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Multistep Transformation Method for Discrete and Continuous Time Enzyme Kinetics

  • Z. Vosika
  • G. Lazović
  • Vojislav Mitic
  • Lj. Kocić
Conference paper

Abstract

In this paper we develop the new physical-mathematical time scale kinetic approach-model applied on organic and non-organic particles motion. Concretely, here, at first, this new research approach is based on enzyme particles dynamics results. At the beginning, a time scale is defined to be an arbitrary closed subset of the real numbers R, with the standard inherited topology. Mathematical examples of time scales include real numbers R, natural numbers N, integers Z, the Cantor set (i.e. fractals), and any finite union of closed intervals of R. Calculus on time scales (TSC) was established in 1988 by Stefan Hilger. TSC, by construction, is used to describe the complex process. This method may utilized for description of physical (classical mechanics), material (crystal growth kinetics, physical chemistry kinetics—for example, kinetics of barium-titanate synthesis), (bio)chemical or similar systems and represents major challenge for contemporary scientists. In this sense, the Michaelis-Menten (MM) mechanism is the one of the best known and simplest nonlinear biochemical network which deserves appropriate attention. Generally speaking, such processes may be described of discrete time scale. Reasonably it could be assumed that such a scenario is possible for MM mechanism. In this work, discrete time MM kinetics (dtMM) with time various step h, is investigated. Instead of the first derivative by time used first backward difference h. Physical basics for new time scale approach is a new statistical thermodynamics, natural generalization of Tsallis non-extensive or similar thermodynamics. A reliable new algorithm of novel difference transformation method, namely multi-step difference transformation method (MSDETM) for solving system of nonlinear ordinary difference equations is proposed. If h tends to zero, MSDETM transformed into multi-step differential transformation method (MSDTM). In the spirit of TSC, MSDETM describes analogously MSDTM.

Keywords

Kinetics Enzymes Materials Ceramics BaTiO3 MSDETM 

References

  1. 1.
    M. Bohner, A. Peterson, Dynamic Equations on Time Scales (Birkhauser, Boston, 2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    M. Bohner, A. Peterson (eds.), Advances in Dynamic Equations on Time Scales (Birkhauser, Boston, 2003)zbMATHGoogle Scholar
  3. 3.
    F.M. Atici, S. Sengul, Modeling with fractional difference equations. J. Math. Anal. Appl. 369(1), 1–9 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M.Z. Odibat, C. Bertelle, M.A. Aziz-Alaoui, G.H.E. Duchamp, A multi-step differential transform method and application to nonchaotic or chaotic systems. Comput. Math Appl. 59(4), 1462–1472 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. Alawneh, Application of the multistep generalized differential transform method to solve a time-fractional enzyme kinetics. Discrete Dyn. Nat. Soc. 592938, 7 (2013)Google Scholar
  6. 6.
    E.T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge University Press, Cambridge, 1993)Google Scholar
  7. 7.
    E.M. Lifshitz, L.P. Pitaevskii, J.B. Sykes, R.N. Franklin, Physical Kinetics (Butterworth-Heinemann, Oxford, 1981)Google Scholar
  8. 8.
    C. Tsallis, Nonextensive statistics: theoretical, experimental and computational evidences and connections. Braz. J. Phys. 29 (1999)Google Scholar
  9. 9.
    B.V. Alexeev, Generalized Boltzmann Physical Kinetics (Elsevier, Amsterdam, 2004)Google Scholar
  10. 10.
    G.E. Gorelik, N.V. Pavlyukevish, V.V. Levdansky, V.G. Leitsina, G.I. Rudin, Physical Kinetics and Transfer Processes in Phase Transitions (Begell House, Danbury, 1995)Google Scholar
  11. 11.
    V.P. Krainov, K. Hendzel, Qualitative Methods in Physical Kinetics and Hydrodynamics (Springer, Berlin, 1992)Google Scholar
  12. 12.
    J.I. Steinfeld, J.S. Francisco, W.L. Hase, Chemical Kinetics and Dynamics, 2nd edn. (Prentice-Hall, Upper Saddle River, 1999)Google Scholar
  13. 13.
    H. Resat, L. Petzold, M.F. Pettigrew, Kinetic modeling of biological systems. Methods Mol. Biol. 541, 311–335 (2009)CrossRefGoogle Scholar
  14. 14.
    W. Hertl, Kinetics of barium titanate synthesis. J. Am. Cerum. Soc. 71(10), 879–883 (1988)CrossRefGoogle Scholar
  15. 15.
    A. Testino, V. Buscaglia, M.T. Buscaglia, M. Viviani, P. Nanni, Kinetic modeling of aqueous and hydrothermal synthesis of barium titanate (BaTiO3). Chem. Mater. 17, 5346–5356 (2005)CrossRefGoogle Scholar
  16. 16.
    M.M. Vijatović, J.D. Bobić, B.D. Stojanović, History and challenges of barium titanate: part I. Sci. Sinter. 40, 155–165 (2008)CrossRefGoogle Scholar
  17. 17.
    M.M. Vijatović, J.D. Bobić, B.D. Stojanović, History and challenges of barium titanate: part II. Sci. Sinter. 40, 235–244 (2008)CrossRefGoogle Scholar
  18. 18.
  19. 19.
    V.V. Mitić, Z.S. Nikolić, M.M. Ristić, The influence of pressing pressure of ferroelectric characteristics of BaTiO 3 -ceramics. Annual Meeting of the American Ceramic Society, Ohio, Cincinnati, 1–3 May 1995Google Scholar
  20. 20.
    S.K. Chiang, W.E. Lee, D.W. Readey, Evolution of the core-shell grain structure in temperature-stable doped BaTiO3, in Proceedings of Industry University Advanced Materials Conference (Denver C., 1989)Google Scholar
  21. 21.
    B. Jordović, V. Mitić, Z.S. Nikolić, Effects of sintering time and temperature on BaTiO3-ceramic microstructured characteristics. Acta Stereol. 13(2), 381–388 (1994)Google Scholar
  22. 22.
    V. Mitić , Z.S. Nikolić, M.M. Ristić, The frequent characteristics of BaTiO3-ceramics as a function of sintering temperature, in International Conference on the Science, Technology and Applications of Sintering (Penn State of University, 24–27 September 1995)Google Scholar
  23. 23.
    J.E. Marsden, L. Sirovich, S. Wiggins (eds.), The Geometry of Biological Time, 2nd edn. (Springer Verlag, New York, 2001)Google Scholar
  24. 24.
    R. Gatenby, T. Vincent, An evolutionary model of carcinogenesis. Cancer Res. 63, 6212–6220 (2003)Google Scholar
  25. 25.
    A. Deutsch, L. Brusch, H. Byrne, G. de Vries, H. Herzel, Mathematical Modeling of Biological Systems, vol. I. (Birkhauser Boston, 2007), pp. 193–203Google Scholar
  26. 26.
    A. Lehninger, D.L. Nelson, M.M. Cox, Lehninger-“Principles of Biochemistry” (W. H. Freeman, New York, 2008)Google Scholar
  27. 27.
    G. Jaroszkiewicz, Principles of Discrete Time Mechanics (University Printing House, Cambridge CB2 8BS, United Kingdom, 2014)Google Scholar
  28. 28.
    G. Katzung, Basic and Clinical Pharmacology: Introduction to Autonomic Pharmacology, 8th edn. (The McGraw Hill Companies, New York, 2001), pp. 75–91Google Scholar
  29. 29.
    P. Taylor, Z. Radić, The cholinesterases: from genes to proteins. Annu. Rev. Pharmacol. Toxicol. 34, 281–320 (1994)CrossRefGoogle Scholar
  30. 30.
    P. Zdrazilova et al., Kinetics of total enzymatic hydrolysis of acetylcholine and acetylthiocholine. Z. Naturforsch. 61(3–4), 289–294 (2006)Google Scholar
  31. 31.
    M.B. Čolović, D.V. Bajuk-Bogdanović, N.S. Avramović, I.D. Holclajtner-Antunovic, N.S. Bošnjaković-Pavlović, V.M. Vasić, D.Z. Krstić, Inhibition of rat synaptic membrane Na+/K+-ATPase and ecto-nucleoside triphosphate diphosphohydrolases by 12-tungstosilicic and 12-tungstophosphoric acid. Bioorg. Med. Chem. 19, 7063–7069 (2011)CrossRefGoogle Scholar
  32. 32.
    M.B. Čolović, V.M. Vasić, N.S. Avramović, M.M. Gajić, D.M. Djurić, D.Z. Krstić, In vitro evaluation of neurotoxicity potential and oxidative stress responses of diazinon and its degradation products in rat brain synaptosomes. Toxicol. Lett. 233, 29–37 (2015)CrossRefGoogle Scholar
  33. 33.
    S. Dimitrov, G. Velikova, V. Beschkov, S. Markov, On the numerical computation of enzyme kinetic parameters. Biomath Commun. 1, 2 (2014)Google Scholar
  34. 34.
  35. 35.
    G. Liu, R. Zhong, R. Hu, F. Zhang, Applications of ionic liquids in biomedicine. Biophys. Rev. Lett. 7(3 & 4), 121–134 (2012)Google Scholar

Copyright information

© Atlantis Press and the author(s) 2017

Authors and Affiliations

  • Z. Vosika
    • 1
  • G. Lazović
    • 1
  • Vojislav Mitic
    • 2
    • 3
  • Lj. Kocić
    • 2
  1. 1.Faculty of Mechanical EngineeringUniversity of BelgradeBelgradeSerbia
  2. 2.Faculty of Electronic EngineeringUniversity of NišNišSerbia
  3. 3.Institute of Technical Sciences of SASABelgradeSerbia

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