Numerical solution of an autocatalytic bio-chemical system F.N.M. Al-Showaikh Department of Mathematics, University of Bahrain, P.O. Box 32038, Isa Town, Bahrain Abstract A glycolytic model, which is an autocatalytic biochemical system, is solved numerically using two numerical methods based on finite-difference schemes. Method 1, the well known Euler method, is an explicit method, whereas method 2 is implicit. Although the implicit method, method 2, is first-order accurate in time it converges to the fixed point(s) for large time step, ‘. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Autocatalytic biochemical system; Implicit method; Finite-difference method 1. Introduction The first observation indicating the existence of oscillatory behaviour in glycolysis was made by Duysen and Amesz in 1957, see Glodbeter [4], who reported, by studying the fluorescence of some glycolytic interme- diates in yeast, that one of these underwent damped oscillations in the course of time. The essential property of glycolytic oscillations is observed only in a precise range of substrate injection rates. This observation, carried out in yeast extracts, was confirmed in suspensions of intact yeast cells (Von Klitzing and Betz [15]). Below a critical value of the substrate injection rate, the system reaches a stable steady state. When this rate increases, oscillations occur, but they disappear when the substrate injection rate exceeds a second, higher, critical value. This disappearance is reversible. The period of glycolytic oscillations is of the order of several minutes and diminishes as the substrate injection rate increases as given in Goldbeter [4]. The source of oscillations within the glycolytic system has been identified (Ghosh and Chance [3]; Hess and Boiteux [6]). Glycolysis represents a chain of enzyme reactions which in yeast transforms a sugar such as glu- cose or fructose into ethanol and CO 2 . When a hexose such as glucose 6-phosphate or fructose 6-phosphate (F6P) is taken as the glycolytic substrate, periodic behaviour is observed. This observation indicates that the source of oscillations lies beyond the first two enzymes of the chain, hexokinase and glucose–phosphate isom- erase. When the phosphofructokinase (PFK) step is bypassed by injecting fructose 1,6-bisphosphate (FBP) as 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.064 E-mail address: faisal@sci.uob.bh Applied Mathematics and Computation 176 (2006) 177–193 www.elsevier.com/locate/amc