Research Article  Open Access
Application of the Multistep Generalized Differential Transform Method to Solve a TimeFractional Enzyme Kinetics
Abstract
The multistep differential transform method is first employed to solve a timefractional enzyme kinetics. This enzymesubstrate reaction is formed by a system of nonlinear ordinary differential equations of fractional order. The fractional derivatives are described in the Caputo sense. A comparative study between the new algorithm and the classical RungeKutta method is presented in the case of integerorder derivatives. The results demonstrate reliability and efficiency of the algorithm developed.
1. Introduction
Mathematical modeling of complex processes is a major challenge for contemporary scientists. In contrast to simple classical systems, where the theory of integerorder differential equations is sufficient to describe their dynamics, complex systems are characterized by the variability of structures in them, multiscale behavior, and nonlinearity in the mathematical description of the mutual relationship between parameters [1]. Fractional derivatives provide an excellent instrument for the description of dynamical behavior of various complex materials and systems. There are two important benefits: (1) we have more degrees of freedom in the model considered and (2) a “memory” is enclosed in the model. Therefore, the number of scientific and engineering problems involving fractional derivatives is already very large and still growing and perhaps the fractional calculus (i.e., derivatives and integrals of any real or complex order) will be the calculus of the twentyfirst century [1].
Recent investigations have shown that many complex biological systems can be represented more accurately through fractional derivative formulation. Magin [2] was the first who used fractional derivatives and fractional integrals in order to model stressstrain relationship in biomaterials. Craiem et al. [3] applied fractional calculus to model arterial viscoelasticity. Abdullah [4] used fractional differential equations to model the MichaelisMenten reaction in a 2d region containing obstacles. He investigated the phenomenon of subdiffusion, which occurs when there is molecular crowding, by proposing a continuous spatial model involving fractional differential equations. The MichaelisMenten mechanism is the simplest chemical network which models the formation of a product through an enzymatic catalysis of a substrate. In particular, an enzyme reacts with the substrate and reversibly forms an intermediate complex, which then decays into the product and original enzyme. This enzyme reaction model is given by (for more details on the enzyme dynamics, see [5–9] and the references therein) where is the enzyme, the substrate, the enzymesubstrate intermediate complex, and the product. The parameters , , and are positive rate constants and denote the rates of reaction of these three processes. Note that substrate binding is reversible but product release is not. The concentration of the reactants in (1) is denoted by lower case letters: Applying the law of mass action, which states that reaction rates are proportional to the concentrations of the reactants, the time evolution of scheme (1) can be determined from the solution of the system of coupled nonlinear ordinary differential equations (ODEs) [6, 7]: subject to the initial conditions Systems (3) can be reduced to only two equations for and , namely [6], subject to the initial conditions and By introducing the following parameters:
Then, (5) and the initial conditions (4) can be represented in dimensionless form as follows: where , , and are dimensionless parameters.
Now, we introduce fractional order into enzyme reaction model. The new system is described by the following set of fractional differential equations: where the fractional derivatives , , and are considered in the Caputo sense, and , , are parameters describing the order of the fractional timederivatives in the Caputo sense. The general response expression contains parameters describing the order of the fractional derivatives that can be varied to obtain various responses. Obviously, the classical integerorder enzyme reaction model can be viewed as a special case from the fractionalorder enzyme reaction model by setting . In other words, the ultimate behavior of the fractional system response must converge to the response of the integerorder version of the equation [10].
Our motivation for this work is to obtain the approximate solution of the timefractional enzyme kinetics (8)–(10) using the multistep generalized differential transform method (MSGDTM). This method is only a simple modification of the generalized differential transform method (GDTM) [11–14], in which it is treated as an algorithm in a sequence of small intervals (i.e., time step) for finding accurate approximate solutions to the corresponding systems. The approximate solutions obtained by using GDTM are valid only for a short time, while the ones obtained by using the MSGDTM [15, 16] are more valid and accurate during a long time and are in good agreement with the RK45 numerical solution when the order of the derivative , .
This paper is organized as follows. In Section 2, we present some necessary definitions and notations related to fractional calculus and differential transform method. In Section 3, the proposed method is described. In Section 4, the method is applied to the problems (8)–(10), and numerical simulations are presented graphically. Finally, the conclusions are given in Section 5.
2. Preliminaries
In this section, we give some basic definitions and properties of the fractional calculus theory and differential transform method which are used further in this paper [1, 10, 17–25].
2.1. Fractional Calculus
Definition 1. A function is said to be in the space if it can be written as for some , where is continuous in , and it is said to be in the space if , .
Definition 2. The RiemannLiouville integral operator of order with is defined as Properties of the operator can be found in [1, 10]. We only need here the following: for , , and , we have where is the incomplete beta function which is defined as The RiemannLiouville derivative has certain disadvantages when trying to model realworld phenomena with fractional differential equations. Therefore, we will introduce a modified fractional differential operator proposed by Caputo in his work on the theory of viscoelasticity.
Definition 3. The Caputo fractional derivative of of order with is defined as for , .
The Caputo fractional derivative was investigated by many authors; for , and , we have For mathematical properties of fractional derivatives and integrals, one can consult the mentioned references.
2.2. The Definitions and Operations of Differential Transform
The differential transform technique is one of the seminumerical analytical methods for ordinary and partial differential equations that uses the form of polynomials as approximations of the exact solutions that are sufficiently differentiable. The basic definition and the fundamental theorems of the differential transform method (DTM) and its applicability for various kinds of differential equations are given in [21–23]. For convenience of the reader, we present a review of the DTM. The differential transform of the th derivative of function is defined as follows: where is the original function and is the transformed function. The differential inverse transform of is defined as: From (17) and (18), we get which implies that the concept of differential transform is derived from Taylor series expansion, but the method does not evaluate the derivatives symbolically. However, relative derivatives are calculated by an iterative way which is described by the transformed equations of the original function. For implementation purposes, the function is expressed by a finite series, and (18) can be written as Here, is decided by the convergence of natural frequency. The fundamental operations performed by differential transform can readily be obtained and are listed in Table 1. The main steps of the DTM, as a tool for solving different classes of nonlinear problems, are the following. First, we apply the differential transform (17) to the given problem (integral equation, ordinary differential equation, or partial differential equations), and then the result is a recurrence relation. Second, solving this relation and using the differential inverse transform (18), we can obtain the solution of the problem.

3. Solving the Systems (8)–(10) Using the Multistep Generalized Differential Transform Method (MSGDTM)
It has been shown that the approximated solutions obtained using DTM are not valid for large for some systems [11–14]. Therefore, we use the multistep generalized differential transform method (MSGDTM) to solve the system of equations (8)–(10). The method is only a simple modification of the generalized differential transform method, in which it is treated as an algorithm in a sequence of small intervals (i.e., time step) for finding accurate approximate solutions to the corresponding systems [15, 16]. This MSGDTM offers accurate solutions over a longer time frame (more stable) compared to the standard GDTM.
Taking the differential transform of (8)–(10) with respect to time gives where , , and are the differential transformation forms of , , and , respectively. The differential transform forms of the initial conditions are given by , , and . In view of the differential inverse transform, the differential transform series solution for the systems (8)–(10) can be obtained as Now, according to the multistep generalized differential transform method, the series solution for the systems (8)–(10) is suggested by where , , and for satisfy the following recurrence relations: such that , , and . Finally, if we start with , , and , using the recurrence relation given in (24), then we can obtain the multistep solution given in (23).
This multistage MSGDTM offers accurate solutions over a longer time frame (more stable) compared to the standard GDTM [11–14]. This distinctive strategy grants the iterative algorithm a timemarching scheme which significantly drives forward the convergence of the solutions precisely with great rapidity. The value of increases according to the designated timestep size in each iteration computation. Each amplified time step will produce a new approximation value for the desired iteration step, and the process is continued until the targeted time frame is achieved. Errors are therefore minimized and depend on the time step , but the main disadvantage lies in having longer computational time. Generally, the convergence is achieved more quickly as the iteration steps increase.
4. Numerical Results
In this work, we carefully propose the MSGDTM, a reliable modification of the GDTM, that improves the convergence of the series solution. The method provides immediate and visible symbolic terms of analytic solutions, as well as numerical approximate solutions to both linear and nonlinear differential equations. To demonstrate the effectiveness of the proposed algorithm as an approximate tool for solving the nonlinear system of fractional differential equations (8)–(10) for larger , we apply the proposed algorithm on the interval . We divide this interval to subintervals with time step or and the temnumber of GDTM series solutions is fixed, . All the results are calculated by using the computer algebra package Mathematica.
Figures 1 and 2 show the approximate solutions obtained using the MSGDTM and the fourthorder RungeKutta method of concentrations of substrate , enzymesubstrate complex , and product for various values of dimensionless reaction parameters , , and when . It can be seen that the results from the MSGDTM match the results of the RungeKutta method very well, which implies that the MSGDTM can predict the behavior of these variables accurately for the region under consideration. Therefore, the proposed method is a very efficient and accurate method that can be used to provide analytical solutions for nonlinear systems of differential equations.
Next, we consider how the concentrations of substrate , enzymesubstrate complex , and product depend upon the magnitude of the order of fractional derivatives. We fix the dimensionless reaction parameters , , and and perform the numerical simulation for different values of , , and . Simulation results are presented in Figures 3, 4, 5, and 6, and it is clear that the concentrations of the substrate , enzymesubstrate complex , and product continuously depend on the fractional derivatives. For example, the concentration of substrate decreases gradually from its initial value of the concentration and converges to zero when and decreases slowly to zero when , , and . However, the product increases rapidly for the integerorder derivatives and increases slowly in the case of fractionalorder derivatives as shown in Figures 3–6. The key finding of these graphs is that the generalized enzyme kinetics model with fractional derivatives has more degrees of freedom and therefore can be varied to obtain various responses of the concentrations of substrate , enzymesubstrate complex , and product .
5. Conclusions
In this paper, the application of MSGDTM was extended to obtain explicit and numerical solutions of a timefractional enzyme kinetics. The MSGDTM was clearly a very efficient and powerful technique in finding the solutions of the proposed equations. The obtained results demonstrate the reliability of the algorithm and its wider applicability to fractional nonlinear evolution equations.
Acknowledgment
The author gratefully acknowledges the financial support received from The University of Jordan during the sabbatical leave from Department of Mathematics, The University of Jordan, Amman, Jordan.
References
 J. S. Leszczynski, An Introduction to Fractional Mechanics, Czestochowa University of Technology, 2011.
 R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Redding, Calif, USA, 2006.
 D. O. Craiem, F. J. Rojo, J. M. Atienza, G. V. Guinea, and R. L. Armentano, “Fractional calculus applied to model arterial Viscoelasticity,” Latin American Applied Research, vol. 38, no. 2, pp. 141–145, 2008. View at: Google Scholar
 F. A. Abdullah, “Using fractional differential equations to model the MichaelisMenten reaction in a 2d region containing obstacles,” ScienceAsia, vol. 37, no. 1, pp. 75–78, 2011. View at: Publisher Site  Google Scholar
 M. A. Savageau, “MichaelisMenten mechanism reconsidered: implications of fractal kinetics,” Journal of Theoretical Biology, vol. 176, no. 1, pp. 115–124, 1995. View at: Publisher Site  Google Scholar
 S. Schnell and C. Mendoza, “Closed form solution for timedependent enzyme kinetics,” Journal of Theoretical Biology, vol. 187, no. 2, pp. 207–212, 1997. View at: Publisher Site  Google Scholar
 A. Meena, A. Eswari, and L. Rajendran, “Mathematical modelling of enzyme kinetics reaction mechanisms and analytical solutions of nonlinear reaction equations,” Journal of Mathematical Chemistry, vol. 48, no. 2, pp. 179–186, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. M. Goh, M. S. M. Noorani, and I. Hashim, “Introducing variational iteration method to a biochemical reaction model,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2264–2272, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 I. Hashim, M. S. H. Chowdhury, and S. Mawa, “On multistage homotopyperturbation method applied to nonlinear biochemical reaction model,” Chaos, Solitons and Fractals, vol. 36, no. 4, pp. 823–827, 2008. View at: Publisher Site  Google Scholar
 I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. View at: MathSciNet
 Z. Odibat, S. Momani, and V. S. Erturk, “Generalized differential transform method: application to differential equations of fractional order,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 467–477, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. Momani and Z. Odibat, “A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor's formula,” Journal of Computational and Applied Mathematics, vol. 220, no. 12, pp. 85–95, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 V. S. Erturk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multiorder fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1642–1654, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. M. Odibat, C. Bertelle, M. A. AzizAlaoui, and G. H. E. Duchamp, “A multistep differential transform method and application to nonchaotic or chaotic systems,” Computers and Mathematics with Applications, vol. 59, no. 4, pp. 1462–1472, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 V. S. Ertrk, Z. M. Odibat, and S. Momani, “An approximate solution of a fractional order differential equation model of human Tcell lymphotropic virus i (HTLVI) infection of CD^{4+ } Tcells,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 996–1002, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. Caputo, “Linear models of dissipation whose Q is almost frequency independentII,” Geophysical Journal of the Royal Astronomical Society, vol. 13, no. 5, pp. 529–539, 1967. View at: Publisher Site  Google Scholar
 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. View at: MathSciNet
 R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, New Jersey, NJ, USA. View at: MathSciNet
 R. L. Bagley and P. J. Torvik, “A theoretical basis for the application of fractional calculus,” Journal of Rheology, vol. 27, no. 3, pp. 201–210, 1983. View at: Publisher Site  Google Scholar
 O. Özkan, “Numerical implementation of differential transformations method for integrodifferential equations,” International Journal of Computer Mathematics, vol. 27, no. 3, 10 pages. View at: Publisher Site  Google Scholar
 C. W. Bert and H. Zeng, “Analysis of axial vibration of compound bars by differential transformation method,” Journal of Sound and Vibration, vol. 275, no. 3–5, pp. 641–647, 2004. View at: Publisher Site  Google Scholar
 B. Jang, “Solving linear and nonlinear initial value problems by the projected differential transform method,” Computer Physics Communications, vol. 181, no. 5, pp. 848–854, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 E. Heinsalu, M. Patriarca, I. Goychuk, G. Schmid, and P. Hänggi, “Fractional fokkerplanck dynamics: numerical algorithm and simulations,” Physical Review E, vol. 73, no. 4, Article ID 046133, 9 pages, 2006. View at: Publisher Site  Google Scholar
 E. Heinsalu, M. Patriarca, I. Goychuk, and P. H. Hanggi, “Use and abuse of a fractional fokkerplanck dynamics for timedependent driving,” Physical Review Letters, vol. 99, no. 12, Article ID 120602, 4 pages, 2007. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2013 Ahmed Alawneh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.