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Springer Series on ChemicalSensors and BiosensorsMethods and Applications
Series Editor: G. Urban
9
Springer Series on Chemical Sensors and BiosensorsSeries Editor: G. Urban
Recently Published and Forthcoming Volumes
Hydrogel Sensors and Actuators
Volume Editors: Gerlach G., Arndt K. -F.
Vol. 6, 2009
Piezoelectric Sensors
Volume Editors: Steinem C., Janshoff A.
Vol. 5, 2006
Surface Plasmon Resonance Based Sensors
Volume Editor: Homola J.
Vol. 4, 2006
Frontiers in Chemical Sensors
Novel Principles and Techniques
Volume Editors: Orellana G., Moreno-Bondi M. C.
Vol. 3, 2005
Ultrathin Electrochemical
Chemo- and Biosensors
Technology and Performance
Volume Editor: Mirsky V. M.
Vol. 2, 2004
Optical Sensors
Industrial, Environmental
and Diagnostic Applications
Volume Editors:
Narayanaswamy R., Wolfbeis O. S.
Vol. 1, 2003
Volume Editors: Mohammed Z.,
Lakhtakia A.
Vol. 8, 2009
Optical Guided-wave Chemical
and Biosensors II
Volume Editors: Mohammed Z.,
Lakhtakia A.
Vol. 7, 2009
Optical Guided-wave Chemical
and Biosensors I
Volume Authors: Romas Baronas Feliksas Ivanauskas
Mathematical Modeling of Biosensors
An Introduction for Chemists and Mathematicians
123
·Juozas Kulys
Chemical sensors and biosensors are becoming more and more indispensable tools in life science,medicine, chemistry and biotechnology. The series covers exciting sensor-related aspects of chemistry,biochemistry, thin film and interface techniques, physics, including opto-electronics, measurementsciences and signal processing. The single volumes of the series focus on selected topics and will beedited by selected volume editors. The Springer Series on Chemical Sensors and Biosensors aims topublish state-of-the-art articles that can serve as invaluable tools for both practitioners and researchersactive in this highly interdisciplinary field. The carefully edited collection of papers in each volume willgive continuous inspiration for new research and will point to existing new trends and brand newapplications.
ISSN 1612-7617ISBN 978-90-481-3242-3 e-ISBN 978-90-481-3243-0DOI: 10.1007/978- - - -
Springer Heidelberg London New York
Library of Congress Control Number: 20099
90 481 3243 0
Dordrecht
31246
c©No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or byany means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without writtenpermission from the Publisher, with the exception of any material supplied specifically for the purposeof being entered and executed on a computer system, for exclusive use by the purchaser of the work.
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Springer is part of Springer Science+Business Media (www.springer.com)
Springer Science+Business Media B.V. 2010
Series Editor
Prof. Dr. Gerald Urban
IMTEK - Laboratory for Sensors
Institute for Microsystems Engineering
Albert-Ludwigs-University
Georges-Kohler-Allee 103
79110 Freiburg
Germany
Volume Authors
Romas Baronas
Vilnius University
Dept. Mathematics & Informatics
Naugarduko 24
LT-03225 Vilnius
Lithuania
Juozas Kulys
Vilnius Gediminas Technical University
Fac. Fundamental Sciences
Dept. Chemistry & Bioengineering
Sauletekio Ave. 11
LT-10223 Vilnius
Lithuania
Feliksas Ivanauskas
Preface
Biosensors are analytical devices in which specific recognition of the chemicalsubstances is performed by biological material. The biological material that servesas recognition element is used in combination with a transducer. The transducertransforms concentration of substrate or product to electrical signal that is ampli-fied and further processed. The biosensors may utilize enzymes, antibodies, nucleicacids, organelles, plant and animal tissue, whole organism or organs. Biosensorscontaining biological catalysts (enzymes) are called catalytical biosensors. Thesetype of biosensors are the most abundant, and they found the largest application inmedicine, ecology, and environmental monitoring.
The action of catalytical biosensors is associated with substrate diffusion intobiocatalytical membrane and it conversion to a product. The modeling of biosen-sors involves solving the diffusion equations for substrate and product with a termcontaining a rate of biocatalytical transformation of substrate. The complicationsof modeling arise due to solving of partially differential equations with non-linearbiocatalytical term and with complex boundary and initial conditions.
The book starts with the modeling biosensors by analytical solution of partialdifferential equations. Historically this method was used to describe fundamentalfeatures of biosensors action though it is limited by substrate concentration, andis applicable for simple biocatalytical processes. Using this method the action ofbiosensors was analyzed at critical concentrations of substrate and enzyme activity.The substrates conversion in single and multienzyme membranes was studied. Thedifferent schemes of substrates conversion which found practical application forbiosensors construction were analyzed. The biosensors dynamics was considered atthe simplest scheme of biocatalyzer action.
The other part of the book covers digital modeling of biosensors. The biosensorsbased on amperometric as well as potentiometric transducers are considered. The ac-tion of biosensors containing single and multienzymes were modeled using the finitedifference technique at nonstationary and steady state. Special emphasis was placedto model biosensors utilizing a complex biocatalytical conversion and biosensorswith multipart transducers geometry and biocatalytical membranes structure.
The final part of the book is dedicated to the basic concepts of the theory of thedifference schemes for the digital solving of linear diffusion equations which arebasis for biosensors modeling.
vii
viii Preface
The book can be recommended for the master and doctoral studies as well as forspecial studies of biosensors modeling. The Part 3 can also be used for independentstudy of digital solution of differential equations.
The book was prepared for the period of students teaching by R. Baronas andF. Ivanauskas at Vilnius University and by J. Kulys at Vilnius Gediminas TechnicalUniversity. The authors acknowledge particular universities for the support of themanuscript preparation. The contribution of the coauthors of the cited publicationsis highly appreciated.
Vilnius, Romas BaronasFebruary 2009 Feliksas Ivanauskas
Juozas Kulys
Acknowledgements
The authors acknowledge Vilnius University (Romas Baronas and FeliksasIvanauskas) and Vilnius Gediminas Technical University (Juozas Kulys) for thesupport of monograph preparation.
ix
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Part I Analytical Modeling of Biosensors
Biosensor Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Kinetics of Biocatalytical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Transducer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Scheme of Biosensor Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Modeling Biosensors at Steady State and Internal Diffusion Limitations . . 91 Biosensors Containing Single Enzyme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Biosensors Containing Multienzymes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Consecutive Substrates Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Parallel Substrates Conversion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Biosensors Utilizing Cyclic Substrates Conversion . . . . . . . . . . . . . . . . . 15
3 Biosensors Utilizing Synergistic Substrates Conversion .. . . . . . . . . . . . . . . . . . . 164 Biosensors Based on Chemically Modified Electrodes . . . . . . . . . . . . . . . . . . . . . 18
Modeling Biosensors at Steady State and External DiffusionLimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Biosensor Using Single Enzyme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Biosensors with Multienzymes .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Biosensor Utilizing Non Michaelis–Menten Enzyme .. . . . . . . . . . . . . . . . . . . . . . 23
Modeling Biosensors Utilizing Microbial Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Metabolite Biosensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 BOD Biosensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Modeling Nonstationary State of Biosensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Potentiometric Biosensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 Amperometric Biosensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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xii Contents
Part II Numerical Modeling of Biosensors
Mono-Layer Mono-Enzyme Models of Biosensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 Mathematical Model of an Amperometric Biosensor . . . . . . . . . . . . . . . . . . . . . . . 44
1.1 Governing Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.2 Initial and Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.3 Dimensionless Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2 Characteristics of the Biosensor Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.1 Biosensor Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.2 Biosensor Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.3 Maximal Gradient of the Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.4 Response Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Finite Difference Solution .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 Numerical Approximation of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2 Calculation Procedure .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Validation of Numerical Solution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4 Numerical Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Peculiarities of the Biosensor Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.1 Effect of the Enzyme Membrane Thickness . . . . . . . . . . . . . . . . . . . . . . . . . 604.2 Stability of the Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 The Response Versus the Substrate Concentration . . . . . . . . . . . . . . . . . . 644.4 The Response Versus the Maximal Enzymatic Rate. . . . . . . . . . . . . . . . . 664.5 Choosing the Enzyme Membrane Thickness . . . . . . . . . . . . . . . . . . . . . . . . 684.6 Biosensor Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.7 Maximal Gradient of the Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5 Flow Injection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2 Numerical Solution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3 Biosensor Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.4 Peculiarities of the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.5 Sequential Injection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6 Biosensors with Chemical Amplification .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.2 Finite Difference Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.3 Concentration Profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.4 Peculiarities of the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7 Potentiometric Biosensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917.2 Biosensor Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.3 Finite Difference Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.4 Validation of Numerical Solution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.5 Simulated Biosensor Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.6 Peculiarities of the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Contents xiii
8 Enzyme Inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038.1 Substrate Inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038.2 Effect of Substrate Inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058.3 Product Inhibition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.4 Effect of Product Inhibition .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
One-Layer Multi-Enzyme Models of Biosensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131 Biosensors Response to Mixture of Compounds .. . . . . . . . . . . . . . . . . . . . . . . . . . . 114
1.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1141.2 Solution of the Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1161.3 Generation of Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1171.4 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
2 Biosensors Acting in Trigger Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202.1 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212.2 Finite Difference Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1262.3 Simulated Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1282.4 Peculiarities of the Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1292.5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Multi-Layer Models of Biosensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 Multi-Layer Approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
1.1 Mathematical Model of Multi-Layer System . . . . . . . . . . . . . . . . . . . . . . . . 1411.2 Numerical Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1431.3 Three-Layer Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
2 Two-Compartment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1472.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1472.2 Transient Numerical Solution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1502.3 Validation of Numerical Solution.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522.4 Simulated Biosensor Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1532.5 Effect of the Diffusion Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1552.6 The Nernst Diffusion Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1572.7 Dimensionless Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1602.8 Impact of the Diffusion Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3 Biosensors with Outer Porous Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1643.2 Numerical Simulation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1663.3 Effect of the Porous Membrane.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1683.4 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4 Biosensors with Selective and Outer Perforated Membranes . . . . . . . . . . . . . . . 1724.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1734.2 Numerical Simulation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5 Biosensors Based on Chemically Modified Electrode . . . . . . . . . . . . . . . . . . . . . . 1785.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.2 Numerical Simulation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1825.3 Dimensionless Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
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5.4 Simulated Biosensor Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.5 Impact of the Diffusion Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1895.6 Impact of the Substrate Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.7 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6 Optical and Fluorescence Biosensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1936.2 Numerical Simulation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1976.3 Simulated Biosensor Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.4 Impact of the Substrate Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Modeling Biosensors of Complex Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2031 Biosensor Based on Heterogeneous Microreactor. . . . . . . . . . . . . . . . . . . . . . . . . . . 204
1.1 Structure of Modeling Biosensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2041.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2061.3 Numerical Simulation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2111.4 Effect of the Tortuosity of the Microreactor Matrix . . . . . . . . . . . . . . . . . 2141.5 Effect of the Porosity of the Microreactor Matrix . . . . . . . . . . . . . . . . . . . 2141.6 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
2 Biosensor Based on Array of Microreactors.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2172.1 Principal Structure of Biosensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2172.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2192.3 Numerical Simulation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2222.4 Effect of the Electrode Coverage with Enzyme .. . . . . . . . . . . . . . . . . . . . . 2252.5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
3 Plate-Gap Biosensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2283.1 Principal Structure of Biosensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2283.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2293.3 Governing Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2303.4 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2313.5 Boundary and Matching Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2313.6 Biosensor Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2323.7 Numerical Simulation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2333.8 Effect of the Gaps Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2363.9 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
4 Biosensors with Selective and Perforated Membranes .. . . . . . . . . . . . . . . . . . . . . 2374.1 Principal Structure of Biosensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2384.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2394.3 Numerical Simulation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2424.4 Effect of the Perforation Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2454.5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
Contents xv
Part III Numerical Methods for Reaction–Diffusion Equations
The Difference Schemes for the Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2491 The Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
1.1 An Equidistant Grid in the Straight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.2 A Non-equidistant Grid in a Straight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.3 The Equidistant Grid in a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2521.4 The Non-equidistant Grid in a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2531.5 The Grid in a Multidimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
2 The Approximation of the Function Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2542.1 The Derivative of the First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2542.2 The Approximation of the Second Order Derivative . . . . . . . . . . . . . . . . 2562.3 The Approximation of the Second Order Derivative
on a Non-equidistant Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2563 The Explicit Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
3.1 The Calculation of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2613.2 The Convergence and the Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
4 The Implicit Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2644.1 The Convergence and the Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2664.2 The Calculation of a Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
5 The Elimination Method for the System of Linear Equations . . . . . . . . . . . . . . 2685.1 Stability of the Elimination Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
6 The Crank–Nicolson Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2716.1 The Calculation of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2736.2 The Convergence and the Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
7 The Difference Scheme with the Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2747.1 The Convergence and the Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
8 The Crank–Nicolson Difference Schemeon Non-equidistance Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2768.1 The Calculation of a Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2788.2 The Convergence and the Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9 The Explicit Difference Schemein the Cylindrical Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2809.1 The Calculation of a Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2819.2 The Convergence and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
10 The Crank–Nicolson Difference Scheme in the Cylindrical Coordinates . . 28210.1 The Calculation of a Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28310.2 The Convergence and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
11 The Discontinuous Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28512 The Explicit Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
12.1 The Calculation of a Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28813 The Crank–Nicolson Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
13.1 The Calculation of a Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
xvi Contents
The Difference Schemes for the Reaction–Diffusion Equations . . . . . . . . . . . . . 2931 The Boundary-Value Problem for the System of Reaction–
Diffusion Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2932 The Explicit Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
2.1 The Calculation of a Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2962.2 The Convergence and the Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
3 The Non-linear Crank–Nicolson Type Difference Scheme . . . . . . . . . . . . . . . . . 2983.1 Calculation of a Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2993.2 The Convergence and the Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
4 The Linear Crank–Nicolson Type Difference Scheme . . . . . . . . . . . . . . . . . . . . . . 3034.1 The Calculation of a Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3044.2 The Convergence and the Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
5 Law of Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3076 The Alternating Directions Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
6.1 Calculation of a Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3116.2 The Convergence and the Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
7 The Explicit Method for the Multidimensional Problems . . . . . . . . . . . . . . . . . . 3127.1 The Calculation of a Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3147.2 The Convergence and the Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
About Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
Introduction
The action of biocatalytical biosensors can be modeled with partial differentialequations (PDE) of substrates and products diffusion and conversion in biocatalyt-ical membranes. This book deals with biosensors modeling using analytical anddigital solution of the PDE. The intrinsic logics of the book is to evaluate criti-cal parameters and conditions that determinate the biosensors response. Since theanalytical solutions of PDE describing biosensors action is possible at limited condi-tions the modeling of complex biosensor action are performed using digital solutionof PDE.
The first part of the book is dedicated to the modeling biosensors by analyticalsolution of partial differential equations. First chapter of Part I contains tutorial in-troduction of kinetics of biocatalytical reactions, transducer function of biosensorsand a general scheme of biosensor action. In second chapter of Part I the modelingbiosensors at steady state and internal diffusion limitation is considered with specialcontribution to varies schemes of enzymes action. Third chapter of Part I concernsthe modeling of biosensors at steady state and external diffusion limitations. Theaction of biosensor containing single enzyme, biosensors with multienzyme andbiosensor utilizing non Michaelis–Menten enzyme kinetics was analyzed. Fourthchapter of Part I contains results of modeling biosensors utilizing microbial cellsacting as specific biocatalytical or unspecific biochemical oxygen demand microre-actor. The main task of fifth chapter of Part I is to analyze limited cases of biosensorsmodeling at nonstationary state at some critical concentrations of substrate whenanalytical solution of PDE was performed. The non stationary response of ampero-metric as well as potentiometric biosensor was analyzed.
At the end of the first part advantages and disadvantages of analytical modelingof biosensors are shown. The largest advantage of aproximal analytical solution isa possibility to get analytical solution of PDE. The disadvantages include limitedconcentration interval of reactive components, not applicable to biosensors withcomplex biocatalytical schemes, very complex solution of non stationary state, lackof analytical solution for complex initial and boundary conditions.
In the second part of the book the corresponding reaction–diffusion problemsare solved using digital modeling. The solving PDE was performed using thefinite difference technique. First chapter of Part II covers mathematical modelswith nonlinear reaction kinetics. The biosensors are assumed to be flat electrodes
xvii
xviii Introduction
having a mono-layer of an enzyme applied onto the electrode surface. Couplingthe enzyme-catalyzed reaction in the enzyme layer (enzyme membrane) with theone-dimensional-in-space diffusion, the mathematical models are described by thenon-stationary reaction-diffusion equations. The biosensors based on amperometricas well as potentiometric transducers are considered. The batch and the injec-tion modes of the biosensor operation are modeled in this chapter. The biosensorsutilizing the amplification by the conjugated electrochemical and the enzymatic sub-strates conversion are also investigated. This chapter ends with the modeling of thebiosensors with the substrate as well as the product inhibition. The initial boundaryvalue problems are solved numerically by applying the finite difference technique.
Second chapter of Part II deals with the mathematical models of two types of am-perometric multi-enzyme biosensors. One type of the biosensors utilizes enzymaticreactions assuming no interaction between the analyzed substrates and the reactionproducts. The mathematical model of such biosensors is to simulate the biosensorresponse to a mixture of compounds (substrates). The second type of the biosensorsutilizes the enzymatic reaction followed by a cyclic product conversion. Two kindsof the product regeneration in the two-enzyme biosensors are analyzed: enzymaticand electrochemical.
Third chapter of Part II covers multi-layer mathematical models. The biosensorsacting in slightly-stirred buffer solutions are described by two-compartment math-ematical models. The biosensor operation is analyzed with a special emphasis tothe Nernst diffusion layer. This chapter also discusses the multienzyme systems,were the enzymes are immobilized separately in different active layers packed ina sandwich like multi-layer arrangement. The effect of the diffusion limitation tothe substrate is investigated when inert outer membranes are applied to stabilize theenzyme layer and to prolong the calibration curve of the biosensor. This chapteralso presents the mathematical models of the amperometric biosensor based on thechemically modified electrode as well as of the peroxidase-based optical biosensor.
Fourth chapter of Part II considers modeling of biosensors for which a two-dimensional-in space domain is required to describe mathematically the biosensoraction. Firstly, an amperometric biosensor based on a carbon paste electrode en-crusted with a single microreactor is considered. Then, an analytical system basedon an array of enzyme microreactors immobilized on a single electrode is investi-gated. Carbon paste porous electrodes are also investigated by applying a plate-gapmodel. The last section of the this chapter focuses on the modeling of a practi-cal amperometric biosensor containing the selective and the perforated membranes.The perforated membrane is analyzed with a special emphasis to the geometry ofthe membrane perforation.
Contemporary numerical methods for solving problems of the mathematicalchemistry are gaining increasing popularity. The aim of first chapter of Part III isto introduce the reader with the relevant facts about the basic concepts of the theoryof the difference schemes for the linear diffusion equations. The linear diffusionequations play an important and crucial role in most models of a biosensor theory.The most popular simple and together effective difference schemes for the linear dif-fusion equations are presented here. This method is being frequently used in solving
Introduction xix
applied problems not only by professional mathematicians, but also by laymen. Theconcepts presented below are of a primary nature and are sufficient for the solutionof the problems of the biosensor. In this book the notations of [222] are mainlyapplied. The many aspects of the numerical methods for the solution of the partialdifferential equations are presented in [5, 12, 187, 216].
The difference schemes are extensively applied to the solution of a biosensorproblems in second chapter of Part III. This chapter is devoted to various differ-ence approximations of the reaction–diffusion equations. The difference technique,developed in a previous chapter, is employed for the construction of the differenceschemes. The main subject of investigation is the system of two nonlinear reaction–diffusion equations in one and two dimensional in space cases.
Part IAnalytical Modeling of Biosensors
Abstract This is part one of the book Mathematical Modeling of Biosensors. Thepart is dedicated to the modeling biosensors by analytical solution of partial differ-ential equations. This part contains tutorial introduction to kinetics of biocatalyticalreactions, transducer function of biosensors and a general scheme of biosensor ac-tion. A special emphasis is placed to the modeling biosensors at steady state andinternal or external diffusion limitation with special contribution to varies schemesof enzymes action, the modeling of biosensors utilizing microbial cells acting asspecific biocatalytical or unspecific biochemical oxygen demand microreactor andthe modeling biosensors at nonstationary state at some critical concentrations ofsubstrate when analytical solution of PDE is performed.
Keywords Biocatalyzer � Biosensor � Diffusion � Kinetics � PDE
Biosensor Action
Contents
1 Kinetics of Biocatalytical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Transducer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Scheme of Biosensor Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1 Kinetics of Biocatalytical Reactions
The biosensors contain immobilized enzymes or other biological catalysts [128,131, 258]. The biocatalyst catalyzes the conversion of the substrate to the product.Biological catalysts (enzymes) show high activity and specificity. The activityof enzymes may exceed the rate of chemically catalyzed reaction by a factor4:6 � 105 � 1:4 � 1017 [58]. The enzymatic activity of the enzymes depends onmany factors, i.e. the free energy of reaction, the substrate docking in the activecenter of enzyme, the proton tunneling and other factors [88, 124, 132, 143].
The general principles of catalytic activity of enzymes are known, but particu-lar factors that determine high enzyme activity are often not established [178]. Thespecificity of enzymes depends on the enzyme type [65, 81]. There are enzymeswhich catalyze the conversion of just one substrate. Other enzymes show broadsubstrates specificity. Oxidoreductases, i.e. enzymes that catalyze electron transfer,may catalyze, for example, the oxidation or reduction of many substrates. To char-acterize the substrates with diverse activity a slang expression “good substrate” and“bad substrate” is used.
The following scheme of biocatalyzer action was postulated by Henri in1902 [109]:
EC S � ES! EC P ; (1)
where E, S, ES and P correspond to the enzyme, the substrate, the enzyme–substrates complex and the product, respectively. In biochemistry the concentrationsare expressed as mol=dm3 � M, whereas in models the concentrations of compo-nents are typically expressed in mol=cm3.
R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on ChemicalSensors and Biosensors 9, DOI 10.1007/978-90-481-3243-0 1,c� Springer Science+Business Media B.V. 2010
3
4 Biosensor Action
Michaelis and Menten confirmed this scheme of enzymes action using acetate tokeep pH of solution [185]. Following the scheme (1) the change of concentration ofeach component can be expressed by ordinary differential equation (ODE):
dE
dtD �k1ES C k�1ES C k2ES ;
dS
dtD �k1ES C k�1ES ;
dES
dtD k1ES � k�1ES � k2ES ;
dP
dtD k2ES ;
(2)
where t is time, E , S , ES , P correspond to the concentrations of the enzyme,the substrate, the enzyme–substrates complex and the product, respectively, andkinetic constants k1, k�1 and k2 correspond to the respective reactions: the enzymesubstrate interaction, the reverse enzyme substrate decomposition and the productformation.
To solve the system of ODE (2), Briggs and Haldane applied quasi-steady stateapproach (QSS) to ES which means that dES=dt � 0 [55]. The calculated “initialrate” of the steady state reaction rate was expressed (S is equal to the initial con-centration S0):
V.S/ D �dS
dtD VmaxS
KM C S ; (3)
where Vmax D k2E is the maximal enzymatic rate, andKM D .k�1Ck2/=k1, andit is called the Michaelis constant. The Michaelis constant is the concentration ofthe substrate at which half the maximum velocity of an enzyme-catalyzed reaction isachieved [55,185]. Typical values of constants of the enzymes which are used for thebiosensor preparation are: k1 D 106�108 M�1s
�1and k�1� k2D 100�1;000 s�1.
Calculations show that during the enzymes action the quasi-steady state is es-tablished during 4:0 � 0:1ms at enzyme and substrate concentration 10�8 M and10�3 M, respectively. It is sufficient to establish a quasi-steady state in the mem-branes of biosensors with the thickness more than 2 � 10�4 cm since the thicknessıd of the effective diffusion layer calculated using the Cottrell equation is [19]:
ıd Dp�Dt; (4)
where the diffusion coefficientD for low molecular weight molecules is about 3 �10�6 cm2=s.
For a more complex biocatalytical process, the establishment of the quasi-steadystate requires much longer period of time. It was shown, for example, that for thesynergistic reactions, involving cyclic mediators conversion, the time of the QSSestablishing can be as large as 180 s [147]. Therefore the expression for the “initialrate” is no longer valid, and the modeling should include the rates of all individualreactions.
2 Transducer Function 5
2 Transducer Function
The purpose of the transducer is to convert the biochemical recognition into anelectronic signal. The transducer is a device that responds selectively to the sub-strate, the product, the mediator or other compound the concentration of which isrelated to the analyte under determination [128, 131]. The transducer should showhigh selectivity since the biosensor selectivity depends on the specificity of the bio-catalytical process and the selectivity of the transducer.
The transducers include amperometric and ion-selective electrodes, opticalsystems and other physical devices realizing different physical phenomena. Thebiocatalytical membrane is located at close proximity to transducer. There are twofundamental categories of transducers in respect of their response. The transducerof the first type, i.e. the amperometric electrode, is monitoring faradaic currentwhich arises when the electrons are transferred between the substrates, the productor the enzyme active center and an electrode. As a result of electrochemical reactionthe concentration of oxidized (reduced) compound at surface of the transducerdrops down. The transducers of the second type, i.e. ion-selective electrodes, op-tical fibber, do not perturb the concentration of the determining compound at thesurface. The difference between the transducer types produces different boundaryconditions for the modeling of the biosensors.
The boundary condition for the first category of transducer can be written
P D 0 or S D 0 at x D 0; (5)
where x stands for space, P and S are the concentrations of the product and thesubstrate at the transducer surface, respectively. This boundary condition means thatthe kinetics of electron transfer is fast, and the potential of the transducer is highenough to keep a current at diffusion limiting condition. If the kinetics of electrontransfer is slow then the transducer current depends on the electrode potential andis obtained from the Butler–Volmer expression [56]. The modeling of biosensorsat this type of boundary conditions has not been performed due to this uncommonstate for the real biosensors.
The boundary condition of the transducer of the second category is
dP
dxD 0 or
dS
dxD 0 at x D 0; (6)
For the ion-selective electrodes this corresponds to the Nerstian boundary condition[56]. For the optical transducer and other transducers this condition means non-leakage (zero flux) of the product or the substrate on the boundary between thetransducer and the biocatalytical membrane.
6 Biosensor Action
3 Scheme of Biosensor Action
The biosensor produces a signal when the analyte under determination diffuses fromthe bulk solution into the biocatalytical membrane. The biocatalyst catalyzes thesubstrate conversion to the product, which is determined by the transducer. Theconcentration change of S and P is associated with the diffusion and the enzymaticreaction. Following Fick the compounds concentration change in the biocatalyticalmembrane can be written
@S
@tD De
@2S
@x2� V.S/;
@P
@tD De
@2P
@x2C V.S/; x 2 .0; d/; t > 0:
(7)
where x and t stand for space and time, respectively, S.x; t/ is the concentration ofthe substrate, P.x; t/ is the concentration of the reaction product, d is the thicknessof the enzyme membrane, De is the diffusion coefficient of compounds in the en-zyme membrane, that is typically used the same for the substrate, the product andthe mediator.
The solution of (7) at corresponding initial and boundary conditions produces theconcentration change of S and P in time and membrane thickness.
For the first type of transducers the response R of biosensor can be written
R.t/ D C1
@P
@x
ˇˇˇˇxD0
; (8)
and for the second type of transducers
R.t/ D C2P.0; t/; (9)
orR.t/ D C3 logP.0; t/; (10)
where C1, C2, C3 are the appropriate constants.The logarithmic dependence is characteristic of ion-selective electrodes, whereas
for optical and other transducers linear dependence between the response and theconcentration is realized.
Simple analytical solution of (7) is impossible even for the simplest initial andboundary conditions due to the hyperbolic function of the enzymatic rate depen-dence on the substrate concentration (3). Therefore the description of biosensorsaction is divided into the simplest cases for which analytical solutions still exist.This approach was used widely, especially at the beginning of the development ofbiosensors, to recognize the principles of the biosensors action. The aproximal an-alytical solution gives information about the critical cases. They are also useful totest the correctness of numerical calculations found at initial and boundary limitingconditions.
3 Scheme of Biosensor Action 7
When the concentration S0 to be measured is very small in comparison with theMichaelis constantKM ,
8x; t W x 2 Œ0; d �; t > 0 W 0 < S.x; t/ < S0 � KM ; (11)
the nonlinear function V.S/ simplifies to that of the first order,
V.S/ D VmaxS
S CKM
� Vmax
KM
S: (12)
Practically, the enzyme reaction can be considered first-order when the concen-tration of the detected species is below one-fifth of KM , i.e. S0 < 0:25KM , [99].This case is rather typical for the biosensors with a high enzyme loading factor.
The nonlinear reaction–diffusion system (7) reduces to a linear one,
@S
@tD De
@2S
@x2� kS;
@P
@tD De
@2P
@x2C kS; x 2 .0; d/; t > 0:
(13)
where k is the first-order reaction constant (linear enzyme kinetic coefficient),
k D Vmax
KM
: (14)
Analytical solutions are typically made at steady state and external and internaldiffusion limiting conditions.
The steady state (stationary) conditions mean that
@S
@tD 0; @P
@tD 0: (15)
The external diffusion limitation indicates that the substrates transport throughthe diffusion (stagnant) layer [134] is a rate limiting process. At internal diffusionlimitation the substrates diffusion through external diffusion layer is fast and theprocess is limited by the diffusion inside an enzyme membrane. The disadvantageof these approximate solutions is an error at the boundaries between the differentapproximate treatments. It is helpful to illustrate this approach by reference to atrivial problem of the substrate conversion in the biocatalytical membrane of thebiosensor and at the concentration of the substrate less than KM . The calculatedprofile of the substrate concentration at the steady state or stationary conditions isshown in Fig. 1.
It is possible to identify an abrupt of concentration change of the substrate at theboundary of biocatalytical membrane/stagnant layer as well as at the boundary stag-nant layer/bulk solution. This comes from approximate solutions at the boundaries
8 Biosensor Action
0.00 0.01 0.02 cm
S0
SS
S
TransducerBulksolution
Stagnantlayer
Enzymemembrane
Fig. 1 The substrate concentration profile in a biosensor at steady state conditions. The concen-tration profile was calculated with the boundary conditions @S=@x D 0 at x D 0 and S D S0 atx � d C ı. The diffusion coefficient (D0) in a stagnant layer is 3 � 10�6 cm2=s, in membraneis De D 10�6 cm2=s, Vmax D 5 � 10�7 mol=cm3s, KM D 10�5 mol=cm3, d D ı D 0:01 cm,S0 D 10�6 mol=cm3
during different approximate treatments. The change of the steady state concentra-tion of the substrate in membrane can be calculated as
S
Ss
D cosh .˛x/
cosh .˛d/; (16)
where S; Ss is substrate concentration at transducer surface and at the boundary ofmembrane and stagnant solution, respectively,
˛2 D Vmax
KMDe
: (17)
On the other hand, at the steady state a substrate flux through the boundary ofstagnant layer/bulk solution is equal to the flux through the boundary of biocatalyt-ical membrane/stagnant layer:
D0
S0 � Ss
ıD De
@S
@x
ˇˇˇˇxDd
D De˛ tanh.˛d/Ss : (18)
A combination of these two solutions (16), (18) produces the concentration pro-file of the substrate in the biocatalytical membrane and the stagnant layer (Fig. 1). Itis possible to notice that the greatest error of calculations is at x D d and x D dCı.However, at the limiting (the internal or the external diffusion limitation) cases thetwo expressions produce very good approximations to the full equation. Therefore,the modeling of the biosensors at two limiting cases was used to solve differentbiosensors problems.
Modeling Biosensors at Steady Stateand Internal Diffusion Limitations
Contents
1 Biosensors Containing Single Enzyme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Biosensors Containing Multienzymes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Consecutive Substrates Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Parallel Substrates Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Biosensors Utilizing Cyclic Substrates Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Biosensors Utilizing Synergistic Substrates Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Biosensors Based on Chemically Modified Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1 Biosensors Containing Single Enzyme
The most popular glucose biosensor is based on glucose oxidase (GO) that catalyzesˇ-D-glucose oxidation with oxygen [261, 273],
ˇ-D-glucoseC O2
D-glucose oxidase�! D-glucono-ı-lactoneCH2O2 (1)
The hydrogen peroxide produced is oxidized on platinum electrode acting as a trans-ducer. One of the first tasks of modeling of this type of the biosensors was devotedto evaluate the dependence of biosensors response on enzymatic parameters [128].The action of the biosensors was analyzed at the internal diffusion limiting condi-tions and at the steady state conditions.
The biosensor response (the current density) was calculated as
i.t/ D neDeF@P
@x
ˇˇˇˇxD0
; (2)
where ne – the number of electrons (for hydrogen peroxide n D 2), F – the Faradaynumber, De – the diffusion coefficient of the substrate and the product in the bio-catalytical membrane.
R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on ChemicalSensors and Biosensors 9, DOI 10.1007/978-90-481-3243-0 2,c� Springer Science+Business Media B.V. 2010
9
10 Modeling Biosensors at Steady State and Internal Diffusion Limitations
To calculate the biosensor response the change of the product concentration atthe transducer surface as indicated in (2) was evaluated. Solving (Chapter 1, eq. 13)with the boundary conditions @S=@x D 0, P D 0 at x D 0, S D S0, P D 0 atx � d and at S0 � KM (the concentration of oxygen is taken in access) producesthe stationary biosensor response (the steady state current density) as:
I D neFDe
S0
d
�
1 � 1
cosh.˛d/
�
; (3)
The solution shows that the biosensor response is a linear function of the sub-strate concentration. The sensitivity of the biosensors expressed as dI=dS0 does notdepend on the enzyme activity if the diffusion module (˛d ) is larger than 1 since
1 � 1
cosh.˛d/� 1: (4)
At ˛d < 1 the approximal solution of (13) is
I � neFDeS0
˛2d
2D neFS0
Vmaxd
2KM
: (5)
In this case the biosensor sensitivity is determined by the enzyme parametersVmax andKM .
The inactivation of an enzyme following the diffusion module change from(˛d > 1) to (˛d < 1) produces a wrong interpretation of an enzyme stability in thebiocatalytical membranes. Simple calculations show that if the enzyme inactivatesin a solution following the first order reaction with a rate constant kin D 0:1 h�1 thehalf-time (�) of enzyme inactivation is 6.9 h (� D ln.2/=kin). If the same enzyme isused for the biocatalytical membranes preparation, and at the beginning of activa-tion the diffusion module of the biosensors is, for example 100 (d D 0:01 cm), theresponse of biosensor will decrease two times only after 87 h.
2 Biosensors Containing Multienzymes
2.1 Consecutive Substrates Conversion
The analysis of the biosensor action containing consecutive substrate conversionwith two enzymes has also been analyzed at internal diffusion limitation and steadystate conditions [130]. These consecutive reactions occur in the bienzyme elec-trode based on immobilized D-glucose oxidase and peroxidase. Under the actionof D-glucose oxidase (1), D-glucose is oxidized with the production of hydrogen
2 Biosensors Containing Multienzymes 11
peroxide. During the second stage, hydrogen peroxide is reduced by ferrocyanideion (6). This reaction is catalyzed by peroxidase
H2O2 C 2Fe.CN/4�6 C 2HC �! 2Fe.CN/3�
6 C 2H2O (6)
Under the stationary conditions at excess concentrations of oxygen and ferro-cyanide, when reactions (1) and (6) are the first order, the change of concentra-tion within the bienzyme membrane is described by the system of the followingequations:
d2S
dx2D VmaxS
KMDe
D ˛21S ;
d2P1
dx2D � VmaxS
KMDe
C V 0maxP1
K 0MDe
D �˛21S C ˛2
2P1 ;
d2P2
dx2D �2V
0maxP1
K 0MDe
D �2˛22P1 ;
(7)
where S , P1 and P2 are the concentrations of glucose, hydrogen peroxide and fer-ricyanide, respectively; De – the diffusion coefficients of S , P1 and P2, which aretaken equal; Vmax , V 0
max , KM and K 0M are the corresponding parameters of the
enzyme reactions (1) and (6).The solution of the system (7) taking into consideration the boundary conditions
S D S0, P1 D 0, P2 D 0 at x � d and dS=dx D dP1=dx D 0, P2 D 0 at x D 0
gives the dependence of electrode current density I on the kinetic and diffusiveparameters (˛1 ¤ ˛2):
I D FDe
dP2
dx
ˇˇˇˇxD0
D 2FDe˛21
d.˛22 � ˛2
1/
��
1 � 1
cosh˛2d
�
� ˛22
˛21
�
1 � 1
cosh˛1d
��
S0 : (8)
Hence, it follows that the current of the bienzyme electrode is proportional to thesubstrate (glucose) concentration. The current is determined by means of diffusionmodule ˛1d and ˛2d . When the rate of enzyme reaction is great (˛1d > 1 and˛2d > 1) the response reaches its maximal value and is determined by the substratediffusion,
I D 2FDe
dS0 (9)
When the activity of peroxidase is considerably greater than the activity ofD-glucose oxidase (˛2d > ˛1d > 1) the biosensor response is determined bythe D-glucose oxidase parameters
I D 2FDe
d
�
1 � 1
cosh˛1d
�
S0 : (10)
12 Modeling Biosensors at Steady State and Internal Diffusion Limitations
Under the kinetic control (˛1d � 1), (10) is transformed to:
I D FVmaxd
KM
S0 : (11)
Due to high molecular activity of the peroxidase, these bienzyme biosensors op-erate in the mode controlled by the D-glucose oxidase reaction. Sensitivity, as wellas the stability of electrodes are close to that of mono-enzyme D-glucose electrode.
The modeling of trienzyme biosensor utilizing consecutive substrates conversionwith three enzymes was completed at internal diffusion limitation and steady stateconditions [145]. The example of successful application of three enzymes mightbe sensitive to the creatinine biosensor [248]. In the membrane of this biosensorcreatininase (E1) hydrolyses creatinine (S) to creatine (P1). The creatine is furtherhydrolyzed with creatinase (E2) to sarcosine (P2). The oxidation of sarcosine withsarcosine oxidase (E3) produces hydrogen peroxide (P3) that is determined amper-ometrically:
SE1�! P1
E2�! P2E3�! P3 (12)
The rate of each reaction (Vi .S/) can be characterized by the standard enzymeparameters V .i/
max and K.i/M , where i D 1; 2 and 3, for E1, E2 and E3 catalyed pro-
cess, respectively. At concentration of S , P1 andP2 less than the Michaelis–Mentenconstants (K.i/
M ), V1 D V .1/maxS=K
.1/M , V2 D V .2/
maxP1=K.2/M , V3 D V .3/
maxP2=K.3/M .
At substrates concentration less thanK.i/M and at a constant diffusion coefficients
the diffusion equations and the enzymatic conversions take a form
1
De
@S
@tD @2S
@x2� ˛2
1S ;
1
De
@P1
@tD @2P1
@x2C ˛2
1S � ˛22P1 ;
1
De
@P2
@tD @2P2
@x2C ˛2
2P1 � ˛23P2 ;
1
De
@P3
@tD @2P3
@x2C ˛2
3P2 ;
(13)
where De – the diffusion coefficient of all compounds in the enzyme membrane,˛i D .V .i/
max.i/=K
.i/M De/
1=2, i D 1; 2; 3.The biosensor response (the current density) was calculated as
i.t/ D 2FDe
@P3
@x
ˇˇˇˇxD0
: (14)
The solution of (13) was found at the steady state conditions (@S=@t D@P1=@t D @P2=@t D @P3=@t D 0) with the boundary conditions: S D S0,P1 D 0, P2 D 0, P3 D 0 at x � d , @S=@x D 0, @P1=@x D 0, @P2=@x D 0,P3 D 0 at x D 0, where d – the membrane thickness.
2 Biosensors Containing Multienzymes 13
0.000 0.002 0.004 0.006 0.008 0.0100.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Con
cent
ration
, μm
ol/c
m3
Membrane thickness, cm
SP1
P2
P3
Fig. 1 The profile of compounds concentration in trienzyme membrane of the biosensor. For cal-culations S0 D 10�6 mol=cm3, ˛1d D 10:0, ˛2d D 10:1, ˛3d D 10:3 and d D 0:01 cm were used
Calculations show that a significant concentration of products in a membrane isproduced if all the diffusion modules (˛id ) are larger than 1 (Fig. 1).
To prove the correctness of the calculations the distribution of compounds in amembrane was also determined at the boundary condition @P3=@x D 0.x D 0/. Inthis case the sum of the compounds was equal to S0 at all x values. At ˛1 ¤ ˛2 ¤ ˛3
the expression of three enzyme biosensors response (the current density) is
I D 2FDe
�˛2
2˛23
.˛22 � ˛2
1/.˛23 � ˛2
1/.1 � cosh.˛1d//
� ˛21˛
23
.˛22 � ˛2
1/.˛23 � ˛2
2/.1 � cosh.˛2d//
C ˛21˛
22
.˛23 � ˛2
1/.˛23 � ˛2
2/.1 � cosh.˛3d//
�S0
d:
(15)
It is impossible in practice to achieve equal values of the diffusion modules for allenzymes. Therefore the biosensor response has not been derived at ˛1 D ˛2 D ˛3.
The dependence of the response of the biosensor on the diffusion module ofthe least active enzymes E1 and E2 is shown in Fig. 2. It is easy to notice thatthe response is very small, still diffusion modules are less than 1. The maximalbiosensor response of 6 � 10�5A=cm2 is achieved when the diffusion modules aregreater than 10.
Experiments show that among three immobilized enzymes the lowest stabilitydemonstrates creatininase (E1). The model permits to predict sensitivity change ofthe biosensor during the enzyme inactivation. If the inactivation follows exponen-tial decay, for example, with half-time 2 days, and at the beginning the biosensorcontains large catalytic activities (˛id � 10), the response decreases just 34:7%during 10 days (Fig. 3). The apparent half-time of biosensor inactivation increasesup to 11:6 days. In fact, this biosensor can be used even longer, i.e. during 15 dayswith permanent calibration.
14 Modeling Biosensors at Steady State and Internal Diffusion Limitations
Fig. 2 The dependence ofthe biosensor response on thediffusion modules ˛1d and˛2d . For calculationsS0 D 10�6 mol=cm3 and˛3d D 10:3 were used
2 3 4 5 6 7 8 910
0
10
20
30
40
50
60
12
34
5678910
Res
pons
e, m
A/c
m2
a 2d
x 1.6
a1 d x 3.3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150
10
20
30
40
50
60
Para
met
er
Days
Response, μA/cm2
Diffusion module
Fig. 3 The changes of the biosensor response and the diffusion module of trienzyme biosen-sor during the enzyme inactivation. The half-time of enzyme inactivation is 2 days, S0 D10�6 mol=cm3, ˛1d D 10:0, ˛2d D 10:1, ˛3d D 10:3 and d D 0:01 cm were used
2.2 Parallel Substrates Conversion
In the presence of adenosine triphosphate (ATP), D-glucose oxidation is paralleledby the reaction of glucose phosphorylation in the bienzyme biosensors, the catalyticmembrane of which is made of D-glucose oxidase and hexokinase [130],
D-glucoseC ATPhexokinase�! D-glucose-6-phosphateC ADP (16)
The increase in ATP concentration leads to the decrease of the response occurringas a result of the D-glucose oxidase reaction (1). For the calculation of the depen-dence of the biosensor response it is assumed that the oxidation and phosphorylationare first-order for D-glucose and ATP, respectively. The solution of equations of
2 Biosensors Containing Multienzymes 15
diffusion and enzyme reactions with boundary conditions: S D S0, S1 D S10,P1 D 0 at x � d , and dS=dx D dS1=dx D 0, P1 D 0 at x D 0 gives the de-pendence of the density I of the biosensor current on the D-glucose (S0) and ATP(S10) concentration (˛1 ¤ ˛2),
I D 2FDe
dP1
dx
ˇˇˇˇxD0
D 2FDe
d
"
S0
�
1 � 1
cosh˛1d
�
�
S10
�˛2
2
˛22 � ˛2
1
�
1 � 1
cosh˛1d
�
� ˛21
˛22 � ˛2
1
�
1 � 1
cosh˛2d
��#
; (17)
where ˛2 D .Vmax=KMDe/1=2 is related to hexokinase reaction.
Thus, it follows that at a high rate of both enzyme reactions (˛1d >1 and˛2d > 1) the density of the biosensor current is determined by the difference be-tween the D-glucose and ATP concentrations,
I D 2FDe
d.S0 � S10/ ; (18)
i.e. the decrease in the biosensors current is proportional to the concentration ofcoenzyme (ATP),
�I D 2FD
dS10 : (19)
If the reaction of D-glucose oxidase proceeds rapidly (˛1d > 1), and the phos-phorylation is at low rate (˛2d < 1), then
�I D FDeV0
maxd
K 0M
S10 : (20)
The sensitivity of such a biosensor is directly proportional to the activity of hex-okinase. The experimental results indicate that the action of the ATP electrode isdetermined by the activity of this enzyme [130]. The hexokinase inactivation re-sults in a quick loss of the biosensors sensitivity.
2.3 Biosensors Utilizing Cyclic Substrates Conversion
Rich biocatalytical possibilities permit to construct different systems utilizing thecyclic substrates conversion. The cyclic conversion substrates in an enzyme mem-brane may considerably increase the sensitivity of the biosensor.
S˛1�! P1
˛2�! SC P2 (21)
where ˛1d and ˛2d are the diffusion modules of the corresponding reactions.
16 Modeling Biosensors at Steady State and Internal Diffusion Limitations
If P2 is considered to be an electrode-active compound, the biosensorsresponse is
I D neFDe
dP2
dx
ˇˇˇˇxD0
D neFDe
d
˛21˛
22
˛21 C ˛2
2
2
64
d 2
2� 1
˛21 C ˛2
2
0
B@1 � 1
cosh.q
˛21 C ˛2
2d/
1
CA
3
75S0 (22)
where the boundary conditions are: S D S0, P1 D 0, P2 D 0, when x � d ;dS=dx D dP1=dx D 0, P2 D 0, when x D 0.
Two important conclusions can be drawn from Eq. 22:
(i) Under the kinetic control of the first or second reaction (˛1d < 1 or ˛2d < 1)the amplification of the signal does not take place.
(ii) At a high enzymatic activity (˛1d > 1 and ˛2d > 1) the response of thebiosensor increases by a value which is directly proportional to the square ofthe membrane thickness:
I D Id
˛21˛
22d
2
2.˛21 C ˛2
2/; (23)
where Id corresponds to the diffusion controlled response of the biosensors con-taining single enzyme
Id D neFDe
S0
d: (24)
The amplification is rapidly increased by the rise of the enzymatic activity. Forexample, at ˛1d D ˛2d D 4, the 4-fold increase of the sensitivity occurs. At˛1d D ˛2d D 10, the amplification enlarges 25-fold.
The possibility of a considerable increase in the sensitivity of the biosensors bymeans of chemical amplification was demonstrated using alcohol dehydrogenasewith cyclic coenzyme (NAD) conversion and in other biocatalytical systems [136].
Biocatalytical systems may utilize other substrates conversion in addition tocyclic conversion. A biosensor showing submicromolar sensitivity to hydrogenperoxide was developed utilizing fungal peroxidase and pyrroloquinoline quinone-dependent glucose dehydrogenase. High sensitivity of the biosensor was achievedby triggering the initiator conversion with peroxidase following the signal amplifi-cation by cyclic conversion of the mediator formed [149].
3 Biosensors Utilizing Synergistic Substrates Conversion
Biocatalytical reactions in biocatalytical membranes may be conjugated with chem-ical conversions. These synergistic reactions allow to generate high sensitive biosen-sors. High sensitive biosensors for heterocyclic compounds determination were built
3 Biosensors Utilizing Synergistic Substrates Conversion 17
using the oxidases-catalyzed hexacyanoferrate(III) reduction [146]. The detectionlimit of some heterocyclic compounds determination was 2 � 10�10 mol=cm3. Thesensitivity of the biosensors was 300–10,000 times larger in comparison to the de-termination of hexacyanoferrate(III).
The steady state current of the biosensor was calculated using the synergisticscheme of oxidases action. Following the scheme the oxidized glucose oxidase(GOox ) is reduced with glucose ant reduction of hexacyanoferrate(III) (Fer) is cat-alyzed by reduced glucose oxidase (GOred),
GOox C D-glucosekred�! GOred C P (25)
GOred C 2Ferkf�! GOox C 2Ferred (26)
In the presence of heterocyclic compounds that act as mediators (M) they are oxi-dized with hexacyanoferrate(III) to cation radicals. The cation radical (Mox) formedreacts with reduced oxidase. The reduced mediator (Mred) is further oxidized withhexacyanoferrate(III),
GOred C 2Moxkox�! GOox C 2Mred (27)
Mred C Ferkexc�! Mox C Ferred (28)
Chemical reaction (28) increases Ferred production rate, therefore the rate of overallprocess is larger than the reactions (26) and (27).
Since the electrochemically active compound is hexacyanoferrate(II) (Ferred), thesteady state response can be calculated like the current density of the biosensor witha chemical amplification [136]:
I D FDe
d
˛21˛
22
ˇ2
�d 2
2� 1
ˇ2
�
1 � 1
cosh.ˇd/
��
M0 ; (29)
where ˛1d and ˛2d are the diffusion modules, ˇ2 D ˛21 C ˛2
2 , d is the enzymemembrane thickness, M0 corresponds to the total mediator. The diffusion moduleswere calculated as ˛1d D .koxE0=De/
1=2d , ˛2d D .kexcFer=De/1=2d , where E0
and Fer stand for the total enzyme and the hexacyanoferrate(III) concentrations, re-spectively, kexc is the bimolecular electron exchange constant between the mediatorand hexacyanoferrate (III).
The density I0 of the steady state current of the biosensor in the absence of themediator was calculated as
I0 D FDe
d
�
1 � 1
cosh.�d/
�
Fer ; (30)
where � D .kfE0=De/1=2, kf is the constant of the reaction of the hexacyanofer-
rate(III) with the reduced glucose oxidase.
18 Modeling Biosensors at Steady State and Internal Diffusion Limitations
Fig. 4 The dependence ofthe biosensor relativesensitivity Sr on the diffusionmodules ˛1d and ˛2d
12 3 456 7 8 91011121314
200
400
600
800
1000
1200
1 23
4 5 6 7 8 9 1011
Rel
ativ
e se
nsiti
vity
x 0
.1
α 2d x 0.1α
1 d
The analysis of the dependence of the relative sensitivity (Sr D I=I0) on thediffusion modules reveals that the Sr is larger than 1 if ˛1d and ˛2d are larger than0.5 (Fig. 4).
At ˛1d D ˛2d D 1, the relative sensitivity Sr of the biosensor is 12:9. It in-creases if both diffusion modules are larger than 1. The calculations show that for thebiosensor containing 1:3 � 10�7 mol=cm3 of glucose oxidase ˛1d is 14.5 and ˛2d
is 113.5. In contrast, the diffusion module (�d ) of the biosensor acting with the purehexacyanoferrate(III) is 0.13 due to the low constant of the reduced enzyme. Since�d is less than 1 it means that the biosensor acts in a kinetic regime. It is easy tonotice that at ˛2d > ˛1d > 1 and �d < 1 the relative sensitivity equals
I
I0
D kox
kf
: (31)
The comparison of calculated and experimentally determined values reveals thatthe calculated relative sensitivity of the biosensors is about three times larger thanthe experimentally determined. This deference can be caused by the limited stabil-ity of the oxidized heterocyclic compounds, uncounted parallel reaction of reducedenzyme with oxygen and external diffusion limitation of hexacyanoferrate(III) andglucose.
4 Biosensors Based on Chemically Modified Electrodes
Chemically modified electrodes (CME) are produced by modifying carbonelectrodes with redox active component (mediator) which reacts with an enzyme[138]. For modification of electrodes adsorption or covalent immobilization of themediator is used. The peculiarities of CME based biosensors modeling arise due tothe mediator location on the electrode that produces special boundary conditions.
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