1. Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India; 2. Department of Mathematics, Dinabondhu Andrews College, Kolkata 700084, India; 3. College of Science, Beijing Technology and Business University, Beijing 100037, China

Abstract In this work, to study the effect of memory on a bi-substrate enzyme kinetic reaction, we have introduced an approach to fractionalize the system, considering it as a threecompartmental model. Solutions of the fractionalized system are compared with the corresponding integer-order model. The equilibrium points of the fractionalized system are derived analytically. Their stability properties are discussed from numerical aspect. We determine the changes of the substances due to the changes of "memory effect". The effect is discussed critically from the perspective of product formation. We have also analyzed the memory induced system with a control measure in view of optimizing the product. Our numerical result reveals that the solutions of the fractionalized system, when it is free from memory, are in good agreement with the integer-order system. It is noticed that the effect of memory influences the reaction in the forward direction and assists in yielding the product more quickly. However, an extensive use of memory makes the system slower, but introduction of a control input makes the reaction faster. It is possible to overcome the slowness of the reaction due to the undue effect of memory by appropriate use of a control measure.

[1] R. Dutta, Fundamentals of Biochemical Engineering. Berlin:Springer, 2008. [2] I. Belgacem and J. L. Gouzé, "Global Stability of full open reversible michaelis-menten reactions," IFAC Proc., vol. 45, no. 15, pp. 591-596, 2012. [3] D. L. Nelson and M. M. Cox, Lehninger Principles of Biochemistry, 6th ed. Basingstoke:Macmillan Education, 2013. [4] S. Sirin, D. A. Pearlman, and W. Sherman, "Physics-based enzyme design:predicting binding affinity and catalytic activity," Proteins:Struct. Funct. Bioinform., vol. 82, pp. 3397-3409, Dec. 2014. [5] D. Vasic-Racki, U. Kragl, and A. Liese, "Benefits of enzyme kinetics modelling," Chem. Biochem. Eng. Quart., vol. 17, no. 1, pp. 7-18, Mar. 2003. [6] R. Roskoski Jr, "The ErbB/HER family of protein-tyrosine kinases and cancer," Pharmacol. Res., vol. 79, pp. 34-74, Jan. 2014. [7] P. K. Roy, S. Datta, S. Nandi, and F. Al Basir, "Effect of mass transfer kinetics for maximum production of biodiesel from Jatropha Curcas oil:a mathematical approach," Fuel, vol. 134, pp. 39-44, Oct. 2014. [8] S. D. Thiberville, N. Moyen, L. Dupuis-Maguiraga, A. Nougairede, E. A. Gould, P. Roques, and X. De Lamballerie, "Chikungunya fever:epidemiology, clinical syndrome, pathogenesis and therapy," Antiv. Res., vol. 99, no. 3, pp. 345-370, Sep. 2013. [9] Y. L. Qi, D. G. Musson, B. Schweighardt, T. Tompkins, L. Jesaitis, A. J. Shaywitz, K. Yang, and C. A. O'Neill, "Pharmacokinetic and pharmacodynamic evaluation of Elosulfase Alfa, an enzyme replacement therapy in patients with morquio a syndrome," Clin. Pharmacokinet., vol. 53, no. 12, pp. 1137-1147, Dec. 2014. [10] J. D. Murray, Mathematical Biology:I. An Introduction, 3rd ed. New York:Springer, 2002. [11] L. A. Segel, Mathematical Models in Molecular and Cellular Biology. Cambridge:Cambridge University Press, 1980. [12] G. Varadharajan and L. Rajendran, "Analytical solution of coupled non-linear second order reaction differential equations in enzyme kinetics," Nat. Sci., vol. 3, no. 6, pp. 459-465, May 2011. [13] A. Meena, A. Eswari, and L. Rajendran, "Mathematical modelling of enzyme kinetics reaction mechanisms and analytical solutions of non-linear reaction equations," J. Math. Chem., vol. 48, no. 2, pp. 179-186, Aug. 2010. [14] P. T. Benavides and U. Diwekar, "Optimal control of biodiesel production in a batch reactor:Part I:deterministic control," Fuel, vol. 94, pp. 211-217, Apr. 2012. [15] P. K. Roy, S. Nandi, and M. K. Ghosh, "Modeling of a control induced system for product formation in enzyme kinetics," J. Math. Chem., vol. 51, pp. 2704-2717, Nov. 2013. [16] F. A. Basir, R. Bhattacharyya, and P. K. Roy, "Delay induced oscillation in a biochemical model and its control," Nonlin. Stud., vol. 22, no. 3, pp. 453-472, Aug. 2015. [17] R. A. Azizyan, A. E. Gevorgyan, V. B. Arakelyan, and E. S. Gevorgyan, "Mathematical modeling of uncompetitive inhibition of Bi-substrate enzymatic reactions," Int. Schol. Sci. Res. Innov., vol. 7, no. 10, pp. 974-977, 2013. [18] S. Westerlund, "Dead matter has memory!," Phys. Scrip., vol. 43, no. 2, pp. 174-179, 1991. [19] V. E. Tarasov, "Review of some promising fractional physical models," Int. J. Mod. Phys. B, vol. 27, no. 9, pp. Article No. 1330005, Mar. 2013. [20] A. A. Stanislavsky, "Memory effects and macroscopic manifestation of randomness," Phys. Rev. E, vol. 61, no. 5, pp. 4752-4759, May 2000. [21] J. K. Popović, M. T. Atanacković, A. S. Pilipović, M. R. Rapaić, S. Pilipović, and T. M. Atanacković, "A new approach to the compartmental analysis in pharmacokinetics:fractional time evolution of diclofenac," J. Pharmacokinet. Pharmacodyn., vol. 37, no. 2, pp. 119-134, Apr. 2010. [22] R. Toledo-Hernandez, V. Rico-Ramirez, G. A. Iglesias-Silva, and U. M. Diwekar, "A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I:fractional models for biological reactions," Chem. Eng. Sci., vol. 117, pp. 217-228, Sep. 2014. [23] E. Ahmed and A. S. Elgazzar, "On fractional order differential equations model for nonlocal epidemics," Phys. A:Statist. Mechan. Appl., vol. 379, no. 2, pp. 607-614, Jun. 2007. [24] M. L. Du, Z. H. Wang, and H. Y. Hu, "Measuring memory with the order of fractional derivative," Sci. Rep., vol. 3, pp. Article No. 3431, Dec. 2013. [25] M. El-Shahed and A. Alsaedi, "The fractional SIRC model and influenza A," Math. Probl. Eng., vol. 2011, pp. Article No. 480378, Aug. 2011. [26] S. Abbas, M. Benchohra, G. M. NǴuérékata, and B. A. Silmani, "Darboux problem for fractional-order discontinuous hyperbolic partial differential equations in Banach algebras," Compl. Variabl. Elliptic Equat.:Int. J., vol. 57, no. 2-4, pp. 337-350, 2012. [27] I. Podlubny, Fractional Differential Equations. San Diego:Academic Press, 1999. [28] F. A. Abdullah, "Using fractional differential equations to model the Michaelis-Menten reaction in a 2-d region containing obstacles," Science Asia, vol. 37, no. 1, pp. 75-78, 2011. [29] A. Alawneh, "Application of the multistep generalized differential transform method to solve a time-fractional enzyme kinetics," Discr. Dyn. Nat. Soc., vol. 2013, pp. Article No. 592938, 2013. [30] A. Dokoumetzidis, R. Magin, and P. Macheras, "Fractional kinetics in multi-compartmental systems," J. Pharmacokinet. Pharmacodyn., vol. 37, no. 5, pp. 507-524, Oct. 2010. [31] T. Sardar, S. Rana, and J. Chattopadhyay, "A mathematical model of dengue transmission with memory," Commun. Nonlin. Sci. Numer. Simulat., vol. 22, no. 1-3, pp. 511-525, May 2015. [32] B. S. Chen, C. Y. Li, B. Wilson, and Y. J. Huang, "Fractional modeling and analysis of coupled MR damping system," IEEE/CAA J. of Autom. Sinica, vol. 3, no. 3, pp. 288-294, Jul. 2016. [33] S. Rana, S. Bhattacharya, J. Pal, G. M. NǴuérékata, and J. Chattopadhyay, "Paradox of enrichment:a fractional differential approach with memory," Phys. A:Statist. Mechan. Appl., vol. 392, no. 17, pp. 3610-3621, Sep. 2013. [34] M. K. Ghosh, J. Pal, and P. K. Roy, "How memory regulates drug resistant pathogenic bacteria? a mathematical study," Int. J. Appl. Comput. Math., vol. 3, no. S1, pp. 747-773, Dec. 2017. [35] V. E. Tarasov, "No violation of the Leibniz rule. No fractional derivative," Commun. Nonlin. Sci. Numer. Simulat., vol. 18, no. 11, pp. 2945-2948, Nov. 2013. [36] M. S. Tavazoei, and M. Haeri, "Chaotic attractors in incommensurate fractional order systems," Phys. D, vol. 237, no. 20, pp. 2628-2637, Oct. 2008. [37] S. Nandi, M. K. Ghosh, R. Bhattacharya, and P. K. Roy, "Mathematical modeling to optimize the product in enzyme kinetics," Control Cybern., vol. 42, no. 2, pp. 431-442, 2013. [38] S. Schnell and P. K. Maini, "Enzyme kinetics at high enzyme concentration," Bull. Math. Biol., vol. 62, no. 3, pp. 483-499, May 2000. [39] S. Schnell and P. K. Maini, "A century of enzyme kinetics:reliability of the K_{M} and v_{max} estimates," Comm. Theoret. Biol., vol. 8, no. 2-3, pp. 169-187, Mar-Jan. 2003.