Numerical modeling for nonlinear biochemical reaction networks

Document Type: Research Paper

Authors

1 Lecturer, Department of Computer Science, University of Central Punjab, Lahore, Pakistan.

2 Assistant Professor, Department of Mathematics, University of Engineering & Technology, KSK Campus, Pakistan

3 Professor, University of Engineering and Technology, Lahore Campus, Lahore, Pakistan.

4 Assistant Professor, Faculty of Electrical Engineering, University of Central Punjab, Pakistan

Abstract

Nowadays, numerical models have great importance in every field of science, especially for solving the nonlinear differential equations, partial differential equations, biochemical reactions, etc. The total time evolution of the reactant concentrations in the basic enzyme-substrate reaction is simulated by the Runge-Kutta of order four (RK4) and by nonstandard finite difference (NSFD) method. A NSFD model has been constructed for the biochemical reaction problem and numerical experiments are performed for different values of discretization parameter ‘h’. The results are compared with the well-known numerical scheme, i.e. RK4. Unlike RK4 which fails for large time steps, the developed scheme gives results that converge to true steady states for any time step used.

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Main Subjects


1. S. Schnell andC. Mendoza, Closed form solution for time dependent enzyme
kinetics, J Theor Biol.187 (1997) 207–212.
2. A. K. Sen, An application of the Adomian decomposition method to the transient
behavior of a model biochemical reaction, J Math Anal Appl. 131 (1988) 232–245.
3. P. Pongsumpun, Mathematical model of Dengue disease with incubation period of
virus, World Acad. Sci. Eng. Technol. 44 (2008) 328–332.
4. M. Rafiq, M.O.Ahmed, S. Ahmed, R. Siddique and A. Pervaiz, Some finite
Difference Methods for One Dimensional Burgers Equation for Irrotational
Incompressible Flow Problem, Pak. J. Engg. & Appl. Sci. 9 (2011) 13–16.
5. Z. Zafar, M. O. Ahmad, A. Pervaiz and M. Rafiq, Fourth Order Compact Method
for One Dimensional Inhomogeneous Telegraph Equation with 􀜱(ℎ􀬸, 􀝇􀬷), Pak. J.
Engg. & Appl. Sci. 14 (2014) 96–101.
6. R. E. Mickens, Numerical Integration of population models satisfying conservation
laws: NSFD METHODS, Biological Dynamics 1 (2007) 427–436.
7. R. E. Mickens, Dynamical consistency: a fundamental principle for constructing
Non–standard finite difference schemes for differential equations, J. Differ. Equ.
Appl. 11 (2005) 645–653.

8. A. A. M. Arafa, S. Z. Rida and H. Mohamed, An application of the homotopy
analysis method to the transient behavior of a biochemical reaction model, Inf. Sci.
Lett. 3 (2014) 29–33.
9. 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, Int. J. Math. Anal. 9 (2015) 403–416.
10. F. Brauer and C. Castillo Chavez, Mathematical Models in Population Biology and
Epidemology, Springer–Verlag, 2012.
11. I. Hashim, M. S. H. Chowdhury and S. Mawa, On multistage homotopy
perturbation method applied to non–linear biochemical reaction model, Chaos
Soltion & Fractals 36 (2008) 823–827.
12. Z. Zafar, K. Rehan and M. Mushtaq, Fractional-order scheme for bovine babesiosis
disease and tick populations, Adv. Difference Equ. (2017) 2017:86.