2. Department of Mathematics, Dinabondhu Andrews College, Kolkata 700084, India;
3. College of Science, Beijing Technology and Business University, Beijing 100037, China
Enzymes are biological catalysts that are necessary in almost every biochemical reaction [1]. These enzymes are proteins synthesized by genes [2]. The main function of an enzyme is to catalyze the making and breaking of chemical bonds depending on an accurate sequence of amino acids and its complicated tertiary structure. The catalytic ability of enzymes increases the rate of a reaction. The enzyme is not used up in the reactions and, it does not change the equilibria of the processes [3]. This raises a new dimension of thinking towards various fields viz. physics [4], chemistry [5], biology [6], ecology [7], epidemiology [8], pharmacokinetics [9] etc. A lot of research has been done about enzymatic processes of different chemical and biochemical transformations. Enzyme kinetics is the study of rates of these reactions to optimize the velocity of reactions, rate of intermediate complexes and products.
For a better understanding of the reaction kinetics, many authors have implemented different techniques to obtain approximate analytical solutions of the enzymatic systems [10][13]. Modern day literature related to enzyme activity in enzymatic processes consist of mathematical approaches to study system dynamics for optimization and quantification of product [14]. Single substrate or double substrate biochemical reactions make the approaches more interesting, of which the latter is more reasonable and important [15][17].
Westerlund stated in [18] that every matter has memory. Although it is debatable, a large number of theoretical physicists considered the memory function as an embedded characteristic of molecular properties, which is discussed in various domains of science and engineering branches [19][21]. ToledoHernandez et al. mentioned in [22] that biochemical reactions involve the participation of living organisms viz. enzymes. The dynamic behavior of living microorganisms not only depends on their current state conditions (e.g., substrate concentration, medium condition, etc.), but also on their previous states. They have explained this phenomena as the dynamics of the reactions that involve memory effects. Now, it is to be noted that integerorder (IO) derivatives consider only local properties (at time
The conception of fractional calculus is first projected by Leibniz [25] in
In this article, we have introduced fractionalization of a twosubstrate enzymatic reaction to study the effect of memory on it. Nonlinear FDEs cannot, in general, be solved analytically [27], but can be solved by numerical techniques [31]. The numerical solutions of the system have been studied here and compared with the integerorder system. We have also observed the dynamics of the different substances of the system by varying the order of the fractional derivatives (which signifies a measure of memory effect in a system [24]). We formulate a control based mathematical model involving the memory effect to conquer the negative effect of the extensive use of memory.
We have organized the rest of the paper as follows. In Section Ⅱ, we formulate the model of a bisubstrate enzymatic reaction involving the memory effect. Some basic theoretical properties, and the existence and stability of equilibrium points are studied in Section Ⅲ. In Section Ⅳ, a control theoretic approach is introduced towards the fractionalorder model. The numerical results are illustrated in Section Ⅴ. Finally, we have completed our article with a discussion and conclusion of the study in Section Ⅵ.
Ⅱ. THE FRACTIONALORDER MODELThe schematic diagram of a twosubstrate enzyme kinetic reaction, as described by Roy et al. [15], is given by
$ \begin{eqnarray}\label{model0} E+S_{1} {\large\overset{k_{1}}{\underset{k_{1}}{\rightleftharpoons}}} ES_{1}+S_{2} {\large\overset{k_{2}}{\underset{k_{2}}{\rightleftharpoons}}} ES_{1}S_{2} \xrightarrow{k_{3}} E+P \end{eqnarray} $  (1) 
where
Let us denote the concentrations
$ \frac{ds_1}{dt} = k_1es_1+k_{1}c_1\nonumber\\ \frac{ds_2}{dt}=k_2c_1s_2+k_{2}c_2\nonumber\\ \frac {de}{dt} = \ k_1es_1+k_{1}c_1+k_3c_2 \\ \frac {dc_1}{dt}=k_1es_1k_{1}c_1k_2c_1s_2+k_{2}c_2\nonumber\\ \frac {dc_2}{dt}=k_2c_1s_2k_{2}c_2k_3c_2\nonumber\\ \frac {dp}{dt}= \ k_3c_2 $  (2) 
with the initial conditions
$ \begin{eqnarray}\label{inicon} s_1(0) = s_{10}, \; s_2(0) = s_{20}, \; e(0) = e_0\nonumber\\ c_1(0) = 0, \; c_2(0) = 0,\; \mbox{and} \; p(0) = 0. \end{eqnarray} $  (3) 
From system
$ \begin{eqnarray}\label{reln} \frac{ds_1}{dt}\frac{ds_2}{dt}+\frac{dc_1}{dt}\!\!&\;=\;&\!\!0\nonumber\\ \frac{ds_2}{dt}+\frac{dc_2}{dt}+\frac{dp}{dt}\!\!&\;=\;&\!\!0\nonumber\\ \frac{ds_2}{dt}+\frac{dc_2}{dt}+\frac{de}{dt}\frac{ds_1}{dt}\;\; &\;=\;&\;\; 0. \end{eqnarray} $  (4) 
Using the initial conditions
$ \begin{eqnarray}\label{cons} s_1s_2+c_1\!\!&\;=\;&\!\! s_{10}s_{20}\nonumber\\ s_2+c_2+p\!\!&\;=\;&\!\!s_{20}\nonumber\\ s_2+c_2+es_1\!\!&\;=\;&\!\!s_{20}+e_0s_{10}. \end{eqnarray} $  (5) 
With the help of the relations
$ \begin{eqnarray}\label{model1.1} \frac{ds_1}{dt}\!\!&\;=\;&\!\!k_1(s_1s_2c_2+s_{20}s_{10}+e_0)s_1\nonumber\\ &+k_{1}(s_2s_1+s_{10}s_{20}), \nonumber\\ \frac{ds_2}{dt}\!\!&\;=\;&\!\!k_2(s_2s_1+s_{10}s_{20})s_2+k_{2}c_2, \nonumber\\ \frac {dc_2}{dt}\;\; &\;=\;&\;\; k_2(s_2s_1+s_{10}s_{20})s_2k_{2}c_2k_3c_2 \end{eqnarray} $  (6) 
with initial conditions,
$ \begin{eqnarray}\label{ini1} s_1(0)=s_{10}, \;\;\; s_2(0)=s_{20}, \;\;\;c_2(0)=0. \end{eqnarray} $  (7) 
The schematic diagram
$ \begin{eqnarray}\label{modelrl} \frac{ds_1}{dt}\!\!&\;=\;&\!\!k_1(s_1s_2c_2+s_{20}s_{10}+e_0)s_1\nonumber\\ &+k_{1}(s_2s_1+s_{10}s_{20}), \nonumber\\ \frac{ds_2}{dt}\!\!&\;=\;&\!\!k_2(s_2s_1+s_{10}s_{20})s_2+{k^{\alpha}_{2}}^{R}_0D^{1\alpha}_t(c_2), \nonumber\\ \frac {dc_2}{dt}\!\!&\;=\;&\!\!k_2(s_2s_1+s_{10}s_{20})s_2\nonumber\\ &{k^{\alpha}_{2}}^{R}_0D^{1\alpha}_t(c_2)k_3c_2 \end{eqnarray} $  (8) 
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Fig. 1 Threecompartment model corresponding to the schematic diagram 
where
The unit on lefthand side of all the subequations of system
In order to use standard initial conditions, the RiemannLioville derivatives must be redefined as Caputo fractional derivatives [22]. The relation between RL and Caputo's derivatives is given by the following equation:
$ \begin{eqnarray}\label{rlcaputorlsn} ^{R}_0D^{1\alpha}_t f(t)\!\!&\;=\;&\!\!^{C}_0D^{\alpha}_t f(t)+\frac{f(0)t^{\alpha1}}{\Gamma(\alpha)}. \end{eqnarray} $  (9) 
The system
$ \begin{eqnarray}\label{modelcp} \frac{ds_1}{dt}\!\!&\;=\;&\!\!k_1(s_1s_2c_2+s_{20}s_{10}+e_0)s_1\nonumber\\ &+k_{1}(s_2s_1+s_{10}s_{20}), \nonumber\\ \frac{ds_2}{dt}\!\!&\;=\;&\!\!k_2(s_2s_1+s_{10}s_{20})s_2\nonumber\\ &+{k^{\alpha}_{2}}^{C}_0D^{1\alpha}_t(c_2)+\frac{A_1 t^{\alpha1}}{\Gamma(\alpha)}, \nonumber\\ \frac {dc_2}{dt}\!\!&\;=\;&\!\!k_2(s_2s_1+s_{10}s_{20})s_2\nonumber\\ &{k^{\alpha}_{2}}^{C}_0D^{1\alpha}_t(c_2)k_3c_2\frac{A_1 t^{\alpha1}}{\Gamma(\alpha)} \end{eqnarray} $  (10) 
where
$ \begin{eqnarray}\label{a1a2} A_1 = k^{\alpha}_{2} c_2(0)=0\nonumber \end{eqnarray} $ 
Therefore, system
$ \begin{eqnarray}\label{modelcpa1a20} \frac{ds_1}{dt}\!\!&\;=\;&\!\!k_1(s_1s_2c_2+s_{20}s_{10}+e_0)s_1\nonumber\\ &+k_{1}(s_2s_1+s_{10}s_{20}), \nonumber\\ \frac{ds_2}{dt}\!\!&\;=\;&\!\!k_2(s_2s_1+s_{10}s_{20})s_2+{k^{\alpha}_{2}}^{C}_0D^{1\alpha}_t(c_2), \nonumber\\ \frac {dc_2}{dt}\!\!&\;=\;&\!\!k_2(s_2s_1+s_{10}s_{20})s_2\nonumber\\ &{k^{\alpha}_{2}}^{C}_0D^{1\alpha}_t(c_2)k_3c_2. \end{eqnarray} $  (11) 
In this section, we have determined the equilibrium points of model
It is not possible to understand the stability of the equilibrium points of the system
$ \begin{eqnarray}\label{trns} X\!\!&\;=\;&\!\!D^{\alpha} s_1, \nonumber\\ Y\!\!&\;=\;&\!\!D^{\alpha} s_2k^{\alpha}_{2} c_2, \nonumber\\ Z\!\!&\;=\;&\!\!D^{\alpha} c_2+k^{\alpha}_{2} c_2. \end{eqnarray} $  (12) 
Using the above transformation
$ \begin{eqnarray}\label{trnseq} D^{1\alpha} X\!\!&\;=\;&\!\!k_1(s_1s_2c_2+s_{20}s_{10}+e_0)s_1\nonumber\\ &+k_{1}(s_2s_1+s_{10}s_{20}), \nonumber\\ D^{1\alpha} Y\!\!&\;=\;&\!\!k_2(s_2s_1+s_{10}s_{20})s_2, \nonumber\\ D^{1\alpha} Z\!\!&\;=\;&\!\!k_2(s_2s_1+s_{10}s_{20})s_2k_3 c_2, \nonumber\\ D^{\alpha} s_1\!\!&\;=\;&\!\!X, \nonumber\\ D^{\alpha} s_2\!\!&\;=\;&\!\!Y+k^{\alpha}_{2} c_2, \nonumber\\ D^{\alpha} c_2\!\!&\;=\;&\!\!Zk^{\alpha}_{2} c_2. \end{eqnarray} $  (13) 
It is sufficient to study the stability properties of system
System (13) has the equilibrium points
$ \begin{eqnarray}\label{s1qdeq} k_1 s^{*^{2}}_1+[k_1(s_{20}s_{10}+e_0)+k_{1}]s^{*}_1\nonumber\\ +k_{1}(s_{20}s_{10})=0, \end{eqnarray} $  (14) 
i.e.,
$ \begin{eqnarray}\label{s1ab} A\!\!&\;=\;&\!\!k_1(s_{20}s_{10}+e_0)+k_{1}, \nonumber\\ B\!\!&\;=\;&\!\!4 k_1 k_{1} (s_{10}s_{20}). \nonumber \end{eqnarray} $ 
The Jacobian matrix
$ \begin{eqnarray}\label{jacobe1} \left( \begin{array}{cccccc} 0&0&0&(k_{1}+k_1 e_0)&k_{1}&0 \\ 0&0&0&k_2\delta&k_2\delta&0 \\ 0&0&0&k_2\delta&k_2\delta&k_3 \\ 1&0&0&0&0&0 \\ 0&1&0&0&0&k^{\alpha}_{2} \\ 0&0&1&0&0&k^{\alpha}_{2} \\ \end{array} \right) \end{eqnarray} $  (15) 
where
Since
Therefore, the characteristic equation of the matrix
$ \begin{eqnarray}\label{jacobe1charlame1} \triangle(J_{E^{*}_{1}}\mbox{diag}([\lambda^{NM}, \lambda^{NM}, \lambda^{NM}, \nonumber\\ \lambda^{M}, \lambda^{M}, \lambda^{M}]))=0, \end{eqnarray} $  (16) 
where "
Expanding the characteristic equation
$ \begin{eqnarray}\label{jacobe1chareqe1} \lambda^{3N}+\Lambda_{11} \lambda^{2N}+\Lambda_{12} \lambda^{N}+\Lambda_{13} \lambda^{N(3\alpha)}\nonumber\\ +\Lambda_{14} \lambda^{N(2\alpha)} + \Lambda_{15}=0 \end{eqnarray} $  (17) 
where
$ \begin{eqnarray}\label{jacobe1charcoeffe1} \Lambda_{11}\!\!&\;=\;&\!\!k_3+k_2\delta+k_{1}+k_1e_0, \nonumber\\ \Lambda_{12}\!\!&\;=\;&\!\!k_2k_3\delta+k_3(k_{1}+k_1e_0)+k_1k_2e_0\delta, \nonumber\\ \Lambda_{13}\!\!&\;=\;&\!\!k^{\alpha}_{2}, \Lambda_{14} = (k_{1}+k_1e_0)k^{\alpha}_{2}\; \mbox{and}\nonumber\\ \Lambda_{15}\!\!&\;=\;&\!\!k_1k_2k_3e_0\delta. \nonumber \end{eqnarray} $ 
For
$ \begin{eqnarray}\label{intordrchareqne1} \lambda^{3}+\eta_1\lambda^{2}+\eta_2\lambda+\eta_3=0 \end{eqnarray} $  (18) 
where
$ \begin{eqnarray}\label{coeffalphaeq1e1} \eta_1\!\!&\;=\;&\!\!k_{2}+k_{1}+k_3+k_1e_0+k_2\delta, \nonumber\\ \eta_2\!\!&\;=\;&\!\!(k_3+k_{2})(k_{1}+k_1e_0)+k_2\delta(k_3+k_1e_0), \nonumber\\ \eta_3\!\!&\;=\;&\!\!k_1k_2k_3e_0\delta.\nonumber \end{eqnarray} $ 
Equation
The Jacobian matrix
$ \begin{eqnarray}\label{jacobe2} \left( \begin{array}{cccccc} 0&0&0&\sqrt{A^{2}+B}&k_1s^*_1+k_{1}&k_1s^*_1 \\ 0&0&0&0&k_2(\delta+s^*_1)&0 \\ 0&0&0&0&k_2(\delta+s^*_1)&k_3 \\ 1&0&0&0&0&0 \\ 0&1&0&0&0&k^{\alpha}_{2} \\ 0&0&1&0&0&k^{\alpha}_{2} \\ \end{array} \right).\nonumber \end{eqnarray} $ 
Proceeding as above, we have the characteristic equation of
$ \begin{eqnarray}\label{jacobe1chareqe2} (\lambda^{N}+\sqrt{A^{2}+B})(\lambda^{2N}+\Lambda_{21} \lambda^{N}\nonumber\\ +\Lambda_{22} \lambda^{N(2\alpha)}+\Lambda_{23})=0 \end{eqnarray} $  (19) 
where
$ \begin{eqnarray}\label{jacobe1charcoeffe2} \Lambda_{21}\!\!&\;=\;&\!\!k_3k_2(\delta+s^{*}_1), \nonumber\\ \Lambda_{22}\!\!&\;=\;&\!\!k^{\alpha}_{2}, \nonumber\\ \Lambda_{23}\!\!&\;=\;&\!\!k_2k_3(\delta+s^{*}_1). \nonumber \end{eqnarray} $ 
Arguments of the roots of the first factor of
Therefore, stability of
$ \begin{eqnarray}\label{jacobe1dtrmnstceqe2} \lambda^{2N}+\Lambda_{21} \lambda^{N}+\Lambda_{22} \lambda^{N(2\alpha)}+\Lambda_{23}=0. \end{eqnarray} $  (20) 
For
$ \begin{eqnarray}\label{intordrchareqne2} (\lambda+\sqrt{A^{2}+B})[\lambda^{2}+[k_{2}+k_3k_2(\delta+s^{*}_1)]\lambda\nonumber\\ k_2k_3(\delta+s^{*}_1)]=0 \end{eqnarray} $  (21) 
which is same as the characteristic equation of the integerorder system
To determine the effect of memory towards the system, we introduce control parameter
$ \begin{eqnarray}\label{modelcpa1a20con} \frac{ds_1}{dt}\!\!&\;=\;&\!\!k_1(s_1s_2c_2+s_{20}s_{10}+e_0)s_1\nonumber\\ &+k_{1}(s_2s_1+s_{10}s_{20}), \nonumber\\ \frac{ds_2}{dt}\!\!&\;=\;&\!\!k_2(s_2s_1+s_{10}s_{20})s_2\nonumber\\ &+(1u(t)){k^{\alpha}_{2}}^{C}_0D^{1\alpha}_t(c_2), \nonumber\\ \frac {dc_2}{dt}\!\!&\;=\;&\!\!k_2(s_2s_1+s_{10}s_{20})s_2\nonumber\\ &(1u(t)){k^{\alpha}_{2}}^{C}_0D^{1\alpha}_t(c_2)k_3c_2 \end{eqnarray} $  (22) 
where
Here the control measure basically stands for temperature, pressure, concentrations of the substances [7], [14] etc.. We study the effect of the control input on the system
In this section, dynamics of reaction kinetics have been analyzed with the help of numerical methods. There are various methods to solve a system of fractionalorder differential equations. We have used the numerical scheme given in [31] and solved our system of equations using the Matlab subroutine "lsqnonlin" and called this method as NSlsq. Here, we have observed the solutions of the fractionalorder system for
Fig. 2 represents the behavioral pattern of the substances for the integerorder (IO) model
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Fig. 2 Concentration profiles of the substances for integerorder system 
The stability regions of the equilibrium points of model system
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Fig. 3 Stability regions of the equilibrium points 
We have compared the dynamic profiles of the substances obtained by decreasing
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Fig. 4 Concentration profiles of the substances for the fractionalorder system 
As the value of
To study the effect of changes of the time taken for formation of the product due to the changes in
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Fig. 5 Concentration profiles of the product 
Here, we investigate the effect due to the changes of control parameter
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Fig. 6 Concentration profiles of the substances of 
We vary the values of the control parameter as
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Fig. 7 Concentration profiles of the product 
In this study, we have presented an approach of fractionalizing a bisubstrate enzyme kinetic reaction. The fractionalorder system is solved numerically, as the system is unlikely to have analytical solutions. Our numerical results reveal that the solutions of the fractionalorder system for
In Section Ⅱ, we have calculated the equilibrium points of the FO system and discussed their stability regions. Our study shows that, similar to the integerorder system, equilibrium points remain stable for a fractionalorder system with a realistic range of parameters.
We have studied the changes of the concentration profiles of the substances due to the changes in
We have focused on the changes of concentration of the product due to the change in
The dynamical behavior of the substances is observed by varying the control input. Presence of the control parameter corresponds to a quicker reaction. The negative effect of the extensive use of memory can be recovered by proper use of a control measure.
The model can be extended by considering memory in both the backward reaction steps. One can consider the mass
[1]  R. Dutta, Fundamentals of Biochemical Engineering. Berlin: Springer, 2008. 
[2]  I. Belgacem and J. L. Gouzé, "Global Stability of full open reversible michaelismenten reactions, " IFAC Proc., vol. 45, no. 15, pp. 591596, 2012. https://www.researchgate.net/publication/280756540_Global_Stability_of_Full_Open_Reversible_MichaelisMenten_Reactions 
[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, "Physicsbased enzyme design: predicting binding affinity and catalytic activity, " Proteins: Struct. Funct. Bioinform., vol. 82, pp. 33973409, Dec. 2014. http://med.wanfangdata.com.cn/Paper/Detail?id=PeriodicalPaper_PM25243583 
[5]  D. VasicRacki, U. Kragl, and A. Liese, "Benefits of enzyme kinetics modelling, " Chem. Biochem. Eng. Quart., vol. 17, no. 1, pp. 718, Mar. 2003. https://www.researchgate.net/publication/237254930_Benefits_of_Enzyme_Kinetics_Modelling 
[6]  R. Roskoski Jr, "The ErbB/HER family of proteintyrosine kinases and cancer, " Pharmacol. Res., vol. 79, pp. 3474, Jan. 2014. http://www.sciencedirect.com/science/article/pii/S1043661813001771 
[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. 3944, Oct. 2014. http://www.sciencedirect.com/science/article/pii/S0016236114004682 
[8]  S. D. Thiberville, N. Moyen, L. DupuisMaguiraga, 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. 345370, Sep. 2013. http://www.ncbi.nlm.nih.gov/pubmed/23811281 
[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. 11371147, Dec. 2014. http://europepmc.org/articles/PMC4243006 
[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 nonlinear second order reaction differential equations in enzyme kinetics, " Nat. Sci., vol. 3, no. 6, pp. 459465, May 2011. http://www.oalib.com/paper/4765 
[13]  A. Meena, A. Eswari, and L. Rajendran, "Mathematical modelling of enzyme kinetics reaction mechanisms and analytical solutions of nonlinear reaction equations, " J. Math. Chem., vol. 48, no. 2, pp. 179186, Aug. 2010. http://link.springer.com/article/10.1007/s1091000996595 
[14]  P. T. Benavides and U. Diwekar, "Optimal control of biodiesel production in a batch reactor: Part Ⅰ: deterministic control, " Fuel, vol. 94, pp. 211217, 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. 27042717, Nov. 2013. https://link.springer.com/article/10.1007/s109100130232x 
[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. 453472, Aug. 2015. https://www.researchgate.net/publication/281549913_Delay_induced_oscillation_in_a_biochemical_model_and_its_control 
[17]  R. A. Azizyan, A. E. Gevorgyan, V. B. Arakelyan, and E. S. Gevorgyan, "Mathematical modeling of uncompetitive inhibition of Bisubstrate enzymatic reactions". Int. Schol. Sci. Res. Innov. , vol.7, no.10, pp.974–977, 2013, 2013. 
[18]  S. Westerlund, "Dead matter has memory!". Phys. Scrip. , vol.43, no.2, pp.174–179, 1991, 1991. DOI:10.1088/00318949/43/2/011 
[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. https://www.worldscientific.com/doi/abs/10.1142/S0217979213300053 
[20]  A. A. Stanislavsky, "Memory effects and macroscopic manifestation of randomness, " Phys. Rev. E, vol. 61, no. 5, pp. 47524759, May 2000. http://www.europepmc.org/abstract/MED/11031515 
[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. 119134, Apr. 2010. https://link.springer.com/article/10.1007/s1092800991473 
[22]  R. ToledoHernandez, V. RicoRamirez, G. A. IglesiasSilva, 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. 217228, Sep. 2014. http://www.sciencedirect.com/science/article/pii/S0009250914003212 
[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. 607614, Jun. 2007. http://www.sciencedirect.com/science/article/pii/S0378437107000623 
[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. http://www.ncbi.nlm.nih.gov/pubmed/24305503 
[25]  M. ElShahed and A. Alsaedi, "The fractional SIRC model and influenza A, " Math. Probl. Eng., vol. 2011, pp. Article No. 480378, Aug. 2011. http://www.ams.org/mathscinetgetitem?mr=2853306 
[26]  S. Abbas, M. Benchohra, G. M. NǴuérékata, and B. A. Silmani, "Darboux problem for fractionalorder discontinuous hyperbolic partial differential equations in Banach algebras". Compl. Variabl. Elliptic Equat.:Int. J. , vol.57, no.24, pp.337–350, 2012, 2012. DOI:10.1080/17476933.2011.555542 
[27]  I. Podlubny, Fractional Differential Equations. San Diego: Academic Press, 1999. 
[28]  F. A. Abdullah, "Using fractional differential equations to model the MichaelisMenten reaction in a 2d region containing obstacles". Science Asia , vol.37, no.1, pp.75–78, 2011, 2011. DOI:10.2306/scienceasia15131874.2011.37.075 
[29]  A. Alawneh, "Application of the multistep generalized differential transform method to solve a timefractional enzyme kinetics". Discr. Dyn. Nat. Soc., vol. 2013, pp. Article No. 592938 , vol.2013, no.592938, 2013. 
[30]  A. Dokoumetzidis, R. Magin, and P. Macheras, "Fractional kinetics in multicompartmental systems, " J. Pharmacokinet. Pharmacodyn., vol. 37, no. 5, pp. 507524, Oct. 2010. http://www.ncbi.nlm.nih.gov/pubmed/20886267 
[31]  T. Sardar, S. Rana, and J. Chattopadhyay, "A mathematical model of dengue transmission with memory, " Commun. Nonlin. Sci. Numer. Simulat., vol. 22, no. 13, pp. 511525, May 2015. http://www.sciencedirect.com/science/article/pii/S100757041400392X 
[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. 288294, Jul. 2016. http://kns.cnki.net/KCMS/detail/detail.aspx?filename=ZDHB201603009&dbname=CJFD&dbcode=CJFQ 
[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. 36103621, Sep. 2013. http://www.ams.org/mathscinetgetitem?mr=3069186 
[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. 747773, Dec. 2017. http://link.springer.com/10.1007/s408190170339z 
[35]  V. E. Tarasov, "No violation of the Leibniz rule. No fractional derivative, " Commun. Nonlin. Sci. Numer. Simulat., vol. 18, no. 11, pp. 29452948, Nov. 2013. http://www.sciencedirect.com/science/article/pii/S1007570413001457 
[36]  M. S. Tavazoei, and M. Haeri, "Chaotic attractors in incommensurate fractional order systems, " Phys. D, vol. 237, no. 20, pp. 26282637, Oct. 2008. http://www.sciencedirect.com/science/article/pii/S0167278908001310 
[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, 2013. 
[38]  S. Schnell and P. K. Maini, "Enzyme kinetics at high enzyme concentration, " Bull. Math. Biol., vol. 62, no. 3, pp. 483499, May 2000. http://www.ncbi.nlm.nih.gov/pubmed/10812718 
[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. 23, pp. 169187, MarJan. 2003. http://www.ingentaconnect.com/content/tandf/gctb/2003/00000008/F0020002/art00003 