Abstract
This chapter introduces mathematical modeling of catalytic biosensors. After a brief tutorial consideration of kinetics of biocatalytic reactions, transducer function of biosensors and a general scheme of biosensor action, a detail mathematical model is then presented for an amperometric biosensor based on a mono-layer of an enzyme immobilized onto the surface of the electrode. The biosensor is modeled by two-component (substrate and product) reaction–diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetics of an enzyme reaction . A few modifications of the mathematical model describing the action of potentiometric, optical and fluorescence biosensors are discussed, too. A special emphasis is placed on the modeling biosensors at steady state and internal or external diffusion limitation with a contribution to the modeling biosensors at non-stationary state at some critical concentrations of the substrate when analytical solution of the governing equations is performed. Using numerical simulation, the influence of the model parameters on the biosensor response is investigated. The simulation of the biosensor operation particularly showed a non-monotonous change of the steady state biosensor current versus the membrane thickness at the various maximal enzymatic rates.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Andrés I (2008) Enzyme biocatalysis: principles and applications. Springer, Dordrecht
Aris R (1975) The mathematical theory of diffusion and reaction in permeable catalysts: vol. 1: the theory of the steady state. Oxford studies in physics. Oxford University Press, Oxford
Aris R (1999) Mathematical modeling: a chemical engineer’s perspective. Academic Press, London
Ašeris V, Baronas R, Petrauskas K (2016) Computational modelling of three-layered biosensor based on chemically modified electrode. Comp Appl Math 35(2):405–421
Baldini F, Chester A, Homola J, Martellucci S (2006) Optical chemical sensors. Springer, Amsterdam
Banica FG (2012) Chemical sensors and biosensors: fundamentals and applications. Wiley, Chichester
Bard A, Faulkner L (2001) Electrochemical methods. fundamentals and applications, 2 edn. Wiley, New York
Baronas R (2017) Nonlinear effects of diffusion limitations on the response and sensitivity of amperometric biosensors. Electrochim. Acta 240:399–407
Baronas R, Gaidamauskaitė E, Kulys J (2007) Modeling a peroxidase-based optical biosensor. Sensors 7(11):2722–2740
Baronas R, Ivanauskas F, Kaunietis I, Laurinavicius V (2006) Mathematical modeling of plate-gap biosensors with an outer porous membrane. Sensors 6(7):727–745
Baronas R, Ivanauskas F, Kulys J (2002) Modelling dynamics of amperometric biosensors in batch and flow injection analysis. J Math Chem 32(2):225–237
Baronas R, Ivanauskas F, Kulys J (2003) The influence of the enzyme membrane thickness on the response of amperometric biosensors. Sensors 3(7):248–262
Baronas R, Ivanauskas F, Kulys J (2007) Computational modelling of the behaviour of potentiometric membrane biosensors. J Math Chem 42(3):321–336
Baronas R, Kulys J (2014) Modeling and simulation of biosensors. In: Kreysa G, Ota K, Savinell R (eds) Encyclopedia of applied electrochemistry. Springer, New York, pp 1304–1309
Bartlett P, Birkin P, Wallace E (1997) Oxidation of β-nicotinamide adenine dinucleotide (NADH) at poly(aniline)-coated electrodes. J Chem Soc Faraday Trans 93(10):1951–1960 (1997)
Bartlett P, Pratt K (1995) Theoretical treatment of diffusion and kinetics in amperometric immobilized enzyme electrodes. Part I: Redox mediator entrapped within the film. J Electroanal Chem 397(1–2):61–78
Bartlett P, Whitaker R (1987) Electrochemical imobilisation of enzymes: part 1. Theory. J Electroanal Chem 224:27–35
Bartlett PN (2008) Bioelectrochemistry: fundamentals, experimental techniques and applications. Wiley, Chichester
Bieniasz L (2017) A specialised cyclic reduction algorithm for linear algebraic equation systems with quasi-tridiagonal matrices. J Math Chem 55(9):1793–1807
Bieniasz L, Britz D (20004) Recent developments in digital simulation of electroanalytical experiments. Pol J Chem 78(9):1195–1219
Bisswanger H (2008) Enzyme kinetics: principles and methods, 2 edn. Wiley-Blackwell, Weinheim
Blaedel W, Kissel T, Boguslaski R (1972) Kinetic behavior of enzymes immobilized in artificial membranes. Anal Chem 44(12):2030–2037
Bosch M, Sánchez A, Rojas F, Ojeda C (2007) Recent development in optical fiber biosensors. Sensors 7(6):797–859
Briggs G, Haldane J (1925) A note on the kinetics of enzyme action. Biochem J 19:338–339
Britz D, Baronas R, Gaidamauskaitė E, Ivanauskas F (2009) Further comparisons of finite difference schemes for computational modelling of biosensors. Nonlinear Anal Model Control 14(4):419–433
Britz D, Strutwolf J (2016) Digital simulation in electrochemistry, 4 edn. Monographs in Electrochemistry. Springer, Cham
Bruice T (2006) Computational approaches: Reaction trajectories, structures, and atomic motions. Enzyme reactions and proficiency. Chem Rev 106(8):3119–3139
Buerk D (1995) Biosensors: theory and applications. CRC Press, Lancaster
Carr P, Bowers L (1980) Immobilized enzymes in analytical and clinical chemistry. Wiley, New York
Carr PW (1977) Fourier analysis of the transient response of potentiometric enzyme electrodes. Anal Chem 49(6):799–802
Chaplin M, Bucke C (1990) Enzyme technology. Cambridge University Press, Cambridge
Chaubey A, Malhotra BD (2002) Mediated biosensors. Biosens Bioelectron 17(6):441–456
Choi M (2004) Progress in enzyme-based biosensors using optical transducers. Microchim Acta 148(3–4):107–132
Clark L, Lyons C (1962) Electrode systems for continuous monitoring in cardiovascular surgery. Ann NY Acad Sci 102:29–45
Crank J (1975) The mathematics of diffusion. Oxford University Press, London
de Gracia J, Poch M, Martorell D, Alegret S (1996) Use of mathematical models to describe dynamic behaviour of potentiometric biosensors: comparison of deterministic and empirical approaches to an urea flow-through biosensor. Biosens Bioelectron 11(1–2):53–61
del Barrio M, Cases R, Cebolla V, Hirsch T, de Marcos S, Wilhelm S, Galbán J (2016) A reagentless enzymatic fluorescent biosensor for glucose based on upconverting glasses, as excitation source, and chemically modified glucose oxidase. Talanta 160:586–591
Dixon M, Webb E, Thorne C, Tipton K (1979) Enzymes, 3rd edn. Longman, London
Eggenstein C, Borchardt M, Diekmann C, Grundig B, Dumschat C, Cammann K, Knoll M, Spener F (1999) A disposable biosensor for urea determination in blood based on an ammonium-sensitive transducer. Biosens Bioelectron 14(1):33–41
Eggins B (2002) Chemical sensors and biosensors. Analytical techniques in the Sciences. Wiley, Chichester
Forrow NJ, Sanghera GS, Walters SJ (2002) The influence of structure in the reaction of electrochemically generated ferrocenium derivatives with reduced glucose oxidase. J Chem Soc Dalton Trans 2002(16):3187–3194
Gaidamauskaitė E, Baronas R (2008) Modeling a peroxide-based fluorescent biosensor. In: Proceedings, 22nd European conference on modeling and simulation ECMS 2008. ECMS, Nicosia, pp 152–156
Grieshaber D, MacKenzie R, Vörös J, Reimhult E (2008) Electrochemical biosensors—sensor principles and architectures. Sensors 8(3):1400–1458
Guilbault G (1970) Enzymatic methods of analysis. Pergamon Press, Oxford
Guilbault G (1984) Analytical uses of immobilized enzymes. Marcel Dekker, New York
Gutfreund H (1995) Kinetics for the life sciences. Cambridge University Press, Cambridge
Habermüller K, Mosbach M, Schuhmann W (2000) Electron-transfer mechanisms in amperometric biosensors. Fresenius J Anal Chem 366(6):560–568
Harsanyi G (2000) Sensors in biomedical applications: fundamentals, technology and applications. CRC Press, New York
Henri V (1902) Theorie generale de l’action de quelques diastases. Compt Rend Hebd Acad Sci Paris 135(1):916–919
Ivanauskas F, Kaunietis I, Laurinavičius V, Razumienė J, Šimkus R (2008) Apparent Michaelis constant of the enzyme modified porous electrode. J Math Chem 43(4):1516–1526
Knopf G, Bassi A (2007) Smart biosensor technology. CRC Press, New York
Kohen A, PKlinman J (1999) Hydrogen tunneling in biology. Chem Biol 6(7):R191–R198
Kulys J (1981) Analytical systems based on immobilized enzymes. Mokslas, Vilnius
Kulys J (1981) The development of new analytical systems based on biocatalysts. Anal Lett 14(6):377–397
Kulys J (2005) Kinetics of biocatalytical synergistic reactions. Nonlinear Anal Model Control 10(3):223–233
Kulys J, Krikstopaitis K, Ziemys A (2000) Kinetics and thermodynamics of peroxidase- and laccase-catalyzed oxidation of N-substituted phenothiazines and phenoxazines. J Biol Inorg Chem 5:333–340
Kulys J, Razumas V (1986) Bioamperometry. Mokslas, Vilnius
Kulys J., Čenas N (1983) Oxidation of glucose oxidase from Penicillium vitale by one- and two-electron acceptors. Biochim Biophys Acta 744(1):57–63
Leatherbarrow RJ, Edwards PR (1999) Analysis of molecular recognition using optical biosensors. Curr Opin Chem Biol 3(5):544–547
Ligler F, Taitt C (2002) Optical biosensors: present and future. Elsevier Science, Amsterdam
Magner E (1998) Trends in electrochemical biosensors. Analyst 123:1967–1970
Malhotra BD, Pandey CM (2017) Biosensors: fundamentals and applications. Smithers Rapra, Shawbury
Marti S, Roca M, Andres J, Moliner V, Silla E, Tunon I, Bertran J (2004) Theoretical insights in enzyme catalysis. Chem Soc Rev 33(2):98–107
Mell L, Maloy T (1975) A model for the amperometric enzyme electrode obtained through digital simulation and applied to the immobilized glucose oxidase system. Anal Chem 47(2):299–307
Mell L, Maloy T (1976) Amperometric response enhancement of the immobilized glucose oxidase enzyme electrode. Anal Chem 48(11): 1597–1601
Merino S, Grinfeld M, McKee S (1998) A degenerate reaction diffusion system modeling an optical biosensor. Z Angew Math Phys 49(1):46–85
Michaelis L, Menten M (1913) Die kinetik der invertinwirkung. Biochem Z 49
Morf WE, van der Wal PD, Pretsch E, de Rooij NF (2011) Theoretical treatment and numerical simulation of potentiometric and amperometric enzyme electrodes and of enzyme reactors. Part 2: time-dependent concentration profiles, fluxes, and responses. J Electroanal Chem 657(1–2):13–22
Nernst W (1904) Theorie der Reaktionsgeschwindigkeit in heterogenen Systemen. Z Phys Chem 47(1):52–55
Olea D, Viratelle O, Faure C (2008) Polypyrrole-glucose oxidase biosensor: effect of enzyme encapsulation in multilamellar vesicles on analytical properties. Biosens Bioelectron 23(6):788–794
Ă–zisik MN (1980) Heat conduction. Wiley, New York
Pickup J, Thévenot D (1993) European achievements in glucose sensor research. In: Turner A (ed) Advances in biosensors, Supplement 1. JAI Press, London, pp 201–225
Rickus J (2005) Impact of coenzyme regeneration on the performance of an enzyme based optical biosensor: a computational study. Biosens. Bioelectron. 21(6):965–972
Rodriguez-Mozaz S, Marco M, de Alda M, Barcelo D (2004) Biosensors for environmental applications: future development trends. Pure Appl Chem 76(4):723–752
Rogers KR (1995) Biosensors for environmental applications. Biosens Bioelectron 10(6–7):533–541
Ruzicka J, Hansen E (1988) Flow injection analysis. Wiley, New York
Sadana A, Sadana N (2011) Handbook of biosensors and biosensor kinetics. Elsevier, Amsterdam
Scheller FW, Schubert F (1992) Biosensors. Elsevier Science, Amsterdam
Schulmeister T (1990) Mathematical modelling of the dynamic behaviour of amperometric enzyme electrodes. Sel Electrode Rev 12(2):203–260
Schulmeister T, Scheller F (1985) Mathematical treatment of concentration profiles and anodic current for amperometric enzyme electrodes. Anal Chim Acta 170:279–285
Sorochinskii V, Kurganov B (1997) Theoretical principles of the application of potentiometric enzyme electrodes. Appl Biochem Micro 33(2):116–124
Spichiger-Kelle UE (1998) Chemical sensors and biosensors for medical and biological applications. Wiley-VCH, New York
Turner APF, Karube I, Wilson GS (eds) Biosensors: fundamentals and applications. Oxford University Press, Oxford
Updike S, Hicks G (1967) The enzyme electrode. Nature 214:986–988
Vo-Dinh T (2003) Biomedical photonics handbook. CRC Press, New York
Štikonienė O, Ivanauskas F, Laurinavičius V (2010) The influence of external factors on the operational stability of the biosensor response. Talanta 81(4–5);1245–1249
Wilson R, Turner A (1992) Glucose oxidase: an ideal enzyme. Biosens Bioelectron 7(3):165–185
Wollenberger U, Lisdat F, Scheller F (1997) Frontiers in biosensorics, vol 2. Practical applications. Birkhauser, Basel
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Baronas, R., Ivanauskas, F., Kulys, J. (2021). Introduction to Modeling of Biosensors. In: Mathematical Modeling of Biosensors. Springer Series on Chemical Sensors and Biosensors, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-030-65505-1_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-65505-1_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-65504-4
Online ISBN: 978-3-030-65505-1
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)