Dynamics of enzyme kinetic model under the new generalized Hattaf fractional derivative

Catalytic action is one of the most important characteristics of enzymes in chemical reactions.  In this article, we propose and study a mathematical model of chemical kinetic reaction with the memory effect using the new generalized Hattaf fractional derivative.  The existence and uniqueness of the solutions are established by means of fixed point theory and, finally, to support the theoretical results, we end the article with the results of numerical simulations based on a novel numerical scheme that includes the Euler method.

enzymatic reaction
Hattaf fractional derivative
fixed point theory
numerical simulations
  1. Wong J. T.-F.  On the Steady-State Method of Enzyme Kinetics.  Journal of the American Chemical Society.  87 (8), 1788–1793 (1965).
  2. Michaelis L., Menten M. L.  Die Kinetik der Invertinwirkung.  Biochemische Zeitschrift.  49, 333–369 (1913).
  3. Cha S.  Kinetic Behavior at High Enzyme Concentrations: Magnitude of errors of michaelis-menten and other approximations.  Journal of Biological Chemistry.  245 (18), 4814–4818 (1970).
  4. Wald S., Wilke C. R., Blanch H. W.  Kinetics of the enzymatic hydrolysis of cellulose.  Biotechnology and Bioengineering.  26 (3), 221–230 (1984).
  5. Najafpour G. D., Shan C. P.  Enzymatic hydrolysis of molasses.  Bioresource Technology.  86 (1), 91–94 (2003).
  6. Gan Q., Allen S. J., Taylor G.  Kinetic dynamics in heterogeneous enzymatic hydrolysis of cellulose: an overview, an experimental study and mathematical modelling.  Process Biochemistry.  38 (7), 1003–1018 (2003).
  7. Urban P. L., Goodall D. M., Bruce N. C.  Enzymatic microreactors in chemical analysis and kinetic studies.  Biotechnology Advances.  24 (1), 42–57 (2006).
  8. Wong M. K. L., Krycer J. R., Burchfield J. G., James D. E., Kuncic Z.  A generalised enzyme kinetic model for predicting the behaviour of complex biochemical systems.  FEBS Open Bio.  5 (1), 226–239 (2015).
  9. Atangana A.  Modeling the Enzyme Kinetic Reaction.  Acta Biotheoretica.  63, 239–256 (2015).
  10. Miłek J.  Estimation of the Kinetic Parameters for H$_{2}$O$_{2}$ Enzymatic decomposition and for catalase deactivation.  Brazilian Journal of Chemical Engineering.  35 (3), 995–1004 (2018).
  11. Khan M., Ahmed Z., Ali F., Khan N., Khan I., Nisar K. S.  Dynamics of two-step reversible enzymatic reaction with Mittag–Leffler Kernel.  PLoS ONE.  18 (3),  e0277806 (2023).
  12. Atangana A., Baleanu D.  New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model.  Thermal Science.  20 (2), 763–769 (2016).
  13. Hattaf K.  A new generalized definition of fractional derivative with non-singular kernel.  Computation.  8 (2), 49 (2020).
  14. Caputo A., Fabrizio M.  A new definition of fractional derivative without singular kernel.  Progress in Fractional Differentiation and Applications.  1 (2), 73–85 (2015).
  15. Al-Refai M.  On weighted Atangana–Baleanu fractional operators.  Advances in Difference Equations.  2020, 3 (2020).
  16. Djida J. D., Atangana A., Area I.  Numerical Computation of a Fractional Derivative with Non-Local and Non-Singular Kernel.  Mathematical Modelling of Natural Phenomena.  12 (3), 4–13 (2017).
  17. Baleanu D., Fernandez A.  On some new properties of fractional derivatives with Mittag–Leffler kernel.  Communications in Nonlinear Science and Numerical Simulation.  59, 444–462 (2018).
  18. Hattaf K.  On some properties of the new generalized fractional derivative with non-singular kernel.  Mathematical Problems in Engineering.  2021, 1580396 (2021).
  19. Hattaf K.  On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology.  Computation.  10 (6), 97 (2022).