Page - 29 - in Advanced Chemical Kinetics

produced an enzyme-reactant complex. Eq. (11) illustrates theMichaelis-Menten kinetics, in which the enzyme-substrate complex is formed after the enzyme is combinedwith the sub- strate. EþS$k1 k�1 ES!k2EþP (11) AscanbeseenfromEq.(11), theproductP isreleasedbythebindingofsubstrateSwithenzyme E. The product released is not reversible; however, the substrate binding is reversible. The reactants’concentrations inEq. (11)arerepresentedbythefollowingletters: s¼ S½ �, e¼ E½ �, c¼ SE½ �, p¼ P½ � (12) The lawofmassaction leads to thesystemof followingnonlinear reactionequations [31], ds dt ¼�k1esþk�1c (13a) de dt ¼�k1esþ k�1þk2ð Þc (13b) dc dt ¼ k1es� k�1þk2ð Þc (13c) dp dt ¼ k2c (13d) where k1 is the forward rate of ES complex formation and k�1 is the backward rate constant. Theaboveproblemisdiscussed theoreticallybyMeenaetal. [32]. Example3:Michaelis-Mentenmechanismforco-substrateandsubstrate Figure 1 illustratesMichaelis-Menten reactionkinetics scheme for co-substrate and substrate. Limoges et al. [33] reported for a redox enzymatic homogenous system along with one- dimensionalmass transport equationaconcisediscussionandderivation. When the enzyme isbeing solubilized, the electrochemical signal that is producedduring the reaction isgovernedbythe followingsetofnonlinearpartialdifferential equations. ∂Q½ � ∂t ¼DP∂ 2Q½ � ∂x2 � C 0 E 1 k1 S½ �þ 1k1,2þ 1k2,2þ 1k2Q½ � (14) ∂S½ � ∂t ¼DS∂ 2S½ � ∂x2 � C 0 E 1 k1 S½ �þ 1k1,2þ 1k2,2þ 1k2Q½ � (15) whereDP ,DSare thediffusioncoefficientsof co-substrateandsubstrate, respectively;Q ,Sare the concentrations of co-substrate and substrate, respectively; x is the distance from the Mathematical Modeling and Simulation of Nonlinear Process in Enzyme Kinetics http://dx.doi.org/10.5772/intechopen.70914 29