Advertisement

Acta Biotheoretica

, Volume 63, Issue 3, pp 239–256 | Cite as

Modeling the Enzyme Kinetic Reaction

  • Abdon Atangana
Regular Article
  • 301 Downloads

Abstract

The Enzymatic control reactions model was presented within the scope of fractional calculus. In order to accommodate the usual initial conditions, the fractional derivative used is in Caputo sense. The methodologies of the three analytical methods were used to derive approximate solution of the fractional nonlinear system of differential equations. Two methods use integral operator and the other one uses just an integral. Numerical results obtained exhibit biological behavior of real world problem.

Keywords

Enzyme kinetic model Fractional derivative Analytical techniques 

Notes

Acknowledgments

The author will express his deepest thanks to the reviewers for their time spared to read and suggest changes to enhance the quality of this paper

Conflict of interest

There is no conflict of interest in this paper.

References

  1. Atangana A (2013) A note on the triple Laplace transform and its applications to some kind of third-order differential equation. Abstr Appl Anal 2013:1–10Google Scholar
  2. Atangana A, Alabaraoye E (2013) Solving a system of fractional partial differential equations arising in the model of HIV infection of CD4+ cells and attractor one-dimensional Keller-Segel equations. Adv Differ Equ 2013:94Google Scholar
  3. Atangana A, Baleanu D (2013) Nonlinear fractional Jaulent-Miodek and Whitham-Broer-Kaup equations within Sumudu transform. Abstr Appl Anal 2013:1–9Google Scholar
  4. Atangana A, Botha JF (2013) Generalized groundwater flow equation using the concept of variable order derivative. Bound Value Probl 2013:53Google Scholar
  5. Atangana A, Kilicman A (2013) Analytical solutions of the space-time fractional derivative of advection dispersion equation. Math Probl Eng 2013:1–9Google Scholar
  6. Atangana A, Secer A (2013) Time-fractional coupled—the Korteweg-de Vries equations. Abstr Appl Anal 2013:1–8Google Scholar
  7. Babakhani A, Dahiya RS (2001) Systems of multi-dimensional Laplace transforms and a heat equation. In: Proceedings of the 16th conference on applied mathematics. Electronic Journal of Differential Equations, Vol 7, pp 25–36Google Scholar
  8. Baleanu D, Diethelm K, Scalas E, Trujillo JJ (2012) Fractional calculus models and numerical methods, series on complexity, nonlinearity and chaos. World Scientific, SingaporeGoogle Scholar
  9. Caputo M (1967) Linear model of dissipation whose Q is almost frequency independent-II. Geophys J R Astron Soc 13:529–539CrossRefGoogle Scholar
  10. Cornish-Bowden A (2004) Fundamenents of enzyme kinetics, 3rd edn. Portland Press Ltd, LondonGoogle Scholar
  11. Danson M, Eisenthal R (2002) Enzyme assays: a practical approach. Oxford University Press, Oxford [Oxfordshire]Google Scholar
  12. Duffy DG (2004) Transform methods for solving partial differential equations. CRC Press, New YorkCrossRefGoogle Scholar
  13. Johnston WK, Unrau PJ, Lawrence MS, Glasner ME, Bartel DP (2001) RNA-catalyzed RNA polymerization: accurate and general RNA-templated primer extension. Science 292(5520):1319–1325CrossRefGoogle Scholar
  14. Kilbas A, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamGoogle Scholar
  15. Lamb GL Jr (1995) Introductory applications of partial differential equations with emphasis on wave propagation and diffusion. Wiley, New YorkCrossRefGoogle Scholar
  16. Lu H (2004) Single-molecule spectroscopy studies of conformational change dynamics in enzymatic reactions. Curr Pharm Biotechnol 5(3):261–269CrossRefGoogle Scholar
  17. Maréchal E, Bastien O (2014) Modeling of regulatory loops controlling galactolipid biosynthesis in the inner envelope membrane of chloroplasts. J Theor Biol 361:1–13CrossRefGoogle Scholar
  18. Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New YorkGoogle Scholar
  19. Murray JD (1984) Asymptotic analysis. Springer, BerlinCrossRefGoogle Scholar
  20. Murray JD (2002) Mathematical biology. Springer, BerlinGoogle Scholar
  21. Nawaz Y (2011) Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations. Comput Math Appl 61:2330–2341CrossRefGoogle Scholar
  22. Podlubny I (1999) Fractional differential equations. Academic Press, San DiegoGoogle Scholar
  23. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives: theory and applications. Gordon and Breach, YverdonGoogle Scholar
  24. Schnell J, Dyson H, Wright P (2004) Structure, dynamics, and catalytic function of dihydrofolate reductase. Annu Rev Biophys Biomol Struct 33:119CrossRefGoogle Scholar
  25. Watugala GK (1993) Sumudu transform: a new integral transform to solve differential equations and control engineering problems. Int J Math Educ Sci Technol 24:35–43CrossRefGoogle Scholar
  26. Wazwaz AM (2007) The variational iteration method for solving linear and nonlinear systems of PDEs. Comput Math Appl 54:895–902CrossRefGoogle Scholar
  27. Wochner A, Attwater J, Coulson A, Holliger P (2011) Ribozyme-catalyzed transcription of an active ribozyme. Science 332(6026):209–212CrossRefGoogle Scholar
  28. Wrighton MS, Ebbing DD (1993) General chemistry, 4th edn. Houghton Mifflin, BostonGoogle Scholar
  29. Xie XS, Lu HP (1999) Single-molecule enzymology. J Biol Chem 274:15967–15970CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Faculty of Natural and Agricultural Sciences, Institute for Groundwater StudiesUniversity of the Free StateBloemfonteinSouth Africa

Personalised recommendations