Page - 22 - in Advanced Chemical Kinetics

developcertainmethods for improvingfunctionalityof somemolecules inacell.Manyproblems intheoreticalandexperimentalbiology/chemistry involvethesolutionofthesteady-statereaction diffusionequationwithnonlinear chemical kinetics. Suchproblemsalsoarise in the formulation ofsubstrateandproductmaterialbalances forenzymes immobilizedwithinparticles [1,2], in the descriptionofsubstrate transport intomicrobialcells [3–5], inmembranetransport, in thetransfer ofoxygentorespiringtissue [6,7],andintheanalysisofanyartificialkidneysystem[8]. To impose the functionality of somemolecules in a cell, amathematicalmodel of suchmeta- bolic systems must be constructed and simulated. Most of the dynamical systems can be approximatedbyvarious types ofdifferential and integral equations involving finite number ofvariablesandparameters.Thus, the futurebehaviorof thesystemcanbepredicted ifmodel kineticsparameters and initial statesof thevariables areavailable. Inparticular, ordinaryand partial differential equations (ODEs and PDEs) are popular in modeling of the metabolic pathwaysorenzymekinetics. Releasing enzyme-substrate reactions under single-molecule kinetics was reported by Shlomi et al. [9].An integral equationmethodwithMichaelis-Mentenkinetics to solvenonlineardiffu- sionproblems in spherical coordinateswas statedbyTosakaandMiyale [10].Maalmi et al. [11] reportednumericalandsemianalytical solutionsofnonlinearequations,whichcovereddiffusiv- ity, size, bulk concentrationof reactant, binding constant ofMichaelis-Mentenkinetics, and site reactivityvalues.Merchant [12] stated theM-MdecayreactiontermsandtheGray-Scott scheme along with the semianalytical method to nonlinear reaction-diffusion systems. Indira and Rajendran [13] described a homotopy perturbation method to obtain substrate and product concentrationswithintheenzymatic layers.RemovalofsubstratefromMichaelis-Mentenkinetics governed the extravascularpartition inwhich the analytical solution for the steady-state condi- tionwasinvestigatedbyBucoloandTripathi [14].DangDoandGreenfield[15]utilizedthefinite integral transformmethod to elucidate the problem based on the nonlinear reaction diffusion coupledwith the chemical kinetics of ageneral shape solid.Chapwanyaet al. [16] conveyedan epidemiologicalmodelwith theMichaelis-Menten contact rate formulation to investigatevaria- tions in the enzymekineticswith a simple susceptible infected recovered (SIR)model.Napper [17] proposed theMichaelis-Mentenkineticsmodel to investigate theoxygen transport toheart tissue.Regalbutoetal. [18]presentedananalyticalmethodologyforobtainingsolutionsbasedon themaximumprinciple tononlinear reaction-diffusionboundaryvalueproblems. Rajendran and Saravanakumar [19] discussedmediated bioelectrocatalysis in order to build bioreactors,bio fuel cells, andbiosensors. Due to the difficulties in solving nonlinear differential equations in enzyme kinetics, some recent advanced analytical and numerical simulation techniques are used to solve the prob- lems in chemical kinetics. Thus, in this review, all analytical andnumericalworks in enzyme kineticsare summarized. 2.Reactiondiffusionsystems Reaction diffusion system is a mathematical model based on how the concentration of sub- stances/products isdisseminatedoverspacechangesunder the influenceofdiffusionandalocal Advanced Chemical Kinetics22