasi-steady state reductions. J. Math. Chem. 50, 1495–1513 (2012). Scholar
  17. 17.
    Hannemann-tamas, R., Gabor, A., Szederkenyi, G., Hangos, K.M.: Model complexity reduction of chemical reaction networks using mixed-integer quadratic programming. Comput. Math. Appl. 65, 1575–1595 (2013). Scholar
  18. 18.
    Klonowski, W.: Simplifying principles for chemical and enzyme reaction kinetics. Biophys. Chem. 18, 73–87 (1983). Scholar
  19. 19.
    Khoshnaw, S.H.A.: Dynamic analysis of a predator and prey model with some computational simulations. J. Appl. Bioinf. Comput. Biol. 6, 2329–9533 (2017). Scholar
  20. 20.
    Okino, M.S., Mavrovouniotis, M.L.: Simplification of mathematical models of chemical reaction systems. Chem. Rev. 98, 391–408 (1998). Scholar
  21. 21.
    Petrov, V., Nikolova, E., Wolkenhauer, O.: Reduction of nonlinear dynamic systems with an application to signal transduction pathways. IET Syst. Biol. 1, 2–9 (2007). Scholar
  22. 22.
    Rao, S., Van der Schaft, A., Van Eunen, K., Bakker, B.M., Jayawardhana, B.: A model reduction method for biochemical reaction networks. BMC Syst. Biol. 8, 52 (2014)CrossRefGoogle Scholar
  23. 23.
    Schneider, K.R., Wilhelm, T.: Model reduction by extended quasi steady state approximation. J. Math. Biol. 40, 443–450 (2000). Scholar
  24. 24.
    Akgul, A., Khoshnaw, S.H.A., Mohammed, W.H.: Mathematical model for the ebola virus disease. J. Adv. Phys. 7, 190–198 (2018). Scholar
  25. 25.
    Chow, M.L., Troussicot, L., Martin, M., Doumeche, B., Guillière, F., Lancelin, J.M.: Predicting and understanding the enzymatic inhibition of human peroxiredoxin 5 by 4-substituted pyrocatechols by combining funnel metadynamics. Solut. NMR Steady State Kinet. Biochem. 55, 3469–3480 (2016). Scholar
  26. 26.
    Lee, J., Yennawar, N.H., Gam, J., Benkovic, S.J.: Kinetic and structural characterization of dihydrofolate reductase from streptococcus pneumoniae. Biochemistry 49, 195–206 (2009). Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematics, College of Basic EducationUniversity of RaparinRanyaIraq
  2. 2.Department of Mathematics, Faculty of Science and HealthKoya UniversityKoyaIraq

Personalised recommendations