Stability Analysis and Numerical Solution for. the Fractional Order Biochemical Reaction Model

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1 Nonlinear Analysis and Differential Equations, Vol. 4, 16, no. 11, HIKARI Ltd, Stability Analysis and Numerical Solution for the Fractional Order Biochemical Reaction Model A. M. Khan 1 and Lalita Mistri 1 Department of Mathematics, JIET Group of Institutions, Jodhpur 34, India Department of Mathematics, Poornima University Jaipur 3395, India Copyright 16 A. M. Khan and Lalita Mistri. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The present article propose a fractional order biochemical reaction model for the Michaelis- Menten Enzyme Kinetic Model (FMIC-MENEKM) resulting from the enzymes reaction process. The numerical solution for the FMIC-MENEKM is carried out by Mittag-Leffler method that are in excellent agreement with those from the classical integer order model. These results show that the fractional modelling has more advantage than classical integer model. We also deal with the stability analysis of equilibrium points. The results presents the complexity of enzymes process, which varies for different fractional order derivatives that are shown graphically. Mathematical Subject Classification: 9C45, 6A33, 4G Keywords: Fractional order biochemical reaction model, Mittag-Leffler method, Non-linear systems, Enzyme process 1. Introduction The basic Michaelis - Menten Enzyme kinetic model resulting from reaction scheme first proposed by Michaelis and Menten [1]. Mathematical Model have been given significant results in understanding the kinetic of the enzyme processes [, 3]. Briggs and Haldane [4] derived the Michaelis - Menten equation from the following reaction scheme

2 5 A. M. Khan and Lalita Mistri ξ + H λ + λ U λ++ ξ + V where ξ is the the enzyme, H is the substrate, U is the enzyme-substrate complex, V is the product of H when metamorphosed, λ + is the first order rate constant, λ + + is the second order rate constant and λ is the first order rate for the reverse reaction. Edeki, Owoloko and other authors [5] investigated the numerical solution of nonlinear Biochemical model using Hybrid numerical analytical techniques. Khan [6] proposed approximate solution of the fractional Susceptible-Infected- Recovered model by modified variational iteration method. Further Sen [7] studied an application of the Adomian decomposition method to the transient behaviour of the model biochemical reaction in the following form η dn = m αn mn dt dm dt = m + (α β)n + mn. Subjected to m () =1, n () =, where parameter η, α, β are dimensionless, m is the dimensionless form of substrate concentration and n is intermediate complex between ξ & H. Fractional calculus has been widely applied in many fields [8] and growing very fast in developing biological models due to its relation with memory and fractals which are abundant in biological systems [9] using time fractional derivatives of f (t) at t = t 1, also fractional order modelling reduces the errors that arising from neglected parameters in real life [1]. In biochemical reaction the membranes of enzyme cell of biochemical organism have fractional order electrical conductance [11]. Hence in this paper we propose a fractional order biochemical reaction model. We use fractional derivative in Caputo sense due to its advantage of dealing initial value cases efficiently.. Fractional Calculus We recall here some basic definitions of fractional calculus Definition 1. The Riemann-Liouville fractional integral operator [1] of order α > of function f : R + R is defined as I α f (x) = 1 x Γ(α) (x t) α 1 f(t)dt (.1) Definition. The Caputo fractional derivative [13] of order α >, n 1 < α n, n N is defined as

3 Stability analysis and numerical solution 53 D α f (x) = I n α D n f (x) = 1 Γ(n α) (x t) n α 1 f (n) (t)dt (.) where f (t) has absolute continuous derivatives up to order (n-1). Lemma 3. The equilibrium points of the following system [1] x D α (x) = f (x), x() = x with < α 1 and x R n (.3) are calculated by solving f (x) =. These points are locally asymptotically stable if all eigenvalues λ i of Jacobian matrix J= f/ x which are evaluated at the equilibrium points must satisfy arg λ i > α(π/) (.4) Now we introduce the fractional order biochemical reaction (FBCR) model in terms of the following equations D α 1(x) = η 1 (y αx xy) D α (y) =(α β)x + xy y (.5) Subjected to x () =, y () = 1, (.6) where α 1, α > and parameter η, α and β are dimensionless, x is the intermediate complex between ξ & H whereas y denotes the dimensionless form of substrate concentration. 3. Mittag-Leffler function method Rida and Arafa [14] discussed the Mittag-Leffler function for solving linear functional differential equations by using Mittag-Leffler function E α(z) and E α,β by decomposing y i (t) and D α y i (t) in terms of infinite series of components as follows y i (t) = E α(a i t α ) = n t a nα n= i (3.1) D α y i (t) = n t a (n 1)α n=1 i, (3.) Γ(n 1)α+1)

4 54 A. M. Khan and Lalita Mistri where E α(z) and E α,β are Mittag-Leffler functions [15] defined as E α = z n n=, z n E α,β = n=, i = 1,, 3.. α, β >. (3.3) Γ(nα+β) 4. Numerical Solution of Fractional Biochemical Reaction model In this section we apply Mittag-Leffler method in fractional biochemical reaction model D α 1(x) = η 1 (y αx xy) (4.1) D α (y) =(α β)x + xy y where x () =, y () = 1 and α 1, α >. (4.) Using Mittag-Leffler function method we put x (t) = E α(at α ) = a n t nα n= y (t) = E α(bt α ) = b n t nα n= (4.3) D α y i (t) = n t a (n 1)α i n=1 Γ(n 1)α+1) If α 1, α = α and using (4.3) in (4.1) n η t(n 1)α n=1 a b n t nα n= + α a n t nα n= + n= c n 1 t nα = Γ(n 1)α+1) (4.4) b n t (n 1)α n=1 + b n t nα Γ(n 1)α+1) n= (α β) a n t nα n= n= c n 1 t nα = (4.5) n a k b n k where c n 1 = k= (4.6) Γ(n k)α+1)γ(kα+1) Equating the coefficient of t nα equal to zero, we get a n+1 η b n + αa n n + c 1 Γ(nα + 1) = (4.7)

5 Stability analysis and numerical solution 55 b n+1 + b n (α β)a n c 1 n Γ(nα + 1) = (4.8) On putting n= we obtain a 1 η b + αa + c 1 Γ(1) = (4.9) b 1 + b (α β)a c 1 Γ(1) = (4.1) c 1 = a b Γ(1)Γ(1). With x () =a =, y () = b = 1, α = 1, β = 3 8, η = 1 1. (4.11) x (t) = a n t nα n= = a + a 1 t α Γ(α+1) + a t α Γ(α+1) + a3 t 3α Γ(3α+1) + (4.1) y (t) = b n t nα n= = b + b 1 t α + Γ(α+1) b t α + Γ(α+1) b3 t 3α Γ(3α+1) + (4.13) Using (4.11) and putting n =1,, 3... in (4.7), (4.8) we get tα x (t) = 1 1 Γ(α+1) tα Γ(α+1) 8 t 3α Γ(3α+1) (4.14) y (t) = 1 t α t α 38 Γ(α+1) 8 Γ(α+1) 8 t 3α Γ(3α+1) + (4.15) Remark: When α = 1, (4.14) and (4.15) reduces to known result due to S. O. Edeki et al. [5, eq. 3, 4, pp. 41]. x (t) = 1 t 15 t t3 (4.16) y (t) = 1 t t t3 + (4.17) Further the basic reproductive number of tissues for model (.5) is given by K where K= (1 + α η ) 4 β η, α η. R = (1+ α, (4.18) η )

6 56 A. M. Khan and Lalita Mistri Theorem 4.1 Consider the FBCR model (.5) (i) The transition free equilibrium state E is locally asymptotically stable if 1 < R < 1. (ii) If 1 > R > 1, the equilibrium state E is unstable, and if R = 1 or 1, it is critical case. Proof. The characteristic equation for equilibrium state at (, ) is given as follows det ( α η λ 1 η ) =, (4.19) α β 1 λ which gives the following eigen values λ 1 = (1+α η )+ K, λ = (1+α η ) K (4.) where K= (1 + α η ) 4 β η. For K, it is clear that the characteristic roots λ <, which means arg λ = π > α(π/). Since (1 + α η ) K < this gives (1 + α η ) + K >. 1 + K (1+ α η ) >, 1 + R >, Hence for R > 1, Further if R < 1 implies K < 1 + α gives λ η 1 < which means arg λ 1 = π > α(π/), which clearly ensure the transition free equilibrium state E is locally asymptotically stable, if 1 < R < 1. If R = - 1, λ = and R = 1, λ 1 = both are critical cases..5 x 16 x(t) y(t) Order of Derivative =.5 solutions x(t), y(t) time t (Figure 1. Solution x (t), y (t) for fractional time derivative.5)

7 Stability analysis and numerical solution x x(t) y(t) Order of derivative =.75 solutions x(t), y(t) time t (Figure. Solution x (t), y (t) for fractional time derivative.75) 4.5 x x(t) y(t) Order of Derivative = 1.5 solutions x(t), y(t) time t (Figure 3. Solution x (t), y (t) for fractional time derivative 1.5)

8 58 A. M. Khan and Lalita Mistri 1 x x(t) y(t) Order of Derivative = 1 solutions x(t), y(t) time t (Figure 4. Solution x (t), y (t) for integer time derivative 1) Conclusion In this paper we introduced a fractional order biochemical reaction model for the Michaelis- Menten Enzyme Kinetic Model (FMIC-MENEKM) and dealt with the mathematical behaviour of the model by solving through Mittag-Leffler method and also investigate stability analysis of their equilibrium states according to the relation between system parameters which are very significantly and with original one. This method gives better realistic series solutions which converge rapidly and results obtained are in excellent agreement with results given by different authors for integer order. The graphical results reveal that solution continuously depends on time fractional derivatives and valid for long time in integer case. We found that the stability of the transition free equilibrium state E of the FMIC-MENEKM model which is stable when the basic reproductive number (R ) is lies between negative one to positive one. However when (R ) is out of this range the system will be unstable. We hope due to providing exact solution and allowing greater degree of freedom and understanding of dynamical behaviour of the model by proposed method, there is vast scope of further study of this kind of problems that will certainly motivate researchers who are working in the field of fractional models.

9 Stability analysis and numerical solution 59 References [1] L. Michaelis and M. Menten, Die Kinetik der invertinwirkun, Biochemistry Zeitung, 49 (1913), [] I. H. Segel, Enzyme Kinetics: Behaviour and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems, John Wiley & Sons, New York, [3] H. Qian, A Guide for Michaelis - Menten Enzyme Kinetic Models (MICMEN),. MICMEN/index.html [4] G. E. Briggs and J. B. Haldane, A Note on the Kinetics of Enzyme Action, Biochem. J., 19 (195), [5] S. O. Edeki, E. A. Owoloko, A. S. Osheku, A. A. Opanuga, H. I. Okagbue and G. O. Akinlabi, Numerical Solutions of Nonlinear Biochemical Model Using a Hybrid Numerical-Analytical Technique, International Journal of Mathematical Analysis, 9 (15), no. 8, [6] A. M. Khan, Amit Chouhan and Pankaj Ramani, Approximate Solution of their fractional susceptible infected recovered model by modified variation iteration method, Journal of Fractional Calculus and Applications, 7 (16), no. 1, [7] A. K. Sen, An Application of the Adomian decomposition method to the transient behaviour of a model biochemical reaction, Journal of Mathematical Analysis and Applications, 131 (1988), [8] J. H. He, Some applications of nonlinear fractional differential equations and their approximations, Bulletin of Sciences Technology & Society, 15 (1999), [9] J. Li, K. Yang, C. Clinton, Modelling the glucose insulin regulatory system and ultradian insulin secretary oscillations with two explicit time delays, Journal of Theoretical Biology, 4 (6), [1] Y. Ding Haiping Ye, A fractional order differential equation model of HIV infection of CD 4 + T- cells, Mathematical and Computer modelling, 5 (9), [11] K. S. Cole, Electric Conductance of biological systems, Proc. Cold Spring Harbour Symp. Quant. Biol Cold Spring Harbor, New York, (1993),

10 53 A. M. Khan and Lalita Mistri [1] I. Petras, Fractional Order Nonlinear Systems: Modelling, Analysis and Simulation, Springer Berlin Heidelberg, [13] Li. J, Yang K., Clinton C., Modelling the glucose insulin regulatory system and ultradian insulin secretary oscillations with two explicit time delays, Journal of Theoretical Biology, 4 (6), [14] S. Z. Rida, A. A. M. Arafa, New Method for Solving Linear Fractional Differential Equations, International Journal of Differential Equation, 11 (11), 1-8, Article ID [15] I. Podlubny, Fractional Differential Equations, Academic Press, New York, Received: June 4, 16; Published: September, 16