Numerical analysis of coupled systems of ODEs and applications to enzymatic competitive inhibition by product

Öz

Enzymatic inhibition is one of the key regulatory mechanisms in cellular metabolism, especially the enzymatic competitive inhibition by product. This inhibition process helps the cell regulate enzymatic activities. In this paper, we derive a mathematical model describing the enzymatic competitive inhibition by product. The model consists of a coupled system of nonlinear ordinary differential equations for the species of interest. Using nondimensionalization analysis, a formula for product formation rate for this mechanism is obtained in a transparent manner. Further analysis for this formula yields qualitative insights into the maximal reaction velocity and apparent Michaelis-Menten constant. Integrating the model numerically, the effects of the model parameters on the model output are also investigated. Finally, a potential application of the model to realistic enzymes is briefly discussed.

Anahtar Kelimeler

• Numerical analysis
• mathematical model
• product inhibition
• hexokinase

Kaynakça

1. [1] Albert L Lehninger, David L Nelson, Michael M Cox, Michael M Cox, et al. Lehninger principles of biochemistry. Macmillan, 2005.
2. [2] Tim DH Bugg. Introduction to enzyme and coenzyme chemistry. John Wiley & Sons, 2012.
3. [3] Perry A Frey and Adrian D Hegeman. Enzymatic reaction mechanisms. Oxford University Press, 2007.
4. [4] Athel Cornish-Bowden and Athel Cornish-Bowden. Fundamentals of enzyme kinetics, volume 510. Wiley-Blackwell Weinheim, Germany, 2012.
5. [5] Kenneth B Taylor. Enzyme kinetics and mechanisms. Springer Science & Business Media, 2002.
6. [6] John E Wilson. Isozymes of mammalian hexokinase: structure, subcellular localization and metabolic function. Journal of Experimental Biology, 206(12):2049–2057, 2003.
7. [7] Xiaofeng Liu, Chang Sup Kim, Feruz T Kurbanov, Richard B Honzatko, and Herbert J Fromm. Dual mechanisms for glucose 6-phosphate inhibition of human brain hexokinase. Journal of Biological Chemistry, 274(44):31155–31159, 1999.
8. [8] Kenneth B Taylor. Enzyme kinetics and mechanisms, volume 64. Springer Science & Business Media, 2002.

9. [9] Irwin H Segel. Enzyme kinetics: behavior and analysis of rapid equilibrium and steady state enzyme systems. Wiley New York, 1993.
10. [10] Brian P Ingalls. Mathematical modeling in systems biology: an introduction. MIT press, 2013.
11. [11] David Fell and Athel Cornish-Bowden. Understanding the control of metabolism, volume 2. Portland press London, 1997.
12. [12] Vinh Q Mai, Tuoi T Vo, and Martin Meere. Modelling hyaluronan degradation by streptococcus pneumoniae hyaluronate lyase. Mathematical biosciences, 303:126–138, 2018.
13. [13] The odeint solver in the integrate module of the scipy library. https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.odeint.html. Accessed: 2020-9-25.
14. [14] Scipy open source python library. https://www.scipy.org/. Accessed: 2020-9-25.
15. [15] Python software foundation. https://www.python.org/. Accessed: 2020-9-25.
16. [16] Howard M Katzen and Robert T Schimke. Multiple forms of hexokinase in the rat: tissue distribution, age dependency, and properties. Proceedings of the National Academy of Sciences, 54(4):1218–1225, 1965.
17. [17] Vilberto Stocchi, Mauro Magnani, Franco Canestrari, Marina Dacha, and Giorgio Fornaini. Multiple forms of human red blood cell hexokinase. preparation, characterization, and age dependence. Journal of Biological Chemistry, 257(5):2357–2364, 1982.
18. [18] JE Wilson. Hexokinases. Reviews of physiology, biochemistry and pharmacology, 126:65, 1995.
19. [19] Tracy K White and John E Wilson. Isolation and characterization of the discrete n-and c-terminal halves of rat brain hexokinase: retention of full catalytic activity in the isolated c-terminal half. Archives of Biochemistry and Biophysics, 274(2):375–393, 1989.
20. [20] Krishan K Arora, Charles R Filburn, and Peter L Pedersen. Structure/function relationships in hexokinase. site-directed mutational analyses and characterization of overexpressed fragments implicate different functions for the n-and c-terminal halves of the enzyme. Journal of Biological Chemistry, 268(24):18259–18266, 1993.
21. [21] James Ning, Daniel L Purich, and Herbert J Fromm. Studies on the kinetic mechanism and allosteric nature of bovine brain hexokinase. Journal of Biological Chemistry, 244(14):3840–3846, 1969.
22. [22] Gerhard Gerber, Heidemarie Preissler, Reinhart Heinrich, and Samuel M Rapoport. Hexokinase of human erythrocytes: purification, kinetic model and its application to the conditions in the cell. European journal of biochemistry, 45(1):39–52, 1974.
23. [23] Chenbo Zeng and Herbert J Fromm. Active site residues of human brain hexokinase as studied by site-specific mutagenesis. Journal of Biological Chemistry, 270(18):10509–10513, 1995.
24. [24] Tsuei-Yun Fang, Olga Alechina, Alexander E Aleshin, Herbert J Fromm, and Richard B Honzatko. Identification of a phosphate regulatory site and a low affinity binding site for glucose 6-phosphate in the n-terminal half of human brain hexokinase. Journal of Biological Chemistry, 273(31):19548–19553, 1998.
25. [25] Zhiwei Zhang, Fenghua Zhang, Liya Song, Ning Sun, Weishi Guan, Bo Liu, Jian Tian, Yuhong Zhang, and Wei Zhang. Site-directed mutation of β-galactosidase from aspergillus candidus to reduce galactose inhibition in lactose hydrolysis. 3 Biotech, 8(11):452, 2018.
26. [26] Jayshree PATİL, Archana CHAUDHARİ, ABDO Mohammed, and Basel HARDAN. Upper and lower solution method for positive solution of generalized caputo frac- tional differential equations. Advances in the Theory of Nonlinear Analysis and its Application, 4(4):279–291.
27. [27] Saleh S Redhwan, Sadikali L Shaikh, and Mohammed S Abdo. Some properties of sadik transform and its applications of fractional-order dynamical systems in control theory. arXiv preprint arXiv:1912.11484, 2019.