Mathematical Modeling of Biosensors [2 ed.] 3030655040, 9783030655044

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Springer Series on Chemical Sensors and Biosensors  9 Series Editor: Gerald Urban

Romas Baronas Feliksas Ivanauskas Juozas Kulys

Mathematical Modeling of Biosensors Second Edition

Springer Series on Chemical Sensors and Biosensors Methods and Applications Volume 9

Series Editor Gerald Urban, Institute for Microsystems Engineering, IMTEK - Laboratory for Sensors, Albert-Ludwigs-University, Freiburg, Germany

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, measurement sciences and signal processing. The single volumes of the series focus on selected topics and will be edited by selected volume editors. The Springer Series on Chemical Sensors and Biosensors aims to publish state-of-the-art articles that can serve as invaluable tools for both practitioners and researchers active in this highly interdisciplinary field. The carefully edited collection of papers in each volume will give continuous inspiration for new research and will point to existing new trends and brand new applications.

More information about this series at http://www.springer.com/series/5346

Romas Baronas • Feliksas Ivanauskas • Juozas Kulys

Mathematical Modeling of Biosensors Second Edition

Romas Baronas Faculty of Mathematics & Informatics Institute of Computer Science Vilnius University Vilnius, Lithuania

Feliksas Ivanauskas Faculty of Mathematics & Informatics Institute of Computer Science Vilnius University Vilnius, Lithuania

Juozas Kulys Life Sciences Center Vilnius University Vilnius, Lithuania

ISSN 1612-7617 Springer Series on Chemical Sensors and Biosensors ISBN 978-3-030-65504-4 ISBN 978-3-030-65505-1 (eBook) https://doi.org/10.1007/978-3-030-65505-1 © Springer Nature Switzerland AG 2010, 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This is the second edition of the monography “Mathematical Modeling of Biosensors” prepared by R. Baronas, F. Ivanauskas and J. Kulys and published by Springer in 2010. In 2012, the monograph was translated into Arabic Language and published at the King Saud University (King Saud University Translation Center, Riyadh, Saudi Arabia, 2012, 544 p., ISBN 978-603-507-033-1). The second edition extends previous achievements of mathematical modeling of biosensors and includes novel discoveries in the area that appeared during the last 10 years. The material showed in the second edition is analysed following functionality of biosensors rather than methods used. The book presents the development and modeling of biosensors from both a chemical and a mathematical point of view. It contains unique modeling methods for amperometric, potentiometric and optical biosensors based mainly on biocatalysts. The book examines processes that occur in the sensors’ layers and at their interface, and it provides analytical and numerical methods to solve equations of conjugated enzymatic (chemical) and diffusion processes. The action of single enzyme as well as polyenzyme biosensors and biosensors based on chemically modified electrodes is studied. The modeling of biosensors that contain perforated membranes and multipart mass transport profiles is critically investigated. Furthermore, it is fully described how signals can be biochemically amplified, how cascades of enzymatic substrate conversion are triggered and how signals are processed via a chemometric approach and artificial neuronal networks. The results of digital modeling are compared with both proximal analytical solutions and experimental data. Completely novel are chapters dedicated to bacterial self-organization as a novel principle of biosensing, modeling carbon nanotube-based biosensors and application of mathematical modeling to optimal design of biosensors. The book was also significantly modified with particular attention to autonomy of the chapters’ reading and shipping. Therefore, each chapter contains the introduction, the concluding remarks and the references. The book consists of 13 chapters, the references and the index. The book begins with Chapter “Introduction to Modeling of Biosensors”. The chapter considers general description of biosensor action, kinetics of biocatalytical reactions, transducer v

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function, schemes of biosensors and principles of modeling of biosensors’ action by analytical and digital solution of partial differential equations. Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors” analyses the dependencies of the internal and external diffusion limitations on the response and sensitivity of amperometric biosensors. Chapter “Biosensors Utilizing Consecutive and Parallel Substrates Conversion” is dedicated to the modeling and analysis of biosensors utilizing consecutive and parallel substrates conversion at stationary and transient conditions. Chapter “Biosensors Response Amplification with Cyclic Substrates Conversion” analyses the mathematical models of different types of amperometric biosensors utilizing the cyclic conversion Chapter “Biosensors Utilizing Synergistic Substrates Conversion” is dedicated to the modeling of biosensors containing glucose oxidase, carbohydrate oxidase and laccase and utilizing a few synergistic schemes of substrates conversion. Chapter “Biosensors Acting in Injection Mode” analyses the sensitivity of an amperometric biosensor acting in the flow injection mode when it contacts an analyte. Chapter “Chemically Modified Enzyme and Biomimetic Catalysts Electrodes” is devoted to chemically modified enzyme electrodes and biomimetic catalysts-based electrodes that exhibit numerous remarkable properties. Chapter “Biosensors with Porous and Perforated Membranes” studies amperometric biosensors with inert and selective membranes, which are mathematically modeled by nonlinear reaction–diffusion equations. Chapter “Biosensors Utilizing Non-Michaelis–Menten Kinetics” analyses the modeling of biosensors utilizing non-Michaelis–Menten kinetics, which reveals the complex behaviour of the biosensor response. Chapter “Biosensors Based on Microreactors” considers the modeling of amperometric biosensors based on microreactors mathematically and numerically in two-dimensional space at transient conditions. Chapter “Modeling Carbon Nanotube Based Biosensors” is dedicated to the modeling of amperometric biosensors based on an enzyme-loaded carbon nanotube layer deposited on the perforated membrane. Chapter “Modeling Biosensors Utilizing Microbial Cells” examines the modeling of biosensors in which the biological component consists of microbial cells. It considers metabolite, BOD sensors and bacterial self-organization as a novel principle of biosensing. Chapter “Application of Mathematical Modeling to Optimal Design of Biosensors” presents a method combining mathematical modeling, chemometrics, multi-objective optimization and multi-dimensional visualization intended for the design and optimization of biosensors. The book can be recommended for the master and doctoral studies, as well as for special studies of biosensors’ modeling. The book was prepared for the period of students teaching and scientific work by Romas Baronas, Feliksas Ivanauskas and Juozas Kulys at the Vilnius University.

Preface

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The authors acknowledge the Vilnius University for the support of the manuscript preparation. The contribution of the co-authors of the cited publications is highly appreciated. Vilnius, Lithuania Vilnius, Lithuania Vilnius, Lithuania September 2020

Romas Baronas Feliksas Ivanauskas Juozas Kulys

Acknowledgements

The book is dedicated to 440-year celebration of the Vilnius University. Authors (Romas Baronas, Feliksas Ivanauskas and Juozas Kulys) acknowledge the Vilnius University for the support of monograph preparation.

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Contents

Introduction to Modeling of Biosensors . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Biosensor Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.1 Kinetics of Biocatalytic Reactions . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.2 Transducer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.3 Scheme of Biosensor Action . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.4 Biosensor Response .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3 Mono-Layer Model of an Amperometric Biosensor. . . . .. . . . . . . . . . . . . . . . . . . 3.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.2 Dimensionless Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.3 Characteristics of the Biosensor Response. . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.4 Solution for the First Order Kinetics . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.5 Solution for the Zero Order Kinetics . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.6 Solution for Michaelis–Menten Kinetics . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4 Peculiarities of the Biosensor Response.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.1 Effect of the Enzyme Membrane Thickness . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Stability of the Response . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.3 The Response Versus the Substrate Concentration . . . . . . . . . . . . . . . . . . . 4.4 The Response Versus the Maximal Enzymatic Rate . . . . . . . . . . . . . . . . . 4.5 Choosing the Enzyme Membrane Thickness . . . . . .. . . . . . . . . . . . . . . . . . . 4.6 The Response Gradient with Respect to the Membrane Thickness .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.7 Maximal Gradient of the Transient Current . . . . . . .. . . . . . . . . . . . . . . . . . . 5 Modeling Potentiometric and Optical Biosensors . . . . . . .. . . . . . . . . . . . . . . . . . . 5.1 Potentiometric Biosensors .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5.2 Optical Biosensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5.3 Fluorescence Biosensors .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 6 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

1 2 3 3 5 6 9 10 10 11 13 15 18 20 25 25 28 29 32 34 36 38 39 39 41 42 43 44

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Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Multi-layer Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.1 Mathematical Model of Multi-layer System . . . . . . .. . . . . . . . . . . . . . . . . . . 2.2 Numerical Approximation .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3 Three-Layer Model.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4 Two-Compartment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Solving the Mathematical Model.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.3 Simulated Biosensor Responses . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.4 Dimensionless Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.5 Effect of the Diffusion Layer .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.6 The Nernst Diffusion Layer . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.7 Impact of the Diffusion Module .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

49 50 51 51 54 55 58 58 62 67 70 74 77 79 81 81

Biosensors Utilizing Consecutive and Parallel Substrates Conversion .. . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Consecutive and Parallel First Order Conversion of Substrates .. . . . . . . . . . . 2.1 Consecutive Substrates Conversion . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.2 Parallel Substrate Conversion . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3 Modeling Dual Catalase-Peroxidase Bioelectrode.. . . . . .. . . . . . . . . . . . . . . . . . . 3.1 Reaction Scheme .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.2 Reaction Rates at Quasi-Steady State . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.3 Two Compartment Reaction–Diffusion Model . . . .. . . . . . . . . . . . . . . . . . . 3.4 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.5 Dynamics of the Biosensor Current .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.6 Biosensor Response to the Second Substrate . . . . . .. . . . . . . . . . . . . . . . . . . 3.7 Impact of the Reaction Rates . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.8 Model Modifications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4 Biosensors Response to Mixture of Compounds . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.1 Reaction Scheme .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Mathematical Model with No Substrates Interaction.. . . . . . . . . . . . . . . . 4.3 Solution of the Problem.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.4 Generation of Pseudo-Experimental Data . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.5 Modeling Substrates Interaction.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

85 86 87 87 95 97 97 98 100 103 104 105 106 107 109 109 109 111 112 114 116 117

Biosensors Response Amplification with Cyclic Substrates Conversion . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Biosensors Utilizing Cyclic Enzymatic Substrates Conversion .. . . . . . . . . . .

121 122 123

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3 Biosensors with Electrochemical and Enzymatic Conversion . . . . . . . . . . . . . 3.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.2 Finite Difference Solution . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.3 Concentration Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.4 Peculiarities of the Biosensor Response . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4 Biosensors Acting in Trigger Mode . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.1 Mathematical Models .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Finite Difference Solution . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.3 Simulated Response.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.4 Peculiarities of the Response . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

126 126 127 128 129 135 135 140 142 144 152 153

Biosensors Utilizing Synergistic Substrates Conversion .. . . . . . . . . . . . . . . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Steady State Current of Biosensors with Synergistic Biocatalytical Scheme. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3 Modeling Glucose Dehydrogenase-Based Amperometric Biosensor .. . . . . 3.1 Reaction Scheme .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.3 Digital Simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.4 Model Validation with Experimental Data . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.5 Biosensor Sensitivity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4 Modeling Laccase-Based Amperometric Biosensor .. . . .. . . . . . . . . . . . . . . . . . . 4.1 Reaction Scheme .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.3 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.4 Digital Simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.5 Limits of the Synergistic Effect . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

155 156 157 159 159 161 165 165 168 170 171 173 174 176 177 178 179

Biosensors Acting in Injection Mode. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Flow Injection Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.2 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.3 Dynamics of the Biosensor Response .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.4 Peculiarities of the Biosensor Response . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3 Sequential Injection Analysis .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4 Effect of Diffusion Limitations on the Biosensor Response . . . . . . . . . . . . . . . 4.1 Two-Compartment Model . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Dynamics of the Biosensor Response .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.3 Impact on the Apparent Michaelis Constant . . . . . .. . . . . . . . . . . . . . . . . . . 5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

183 184 184 185 186 188 190 194 195 196 198 199 202 203

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Chemically Modified Enzyme and Biomimetic Catalysts Electrodes . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Modeling Biosensors Utilizing First Order Kinetics . . . .. . . . . . . . . . . . . . . . . . . 3 Modeling Biosensors Utilizing Michaelis–Menten Kinetics . . . . . . . . . . . . . . . 3.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.2 Solving the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.3 Dimensionless Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.4 Simulated Biosensor Action .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.5 Impact of the Diffusion Module .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.6 Impact of the Substrate Concentration . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4 Modeling Multi-Layer CME-Based Biosensor . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.1 Three-Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Effective Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.3 Concentration Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5 Analysis of the Electrocatalysis Using Michaelis–Menten and Second-Order Reaction Schemes . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5.2 Characteristics of the Response . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5.3 Numerical Simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5.4 Effect of the Combination of Two Types of Kinetics. . . . . . . . . . . . . . . . . 5.5 Impact on the Apparent Michaelis Constant . . . . . . .. . . . . . . . . . . . . . . . . . . 6 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

207 208 209 210 211 215 218 220 223 225 227 227 229 230

Biosensors with Porous and Perforated Membranes . . . . .. . . . . . . . . . . . . . . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Biosensors with Outer Porous Membrane.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.2 Numerical Simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.3 Effect of the Porous Membrane . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3 Biosensors with Selective and Outer Perforated Membranes . . . . . . . . . . . . . . 3.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.2 Numerical Simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.3 Effect of the Selective Membrane .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4 Two-Dimensional Modeling of Biosensors with Selective and Perforated Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.1 Principal Structure of Biosensor.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.3 Numerical Simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.4 Effect of the Perforation Topology .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

243 244 245 245 248 250 254 254 258 259

Biosensors Utilizing Non-Michaelis–Menten Kinetics . . .. . . . . . . . . . . . . . . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Steady State Modeling of Substrate Inhibition . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

275 276 278

231 232 234 235 235 237 238 239

261 262 262 266 269 271 272

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3 Transient Modeling of Substrate and Product Inhibition .. . . . . . . . . . . . . . . . . . 3.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.2 Characteristics of the Biosensor Response. . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.3 Numerical Simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.4 Dimensionless Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.5 Effect of Substrate Inhibition .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.6 Effect of Product Inhibition . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4 Transient Modeling of Allostery.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Transient Kinetics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.3 Effect of Cooperativity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

281 281 284 285 287 289 292 293 293 296 297 298 298

Biosensors Based on Microreactors . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Biosensor Based on Heterogeneous Microreactor .. . . . . .. . . . . . . . . . . . . . . . . . . 2.1 Structure of Modeling Biosensor .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.3 Numerical Simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.4 Effect of the Tortuosity of the Microreactor Matrix .. . . . . . . . . . . . . . . . . 2.5 Effect of the Porosity of the Microreactor Matrix .. . . . . . . . . . . . . . . . . . . 3 Biosensor Based on Array of Microreactors . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.1 Principal Structure of Biosensor.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.3 Numerical Simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.4 Effect of the Electrode Coverage with Enzyme . . .. . . . . . . . . . . . . . . . . . . 4 Plate–Gap Biosensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.1 Principal Structure of Biosensor.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.3 Effect of the Gaps Geometry . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

303 304 305 305 307 313 314 316 319 319 321 324 326 330 330 331 338 338 340

Modeling Carbon Nanotube Based Biosensors . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Carbon Nanotube Based Mediated Biosensor . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.1 Principal Structure of the Biosensor . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.3 Numerical Simulation and Model Validation .. . . . .. . . . . . . . . . . . . . . . . . . 2.4 Effect of the Structural Anisotropy of the CNT Mesh . . . . . . . . . . . . . . . 2.5 Effect of the Partition Coefficient . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3 One-Dimensional Modeling of CNT Based Mediated Biosensor.. . . . . . . . . 3.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.2 Numerical Simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

345 346 347 347 349 355 356 358 359 359 360

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3.3 Impact of the Perforation Level . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.4 Impact of the Tortuosity in the Perforated Membrane.. . . . . . . . . . . . . . . 4 Carbon Nanotube Based Unmediated Biosensor . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.1 Principal Structure of the Biosensor . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.3 Numerical Simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.4 Experimental Model Validation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.5 Impact of Enzyme Concentration . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.6 Impact of Electrochemical Reaction Rate . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

361 362 363 364 365 368 369 371 372 373 374

Modeling Biosensors Utilizing Microbial Cells . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Metabolite Biosensor.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3 BOD Biosensor .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4 Modeling Bacterial Self-Organization . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.1 3D Mathematical Model .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.2 Dimensionless Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.3 3D Simulation of Population Dynamics . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.4 Population Dynamics Near the Top Surface . . . . . . .. . . . . . . . . . . . . . . . . . . 4.5 Population Dynamics Near the Contact Line . . . . . .. . . . . . . . . . . . . . . . . . . 4.6 Population Dynamics Near the Lateral Surface . . .. . . . . . . . . . . . . . . . . . . 5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

377 378 379 382 384 384 387 389 391 394 397 400 401

Application of Mathematical Modeling to Optimal Design of Biosensors .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2 Optimization of Bi-Layer Biosensors: Trade-off Between Sensitivity and Enzyme Amount .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.2 Biosensor Characteristics .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.3 Bi-Objective Optimization Problem.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 2.4 Results of Computational Experiments . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3 Applying Multi-Objective Optimization and Decision Visualization .. . . . . 3.1 Modeling Biosensor.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.2 Biosensor Characteristics .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.3 Optimal Design of the Biosensor .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 3.4 Visualization of the Optimization Results . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4 Optimization of the Analytes Determination with Biased Biosensor Response .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 4.1 Biosensors Array for Long-Term Glucose Measurement . . . . . . . . . . . . 4.2 Multianalyte Determination with Biased Biosensor Response . . . . . .

405 406 407 408 409 411 412 416 416 417 418 422 424 425 426

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5 Neural Networks for an Analysis of the Biosensor Response.. . . . . . . . . . . . . 5.1 Generation of Data Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5.2 Prediction of Concentrations Using Neural Networks . . . . . . . . . . . . . . . 5.3 Input Data Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 5.4 Locally Weighted Neural Network Setup .. . . . . . . . .. . . . . . . . . . . . . . . . . . . 5.5 Biosensor Calibration Results . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 6 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . References . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

434 435 435 435 436 439 440 441

Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .

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5

6

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biosensor Action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Kinetics of Biocatalytic Reactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Transducer Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Scheme of Biosensor Action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mono-Layer Model of an Amperometric Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Dimensionless Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Characteristics of the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Solution for the First Order Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Solution for the Zero Order Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Solution for Michaelis–Menten Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peculiarities of the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Effect of the Enzyme Membrane Thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stability of the Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Response Versus the Substrate Concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Response Versus the Maximal Enzymatic Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Choosing the Enzyme Membrane Thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The Response Gradient with Respect to the Membrane Thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Maximal Gradient of the Transient Current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Potentiometric and Optical Biosensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Potentiometric Biosensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Optical Biosensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Fluorescence Biosensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 3 5 6 9 10 10 11 13 15 18 20 25 25 28 29 32 34 36 38 39 39 41 42 43

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 © Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_1

1

2

Introduction to Modeling of Biosensors

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 nonstationary 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. Keywords Biosensor · Transducer · Biocatalysis · Diffusion · Partial differential equation · Finite difference technique · Mathematical model

1 Introduction Biosensors are analytical devices that are based on the direct coupling of an immobilized biologically active compound with a signal transducer and an electronic amplifier [34, 44, 62]. The biosensors may utilize enzymes, antibodies, nucleic acids, organelles, plant and animal tissue, whole organism or organs [28, 78, 83]. Biosensors containing biological catalysts (enzymes) are called catalytic biosensors. Biosensors of this type are the most abundant, and they found the largest application in medicine, ecology, environmental monitoring and industry [39, 48, 75, 81, 82, 88]. Electrochemical biosensors usually monitor the current at a fixed voltage (amperometry) or the voltage at zero current (potentiometry), or measure conductivity or impedance changes [62, 78, 83]. Starting from the publication of Clark and Lyons in 1962 [34], the amperometric biosensors became one of the popular and perspective trends of biosensorics [61, 74]. The amperometric biosensors measure the changes of the current of indicator electrode by direct electrochemical oxidation or reduction of the products of the biochemical reaction. In amperometric biosensors, the potential at the electrode is held constant, while the current is measured [1, 47, 77]. The potentiometric biosensors operate under conditions of near-zero current flow and measure the difference in potential between the working electrode and a reference electrode. Ion-selective electrodes, of which the pH electrode is a wellknown example, are the most important of this class of transducers [62, 78, 83]. The prediction of geometric, kinetic and catalytic parameters is of crucial importance for solving analytical problems and development of novel biosensors. Because it is not generally possible to measure the concentration of substrate inside enzyme membranes, starting from seventies various mathematical models of catalytic biosensors have been developed and used as an important tool to study and optimize analytical characteristics of actual biosensors [44, 64, 65]. Using

2 Biosensor Action

3

mathematical and computational modeling to characterize the biosensor response in a wide range of input parameters can guide the experimental work, thus reducing development time and costs [16, 63, 68, 79]. The action of catalytic biosensors is associated with substrate diffusion into biocatalytic membrane and its conversion to a product. The simulation of biosensor action includes solving the diffusion equations for substrate and product with a term containing a rate of biocatalytic transformation of substrate. The complications of modeling arise due to solving partially differential equations with nonlinear biocatalytic term and with complex boundary and initial conditions [3, 14, 46]. This chapter introduces mathematical modeling of catalytic biosensors. After a short description of main concepts of biosensor action [18, 46, 53, 57], a detail mathematical model is presented for an amperometric biosensor based on a layer of an enzyme immobilized onto the surface of the probe [11, 12]. 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 [46, 79]. A few modifications of the mathematical model describing the action of potentiometric, optical and fluorescence biosensors are discussed, too [9, 13, 42, 53].

2 Biosensor Action 2.1 Kinetics of Biocatalytic Reactions The biosensors contain immobilized enzymes or other biological catalysts [1, 53, 83]. The biocatalyst catalyses the conversion of the substrate to the product. Biological catalysts (enzymes) show high activity and specificity. The activity of enzymes may exceed the rate of chemically catalysed reaction by a factor 4.6 × 105 –1.4 × 1017 [27]. The activity of the enzymes depends on many factors, i.e. the free energy of reaction, the substrate docking in the active centre of enzyme, the proton tunnelling and other factors [41, 52, 56, 58]. The general principles of catalytic activity of enzymes are known, but particular factors that determine high enzyme activity are often not established [63]. The specificity of enzymes depends on the enzyme type [31, 38]. There are enzymes that catalyse the conversion of just one substrate. Other enzymes show broad substrates specificity. Oxidoreductases, i.e. enzymes that catalyse electron transfer, may catalyse, for example, the oxidation or reduction of many substrates. To characterize the substrates with diverse activity, a slang expression “good substrate” and “bad substrate” is used. The following scheme of biocatalyser action was postulated by Henri in 1902 [49]: k2 k1 E + S FGGGGGB GGGGG ES GGGA E + P, k−1

(1)

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Introduction to Modeling of Biosensors

where E, S, ES and P correspond to the enzyme, the substrate, the enzyme– substrates complex and the product, respectively. In biochemistry, the concentrations are expressed as mol/dm3 ≡ M, whereas in models the concentrations of components are typically expressed in mol/cm3 , and kinetic constants k1 , k−1 and k2 correspond to the respective reactions: the enzyme–substrate interaction, the reverse enzyme–substrate decomposition and the product formation. Michaelis and Menten confirmed this scheme of enzymes action using acetate to keep the pH of solution [67]. Following the scheme (1), the change of concentration of each component can be expressed by ordinary differential equation (ODE): dE = −k1 ES + k−1 ES + k2 ES , dt dS = −k1 ES + k−1 ES , dt dES = k1 ES − k−1 ES − k2 ES , dt dP = k 2 ES , dt

(2)

where t is the time, E, S, ES and P correspond to the concentrations of the enzyme, the substrate, the enzyme–substrate complex and the product, respectively [18, 46]. To solve the system (2) of ODE, Briggs and Haldane applied quasi-steady state (QSS) approach (QSSA) to complex ES, which means that dES /dt ≈ 0 [24]. The calculated “initial rate” of the steady state reaction rate was expressed (S is equal to the initial concentration S0 ): V (S) = −

Vmax S dS = , dt KM + S

(3)

where Vmax = k2 E0 is the maximal enzymatic rate, E0 is the initial enzyme concentration, and KM = (k−1 + k2 )/k1 , and it is called the Michaelis constant. The Michaelis constant is the concentration of the substrate at which half the maximum velocity of an enzyme-catalysed reaction is achieved [24, 67]. Typical values of constants of the enzymes that are used for the biosensor preparation are −1 k1 = 106 –108 M−1 s and k−1 ≈ k2 = 100–1000 s−1 . Calculations show that during the enzymes action the quasi-steady state is established during 4.0–0.1 ms at enzyme and substrate concentrations 10−8 M and 10−3 M, respectively. It is sufficient to establish a quasi-steady state in the membranes of biosensors with the thickness more than 2 × 10−4 cm since the thickness δd of the effective diffusion layer calculated using the Cottrell equation is [7]: δd =

√ πDt ,

(4)

2 Biosensor Action

5

where the diffusion coefficient D for low molecular weight molecules is about 3 × 10−6 cm2 /s. For a more complex biocatalytic process, the establishment of the quasi-steady state requires much longer period of time. It was shown, for example, that for the synergistic reactions, involving cyclic mediators conversion, the time of the QSS establishing can be as large as 180 s [55]. Therefore, the expression for the “initial rate” is no longer valid, and the modeling should include the rates of all individual reactions.

2.2 Transducer Function The purpose of the transducer is to convert the biochemical recognition into an electronic signal. The transducer is a device that responds selectively to the substrate, the product, the mediator or other compounds, the concentration of which is related to the analyte under determination [53]. The transducer should show high selectivity since the biosensor selectivity depends on the specificity of the biocatalytic process and the selectivity of the transducer [6, 7, 43]. The transducers include amperometric and ion-selective electrodes, optical systems and other physical devices realizing different physical phenomena. The biocatalytic membrane is located at close proximity to transducer. There are two fundamental categories of transducers in respect of their response. The transducer of the first type, i.e. the amperometric electrode, is monitoring the faradaic current that arises when the electrons are transferred between the substrates, the product or the enzyme active centre and an electrode. As a result of electrochemical reaction, the concentration of oxidized (reduced) compound at the surface of the transducer drops down. The transducers of the second type, i.e. ion-selective electrodes and optical fibre, do not perturb the concentration of the determining compound at the surface. The difference between the transducer types produces different boundary conditions for the modeling of the biosensors [18, 46, 62]. The boundary condition for the first category of transducer can be written as P =0

or S = 0

at x = 0,

(5)

where x stands for space and P and S are the concentrations of the product and the substrate at the transducer surface, respectively. This boundary condition means that the kinetics of electron transfer is fast, and the potential of the transducer is high enough to keep a current at diffusion limiting condition. If the kinetics of electron transfer is slow, then the transducer current depends on the electrode potential and is obtained from the Butler–Volmer expression [26]. The modeling of biosensors at this type of boundary conditions has not been performed due to this uncommon state for the real biosensors [7, 18, 46].

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Introduction to Modeling of Biosensors

The boundary condition of the transducer of the second category is dP =0 dx

or

dS = 0 at x = 0. dx

(6)

For the ion-selective electrodes, this corresponds to the Nernstian boundary condition [26, 69]. For the optical transducer and other transducers, this condition means non-leakage (zero flux) of the product or the substrate on the boundary between the transducer and the biocatalytic membrane.

2.3 Scheme of Biosensor Action The biosensor produces a signal when the analyte under determination diffuses from the bulk solution into the biocatalytic membrane. The biocatalyst catalyses the substrate conversion to the product, which is determined by the transducer [18, 46, 62]. The concentration change of S and P is associated with the diffusion and the enzyme reaction. Following Fick, the compounds concentration change in the biocatalytic membrane can be written as ∂ 2S ∂S = DS 2 − V (S), ∂t ∂x ∂P ∂ 2P + V (S), = DP ∂t ∂x 2

(7) x ∈ (0, d),

t > 0,

where x and t stand for space and time, respectively, S(x, t) is the concentration of the substrate, P (x, t) is the concentration of the reaction product, d is the thickness of the enzyme membrane, DS and DP are the diffusion coefficients of compounds in the enzyme membrane, that is typically used as the same for the substrate, the product and the mediator [18, 46, 62]. The solution of (7) at the corresponding initial and boundary conditions produces the concentration change of S and P in time and membrane thickness. For the first type of transducers, the response R of biosensor can be written as  ∂P  R(t) = C1 , (8) ∂x x=0 and for the second type of transducers R(t) = C2 P (0, t),

(9)

R(t) = C3 log P (0, t),

(10)

or

where C1 , C2 and C3 are the appropriate constants.

2 Biosensor Action

7

The logarithmic dependence is characteristic of ion-selective electrodes, whereas for optical and other transducers linear dependence between the response and the concentration is realized. Simple analytical solution of (7) is impossible even for the simplest initial and boundary conditions due to the hyperbolic function of the enzymatic rate dependence on the substrate concentration (3). Therefore, the description of biosensors action is divided into the simplest cases for which analytical solutions still exist. This approach was used widely, especially at the beginning of the development of biosensors, to recognize the principles of the biosensors action. The aproximal analytical solution gives information about the critical cases. They are also useful to test the correctness of numerical calculations found at initial and boundary limiting conditions. When the concentration S0 to be measured is very small in comparison with the Michaelis constant KM , ∀x, t : x ∈ [0, d], t > 0 :

0 < S(x, t) < S0  KM ,

(11)

the nonlinear function V (S) simplifies to that of the first order, V (S) =

Vmax Vmax S ≈ S = kS, S + KM KM

(12)

where k is the first order reaction constant (linear enzyme kinetic coefficient), k = Vmax /KM . Practically, the enzyme reaction can be considered first order when the concentration of the detected species is below one-fifth of KM , i.e. S0 < 0.25KM [44]. 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, ∂ 2S ∂S = DS 2 − kS, ∂t ∂x ∂P ∂ 2P = DP + kS, ∂t ∂x 2

(13) x ∈ (0, d),

t > 0.

Analytical solutions are typically made at steady state and external and internal diffusion limiting conditions. The steady state (stationary) conditions mean that ∂S = 0, ∂t

∂P = 0. ∂t

(14)

The external diffusion limitation indicates that the substrates transport through the diffusion (stagnant) layer is a rate limiting process [57]. At internal diffusion limitation, the substrates diffusion through external diffusion layer is fast and the process is limited by the diffusion inside an enzyme membrane. The disadvantage

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Introduction to Modeling of Biosensors

Concentration

S0

SS

S

Transducer

0.00 Enzyme membrane

0.01

Stagnant layer

0.02

cm >

Bulk solution

Fig. 1 The substrate concentration profile in a biosensor at steady state conditions. The concentration profile was calculated with the boundary conditions ∂S/∂x = 0 at x = 0 and S = S0 at x ≥ d + δ. The diffusion coefficient DSb in a stagnant layer is 3 × 10−6 cm2 /s, in membrane is DS = 10−6 cm2 /s, Vmax = 5 × 10−7 mol/cm3 s, KM = 10−5 mol/cm3 , d = δ = 0.01 cm, S0 = 10−6 mol/cm3

of these approximate solutions is an error at the boundaries between the different approximate treatments. It is helpful to illustrate this approach by reference to a trivial problem of the substrate conversion in the biocatalytic membrane of the biosensor and at the concentration of the substrate less than KM . The calculated profile of the substrate concentration at the steady state or stationary conditions is shown in Fig. 1. It is possible to identify an abrupt of concentration change of the substrate at the boundary of biocatalytic membrane/stagnant layer as well as at the boundary stagnant layer/bulk solution. This comes from approximate solutions at the boundaries during different approximate treatments. The change of the steady state concentration of the substrate in membrane can be calculated as S cosh (αx) = , Ss cosh (αd)

(15)

where S, Ss is the substrate concentration at the transducer surface and at the boundary of membrane and stagnant solution, respectively, α2 =

Vmax . KM DS

(16)

On the other hand, at the steady state, a substrate flux through the boundary of stagnant layer/bulk solution is equal to the flux through the boundary of biocatalytic membrane/stagnant layer: DSb

 S0 − Ss ∂S  = DS α tanh(αd)Ss . = DS  δ ∂x x=d

(17)

2 Biosensor Action

9

A combination of these two solutions (15) and (17) produces the concentration profile of the substrate in the biocatalytic membrane and the stagnant layer (Fig. 1). It is possible to notice that the greatest error of calculations is at x = d and x = d+δ. However, at the limiting (the internal or the external diffusion limitation) cases, the two expressions produce very good approximations to the full equation. Therefore, the modeling of the biosensors at two limiting cases was used to solve different biosensors problems.

2.4 Biosensor Response The most popular glucose biosensor is based on glucose oxidase (GOx) that catalyses β-D-glucose oxidation with oxygen [84, 87] β-D-glucose + O2

D-glucose oxidase

−→

D-glucono-δ-lactone + H2 O2 .

(18)

The hydrogen peroxide produced is oxidized on platinum electrode acting as a transducer. One of the first tasks of modeling of this type of the biosensors was devoted to evaluate the dependence of biosensors response on enzymatic parameters [53]. The action of the biosensors was analysed at the internal diffusion limiting conditions and at the steady state conditions. The measured anodic or cathodic current is usually accepted as a response of the biosensor. The output current depends upon the flux of the electro-active substance (product) at the electrode surface, i.e. on the border x = 0. In the case of the amperometry, the biosensor current is also directly proportional to the area of the electrode surface. The current iA (t) of the amperometric biosensor at time t can be obtained explicitly from the Faraday and the Fick laws iA (t) = ne FADP

 ∂P  , ∂x x=0

(19)

where ne is the number of electrons involved in a charge transfer (for hydrogen peroxide ne = 2). F is the Faraday constant, F = 96485 C/mol, and DP is the diffusion coefficient of the product in the biocatalytic membrane. Due to the direct proportionality, the current is normalized with the area of that surface. Consequently, the density i(t) of the biosensor current at time t is expressed as follows:  iA ∂P  = ne F DP i(t) = . (20) A ∂x x=0

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Introduction to Modeling of Biosensors

The analytical system usually approaches the steady state as t → ∞ I = lim i(t).

(21)

t →∞

I is taken as the density of the steady state biosensor current.

3 Mono-Layer Model of an Amperometric Biosensor Amperometric biosensors were the first type to be developed and have been in use as glucose biosensors for over many years [6, 28, 62]. Consider an amperometric biosensor as the enzyme electrode having a layer of the enzyme immobilized onto the surface of the probe [40, 78, 83].

3.1 Mathematical Model Assuming the symmetrical geometry of the electrode and homogeneous distribution of the immobilized enzyme in the enzyme layer of a uniform thickness, a mathematical model of the biosensor action can be expressed by a system of two-component (substrate and product) reaction–diffusion equations. Coupling the enzyme-catalysed reaction (1) in the enzyme layer (enzyme membrane) with the one-dimensional-in-space diffusion, described by Fick’s law, and applying QSSA lead to the following equations [17, 46, 79]: ∂S ∂ 2S Vmax S = DS 2 − , ∂t ∂x KM + S ∂P Vmax S ∂ 2P = DP , + 2 ∂t ∂x KM + S

(22) x ∈ (0, d),

t > 0,

where x and t stand for space and time, respectively, S(x, t) is the concentration of the substrate, P (x, t) is the concentration of the reaction product, d is the thickness of the enzyme layer, DS and DP are the diffusion coefficients, Vmax is the maximal enzymatic rate and KM is the Michaelis constant. The Michaelis constant KM is the concentration of the substrate at which half the maximum velocity of the enzymecatalysed reaction is achieved [24, 67]. Let x = 0 represent the electrode surface, while x = d is the boundary between the analysed solution and the enzyme membrane. Initially, no substrate as well as product appears inside the enzyme layer. The operation of the biosensor starts when the substrate appears over the surface of the enzyme membrane. This is expressed in the initial conditions (t = 0) S(x, 0) = 0, S(d, 0) = S0 ,

P (x, 0) = 0, P (d, 0) = P0 ,

x ∈ [0, d),

(23)

3 Mono-Layer Model of an Amperometric Biosensor

11

where S0 and P0 are the concentrations of the substrate and the product in the bulk solution, respectively. Usually, the zero concentration of the reaction product in the bulk is assumed, P0 = 0. In the scheme (1), the product (P) is an electro-active substance. In the case of the amperometric biosensors, due to the electrode polarization, the concentration of the reaction product at the electrode surface (x = 0) is being permanently reduced to zero, P (0, t) = 0,

t > 0.

(24)

At the electrode surface, the substrate does not react. Because of this, at the electrode surface, the non-leakage (zero flux) boundary condition is defined for the substrate, DS

∂S  = 0,  ∂x x=0

t > 0.

(25)

Assuming the finite diffusivity DS of the substrate, the boundary condition (25) reduces to the following one: ∂S  = 0,  ∂x x=0

t > 0.

(26)

The concentration of the substrate as well as of the product over the enzyme surface (bulk solution/membrane interface) is assumed constant during the biosensor operation, S(d, t) = S0 ,

P (d, t) = P0 ,

t > 0.

(27)

3.2 Dimensionless Model In order to define the main governing parameters of the mathematical model (22)– (27), the dimensional variables (x and t) and unknown concentrations (S and P ) are replaced with the following dimensionless parameters: xˆ =

x , d

tˆ =

DS t , d2

S Sˆ = , KM

P Pˆ = , KM

DS Dˆ P = , DP

(28)

where xˆ stands for the dimensionless distance from the electrode surface, tˆ is the dimensionless time, Sˆ and Pˆ are the dimensionless concentrations of the substrate and the product, respectively [3, 35, 71].

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Introduction to Modeling of Biosensors

The governing equations (22) in the dimensionless coordinates xˆ and tˆ are expressed as follows: Sˆ ∂ Sˆ ∂ 2 Sˆ − σ2 = , 2 ∂ xˆ ∂ tˆ 1 + Sˆ Sˆ ∂ 2 Pˆ ∂ Pˆ + σ2 = Dˆ P , 2 ∂ xˆ ∂ tˆ 1 + Sˆ

(29) tˆ > 0 ,

xˆ ∈ (0, 1),

where σ 2 is the dimensionless diffusion module, which is also known as the Damköhler number [2], σ2 =

Vmax d 2 = α2 d 2 , KM DS

σ = αd.

(30)

The diffusion module σ 2 compares the rate of enzyme reaction (Vmax /KM ) with the mass transport through the enzyme layer (DS /d 2 ). The biosensor response is controlled by the diffusion when σ 2  1. If σ 2  1, then the enzyme kinetics determines mainly the biosensor response. Assuming (28), the initial conditions (23) transform to the following conditions: ˆ x, S( ˆ 0) = 0,

Pˆ (x, ˆ 0) = 0,

ˆ S(1, 0) = Sˆ0 ,

Pˆ (1, 0) = Pˆ0 ,

xˆ ∈ [0, 1) ,

(31)

where Sˆ0 and Pˆ0 are the dimensionless concentrations of the substrate and the product in the bulk, respectively, Sˆ0 = S0 /KM , Pˆ0 = P0 /KM . The boundary conditions (24), (26) and (27) are then rewritten as follows (tˆ > 0): ∂ Sˆ  = 0,  ˆ ∂ xˆ x=0 ˆ tˆ) = Sˆ0 , Pˆ (1, tˆ) = Pˆ0 . S(1,

Pˆ (0, tˆ) = 0 ,

(32)

The dimensionless density iˆ of the current (flux) and the corresponding dimensionless density Iˆ of the steady state current are defined as follows: ˆ  i(t) d ˆ tˆ) = Dˆ P ∂ Pe  = , i( ˆ ∂ xˆ x=0 ne F DS KM

ˆ tˆ) . Iˆ = lim i( tˆ→∞

(33)

Assuming the same diffusion coefficients for both species (the substrate and the product, DS = DP , Dˆ P = 1) and the zero concentration of the product in the bulk (Pˆ0 = 0), only the following two dimensionless parameters remain in the dimensionless mathematical model (29)–(32): the dimensionless substrate concentration Sˆ0 in the bulk solution and the diffusion module σ 2 . The advent

3 Mono-Layer Model of an Amperometric Biosensor

13

of the diffusion module σ 2 is one of the most important outcomes of defining a dimensionless model of the biosensor action. The diffusion module is the main parameter expressing all internal characteristics of the biosensor.

3.3 Characteristics of the Biosensor Response 3.3.1 Biosensor Sensitivity The sensitivity is one of the most important characteristics of the biosensors [32, 45, 51, 78, 83, 88]. The biosensor sensitivity can be expressed as the gradient of the steady state current with respect to the substrate concentration. Since the biosensor current as well as the substrate concentration varies even in orders of magnitude, especially when comparing different sensors, another useful parameter to consider is a dimensionless sensitivity. The dimensionless sensitivity for the substrate concentration S0 is given by BS (S0 ) =

dI (S0 ) S0 × , I (S0 ) dS0

(34)

where BS stands for the dimensionless sensitivity of the amperometric biosensor and I (S0 ) is the density of the steady state biosensor current calculated at the substrate concentration S0 . BS varies between 0 and 1.

3.3.2 Maximal Gradient of the Current The maximal gradient of the biosensor current calculated with respect to the time is another common characteristic of the biosensor action [4, 10, 78]. This type of measurement is rather often used in the biosensors operating in the transient mode. Since the biosensor current as well as the time varies even in orders of magnitude, the dimensionless maximal gradient is used to compare different sensors. The dimensionless maximal gradient that varies between 0 and 1 is given by  BG = max

i(t )>0

 di(t) t × , i(t) dt

(35)

where BG is the dimensionless maximal gradient of the biosensor current with respect to the time. Due to a possible delay in rising the current, the condition i(t) > 0 avoids a possible indeterminacy when calculating the maximal gradient BG .

14

Introduction to Modeling of Biosensors

3.3.3 Apparent Michaelis Constant The Michaelis constant KM has been defined as the substrate concentration at which the reaction rate is the half of its maximal value [24, 67]. The effect of halving of the maximal current is valid when the biosensor response is under enzyme kinetics control, i.e. when the diffusion module σ is significantly less than the unity. In such a case, the Michaelis constant KM is approximately equal to the apparent Michaelis app constant KM , which is used as one of the main characteristics of the sensitivity and app the calibration (saturation) curve of biosensors [46, 50, 70, 78, 86]. KM is assumed to be the concentration of the substrate at which the biosensor response reaches a half of the maximal response when the substrate concentration is extrapolated to infinity keeping the other model parameters constant,  app KM

=

S0∗

:

I (S0∗ )

 = 0.5 lim I (S0 ) , S0 →∞

app Kˆ M =

app

KM , KM

(36)

where I (S0 ) is the density of the steady state biosensor current calculated at app the substrate concentration S0 , and Kˆ M is the dimensionless apparent Michaelis constant. The apparent Michaelis constant is also known as the half maximal effective concentration of the substrate to be determined [21]. app app For the biosensor of a concrete configuration, the KM as well as Kˆ M can be rather easily calculated by multiple calculations of the maximal response changing the substrate concentration S0 [8].

3.3.4 Response Time The time interval from the beginning of the biosensor action up to the moment of the current measured is called the biosensor response time [6, 62, 78]. The moment of the measurement depends on the type of the device. Devices operating in the stationary mode usually use the time when the absolute current slope value falls below a given small value. Since the biosensor current varies even in orders of magnitude, the current is usually normalized with the current value. In other words, the time T needed to achieve a given dimensionless decay rate ε is accepted as the response time,   t  di(t)  t:  0 i(t) dt

T = min

(37)

However, the response time T as the approximate time of the steady state is very sensitive to the decay rate ε, T → ∞ when ε → 0. Because of this, the time

3 Mono-Layer Model of an Amperometric Biosensor

15

of the partial stationary current is usually used. The resultant relative output signal function i ∗ (t) is expressed as follows: i ∗ (t) =

i(t) . I

(38)

Let us notice that for the amperometric biosensors ∀t : t > 0 :

0 ≤ i ∗ (t) ≤ 1;

i ∗ (0) = 0,

i ∗ (T ) ≈ 1 .

(39)

The half-time T0.5 of the steady state or the half of the time moment of occurrence of the steady state current is often used instead of the time T to evaluate the dynamics of the biosensor operation,   T0.5 = t : i ∗ (t) = 0.5 .

(40)

The times T0.9 and T0.95 needed to reach 90% and 95% of the stationary current, respectively, are also rather often used as the response times, i ∗ (T0.9 ) = 0.9, i ∗ (T0.95 ) = 0.95. In the case of the transient operational mode, the slope di(t)/dt is used to find out the maximal gradient (35) of the current [78]. Then, the time TG of the maximal derivative of i(t) is accepted as the response time,   di(t) t × = BG , TG = t : i(t) dt

(41)

where BG is the maximal gradient defined by (35).

3.4 Solution for the First Order Kinetics When the concentration S0 to be measured is very small in comparison with the Michaelis constant KM as defined by (11), the nonlinear Michaelis–Menten function V (S) simplifies to that of the first order (12). Assuming the approximation (12), the initial boundary value problem (22)–(27) can be solved analytically [3, 35, 71, 79, 80]. The concentration profiles for both species, the substrate and the product, are expressed as follows [79]:  S(x, t) = S0

∞ 4  sin ((2n + 1)(d − x)π/(2d)) k + ue−(k+u)t 1− × π 2n + 1 k+u

,

n=0

(42)

16

Introduction to Modeling of Biosensors

with u = DS (2n + 1)2 π 2 /(4d 2), P (x, t) = S0

 ∞

2k  1 − (−1)m (1 − ewt ) sin (m(d − x)π/d) π mw m=1

∞  4(−1)m  (−1)n + π (2n + 1)(k + u) n=0  4m2 k(1 − e−wt ) u(e−(k+u)t − e−wt ) + × × , w w−k−u 4m2 − (2n + 1)2 (43)

with w = DP m2 π 2 /d 2 . The corresponding steady state concentrations are [2, 29, 53, 79] Sss (x) = lim S(x, t) = S0 × t →∞

cosh(αx) , cosh(αd)

Pss (x) = lim P (x, t) =

(44)

t →∞

= S0 ×

DS 1 − cosh(αx) + (cosh(αd) − 1)x/d × , DP

√ √ where α = k/DS = Vmax /(KM DS ) as introduced in (16). Having the concentration of the product, the density of the biosensor current has been calculated as follows [80]: ∞

 S0  (−1)m − 1  i(t) = − ne F DP 2k × 1 − ewt d w m=1

∞  

4m 4m (−1)n × 2 π (2n + 1)(k + u) 4m − (2n + 1)2 n=0  k(1 − e−wt ) u(e−(k+u)t − e−wt ) + × . w w−k−u +

(45)

Assuming t → ∞, the corresponding density of the steady state current of the amperometric biosensors is calculated from i(t) [53], I = ne F DS

S0 d

 1−

1 cosh(αd)

= ne F DS

S0 d

 1−

1 cosh(σ )

.

(46)

The solution shows that the biosensor response is a linear function of the substrate concentration. The biosensor sensitivity BS does not depend on the

3 Mono-Layer Model of an Amperometric Biosensor

17

enzyme activity if the diffusion module σ is larger than 1 since 1−

1 ≈ 1. cosh(σ )

(47)

At σ = αd  1, the approximate solution of (22)–(27) leads to I ≈ ne F DS S0

α2 d σ2 Vmax d = ne F DS S0 = ne F S0 . 2 2d 2KM

(48)

In this case, the biosensor sensitivity is determined by the enzyme parameters Vmax and KM . The inactivation of an enzyme following the diffusion module change from (σ > 1) to (σ < 1) produces a wrong interpretation of an enzyme stability in the biocatalytic membranes. Simple calculations show that if the enzyme inactivates in a solution following the first order reaction with a rate constant kin = 0.1h−1 , the half-time (τ ) of enzyme inactivation is 6.9 h (τ = ln(2)/kin ). If the same enzyme is used for the biocatalytic membranes preparation, and at the beginning of activation the diffusion module of the biosensors is, for example, 100 (d = 0.01 cm), the response of biosensor will decrease two times only after 87 h. Figures 2 and 3 show concentration profiles and evolutions of the output current, respectively, typical for the first order enzyme kinetics at the following values of the model parameters: DS = DP = 300 μm2 /s, KM = 100 μM,

ne = 2,

Vmax = 100 μM/s.

d = 0.1 mm

(49)

One can see in Fig. 2 that at relatively large time t = 3 s (curve 3) the profile of the substrate concentration practically coincides with that at the steady state conditions (curve 4), while the corresponding concentrations of the product differ notably. The evolution of the density i(t) of the biosensor current is presented in Fig. 3. The biosensor response was modeled for the biosensors having different enzyme membrane thicknesses d: 0.01, 0.015, 0.1 and 0.15 mm. One can see in Fig. 3 that the biosensor current appears with some delay at relatively thick enzyme layers. This delay increases with the increase of the enzyme membrane thickness. When comparing the evolution of the biosensor current in two cases of relatively thin (d = 0.01 and 0.015 mm) membranes, one can see that the biosensor response is noticeably higher at the thicker membrane (d = 0.015 mm) than at the thinner one (d = 0.01 mm). However, when comparing the biosensor responses in the other two cases of the ten times thicker (d = 0.1 and 0.15 mm) membranes, one can see the opposite tendency, i.e. the biosensor of the thicker (d = 0.15 mm) membrane generates lower response than the thinner one (d = 0.1 mm).

18

Introduction to Modeling of Biosensors 1.0 0.9 0.8

S, μM

0.7 0.6 0.5 0.4

4

0.3

3

2

0.2

1

0.1 0.0 0.00

0.01

0.02

0.03

0.04

a)

0.05

0.06

0.07

0.08

0.09

0.10

x, mm 0.6 0.5

4

P, μM

0.4 0.3

3 0.2

2

0.1

1 0.0 0.00

0.01

0.02

b)

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

x, mm

Fig. 2 Concentration profiles of the substrate (a) and the product (b) at the following values of time t: 0.5 (1), 1 (2), 3 (3) s and the steady state (4) in the case of first order reaction rate. The values of the model parameters are defined in (49)

3.5 Solution for the Zero Order Kinetics At low enzyme loading factor and large substrate concentrations, the nonlinear Michaelis–Menten function V (S) reduces to that of the zero order, V (S) =

Vmax S ≈ Vmax , S + KM

S0  KM .

(50)

3 Mono-Layer Model of an Amperometric Biosensor

19

1.2 1.1

2

1.0

1

i, nA/mm2

0.9 0.8 0.7

3

0.6 0.5

4

0.4 0.3 0.2 0.1 0.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

t, s Fig. 3 Dynamics of the biosensor current at four values of the thickness d: 0.01 (1), 0.015 (2), 0.1 (3), 0.15 (4) mm. The other parameters are the same as in Fig. 2

In this case, the analytical solution of the initial boundary value problem (22)– (27) is [3, 29, 71, 79] ∞  4  sin((2n + 1)(d − x)π/(2d)) S(x, t) = S0 − π 2n + 1 n=0 (51)   1 − e−wt + S0 e−wt , × Vmax w ∞  4Vmax  sin((2n + 1)(d − x)π/(2d)) 1 − e−vt P (x, t) = × , π 2n + 1 v

(52)

n=0

where w = DS (2n + 1)2 π 2 /(4d 2 ) ,

(53)

v = DP (2n + 1)2 π 2 /d 2 .

(54)

The corresponding stationary concentrations are [79]   Vmax x 2 − d 2 Sss (x) = S0 + , 2DS Pss (x) =

Vmax x(d − x) . 2DP

(55)

(56)

20

Introduction to Modeling of Biosensors

The densities of the current and of the corresponding stationary current are [79] i(t) = ne F DP

∞ 4Vmax  1 − e−vt , d v

(57)

n=0

I=

ne F Vmax d . 2

(58)

Figure 4 shows typical concentration profiles calculated by (51)–(56). When comparing the concentration profiles obtained at the zero order reaction rate (Fig. 4) with those obtained at the first order rate (Fig. 2), one can see a noticeable difference in the shape of the curves. The evolution of the density i(t) of the biosensor current is presented in Fig. 5. The biosensor response was modeled for the biosensors having the same values of the thickness d as in the case of the first order reaction rate (see Fig. 3). The biosensor current appears with some delay at relatively thick enzyme layers. This delay increases with the increase in the enzyme membrane thickness. When comparing the evolution of the biosensor current for different thicknesses of the enzyme membrane, one can see that the steady state response is always higher at the thicker membrane than at the thinner one. The direct proportionality of the steady state current on the membrane thickness is defined in (58). In the case of zero order reaction rate (Fig. 5), the behaviour of the biosensor current importantly differs from that in the case of first order reaction rate (Fig. 3).

3.6 Solution for Michaelis–Menten Kinetics For many nonlinear reaction–diffusion problems, the exact analytical solutions are practically impossible. Therefore, at the transient conditions, the nonlinear initial boundary value problem (22)–(27) was solved numerically by applying the finite difference technique [25, 26]. To find a numerical solution of the problem in the domain [0, d] × [0, T ], a discrete grid is introduced. The easiest way is to use a uniform discrete grid ωh = {xi : xi = ih, i = 0, 1, . . . , N, hN = d}, ωτ = {tj : tj = j τ, j = 0, 1, . . . , M; τ M = T },

(59)

where ωh and ωτ are the uniform discrete grids for space and time, respectively.

3 Mono-Layer Model of an Amperometric Biosensor

21

100 90

4

80

S, mM

70 60 50

3

40 30

2

20

1

10 0 0.00

0.01

0.02

0.03

0.04

a)

0.05

0.06

0.07

0.08

0.09

0.10

0.09

0.10

x, mm 0.5

0.4

P, mM

4 0.3

3 0.2

2

0.1

b)

0.0 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1

0.08

x, mm

Fig. 4 Concentration profiles for the substrate (a) and product (b) at the following values of time t: 0.5 (1), 1 (2), 3 (3) s and steady state (4) in the case of zero order reaction rate. The values of the model parameters are defined in (49)

The following notation is used in the finite difference approximations presented below: j

Si = S(xi , tj ), i = 0, . . . , N,

j

Pi = P (xi , tj ), j = 0, . . . , M.

ij = i(tj ),

(60)

Introduction to Modeling of Biosensors

i, μA/mm2

22 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

4

3

2

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

t, s Fig. 5 Dynamics of the biosensor current at four values of the thickness d: 0.01 (1), 0.015 (2), 0.1 (3), 0.15 (4) mm in the case of zero order reaction rate. The other parameters are the same as in Fig. 4

There are several kinds of the finite difference schemes available for the approximation of the reaction–diffusion equations. The degree of the implicity is the main difference between them. The implicit scheme and the explicit one are two widely used schemes in practice [20, 25, 26].

3.6.1 Explicit Scheme The partial differential equations (22) can be approximated with the following explicit scheme: j +1

j

j

j

j j S − 2Si + Si−1 Vmax Si − Si = DS i+1 − , j τ h2 KM + Si

Si

j +1

Pi

j

j

j

j j P − 2Pi + Pi−1 Vmax Si − Pi = DP i+1 + , j τ h2 KM + Si

i = 1, . . . , N − 1,

(61)

j = 0, . . . , M − 1 .

The explicit difference scheme is popular because of the programming simplicity. However, the explicit method is conditionally stable and converges with the rate

3 Mono-Layer Model of an Amperometric Biosensor

23

O(τ + h2 ). The differential equations are nonlinear, and it is possible to formulate the sufficient stability conditions as τ max{DS , DP } 1 ≤ , h2 4

τ Vmax 1 ≤ . KM 2

(62)

3.6.2 Semi-Implicit Scheme When approximating the reaction–diffusion equations (22) with a semi-implicit scheme, the following finite difference equations are obtained: j +1

j +1

j +1

j S − 2Si − Si = DS i+1 τ h2

Si

j +1

j +1

j +1

+ Si−1

j +1

j P − 2Pi − Pi = DP i+1 τ h2

Pi

i = 1, . . . , N − 1,

j +1

−

Vmax Si

j

KM + Si

j +1

+ Pi−1

,

j +1

Vmax Si

+

KM +

j Si

(63) ,

j = 0, . . . , M − 1 .

The semi-implicit scheme is unconditionally stable and converges with the rate O(τ + h2 ).

3.6.3 Approximation of Initial and Boundary Conditions The initial conditions (23) are approximated as follows: Si0 = 0,

Pi0 = 0,

0 SN = S0 ,

i = 0, . . . , N − 1,

PN0 = P0 .

(64)

The approximation of boundary conditions (24)–(27) leads to the following difference equations: j

j

S0 = S1 ,

j

SN = S0 ,

j

P0 = 0,

j

PN = P0 ,

j = 1, . . . , M.

(65)

3.6.4 Calculation Procedure The calculation of the numerical solution starts at the layer t = t0 = 0 with applying Equations (64). When the solution on the layer tj (j = 0, 1, . . . , M − 1) has been calculated, the solution on the next layer t = tj +1 can be calculated by using (61) or (63). When using the implicit scheme (63), the systems of linear algebraic equations

24

Introduction to Modeling of Biosensors

can be efficiently solved in both steps because of the tridiagonality of the matrices of the systems [19, 26]. Having the numerical solution of the problem, the biosensor current at time t = tj is calculated by j

i(tj ) ≈ ij = ne F DP P1 / h,

j = 1, . . . , M.

(66)

In the numerical simulation, the response time T was assumed as the time when the absolute biosensor current slope value falls below a given small dimensionless decay rate ε normalized with the current value,  |ij − ij −1 | t 0 ij τ 

(67)

In calculations, ε = 10−2 can be successfully used. The half-time T0.5 of the steady state or a half of the time moment of occurrence of the steady state current can be calculated after the approximate steady state was reached. Having the calculated stationary current I and the transitional currents at all the times tj , tj < T , j = 1, . . . , M, T0.5 is calculated as follows: T0.5 ≈ min

1≤j ≤M

  tj : ij ≥ 0.5I .

(68)

T0.9 , T0.95 and any other time Tα are calculated very similarly, Tα ≈ min

1≤j ≤M

  tj : ij ≥ αI ,

(69)

where α is between 0 and 1. The maximal gradient BG of the current is calculated as  BG ≈

max

1≤j ≤M, ij >0

ij − ij −1 tj × ij τ



 =

max

1≤j ≤M, ij >0

 j (ij − ij −1 ) . ij

(70)

Having the maximal gradient BG , the corresponding time TG can be obtained from   tj ij − ij −1 = BG , ij > 0, j = 1, . . . , M . TG ≈ tj : × (71) ij τ The calculation of the biosensor sensitivity by (34) requires the simulation of the biosensor for at least two values of the substrate concentration S0 . To find out the

4 Peculiarities of the Biosensor Response

25

biosensor sensitivity at appropriate value of S0 , the response has to be calculated for the other concentrations close to the first one, BS (S0 ) ≈

I (S0 + o(S0 )) − I (S0 ) S0 × , I (S0 ) o(S0 )

(72)

where o(S0 ) is a relatively small part of S0 , e.g. 5% from the concentration to be analysed, i.e. o(S0 ) = 0.05S0 . The calculation of the other characteristics of the biosensor, such as the steady state current (21), the maximal gradient (35), the response times (40) and (41), is rather easy.

3.6.5 Validation of Numerical Solution Before applying the numerical solution to proceed with an investigation of the modeling process, the solution has to be validated. The usual method of validating the numerical solution is the use of some special limits of the input parameters for which analytic solutions are available. When applying this method, the numerical solution in the limits is tested against the associated analytical solutions. Solutions for the first and zero order kinetics presented above in this chapter were used for the validation of the numerical solution [11, 12, 25]. The first order enzyme kinetics was modeled accepting S0 = 1 μM = 0.01KM (Figs. 2 and 3), while the zero order kinetics was modeled at S0 = 100 mM = 1000KM (Figs. 4 and 5). No noticeable difference between the numerical and the corresponding analytical solutions was achieved at the model parameter values defined in (49).

4 Peculiarities of the Biosensor Response Using numerical simulation, the influence of enzyme membrane thickness and other peculiarities of the biosensor response extend the investigations performed in [11, 12].

4.1 Effect of the Enzyme Membrane Thickness Figure 6 shows the density of the steady state current (I ), while Fig. 7 presents the half-time T0.5 of the steady state versus the thickness d of the enzyme membrane. Figure 6 also presents the corresponding densities of the steady state current calculated analytically using (46).

26

Introduction to Modeling of Biosensors

40

1 2 3 4

I, nA/mm2

30

20

10

0 1

10

100

d, μm Fig. 6 The dependence of the steady state biosensor current I on the thickness d of the enzyme membrane at four maximal enzymatic rates Vmax : 1 (1), 10 (2), 100 (3) and 1000 (4) μM/s, S0 = 10 μM. Symbols are numerical solutions, while curves are analytical ones (formula (46)). The other parameters are defined in (49)

102

1 2 3 4

T0.5, s

101

100

10-1

10-2

10-3 1

10

d, μm

100

Fig. 7 The dependence of the half-time T0.5 of the steady state on the thickness d of the enzyme membrane at the same values of the model parameters as in Fig. 6

4 Peculiarities of the Biosensor Response

27

One can see in Fig. 6 that the density of the steady state biosensor current is a non-monotonous function of d at all values of the maximal enzymatic rate Vmax . The higher maximal enzymatic rate Vmax corresponds to the greater value of I . The results obtained by the numerical simulation show that the maximum of I equals about 394 μA/mm2 at Vmax = 1 mM/s, while the maximal I is approximately equal to 125 μA/mm2 at ten times smaller value of Vmax = 100 μM/s. The higher maximum of I corresponds to the thinner enzyme membrane. In the case of Vmax = 1 mM/s, the maximum of I (d) gains at d ≈ 8.6 μm, while in the case of Vmax = 100 μM/s, the maximum of I gains at d ≈ 27 μm. Since S0 = 10 μM = 0.1KM , formula (46) can used to find analytically the membrane thickness d at which the steady state current gains the maximum at given values of ne , DS , S0 , Vmax , KM , where S0  KM . At first, a derivative of I (d) with respect to the thickness d is calculated [12], dI (d) − cosh2 (σ ) + cosh(σ ) + σ sinh(σ ) = ne F DP S0 . dd d 2 cosh2 (σ )

(73)

Then the value of σ at which that derivative gets zero is found, − cosh2 (σ ) + cosh(σ ) + σ sinh(σ ) = 0.

(74)

Equation (74) was solved numerically. The unique non-zero solution σ = σ max ≈ 1.5055 was obtained [12]. Sequentially, I gains the maximum at the membrane thickness dmax , where  dmax =

1 σmax

DS KM , Vmax

σ max = 1.5055.

(75)

Accepting (49) and (75), one can calculate that dmax ≈ 8.25 μm and I ≈ 400 μA/mm2 at Vmax = 1 mM/s; dmax ≈ 26.1 μm and I ≈ 128 μA/mm2 at Vmax = 100 μM/s. These values compare sufficiently well with the corresponding values obtained by the numerical simulation of the biosensor operation. The corresponding values of the density of the steady state current as well as thickness d vary by about 5 %. The difference in these values appears because of the substrate concentration S0 = 0.1KM . The analytical solution (46) is valid at S0  KM only, while the numerical one does not have such kind of restrictions at all. Because of this, values of dmax obtained by using the computer simulation at S0 = 10 μM and (49) are, in general, more accurate than the analytical ones. Consequently, the steady state biosensor current as a function (46) of the membrane thickness d gains the maximum when the diffusion module σ equals σmax = 1.5055. According to (30) and (75), the thichness dmax does not depend on the substrate concentration S0 . Nevertheless, using the numerical simulation, the following values of σmax were obtained at four values of S0 : σmax ≈ 1.51 at S0 = 0.01 μM, σmax ≈ 1.52 at S0 = 1 μM, σmax ≈ 1.57 at S0 = 10 μM and σmax ≈ 2.1 at

28

Introduction to Modeling of Biosensors

S0 = 100 μM. The diffusion module σmax is approximately constant at S0  KM , so that it is about coincident with the value obtained from the analytical solution (46). dmax increases with the increase in the substrate concentration S0 . That is especially notable at the substrate concentrations S0 ≈ KM . In the case of S0  KM , the stationary current increases linearly with the increase in the thickness d (see (58)), i.e. I → ∞ when dmax → ∞ and S0  KM . The dependence of dmax on Vmax is fairly low, and dmax varies by less than 3.5% when Vmax changes from 1 to 1000 μM/s at any value of S0 mentioned above. Figure 7 shows that the half-time T0.5 of the steady state increases notable with the increase in the thickness d of the enzyme membrane. That increase is slightly exponential. The half-time T0.5 reaches even one hundred seconds at d = 400 μm, while T0.5 varies about only 0.1 s at d = 10 μm. The effect of Vmax on the response time is rather slight. A greater enzyme activity Vmax , i.e. a higher reaction rate, corresponds to a shorter response time. T0.5 equals 11.4 s at Vmax = 1 μM/s, while T0.5 = 4.7 s at Vmax = 1000 μM/s.

4.2 Stability of the Response The stability of the response is one of the most critical characteristics of the biosensors [72]. It is very important to keep the analytical capability of the biosensors for as long as possible period. Usually, the maximal enzymatic rate Vmax decreases permanently due to the enzyme inactivation. In general, the biosensor response is sensitive to changes in Vmax . Figure 6 shows that the maximal biosensor current differs by orders of magnitude when changing Vmax . The variation is especially notable in the cases of relatively thin enzyme membranes. In the case of relatively thick enzyme membrane, the steady state current does not practically vary by changing Vmax . Consequently, the biosensor containing the thicker enzyme layer gives more stable response than the biosensor with the thinner layer. However, the thick membrane-based biosensors have very durable response time (see Fig. 7). Because of this, the relatively thick biosensors are of limited applicability, e.g. in the flow injection systems, which are widely used for determination of various compounds [76]. Thus, the problem of the membrane thickness optimization arises. The task is to find the thickness d of the membrane as small as possible, ensuring the stability of the biosensor response at a range of Vmax as wide as possible. Let V1 and V2 be two values of the maximal enzymatic rate, for which a stable biosensor response to the substrate of the concentration of S0 is required. The minimal membrane thickness dε (V1 , V2 , S0 ) is introduced as the relative difference R(d, V1 , V2 , S0 ) between the biosensor response (the density of the steady state biosensor current) at the thickness

4 Peculiarities of the Biosensor Response

29

d = dε , Vmax = V1 , and another response at d = dε , Vmax = V2 , is less than the dimensionless decay rate ε, |I (d, V1 , S0 ) − I (d, V2 , S0 )| , I (d, V1 , S0 )

(76)

dε (V1 , V2 , S0 ) = min{d : Rd (d, V1 , V2 , S0 ) < ε},

(77)

Rd (d, V1 , V2 , S0 ) =

d>0

where I (d, Vmax , S0 ) is the density of the steady state biosensor current calculated at the membrane thickness d, the maximal enzymatic rate Vmax and the substrate concentration S0 [12]. Let us assume S0 = 10 μM, V1 = 100 μM/s, V2 = 1000 μM/s and ε = 0.01. The numerical results presented in Fig. 6 and some additional calculations show that dε approximately equals 92 μm. To evaluate the biosensor stability at a wide range of the substrate concentration S0 , the response of the biosensor based on the membrane of the thickness d = dε (V1 , V2 , S0 ) = 92 μm was calculated at different values of S0 from this range. The following sections discuss this in detail.

4.3 The Response Versus the Substrate Concentration Figure 8 shows the density of the steady state current of the biosensor of the membrane thickness d = 92 μm versus the substrate concentration S0 at the 101

I, μA/mm2

100

4 5

1 2 3

10-1 10-2 10-3 10-4

KM 10-5 10-6

10-5

10-4

10-3

10-2

10-1

S0 , M Fig. 8 The dependence of the density of the steady state current on the concentration S0 of the substrate at five maximal enzymatic rates Vmax : 0.1 (1), 1 (2), 10 (3), 100 (4) and 1000 (5) μM/s, d = dε (100 μM/s, 1000 μM/s, 10 μM) = 92 μm, calculated assuming ε = 0.01

30

Introduction to Modeling of Biosensors 12 11

1 2 3 4 5

10

T0.5 , s

9 8 7 6 5 4 3 2 1 10-6

10-5

10-4

S0 , M

10-3

10-2

10-1

Fig. 9 The dependence of the half-time T0.5 of steady state biosensor response on the concentration S0 of the substrate at five maximal enzymatic rates Vmax : 0.1 (1), 1 (2), 10 (3), 100 (4) and 1000 (5) μM/s. The other parameters are the same as in Fig. 8

following five values of Vmax : 0.1 (1), 1 (2), 10 (3), 100 (4) and 1000 (5) μM/s. In this figure, one can see no noticeable difference between the values of I calculated at two values of Vmax : 100 and 1000 μM/s when the substrate concentration S0 is less than about 10−3 M. Figure 9 explicitly shows the stable response of the biosensor based on the enzyme membrane of the thickness d = 92 μm when the maximal enzymatic rate reduces ten times: from 1000 to 100 μM/s. Although the membrane thickness dε of 92 μm has been calculated at the substrate concentration S0 = 10−5 M, the biosensor response is sufficiently stable to the substrate of the concentration being up to about 10−3 M. The dependence of dε on the substrate concentration has already been noticed above. The biosensor response is especially sensitive to changes in Vmax at high concentrations of the substrate. Figure 9 shows that the response of the biosensor of thickness d = 92 μm is approximately constant at the concentration higher than about 10−2 M. Because of this, such a biosensor is not practically useful to determine the substrate of the concentration larger than 10−2 M. Figure 9 shows the effect of the substrate concentration S0 on the half-time T0.5 of the steady state biosensor response. The thickness d of the enzyme membrane is the same as above, i.e. d = dε = 92 μm. One can see in Fig. 9 that T0.5 is a monotonously decreasing function of S0 at Vmax = 0.1, 1, 10 μM/s, and T0.5 is a non-monotonic function of S0 at two largest values of Vmax : 100 and 1000 μM/s. At S0 being between 10−4 and 10−2 M, a shoulder on the curve appears for those two enzymatic rates. It seems possible that the shoulder on the curve arises because of the high Vmax . At the substrate concentration S0  KM , the reaction kinetics is of the zero order throughout the membrane, whereas for S0  KM , the kinetics is of

4 Peculiarities of the Biosensor Response

31

the first order throughout. At intermediate values of the substrate concentration S0 , the kinetics changes from the zero order to the first order across the membrane [15, 17, 44, 54]. The biosensor based on the enzyme membrane of the thickness of 92 μm, gives a very stable response (Fig. 8) in a sufficiently short time (Fig. 9) when Vmax is between 100 and 1000 μM/s as well as the substrate concentration S0 is less than about 10−3 M. The Michaelis constant KM is known to be the substrate concentration at which the reaction rate is half of its maximal value. Figure 8 shows the halving effect for two values of Vmax : 0.1 and 1 μM/s. The half of the maximum of the steady state current is reached at substrate concentration of about KM = 0.1 mM for both values of Vmax . The relative difference between the half of the maximum of steady state current and the maximal current at KM does not exceed 0.3% at Vmax = 0.1 and 3% at Vmax = 1 μM/s. The maximum of the density of the steady state current is equal to about 0.89 and to about 8.9 nA/mm2 at the maximal enzymatic rates of 0.1 and 1 μM/s, respectively. The effect of halving is not valid for the higher maximal enzymatic rates. Accepting the thickness d of 92 μm the diffusion module σ equals unity at Vmax = 3.5 μM/s. In the cases when Vmax > 3.5 μM/s, the biosensor response is under the diffusion control. The effect of halving is not valid when the biosensor response is under the diffusion control. For example, in the case of Vmax = 10 μM/s(σ ≈ 1.7), the half of the maximum of the steady state current is reached at S0 ≈ 160 μM = 1.6KM . If Vmax = 100 μM/s (σ ≈ 5.3), then the half of the maximum of steady state current is reached at S0 ≈ 7.6 KM . Reaching the half of the maximum of steady state current at larger concentrations of the analyte usually means the longer calibration curve of the biosensor response. This is also expressed by the biosensor sensitivity presented in Fig. 10. One can see in this figure that the higher maximal enzymatic rate corresponds to the longer line close to the unity. The zero biosensor sensitivity (BS = 0) at the concentration S0 of the substrate means that the change in the concentration does not affect the biosensor current, i.e. the current does not change at changing concentrations of the analyte. If the normalized biosensor sensitivity BS is close to zero at the substrate concentration S0 , then the biosensor is not practically useful to determinate the analyte of such a concentration. Figures 8 and 10 show that the biosensors based on the enzyme of the higher activity Vmax are applicable to a wider range of substrate concentrations rather than those of the lower activity. This peculiarity of the amperometric biosensors can be reformulated in a more common way as follows: the biosensors characterized with the greater diffusion module are applicable to a wider range of the substrate concentrations rather than those of the smaller diffusion module.

32

Introduction to Modeling of Biosensors 1.0 0.9 0.8 0.7

BS

0.6 0.5 0.4 0.3 0.2

1 2 3 4 5

0.1 0.0 10-6

10-5

10-4

10-3

10-2

10-1

S0 , M Fig. 10 The normalized biosensor sensitivity BS versus the substrate concentration S0 . The parameters and notations are the same as in Fig. 8

4.4 The Response Versus the Maximal Enzymatic Rate The effect of the maximal enzymatic rate Vmax on the biosensor response at different concentrations (S0 ) of the substrate was also investigated. The computer simulation results are depicted in Figs. 11 and 12. The calculations were done at the thickness d = 100 μm of the enzyme layer and the following five values of the concentration S0 : 10−6 (1), 10−5 (2), 10−4 (3), 10−3 (4) and 10−2 (5) M [12]. As one can see in Fig. 11, the steady state current is approximately a linear function of Vmax at a low activity of the enzyme. The range of the linearity increases with the increase in the substrate concentration S0 . In the case of the zero order reaction rate, i.e. at very high concentrations of the substrate (S0  KM , curve 5), the current density I is practically the linear function of Vmax at all values of the Vmax . The linear dependence of I on the Vmax also follows from the formulae (58). In the case of the first order reaction rate (S0  KM , curves 1 and 2), the dependence is nonlinear as defined in (46). The density I of the steady state current is the linear function of Vmax in all the cases when the diffusion module is less than unity (σ < 1). Let us notice that σ = 1 at Vmax = 3 μM/s and values (49). Figure 12 shows the dependence of the half-time T0.5 of the steady state. At low concentrations of the substrate (S0  KM , curves 1 and 2), the half-time T0.5 is the monotonously decreasing function of Vvax . The response time changes slightly at moderate values of the substrate concentration S0 (curves 3 and 4). At high concentrations of the substrate (S0  KM , curve 5), the half-time T0.5 is the monotonously increasing function of the Vvax . However, at a very high enzyme activity (Vmax >≈ 0.8 mM/s, σ 2 >≈ 30), T0.5 depends neither on the S0 nor on the

4 Peculiarities of the Biosensor Response

33

101 0

I, μA/mm2

10

10-1

1 2 3 4 5

10-2 10-3 10-4 10-5 10-7

10-6

10-5

10-4

10-3

Vmax , M/s

T0.5 , s

Fig. 11 The density I of steady state current versus the maximal enzymatic rate Vmax at five concentrations S0 of the substrate: 10−6 (1), 10−5 (2), 10−4 (3), 10−3 (4) and 10−2 (5) M, d = 100 μm

14 13 12 11 10 9 8 7 6 5 4 3 2 1 10-7

1 2 3 4 5

10-6

10-5

10-4

10-3

Vmax , M/s Fig. 12 The half-time T0.5 of the steady state versus the maximal enzymatic rate Vmax . The other parameters and the notation are the same as in Fig. 11

34

Introduction to Modeling of Biosensors

Vmax . Let us notice that at the substrate concentration S0 of 10KM and the enzyme activity Vmax being between 10 and 100 μM/s, a shoulder on curve 4 appears.

4.5 Choosing the Enzyme Membrane Thickness Formulas (76) and (77) introduce the concept of the minimal membrane thickness dε (V1 , V2 , S0 ), at which the relative difference Rd (d, V1 , V2 , S0 ) of the biosensor response is less than the decay rate ε. That concept can be considered as a framework to be used for the determination of the membrane thickness in the design of the biosensors producing a highly stable response to the substrate of concentration S0 , while the enzymatic rate changes from V1 to V2 . In this case, the minimal thickness dε needs to be calculated at concrete characteristics of the biosensor operation: the diffusion coefficients DS , DP , the number of electrons ne , the Michaelis constant KM and the substrate concentration S0 approximate to the expected one [12]. Rather often the concentration of the analyte to be analysed varies within a known interval. Since the biosensor response is usually more stable at lower concentrations of the substrate (Fig. 8) than at higher ones, the larger value of the range of the expected concentrations should be used in the calculation of dε to ensure the stable response in the entire interval of the expected concentrations. In the case when S0  KM , the density I of the steady state current may be calculated analytically from (46); otherwise, the computer simulation is infinitely preferable for the calculation of I (d, Vmax , S0 ), used in the framework and expressed by the formulas (76) and (77). To be sure that the framework, based on the definitions (76) and (77), really helps to find the membrane thickness at which the biosensor gives the relatively stable response, the biosensor response is also calculated in the case of the significantly thinner membrane. Figure 13 shows the density I of the steady state current versus the substrate concentration S0 at the same values of the enzymatic rate Vmax as in Fig. 8; however, the enzyme membrane is more than eight times thinner, d = 10 μm. One can see in Fig. 13 that the biosensor response is very sensitive to changes of Vmax . For example, in the case of S0 = 1 μM, the current density I at Vmax = 1000 (μM/s) is about 4.7 times higher than I at Vmax = 100 (μM/s) (Fig. 8), while the corresponding values of I are approximately the same in the case of the membrane thickness d = dε (100(μM/s), 1000(μM/s), 10(μM) = 92 μm (Fig. 8). Let us notice (Fig. 8) that at d = 10 μm, the relative difference Rd (formula (76)) between I at Vmax = 10 μM/s and another one value of I at Vmax = 1000 μM/s equals approximately 0.59 when S0 = 10 μM. This difference keeps approximately unchanged at all S0 less than about 100 μM. As one more example of the framework application, let us choose the enzyme membrane thickness d to reduce that difference. Accepting V1 = 10 μM/s, V2 = 100V1 = 1000 μM/s, ε = 0.1 and using the definition (77) as well as the results presented in Fig. 6, the value of dε (V1 , V2 , 10 (μM)) ≈ 170 μM was found. Figure 14 plots the density I of

4 Peculiarities of the Biosensor Response

100

I, μA/mm2

10-1

35

4 5

1 2 3

10-2 10-3 10-4 10-5 10-6 10-6

KM 10-5

10-4

10-3

10-2

10-1

S0 , M Fig. 13 The dependence of the density I of the steady state current on the substrate concentration S0 at the enzyme membrane thickness d = 10 μm and five maximal enzymatic rates Vmax : 0.1 (1), 1 (2), 10 (3), 100 (4) and 1000 (5) μM/s

101

I, μA/mm2

100 10-1

1 2 3 4 5

10-2 10-3 10-4 10-5 10-6

KM 10-5

10-4

10-3

10-2

10-1

S0 , M Fig. 14 The dependence of the density I of the steady state current on the concentration S0 of the substrate at d = dε (10 μM/s, 1000 μM/s, 10 μM) = 170 μm, ε = 0.1. The other parameters and the notations are the same as in Fig. 8

36

Introduction to Modeling of Biosensors

the steady state current versus S0 at d = 170 μM at the same values of Vm ax as in Figs. 8 and 13. No notable difference is observed between values of I , calculated at the three values of Vmax , 10, 100 and 1000 μM/s, when the substrate concentration S0 is less than about 5KM = 500 μM. Figure 14 presents the stable response of the biosensor based on the enzyme membrane of the thickness d = 170 μM, when the maximal enzymatic rate reduces 100 times, from 1000 to 10 μM/s, while analysing the substrate of the concentration less than 100 μM [12].

4.6 The Response Gradient with Respect to the Membrane Thickness Figure 6 shows the significant influence of the membrane thickness on the biosensor response. However, the significance of the influence is different at different membrane thicknesses. The normalized gradient BM of the biosensor steady state current I with respect to the membrane thickness d was introduced as a measure of the biosensor resistance (sensitivity) to changes in the membrane thickness, BM =

dI (d) d × , I (d) dd

(78)

where I (d) is the density of the steady state biosensor current calculated at the membrane thickness of d [12, 13]. Since I is in general a non-monotonous function of d (Fig. 6), the BM varies between −1 and 1. The cases when BM is close to 1 or −1 correspond to the biosensors and the response of which is very sensitive to changes in the thickness d of the enzyme membrane. If the response gradient BM is about to zero, then the corresponding biosensors are very resistant to relatively small changes in d. In the case of the first order reaction rate when the substrate concentration S0 is significantly less than the Michaelis constant KM (S0  KM ), the normalized response gradient BM can be calculated analytically from Eqs. (46) and (73) as follows: BM =

σ tanh(σ ) − 1. cosh(σ ) − 1

(79)

One can see in the formula that the gradient BM of the biosensor response does not depend on the substrate concentration. However, (79) is valid for only small substrate concentrations, S0  KM .

4 Peculiarities of the Biosensor Response

37

1.0 0.8 0.6 0.4

BM

0.2 0.0

1 2 3 4

-0.2 -0.4 -0.6 -0.8 -1.0 1

10

100

d, μm Fig. 15 The dependence of the normalized gradient BM of the biosensor response on the thickness d of the enzyme membrane. The parameters and the notations are the same as in Fig. 6. Symbols are the numerically calculated solutions, while curves are analytical solutions (formula (79))

Figure 15 plots the resistance gradient BM versus the membrane thickness d. The substrate concentration S0 and the other parameters are the same as in Fig. 6. In Fig. 15, the symbols are the numerically calculated solutions of the model, while the lines are the analytical ones (formula (79)). One can see in Fig. 15 that the shape of all the curves of the normalized gradient is very similar. The higher maximal enzymatic rate Vmax corresponds to the thicker enzyme membrane at which BM = 0. When the enzyme activity Vmax equals 1 μM/s, the biosensor based on the enzyme membrane of the thickness d ≈ 0.26 mm is mostly resistant to changes in d. For the tenfold greater Vmax , BM falls about zero at d ≈ 0.82 mm. In general, the hundredfold increase in Vmax corresponds to the tenfold thicker enzyme membrane at which BM = 0. From (79), the value of σ at which BM = 0 can be calculated as a solution of the following equation: σ tanh(σ ) − 1 = 0. cosh(σ ) − 1

(80)

Rewriting terms shows that Eq. (80) is equivalent to (74). The biosensors satisfying σ ≈ 1.51 are mostly resistant to changes in the enzyme membrane thickness.

38

Introduction to Modeling of Biosensors

In Fig. 15, the relative difference between the numerical solutions and the analytical ones reaches about 20%. The largest difference is notable at the thickest enzyme membranes. This difference can be explained by the substrate concentration S0 = 0.1KM used in the numerical simulation. The analytical expression (79) is valid for S0  KM .

4.7 Maximal Gradient of the Transient Current When changing the substrate concentration S0 as well as the maximal enzymatic rate Vmax , the normalized maximal gradient BG of the biosensor current stays practically unchanged [12]. In all the numerical experiments, the results of which are depicted in Fig. 8, and the maximal gradient BG varies only between 44.6 and 44.8. The time TG of the maximal gradient equals about 0.08 s in all these calculations. The thickness d of the enzyme layer has a notable effect on the maximal gradient BG . The dependence of BG on the thickness d is presented in Fig. 16. Figure 17 shows the corresponding times of the maximal gradients. Since no noticeable difference was observed when changing Vmax and S0 , both Figs. 16 and 17, present the results only for the fixed values of Vmax and S0 . As one can see in these figures, the maximal gradient GD and the time TG vary nonlinearly in a few orders of magnitude when changing the thickness d. However, the time TG stays very short even for thick enzyme membranes, e.g. TG = 0.37 s in the case of d = 0.5 mm [4, 10]. 110 100 90 80

BG

70 60 50 40 30 20 10 0 1

10

100

d, μm Fig. 16 The normalized maximal gradient BG versus the thickness d of the enzyme membrane at Vmax = 100 μM/s. The other parameters are the same as in Fig. 6

5 Modeling Potentiometric and Optical Biosensors

39

0.35 0.30 0.25

TG

0.20 0.15 0.10 0.05 0.00 1

10

100

d, μm Fig. 17 The time TG of the maximal gradient of the biosensor current versus the thickness d of the enzyme membrane. All the parameters are the same as in Fig. 16

5 Modeling Potentiometric and Optical Biosensors When modeling biosensors based on transducers of the second type, the mathematical model (22)–(27) has to be adjusted by replacing the boundary condition (24) with the non-leakage (zero flux) boundary condition DP

∂P  = 0,  ∂x x=0

t > 0.

(81)

Since the model dependence on the transducer type is slight, the same methods of the analytical and numerical solutions can be applied for solving the corresponding mathematical models at transient and steady state conditions [13, 30, 36, 68, 73, 81].

5.1 Potentiometric Biosensors Potentiometric sensors operate under conditions of near-zero current flow and measure the difference in potential between the working electrode and a reference electrode. The analytical information is obtained by converting the recognition process into the potential, which is proportional (in a logarithmic fashion) to the concentration of the reaction product. Ion-selective electrodes, of which the pH electrode is a well-known example, are the most important of this class of transducers. These devices have been widely used in environmental, medical and industrial applications [39, 48, 75, 81, 82, 88].

40

Introduction to Modeling of Biosensors

Typically, the change of the potential caused by the reaction product concentration change is measured. The potential is proportional in the logarithmic fashion to the concentration of the reaction product at the electrode surface (x = 0) as described by the Nernst equation, E(t) = E0 +

Rc TK ln P (0, t), zF

(82)

where E is the measured potential (in volts), E0 is the characteristic constant for the ion-selective electrode, Rc is the universal gas constant, Rc = 8.314 J/mol K, TK is the absolute temperature (K), z is the signed ionic charge and F is the Faraday constant [22, 53]. The potentiometric system mathematically described by (22)–(23), (26), (27) and (81) approaches a steady state as t → ∞, Ess = lim E(t),

(83)

t →∞

where Ess is assumed to be the steady state biosensor potential. When the substrate concentration S0 to be measured is very small in comparison with the Michaelis constant KM , the biochemical reaction takes place under the first order reaction rate. Accepting the substrate and the product equal diffusivities and a relatively low concentration of the substrate, S0  KM , the product concentration at the electrode surface was found by Carr using the Fourier transformation [30]  1−

1 cosh(σ )   π 2 Dt 4 4 + σ 2 π 2 (1 − exp(−(Vmax /KM )t)) − exp − , π 4d 2 4 + σ 2π 2

P (0, t) = S0

(84)

where D = DS = DP . The analysis of (84) reveals that if enzyme activity is high the exponential expression containing the Vmax term is approaching zero. Therefore, in the case of the high enzyme activity Vmax , Eq. (84) may be significantly reduced [53]  P (0, t) = S0 1−

1 cosh(σ )



 π 2 Dt 4 . − exp − π 4d 2

(85)

Introducing (89) into the Nernst equation (82) produces the potentiometric biosensor response (at 25◦ C): E = E0 + 0.0591 ln P (0, t)   = E0 + 0.0591 ln S0 1 −

1 cosh(αd)

−

 4 π 2 Dt exp − . π 4d 2

(86)

5 Modeling Potentiometric and Optical Biosensors

41

This shows that the dynamics of the biosensor response weekly depends on the enzyme activity. If the diffusion module is σ  1, the biosensor will achieve the difference of 0.1 or 1.0 mV to the steady state response during a time (τ0.1 or τ1.0 ): τ0.1 =

2.35d 2 , D

τ1.0 =

1.42d 2 . D

(87)

As an example, the response of a potentiometric biosensor that is different by 0.1 mV from the steady state will be achieved during 78 s at the membrane thickness of 0.01 cm and D = 3×10−6 cm2 /s. The double increase of the membrane thickness will increase this time up to 313 s. The response time of the real biosensors, with the diffusion module about 1, is 18–26 % longer. Assuming t → ∞, the corresponding product concentration at the steady state conditions can be obtained from formulae (84):  P (0, t) = S0 1 −

1 . cosh(σ )

(88)

Having (88), the steady state potential Ess is easily calculated as Ess = E0 +

  Rc TK 1 ln S0 1 − . zF cosh(σ )

(89)

In the case of the especially high substrate concentration (the zero order reaction rate), S0  KM , the product concentration at the electrode surface was also found analytically by [30, 53] P (0, t) =

Vmax d 2 D



16 1 − 3 2 π



 π 2 Dt exp − . 4d 2

(90)

Assuming t → ∞, the corresponding stationary potential E∞ is calculated as follows (S0  KM ): Ess = E0 +

  2 Vmax d 2 σ KM Rc TK Rc TK ln = E0 + ln . zF 2D zF 2

(91)

5.2 Optical Biosensors Optical biosensors are based on the measurement of absorbed or emitted light resulting from the biochemical reaction [23, 33, 60]. Those devices allow real-time analysis of molecular interactions without labelling requirements [59]. The light absorbance is usually assumed as the response of the optical biosensor. The optical signal arises due to the product absorbance in the enzyme and diffusion

42

Introduction to Modeling of Biosensors

layers. The product molecules that escape from the enzyme and diffusion layers do not contribute to the signal. The absorbance A(t) at time t may be obtained as follows: A(t) = εP lef P¯ ,

lef = d + δ,

(92)

where d and δ are the thicknesses of the enzyme and diffusion layers, respectively, εP is the molar extinction coefficient of the product, P¯ is the concentration of the product averaged through the enzyme and diffusion layers, lef is the effective thickness of the enzyme and diffusion layers [66, 73, 85]. For organic compounds, εP varies between 102 and 104 m2 /mol. The corresponding analytical system approaches a steady state as t → ∞, Ass = lim A(t), t →∞

(93)

where Ass is the steady state absorbance. The response of a peroxidase-based optical biosensor was modeled numerically [9]. The influence of the substrate concentration as well as of the thickness of the enzyme layer on the biosensor response was investigated. Calculations showed complex kinetics of the biosensor response, especially at low concentrations of the peroxidase and of the hydrogen peroxide [9].

5.3 Fluorescence Biosensors The reaction product may be fluorescent, and the fluorescence may be measured [23, 60]. By extension of the Beer–Lambert law, the response of a fluorescence FI (t) at time t can be obtained as an inversely exponential function of the average concentration of the product [5, 51, 85],   ¯ FI (t) = I0 φ 1 − 10−εP lef P ,

(94)

where I0 stands for the intensity of excitation light, ϕ is a quantum yield of fluorophore, εP is the molar extinction coefficient of the product, lef is the effective thickness of the enzyme and diffusion layers and P¯ is the concentration of the product averaged through the enzyme and diffusion layers [37, 42, 85]. The quantum yield of the product fluorescence ϕ, defined as the ratio of the number of photons emitted to the number of photons absorbed, practically varies between 0.001 and 1. When the absorbance, which was expressed as εP lef P¯ , is less than 0.1, the fluorescence FI (t) is almost linearly proportional to the averaged concentration of the product P¯ [85], FI (t) = 2.303I0ϕεP lef P¯ .

(95)

6 Concluding Remarks

43

Since the fluorescence FI (t) is directly proportional to the intensity I0 of the excitation light, the measured fluorescence F (t) is usually reported in the relative fluorescence units (RFU), F (t) = FI (t)/I0 .

(96)

The steady state fluorescence Fss is defined as follows: Fss = lim F (t) . t →∞

(97)

Since the optical absorbance is directly proportional to the concentration of the reaction product (see (92)), the fluorescence can be calculated from the corresponding absorbance. A peroxidase-based fluorescent biosensor was modeled numerically using finite difference technique [42]. It was determined that a thicker enzyme layer leads to higher sensitivity and fluorescence values. The thickness d of the enzyme layer especially affects the sensitivity in cases of high outer substrate and low initial hydrogen peroxide concentrations. The higher sensitivity of the peroxidasebased fluorescent biosensor can be achieved by increasing the concentration of the hydrogen peroxide [42].

6 Concluding Remarks The mathematical model (22)–(27) based on two-component reaction–diffusion equations and the corresponding dimensionless model (29)–(32) of an amperometric biosensor can be successfully used to investigate the kinetic regularities of enzyme membrane-based biosensors. The density of the steady state biosensor current is a non-monotonous function of enzyme membrane thickness d at different values of the maximal enzymatic rate Vmax (Fig. 6). When the initial substrate concentration S0 is significantly less than the Michaelis constant KM (S0  KM ), the function I (d) gains the maximum at the membrane thickness d at which the diffusion module σ approximately equals 1.5055. The steady state current I increases with an increase in the membrane thickness d when the enzyme kinetics predominates in the biosensor response, while I decreases when the response is significantly under diffusion control. The higher maximal enzymatic rate Vmax corresponds to a greater maximum of I (d). In the case of extremely high substrate concentrations (S0  KM ), the steady state biosensor current can be calculated from formula (58), while formula (46) may be used for very low substrate concentrations (S0  KM ). The numerical simulation should be applied for the entire domain of substrate concentration S0 . The mathematical model (22)–(27) together with definitions (76) and (77) describe a framework for selecting the membrane thickness d, ensuring the stable

44

Introduction to Modeling of Biosensors

biosensor response. In case S0  KM , the current I to be used in (76), may be calculated analytically from (46); otherwise, it should be calculated numerically.

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52. Kohen A, PKlinman J (1999) Hydrogen tunneling in biology. Chem Biol 6(7):R191–R198 53. Kulys J (1981) Analytical systems based on immobilized enzymes. Mokslas, Vilnius 54. Kulys J (1981) The development of new analytical systems based on biocatalysts. Anal Lett 14(6):377–397 55. Kulys J (2005) Kinetics of biocatalytical synergistic reactions. Nonlinear Anal Model Control 10(3):223–233 56. 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 57. Kulys J, Razumas V (1986) Bioamperometry. Mokslas, Vilnius ˇ 58. Kulys J., Cenas N (1983) Oxidation of glucose oxidase from Penicillium vitale by one- and two-electron acceptors. Biochim Biophys Acta 744(1):57–63 59. Leatherbarrow RJ, Edwards PR (1999) Analysis of molecular recognition using optical biosensors. Curr Opin Chem Biol 3(5):544–547 60. Ligler F, Taitt C (2002) Optical biosensors: present and future. Elsevier Science, Amsterdam 61. Magner E (1998) Trends in electrochemical biosensors. Analyst 123:1967–1970 62. Malhotra BD, Pandey CM (2017) Biosensors: fundamentals and applications. Smithers Rapra, Shawbury 63. 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 64. 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 65. Mell L, Maloy T (1976) Amperometric response enhancement of the immobilized glucose oxidase enzyme electrode. Anal Chem 48(11): 1597–1601 66. Merino S, Grinfeld M, McKee S (1998) A degenerate reaction diffusion system modeling an optical biosensor. Z Angew Math Phys 49(1):46–85 67. Michaelis L, Menten M (1913) Die kinetik der invertinwirkung. Biochem Z 49 68. 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 69. Nernst W (1904) Theorie der Reaktionsgeschwindigkeit in heterogenen Systemen. Z Phys Chem 47(1):52–55 70. 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 71. Özisik MN (1980) Heat conduction. Wiley, New York 72. 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 73. 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 74. 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 75. Rogers KR (1995) Biosensors for environmental applications. Biosens Bioelectron 10(6– 7):533–541 76. Ruzicka J, Hansen E (1988) Flow injection analysis. Wiley, New York 77. Sadana A, Sadana N (2011) Handbook of biosensors and biosensor kinetics. Elsevier, Amsterdam 78. Scheller FW, Schubert F (1992) Biosensors. Elsevier Science, Amsterdam 79. Schulmeister T (1990) Mathematical modelling of the dynamic behaviour of amperometric enzyme electrodes. Sel Electrode Rev 12(2):203–260 80. Schulmeister T, Scheller F (1985) Mathematical treatment of concentration profiles and anodic current for amperometric enzyme electrodes. Anal Chim Acta 170:279–285

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Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

Contents 1 2

3 4

5

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-layer Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mathematical Model of Multi-layer System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Numerical Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three-Layer Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Compartment Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Solving the Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Simulated Biosensor Responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Dimensionless Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Effect of the Diffusion Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The Nernst Diffusion Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Impact of the Diffusion Module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50 51 51 54 55 58 58 62 67 70 74 77 79 81

Abstract This chapter deals with applying a multi-layer approach to the modeling of biosensors. The multi-layer mathematical models of amperometric biosensors are considered at stationary and transient conditions. First, a multi-layer approach is demonstrated for a biosensor having several mono-enzyme layers sandwichlikely applied onto the electrode surface. Then, a two-compartment model involving an enzyme layer, where the enzyme reaction as well as the mass transport by diffusion take place, and a diffusion limiting region, where only the mass transport by diffusion takes place, is analysed. The dependencies of the internal and external diffusion limitations on the response and sensitivity of amperometric biosensors are investigated, and the conditions when the mass transport outside the enzyme membrane may be neglected to simulate the biosensor response accurately in a well-stirred solution are considered. The mathematical models are based on the

© Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_2

49

50

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

reaction–diffusion equations containing a nonlinear term related to Michaelis– Menten kinetics. The computer simulation was carried out using the finite difference technique. Keywords Biosensor · Response · Sensitivity · Internal/external diffusion

1 Introduction There are various reasons for applying a multi-layer approach to the modeling of biosensors [71]. Multi-layer models are usually used in the following cases [18, 48, 71, 72]: • The bulk solution is assumed to be slightly stirred or non-stirred. This assumption leads to two-compartment models [15, 20, 25, 30, 41, 49, 82]. • The enzyme layer is covered with an inert outer membrane [19, 46, 58, 70, 72]. The membrane stabilizes the enzyme layer and creates a diffusion limitation to the substrate, i.e. lowers the substrate concentration in the enzymatic layer and thereby prolongs the calibration curve of the biosensor [51, 57, 69, 74, 78]. • The electrode is covered with a selective membrane [18, 64, 72]. Selective membranes are usually impermeable to certain molecules and permeable to a desired substance. This arrangement can notably increase the biosensor selectivity. The selective layer can also protect the metal interface of the electrode [11, 23, 34, 52]. • In multi-enzyme systems, enzymes are often immobilized separately in different active layers packed in a sandwich-like multi-layer arrangement [8–10, 44, 60, 76]. This approach seems to be a rather fast and cheap method to design biosensors for different purposes. Different combinations of constructive features lead to different multi-layer models of biosensors [71]. In this chapter, multi-layer mathematical models of amperometric biosensors are considered at stationary and transient conditions. First, a multi-layer approach is demonstrated for a biosensor having several mono-enzyme layers sandwich-likely applied onto the electrode surface [65, 70, 71]. To find reciprocity between the layers, the concentration profiles of the substrate and the reaction product are then analysed in the particular case of a three-layer model. Two-compartment models are among the most widely used multi-layer models of biosensors. These models have been extensively investigated starting from 1970s [23, 25, 41, 49]. In a two-compartment model, the bulk solution region is usually considered as a second layer [71], though some authors have considered models with an infinite bulk solution [25, 41]. In this chapter, a two-compartment model with a finite compartment corresponding to the bulk solution is considered [71]. If the bulk solution is well-stirred and in powerful motion, then the diffusion layer over the enzyme layer remains at a constant thickness. Usually, the more intensive stirring corresponds to the thinner diffusion layer. That diffusion layer is known as the Nernst diffusion layer [33, 53, 62, 80]. Although the outer diffusion layer is often neglected and one-layer models are used [27, 70, 71], in practice, the zero thickness of the Nernst layer cannot be achieved [50, 53].

2 Multi-layer Approach

51

In this chapter, the influence of the thickness of the enzyme membrane as well as the outer diffusion layer on the biosensor response is also analysed. The conditions when the mass transport outside the enzyme membrane may be neglected to simulate the biosensor response accurately in a well-stirred solution are considered. The computer simulation was carried out using the finite difference technique [3, 27, 68].

2 Multi-layer Approach According to a multi-layer approach, a biosensor is considered as an electrode, having several layers each with a single enzyme sandwich-likely applied onto the electrode surface [65, 70, 71]. In each enzyme layer, the substrate (S) combines reversibly with an enzyme (Ek ) to form a complex (ESk ). The complex then dissociates into a product (P) and the enzyme is released: S + Ek  ESk → Ek + P,

k = 1, . . . , K,

(1)

where K is the number of the layers. Assuming the quasi-steady state approximation, the concentration of the intermediate complex (ESk ) does not change and may be neglected when simulating the biochemical behaviour of biosensors [69–71, 78]. Consequently, consider in each enzyme layer a scheme where the substrate (S) binds to the enzyme (Ek ) and is converted to the product (P): Ek

S −→ P,

k = 1, . . . , K.

(2)

2.1 Mathematical Model of Multi-layer System Coupling the enzyme-catalysed reaction in each enzyme layer with the onedimensional-in-space diffusion leads to the following equations (t > 0): (k)

∂S (k) ∂ 2 S (k) Vmax S (k) = DS(k) − , (k) ∂t ∂x 2 KM + S (k) (k)

2 (k) Vmax S (k) ∂P (k) (k) ∂ P = DP + , (k) ∂t ∂x 2 KM + S (k)

x ∈ (ak−1 , ak−1 + dk ),

ak = ak−1 + dk ,

(3)

k = 1, . . . , K,

52

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

where x and t stand for space and time, S (k) (x, t) and P (k) (x, t) are the concentrations of the substrate and of the product in k-th layer, dk is the thickness of the k-th (k) (k) (k) (k) enzyme layer, DS and DP are the diffusion coefficients, Vmax and KM denote the maximal enzymatic rate and the Michaelis constant for the enzyme Ek and ak stands for the interface separating two adjacent layers, a0 = 0. The closed interval [a0 , aK ] (as well as [0, aK ]) covers all the enzyme layers. Let x = a0 = 0 represent the electrode surface, while x = aK represents the boundary between the sandwich-like enzyme membrane and the bulk solution. According to the following initial conditions, the biosensor operation starts when the substrate appears over the surface of the upper layer (t = 0): S (k) (x, 0) = 0,

x ∈ [ak−1 , ak ],

S (K) (x, 0) = 0,

x ∈ [aK−1, aK ),

k = 1, . . . , K − 1, (4)

S (K) (aK , 0) = S0 , where S0 is the concentration of the substrate in the bulk solution. Initially, no product appears in the entire domain: P (k) (x, 0) = 0,

x ∈ [ak−1 , ak ],

k = 1, . . . , K.

(5)

At the electrode surface (x = a0 = 0), the boundary conditions depend on the electric activity of the substances. Accepting amperometry leads to the following conditions at the electrode surface (t > 0): DS(1)

∂S (1)  = 0,  ∂x x=0 P

(1)

(6)

(0, t) = 0.

The problem solutions at two adjacent layers have to be continuous (t > 0) S (k) (ak , t) = S (k+1) (ak , t), S (k) (ak , t) = S (k+1) (ak , t),

k = 1, . . . , K − 1.

(7)

On the boundary between two adjacent layers, the mass conservation relations are additionally defined (k) ∂S

DS

(k) 

∂x

(k) ∂P DP

 

(k+1) ∂S

x=ak

(k) 

∂x

 

x=ak

= DS =

(k+1) 

∂x

(k+1) ∂P DP

 

x=ak

, (8)

(k+1) 

∂x

 

x=ak

,

k = 1, . . . , K − 1.

2 Multi-layer Approach

53

The matching conditions (7) and (8) mean that the fluxes of the substrate and product through the (k + 1)-th layer are equal to the corresponding fluxes entering the k-th layer. The partitions of the substrate and the product in the k-th layer versus the (k + 1)-th layer are assumed to be equal [63, 71]. When the bulk solution is well-stirred outside and in the powerful motion, the concentration of the substrate as well as the product over the top layer enzyme surface remains constant during the biosensor operation (t > 0): S (K) (aK , t) = S0 , P (K) (aK , t) = 0.

(9)

Due to conditions (7), the concentrations of the substrate (S) and of the reaction product (P ) can be defined for the entire interval x ∈ [a0 , aK ] as follows (t ≥ 0): ⎧ ⎪ S (1) (x, t), x ∈ [a0 , a1 ], ⎪ ⎪ ⎪ ⎨S (2) (x, t), x ∈ (a , a ], 1 2 S(x, t) = ⎪. . . ⎪ ⎪ ⎪ ⎩ (K) S (x, t), x ∈ (aK−1 , aK ], ⎧ ⎪ P (1) (x, t), ⎪ ⎪ ⎪ ⎨P (2) (x, t), P (x, t) = ⎪ ... ⎪ ⎪ ⎪ ⎩ (K) P (x, t),

(10)

x ∈ [a0 , a1 ], x ∈ (a1 , a2 ],

(11)

x ∈ (aK−1 , aK ].

Both concentration functions (S and P ) are now continuous in the entire domain x ∈ [a0 , aK ]. The biosensor current depends upon the flux of the product at the electrode surface: i(t) = ne F DP(1)

∂P (1)   , ∂x x=0

(12)

where i(t) is the density of the anodic or cathodic current at time t, ne is the number of electrons involved in a charge transfer at the electrode surface and F is the Faraday constant. The system (3)–(9) approaches a steady state as t → ∞: I = lim i(t), t →∞

where I is the density of the steady state biosensor current.

(13)

54

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

2.2 Numerical Approximation Equations (3)–(9) form K initial boundary value problems formulated on each layer. Due to the matching conditions (8), these problems cannot be solved independently of each other [70, 71]. (k) In the case of the first order rate of the enzymatic reactions (S0  KM for ∀k : k = 1, . . . , K), the problems (3)–(9) can be solved analytically. Schulmeister solved this problem by using polygon approximations of the boundary conditions (8) [70]. To find a numerical solution of the nonlinear problem (3)–(9) in the domain [a0 , aK ] × [0, T ], a discrete grid has to be introduced. Since the thicknesses of the enzyme layers may differ even in the orders of magnitude [83], it is reasonable to use an individual step size for each layer. Assuming the step size hk constant through k-th layer, each layer can be uniformly discretized, k = 1, . . . , K:   (k) (k) (k) ωh = xi : xi = ak−1 + ihk , i = 0, 1, . . . , Nk , hk Nk = dk , (14) ωτ = {tj : tj = j τ, j = 0, 1, . . . , M; τ M = T }, where T is the time of the biosensor operation. Let us assume the following notation:   (k),j (k) = S (k) xi , τj , Si   (k),j Pi = P (k) xi(k) , τj ,

(15)

j = 0, . . . , T ,

i = 0, . . . , Nk ,

k = 1, . . . , K.

The governing equations (3), the initial conditions (4), (5) and the boundary conditions (6)–(9) can be approximated using finite differences. Both types of the finite difference schemes, explicit and implicit, are applicable [3, 27, 28, 68]. The matching conditions (8) are approximated as follows [68, 71]: (k),j (k) SNk DS

− SNk −1

(k),j

(k),j (k) PNk DP

− PNk −1

hk

(k+1),j

= DS(k+1)

(k),j

hk

=

S1

(k+1),j

− S0

,

hk+1

(k+1),j (k+1) P1 DP

(k+1),j

− P0

hk+1

(16) ,

k = 1, . . . , K − 1. A union of ωh(1) , . . . , ωh(K) forms a discretization ωh for the entire interval [a0 , aK ]: ωh =

K  k=1

(k)

ωh .

(17)

3 Three-Layer Model

55

The ordered set ωh contains N = (N1 × N2 × · · · × NK ) + 1 points. Let x0 , x1 , . . . , xN be an ordered sequence of all points consisting of ωh , x0 = a0 = 0. In the discrete presentation ωh of the interval [a0 , aK ], K − 1 points correspond to the interfaces separating adjacent media. Let bound(k) be an index of a point in ωh corresponding to a boundary between k-th and (k + 1)-th enzyme layers, k = 1, . . . , K − 1: bound(k) =

k 

Nk .

(18)

i=1

The following relation maps the concentrations of the substrate and the product for each layer on the entire domain (j = 1, . . . , M):   ⎧ (1) x (1) , τj , ⎪ S 0 ≤ i ≤ bound(1), ⎪ ⎪  i  ⎪ ⎪ (2) ⎨S (2) x bound(1) < i ≤ bound(2), j i−bound(1), τj , Si = ⎪ ⎪ ...  ⎪  ⎪ ⎪ ⎩S (K) x (K) , τj , bound(K − 1) < i ≤ bound(K), i−bound(K−1)   ⎧ (1) ⎪ P (1) xi , τj , ⎪ ⎪   ⎪ ⎪ ⎨P (2) x (2) , τj , j i−bound(1) Pi = ⎪. . . ⎪ ⎪  ⎪ ⎪ (K)  (K) ⎩ xi−bound(K−1), τj , P

(19)

0 ≤ i ≤ bound(1), bound(1) < i ≤ bound(2),

(20)

bound(K − 1) < i ≤ bound(K).

Using (19) and (20), the difference equations can be solved to find approximate values of the concentrations S(x, t) and P (x, t) (see descriptions (10) and (11)) in the domain [a0 , aK ] × [0, T ]. Having the numerical solution of the problem, the density of the biosensor current at time t = tj is calculated by j

i(tj ) ≈ ij = ne F DP(1) P1 / h1 ,

j = 1, . . . , M.

(21)

3 Three-Layer Model To find reciprocity between the layers, the multi-layer approach was adopted to three-layer model, and the biosensor response was simulated at different maximal enzymatic rates for different layers. Figures 1 and 2 show the steady state concentration profiles of the substrate S and the product P assuming the same substrate concentration S0 in the bulk. All three enzyme layers were of the same thickness,

56

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors 100 90

5

80

S, μM

70

4

60 50 40 30

3

20

2

10

1

0 0

a)

50

100

150

200

250

300

x, μm 50

2 1

P, μM

40

30

3

4

20

10

5

0 0

50

100

150

200

250

300

x, μm

b)

Fig. 1 Steady state concentration profiles of the substrate (a) and the product (b) in three enzyme (1) (2) layers at the following maximal enzymatic rates: Vmax : 10 (1, 3, 4), 1 (2, 5), Vmax : 10 (1, 2, 4), 1 (3) (3, 5) and Vmax : 10 (1–3), 1 (4, 5) µM/s. The other parameters are as defined in (22). Dashed lines show boundaries between adjacent layers

and the diffusion coefficients as well as the Michaelis constants were also identical for all three layers: (k)

(k)

DS = DP = 300 µm2/s, k = 1, 2, 3,

(k)

KM = 100 µM,

S0 = 100 µM.

dk = 100 µm,

(22)

3 Three-Layer Model

57

100 99

5

98

2

S, μM

97 96 95

3

94

4

93 92

1

91 90 0

50

100

150

a)

200

250

300

200

250

300

x, μm 2

1

P, μM

2

4 1

3 5 0 0

b)

50

100

150

x, μm

Fig. 2 Steady state concentration profiles of the substrate (a) and the product (b) in three enzyme (1) (2) layers at the following maximal enzymatic rates: Vmax : 0.1 (1, 3, 4), 0.01 (2, 5), Vmax : 0.1 (1, 2, (3) 4), 0.01 (3, 5) and Vmax : 0.1 (1–3), 0.01 (4, 5) µM/s. The other parameters are as defined in (22). Dashed lines show boundaries between adjacent layers

Figure 1 shows the steady state concentration profiles in the case when a threelayer biosensor acts under limitation of the diffusion (diffusion module σ 2 > 1), while Fig. 2 corresponds to the limitation of the enzyme kinetics (σ 2 < 1). When comparing the shape of corresponding curves in the figures, one can see different positions of peaks.

58

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

The shape of curves 1–4 in Fig. 1b differs noticeably; however, the gradients of the product concentrations at the electrode surface (x = 0) are comparable. The density of the biosensor current in these four cases differs less than 5%. One can observe a very similar situation in Fig. 2b when comparing curves 1 and 4. In both Figs. 1b and 2b, curves 1 and 4 correspond to different enzymatic activity in the distal (from the electrode) layer and the same enzymatic activities in other two layers. Consequently, the decrease in enzyme activity in a farther layer more considerably affects the biosensor response than in the nearest layer. In Figs. 1 and 2, curves 1 and 5 show the concentration profiles for biosensors with three practically identical layers, i.e. in all three layers even the maximal enzymatic rates are the same. In such cases of identical layers, a multi-layer biosensor becomes practically identical to a mono-layer biosensor with the thickness equal to the sum of the thicknesses of all the layers of the corresponding multi-layer biosensor. This feature of models can be applied to testing the numerical solution of multi-layer model by comparing the solution with the solution of the corresponding mono-layer model (see Chapter “Introduction to Modeling of Biosensors”).

4 Two-Compartment Model An amperometric biosensor to be modeled is considered as an electrode and a relatively thin layer of an enzyme (enzyme membrane) applied onto the electrode surface [12, 38, 69, 78]. In the enzyme layer, we consider the enzyme-catalysed reaction k2 k1 E + S FGGGGGB GGGGG ESGGGA E + P, k−1

(23)

where the substrate (S) combines reversibly with an enzyme (E) to form a complex (ES). The complex then dissociates into the product (P), and the enzyme is regenerated [39, 69]. Assuming the quasi-steady state approximation, the concentration of the intermediate complex (ES) does not change and may be neglected when modeling the biochemical behaviour of biosensors [32, 35, 54, 73]. In the resulting scheme, the substrate (S) is enzymatically converted to the product (P): E

S −→ P.

(24)

4.1 Mathematical Model Assuming the symmetrical geometry of the electrode and the homogeneous distribution of the immobilized enzyme in the enzyme membrane, the dynamics of the

4 Two-Compartment Model

59

concentrations of the substrate as well as the product in the enzyme layer can be described in a one-dimensional-in-space domain [13, 15, 71].

4.1.1 Governing Equations Coupling the enzyme-catalysed reaction (24) in the enzyme layer with the onedimensional-in-space diffusion, described by the Fick law, leads to the following equations of the reaction–diffusion type: ∂Se ∂ 2 Se Vmax Se = DSe − , 2 ∂t ∂x KM + Se ∂Pe ∂ 2 Pe Vmax Se = DPe + , ∂t ∂x 2 KM + Se

(25) x ∈ (0, d),

t > 0,

where x and t stand for space and time, respectively, Se (x, t) and Pe (x, t) are the concentrations of the substrate and the reaction product in the enzyme layer, respectively, d is the thickness of the enzyme membrane, DSe and DPe are the diffusion coefficients, Vmax = k2 E0 is the maximal enzymatic rate, E0 is the total enzyme concentration and KM is the Michaelis constant, KM = (k−1 + k2 )/k1 [4, 22, 29, 39]. In the second layer, only the mass transport by diffusion takes place: ∂Sb ∂ 2 Sb , = DSb ∂t ∂x 2 ∂Pb ∂ 2 Pb = DPb , ∂t ∂x 2

(26) x ∈ (d, d + δ),

t > 0,

where δ is the thickness of the external diffusion layer, Sb (x, t) and Pb (x, t) are the concentrations of the substrate and the reaction product in the diffusion layer, respectively, and DSb and DPb are the diffusion coefficients of the species in the bulk solution.

4.1.2 Initial and Boundary Conditions Let x = 0 represent the electrode surface, while x = d represents the boundary layer between the enzyme membrane and the bulk solution. The biosensor operation starts when some substrate appears in the bulk solution (t = 0): Se (x, 0) = 0,

Pe (x, 0) = 0,

x ∈ [0, d],

Sb (x, 0) = 0,

Pb (x, 0) = 0,

x ∈ [d, d + δ),

Sb (d + δ, 0) = S0 ,

Pb (d + δ, 0) = 0,

where S0 is the concentration of the substrate in the bulk solution.

(27)

60

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

On the boundary between two subregions having different diffusivities, the matching conditions are defined (t > 0): ∂Se  ∂Sb  = DSb , Se (d, t) = Sb (d, t), DSe   ∂x x=d ∂x x=d (28) ∂Pe  ∂Pb  DPe = DPb , Pe (d, t) = Pb (d, t).   ∂x x=d ∂x x=d The external diffusion layer of thickness δ remains unchanged with time. Away from it, the concentration of the substrate as well as of the product remains constant (t > 0): Sb (d + δ, t) = S0 ,

Pb (d + δ, t) = 0.

(29)

At the electrode surface (x = 0), due to the amperometry, the potential is chosen to keep zero concentration of the reaction product: Pe (0, t) = 0,

DSe

∂Se  = 0,  ∂x x=0

t > 0.

(30)

4.1.3 Biosensor Response The system (25)–(30) approaches a steady state as t → ∞. The density i(t) of the biosensor current at time t can be obtained explicitly from the Faraday and the Fick laws [71]: i(t) = ne F DPe

∂Pe   , ∂x x=0

I = lim i(t), t →∞

(31)

where ne is the number of electrons involved in a charge transfer at the electrode surface, F is the Faraday constant and I is the density of the steady state biosensor current. The sensitivity is another very important characteristic of the biosensors [67, 69, 78]: BS (S0 ) =

dI (S0 ) S0 , × dS0 I (S0 )

(32)

where BS (S0 ) is the dimensionless sensitivity and I (S0 ) is the density of the steady state output current calculated at the substrate concentration S0 .

4 Two-Compartment Model

61

4.1.4 Model Modifications For δ → 0, the solution of the two-compartment model approaches the solution of the corresponding one-layer model discussed in Chapter “Introduction to Modeling of Biosensors”. The two-compartment mathematical model (25)–(30) of an amperometric biosensor would become the two-compartment model of the corresponding potentiometric biosensor if the boundary condition Pe (0, t) = 0 was replaced with the following boundary condition: DPe

 ∂Pe  = 0, ∂x x=0

t > 0.

(33)

The initial condition (27) indicates that biosensor operation starts at the substrate of concentration S0 approach the distal boundary of the diffusion layer, i.e. the substrate enters to diffusion layer. These conditions model the infusion of the substrate into the bulk solution. Sometimes, a biosensor is instantly immersed into the bulk solution containing the substrate of concentration S0 [15]. In this case, initial conditions (27) have to be replaced with Se (x, 0) = 0,

x ∈ [0, d),

Se (d, 0) = S0 ,

Pe (x, 0) = 0,

x ∈ [0, d],

Sb (x, 0) = S0 ,

Pb (x, 0) = 0,

(34) x ∈ [d, d + δ].

Let us notice that both initial conditions, (27) and (34), lead to the same steady state solution [4, 27]. Only a transient solution is affected by the initial conditions. The thickness δ of the external diffusion layer depends upon the nature and intensity of the stirring of the buffer solution. The less intense stirring corresponds to the thicker diffusion layer. A non-perfect stirring is usually modeled by a relatively thick external diffusion layer. To model a biosensor action in a non-stirred buffer solution, some authors use a non-leakage (zero flux) boundary condition on the boundary x = d + δ:  ∂Sb  DSb = 0, ∂x x=d+δ

 ∂Pb  DPb = 0, ∂x x=d+δ

t > 0,

(35)

instead of conditions (29), and the initial condition (34) instead of (27) [14, 49, 71]. On the other hand, the diffusion layer can be treated as an inert membrane that is often used to protect the enzyme layer [11, 21, 23, 34]. This membrane can be a polymer membrane possessing electrical charge or neutral. In some cases, the role of this membrane can play thin layer of the unmixed solvent on the surface of the enzyme layer. This outer membrane can be characterized by thickness δ and the diffusion coefficients for the substrate (DSb ) and the product (DPb ) [47, 58, 59].

62

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

The outer membrane can be impermeable for the product. In this case, the product is trapped inside the biosensor, while the substrate diffuses. Such a situation can happen when both the product and the outer membrane are charged [51]. In this case, instead of conditions (29), the following boundary conditions should be used on the boundary x = d + δ [58, 59]: Sb (d + δ, t) = S0 ,

DPb

 ∂Pb  = 0, ∂x x=d+δ

t > 0.

(36)

Rather often mathematical models of biosensors involve both the outer membrane and the diffusion layer [1, 6, 79]. However, the mass transport through several diffusion layers can be rather efficiently approximated by a single diffusion layer with effective diffusion coefficients [43], and a multi-compartment model can be reduced to a two-compartment model at least at the steady state conditions [7]. Therefore, the effects presented here can be applied also to amperometric biosensors modeled by several diffusion layers, including outer membranes. In addition, the substrate and the product degradation can be taken into consideration [36, 47, 58, 59]. When the rate of the substrate and the product degradation is expressed by a first order reaction with rate constants C1 and C2 , respectively, the first governing equation of the system (25) is supplemented by a term −C1 Se and the second equation by −C2 Pe . The governing equations (26) are supplemented by terms −C1 Sb and −C2 Pb , respectively [58, 59].

4.2 Solving the Mathematical Model The nonlinear initial boundary value problem (25)–(30) can be analytically solved only for specific values of the model parameters [5, 63, 71].

4.2.1 First Order Steady State Solution At so low concentrations of the substrate as S0  KM , the nonlinear reaction rate in (25) reduces to the first order reaction rate Vmax Se /KM . At these conditions, the two-compartment model can be solved analytically. The stationary concentrations of the substrate and the product in both layers have been presented by Schulmeister [71]: Se ss (x) = lim Se (x, t) = S0

DSb cosh(σ x/d) , DSb cosh(σ ) + DSe σ (δ/d) sinh(σ )

Sb ss (x) = lim Sb (x, t) = S0

DSb cosh(σ ) + DSe σ ((x − d)/d) sinh(σ ) , DSb cosh(σ ) + DSe σ (δ/d) sinh(σ )

t →∞

t →∞

(37)

4 Two-Compartment Model

63

 DSb − DSe σ sinh(σ )/ cosh(σ ) S0 d +δ Pe ss (x) = lim Pe (x, t) = t →∞ d +δ DSb + DSe σ (δ/d) sinh(σ )/ cosh(σ )

  σ DSe δ sinh(σ ) DSe DPb 1 x × + 1− d cosh(σ ) DPe cosh(σ ) DPb d + DPe δ

DSe 1 − cosh(σ x/d)) , + DPe cosh(σ )  DSb − DSe σ sinh(σ )/ cosh(σ ) S0 Pb ss (x) = lim Pb (x, t) = d+δ t →∞ d +δ DSb + DSe σ (δ/d) sinh(σ )/ cosh(σ )

 

  σ DSe sinh(σ ) 1 × + DSe 1 − DPb d + DPe δ cosh(σ ) cosh(σ ) × (d + δ − x), (38) where σ 2 is the dimensionless diffusion module or the Damköhler number, σ2 =

Vmax d 2 . KM DSe

(39)

The density of the steady state current reaches [71]  DSb − σ DSe sinh(σ )/ cosh(σ ) DPe d+δ d +δ DSb + σ (δ/d)DSe sinh(σ )/ cosh(σ )     σ DSe δ sinh(σ ) DSe DPb 1 + × 1− DPb d + DPe δ . d cosh(σ ) DPe cosh(σ ) (40)

I = ne F S0

4.2.2 Zero Order Steady State Solution When the substrate concentration S0 to be measured is very high in comparison with the Michaelis constant KM (S0  KM ), the reaction term in (25) reduces to the zero order reaction rate Vmax . Then, the linear initial boundary value problem (25)–(30) can be solved easily [5]. The stationary concentrations of the substrate and the product in both layers are as follows: 

dδ d2 − x2 + Sess (x) = lim Se (x, t) = S0 − Vmax t →∞ 2DSe DSb  d +δ−x Sb ss (x) = lim Sb (x, t) = S0 − Vmax d , t →∞ DSb

, (41)

64

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

DPb d + 2DPe δ ×d −x , DPb d + DPe δ  Vmax d 2 d +δ−x Pbss (x) = lim Pb (x, t) = . t →∞ 2 DPb d + DPe δ Vmax x Pe ss (x) = lim Pe (x, t) = t →∞ 2DPe



(42)

Having analytical expression of the product concentration Pe in the enzyme layer, the density of the steady state current is expressed as follows: Vmax d I = ne F 2



DPb d + 2DPe δ DPb d + DPe δ

.

(43)

As one can see from (43), the biosensor response is invariant to the substrate concentration S0 to be determined. So, the enzyme biosensors are not suitable for determination of high concentrations (S0  KM ) of the substrate.

4.2.3 Steady State Solution at External Diffusion Limitations The modeling of the biosensors action at an external diffusion limitation is easer due to the linear gradient of the substrates concentration in a stagnant layer. The analysis of such systems, however, did not receive a lot of attention since the internal diffusion problems are intrinsic for the catalytical biosensors. For a biosensor acting at the external diffusion limitation and at the steady state conditions, the flux of the substrate through a stagnant layer is equal to the enzyme reaction rate on the surface of the transducer: DSb

Vmaxs Ss S0 − Ss = , δ KM + Ss

(44)

where Ss is the substrate concentration at the boundary of the enzyme membrane and stagnant solution, Vmaxs corresponds to the maximal enzyme rate on the surface expressed as mol/cm2s. The solution of this equation is Ss =

1 1 1 S0 − KM − ρKM 2 2 2  1 2 + 2ρK 2 + ρ 2 K 2 , + S 2 + 2S0 KM − 2S0 ρKM + KM M M 2 0

(45)

where ρ=

Vmaxs δ . KM DSb

(46)

4 Two-Compartment Model

65

The biosensor response is I = ne F

Vmaxs Ss , KM + Ss

(47)

where Ss comes from (45). At S0  KM , the expression for the response is much simpler: I = ne F

Vmaxs S0 . KM (1 + ρ)

(48)

At this substrate concentration, the sensitivity of the biosensor depends on the value of the diffusion module. At ρ  1, the sensitivity depends on the kinetic parameters of an enzyme: I = ne F

Vmaxs S0 . KM

(49)

At ρ  1, the biosensor acts in the diffusion controlled regime, and the sensitivity is determined by the diffusion parameter and the thickness of a stagnant layer: I = ne F

S0 DSb . δ

(50)

In this case, the temperature inactivation of an enzyme or the other factors influencing the enzyme activity no longer perturb the sensitivity of the biosensor.

4.2.4 Transient Numerical Solution At the transient conditions, the problems (25)–(30) can be solved numerically using the finite difference technique [27, 28]. To simulate the biosensor action, a discrete grid has to be introduced. In a common case, different step sizes can be used for different compartments. For simplicity, a constant step size in the entire domain can be applied. To find a finite difference solution of the problem in the domain [0, d + δ] × [0, T ], a uniform discrete grid ωh × ωτ is introduced: ωh = {xi : xi = ih, i = 0, . . . , Nd , . . . , N; hNd = d, hN = d + δ}, ωτ = {tj : tj = j τ, j = 0, . . . , M; τ M = T }.

(51)

66

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

As in a general multi-layer model introduced in (10) and (11), the concentrations of both species, S and P , can be defined in the entire domain x ∈ [0, d + δ] as follows (t ≥ 0): S(x, t) =

P (x, t) =

 Se (x, t),

x ∈ [0, d],

Sb (x, t), x ∈ (d, d + δ],  Pe (x, t), x ∈ [0, d],

(52)

Pb (x, t), x ∈ (d, d + δ].

Both concentration functions, S and P , are continuous in the entire domain x ∈ [0, d + δ]. The concentrations of both species are defined on the discrete grid (51): j

Si = S(xi , tj ), i = 0, . . . , N,

j

Pi = P (xi , tj ),

ij = i(tj ),

(53)

j = 0, . . . , M.

To find approximate values of S and P on ωh × ωτ , a semi-implicit finite difference scheme can be used by replacing the differential equations (25) and (25) with the following difference equations: j +1

j +1

j +1

j +1

Pi

j +1

j S − 2Si − Si = DSe i+1 τ h2

Si

j +1

j P − 2Pi − Pi = DPe i+1 τ h2

i = 1, . . . , Nd − 1, j +1

j +1

j +1

j +1

j

−

Vmax Si

j

KM + Si

j +1

+ Pi−1

,

j

+

Vmax Si

j KM + Si

(54) ,

j = 1, . . . , M, j +1

j S − 2Si − Si = DSb i+1 τ h2

Si Pi

j +1

+ Si−1

j +1

+ Si−1

j +1

j P − 2Pi − Pi = DPb i+1 τ h2

,

j +1

+ Pi−1

i = Nd + 1, . . . , N − 1,

(55) ,

j = 1, . . . , M.

The initial conditions (27) are approximated by Si0 = 0,

i = 0, . . . , N − 1,

0 SN = S0 ,

Pi0 = 0,

(56) i = 0, . . . , N.

4 Two-Compartment Model

67

The matching (28) and the boundary conditions (29) and (30) are approximated as follows: j

DSe

j

SNd − SNd −1 h j

DPe

= DSb

j

PNd − PNd −1 h j SN j

DSe

j

j

SNd +1 − SNd h j

= DPb = S0 ,

j

PNd +1 − PNd h j PN

, ,

(57)

= 0,

j

S1 − S0 = 0, h

j

P0 = 0,

j = 1, . . . , M.

The resulting systems of linear algebraic equations are solved efficiently because of the tridiagonality of their matrices [24, 27]. Having a numerical solution of the problem, the density of the biosensor current at time t = tj can be easily calculated by  j j ij ≈ i(tj ) = ne F DSe P1 − P0 / h,

j = 0, . . . , M.

(58)

In the case of momentary immersion of a biosensor into a bulk solution, the initial conditions (27) are approximated by Si0 = 0,

i = 0, . . . , Nd − 1,

Si0 = S0 ,

i = Nd , . . . , N,

Pi0 = 0,

i = 0, . . . , N.

(59)

In a numerical simulation, the biosensor response time T and the half-time T0.5 of the steady state can be calculated as in the case of single-layer model (see Chapter “Introduction to Modeling of Biosensors”). The exact analytical solutions (37)–(43) of the reaction–diffusion problem (25)– (30) were applied to validate the numerical solution.

4.3 Simulated Biosensor Responses Figure 3 shows the concentration profiles of the substrate S and the product P calculated from the two-compartment model (25)–(30) at the following values of

68

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors 1.0 0.9 0.8

S / S0

0.7 0.6

6

0.5

3

5

0.4

4

0.3

2

0.2

1

0.1 0.0

a)

0

50

100

150

200

x, μm

0.20 0.18

6

0.16 0.14

P / S0

5

0.12

4

0.10 0.08

3

0.06

2

0.04 0.02

1

0.00

b)

0

50

100

150

200

x, μm

Fig. 3 Concentration profiles of the substrate (a) and the product (b) in two compartments obtained for t = 6.8 (1), 8.4 (2), 16 (3), 18.3 (4), 27.8 (5) and 124 (6) s. The profiles are normalized with the bulk concentration S0 . The model parameters are as defined in (60). Dashed lines show boundaries between adjacent layers

the model parameters: DSe = DPe = 300 µm2/s, KM = S0 = 100 µM,

DSb = DPb = 600 µm2/s,

Vmax = 10 µM/s.

d = δ = 100 µm.

(60)

4 Two-Compartment Model

69

In Fig. 3, curves 6 show the concentration profiles at the steady state conditions. Curves 4 correspond to the concentrations at time T0.5 of a half of the steady state. The other curves show the concentrations at the intermediate values of time: T0.05 = 6.8 (curve 1), T0.1 = 8.4 (2), T0.25 = 16 (3), T0.75 = 27.8 (5) s. It should be borne in mind that Tα is the time needed to reach α% of the stationary current (see Chapter “Introduction to Modeling of Biosensors” for details). One can see in Fig. 3 linear curves 6 at x ∈ [d, d +δ]. At the steady state conditions, the concentrations approach straight line because of linearity of governing equations (26). The evolution of the density i(t) of the biosensor current accepting different initial conditions is presented in Fig. 4. Figure 4a shows the dynamics of the biosensor current in the case of infusion modeled by initial conditions (27), while Fig. 4b corresponds to the immersion modeled by initial conditions (34). For both modes of the biosensor operation, the response was modeled for biosensors at different thickness δ of the external diffusion layer, keeping the enzyme membrane thickness constant. Although the shape of curves in Fig. 4a notably differs from these in Fig. 4b, the corresponding steady state currents are identical. In the case of the infusion (Fig. 4a), i(t) is a monotonous increasing function for all six thicknesses of the external diffusion layer. In the case of the immersion (Fig. 4b), i(t) is a non-monotonous function for relatively thick external diffusion layers (curves 3–6). As has been mentioned above, the distinction in the initial conditions (27) and (34) makes no influence on the steady state response. Both curves marked by 1 are identical because both of them correspond to zero thickness δ of the diffusion layer. In cases of the non-monotonous density i(t) of the biosensor current (curves 3–6 in Fig. 4b), the maximal current is notably less (about 23%) than the steady state (maximal) current when the diffusion layer is neglected, δ = 0 (curve 1). No notable difference is observed between the maximal currents in all cases of nonmonotonous current density i(t) (curves 3–6). Similar effect was also observed in the case of amperometric biosensors acting with a substrate cyclic conversion [17]. As one can see in Fig. 4, an increase in the thickness δ of the external diffusion layer notably decreases the steady state current and prolongs the response time. At the thickness δ = 5d = 500 µm (curve 6), the density of the steady state current (I = 15.4 nA/mm2) is more than two times less than the density of the steady state current (I = 32.7 nA/mm2) at zero thickness of the diffusion layer. The half (T0.5 ) of these two steady state responses were achieved at 113 and 7.7 s, respectively. However, additional numerical experiments showed that in some cases an increase in the thickness δ can even increase the steady state current. The effect of the external diffusion layer on the biosensor response is investigated in the following sections in detail.

70

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors 35

1

30

i, nA/mm2

2 25

3

20

4 5

15 10

6 5 0 0

a)

50

100

150

200

250

300

t, s 35

1

30

i, nA/mm2

2

3

25

4 20

5

15

6

10 5 0

b)

0

50

100

150

200

250

300

t, s

Fig. 4 The dynamics of the biosensor current at six values of the thickness δ: 0 (1), 100 (2), 200 (3), 300 (4), 400 (5) and 500 (6) µm in the cases of infusion (a, initial conditions (27)) and immersion (b, (34)). The other parameters are the same as in Fig. 3

4.4 Dimensionless Model Each additional layer to be considered in the modeling of the biosensor action includes some additional parameters into a mathematical model. The twocompartment model (25)–(30) in comparison with the corresponding one-layer model contains three additional parameters: the thickness δ of the diffusion layer as well as the diffusion coefficients DSb and DPb . Increasing the number of the

4 Two-Compartment Model

71

model parameters complicates the investigation of the behaviour of the biosensor response. Because of this, it is very important to find the main governing parameters of the mathematical model. Introducing dimensionless parameters is a widely used technique to reduce the number of model parameters [2, 31, 33, 44, 55, 66]. In order to define the main governing parameters of the two-compartment model (25)–(30), the dimensional variables (x and t) and unknown concentrations (Se , Pe , Sb and Pb ) are replaced with the following dimensionless parameters: DS t x , tˆ = 2e , d d S Pe e , Pˆe = , Sˆe = KM KM xˆ =

Sb Sˆb = , KM

Pb Pˆb = , KM

(61)

where xˆ is the dimensionless distance from the electrode surface, tˆ stands for the dimensionless time and Sˆe , Pˆe , Sˆb and Pˆb are the dimensionless concentrations. Having defined dimensionless variables and unknowns, the following dimensionless parameters characterize the domain geometry and the substrate concentration in the bulk: δˆ =

δ , d

S0 Sˆ0 = , KM

T DSe Tˆ = , d2

T0.5 DSe Tˆ0.5 = , d2

(62)

where δˆ is the dimensionless thickness of the external diffusion layer, Sˆ0 is the dimensionless substrate concentration in the bulk solution, Tˆ is the dimensionless response time and Tˆ0.5 is the half-time of the steady state. The dimensionless thickness of enzyme membrane equals one. The governing equations (25) in the dimensionless coordinates are expressed as follows: ∂ 2 Sˆe ∂ Sˆe Sˆe = − σ2 , 2 ∂ xˆ ∂ tˆ 1 + Sˆe ˆ ∂ Pˆe DSe ∂ 2 Pˆe 2 Se + σ , = DPe ∂ xˆ 2 ∂ tˆ 1 + Sˆe

(63) xˆ ∈ (0, 1),

tˆ > 0,

where σ 2 is the dimensionless diffusion module as defined in (39). The governing equations (26) take the following form: ∂ Sˆb DSb ∂ 2 Sˆb , = DSe ∂ xˆ 2 ∂ tˆ ∂ Pˆb DPb ∂ 2 Pˆb , = DSe ∂ xˆ 2 ∂ tˆ

(64) ˆ xˆ ∈ (1, 1 + δ),

tˆ > 0.

72

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

The initial conditions (27) transform to the following conditions: ˆ 0) = 0, Sˆe (x,

Pˆe (x, ˆ 0) = 0,

xˆ ∈ [0, 1],

Sˆb (x, ˆ 0) = 0,

Pˆb (x, ˆ 0) = 0,

ˆ xˆ ∈ [1, 1 + δ),

ˆ 0) = Sˆ0 , Sˆb (1 + δ,

(65)

ˆ 0) = 0. Pˆb (1 + δ,

The matching (28) and the boundary (29) and (30) conditions are rewritten as follows (tˆ > 0): ∂ Sˆe  =  ˆ ∂ xˆ x=1 ∂ Pˆe  =  ˆ ∂ xˆ x=1

DSb ∂ Sˆb   , Sˆe (1, tˆ) = Sˆb (1, tˆ), ˆ DSe ∂ xˆ x=1 DPb ∂ Pˆb   , Pˆe (1, tˆ) = Pˆb (1, tˆ), ˆ DSe ∂ xˆ x=1

ˆ t) = Sˆ0 , Sˆb (1 + δ, Pˆe (0, tˆ) = 0,

(66)

ˆ tˆ) = 0, Pˆb (1 + δ, ∂ Sˆe  = 0.  ˆ ∂ xˆ x=0

(67)

The dimensionless current (flux) iˆ and the corresponding dimensionless stationary current Iˆ are defined as follows: ˆ  i(t) d ˆ tˆ) = ∂ Pe  i( = , ˆ ∂ xˆ x=0 ne F DSe KM

(68)

ˆ tˆ). Iˆ = lim i( tˆ→∞

Assuming the same diffusion coefficients for both species (the substrate and the product), only the following dimensionless parameters remain in the dimensionless ˆ mathematical model (63)–(68): δ—the thickness of the diffusion layer, Sˆ0 —the substrate concentration in the bulk solution, σ 2 —the diffusion module, and Drel — the ratio of the external diffusivity to the internal diffusivity and Drel = DSb /DSe = DPb /DPe . In all the calculations, Drel was equal to 2 as defined in (60). The advent of the diffusion module σ 2 is one of the most important outcomes of defining a dimensionless model of the biosensor action. The diffusion module is the main parameter expressing all internal characteristics of the biosensor. The Biot number is another widely used dimensionless parameter that compares the relative transport resistances, external and internal [4, 12, 40, 42, 56, 81]. Since the mass transport capacity is different for the substrate and product, we introduce

4 Two-Compartment Model

73

two Biot numbers: βS =

DSb /δ DSb d = , DSe /d DSe δ

DPb /δ DPb d = , DPe /d DPe δ

βP =

(69)

where βS and βP are the Biot numbers for the substrate and product, respectively. Large values of Biot number mean that the internal diffusion is slow compared with the external diffusion. This case when a biosensor operates under internal diffusion control is rather typical. The case of relatively low Biot number values implies much slower diffusion within the outer membrane compared with the diffusion within the enzyme layer, which leads to the external diffusion control case [12]. Taking into account definition (69), the formula (40) of the density of the steady state current can be notably reduced [13]: I = ne F S0

DSe d



σ sinh(σ ) + βP cosh(σ ) − βP σ sinh(σ ) + βS cosh(σ )



βS , βP + 1

S0  KM .

(70)

The formula (43) applicable for high concentrations of the substrate reduces slightly [13]: I = ne F

Vmax d (βP + 2) , 2 (βP + 1)

S0  KM .

(71)

For low concentrations of the substrate, the dimensionless current Iˆ can be expressed through the dimensionless substrate concentration Sˆ0 , diffusion module σ and Biot numbers βS and βP : Iˆ = Sˆ0



σ sinh(σ ) + βP cosh(σ ) − βP σ sinh(σ ) + βS cosh(σ )



βS , βP + 1

Sˆ0  1.

(72)

Accordingly, the dimensionless current Iˆ for high concentrations of the substrate can be expressed through the diffusion module σ and Biot number βP : σ 2 (βP + 2) , Iˆ = 2 (βP + 1)

Sˆ0  1.

(73)

To investigate the effect of the Biot number and to understand in which way the limiting conditions of internal or external diffusion can be attained, it is convenient to assume both Biot numbers to be equal [12, 13]. Assuming β = βS = βP leads to an expression of the dimensionless steady state current Iss as a function of only two variables, the diffusion module σ and Biot number β:  Iˆ = Sˆ0 1 −

β σ sinh(σ ) + β cosh(σ )



β , β +1

S0  1.

(74)

74

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

4.5 Effect of the Diffusion Layer To investigate the dependence of the steady state biosensor current on the relative thickness of the diffusion layer, a dimensionless ratio γ is introduced. The relative thickness γ of the diffusion layer is the ratio of the thickness δ of the external diffusion layer to the thickness d of the enzyme layer, γ = δ/d, γ ≥ 0. Since the steady state current is very sensitive to the thickness of the enzyme layer, the steady state biosensor current has to be normalized to evaluate the effect of the ratio γ on the biosensor response. The normalized steady state biosensor current IN is expressed by the steady state current at the thickness δ of the diffusion layer divided by the steady state current assuming the zero thickness d of the diffusion layer IN (d, δ) =

I (d, δ) , I (d, 0)

d > 0, δ ≥ 0,

(75)

where I (d, δ) is the density of the steady state current as defined by (31) and calculated at given thickness d of the enzyme membrane and the thickness δ of the diffusion layer. The biosensor response versus the dimensionless ratio γ = δ/d was calculated at very different values of the membrane thickness d. The results obtained at values of the model parameters (60) are depicted in Fig. 5. One can see in Fig. 5 that the steady state biosensor current notably decreases with an increase in the ratio γ in cases when the enzyme membrane thickness d

1 2 3

1.6 1.4

IN

1.2 1.0 0.8

4 5 6

0.6 0.4 0

1

2

3

4

γ Fig. 5 The normalized maximal biosensor current IN versus ratio γ = δ/d at six values of the thickness d of the enzyme layer: 10 (1), 20 (2), 50 (3), 100 (4), 200 (5) and 500 (6) µm. The other parameters are the same as in Fig. 3

4 Two-Compartment Model

75

is equal or greater than 100 µm (curves 4–6). That decrease is nonlinear. In cases of relatively thin enzyme membranes (d ≤ 20 µm, curves 1 and 2), the biosensor current increases with the increase in the ratio γ . In the case of d = 50 µm, the density I of the steady state current is a non-monotonous function of the ratio γ . Analysing the dependence of the enzyme membrane thickness d on the behaviour of the stationary current as a function of the ratio γ , the diffusion module σ 2 is calculated. Accepting the values of the model parameters defined in (60), σ 2 equals unity at d ≈ 55 µm. This thickness correlates well with the thickness at which IN is a non-monotonous function of γ (curve 3 in Fig. 5). Additional calculations of the response changing the maximal enzymatic rate Vmax confirm a statement that the density I of the stationary current is a monotonous decreasing function of the dimensionless ratio γ if the biosensor response is distinct under the diffusion control (σ 2  1). The stationary current is a monotonous increasing function of γ when the enzyme kinetics controls the biosensor response (σ 2  1). The dependence of the normalized stationary current on the Bio number β = βS = βP has been investigated [45]: IN (β) =

I (β) , I (∞)

(76)

where I (β) is the density of the steady state current calculated at the given Biot number β. I (∞) corresponds to the biosensor response for the zero thickness of the external diffusion layer, δ = 0. The biosensor response versus the Biot number β was investigated at different values of the maximal enzymatic rate Vmax and the membrane thickness d. The results of the calculations obtained at two values of Vmax : 10 and 100 µM/s and various values of the thickness d are depicted in Fig. 6. As one can see in Fig. 6, the steady state biosensor current is a monotonous increasing function of the Biot number β when the response is under the diffusion control (σ >≈ 1.5) [16]. IN is a non-monotonous function of β when the enzyme kinetics controls the biosensor response (σ 10. One can see in Fig. 6 a non-monotonous dependence of the biosensor current on the Biot number β when the enzyme kinetics controls the biosensor response (σ < 1). Similar non-monotonicity of the steady state biosensor current was also observed by others [19, 26, 37, 75]. According to Somasundrum et al. [75], the maximum response corresponds “to the changeover from kinetic to diffusion control” [26]. Having the analytical expression (74) of Iˆ in terms of only two parameters, σ and β, the properties of the biosensor response can be investigated analytically [13]. Particularly, the current Iˆ as a function of β has extremums when ∂ Iˆ/∂β = 0, i.e. when (cosh2 (σ ) − cosh(σ ) − σ sinh(σ ))β 2 + 2σ sinh(σ )(cosh(σ ) − 1)β +

76

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

IN

2

1

4 5 6

1 2 3 0 0.1

1

7 8 9

10

100

Fig. 6 The normalized steady state current IN versus the Biot number β at Vmax = 100 (1–4) and Vmax = 10 µM/s (5–9) and and nine diffusion module σ : 0.18 (5), 0.29 (1), 0.37 (6), 0.58 (2), 0.91 (7), 1.15 (3), 1.83 (8), 2.89 (4) and 3.65 (9). The other parameters are the same as in Fig. 3

σ 2 sinh2 (σ ) = 0. This equation has only one meaningful (σ > 0 and β > 0) root βmax (σ ) =

  √ σ sinh(σ ) cosh(σ ) + 1 − cosh(σ ) + σ sinh(σ ) − 1 cosh(σ ) + σ sinh(σ ) − cosh2 (σ )

.

(77)

However, βmax is positive only when cosh(σ ) + σ sinh(σ ) − cosh2 (σ ) > 0,

(78)

which is valid at the following values: 0 < σ < σmax , where σmax ≈ 1.5 is a positive root of equation cosh(σ ) + σ sinh(σ ) − cosh2 (σ ) = 0 [13]. The same σmax -value has also been calculated when finding the maximum of steady state current for the corresponding one-compartment model [16]. βmax is a value of the Biot number β at which the state–state current Iˆ gains the maximum at given σ . When σ changes from 0 to σmax , βmax monotonously increases starting from zero to infinity. Values of σmax as well as of βmax are invariant to all other model parameters and are applicable for all concentrations S0  1.

4 Two-Compartment Model

77

4.6 The Nernst Diffusion Layer The thickness δ of the external diffusion layer depends upon the nature and the intensity of stirring of the buffer solution. Usually, the more intense stirring corresponds to the thinner diffusion layer. That diffusion layer is also known as the Nernst diffusion layer [33, 53, 62, 80]. According to the Nernst approach, a layer of thickness δ remains stagnant. Away from it, the solution is in motion and uniform in concentration. The thickness of the diffusion layer remains unchanged with time [2, 55, 56]. The thickness of the Nernst diffusion layer practically does not depend upon the enzyme membrane thickness. In the case when the bulk solution is stirred by rotation of the enzyme electrode, the thickness δ of the external diffusion layer can be calculated as follows [33, 53]: δ = 1.61D 1/3ω−1/2 ν 1/6 ,

(79)

where D is the diffusion coefficient of the species, ω is the rotation rate in Hz and ν is the viscosity in m2 /s. According to (79), the thickness δ is inversely dependent on the square root of the rotation rate ω. Theoretically, δ can be as minimal as desirable. Practically, the thickness of the Nernst diffusion layer can be minimized up to δ = 2 µm by increasing the rotation speed [50, 53]. In another frequently used case when the solution is stirred in a magnetic stirrer, it is difficult to achieve the thickness δ less than 20 µm. In the case when an analyte is well-stirred and effectively mixed, the mass transport by diffusion outside the enzyme membrane quite often is neglected [27, 70, 71]. However, in practice, the zero thickness of the Nernst layer cannot be achieved. The biosensor model taking into consideration the Nernst diffusion layer describes the biosensor action more precisely than another model where the Nernst diffusion layer is neglected. In addition, the Nernst diffusion layer of thickness δ may be neglected for a biosensor having membrane thickness d only if the steady state current calculated, considering the Nernst layer, is approximately the same as in the case when the Nernst diffusion layer is neglected. Consequently, the Nernst diffusion layer may be neglected if I (d, δ) ≈ I (d, 0), i.e. IN (d, δ) ≈ 1, where I (d, δ) is the density of the stationary current calculated at the given values of d and δ and IN is the normalized stationary current as defined by (75). The relative error of the biosensor response as a function of the Biot number β is described as follows: R(β) =

|I (β) − I (∞)| . I (β)

(80)

R(β) can be called the relative error of the use of the mathematical model where the diffusion layer of thickness δ is neglected at the Biot number β. This function may also be regarded as a level of reliability of the mathematical model where the Nernst diffusion layer is not taken into account.

78

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors 0.4

1 2 3 4

R

0.3

0.2

0.1

0.0 1

10

100

a) 0.5

1 2 3 4

0.4

R

0.3

0.2

0.1

b)

0.0 0.1

1

10

100

Fig. 7 The relative error R versus the Biot number β at two thicknesses δ of the Nernst layer: 2 (a) and 20 (b) µm. Vmax = 0.1 (1), 1 (2), 10 (3) and 100 (4) µM/s. The other parameters are the same as in Fig. 3

To investigate the effect of the Nernst diffusion layer on the biosensor response when the analyte is well-stirred, the relative error R has to be calculated at practically minimal thickness of the diffusion layer. Since the effect of the diffusion layer on the biosensor response significantly depends upon the diffusion module, the normalized response was calculated by changing in a wide range both the maximal enzymatic rate Vmax and the membrane thickness d. Figure 7 shows the results of the calculation for two values of the minimal thickness δ of the Nernst diffusion

4 Two-Compartment Model

79

layer, corresponding to two types of stirring: by rotating electrode (a, δ = 2 µm) and by magnetic stirrer (b, δ = 20 µm). As one can see in Fig. 7, the effect of the Nernst layer mainly decreases with the increase in the Biot number β as well as in the membrane thickness d. Figure 7 shows that the Nernst diffusion layer should be taken into consideration in all the cases when the enzyme membrane is thinner than about 25δ (β = 50). Ignoring the Nernst diffusion layer in modeling of the biosensor response at β = 1, the simulated density I of the steady state current produces difference even more than 30% (R > 0.3) in comparison to the true stationary current. In all cases when the Biot number is greater than about 50 (d > 25δ), the relative error is less than 2%. The effect of the Nernst diffusion layer becomes slight only in cases when the Biot number is greater than about 50. At higher values of maximal enzymatic activity Vmax (curves 3 and 4), the relative errors are slightly lower than at lower activity (curves 1 and 2). The Nernst diffusion layer should be taken into consideration when an analytical system based on an amperometric biosensor acts under the conditions when the Biot number is less than about 50 (Fig. 7). On the other hand, the Nernst diffusion layer may be neglected when Biot number is greater than 50. When the diffusion layer is neglected, a two-compartment model reduces to a notably simpler one-layer model.

4.7 Impact of the Diffusion Module To investigate the effect of two main parameters of the two-compartment model, the diffusion module σ 2 and the Biot number β, on the biosensor response, the biosensor action was simulated at different values of β, changing the module σ 2 in a few orders of magnitude. Figure 8a shows the dependence of the steady state dimensionless current Iˆ on σ 2 , while Fig. 8b shows the corresponding dependence of the sensitivity BS , introduced in (32). The diffusion module σ 2 is varied by changing exponentially the maximal enzymatic rate Vmax from 0.1 up to 1000 µM/s. Values of all other parameters were kept constant as defined in (60). As one can see in Fig. 8a, at low values of the diffusion module (σ 2 < 1) when the enzyme kinetics controls the biosensor response, the stationary dimensionless current Iˆ is approximately a linear function of σ 2 . At these conditions, the relative thickness β of the external diffusion layer affects the dimensionless current Iˆ only slightly. When the biosensor response is considerably controlled by the diffusion (σ 2 > 10), the stationary current practically does not depend on σ 2 . However, the effect of the Biot number on the current Iˆ is much more significant namely at σ 2 > 10. One can also see in Fig. 8a a longer linear range of curves at greater values of the Biot number rather than at lover ones. Figure 8b shows the dependence of the biosensor sensitivity on the diffusion module σ 2 . The sensitivity BS monotonously increases with an increase in module σ 2 . The shape of all the curves looks very similar. However, the smaller Biot number corresponds to a greater sensitivity BS . This is especially noticeable at

80

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

100

Î

10-1

1 2 3 4 5

10-2

10-3 0.01

0.1

1

a)

10

100

10

100

2

1.0

0.9

BS

0.8

1 2 3 4 5

0.7

0.6

0.5

b)

0.01

0.1

1 2

Fig. 8 The steady state dimensionless current Iˆ (a) and the dimensionless sensitivity BS versus the diffusion module σ 2 at five values of the Biot number β: 0.2 (1), 0.5 (2), 1 (3), 5 (4) and 10 (5). The other parameters are the same as in Fig. 3

moderate values of σ 2 , 0.1 < σ 2 < 10. Let us recall that a smaller β corresponds to a thicker external diffusion layer. The thicker external diffusion layer creates a greater diffusion limitation to the substrate. Figure 8b well illustrates a widely known and a very useful feature of biosensors that an additional diffusion limitation on the substrate increases their sensitivity and prolongs their calibration curve [51, 57, 61, 77].

References

81

5 Concluding Remarks The two-compartment mathematical model (25)–(30) and the corresponding dimensionless model (63)–(67) of the amperometric biosensor operation can be successfully used to investigate the dependence of the internal and external diffusion limitations on the biosensor response, sensitivity and stability of enzyme membranebased biosensors. For low concentrations of the substrate, the dimensionless steady state current Iˆ can be expressed as a function (72) of four variables, the dimensionless substrate concentration Sˆ0 (Sˆ0  1), the diffusion module σ and the Biot numbers βS and βP . When S0  1 and β = βS = βP , the current Iˆ is a non-monotonous function of the Biot number β for 0 < σ < 1.5 and a monotonous increasing function of β for σ > 1.5 (Figs. 5 and 6). At high concentrations of the substrate (Sˆ0  1), the current Iˆ is a function (73) of only two variables, σ and βP . The Nernst diffusion layer should be taken into consideration when modeling an analytical system based on an amperometric biosensor acting under the conditions when the Biot number is less than about 50, and the Nernst diffusion layer may be neglected when Biot number is greater than about 50 (Fig. 7). When the diffusion layer is neglected, a two-compartment model reduces to a simpler one-layer model. The biosensor sensitivity monotonously increases with an increase in module σ 2 . The smaller Biot number corresponds to a greater sensitivity BS . An additional diffusion limitation on the substrate increases their sensitivity and prolongs their calibration curve (Fig. 8).

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60. Meyerhoff M, Duan C, Meusel M (1995) Novel nonseparation sandwich-type electrochemical enzyme immunoassay system for detecting marker proteins in undiluted blood. Clin Chem 41(9):1378–1384 61. Mullen W, Keedy F, Churchouse S, Vadgama P (1986) Glucose enzyme electrode with extended linearity: application to undiluted blood measurements. Anal Chim Acta 183:59–66 62. Nernst W (1904) Theorie der Reaktionsgeschwindigkeit in heterogenen systemen. Z Phys Chem 47(1):52–55 63. Özisik MN (1980) Heat conduction. Wiley, New York 64. Pfeiffer D, Scheller F, Setz K, Schubert F (1993) Amperometric enzyme electrodes for lactate and glucose determinations in highly diluted and undiluted media. Anal Chim Acta 281(3):489–502 65. Romero MR, Baruzzi AM, Garay F (2012) Mathematical modeling and experimental results of a sandwich-type amperometric biosensor. Sens Actuator B Chem 162(1):284–291 66. Rong Z, Cheema U, Vadgama P (2006) Needle enzyme electrode based glucose diffusive transport measurement in a collagen gel and validation of a simulation model. Analyst 131(7):816–821 67. Sadana A, Sadana N (2011) Handbook of biosensors and biosensor kinetics. Elsevier, Amsterdam 68. Samarskii A (2001) The theory of difference schemes. Marcel Dekker, New York-Basel 69. Scheller FW, Schubert F (1992) Biosensors. Elsevier Science, Amsterdam 70. Schulmeister T (1987) Mathematical treatment of concentration profiles and anodic current of amperometric enzyme electrodes with chemically amplified response. Anal Chim Acta 201:305–310 71. Schulmeister T (1990) Mathematical modelling of the dynamic behaviour of amperometric enzyme electrodes. Sel Electrode Rev 12(2):203–260 72. Schulmeister T, Pfeiffer D (1193) Mathematical modelling of amperometric enzyme electrodes with perforated membranes. Biosens Bioelectron 8(2):75–79 73. Segel LA, Slemrod M (1989) The quasi-steady-state assumption: a case study in perturbation. SIAM Rev 31(3):446–477 74. Senda M, Ikeda T, Miki K, Hiasa H (1986) Amperometric biosensors based on a biocatalyst electrode with entrapped mediator. Anal Sci 2(6):501–506 75. Somasundrum M, Aoki K (2002) The steady-state current at microcylinder electrodes modified by enzymes immobilized in conducting or non-conducting material. J Electroanal Chem 530(1–2):40–46 76. Sorochinskii V, Kurganov B (1996) Amperometric biosensors with a laminated distribution of enzymes in their coating. Steady state kinetics. Biosens Bioelectron 11(1–2):45–51 77. Tang L, Koochaki Z, Vadgama P (1990) Composite liquid membrane for enzyme electrode construction. Anal Chim Acta 232:357–365 78. Turner APF, Karube I, Wilson GS (eds) (1990) Biosensors: fundamentals and applications. Oxford University Press, Oxford 79. Velkovsky M, Snider R, Cliffel DE, Wikswo JP (2011) Modeling the measurements of cellular fluxes in microbioreactor devices using thin enzyme electrodes. J Math Chem 49(1):251–275 80. Wang J (2000) Analytical electrochemistry, 2 edn. Wiley, New-York 81. Yang H (2000) Mathematical model for liquid-liquid phase-transfer catalysis. Chem Eng Comm 179(1):117–132 82. Ylilammi M, Lehtinen L (1988) Numerical analysis of a theoretical one-dimensional amperometric enzyme sensor. Med Biol Eng Comput 26(1):81–87 83. Zhao W, Xu J, Chen H (2006) Electrochemical biosensors based on layer-by-layer assemblies. Electroanal 18(18):1737–1748

Biosensors Utilizing Consecutive and Parallel Substrates Conversion

Contents 1 2

3

4

5

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consecutive and Parallel First Order Conversion of Substrates. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Consecutive Substrates Conversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Parallel Substrate Conversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Dual Catalase-Peroxidase Bioelectrode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Reaction Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Reaction Rates at Quasi-Steady State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Two Compartment Reaction–Diffusion Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Numerical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Dynamics of the Biosensor Current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Biosensor Response to the Second Substrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Impact of the Reaction Rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Model Modifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biosensors Response to Mixture of Compounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Reaction Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mathematical Model with No Substrates Interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Solution of the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Generation of Pseudo-Experimental Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Modeling Substrates Interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86 87 87 95 97 97 98 100 103 104 105 106 107 109 109 109 111 112 114 116

Abstract Coupling different enzymes either in sequence or in competition pathway is a way to enhance the range of analyte species accessible to measurement, the selectivity and the sensitivity of biosensors. In this chapter, mathematical models of several types of amperometric multi-enzyme biosensors utilizing consecutive or parallel substrates conversion are modeled and analysed at stationary and transient conditions. A biosensor based on bienzyme electrode with co-immobilized Dglucose oxidase and peroxidase is considered under stationary conditions at excess concentrations of oxygen and ferrocyanide. A trienzyme biosensor utilizing consecutive substrates conversion with three enzymes is modeled at internal diffusion limitation. A biosensor with dual catalase-peroxidase bioelectrode is mathematically

© Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_3

85

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Biosensors Utilizing Consecutive and Parallel Substrates Conversion

modeled by nonlinear reaction–diffusion equations. Finally, multi-enzyme biosensors utilizing parallel and competitive multi-substrate conversion are analysed. Keywords Multi-enzyme biosensor · Catalase · Peroxidase · Substrates consecutive/parallel conversion · Stationary/transient state · Mathematical model

1 Introduction Coupling different enzymes either in sequence or in competition pathway is a way to enhance the range of analyte species accessible to measurement, the selectivity and the sensitivity of biosensors [1, 20, 24, 34, 38]. The application of enzymes that uses the same substrate can open new possibilities for biosensors building [35]. A biosensor can produce molecular oxygen, which can be selectively determined by a sensitive to oxygen electrode, e.g. Clark-type electrode [52, 65]. For this reason the biosensors using this high selective electrode as a transducer are applicable for metabolites determination in complex biological liquids [69]. Mathematical modeling of two-enzyme biosensors started in 1980 [30, 31] with the modeling of an amperometric mono-layer enzyme electrode with two coimmobilized enzymes. It was only the first order reaction rate that was considered in the first mathematical models [30, 31, 35, 47, 59]. The dynamic response of these electrodes was analysed by solving diffusion equations and using Green’s function [35]. Further analysis of dual enzyme biosensors response was performed by Scheller, Schulmeister and others [56, 57]. A comprehensive review of the models developed during the first decade was given in [58]. Later, nonlinear mathematical models have been developed for amperometric two-enzyme biosensors [44, 60, 62, 63]. It should be noted that the basic methods of the problems solution are the same as in the one-layer one-enzyme case. In this chapter, mathematical models of several types of amperometric multienzyme biosensors are considered at stationary and transient conditions. A biosensor based on bienzyme electrode with co-immobilized D-glucose oxidase and peroxidase is considered under stationary conditions at excess concentrations of oxygen and ferrocyanide. In the presence of adenosine triphosphate (ATP), D-glucose oxidation is paralleled by the reaction of glucose phosphorylation in the bienzyme biosensors, the catalytic membrane of which is made of D-glucose oxidase and hexokinase [30]. A trienzyme biosensor utilizing consecutive substrates conversion with three enzymes is modeled at internal diffusion limitation [32]. A biosensor with dual catalase-peroxidase bioelectrode represents the next type of biosensors [6, 37]. The application of catalase and peroxidase allows to construct a biosensor realizing parallel substrate conversion and apply it in pharmaceutical environment. A rather unspecific electrochemical method is usually used [16, 22] for the determination acetaminophen, therefore a specifically designed biosensor is relevant.

2 Consecutive and Parallel First Order Conversion of Substrates

87

Final type of the biosensors utilizes enzymatic reactions assuming no interaction between the analysed substrates and reaction products [9–11]. The mathematical model of such biosensors allows to simulate the biosensor response to a mixture of compounds (substrates).

2 Consecutive and Parallel First Order Conversion of Substrates 2.1 Consecutive Substrates Conversion 2.1.1 Biosensors with Multi-Enzymes The conversion of the substrate, for example, by adsorbed polyenzyme system, is (i) catalysed by the corresponding enzyme with the matching kinetic parameters Vmax,s (i) and KM : S1 −→ S2 −→ . . . Si −→ Sn .

(1)

At the steady state conditions, the substrate flux from the bulk solution, where the concentration is S0 , is equal to the rate of the product (the substrate of the following reactions) generation: (1)

D0

Vmax,s S1 S0 − S1 = (1) , δ KM + S1 (1)

D0

(2)

Vmax,s S1 S2 Vmax,s S2 = (1) − (2) , δ KM + S1 KM + S2

(i−1) (i) Vmax,s Si Si−1 Si Vmax,s D0 = (i−1) − (i) , δ KM + Si−1 KM + Si

(2) i = 3, . . . , n − 1,

(n−1)

D0

Vmax,s Sn−1 Sn = (n−1) , δ KM + Sn−1

where δ is the thickness of the diffusion layer and D0 is the diffusion coefficient in the diffusion layer. (i) (i) (i) At low substrate concentrations (Si  KM ) and ρi = Vmax,s δi /(KM D0 ), n−2 

Sn−1 =

n−1 

ρi

(1 + ρi )

S0

(3)

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Biosensors Utilizing Consecutive and Parallel Substrates Conversion

and n−1 

Sn =

n−1 

ρi

(4)

S0 .

(1 + ρi )

It follows that the degree of the substrate conversion depends on the stadium for that diffusion module ρi and is less than 1. The longer the chain (larger enzyme numbers n), the less the product formation. Even at n = 3, the significant yield of the product is generated if all the diffusion modules are greater than 1. If the rate of the last product formation is proportional to the current, the biosensors response can be expressed as (n−1) Vmax,s

I = ne F

(n−1) KM

n−2 

n−1 

ρi

S0 .

(5)

(1 + ρi )

At ρi > 1 the sensitivity of the biosensor is determined by the kinetic parameters (n−1) (n−1) of the last reaction. At ρi < 1 the sensitivity is a product of Vmax,s /KM and the diffusion modules.

2.1.2 D-Glucose Oxidase-Peroxidase Biosensor The most popular glucose biosensor is based on glucose oxidase (GO) that catalyses β-D-glucose oxidation with oxygen [67, 68, 70], β-D-glucose + O2

D-glucose oxidase

−→

D-glucono-δ-lactone + H2 O2 .

(6)

A more complex biosensor is based on bienzyme electrode with co-immobilized D-glucose oxidase and peroxidase [30]. Under the action of D-glucose oxidase (6), D-glucose is oxidized with the production of hydrogen peroxide. During the second stage, hydrogen peroxide is reduced by ferrocyanide ion (7). This reaction is catalysed by peroxidase + H2 O2 + 2Fe(CN)4− 6 + 2H

peroxidase

−→

2Fe(CN)3− 6 + 2H2 O.

(7)

Under the stationary conditions at excess concentrations of oxygen and ferrocyanide, when reactions (6) and (7) are of the first order, the change of concentration

2 Consecutive and Parallel First Order Conversion of Substrates

89

within the bienzyme membrane is described by the system of the following equations: Vmax S d2 S = = α12 S, dx 2 KM De d2 P1 Vmax S V  P1 2 2 =− + max  D = −α1 S + α2 P1 , 2 dx KM De KM e

(8)

 P Vmax d2 P2 1 2 = −2  D = −2α2 P1 , dx 2 KM e

where S, P1 and P2 are the concentrations of glucose, hydrogen peroxide and ferricyanide, respectively; De —the diffusion coefficient of S, P1 and P2 , which  , K and K  are the corresponding parameters of the are taken equal; Vmax , Vmax M M enzyme reactions (6) and (7), and α12 =

Vmax , KM De

α22 =

 Vmax .  KM De

(9)

The solution of the system (8) taking into consideration the boundary conditions S = S0 , P1 = 0, P2 = 0 at x ≥ d and dS/dx = dP1 /dx = 0, P2 = 0 at x = 0 gives the dependence of electrode current density I on the kinetic and diffusive parameters (α1 = α2 ) [30]:  dP2  dx x=0    2F De α12 α22 1 1 = 1− − 2 1− S0 cosh α2 d cosh α1 d d(α12 − α22 ) α1    2F De σ12 σ22 1 1 1− = − 2 1− S0 , cosh σ2 cosh σ1 d(σ12 − σ22 ) σ1

I = F De

(10)

where S0 is the initial concentration of the substrate, d is the enzyme membrane thickness, F is Faraday constant (F = 96485 C/mol), σ1 and σ2 are the diffusion modules, σ1 = α1 d,

σ2 = α2 d.

(11)

Hence, it follows that the current of the bienzyme electrode is proportional to the substrate (glucose) concentration. The current is determined by means of diffusion modules σ1 and σ2 . When the rate of enzyme reaction is great (σ1  1 and σ2  1)

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Biosensors Utilizing Consecutive and Parallel Substrates Conversion

the response reaches its maximal value and is determined by the substrate diffusion, I=

2F De S0 . d

(12)

When the activity of peroxidase is considerably greater than the activity of Dglucose oxidase (σ2  σ1 ) the biosensor response is determined by the D-glucose oxidase parameters, I =

2F De d

 1−

1 cosh σ1

S0 .

(13)

Under the kinetic control (σ1  1), (13) is transformed to I=

F Vmax d S0 . KM

(14)

Due to high molecular activity of the peroxidase, these bienzyme biosensors operate in the mode controlled by the D-glucose oxidase reaction. Sensitivity as well as the stability of electrodes are close to that of mono-enzyme D-glucose electrode.

2.1.3 Bienzyme Electrode at Transient State The consecutive conversion of the substrates in the bienzyme biocatalytic membranes takes place according to the following scheme: k1

k2

S −→ Z −→ P,

(15)

where S is the substrate, Z is the intermediate, P is the electrochemically active product, and k1 and k2 are the first order rate constants (Vmax /KM ) of the enzymatic conversion of the compounds. Assuming that there exist no outer diffusion limitations, the mass transport and the consecutive substrates conversion (15) in the biocatalytical membrane at a transient state are described by the following system of the reaction–diffusion equations [35]: ∂S ∂ 2S = De 2 − k1 S , ∂t ∂x ∂Z ∂ 2Z = De 2 − k2 Z + k1 S , ∂t ∂x ∂P ∂ 2P = De 2 + k2 Z , ∂t ∂x

(16)

2 Consecutive and Parallel First Order Conversion of Substrates

91

where S, Z, P correspond to the concentrations of the substrate, the intermediate and the product, respectively. Governing equations (16) together with the initial (S = 0, Z = 0, P = 0 when 0 ≤ x < d, t = 0) and the boundary (S = S0 , Z = 0, P = 0 when x ≥ d, t ≥ 0; ∂S/∂x = 0, ∂Z/∂x = 0, P = 0 when x = 0, t ≥ 0) conditions form together the initial boundary value problem. The system (16) together with the initial and the boundary conditions was solved using the Green function [3]. When k1 = k2 the solution of (16) is possible using the L’Hopital rule. The evaluation of the system (15) for k1 = k2 gives the following expression for the biosensor response (the current density) [35]: i(t) = ne F De

 ∂P  4k1k2 = ne F De S0 2  ∂x x=0 d

∞  ∞ 

(−1)m+1 γ2m−1 (t) , × 2 2 2 (k + γ2m−1 De )(k2 + γ2m−1 De )(4γn2 − γ2m−1 ) n=1 m=1 1

(17)

where ne is a number of electrons involved in the electrochemical reaction, F is Faraday constant, γ2m−1 = (2m − 1)π/2d, γn = nπ/2d, and function (t) takes the following form: (t) = 1 − − +

2 2 De )(k2 + γ2m−1 De ) exp(−4γn2 De t) (k1 + γ2m−1 2 2 (4γn2 De − γ2m−1 De − k1 )(4γn2 De − γ2m−1 De − k2 )

2 2 De ) exp(−(k1 + γ2m−1 De )t) 4γn2 De (k2 + γ2m−1

(18)

2 (k2 − k1 )(4γn2 De − γ2m−1 De − k1 ) 2 2 4γn2 De (k1 + γ2m−1 De ) exp(−(k2 + γ2m−1 De )t) 2 (k2 − k1 )(4γn2 De − γ2m−1 De − k2 )

.

It is possibly to notice a kinetic indistinguishability of the constants k1 and k2 for the consecutive conversion. At a high value of k1 the dynamics of the response is identical to that of a mono-enzyme electrode characterized by the rate constant k2 . Generally, the kinetic behaviour of the electrode is determined by the diffusion modules σ1 = α1 d = (k1 /De )1/2 d and σ2 = α2 d = (k2 /De )1/2 d, and at high values the processes proceed at the same rates as the substance transfer through the inert membrane. At a low catalytic activity of one of the enzyme the response decreases more than 4 times [35].

2.1.4 Trienzyme Biosensor The modeling of trienzyme biosensor utilizing consecutive substrates conversion with three enzymes was completed at internal diffusion limitation and steady state

92

Biosensors Utilizing Consecutive and Parallel Substrates Conversion

conditions [32]. The example of successful application of three enzymes might be sensitive to the creatinine biosensor [64]. In the membrane of this biosensor creatininase (E1 ) hydrolyses creatinine (S) to creatine (P1 ). The creatine is further hydrolyzed with creatinase (E2 ) to sarcosine (P2 ). The oxidation of sarcosine with sarcosine oxidase (E3 ) produces hydrogen peroxide (P3 ) that is determined amperometrically, E1

E2

E3

S −→ P1 −→ P2 −→ P3 .

(19)

The rate Vi (S) of each reaction can be characterized by the standard enzyme parameters Vmaxi and KMi , where i = 1, 2 and 3, for E1 , E2 and E3 catalysed process, respectively. At concentration of S, P1 and P2 less than the Michaelis–Menten constants KMi , V1 = Vmax1 S/KM1 , V2 = Vmax2 P1 /KM2 , V3 = Vmax3 P2 /KM3 . At the substrates concentration less than KMi and at constant diffusion coefficients the diffusion equations and the enzymatic conversions take a form ∂ 2S 1 ∂S = − α12 S , De ∂t ∂x 2 ∂ 2 P1 1 ∂P1 = + α12 S − α22 P1 , De ∂t ∂x 2 1 ∂P2 ∂ 2 P2 = + α22 P1 − α32 P2 , De ∂t ∂x 2

(20)

1 ∂P3 ∂ 2 P3 = + α32 P2 , De ∂t ∂x 2 where De —the diffusion coefficient of all compounds in the enzyme membrane, αi = (Vmaxi /KMi De )1/2 , i = 1, 2, 3 [32]. The biosensor response (the current density) was calculated as i(t) = 2F De

 ∂P3  . ∂x x=0

(21)

The solution of (20) was found at the steady state conditions (∂S/∂t = ∂P1 /∂t = ∂P2 /∂t = ∂P3 /∂t = 0) with the boundary conditions: S = S0 , P1 = 0, P2 = 0, P3 = 0 at x ≥ d, ∂S/∂x = 0, ∂P1 /∂x = 0, ∂P2 /∂x = 0, P3 = 0 at x = 0, where d—the membrane thickness. Calculations show that a significant concentration of products in a membrane is produced if all the diffusion modules σi (σi = αi d, i = 1, 2, 3) are larger than 1 (Fig. 1) [32]. To prove the correctness of the calculations the distribution of compounds in a membrane was also determined at the boundary condition ∂P3 /∂x = 0 (x = 0). In this case the sum of the compounds was equal to S0 at all x values. At σ1 = σ2 = σ3

2 Consecutive and Parallel First Order Conversion of Substrates

93

1.0

Concentration, mM

0.9 0.8 0.7

S

0.6

P3

0.5 0.4 0.3

P1

0.2 0.1

P2

0.0 0.00

0.02

0.04

0.06

0.08

0.10

d, mm Fig. 1 The profile of compounds concentration in trienzyme membrane of the biosensor. For calculations S0 = 1 mM, σ1 = 10.0, σ2 = 10.1, σ3 = 10.3 and d = 0.1 mm were used

(α1 = α2 = α3 ) the expression of three enzyme biosensors response (the current density) is I=

σ22 σ32 2F De (1 − cosh(σ1 )) d (σ22 − σ12 )(σ32 − σ12 ) − +

σ12 σ32 (σ22 − σ12 )(σ32 − σ22 ) σ12 α22 (σ32 − σ12 )(σ32 − σ22 )

(1 − cosh(σ2 ))

(22)

(1 − cosh(σ3 )) S0 .

It is impossible in practice to achieve equal values of the diffusion modules for all enzymes. Therefore the biosensor response has not been derived at σ1 = σ2 = σ3 (α1 = α2 = α3 ). The dependence of the response of the biosensor on the diffusion module of the least active enzymes E1 and E2 is shown in Fig. 2. It is easy to notice that the response is very small, still diffusion modules σ1 , σ2 and σ3 are less than 1. The maximal biosensor response of 0.6 µA/mm2 is achieved when the diffusion modules are greater than 10 [32]. Experiments show that among three immobilized enzymes the lowest stability demonstrates creatininase (E1 ). The model permits to predict sensitivity change of the biosensor during the enzyme inactivation. If the inactivation follows exponential decay, for example, with half-time 2 days, and at the beginning the biosensor contains large catalytic activities (σi = αi d ≈ 10), the response decreases just

94

Biosensors Utilizing Consecutive and Parallel Substrates Conversion

0.6

0.4

I, μA/mm

2

0.5

0.3 0.2

0.0 0.5

1.0

1.5

2.0

1

6.0 5.0 4.0 3.0 2.0 1.0 2

0.1

2.5

3.0

0.6

60

0.5

50

0.4

40

0.3

30

I

1

I, μA/mm2

Fig. 2 The dependence of the biosensor response on the diffusion modules σ1 and σ2 . For calculations S0 = 1 mM and σ3 = 10.3 were used

1

0.2

20

0.1

10

0.0 0

1

2

3

4

5

6

7

8

9

0 10 11 12 13 14 15

Days Fig. 3 The changes of the biosensor response and the diffusion module of trienzyme biosensor during the enzyme inactivation. The half-time of enzyme inactivation is 2 days, S0 = 1 mM, σ1 = 10.0, σ2 = 10.1, σ3 = 10.3 and d = 0.1 mm

34.7 % during 10 days (Fig. 3). The apparent half-time of biosensor inactivation increases up to 11.6 days. In fact, this biosensor can be used even longer, i.e. during 15 days with permanent calibration.

2 Consecutive and Parallel First Order Conversion of Substrates

95

2.2 Parallel Substrate Conversion 2.2.1 D-Glucose Oxidase-Hexokinase Biosensor In the presence of adenosine triphosphate (ATP), D-glucose oxidation is paralleled by the reaction of glucose phosphorylation in the bienzyme biosensors, the catalytic membrane of which is made of D-glucose oxidase and hexokinase [30], D-glucose + ATP

hexokinase

−→

D-glucose-6-phosphate + ADP.

(23)

The increase in ATP concentration leads to the decrease of the response occurring as a result of the D-glucose oxidase reaction (6). For the calculation of the dependence of the biosensor response it is assumed that the oxidation and phosphorylation are the first order for D-glucose and ATP, respectively. The solution of equations of diffusion and enzyme reactions with boundary conditions: S = S0 , S1 = S10 , P1 = 0 at x ≥ d, and dS/dx = dS1 /dx = 0, P1 = 0 at x = 0 gives the dependence of the density I of the biosensor current on the D-glucose (S0 ) and ATP (S10 ) concentration (α1 = α2 ), 

 dP1  2F De 1 = I = 2F De S0 1 − dx x=0 d cosh α1 d    α22 α12 1 1 − S10 1− 1− − 2 , cosh α1 d cosh α2 d α22 − α12 α2 − α12

(24)

 /K  D )1/2 are related to glucose where α1 = (Vmax /KM De )1/2 and α2 = (Vmax M e oxidase and hexokinase reaction, respectively [30]. Thus, it follows that at a high rate of both enzyme reactions (α1 d > 1 and α2 d > 1) the density of the biosensor current is determined by the difference between the D-glucose and ATP concentrations,

I=

2F De (S0 − S10 ) , d

(25)

i.e. the decrease in the biosensors current is proportional to the concentration of coenzyme (ATP), I =

2F De S10 . d

(26)

If the reaction of D-glucose oxidase proceeds rapidly (α1 d > 1), and the phosphorylation is at low rate (α2 d < 1), then I =

 d F Vmax S10 .  KM

(27)

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Biosensors Utilizing Consecutive and Parallel Substrates Conversion

The sensitivity of such a biosensor is directly proportional to the activity of hexokinase. The experimental results indicate that the action of the ATP electrode is determined by the activity of this enzyme [30]. The hexokinase inactivation results in a quick loss of the biosensors sensitivity.

2.2.2 Bienzyme Electrode at Transient State The parallel conversion of the substrates in the bienzyme biocatalytic membranes proceeds according to k1

k2

P1 ←− S −→ P2 ,

(28)

where S is the substrate, P1 and P2 are products, k1 and k2 are the first order rate constants (Vmax /KM ) of the enzymatic conversion of the compounds, and P1 is the only electrochemically active product [35]. The following equations describe the parallel substrate conversion (28) [35]: ∂ 2S ∂S = De 2 − (k1 + k2 )S, ∂t ∂x ∂P1 ∂ 2 P1 = De + k1 S, ∂t ∂x 2

(29)

∂P2 ∂ 2 P2 = De + k2 S, ∂t ∂x 2 where P1 and P2 stand for the concentrations of the products P1 and P2 , respectively. The following initial and the boundary conditions were assumed for the system of the parallel substrate conversion: S = 0, P1 = 0, P2 = 0 when 0 ≤ x < d, t = 0; ∂S/∂x = 0, P1 = 0, ∂P2 /∂x = 0 when x = 0, t ≥ 0; S = S0 , P1 = 0, P2 = 0 when x ≥ d, t ≥ 0. The system (29) together with the initial and the boundary conditions was solved using the Green function [3, 35]. The solution of the transient behaviour of the biosensor with a parallel substrate conversion is  ∂P1  i(t) = ne F De ∂x x=0 ∞ ∞ S0 4k1   (−1)m+1 γ2m−1 (t) , = ne F De 2 2 2 d (k + k2 + γ2m−1 De )(4γn2 − γ2m−1 ) n=1 m=1 1

(30)

3 Modeling Dual Catalase-Peroxidase Bioelectrode

97

where (t) = 1 + −

2 De ) exp(−4γn2 De t) (k1 + k2 + γ2m−1 2 4γn2 De − γ2m−1 De − k1 − k2

2 De )t) 4γn2 De exp(−(k1 + k2 + γ2m−1 2 4γn2 De − γ2m−1 De − k1 − k2

(31) .

As it follows from this equation, the response time of the biosensor is equal to that of the mono-enzyme electrodes characterized by the constant value (k1 +k2 ), i.e. the conversion of a part of the substrate into the electrochemically inactive product results in the same decrease of the stationary and non-stationary current [35]. Therefore the response time of the biosensor remains unchanged. The results of mathematical modeling were compared with the experimental data [35]. The comparison showed that the kinetics of the biosensor response depends on the biocatalytical schemes and rates of the biocatalytical conversion as predicted by the models.

3 Modeling Dual Catalase-Peroxidase Bioelectrode A mathematical model of a biosensor with a dual catalase-peroxidase bioelectrode is presented in this section [6, 37]. The developed mathematical model was used to prove a concept of bioelectrode analytical application of such configuration. The application of catalase and peroxidase allows to construct a biosensor realizing parallel substrate conversion and apply it in pharmaceutical environment. A rather unspecific electrochemical method is usually used for the determination acetaminophen [16, 22], therefore a specifically designed biosensor is relevant.

3.1 Reaction Scheme The biosensor containing catalase (E1 ) and peroxidase (E2 ) converts hydrogen peroxide (H2 O2 ). The catalase catalyses hydrogen peroxide splitting with oxygen (O2 ) production in the following process: 2H2 O2 → 2H2 O + O2 .

(32)

98

Biosensors Utilizing Consecutive and Parallel Substrates Conversion

The peroxidase catalyses a variety of substrates (S2 ) oxidation in the following peroxidase-catalysed process: 2H2 O2 + 2S2 + 2H+ → 2H2 O + P2 .

(33)

Further S1 denotes H2 O2 , and P1 -O2 . The remarkable feature of this conjugated system is molecular oxygen production, which can be selectively determined by a sensitive to oxygen electrode [55, 66]. Catalase (E1 ) and peroxidase (E2 ) catalyse S2 conversion in a more complex scheme and typically include intermediates [36].

3.2 Reaction Rates at Quasi-Steady State In case of hydrogen peroxide (S1 ) catalysed process (32), intermediate C1 is introduced: k11 E1 + S1 GGGGGGGA C1 + H2 O,

(34a)

k12 C1 + S1 GGGGGGGA E1 + P1 .

(34b)

The change of E1 and C1 in time t can be described as follows [6]: dE1 = −k11 E1 S1 + k12 C1 S1 , dt dC1 = k11 E1 S1 − k12 C1 S1 . dt

(35a) (35b)

By adding (35a) with (35b) and integrating the sum, we get E1 + C1 = E10 ,

(36)

where E10 is the initial as well as total concentration of the first enzyme. By applying quasi-steady state approximation [3, 55, 66] (dE1 /dt = dC1 /dt = 0) and inserting the expression of E1 , obtained from (36), into (35b) and equalling it to zero (by a), the concentration C1 can be expressed as follows: C1 = when k11 = k12 = k1 .

k11 E10 E10 , = k11 + k12 2

(37)

3 Modeling Dual Catalase-Peroxidase Bioelectrode

99

The change of hydrogen peroxide (S1 ) and oxygen (P1 ) concentrations can be written as follows: dS1 = −k11 E1 S1 − k12 C1 S1 = −k1 (E1 + C1 )S1 = −k1 E10 S1 , dt dP1 E10 = k12 C1 S1 = k12 S1 = 0.5k1E10 S1 . dt 2

(38a) (38b)

In case of acetaminophen (S2 ) catalysed process (33), intermediate C2 is introduced: k21 E2 + S1 GGGGGGGA C2 ,

(39a)

k22 C2 + S2 GGGGGGGA E2 + P2 .

(39b)

The change of E2 and C2 in time t can be described as follows: dE2 = −k21 E2 S1 + k22 C2 S2 , dt dC2 = k21 E2 S2 − k22 C2 S2 . dt

(40a) (40b)

By applying the same approach as for the first enzyme, Eqs. (40) are summed and the resulting sum is integrated, we get E2 + C2 = E20 ,

(41)

where E20 is the initial as well the total concentration of the second enzyme. At the quasi-steady state conditions when dE2 /dt = dC2 /dt = 0, the concentration C2 can be expressed as follows: C2 =

k21 E20 S1 . k21 S1 + k22 S2

(42)

Finally, we get the rate of acetaminophen S2 conversion to the second product (P2 ) concentrations, k21 k22 E20 S1 S2 dS2 = −k22 C2 S2 = − , dt k21 S1 + k22 S2

(43a)

dP2 k21 k22 E20 S1 S2 = k22 C2 S2 = . dt k21 S1 + k22 S2

(43b)

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Biosensors Utilizing Consecutive and Parallel Substrates Conversion

At the quasi-steady state, the reaction scheme utilizing two enzymes (E1 and E2 ) catalysed two substrates (S1 and S2 ) conversion to two products (P1 and P2 ) can be simplified as follows [6, 37]: E1 S1 GGGGGGA P1 ,

E2 S1 + S2 GGGGGGA P2 .

(44)

The molecular oxygen (P1 ) can be effectively detected by applying an oxygensensitive electrode. In the first phase of the biosensors action, only hydrogen peroxide (S1 ) is present in the solution. At the end of the first phase when the biosensor response reaches steady state, the second substrate (S2 , e.g. acetaminophen [37]) is poured into the solution, and the second phase of the biosensor action starts [6]. The relative change between the responses in the both phases is measured as the final response of the biosensor. This relative response indicates the concentration of the second substrate. The first phase is only used for the determination of the first substrate concentration and is not decisive when the concentration is known before the experiment [6, 37].

3.3 Two Compartment Reaction–Diffusion Model The mathematical model of the biosensor utilizing parallel substrate conversion as defined in (44) comprises of two compartments, a layer of two enzymes coimmobilized on the electrode surface and an outer diffusion layer [5, 6, 37].

3.3.1 Governing Equations Coupling the catalase (38) and peroxidase (43) catalysed reactions in the enzymatic layer with the one-dimensional-in-space diffusion, described by Fick’s law, leads to the following reaction–diffusion equations describing the biosensor operation in the enzymatic layer (a0 < x < a1 , t > 0): ∂S1e ∂ 2 S1e = DS1 e − r1 − r2 , ∂t ∂x 2

(45a)

∂S2e ∂ 2 S2e = DS2 e − r2 , ∂t ∂x 2

(45b)

∂P1e ∂ 2 P1e r1 = DP1 e + , ∂t 2 ∂x 2

(45c)

∂P2e ∂ 2 P2e = DP2 e + r2 , ∂t ∂x 2

(45d)

3 Modeling Dual Catalase-Peroxidase Bioelectrode

101

where x and t stand for space and time, respectively, S1e (x, t), S2e (x, t), P1e (x, t) and P2e (x, t) are the molar concentrations of the substrates S1 , S2 and the products P1 , P2 in the enzyme layer of the thickness d1 = a1 − a0 , respectively, DS1e , DS2e , DP1e and DP2e are the constant diffusion coefficients, and r1 and r2 stand for rates of reactions (44), r1 = k1 E10 S11 ,

r2 =

k21 k22 E20 S11 S21 , k21 S11 + k22 S21

(46)

where E10 and E20 are the total concentrations of the first and the second enzymes, respectively [6, 37]. Outside the enzyme layer only the mass transport by diffusion of the substrates and the products takes place. The governing equations for the outer diffusion layer (a1 < x < a2 ) are represented by the assumption that the external mass transport obeys a finite diffusion regime ∂ 2 Ui b ∂Ui b = DUi b , ∂t ∂x 2

U = S, P ,

i = 1, 2,

(47)

where Si b (x, t) and Pi b (x, t) are the molar concentrations of both substrates and both products in the outer diffusion layer (bulk) of the thickness d2 = a2 − a1 , and DSi b and DPi b are the constant diffusion coefficients, i = 1, 2 [6, 37]. 3.3.2 Initial Conditions At the beginning (t = 0) of the biosensor operation only the first substrate S1 is present in the buffer solution [6, 37]. No other substances are present in the enzymatic and diffusion layers, Ui e (x, 0) = 0,

U = S, P ,

i = 1, 2,

a0 ≤ x ≤ a1 ,

(48a)

Ui b (x, 0) = 0,

U = S, P ,

i = 1, 2, a1 ≤ x < a2 ,

(48b)

Pi b (a2 , 0) = 0,

i = 1, 2,

S1b (a2 , 0) = S10 ,  0, S2b (a2 , 0) = S20 ,

(48c) (48d)

if TS2 = 0, if TS2 > 0,

(48e)

where S10 and S20 are the concentrations of the first and the second substrates in the bulk solution, respectively, and TS2 is the time moment when the second substrate appears in the bulk [6, 37].

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Biosensors Utilizing Consecutive and Parallel Substrates Conversion

3.3.3 Boundary and Matching Conditions The substrates and the second product are electro-inactive substances, while the concentration P1 of the first product at the electrode surface is being permanently reduced to zero due to the electrode polarization (t > 0), DSi e

 ∂Si e  = 0, ∂x x=a0

i = 1, 2,

P1e (a0 , t) = 0,  ∂P2e  = 0. DP2e ∂x x=a0

(49a) (49b) (49c)

The concentrations of the substrates and the products in the bulk solution remain constant during the biosensor operation, S1b (a2 , t) = S10 ,  0, S2b (a2 , t) = S20 , Pi 2 (a2 , t) = 0,

(50a) t < TS2 , t ≥ TS2 ,

i = 1, 2.

(50b) (50c)

The fluxes of the substrates and the products through the stagnant external layer are assumed to be equal to the corresponding fluxes entering the surface of the enzyme membrane (t > 0),   ∂Ui e  ∂Ui b  DUi e = DUi b , ∂x x=a1 ∂x x=a1 Ui e (a1 , t) = Ui b (a1 , t),

i = 1, 2,

(51a) U = S, P .

(51b)

3.3.4 Biosensor Response The density of the measured current is usually assumed as the response of amperometric biosensors. The density i(t) of the current is directly proportional to the flux of the first reaction product at the electrode surface as defined by the Faraday and Fick laws [55, 66],  ∂P1e  . i(t) = ne F DP1 e ∂x x=a0

(52)

3 Modeling Dual Catalase-Peroxidase Bioelectrode

103

It is assumed that the system (45)–(51) approaches a steady state as t → ∞, IS1 S2 = lim i(t),

(53)

t →∞

where IS1 S2 is the steady state biosensor current as the steady state response to substrates S1 and S2 . The difference IS2 between the local steady state current IS1 at zero concentration of the substrate S2 and the overall steady state current IS1 ,S2 is preferable for investigating the dependence of the biosensor current on the concentration S20 of the second substrate S2 [6], IS2 = IS1 − IS1 S2 .

(54)

Since the density of the biosensor current varies in orders of magnitude with the concentrations of the substrates, it is reasonable to normalize the difference IS2 with the current IS1 , IS2 r =

IS2 IS − IS1 S2 = 1 , IS1 IS1

(55)

where IS2 r is the relative (dimensionless) steady state biosensor current or the relative response [6, 37].

3.4 Numerical Solution Because of the nonlinearity of the governing equations (45) the initial boundary value problem (45)–(51) can be analytically solved only for a specific set of the model parameters [17, 28, 31]. A uniform discrete grid in both directions (space x and time t) with 600 points in the space direction was introduced to find a numerical solution [6]. A CrankNicolson finite difference scheme has been built as a result of the difference approximation [18, 54]. The resulting system of linear algebraic equations was solved rather efficiently due to the tridiagonality of the matrix. The digital simulator has been programmed in C++ language [50]. In the numerical simulation, the density of the steady state biosensor response was assumed when the change of the current remains a relatively small during a relatively long time. The time TR needed to achieve given dimensionless decay rate ε can be assumed as the response time,     t  di(t)  0 dt i(t)

I (TR ) ≈ IS1 S2 .

(56)

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Biosensors Utilizing Consecutive and Parallel Substrates Conversion

The decay rate ε highly influences the assumed response time, TR → ∞, when ε → 0. In calculations, ε = 10−4 was used[6, 37]. The computational model of the biosensor was validated by known analytical solutions when the enzymatic reactions approach the first order kinetics, and the governing equations (45) reduce to linear equations. At certain concentrations of the substrates, the concentration of the first product P1 becomes invariant to the concentration of the second substrate S2 , and the governing equations (45) can be reduced to the following two linear equations: ∂ 2 S1e ∂S1e = DS1 e − v, ∂t ∂x 2

(57a)

∂P1e ∂ 2 P1e r1 = DP1 e + , 2 ∂t 2 ∂x

(57b)

where v stands for the effective rate of the substrate S1 reduction. The two compartment model based on governing equations (57) was solved analytically [58] (also see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”). The following cases when the reactions (44) approach the first order kinetics were considered: • S20 = 0: r2 = 0 and v = r1 ; • k21 S10  k22 S20 : r2 ≈ k21 E20 S1e and v ≈ r1 + k21 E20 S1e = (k1 E10 + k21 E20 )S1e ; • k21 S10  k22 S20 and k1 E10  k21 E20 : r2  r1 and v = r1 ; • k21 S10  k22 S20 and k1 E10 S10  k21 E20 S20 : r2  r1 and v = r1 . At the listed conditions, the relative difference between the steady state current density IS1 S2 calculated numerically and that obtained analytically was less than 1% at different concentrations S10 , S20 , E10 and E20 . The following values of the model parameter were constant in all the calculations [2, 15, 36, 43]: d2 = 0.1 mm, k1 = 107 M−1 s

ne = 2, −1

−1

, k21 = 7.1 × 106 M−1 s

DSi e = DPi e = 3 × 10−10 m2 /s,

−1

, k22 = 6 × 106 M−1 s

DSi b = DPi b = 2DSi e ,

,

(58)

i = 1, 2.

3.5 Dynamics of the Biosensor Current The evolution of the biosensor current i at six different concentrations S20 of the second substrate S2 is depicted in Fig. 4. The biosensor response was simulated at the thickness d1 = 0.1 mm of the enzyme layer and the concentration S10 = 5 mM

3 Modeling Dual Catalase-Peroxidase Bioelectrode

105

0.7

1

0.6

2 3

i, μA/mm2

0.5 0.4

4

0.3

5

0.2 0.1 0.0 0

50

100

150

200

250

300

350

t, s Fig. 4 The dynamics of the biosensor current density i(t) at six concentrations S20 of the second substrate: 10−6 (1), 10−4 (2), 10−3 (3), 10−2 (4) and 10−1 (5) M, S10 = 5 mM, d1 = 0.1 mm and TS2 = 200 s

of the first substrate. The concentrations of catalase and peroxidase were assumed equal, E10 = E20 = 1 µM. Values of the second substrate concentration S20 varied from 1 µM up to 0.1 M. Values of the other parameters were as defined in (58) [6]. As one can see in Fig. 4, the maximal current density is achieved at t = TS2 = 200 s, when the second substrate was poured into the buffer solution. A greater concentration of the second substrate (S20 > 10−4 M) corresponds to a greater change of the current density (curves 4–6). Smaller concentrations (S20 ≤ 10−4 M) of the second substrate have no noticeable impact on the biosensor response, and the relative difference is less than a few percent [6].

3.6 Biosensor Response to the Second Substrate Figure 5 shows the impact of the second substrate concentration S20 on the biosensor steady state relative current IS2 r . Values of enzyme concentrations (E10 and E20 ) varied from 10−8 up to 10−5 M [6]. As one can see in Fig. 5, all values of the steady state relative current IS2 r are less even than 0.01 when the concentration E10 of the peroxidase is relatively large and the concentration E20 of the catalase is relatively small (curves 4–6). In the opposite case (curves 1–3), the steady state biosensor response varies in several orders of magnitude case. The relative response IS2 r reaches even unity at very low concentration E10 and high concentrations of E20 and S20 (curves 1 and 2) .

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Biosensors Utilizing Consecutive and Parallel Substrates Conversion

1 2 3 4 5 6

100

2

IS r

10-1

10-2

10-3

10-4 10-6

10-5

10-4

10-3

10-2

S2 , M 0

Fig. 5 The steady state relative current IS2 r versus the second substrate concentration S20 at different concentrations of the enzymes E1 : 10−8 (1), 10−7 (2), 10−6 (3 and 5), 10−5 (4 and 6) and E2 : 10−8 (5 and 6), 10−7 (3 and 4), 10−6 (2), 10−5 (1), S10 = 1 mM, d1 = 0.1 mm and the other parameters are as defined in (58)

Additional numerical experiments showed that the length of the linear range of the calibration curve of the catalase-peroxidase biosensor is directly proportional to the concentration S10 of the first substrate [5–7, 37].

3.7 Impact of the Reaction Rates The following dimensionless reaction rate as the ratio ξ between the concentrations of the catalase and peroxidase and their kinetic reaction rates was introduced [6]: ξ=

k1 E10 . k21 E20

(59)

In order to investigate the dependence of the steady state biosensor response on the dimensionless reaction rate ξ , the response was simulated by changing the rate ξ in a few orders of magnitude (from 10−3 to 103 ), at three very different concentrations (10−6, 10−4 and 10−1 M) of the second substrate, two values of the thickness d1 (10−6 and 10−3 m) of the enzyme layer and a moderate concentration (10−3 M) of the first substrate. The simulation results are depicted in Fig. 6 [6]. The rate ξ was varied by changing the concentrations E10 and E20 and keeping the kinetic reaction rates k1 and k21 constant [6].

3 Modeling Dual Catalase-Peroxidase Bioelectrode

107

100 10-1

I S2 r

10-2 10-3 10-4 10-5 10-2

1 4

2 5 10-1

3 6 100

101

102

103

Fig. 6 The steady state relative current IS2 r versus the the dimensionless reaction rate ξ at three concentrations S20 of the second substrate: 10−6 (1, 4), 10−4 (2, 5), 10−1 (3, 6) M, two thicknesses d1 of the enzyme layer: 10−6 (1, 2 and 3), 10−3 (4, 5 and 6) m and the concentration S10 = 1 mM of the first substrate. The other parameters are as defined in (58)

As one can see in Fig. 6, at high values of the dimensionless reaction rate ξ (ξ > 10, k1 E10 > 10k21 E20 ), the steady state relative current IS2 r approximately linearly decreases with the increasing of the rate ξ . However, at low values of the ξ (ξ < 0.1), the current IS2 r practically does not change when changing the rate xi. The change of the steady state response is more noticeable at greater concentrations of the second substrate (S20 = 0.1 M, curves 3 and 6) than at lower ones (S20 = 10−6 M, curves 1 and 4). At S20 = 0.1 M the steady state current IS2 r changes in three orders of magnitude (curves 3 and 6), while at S20 = 10−6 M the IS2 r changes only in one-two orders of magnitude.

3.8 Model Modifications The two compartment mathematical model (45)–(52) describes the dynamics of the amperometric two-enzyme electrode at which surface the reaction product is being permanently reduced to zero. A wide variety of electrodes are used in practice for the determination of the product concentrations [8, 53, 55, 66].

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Biosensors Utilizing Consecutive and Parallel Substrates Conversion

3.8.1 Clark-Type Electrode A widely used Clark-type electrode (CE) measures oxygen on an electrode surface, while the oxygen itself is not consumed to generate current [21]. Alternatively, the electrode can be so polarized that the concentration of the reaction product at the electrode surface is being permanently reduced to zero as defined in the boundary condition (49b). An electrode of this type is called NonClark-type electrode (NCE). So, the model (45)–(52) corresponds to two-enzyme NCE. The mathematical model (45)–(52) can be adopted to the corresponding CE by replacing the boundary condition (49b) with the following equation: DP1e

 ∂P1e  = 0, ∂x x=a0

(60)

and the density of the output current i(t) have to be calculated as follows [55, 66]: i(t) = ks F P1e (a0 , t),

t > 0,

(61)

where ks is the heterogenic constant calculated experimentally. When comparing steady state responses simulated by two different models corresponding to CE and NCE, no significant difference was noticed. At the specific values of the model parameters only the time of the steady state can differ several times due to difference in the electrode type [7].

3.8.2 Introducing Dialysis Membrane The two compartment mathematical model (45)–(52) can be improved by introducing the third compartment corresponding to a dialysis membrane separating the enzyme layer from the bulk solution [37]. Only the mass transport by the diffusion of both substrates and both products can be considered in the dialysis membrane. Modeling of mass transport in the dialysis membrane is discussed in Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”. The improved three-compartment model was applied for simulation of the action of a specific dual catalase-peroxidase biosensor [37]. Acetaminophen (Ac) was used as a model compound for measurements of bioelectrode parameters. Since oxygen was measured by a Clark-type oxygen bioelectrode, the boundary conditions (60) and (61) have been applied instead of conditions (49) and (52). Simulation results have been compared with experimental results of the bioelectrode response. The digital simulation of the catalase-peroxidase biosensor action produced a steady state response that was 11% smaller than experimentally determined. This was explained by the decrease of the membrane thickness (about 19%) in

4 Biosensors Response to Mixture of Compounds

109

the bioelectrode. The simulations of the bioelectrode response revealed that the bioelectrode acts in the diffusion limiting condition at almost all biocatalytical membrane compositions. Only at very low peroxidase concentrations the sensitivity decreased proportionally to the peroxidase concentration indicating a kinetic regime of bioelectrode action [37].

4 Biosensors Response to Mixture of Compounds In this section, mathematical models of amperometric multi-enzyme biosensors are considered. The biosensors utilizing parallel multi-substrate conversion to multiproduct. Two types of enzymatic reactions are considered: with interaction [13, 14, 40] and with no interaction [9–11] between the analysed substrates and reaction products. The mathematical model of such biosensors allows to simulate the biosensor response to a mixture of compounds (substrates).

4.1 Reaction Scheme Let us consider the response of an amperometric biosensor to a mixture of substrates. Assuming no interaction between the substrates (compounds) of the mixture, enzyme-catalysed reactions are considered, Ek

Sk −→ Pk ,

k = 1, . . . , K,

(62)

where the substrate Sk binds to the specific enzyme Ek to form an enzyme–substrate complex, following the substrate Sk conversion to the product Pk , k = 1, . . . , K, K is the number of the substrates present in the mixture [9, 10]. An idealized biosensor is assumed to be an enzyme electrode, having a layer of multi-enzyme immobilized onto the surface of the probe.

4.2 Mathematical Model with No Substrates Interaction Assuming no interaction between the analysed substrates (compounds) of the mixture, the symmetrical geometry of the electrode, the homogeneous distribution of immobilized enzyme in the enzyme membrane, and considering one-dimensionalin-space mass transport by the diffusion, coupling the enzyme reaction with the

110

Biosensors Utilizing Consecutive and Parallel Substrates Conversion

diffusion as described by Fick’s law leads to the following equations (x ∈ (0, d), t > 0): ∂Sk ∂ 2 Sk Vmaxk Sk = DSk − , ∂t ∂x 2 KMk + Sk

(63)

∂ 2 Pk ∂Pk Vmaxk Sk = DPk + , ∂t ∂x 2 KMk + Sk

k = 1, . . . , K,

where x and t stand for space and time, respectively, Sk (x, t) is the concentration of the substrate Sk , Pk (x, t) is the concentration of the reaction product Pk , d is the thickness of the enzyme layer, DSk , DPk are the diffusion coefficients of the substrate Sk and the product Pk , respectively, Vmaxk is the maximal enzymatic rate of the biosensor attainable with that amount of enzyme Ek , when the enzyme is fully saturated with the substrate (compound) Sk , and KMk is the Michaelis constant [9, 10]. The biosensor operation starts when some substrate appears over the surface of the enzyme layer. This is defined by the initial conditions (t = 0) Sk (x, 0) =

 0,

0 ≤ x < d,

Sk 0 ,

Pk (x, 0) = 0,

x = d,

(64)

0 ≤ x ≤ d,

k = 1, . . . , K,

where Sk 0 is the concentration of substrate Sk in the bulk solution. At the electrode surface (x = 0), the boundary conditions depend on the electric activity of the substance. Following the scheme (62) the substrates are electroinactive substances, DSk

∂Sk  = 0,  ∂x x=0

t > 0.

(65)

All the reaction products are assumed to be electro-active substances. In the case of amperometry, the potential at the electrode is chosen to keep zero concentration of the product, Pk (0, t) = 0,

t > 0.

(66)

If the substrate is well-stirred and in a powerful motion, then the diffusion layer (0 < x < d) remains at a constant thickness. Consequently, the concentration of all the substrates as well as the products over the enzyme surface (bulk solution/membrane interface) remains constant during the biosensor action, Sk (d, t) = Sk 0 , Pk (d, t) = 0,

t > 0,

k = 1, . . . , K.

(67)

4 Biosensors Response to Mixture of Compounds

111

The anodic current depends upon the flux of the electro-active analyte (product) at the electrode surface (x = 0). The density ik (t) of the biosensor current, as a result of the reaction of the substrate Sk with the enzyme at time t, is proportional to the concentration gradient of the product Pk at the surface of the electrode as described by the Faraday and the Fick laws [29, 58], ik (t) = nek F DPk

∂Pk   , ∂x x=0

(68)

where nek is a number of electrons involved in a charge transfer at the electrode surface and F is the Faraday constant. Assuming that the overall biosensor response to a mixture represents the total sum of individual responses to each constituent substrate and having values of the current density ik (t) for all compounds, k = 1, . . . , K, the total density i(t) of the biosensor current can be calculated additively [9, 10], i(t) =

K 

ik (t).

(69)

k=1

4.3 Solution of the Problem When analysing the problem (63)–(67), one can notice that there is no direct relationship between pairs of the unknown variables Sk1 , Pk1 and Sk2 , Pk2 , for all k1 , k2 = 1, . . . , K if k1 = k2 . Because of this, the initial boundary value problem (63)–(67) consisting of 8K equations can be split into K problems containing only eight equations (63)–(67) at a given k, k = 1, . . . , K. The problem (63)–(67) formulated for a given compound Sk1 of the mixture, can be solved individually and independently from the problem formulated for another compound Sk2 , k1 , k2 = 1, . . . , K, k1 = k2 [9, 10]. Let us assume the formulation of the problem (63)–(67) for a single substrate S = Sk and the reaction product P = Pk , k = K = 1. Let Vmax be the maximal enzymatic rate of the modeled biosensor, KM is the corresponding Michaelis constant, S is the concentration of the substrate S, and P is concentration of the reaction product P. The problem (63)–(67), formulated for a single substrate S and the reaction product P, can be solved numerically as discussed in Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”. In the common case of K compounds, having responses of the biosensor to each constituent compound, Eq. (69) allows to calculate the common biosensor response to the mixture of K compounds [9]. Consequently, to obtain values I (tj ) of the density of the total biosensor current, it is required [9, 10]:

112

Biosensors Utilizing Consecutive and Parallel Substrates Conversion

1. to run computer simulation K times to obtain values Ik (tj ) of the density of the biosensor current for each compound of the mixture, k = 1, . . . , K, j = 1, . . . , N; 2. to calculate the density of the total biosensor current as defined by (69), where j = 0, . . . , N, t0 = 0, tN = T , T is the full time of the biosensor operation. In step (1) only values of the following parameters: DSk , DPk , Vmaxk , KMk and Sk 0 vary when one computer simulation changes to the next one.

4.4 Generation of Pseudo-Experimental Data An accurate and reliable calibration of an analytical system as well as the proper test of the methods of chemometrics requires a lot of experimental data [4, 19, 45]. Assuming a good enough adequacy of the mathematical model to the physical phenomena, the data synthesized using a computer simulation can be employed instead of the experimental ones [10, 11, 40]. The computational experiments are usually much cheaper and faster than the physical ones. The computer simulation is perfectly reasonable when the biosensors to be used in practice are still in a stage of development. Then the design of smart biosensors to be used in analytical systems and the development of effective methods of data analysis may take place in parallel. The computer simulation can be applied for generating data to be used for a calibration of an amperometric biosensor [9–11]. A system of four (K = 4) compounds was considered. Each compound of eight (M = 8) different concentrations was used in the calibration to have the biosensor response to a wide range of substrate concentrations. It is necessary to solve the problem (63)–(67) for a given component Sk numerically K × M = 4 × 8 = 32 times at 4 different values of the maximal enzymatic rate and 8 values of the substrate concentration. The following values of the parameters of the mathematical model (63)–(67) were assumed constant in all numerical experiments: DSk = DPk = 300 µm2/s, KMk = 100 µM, ne k = 2,

d = 200 µm,

k = 1, . . . , K.

(70)

The enzymatic reaction for each component of the mixture was characterized by the individual maximal enzymatic rate, Vmaxk = 10−k mM/s,

k = 1, . . . , K.

(71)

4 Biosensors Response to Mixture of Compounds

113

To have the biosensor responses for a wide range of the substrate concentrations, the following values of the Sk0 for every of the substrates S1 , . . ., SK of the mixture were taken: Sk 0 ∈ {ςm × S0 , m = 1, . . . , M}, K = 4,

M = 8,

k = 1, . . . , K,

S0 = 10 µM,

ς1 = 1, ς2 = 2, ς3 = 4, ς4 = 8,

(72)

ς5 = 12, ς6 = 16, ς7 = 32, ς8 = 64. In the simulation of the biosensor response for all the values defined in (70)–(72), only the values of three parameters, Vmaxk , KMk and Sk 0 , of the model vary when one computer simulation changes to the next one. Let ik m (tj ) be a value of the density ik (tj ) of the biosensor current calculated at the concentration Sk 0 = Sk 0m of the substrate Sk , m = 1, . . . , M, k = 1, . . . , K, j = 0, . . . , N, t0 = 0, tN = T . Having M numerical solutions (M sets of values of the biosensor current) ik m (tj ), j = 0, . . . , N, for each k = 1, . . . , K (in total K × M solutions), the full factorial ik m1 , . . . , kmK (tj ) of M K = 84 = 4096 solutions can be produced additively [9, 10], im1 ,...,mK (tj ) =

K 

ik mk (tj ),

k=1

(73)

m1 , . . . , mK = 1, . . . , M, j = 0, . . . , N. During an ordinary computer simulation, the values of the biosensor current were stored in a file every second of the simulation. Thus, N = T values of ik m (tj ), tj = j (seconds), j = 1, . . . , N, for each k = 1, . . . , K and m = 1, . . . , M were produced as a result of the computer simulation of the biosensor response (in total K × M × N values). The results of K × M simulations were stored into K × M files for both modes of the analysis [9, 10]. Later, using an additional simple utility of summation, a matrix M K × N of the biosensor response data was produced following (69) and stored in a new file. The calculation results are depicted in Fig. 7 which shows only every 64th of M K (the full factorial of M × K) simulated biosensor responses for K = 4 values of the maximal enzymatic rate and M = 8 substrate concentrations. An evolution of biosensor currents is depicted for the first 80 s of the biosensor action only because of a marginal change of the biosensor current at greater values of time t. The data synthesized using the computer simulation was successfully applied to calibrate and validate the sensor system based on an amperometric biosensor and an artificial neural network [10, 11, 40]. Coupling the biosensors with artificial neural networks is growing in importance as a tool for multi-component analysis [27, 42, 48, 51, 71].

Biosensors Utilizing Consecutive and Parallel Substrates Conversion

i, nA/mm2

114 65 60 55 50 45 40 35 30 25 20 15 10 5 0 0

10

20

30

40

50

60

70

80

t, s Fig. 7 Every 64th biosensor response curve of M K responses at K = 4 values of the maximal enzymatic rate and M = 8 substrate concentrations

Data collected in complex processes contains a wealth of redundant information , since the variables are collinear. The data preprocessing methods can be applied in such situations to enhance the relevant information, to make the resulting models simpler and easier to interpret. The correlation coefficients analysis and the principal component analysis are among the approaches widely used to reduce the dimensionality of the vectors (im1 ,...,mK (t0 ), . . ., im1 ,...,mK (tN )) of input data [41, 46].

4.5 Modeling Substrates Interaction A special case of multi-substrates conversion by single nonspecific enzyme has a big practical interest. If substrates interact with a single enzyme without formation of other complex containing two or more molecules of substrate [49, 70], each substrate acts as a competitive inhibitor to others [13, 14, 40]. The task of this modeling is to calculate the dependence of biosensor response on the concentration of analytes and enzyme specificity. 4.5.1 Reaction Scheme Consider a mono-enzyme catalysed multi-substrate conversion [49, 70], k1 k

k2 k

E + Sk  ESk −→ E + Pk , k−1 k

k = 1, . . . , K,

(74)

4 Biosensors Response to Mixture of Compounds

115

where E denotes the enzyme, Sk is the substrate, ESk stands for the enzyme and substrate complex, Pk is the reaction product, kinetic constants k1k , k−1 k and k2k correspond to the respective reactions: the enzyme–substrate interaction, the reverse enzyme–substrate decomposition and the product formation, and K is the number of substrates to be analysed. When substrates S1 , . . . , SK (K > 1) react with a single enzyme E without formation of any multi-fold complex containing two or more substrates, and the substrates do not combine directly with each other, then in mixtures of S1 , . . . , SK each substrate acts as a competitive inhibitor of the others [26, 33]. Particularly, 5α-androstan-3-one and 5α-androstan-3,16-dione are special case of substrates and 20β-hydroxy steroid-NAD oxidoreductase (EC 1.1.1.53) is a sample of enzyme E [49]. The biosensor to be modeled was intended to analyse a mixture of k substrates (compounds) [13, 14, 40]. Practical mono-enzyme analytical systems are usually limited to determining only a few (often to two) substrates [49]. The reactions in the network (74) are usually of different rates [26, 55]. The large difference of timescales in the reactions creates difficulties for simulating the temporal evolution of the network and for understanding the basic principles of the biosensor operation. To sidestep these problems, the quasi-steady state approach (QSSA) is often applied [25, 39, 61]. According to the QSSA, the concentration of the intermediate complex does not change with time, and the reaction network (74) reduces to the following reaction: E

S1 + S2 + · · · + SK −→ P1 + P2 + · · · + PK .

(75)

4.5.2 Model Equations Applying QSSA and coupling the enzyme-catalysed reactions (75) in the enzyme layer with the one-dimensional-in-space diffusion, described by Fick’s law, lead to the following system of 2K equations of the reaction–diffusion type (t > 0, 0 < x < d): ∂ 2 Sk Vmaxk Sk ∂Sk  , = DSk − ∂t ∂x 2 KMk 1 + K S /K j M j j =1 ∂Pk ∂ 2 Pk Vmaxk Sk   , k = 1, . . . , K, = DPk + 2 ∂t ∂x KMk 1 + K S /K j M j j =1

(76)

where Vmaxk = k2k E0 , KMk = (k−1 k + k2k )/k1k , E0 is the total concentration of the enzyme, k = 1, . . . , K, the other notation is the same as in (63).

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Biosensors Utilizing Consecutive and Parallel Substrates Conversion

The entire formulation of the mathematical model corresponding to reaction (75) can be derived from the model (63)–(69) by replacing only Eq. (63) with Eq. (76) [13, 14, 40].

4.5.3 Numerical Simulation Since the reaction term in (76) includes concentrations of all K substrates, the initial boundary value problem (76), (64)–(67) cannot be solved by splitting the problem to K subproblems solved by the same procedure as it was in solving (63)–(67). Because of the nonlinearity of the governing equations (76), the initial boundary value problem (76), (64)–(67) can be analytically solved only for a specific set of the model parameters [17, 58]. The problem was numerically solved for the particular case of two (K = 2) [13, 14] and four (K = 4) substrates [40]. The finite difference technique was applied for discretization of the mathematical model [17, 54]. A uniform discrete grid in both directions, space x and time t, was introduced to find a numerical solution. An explicit finite difference scheme has been built as a result of the difference approximation of the model equations [18, 23]. The digital simulator has been programmed in C++ language [50]. To make the difference scheme stable the time step size τ was found from the sufficient stability conditions [12, 54]. The size τ = 10−4 s was sufficient for all simulations when dividing the enzyme layer into 200 points.

5 Concluding Remarks The mathematical model (45)–(51) of an amperometric biosensor based on catalaseperoxidase was used to simulate its action, to investigate the kinetic peculiarities of the response and to optimize the configuration of the biosensor. This type of the biosensor was applied for quantitative evaluation of the target (the second substrate) concentrations less than the concentration of the hydrogen peroxide (the first substrate) (Fig. 5). The steady state relative biosensor current is a monotonous decreasing function of the relative enzymatic activity ξ of the catalase to the peroxidase. At high values of the relative activity ξ (ξ > 10), the current IS2 r approximately linearly decreases with increasing ξ , while at low values of ξ (ξ < 0.1), the current IS2 r is practically insensitive to changes in ξ (Fig. 6). Biosensors dedicated to analysis of compounds in complex media were analysed assuming no interaction between the components of a mixture. The mathematical model (63)–(67) describes an operation of these type of biosensors. The initial boundary value problem (63)–(67) was solved for each component independently. The total biosensor current was calculated additively from the individual biosensor responses to each component of the mixture.

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Computer simulation was used also to generate pseudo-experimental biosensor response to mixtures of compounds. If K is a number of mixture component and M is a number of different concentrations of each component, then the biosensor responses for full factorial of mixtures (M K samples) can be synthesized by a simple routine of summation from the results of K × M computer simulations of the response.

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Contents 1 2 3

4

5

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biosensors Utilizing Cyclic Enzymatic Substrates Conversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biosensors with Electrochemical and Enzymatic Conversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Finite Difference Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Concentration Profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Peculiarities of the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biosensors Acting in Trigger Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Mathematical Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Finite Difference Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Simulated Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Peculiarities of the Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122 123 126 126 127 128 129 135 135 140 142 144 152

Abstract The sensitivity of biosensors can be notably increased by a cyclic conversion of substrates or reaction products. In this chapter, mathematical models of different types of amperometric biosensors utilizing the cyclic conversion are modeled and analysed at transient conditions assuming an absence of outer diffusion limitations. A specific type of highly sensitive biosensors utilizing the substrate cyclic conversion in single enzyme membrane has been analytically modeled assuming the first-order reaction kinetics. The amplification of biosensor response by conjugated electrochemical and enzymatic substrate conversions is modeled by reaction–diffusion equations containing a nonlinear term related to Michaelis– Menten kinetic of the enzymatic reaction. The trigger of the response of biosensors utilizing substrate (analyte) conversion following the cyclic product conversion has been modeled and analysed computationally, too. The simulated response of the biosensors acting in two trigger schemes is compared with the response of a single enzyme biosensor utilizing Michaelis–Menten kinetics. The numerical experiments demonstrated significant gain in the biosensor sensitivity when the biosensor response was under diffusion control. © Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_4

121

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Biosensors Response Amplification with Cyclic Substrates Conversion

Keywords Biosensor response · Signal amplification · Cyclic substrates conversion · Stationary/transient state · Mathematical model

1 Introduction The detection limit of the enzyme electrodes depends on the sensitivity of amperometric systems [9, 17, 33, 39]. The sensitivity can be notably increased by a cyclic conversion of the substrate or the product [14, 19, 22, 32, 34]. The cyclic conversion of the substrate and the regeneration of the analyte are typically performed by using a membrane containing two enzymes [19, 30]. A specific type of highly sensitive amperometric electrodes were developed by utilizing the substrate cyclic conversion in single enzyme membranes [12, 20, 23, 27, 34]. In these electrodes the cyclic conversion of the substrate was carried out by a conjugation of the enzymatic reaction with chemical or electrochemical process. Chemical conversion is a highly suitable approach for increasing the sensitivity of biosensors [13, 15, 16, 18, 25, 28, 38]. If a biosensor contains an enzyme that starts analyte conversion following the cyclic product conversion the scheme of the biosensor action can be called as “triggering” [23]. An amperometric detection of alkaline phosphatase based on hydroquinone recycling might be an example of this type of conversion [10]. The substrate of the alkaline phosphatase, i.e. p-hydroxyphenyl phosphate, is usually hydrolysed by alkaline phosphatase to hydroquinone. The hydroquinone, instead of being detected directly, enters into an amplification cycle where it is oxidized to quinone at the electrode surface and then reduced back to hydroquinone by glucose oxidase in the presence of glucose. The consumption-regeneration cycle of the hydroquinone resulted in an amplification factor about eight. Another example of utilizing trigger scheme might be a high sensitive determination of βgalactosidase used as a label in heterogeneous immunoassay [29]. As substrate p-aminophenyl-β-galactopyranoside was used. Produced p-aminophenol that is electrochemically active compound can be detected directly [31]. To increase the sensitivity of the determination, the p-aminophenol is entered into a bioelectrocatalytic amplification cycle using glucose dehydrogenase (GDH). Both presented schemes include enzymatic trigger reactions together with the electrochemical and enzymatic amplification steps. Therefore, by analogy with the electrochemical nomenclature they may be abbreviated as acting following the CEC mechanism. The triggering of the consecutive substrate conversion can also be realized by an enzymatic conversion of substrate (trigger reaction) following the second enzymatic reaction and electrochemical conversion. This scheme can be abbreviated as CCE. The scheme may be realized, for example, by using peroxidase and glucose dehydrogenase. The peroxidase produces an oxidized product that is reduced by the GDH so realizing the cyclic conversion of the product. Mathematical models have been widely employed to investigate the kinetic peculiarities of the amperometric biosensors realizing the response amplification with cyclic conversion of substrates or products [7, 11, 19, 35–37]. Models coupling

2 Biosensors Utilizing Cyclic Enzymatic Substrates Conversion

123

the enzyme-catalysed reaction with the diffusion in an enzyme layer (membrane) are usually used. In cases when the analyte is assumed to be well-stirred and in powerful motion, the mass transport by diffusion outside the enzyme membrane is usually neglected. In this chapter, different amperometric biosensors with cyclic conversion are modeled at transient conditions assuming that there exist no outer diffusion limitations. In the next section, a biosensor utilizing cyclic substrate conversion in a single enzyme membrane is analytically modeled assuming the first-order reaction kinetics [24]. Then, a biosensor with the amplification by conjugated electrochemical and the enzymatic substrate conversions is modeled numerically [4, 6, 23]. And finally, biosensors acting in two trigger schemes, CEC and CCE, are mathematically described and numerically analysed [5, 21]. The simulated response of the trigger biosensors is compared with the response of a single enzyme biosensor utilizing Michaelis–Menten kinetics.

2 Biosensors Utilizing Cyclic Enzymatic Substrates Conversion Rich biocatalytical possibilities permit to construct different systems utilizing the cyclic substrates conversion. The cyclic conversion of substrates in an enzyme membrane may considerably increase the sensitivity of the biosensor. The cyclic substrates conversion takes place according to the following scheme: k1

k2

S −→ Z −→ S + P,

(1)

where S is the substrate, Z is the intermediate, P is the electrochemically active product, and k1 and k2 are the first-order rate constants (Vmax /KM ) of the enzymatic conversion of the compounds. Assuming that there exist no outer diffusion limitations and the conversion of the compounds in the biocatalytic membrane follows the first-order reaction kinetics, the cyclic substrates conversion is described by a system of linear reaction–diffusion equations [24], ∂ 2S ∂S = De 2 − k1 S + k2 Z , ∂t ∂x ∂Z ∂ 2Z = De 2 + k1 S − k2 Z , ∂t ∂x

(2)

∂P ∂ 2P = De 2 + k2 Z , ∂t ∂x where S, Z, P correspond to the concentrations of the substrate, the intermediate and the product, respectively.

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Biosensors Response Amplification with Cyclic Substrates Conversion

Governing equations (2) together with the initial, S(x, t) = 0,

Z(x, t) = 0,

P (x, t) = 0,

0 ≤ x < d, t = 0,

(3)

and the boundary, S(x, t) = S0 , De

Z(x, t) = 0,

∂S(x, t) = 0, ∂x

De

P (x, t) = 0,

∂Z(x, t) = 0, ∂x

x ≥ d, t > 0,

P (x, t) = 0,

x = 0, t > 0,

(4)

conditions form together the initial boundary value problem, which was solved using the Green function [1, 24]. The transient behaviour of the cyclic conversion is expressed by the dynamics of the density i(t) of the output current,  dP  4k1 k2 S0 i(t) = ne F De = ne F De  dx x=0 d2 (5) ∞  ∞  (−1)m+1 (t) , × 2 2 γ (4γn2 De − γ2m−1 De )(γ2m−1 De + k1 + k2 ) n=1 m=1 2m−1 where (t) = 1 − + −

2 2 De + k1 + k2 )4γn2 De exp(−γ2m−1 De t) (γ2m−1 2 (k1 + k2 )(4γn2 De − γ2m−1 De ) 2 2 (γ2m−1 De + k1 + k2 )γ2m−1 De exp(−4γn2 De t)

2 2 (k1 + k2 − 4γn2 De + γ2m−1 De )(4γn2 De − γ2m−1 De ) 2 2 De exp(−(γ2m−1 De + k1 + k2 )t) 4γn2 De γ2m−1 2 (k1 + k2 )(k1 + k2 − 4γn2 De + γ2m−1 De )

.

In limiting cases, the expression of (5) becomes simpler. When k1 + k2 > De /d 2 and m = 1, the current is expressed as   ∞ k1 k2 S0  32 exp −(π/2d)2 De t γn2 De i(t) = I − ne F De , (6) (k1 + k2 )d π(4γn2 De − (π/2d)2 De )2 n=1

where I is the steady state response of the biosensor S0 k1 k2 I = ne F De d k1 + k2



1 d2 − 2De k1 + k2

 1−

1 cosh βd

(7)

and β = ((k1 + k2 )/De )1/2 . The kinetic behaviour of the biosensor is mainly determined by the diffusion modules α1 d = (k1 /De )1/2 d and α2 d = (k2 /De )1/2 d. Two important conclusions

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can be drawn from the Eq. (7): 1. under the kinetic control of the first or second reaction (α1 d < 1 or α2 d < 1) the amplification of the signal does not take place; 2. at a high enzymatic activity (α1 d > 1 and α2 d > 1) the response of the biosensor increases by a value which is directly proportional to the square of the membrane thickness: I = Id

α12 α22 d 2 k1 k2 d 2 = Id , 2De (k1 + k2 ) 2(α12 + α22 )

(8)

where Id corresponds to the diffusion controlled response of the biosensors containing single enzyme Id = ne F De

S0 . d

(9)

The amplification is rapidly increased by the rise of the enzymatic activity. For example, at α1 d = α2 d = 4, the fourfold increase of the sensitivity occurs. At α1 d = α2 d = 10, the amplification enlarges 25-fold. The possibility of a considerable increase in the sensitivity of the biosensors by means of chemical amplification was demonstrated using alcohol dehydrogenase with cyclic coenzyme (NAD) conversion and in other biocatalytical systems [22]. So, it follows that at βd ≥ 20 a 95% response is practically constant and has a minimal value of 1.31d 2/De , i.e. during the cyclic substrate conversion even at high enzyme rate the kinetics of biosensor response is 3.54 times slower than the substance diffusion through the membrane [24]. In the second limiting case when k1 + k2 < De /d 2 , (5) takes the following form: i(t) = I − ne F De

∞ k1 k2 S0  32γn2 De (t) , (k1 + k2 )d π[4γn2 De − (π/2d)2 De ]

(10)

n=1

where (t) =

exp(−(π/2d)2 De t) 4γn2 De − (π/2d)2 De +

(π/2d)2 De exp(−(π/2d)2 De t + k1 t + k2 t) . ((π/2d)2 De + k1 + k2 )((π/2d)2 De + k1 + k2 − 4γn2 De )

Summing over the first 10 terms of the series in this equation indicates that when βd < 0.1 the response is reached more slowly, the time t95 of 95% response equals 1.99d 2/De [24].

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Biosensors Response Amplification with Cyclic Substrates Conversion

3 Biosensors with Electrochemical and Enzymatic Conversion This section presents a model allowing the computer simulation of the response of a biosensor, utilizing the conjugated electrochemical and the enzymatic substrate conversion [4, 6, 26]. The scheme of the substrate (S) electrochemical conversion to the product (P) following catalysed with the enzyme (E) the product conversion to the substrate is considered [23], E

S −→ P −→ S.

(11)

In the case of phenol sensitive biosensors , for example, 1, 2-benzoquinone (S) is electrochemically reduced and pyrocatechol (P) formed is oxidized in the membrane with immobilized laccase [23].

3.1 Mathematical Model Assuming the symmetrical geometry of the electrode and the homogeneous distribution of the immobilized enzyme in the membrane, the dynamics of the biosensor can be described by the reaction–diffusion system, ∂ 2S Vmax P ∂S = DS 2 + , ∂t ∂x KM + P ∂P ∂ 2P Vmax P = DP , − ∂t ∂x 2 KM + P

(12) x ∈ (0, d),

t > 0,

where x and t stand for space and time, respectively, S(x, t) and P (x, t) are the substrate and the reaction product concentrations, respectively, d is the thickness of the enzyme membrane, DS , DP are the diffusion coefficients, Vmax is the maximal enzymatic rate and KM is the Michaelis constant [6]. Let x = 0 represent the electrode surface, and x = d represent the boundary layer between the analysed solution and the enzyme membrane. Assuming the zero concentration of the reaction product in the bulk solution, the initial conditions are the same as in the case of amperometric biosensors with no cyclic conversion, S(x, 0) = 0,

x ∈ [0, d) ,

S(d, 0) = S0 , P (x, 0) = 0,

x ∈ [0, d) ,

P (d, 0) = 0, where S0 is the concentration of the substrate in the bulk solution [6].

(13)

(14)

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127

In the scheme (11) the substrate is an electro-active substance. Due to the amperometry the electrode potential is chosen to keep the zero concentration of the substrate at the electrode surface. During the electrochemical conversion the product is generated, S −→ P. The rate of the product generation at the electrode is proportional to the rate of conversion of the substrate (t > 0), S(0, t) = 0, DP

(15)

∂P  ∂S  = −DS  .  ∂x x=0 ∂x x=0

(16)

When the buffer solution is well-stirred and in a powerful motion, then the diffusion layer (0 < x < d) remains at a constant thickness. Consequently, the concentration of the substrate as well as the product over the enzyme surface (bulk solution/membrane interface) does not change while the biosensor contacts the substrate solution (t > 0). Assuming the zero concentration of the reaction product and S0 as the concentration of the substrate in the bulk solution, the boundary conditions at the solution/membrane interface are identical to those of the amperometric biosensor without the cyclic conversion (t > 0), S(d, t) = S0 ,

P (d, t) = 0.

(17)

The biosensor current depends upon the flux of the electro-active substance (substrate) at the electrode surface, i.e. at the border x = 0. Consequently, the density i(t) of the biosensor current at time t is explicitly obtained from Faraday’s and Fick’s laws i(t) = ne F DS

∂S  ∂P  = −ne F DP   , ∂x x=0 ∂x x=0

(18)

where ne is the number of electrons involved in a charge transfer, F is the Faraday constant. All other characteristics of the biosensor response defined for the amperometric biosensors in Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors” can also be applied to the biosensor with the substrate cyclic conversion.

3.2 Finite Difference Solution Accepting the uniform discrete grid for space and time the partial differential equations (12) can be replaced with the following difference equations [4]: j +1

Si

j +1

j +1

j S − 2Si − Si = DSe i+1 τ h2

j +1

+ Si−1

j

+

Vmax Pi

j

KM + Pi

,

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Biosensors Response Amplification with Cyclic Substrates Conversion j +1

j +1

Pi

j +1

j P − 2Pi − Pi = DPe i+1 τ h2

i = 1, . . . , Nd − 1,

j +1

+ Pi−1

j

−

Vmax Pi

j

KM + Pi

,

(19)

j = 1, . . . , M.

The initial conditions (13) and (14) are approximated by Si0 = 0,

i = 0, . . . , N − 1;

0 SN = S0 ,

(20)

Pi0 = 0,

i = 0, . . . , N.

The boundary conditions (15)–(17) are approximated as follows: j

S0 = 0, j

j

j

SN = S0 ,

PN = 0,

j

j

j

DP (P1 − P0 ) = −DS (S1 − S0 ),

j = 1, . . . , M.

(21)

Having the numerical solution of the problem, the density of the biosensor current at time t = tj can be calculated by j

j

i(tj ) = ne F DS (S1 − S0 )/ h,

j = 0, . . . , M

(22)

or by j

j

i(tj ) = −ne F DP (P1 − P0 )/ h,

j = 0, . . . , M.

(23)

3.3 Concentration Profiles Figure 1 shows the distribution of the substrate and the product for the catalytically active (Vmax > 0) as well as for the inactive (Vmax = 0) electrodes at the steady state condition [6]. In the case of the catalytically inactive electrode (Vmax = 0) the mathematical model (12)–(17) describes the action of the following scheme: S −→ P.

(24)

In scheme (24) no substrate regeneration is observed. It is only the product that is produced as the result of the electrochemical reaction. Comparing the response of the catalytically active electrode (Vmax > 0) with the inactive one (Vmax = 0) allows us to evaluate the effect of the catalytical activity on the response. It is obvious that the gradient of both species is constant in the entire membrane in the case of the absence of the catalytic activity. The nonlinear distribution of

3 Biosensors with Electrochemical and Enzymatic Conversion

129

10 9

S

8

S, P, μM

7

1 2 3

6 5 4

0.5S0

3 2

P

1 0 0

10

20

30

40

50

60

70

80

90

100

x, μm Fig. 1 The concentration profiles of the substrate (S) and the product (P) in the enzyme layer at three maximal enzymatic rates Vmax , 0 (1), 10 (2) and 1000 (3) µM/s, S0 = 10 mM, d = 100 µm

the concentration of both compounds, however, is observed at Vmax > 0. The gradient of the substrate concentration significantly increases with the increase in the maximal enzymatic rate Vmax . Consequentially, the biosensor current also increases. The response is greater at larger maximal enzymatic rate. Because of the boundary conditions (15)–(17), the symmetry with respect to the axis S = P = 0.5S0 can be noticed at all values of Vmax in Fig. 1. In the case of the symmetry, P (x, t) = S0 − S(x, t) at x ∈ [0, d] and t → ∞. This can be observed only at the steady state conditions, i.e. ∂S/∂t = ∂P /∂t = 0.

3.4 Peculiarities of the Biosensor Response 3.4.1 Dependence of Response on the Substrate Concentration Using computer simulation the dependence of the steady state biosensor current as well as the biosensor response time on the substrate concentration S0 has been investigated [6]. To get the results for a wide range of the values of the maximal enzymatic rate the investigation was carried out at the following values of Vmax : 0, 1, 10, 100 and 1000 µM/s. The results of the calculations are depicted in Figs. 2, 3, and 4. In these calculations the biosensor was modeled accepting the relatively thick enzyme layer, d = 150 µm. Figure 2 presents the dependence of the biosensor steady state current on the substrate concentration S0 . As it is possible to notice, the density I of the steady state biosensor current is the monotonous increasing function of S0 at all values of

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Biosensors Response Amplification with Cyclic Substrates Conversion 102

I, μA/mm2

101 100

1 2 3 4 5

10-1 10-2 10-3 10

KM

-4

10-6

10-5

10-4

10-3

10-2

10-1

S0 , M Fig. 2 The dependence of the density of the steady state current on the substrate concentration S0 at five maximal enzymatic rates Vmax : 0 (1), 1 (2), 10 (3), 100 (4) and 1000 (5) µM/s, d = 150 µm 30

1 2 3 4

25

G

20 15 10 5 0 10-6

10-5

-4 KM 10

10-3

10-2

10-1

S0, M

Fig. 3 The dependence of the signal gain GS on the substrate concentration S0 at four maximal enzymatic rates Vmax : 1 (1), 10 (2), 100 (3) and 1000 (4) µM/s, d = 150 µm

Vmax . In the case of the catalytically inactive electrode (Vmax = 0) the current density I is the linear function of S0 in the entire domain of the investigation. The nonlinear biosensor current versus S0 is observed with the catalytically active membrane, i.e. Vmax > 0. The calculations show that at low concentrations of the substrate, S0 < KM , the density I of the steady state current is approximately the linear

3 Biosensors with Electrochemical and Enzymatic Conversion

131

30

1 2 3 4 5

T0.5 , s

25

20

15

10 10-6

10-5

10-4

10-3

10-2

10-1

S0 , M Fig. 4 The half-time T0.5 of the steady state biosensor response versus the substrate concentration S0 at five maximal enzymatic rates Vmax : 0 (1), 1 (2), 10 (3), 100 (4) and 1000 (5) µM/s, d = 150 µm

function of S0 even at Vmax > 0. That is because of the linearity of the Michaelis– Menten equation at S0 < KM . The higher Vmax (Fig. 2) evokes the higher biosensor response. However, at the high concentration of the substrate, when S0 > KM , the amplification of the biosensor response decreases. At all values of Vmax > 0 the current converges to the maximal value, calculated at Vmax = 0, while the substrate concentration S0 increases. The ratio of the steady current increase measured with the enzyme active electrode (Vmax > 0) to the steady state current measured with the catalytically inactive electrode (Vmax = 0) can be considered as the gain of the sensitivity, GS , of the sensor with chemically amplified response [6], GS (Vmax ) =

I (Vmax ) , I (0)

(25)

where I (Vmax ) is the density of the steady state biosensor current calculated at the maximal enzymatic activity of Vmax . Figure 3 shows the dependence of the gain GS of the steady state current on the substrate concentration S0 [6]. As it is possible to notice in Fig. 3, the biosensor response increases up to 26 times (GS ≈ 26) at S0 = 1 µM and Vmax = 1 m M/s. In comparison with the response of the catalytically inactive electrode, this increase of the biosensor sensitivity compares fairly with the increase of the response of the biosensor with the immobilized laccase [23]. As has been mentioned above, the biosensor response is slightly amplified at high concentrations of the substrate. For all Vmax ≤ 1 mM/s, the gain GS of the steady state current is less than 4 at S0 > 100KM . According to the definition (25),

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Biosensors Response Amplification with Cyclic Substrates Conversion

GS → 1 when Vmax → 0, GS (0) = 1, for all values of S0 . Figure 3 distinctly shows that the amplification is especially significant at low concentrations of the substrate. Figure 4 shows the effect of the substrate concentration S0 on the half-time T0.5 of the steady state current [6]. As it is possible to notice in Fig. 4, T0.5 is the monotonous decreasing function of S0 at Vmax > 0. T0.5 is approximately the constant function, T0.5 = 10.5 s, at Vmax = 0. The most significant difference in T0.5 is noticed at low substrate concentrations when the largest amplification appears (see Figs. 3 and 4). For example, at Vmax = 1 mM/s and S0 = 1 µM, T0.5 is about 2.7 (28.6/10.5) times greater than at Vmax = 0 and the same S0 . The variation of T0.5 vs. S0 decreases with the increase in S0 . In Fig. 4 the shapes of curves of T0.5 vs. S0 are very similar for all values of Vmax being between 0.01KM and KM . At S0 being greater than 10KM a shoulder on the curve appears for Vmax = 1 mM/s. This shoulder is rather similar to the other one observed when simulating the response of the amperometric biosensor with no cyclic conversion (see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”) [2, 8]. It seems possible that the reason of the appearance of the shoulders on the curves is the same.

3.4.2 Sensitivity Versus Substrate Concentration As it is possible to notice in Fig. 2 the steady state current as a function of the substrate concentration S0 does not become a constant as it was in the case of biosensors without amplification (see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”). This means that the enzyme electrodes with the substrate cyclic conversion have a long calibration curve. This also means that the enzyme electrodes with the substrate cyclic conversion are sensitive enough even at relatively high concentrations of the substrate. Figure 5 shows the dependence of the biosensor sensitivity for a wide range of the substrate concentrations calculated for several values of the enzyme activity Vmax . One can see in Fig. 5 that at a wide range of the enzymatic activities Vmax , the biosensor sensitivity never approaches zero when the substrate concentration changes from 1 µM to 0.1 M. The highest sensitivity is proper to the catalytically inactive electrode. The sensitivity of the catalytically active enzyme electrodes is the non-monotonous function of substrate concentration S0 . The cave sharpness on the curve depends on the enzymatic activity Vmax . The concentration of the minimal sensitivity differs. In the case of a very low enzymatic activity (Vmax = µM/s), the minimum appears at S0 ≈ KM = 100 µM. For a higher value of Vmax the minimum of the sensitivity gains at a higher concentration S0 .

3.4.3 Effect of the Enzyme Membrane Thickness To investigate the influence of the thickness d of the enzyme layer on the dynamics of the biosensor response the density I of the steady state current is calculated at a

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133

1.0 0.9

BS

0.8

1 2 3 4 5

0.7 0.6 0.5 0.4 10-6

10-5

10-4

10-3

10-2

10-1

S0, M Fig. 5 The normalized biosensor sensitivity BS versus the substrate concentration S0 . The parameters and the notations are the same as in Fig. 2

I, μA/mm2

100

10-1

1 2 3 4 5

10-2

dσ(1000)

dσ(100)

dσ(10)

dσ(1)

10-3 1

10

d, μm

100

Fig. 6 The dependence of the density I of the steady state current on the membrane thickness d at five enzymatic rates Vmax : 0 (1), 1 (2), 10 (3), 100 (4) and 1000 (5) µM/s, and the substrate concentration S0 = 10 µM/s

wide range of the membrane thickness d: from 1 to 500 µm and different values of the maximal enzymatic rate Vmax [6]. The results of the calculations are depicted in Fig. 6. No notable signal amplification occurs in the case of a very thin enzyme layer, d < 3 µm, at all used values of the maximal enzymatic rate Vmax (Fig. 6). Let us

134

Biosensors Response Amplification with Cyclic Substrates Conversion

notice that a noticeable amplification starts at the thickness d at which the diffusion module σ 2 is approximately equal to the unity. The diffusion coefficient DS and the Michaelis–Menten constant KM were constant in all the numerical experiments, DS = 300 µm2/s,

KM = 100 µM .

(26)

The enzyme layer thickness dσ , at which σ 2 = 1, can be defined as a function of Vmax ,  dσ (Vmax ) =

DS KM d = , Vmax σ

σ2 =

Vmax d 2 . DS KM

(27)

In Fig. 6 the values of the thickness dσ (Vmax ) are presented for all non-zero enzymatic rates. The figure shows a fair correlation between the values of dσ and the enzyme layer thickness, at which the amplification starts. From the thickness d considerably greater than dσ , the steady state I becomes approximately constant. For example, in the case of Vmax√= V5 = 1 mM/s, the current density I = 106 µA/cm2 at d = 11.6 µM ≈ 2dσ (dσ = 30 µM at d = 1 µm) differs from the other one I = 88.7 µA/cm2 at about 45 times thicker membrane, d = 500 µm ≈ 91dσ , only by about 16.3%. The steady state biosensor current decreases linearly with the increase in the membrane thickness d when the enzyme reaction rate distinctly (σ < 0.5) controls the biosensor response, and it changes slightly only when the biosensor response is significantly under the diffusion control (σ > 2) (Fig. 6). The approximately linear function I of d turns to the approximately constant one for all values of Vmax [6].

3.4.4 Effect of the Reaction Rate The results of the investigation of the effect of the maximal enzymatic rate Vmax on the response gain GS are presented in Fig. 7 at two values of the substrate concentration S0 : 10, 100 µM and two values of the thickness d: 15 and 150 µm [6]. As it is possible to notice in Fig. 7, the increase of the gain GS is approximately linear at relatively high maximal enzymatic rates. The significant increase of GS starts at Vmax at which the diffusion module approximately equals one. Similarly to the function dσ , the maximal enzymatic activity Vσ is introduced as the function of the thickness d, at which σ 2 = 1, Vσ (d) =

DS KM . d2

(28)

The increase of GS starts at Vmax ≈ Vσ . The gain GS becomes approximately the linear increasing function of Vmax at Vmax >≈ 4Vσ , i.e. when σ >≈ 2. This property is valid for both concentrations of the substrate and both thicknesses of

4 Biosensors Acting in Trigger Mode

135

GS

100

10

1 2 3 4

Vσ(150) 1 10-7

10-6

Vσ(15) 10-5

10-4

10-3

Vmax, M/s Fig. 7 The response gain GS versus the maximal enzymatic rate Vmax at two values of the substrate concentration S0 : 10 (1, 2), 100 (3, 4) µM, and two values of the thickness d: 15 (1, 3) and 150 (2, 4) µm

the enzyme layer. The change of the substrate concentration S0 by 10 times slightly influences the behaviour of GS vs. Vmax .

4 Biosensors Acting in Trigger Mode This section presents mathematical models of biosensors acting in a trigger mode. One mathematical model describes the dynamics of the response of biosensors utilizing a trigger enzymatic reaction following the electrochemical and enzymatic product cyclic conversion (CEC scheme), while the other model describes the behaviour of biosensors utilizing a trigger enzymatic reaction following enzymatic and electrochemical conversion of the product (CCE scheme) [5]. The modeled response of the trigger biosensors is compared with the response of a single enzyme mono-layer biosensor.

4.1 Mathematical Models The biosensor is considered as an electrode, containing a mono-layer membrane with immobilized two enzymes applied onto the surface of the electrochemical transducer. The symmetrical geometry of the electrode and homogeneous distribution of immobilized enzymes in the enzyme membrane of a uniform thickness are assumed.

136

Biosensors Response Amplification with Cyclic Substrates Conversion

4.1.1 Modeling Biosensor Acting in CEC Mode Consider the CEC scheme E1

S −→ P1 ,

(29)

P1 −→ P2 ,

(30)

E2

P2 −→ P1 ,

(31)

where the substrate (S) is enzymatically (E1 ) converted to the product (P1 ) followed by the electrochemical conversion of the product (P1 ) to another product (P2 ) that in turn is enzymatically (E2 ) converted back to P1 [5]. Coupling the enzyme-catalysed reactions (29) and (31) with the onedimensional-in-space diffusion described by Fick’s law leads to the following system of equations [5]: (1) ∂S ∂ 2S S Vmax = DS 2 − (1) , ∂t ∂x KM + S (1)

(2)

∂ 2 P (1) Vmax S Vmax P (2) ∂P (1) = DP(1) + , + (1) (2) ∂t ∂x 2 KM + S KM + P (2)

(32)

(2)

2 (2) Vmax P (2) ∂P (2) (2) ∂ P = DP − , (2) ∂t ∂x 2 KM + P (2)

x ∈ (0, d),

t > 0,

where x and t stand for space and time, respectively, S(x, t) and P (i) (x, t) denote (i) the concentrations of the substrate S and product Pi , respectively, Vmax is the (i) maximal enzymatic rate, KM is the Michaelis constant, d is the thickness of the enzyme membrane, DS and DP(i) are the diffusion coefficients, i = 1, 2. The model (1) (1) (2) parameters Vmax and KM stand for the enzyme-catalysed reaction (29), while Vmax (2) and KM —for the next enzymatic reaction (31). Let x = 0 represent the electrode surface and x = d—the bulk solution/enzyme membrane interface. Assuming the substrate concentration of S0 and zero concentration of the reaction product in the bulk solution, the initial conditions are defined as follows (t = 0): S(x, 0) = 0,

x ∈ [0, d),

S(d, 0) = S0 , P

(i)

(x, 0) = 0,

(33) x ∈ [0, d], i = 1, 2.

4 Biosensors Acting in Trigger Mode

137

The electrode potential is chosen to keep zero concentration of the product P1 at the electrode surface, P (1) (0, t) = 0,

t > 0.

(34)

The substrate is electro-inactive substance,  ∂S  DS  = 0, t > 0. ∂x x=0

(35)

Due to the electrochemical reaction (30), the generation rate of the product P2 at the electrode surface is proportional to the generation rate of the product P1 , DP(2)

∂P (2)  ∂P (1)  = −DP(1)   , ∂x x=0 ∂x x=0

t > 0.

(36)

When the bulk solution is well-stirred outside and in a powerful motion, the diffusion layer remains at a constant thickness (0 < x < d). Thus, the concentrations of the substrate and of both products over the enzyme surface (bulk solution/membrane interface) remain constant while the biosensor contacts the solution of the substrate (t > 0), S(d, t) = S0 ,

P (1) (d, t) = 0,

P (2) (d, t) = 0.

(37)

The biosensor current depends upon the flux of the product P1 at the electrode surface, i.e. at the border x = 0. The density iCEC (t) of the current at time t is obtained explicitly from Faraday’s and Fick’s laws using the flux of the product P1 at the surface of the electrode, iCEC (t) = ne F DP(1)

∂P (1)   , ∂x x=0

(38)

where ne is the number of electrons involved in a charge transfer at the electrode surface, and F is the Faraday constant. Taking into account (36), iCEC (t) can also be calculated as follows: (2) ∂P

iCEC (t) = −ne F DP

(2) 

∂x

 

x=0

.

(39)

The system (32)–(37) approaches a steady state as t → ∞: ICEC = lim iCEC (t), t →∞

(40)

where ICEC is taken as the density of the steady state current of the biosensor acting in CEC mode.

138

Biosensors Response Amplification with Cyclic Substrates Conversion

4.1.2 Modeling Biosensor Acting in CCE Mode In the CCE scheme, the substrate S is enzymatically (E1 ) transformed into the product P1 followed by the enzymatic (E2 ) conversion of P1 into another product P2 that is electrochemically converted back to the product P1 , E1

S −→ P1 ,

(41)

E2

P1 −→ P2 ,

(42)

P2 −→ P1 .

(43)

Assuming the same geometry of the enzyme membrane as in the CEC mode, coupling the reactions (41)–(42) with the diffusion leads to the following Eqs. [5]: (1)

∂ 2S Vmax S ∂S = DS 2 − (1) , ∂t ∂x KM + S (1)

(2)

2 (1) Vmax P (1) Vmax S ∂P (1) (1) ∂ P = DP − + , (1) (2) ∂t ∂x 2 KM +S KM + P (1) (2) (1) 2 (2) Vmax ∂P (2) P (2) ∂ P + , = DP 2 (2) ∂t ∂x KM + P (1)

(44)

x ∈ (0, d),

t > 0.

Here and below, the notations are the same as in the model of a biosensor acting in CEC mode (Sect. 4.1.1). The initial conditions are the same as in the case of CEC scheme, (33). When the biosensor acts in the CCE mode, the electrode potential is chosen to keep zero concentration of the product P2 at the electrode surface, P (2) (0, t) = 0,

t > 0.

(45)

The rate of the product P1 generation at the electrode surface is proportional to the rate of conversion of the product P2 . Thus, the boundary conditions (35)–(37) are also applicable to the system (41)–(43). The density iCCE (t) of the biosensor current is proportional to the concentration gradient of the product P2 at the surface of the electrode: (2) ∂P

iCCE (t) = ne F DP

(2) 

∂x

 

(1) ∂P

x=0

= −ne F DP

(1) 

∂x

 

x=0

.

(46)

The density ICCE of the steady state current of the biosensor acting in the CCE mode is calculated as follows: ICCE = lim iCCE (t). t →∞

(47)

4 Biosensors Acting in Trigger Mode

139

4.1.3 Modeling Biosensor Acting in CE Mode To compare the responses of trigger and ordinary amperometric biosensors, the action of the CE biosensor was analysed. A CE biosensor contains a monolayer membrane with only one immobilized enzyme applied onto the surface of the electrochemical transducer. The number of immobilized enzymes is the only fundamental difference between the CE biosensor and the trigger one. In accordance with the CE scheme, the substrate S is enzymatically (E1 ) converted to the product P1 followed by the electrochemical product (P1 ) conversion to another product (P2 ): E1

S −→ P1 ,

(48)

P1 −→ P2 .

(49)

In this scheme only the product P1 is the electro-active substance. The mathematical model of the biosensor acting in the CE mode can be derived from the model (32)–(37) of the biosensor acting in the CEC mode by accepting the (2) inactive enzyme E2 , i.e. Vmax = 0. Assuming iCE (t) to be the density of the anodic current, the transient current iCE (t) and the corresponding stationary current ICE of the biosensor acting in the CE mode can be calculated as of the CEC biosensor using (38) and (50), respectively, (1) ∂P

iCE (t) = ne F DP

(1) 

∂x

 

x=0

,

ICE = lim iCE (t). t →∞

(50)

Let us notice, that following the schemes (48) and (49), the product P2 has no effect on the response of the CE biosensor. Consequently, the response of that biosensor is identical to the response of a biosensor utilizing only the scheme (48). This type of the biosensors has been in detail analysed, particularly, in Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”. 4.1.4 Enzymatic Amplification Both of presented trigger schemes (CEC and CCE) include enzymatic reactions together with electrochemical and enzymatic amplification steps [5, 10, 29]. The response amplification is one of the most important characteristics of the trigger biosensors. To compare the amplified biosensor response with the response without amplification, the gain of the sensitivity is defined as the ratio of the steady state current of the trigger biosensor to the steady state current of the corresponding CE biosensor [5], GCEC =

ICEC , ICE

GCCE =

ICCE . ICE

(51)

140

Biosensors Response Amplification with Cyclic Substrates Conversion

When comparing the CEC and CCE schemes with the CE scheme and the corresponding mathematical models, one can see that the enzymatic activity of the enzyme E2 is the determinant characteristics. Because of this, it is reasonable to analyse the signal gains GCEC and GCCE as functions of the enzymatic activity (2) Vmax assuming all other parameters as being identical [5], (2) GCEC (Vmax )=

(2) GCCE (Vmax ) (2)

(2)

(2)

(2)

(2)

ICEC (Vmax ) ICEC (Vmax ) , = ICE ICEC (0)

(52)

ICCE (Vmax ) ICCE (Vmax ) , = = ICE ICEC (0) (2)

where ICEC (Vmax ) and ICCE (Vmax ) are the densities of the steady state currents of the trigger biosensors acting in the CEC and the CCE mode, respectively, at the (2) maximal activity Vmax of the enzyme E2 . Since at zero activity of the enzyme E2 , the response of the CEC biosensor becomes identical to the response of CE biosensor, ICEC (0) equals ICE .

4.2 Finite Difference Solution Since the type of equations of the mathematical models of trigger biosensors are the same as in the one-layer one-enzyme case, the corresponding initial boundary value problems can be solved similarly. To find a numerical solution of the problem in the domain [0, d]×[0, T ] a discrete grid has to be introduced. Let ωh × ωτ be a uniform discrete grid defined as follows: ωh = {xi : xi = ih, i = 0, 1, . . . , N, hN = d}, ωτ = {tj : tj = j τ, j = 0, 1, . . . , M; τ M = T }.

(53)

The following notation is used in the finite difference approximations presented below: j

Si = S(xi , tj ), i = 0, . . . , N,

(1),j

Pi

= P (1) (xi , tj ),

(2),j

Pi

= P (2) (xi , tj ),

j = 0, . . . , M.

(54)

4.2.1 CEC Mode Partial differential equations (32) can be approximated by the following implicit finite difference scheme: j +1

Si

j +1

j +1

j S − 2Si − Si = DS i+1 τ h2

j +1

+ Si−1

(1)

−

j

Vmax Si (1)

j

KM + Si

,

4 Biosensors Acting in Trigger Mode (1),j +1

Pi

(1),j

− Pi τ

=

141

(1),j +1 (1) Pi+1 DP

(1),j +1

− 2Pi

h2

j

(2),j +1

Pi

+

j

(2),j +1

− 2Pi

(2),j

(2)

KM + Pi (2),j +1

Pi+1

= DP(2)

(2) Vmax Pi

(1)

KM + Si

(2),j

− Pi τ

(2),j

(1) Vmax Si

+

(1),j +1

+ Pi−1

,

(2),j +1

+ Pi−1

h2

i = 1, . . . , N − 1,

(2),j

−

(2) Vmax Pi

(2),j

(2) KM + Pi

j = 0, . . . , M − 1.

, (55)

The initial conditions (33) are approximated as follows: Si0 = 0,

i = 0, . . . , N − 1,

0 SN = S0 , (1),0

(56)

(2),0

= Pi

Pi

= 0,

i = 0, . . . , N.

The boundary conditions (34)–(37) can be approximated as follows: (1),j

P0

(2),j

DP(2) (P1 (1),j

PN

j

= 0,

j

j

S0 = S1 , (2),j

− P0 (2),j

= PN

SN = S0 , (1),j

(1),j

) = −DP(1) (P1

= 0,

− P0

),

(57)

j = 1, . . . , M.

The density of the biosensor current can easily be calculated, (1),j

iCEC (tj ) ≈ iCEC,j = ne F DP P1

j = 1, . . . , M .

/ h,

(58)

4.2.2 CCE Mode In the case of CCE mode, the governing equations (44) can be approximated similarly as in the case of the CEC mode, j +1

j +1

Si (1),j +1

Pi

j +1

j S − 2Si − Si = DS i+1 τ h2 (1),j +1

(1),j

− Pi τ

= DP(1)

Pi+1

(1),j +1

− 2Pi

(1)

−

j

(2)

Vmax Si (1)

j

KM + Si

−

(1)

(1),j +1

+ Pi−1 (1),j (1),j

KM + Pi

j

KM + Si

Vmax Pi (2)

j

Vmax Si

h2 (1)

+

j +1

+ Si−1

,

,

142

Biosensors Response Amplification with Cyclic Substrates Conversion (2),j +1

Pi

(2),j

− Pi τ

=

(2),j +1 (2) Pi+1 DP

(2),j +1

− 2Pi

i = 1, . . . , N − 1,

(2),j +1

+ Pi−1

h2

(1),j

+

(2) Vmax Pi

(1),j

(2) KM + Pi

j = 0, . . . , M − 1 .

, (59)

Since the initial conditions are the same as in the case of the CEC scheme, the same approximation (56) can be used. The boundary condition (45) is approximated as follows: (2),j

P0

= 0,

j = 1, . . . , M .

(60)

All other boundary conditions (35)–(37) are common for both cases of triggering: the CEC and the CCE. Thus, the same approximation can be used for the boundary conditions. The density of the biosensor current iCEC can be calculated as follows: (2),j

iCCE (tj ) ≈ iCCE,j = ne F DP P1

/ h,

j = 1, . . . , M.

(61)

4.3 Simulated Response Figures 8 and 9 show the concentration profiles of the substrate and products in the enzyme layer for biosensors acting in the CEC and the CCE modes calculated at the following values of the parameters: DS = DP = 300 µm2/s,

d = 100 µm,

S0 = 100 µM ,

(1) (2) = KM = 100 µM, KM

(1) (2) Vmax = Vmax = 100 µM/s .

(62)

Figures 8 and 9 show the concentration profiles at the time when the steady state as well as 50% of the steady state response has been reached. It is easy to notice in the figures that the concentrations of the substrate at steady state conditions are approximately the same for both biosensors [5]. At the time when the half of the steady state response was reached, no significant difference has also been observed in the entire enzyme layer, x ∈ [0, d]. The similarity of the substrate concentrations in the both modes (CEC and CCE) of the biosensor operation can be explained analytically by the identity of the equations describing the dynamics of the substrate concentration [5]. The density of the steady state current and the time of steady state are very similar in both types of biosensors, ICEC ≈ iCEC (120) ≈ 0.26 µA/mm2 and ICCE ≈ iCCE (120) ≈ 0.25 µA/mm2 . At the steady state conditions, i.e. when ∂S/∂t = ∂P (1) /∂t = ∂P (2) /∂t = 0, the equality S(x, t) + P (1) (x, t) + P (2) (x, t) = S0 is valid for all x ∈ [0, d] when t → ∞. This can be observed in both Figs. 8 and 9 [5].

4 Biosensors Acting in Trigger Mode

143

100

S , P(1) , P(2) , μM

90

1 2

80 70 60

P(2)

P(1)

50 40

S

30 20 10 0 0

10

20

30

40

50

60

70

80

90

100

x, μm Fig. 8 Concentration profiles of the substrate (S) and products (P (1) and P (2) ) in the enzyme layer of the CEC biosensor at the steady state time T = 120 s (1) when the steady state is reached and at the half of it, T0.5 = 10.5 s (2). Model parameters are defined in (62)

1.0

S , P(1) , P(2) , μM

0.9

1 2

0.8 0.7

S

0.6

P

0.5

(1)

P(2)

0.4 0.3 0.2 0.1 0.0 0

10

20

30

40

50

60

70

80

90

100

x, μm Fig. 9 Concentration profiles of the substrate and products in the enzyme layer of the CCE biosensor at T = 120 s (1) and T0.5 = 11 s (2). The other parameters and notation are the same as in Fig. 8

144

Biosensors Response Amplification with Cyclic Substrates Conversion

4.4 Peculiarities of the Response 4.4.1 Dependence of the Steady State Current on the Reactions Rates Figures 10 and 11 show the dependence of steady state current on the activity of both enzymes (E1 and E2 ) for both action modes: the CEC and the CCE [5]. In (1) (2) calculations, the maximal enzymatic rates Vmax and Vmax varied in four order of magnitude: from 10−7 up to 10−3 M/s, the substrate concentration S0 was equal to (1) (2) 100 µM = KM = KM , the thickness of the enzyme layer was equal to 100 µm. One can see in Figs. 10 and 11 that ICEC as well as ICCE are monotonously increasing (1) (2) functions of both enzymatic rates: Vmax and Vmax . (2) In the case of the CEC mode, the activity of the enzyme E2 (Vmax > 0) stimulates a notable increase of the biosensor current [5]. When the enzyme membrane (layer) (2) contains no enzyme E2 (Vmax = 0), the biosensor acting in CEC mode still generates (1) the current only if Vmax > 0. In the case of CCE mode, the appearance of an active (2) enzyme E2 (Vmax > 0) is a critical factor for the biosensor current. ICCE = 0 if (2) (1) Vmax = 0 even if the activity of the enzyme E1 is very high (Vmax  0). Because (2) of this, at low values of Vmax the density ICCE of the steady state current increases (2) . That effect is observed in Figs. 10 and 11 very sharply with the increase of Vmax and is called the surface salience. The salience of the surface ICCE (Fig. 11) is more (2) noticeable than the salience of the surface ICEC (Fig. 10). Consequently, at Vmax →

0

10

-2

10

-3

10

ICEC , μA/mm 2

-1

10

-3

10 -4

10 -3

-5

max

(1)

10

, M /s

M

10

-6

,

10

(2 )

-6

-5

V

-7

10

-7

10

ma x

10

/s

10

-4

V

10

(1) Fig. 10 The density ICEC of the steady state current versus the maximal enzymatic rates Vmax (2) and Vmax of the biosensor acting in CEC mode. The values of the other parameters are the same as in Fig. 8

4 Biosensors Acting in Trigger Mode

145 0

10

-1

-2

10

-3

10

-4

10

-5

10

ICCE , μA/mm 2

10

-3

10 -4

10 -3

-5

max

(1)

, M /s

M

10

-6

10

,

10

(2 )

-6

-5

V

-7

10

-7

10

ma x

10

/s

10

-4

V

10

(1) Fig. 11 The density ICCE of the steady state current versus the maximal enzymatic rates Vmax (2) and Vmax of the biosensor acting in CCE mode. The values of the model parameters are the same as in Fig. 8

(1)

0 and Vmax > 0, ICCE → 0 in CCE mode of the biosensor operation and ICEC → ICE in another mode (CEC) of triggering. On the other hand, Figs. 10 and 11 show, (2) that ICCE ≈ ICEC at very high values of the maximal enzymatic rate Vmax [5].

4.4.2 Effect of the Reaction Rates on the Amplification To investigate the effect of the amplification the density ICE of the stationary current has to be calculated at the same conditions as those given above. Having ICEC , ICCE and ICE the gains GCEC and GCCE can be easily calculated [5]. The results of calculations are depicted in Figs. 12 and 13. One can see in both figures that the signal gain increases with the increase of the (2) (2) enzymatic rate Vmax . The increase is especially notable at high values of Vmax . The (1) variation of Vmax on the response gain is only slight. The gain varies from 15 to 19 (1) at Vmax = 1 mM/s in both modes of the biosensor operation: the CEC and the CCE. When comparing the gain in the CEC mode (Fig. 12) with the gain in the CCE (2) mode (Fig. 13), one can notice a significant difference at low values of Vmax . The (2) gain GCEC starts to increase from about unity, while GCCE at low values of Vmax (2) (Vmax < 1 µM/s)) is even less than unity. It means that in the case of low activity of the enzyme E2 , the steady state current of the biosensor acting in the CCE mode is even less than the steady state current of a biosensor acting in the CE mode under the same conditions. It follows from the model of the CCE biosensor that P (2) (x, t) ≈ 0

146

Biosensors Response Amplification with Cyclic Substrates Conversion

GCEC

20 18 16 14 12 10 8 6 4 2 -3 0 10 -4

10

-3

max

-7

-7

10 10

,

10

-6

, M/ 10 s

(2 )

(1)

-6

ma x

-5

10

M

10

-4

10

V

/s

-5

V

10

(1)

(2)

Fig. 12 The signal gain GCEC versus the maximal enzymatic rates Vmax and Vmax of the biosensor acting in the CEC mode under the same conditions as in Fig. 8

(2) (2) (1) when Vmax → 0. Thus, when Vmax → 0 at all positive values of Vmax , GCCE → 0 and GCEC → 1. On the other hand, Figs. 12 and 13 show that GCEC ≈ GCCE at (2) (2) high maximal enzymatic rate Vmax , e.g. at Vmax = 1 mM/s [5].

4.4.3 The Amplification Versus the Substrate Concentration To investigate the dependence of the signal gain on the substrate concentration S0 the biosensor response has to be simulated changing S0 in a wide range, e.g. from 1 µM to 100 mM [5]. Since the signal gain of trigger biosensors is only significant (2) at relatively high maximal rate Vmax of the enzyme E2 (see Figs. 12 and 13), the (2) following two values of Vmax were used: 0.1 and 1 mM/s. In numerical simulation, (1) (2) two values of the maximal rate Vmax of the enzyme E1 were the same as of Vmax . (1) (2) Assuming KM = KM = KM = 100 µM, the signal gains GCEC and GCCE are depicted in Fig. 14 as functions of the normalized substrate concentration S0 /KM . One can see in Fig. 14 that the behaviour of the signal gain versus the substrate concentration is very similar in both modes of the biosensor action: the CEC and the CCE. Some noticeable difference between values of GCEC and GCCE is observed only at high substrate concentrations, S0 > KM . However, in the case of a higher (2) (2) (1) (1) value of Vmax , Vmax = 1 mM/s, and a lower Vmax , Vmax = 0.1 mM/s, no noticeable difference is observed between values of GCEC (curve 5 in Fig. 14) and GCCE (curve 6 in Fig. 14) in the entire domain of the substrate concentration [5].

4 Biosensors Acting in Trigger Mode

147

GCCE

20 18 16 14 12 10 8 6 4 2 -3

10 -4

10

-3

/s

-5

max

, M/ s

-6

10

-6

10

-7

-7

10 10

,

(1)

(2 )

-5

10

V

ma x

10

M

10

-4

V

10

(1)

(2)

Fig. 13 The signal gain GCCE versus the maximal enzymatic rates Vmax and Vmax of the biosensor acting in the CCE mode under the same conditions as in Fig. 8

GCEC , GCCE

10

1

1 2 3 4

5 6 7 8

0.1 10-2

10-1

100

101

102

103

S0 / K M Fig. 14 The signal gains GCEC (1, 3, 5, 7) and GCCE (2, 4, 6, 8) versus the normalized substrate (1) (2) : 1 (1–4), 0.1 (5–8) and Vmax : concentration S0 /KM at the following maximal enzymatic rates Vmax 1 (1, 2, 5, 6), 0.1 (3, 4, 7, 8) mM/s. The other parameters are the same as in Fig. 8

(2) Figure 14 shows the significant importance of the maximal enzymatic rate Vmax for both signal gains: GCEC and GCCE . That importance is especially perceptible at (2) low and moderate concentrations of substrate, S0 < KM . At S0 < 0.1KM and Vmax

148

Biosensors Response Amplification with Cyclic Substrates Conversion

= 1 mM/s, due to the amplification, the steady state current increases up to about 18 times (GCEC ≈ GCCE ≈ 18). However, at the same S0 and tenfold lower value of (2) Vmax , the gain is about three times less, GCEC ≈ GCCE ≈ 5.7. Consequently, at low substrate concentrations, S0 < 0.1KM , and a wide range of the maximal enzymatic (1) (2) , the tenfold reduction of Vmax reduces the signal gain about three times. rate Vmax That property is valid for both modes of triggering: the CEC and the CCE [5]. An increase in the substrate concentration causes a decrease in the signal gain (1) (2) when S0 becomes greater than KM (Fig. 14), i.e. when S0 > KM and S0 > KM . The decrease is more marked in cases of the higher values of the enzymatic rate (1) Vmax (curves 1–4 in Fig. 14) rather than of the lower ones (curves 5–8). Additional (1) calculations showed that at a very low activity of enzyme E1 when Vmax = 100 nM/s, both signal gains practically do not vary when changing the substrate concentration in the entire domain [5]. Because of a very stable amplification of the biosensor signal at wide range of substrate concentration, the use of the biosensors acting in a trigger mode is (1) especially reasonable at relatively low maximal enzymatic activity (rate Vmax ) of (2) enzyme E1 and high activity (rate Vmax ) of enzyme E2 . In the case of relatively (1) high maximal enzymatic activity Vmax the signal amplification is stable only for low concentrations of the substrate [5]. Additional calculations showed that the signal gains vanish rapidly with the (2) decrease of the enzymatic activity Vmax of enzyme E2 . For example, in the case (2) of Vmax = 1 µM/s the gains become less than 2 even at low substrate concentration, GCEC ≈ 1.9 and GCCE ≈ 1.3 at S0 = 0.01KM [5]. That effect is also observed in Figs. 12 and 13. Numerical simulation confirmed the property that the tenfold (2) reduction in Vmax reduces the signal gains GCCE and GCCE about three times at (2) wide range also of Vmax [5]. Similar dependence of the signal gain on the substrate concentration was observed in the case of an amperometric enzyme electrode acting under chemical amplification by cyclic substrate conversion discussed in Sect. 3. In the case of the biosensor with substrate cyclic conversion the signal gain of 26 times was observed at the maximal enzymatic rate of 1 mM/s and the thickness of 150 µm of the enzyme layer. In order to compare that gain with the gain achieved in the trigger mode, the gains GCEC and GCCE were calculated for the enzyme membrane of thickness 150 µm. The numerical simulation of the action of the trigger biosensors showed (1) (2) very similar amplification, GCEC ≈ GCCE ≈ 22 at Vmax = Vmax = 1 mM/s [5].

4.4.4 Effect of the Enzyme Membrane Thickness on the Amplification As has been mentioned in the previous chapters, the steady state current of membrane biosensors significantly depends on the thickness of enzyme layer. The steady state time varies even in orders of magnitude. To investigate the dependence of the signal gain of the trigger biosensors on the enzyme membrane thickness d the

4 Biosensors Acting in Trigger Mode

149

102

GCEC , GCCE

101 100

1 2 3 4 5 6

10-1 10-2 10-3

d (1)

d (0.1)

d (0.01)

10-4 1

10

100

d, μm

Fig. 15 The signal gains GCEC (1–3) and GCCE (4–6) versus the enzyme membrane thickness d (1) (2) = 1 mM/s and three maximal enzymatic rates Vmax : 1 (1, 4), 0.1 (2, 5), 0.01 (3, 6) mM/s; at Vmax The other parameters are the same as in Fig. 8

response of biosensors was simulated varying d from 1 up to 500 µm at different (1) (2) maximal enzymatic rate Vmax of the enzyme E1 and rate Vmax of the enzyme E2 [5]. Figure 15 shows the signal gains GCEC and GCCE versus the membrane (1) thickness d at the maximal enzymatic rate Vmax = 1 mM/s and three values of the (2) rate Vmax : 0.01, 0.1 and 1 mM/s. When comparing the gain GCEC with the gain GCCE , one can notice a valuable difference in the behaviour of the signal gains. In the case of CEC biosensor action, no notable amplification is observed for the thin enzyme membranes (d < 10 µm). A more distant increase in the thickness d causes an increase of the gain GCEC . The thickness at which GCEC starts to (2) increase depends on the maximal enzymatic rate Vmax [5]. 2 The diffusion module σ is one of the principal parameters controlling the biosensor behaviour. Since the diffusion coefficients and the Michaelis–Menten constant were the same in all the numerical experiments as defined in (62), and the behaviour of the biosensors acting in the trigger modes is mainly determined by (2) the enzymatic rate Vmax (Figs. 12 and 13), the thickness dσ of the enzyme layer is (2) introduced as a function of Vmax at which the diffusion module equals unity, σ 2 = 1,  dσ

(2) Vmax



 =

(2)

DS KM (2)

.

(63)

Vmax

The following three values of dσ (Vmax ) are depicted in Fig. 15: dσ (1 mM/s) ≈ 5.48 µm, dσ (0.1 mM/s) ≈ 17.3 µm and dσ (0.01 mM/s) ≈ 54.8 µm. These values were calculated at constant values of DS and KM defined in (62). When comparing

150

Biosensors Response Amplification with Cyclic Substrates Conversion

value dσ (1 mM/s) with the membrane thickness at which the gain GCEC starts to (2) increase at corresponding Vmax of 1 mM/s (curve 1 in Fig. 15), one can see that the amplification becomes noticeable when the mass transport by diffusion starts to control the biosensor response. As one can see in Fig. 15, this effect is also valid for (2) two other values of the maximal enzymatic rate Vmax : 0.1 and 0.01 mM/s. However, this is valid only in the case of the biosensor acting in the CEC mode [5]. In the case of the CCE mode, the signal gain GCCE permanently increases with an increase in the thickness d. GCCE is approximately a linear increasing function of the enzyme layer thickness d. However, practical amplification (GCCE > 1) takes place only in cases of relatively thick membranes (d >≈ 2dσ ). As has been noted above (see Fig. 13), the steady state current of the biosensor acting in the CCE mode may be even significantly less than the steady state current of the corresponding biosensor acting in the CE mode at the same conditions. In the case of relatively thick enzyme membrane the gain GCCE is equal approximately to GCEC [5]. (2) Using computer simulation for different enzymatic rates Vmax the thickness dG (1) of the enzyme membrane at which GCCE = 1 was calculated. Accepting Vmax = (2) 1 mM/s the following values were found: dG ≈ 9 µm at Vmax = 1 mM/s, dG ≈ (2) (2) 30 µm at Vmax = 0.1 mM/s and dG ≈ 90 µm at Vmax = 0.01 mM/s. These values of the membrane thickness compare favourably with values of the thickness dmax at which the steady state current as a function of the membrane thickness d gains the maximum [2, 3, 5]. Consequently, for low substrate concentrations, the thickness dG of the enzyme membrane at which GCCE = 1 can be rather precisely expressed as dG ≈ 1.5dσ , where dσ was introduced in (63). Additional calculations showed (1) that this property is valid for wide ranges of both maximal enzymatic rates, Vmax (2) and Vmax , if only the substrate concentration S0 is less than the Michaelis–Menten constant KM [5].

4.4.5 Effect of the Membrane Thickness on the Response Time From a practical point of view it is very important to have the biosensor response time as short as possible. To compare the time of steady state amplified biosensor response with the steady state time of the response without amplification, the prolongation of the response time is introduced as the ratio of the steady state time of the trigger biosensor to the steady state time of the corresponding CE biosensor [5], (2)

(2) LCEC (Vmax )=

(2) LCCE (Vmax ) (2)

(2)

TCEC (Vmax ) TCEC (Vmax ) , = TCE TCEC (0)

(2) (2) TCCE (Vmax TCCE (Vmax ) ) = = , TCE TCEC (0) (2)

(64)

where TCEC (Vmax ) and TCCE (Vmax ) are the steady state times of the trigger biosensors acting in the CEC and the CCE mode, respectively, calculated at

4 Biosensors Acting in Trigger Mode

151

3.5

LCEC , LCCE

3.0

1 2 3 4 5 6

2.5 2.0 1.5 1.0

d (1)

d (0.1)

d (0.01)

0.5 1

10

d, μm

100

Fig. 16 The response time prolongations LCEC (1–3) and LCCE (4–6) versus the enzyme membrane thickness d. The parameters and the notation are the same as in Fig. 15

(2)

maximal enzymatic rate Vmax of the enzyme E2 , TCE is the steady state time of the corresponding CE biosensor. Since the action of the CE biosensor can be simulated (2) as an action of the CEC biosensor accepting Vmax = 0, TCE = TCEC (0) is assumed. Figure 16 shows the change of the response time versus the membrane thickness (1) (2) d at Vmax = 1 mM/s and different values of Vmax . One can see in Fig. 16 that in all of the presented cases the prolongation of the response time (LCEC as well as LCCE ) is a non-monotonous function of the thickness d of the enzyme layer [5]. The shoulder on the curves is especially noticeable at high maximal enzymatic rates. Similar effect was noticed in the case of biosensors with substrate cyclic conversion [6] and during the oxidation of β-nicotinamide adenine dinucleotide (NADH) at poly(aniline)-coated electrodes [8]. In cases of the thin enzyme membranes (d < 10 µm), the prolongation of the response time is insignificant. However, an increase in the membrane thickness d prolongs the response time up to 3.4 times in both modes of the triggering: the CEC and the CCE [5]. (2) In the case of CEC mode, the slight influence of the maximal enzymatic rate Vmax (2) on LCEC can be noticed in Fig. 16, while no notable influence of Vmax on LCCE is observed in the case of the CCE action mode. Additional calculations showed that the response time prolongation slightly depends on the substrate concentration S0 (1) as well as on the maximal activity Vmax of the enzyme E1 [5].

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Biosensors Response Amplification with Cyclic Substrates Conversion

5 Concluding Remarks The steady state biosensor current (I ) with substrate cyclic conversion is a monotonous increasing function of substrate concentration S0 at all values of maximal enzymatic rate Vmax (Fig. 2). In the case of catalytically inactive membrane (Vmax = 0) the steady state current is approximately a linear function of S0 . The nonlinear current I versus S0 is observed with the active enzyme, Vmax > 0. However, at the substrate concentration S0 less than Michaelis constant KM , S0 < KM , the steady state current is approximately a linear function also at Vmax > 0. Due to the cyclic conversion, the gain GS in sensitivity of a biosensor can increase in some tens of times. The increase of sensitivity highly depends on the maximal enzymatic rate Vmax , substrate concentration S0 as well as the thickness d of the enzyme layer (Figs. 3 and 7). Significant gain in sensitivity is observed when the biosensor response is under diffusion control only, i.e. when σ 2 > 1. The gain GS is approximately a linear function of Vmax when the biosensor response is significantly (σ 2 > 1) under diffusion control (Fig. 7). The steady state current I decreases with increasing the membrane thickness d in approximately linear dependence when the reaction rate distinctly controls the biosensor response (σ 2 < 0.5) and I changes slightly with increase of d when the biosensor response is significantly (σ 2 > 2) under diffusion control (Fig. 6). The mathematical model (32)–(37) of the biosensor operation can be used to investigate the dynamics of the response of biosensors utilizing a trigger enzymatic reaction following electrochemical and enzymatic product cyclic conversion (CEC scheme (29)–(31)), while the model (44), (33), (35)–(37), (45) can be applied as a framework to investigate the behaviour of biosensors utilizing a trigger enzymatic reaction following enzymatic and electrochemical conversion of the product (CCE scheme (41)–(43)). The signal gains in sensitivity, GCEC and GCCE , of trigger biosensors are mainly (2) determined by the maximal enzymatic activity Vmax of the enzyme E2 (Figs. 12 (2) and 12). The enzymatic activity Vmax is a critical factor for the biosensor current (2) in the case of the CCE mode, GCCE → 0 if Vmax → 0. In the case of the CEC (2) biosensor, a decrease of activity Vmax causes a decrease in gain GCEC , however, (2) → 0. GCEC stays greater than unity, GCEC → 1 if Vmax Both signal gains, GCEC and GCCE , are mostly valuable when the concentration of the substrate is less than the Michaelis–Menten constant (Fig. 14). A valuable amplification (up to dozens of times) at wide range of substrate concentration is achieved only in the case of relatively low maximal enzymatic activity of enzyme E1 and high activity of enzyme E2 . In both biosensor action modes, the CEC and the CCE, a significant amplification of the signal is observed if the response is under the mass transport control, i.e. if σ 2  1 (Fig. 15). In cases when the valuable amplification of the signal of a triggering biosensor is achieved, the response time is up to several times longer than the response time of the corresponding biosensor acting without triggering (Fig. 16) [5].

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24. Kulys J, Sorochinskii V, Vidziunaite R (1986) Transient response of bienzyme electrodes. Biosensors 2(3):135–146 25. Limoges B, Marchal D, Mavré F, Savéant JM (2008) Theory and practice of enzyme bioaffinity electrodes. Chemical, enzymatic, and electrochemical amplification of in situ product detection. J Am Chem Soc 130(23):7276–7285 26. Litvinas L, Baronas R, Žilinskas A (2017) Application of two phase multi-objective optimization to design of biosensors utilizing cyclic substrate conversion. In: Paprika Z, Horák P, Váradi K, Zwierczyk PT, Vidovics-Dancs Á (eds) Proceedings, 31st European conference on modelling and simulation ECMS 2017. ECMS, Budapest, pp 469–474 27. Malinauskas A, Kulys J (1978) Alcohol, lactate and glutamate sensors based on oxidoreductases with regeneration of nicotinamide adenine dinucleotide. Anal Chim Acta 98(1):31–37 28. Mizutani F, Kato D, Kurita R, Mie Y, Sato Y, Niwa O (2011) Highly-sensitive biosensors with chemically-amplified responses. Electrochem. 76(8):515–521 29. Nistor C, Rose A, Wollenberger U, Pfeiffer D, Emnéus JA (2002) A glucose dehydrogenase biosensor as an additional signal amplification step in an enzyme-flow immunoassay. Analyst 127(8) 30. Popovtzer R, Natan A, Shacham-Diamand Y (2007) Mathematical model of whole cell based bio-chip: an electrochemical biosensor for water toxicity detection. J Electroanal Chem 602(1):17–23 31. Razumas V, Kulys J, Malinauskas A (1980) Kinetic amperometric determination of hydrolase activity. Anal Chim Acta 117:387–590 32. Scheller F, Renneberg R, Schubert F (1988) Coupled enzyme reactions in enzyme electrodes using sequence, amplification, competition, and antiinterference principles. In: Mosbach K (ed) Methods in enzymology, vol 137. Academic Press, New-York, pp 29–43 33. Scheller FW, Schubert F (1992) Biosensors. Elsevier Science, Amsterdam 34. Schubert F, Kirstein D, Schröder K, Scheller F (1985) Enzyme electrodes with substrate and co-enzyme amplification. Anal Chim Acta 169:391–396 35. Schulmeister T (1987) Mathematical treatment of concentration profiles and anodic current of amperometric enzyme electrodes with chemically amplified response. Anal Chim Acta 201:305–310 36. Schulmeister T, Rose J, Scheller F (1997) Mathematical modelling of exponential amplification in membrane-based enzyme sensors. Biosens Bioelectron 12(9–10):1021–1030 37. Sorochinskii V, Kurganov B (1997) Theoretical principles of the application of potentiometric enzyme electrodes. Appl Biochem Micro 33(2):116–124 38. Streffer K, Kaatz H, Bauer C, Makower A, Schulmeister T, Scheller F, Peter M, Wollenberger U (1998) Application of a sensitive catechol detector for determination of tyrosinase inhibitors. Anal Chim Acta 362(1):81–90 39. Turner APF, Karube I, Wilson GS (eds) (1990) Biosensors: fundamentals and applications. Oxford University Press, Oxford

Biosensors Utilizing Synergistic Substrates Conversion

Contents 1 2 3

4

5

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady State Current of Biosensors with Synergistic Biocatalytical Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Glucose Dehydrogenase-Based Amperometric Biosensor. . . . . . . . . . . . . . . . . . . . . . 3.1 Reaction Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Digital Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Model Validation with Experimental Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Biosensor Sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Laccase-Based Amperometric Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Reaction Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mathematical Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Initial and Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Digital Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Limits of the Synergistic Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

156 157 159 159 161 165 165 168 170 171 173 174 176 177 178

Abstract Biosensors containing glucose oxidase, carbohydrate oxidase and laccase and utilizing a few synergistic schemes of substrates conversion are modeled at steady state and transient conditions. A glucose dehydrogenase-based bioelectrocatalytical system, where ferricyanide is converted to ferrocyanide in the presence of highly reactive organic electron transfer compounds, and a laccase-based bioelectrode utilizing synergistic N-substituted phenothiazine and phenoxazine oxidation in the presence of hexacyanoferrate (II) are modeled mathematically by nonlinear reaction–diffusion equations. The modeling biosensors comprise three compartments, an enzyme layer, a dialysis membrane and an outer diffusion layer. The digital simulation was carried out using the finite difference technique. By changing the input parameters, the action of biosensors was analysed with a special emphasis to the influence of the kinetic constants and reagents concentrations on the synergy of the simultaneous substrates conversion. The digital simulation of the system confirmed that the high sensitivity of the bioelectrode achieved in the presence © Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_5

155

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of organic mediators is due to the synergistic substrates conversion demonstrated experimentally. Keywords Biosensor response · Synergistic substrates conversion · Stationary/transient state · Mathematical model

1 Introduction Kinetic analysis performed in publication [18] demonstrated that biocatalytical reactions may be conjugated with chemical processes. If chemical conversion acts in synergy with the biocatalytical process, the conversion rate increases. The fitting of synergistic reactions has been described in [23]. The synergistic reactions were applied for the intensification of biological processes such as carbohydrates transformation [22], laccase-catalysed bisphenol A (BisA) and other xenobiotics conversion [15, 19]. Performed analysis showed that the synergistic processes can be enhanced more than ten thousand times. The synergistic schemes of the substrates conversion can also be applied for producing powerful biofuel cells [40]. The utilization of synergistic processes in biocatalytical membranes permits to build high sensitive biosensors [18]. High sensitive biosensors for the determination of heterocyclic compounds were built using the oxidases-catalysed hexacyanoferrate (III) reduction. The detection limit of some heterocyclic compounds determination was 0.2 µM. The sensitivity of the biosensors was 300–10,000 times larger in comparison to the determination of hexacyanoferrate (III). A bioelectrode utilizing a synergistic scheme of substrate conversion was built using glucose dehydrogenase from Acinetobacter calcoaceticus immobilized on the surface of a graphite electrode. The response of the bioelectrode increased up to 34,000 times in the presence of high reactive organic electron acceptors (mediators). The comparison of the bioelectrode sensitivity with kinetic parameters of enzyme action in homogeneous solution revealed good correlation between the sensitivity of the bioelectrode and the predicted value from the kinetic scheme of the reactivity of mediators [16]. Bioelectrodes employing the conversion of N-substituted phenothiazines and hexacyanoferrate (II) or N-substituted phenoxazines and hexacyanoferrate (II) that act as synergistic substrates were built using graphite electrode and recombinant laccase. The response of the bioelectrodes to low reactive substrate (hexacyanoferrate (II)) increases 2.9–97 times in the presence of high reactive phenothiazine or phenoxazine mediator [21]. The action of synergistic biosensors was modeled at steady state and transient conditions.

2 Steady State Current of Biosensors with Synergistic Biocatalytical Scheme

157

2 Steady State Current of Biosensors with Synergistic Biocatalytical Scheme Modeling the biosensors containing glucose oxidase, carbohydrate oxidase and laccase and utilizing a few synergistic schemes of substrates conversion has been performed at steady state conditions by solving differential equations [16– 18, 21, 34]. Following the scheme, the oxidized glucose oxidase (GOox ) is reduced with glucose, and the reduction of hexacyanoferrate (III) (Fer) is catalysed by reduced glucose oxidase (GOred ), kred

GOox + D-glucose −→ GOred + P,

(1)

kf

GOred + 2Fer −→ GOox + 2Ferred .

(2)

In the presence of heterocyclic compounds that act as mediators (M), they are oxidized with hexacyanoferrate (III) to cation radicals. The cation radical (Mox ) formed reacts with reduced oxidase. The reduced mediator (Mred ) is further oxidized with hexacyanoferrate (III), kox

GOred + 2Mox −→ GOox + 2Mred ,

(3)

kexc

Mred + Fer −→ Mox + Ferred .

(4)

Chemical reaction (4) increases Ferred production rate, and therefore the rate of overall process is larger than reactions (2) and (3). Since the electrochemically active compound is hexacyanoferrate (II) (Ferred ), the steady state response can be calculated like the current density of the biosensor with a chemical amplification [20]: I =

F De α12 α22 d β2



1 d2 − 2 2 β

 1−

1 cosh(βd)

M0 ,

(5)

where σ1 = α1 d and σ2 = α2 d are the diffusion modules, β 2 = α12 + α22 , d is the enzyme membrane thickness and M0 corresponds to the total mediator. The diffusion modules were calculated as σ1 = (kox E0 /De )1/2 d, σ2 = (kexc F er/De )1/2 d, where E0 and F er stand for the total enzyme and the hexacyanoferrate (III) concentrations, respectively, and kexc is the bimolecular electron exchange constant between the mediator and hexacyanoferrate (III). The density I0 of the steady state current of the biosensor in the absence of the mediator is calculated as  F De 1 1− F er , (6) I0 = d cosh(γ d)

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Biosensors Utilizing Synergistic Substrates Conversion

1.2x104 1.0x104

Sr

8.0x103 6.0x103 4.0x103 2.0x103 2

100 4

80 6

60 8

10

1

40 20

12

2

14

Fig. 1 The dependence of the biosensor relative sensitivity Sr on the diffusion modules σ1 and σ2

where γ = (kf E0 /De )1/2 and kf is the constant of the reaction of the hexacyanoferrate (III) with the reduced glucose oxidase. The analysis of the dependence of the relative sensitivity (Sr = I /I0 ) on the diffusion modules reveals that Sr is larger than 1 if σ1 and σ2 are larger than 0.5 (Fig. 1). At α1 d = α2 d = 1, the relative sensitivity Sr of the biosensor is 12.9. It increases if both diffusion modules are larger than 1. The calculations show that for the biosensor containing 0.13 mM of glucose oxidase and F er = 8 mM, α1 d is 14.5 and α2 d is 113.5. In contrast, the diffusion module (γ d) of the biosensor acting with the pure hexacyanoferrate (III) is 0.13 due to the low constant of the reduced enzyme. Since γ d is less than 1, it means that the biosensor acts in a kinetic regime. It is easy to notice that at α2 d > α1 d > 1 and γ d < 1, the relative sensitivity equals Sr =

I kox = . I0 kf

(7)

The comparison of calculated and experimentally determined values reveals that the calculated relative sensitivity of the biosensors is about 3 times larger than the experimentally determined one. This deference can be caused by the limited stability of the oxidized heterocyclic compounds, uncounted parallel reaction of reduced enzyme with oxygen and external diffusion limitation of hexacyanoferrate (III) and glucose.

3 Modeling Glucose Dehydrogenase-Based Amperometric Biosensor

159

3 Modeling Glucose Dehydrogenase-Based Amperometric Biosensor At transient conditions, the biosensors utilizing synergistic schemes of substrates conversion have been modeled using different schemes for the nonlinear reaction– diffusion equations [13, 38]. A scheme of a glucose dehydrogenase-based bioelectrocatalytical system where ferricyanide is converted to ferrocyanide in the presence of highly reactive organic electron transfer compounds has also been investigated numerically [1]. This section shows the impact of the synergistic effect on the biosensor response and the sensitivity at different concentrations of glucose, which provides a strong insight on the prospect of practical application of the synergistic effect when developing glucose dehydrogenase biosensors. A computational model was developed for simulating the glucose dehydrogenase-based bioelectrode action [1, 16] and determining the influence of the physical and kinetic parameters on the biosensor sensitivity. The model was then applied to the industrially relevant optimization of the biosensors parameters [5]. The following three objectives were optimized: the apparent Michaelis constant, the output current and the enzyme amount.

3.1 Reaction Scheme The GDH biosensor being modeled is assumed to be composed of a graphite electrode covered with the nylon net (160 mesh, the thread thickness 100 µm) and the enzyme solution. The enzyme layer is separated from the bulk solution by means of the dialysis membrane. The diffusion layer where the flux of the substances takes place is also considered. The schematic view of the modeled biosensor is presented in Fig. 2 [1]. The scheme of GDH action is rather complex. The effect of substrate inhibition and cooperativity on the electrochemical response of GDH at steady state conditions was investigated in [12]. For simplification of modeling, a ping-pong scheme of mediator oxidation at high glucose concentration describing GDH action was used as described in [35]. The scheme of the GDH-based bioelectrocatalytical system involves GDH reaction (8a) with glucose followed by the reduced GDH oxidation (8b) with ferricyanide and (8c) with the oxidized mediator as well as a cross reaction (8d) of the ferricyanide and the reduced mediator [1, 12, 35], kred

GDHox + glucose −→ GDHred + P, kf

GDHred + 2Fox −→ GDHox + 2Fred ,

(8a) (8b)

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Biosensors Utilizing Synergistic Substrates Conversion

Fig. 2 The schematic view of the GDH amperometric biosensor. The thicknesses of the layers are indicated next to the boundaries of the layers

Enzyme layer

e-

Dialysis Diffusion layer membrane

GDHred

Fox

Fox

Fred

Mox

Mox

δ-glucolactone Buffer solution

Elec trode

Fred

Fox Mox

eglucose

Mred

Mred

GDHox d2

d1 a1

a0

kox

GDHred + 2Mox −→ GDHox + 2Mred , kex(d)

Fox + Mred ←−→ Fred + Mox , kex(r)

Fred Mred d3

a2

a3

x

(8c) (8d)

where GDHox and GDHred are the oxidized and reduced glucose dehydrogenase, P is the reaction product (δ-gluconolactone), Fox and Fred are ferricyanide and ferrocyanide and Mox and Mred stand for the oxidized and reduced mediators, respectively. The reaction rate constants kred , kf and kox correspond to the respective biocatalytical processes, and kex(d) and kex(r) belong to the electron exchange reactions. The biocatalytical current is produced during the oxidation of ferrocyanide and reduced mediator on the electrode surface as both, the mediators and hexacyanoferrates, are the redox active compounds, Fred − e− −→ Fox ,

(9a)

Mred − e− −→ Mox .

(9b)

In terms of substrates and products, the reaction scheme (8) and (9) can be rewritten as follows: k1

Eox + G −→ Ered + P, k2

Ered + 2S1 −→ Eox + 2P1 ,

(10a) (10b)

3 Modeling Glucose Dehydrogenase-Based Amperometric Biosensor k3

Ered + 2S2 −→ Eox + 2P2 , k4

161

(10c)

S1 + P2 ←→ P1 + S2 ,

(10d)

P1 − e− −→ S1 ,

(11a)

P2 − e− −→ S2 ,

(11b)

k5

where Ered and Eox correspond to the reduced and oxidized GDH, S1 and S2 are the substrates—ferricyanide and oxidized mediator and P1 and P2 stand for the products (ferrocyanide and reduced mediator) of the reactions.

3.2 Mathematical Model The mathematical model of the biosensor in a one-dimensional domain involves the following regions [1, 16]: 1. An enzyme-loaded nylon net (a0 < x < a1 , see Fig. 2). Due to the relatively small volume of the nylon net in comparison with the volume of the enzyme, the enzyme-loaded mesh is assumed as a periodic media, and the homogenization process is applied to the enzyme-loaded mesh [39]. In the enzyme layer, the enzymatic reactions (10a)–(10c), the cross reaction (10d) and the mass transport by the diffusion of all compounds take place. 2. A dialysis membrane (a1 < x < a2 ), where only the reactions (10d) and the mass transport of low molecular weight compounds (substrates, products and glucose) take place. 3. An outer diffusion limiting region (a2 < x < a3 ), where the cross reaction (10d) and the mass transport by the diffusion take place. This layer is modeled according to the Nernst approach [27]. 4. A convective region (x > a3 ), where the analyte concentration is maintained constant. These assumptions lead to a three-compartment model. The homogenized enzyme layer corresponds to the first compartment of the mathematical model. The dialysis membrane and the outer diffusion are the next two compartments.

3.2.1 Governing Equations Assuming the symmetrical geometry of the biosensor and homogeneous distribution of the immobilized enzyme, the mass transport and the reaction kinetics in the enzyme layer can be described by the following system of the reaction–diffusion

162

Biosensors Utilizing Synergistic Substrates Conversion

equations (a0 < x < a1 , t > 0): ∂ 2 Ered ∂Ered = DEred + k1 Eox G1 − 2k2 Ered S1,1 − 2k3 Ered S2,1 , ∂t ∂x 2

(12a)

∂Eox ∂ 2 Eox = DEox − k1 Eox G1 + 2k2Ered S1,1 + 2k3Ered S2,1 , ∂t ∂x 2

(12b)

∂S1,1 ∂ 2 S1,1 = DS1,1 − 2k2 Ered,1S1,1 − k4 S1,1 P2,1 + k5 P1,1 S2,1 , ∂t ∂x 2

(12c)

∂S2,1 ∂ 2 S2,1 = DS2,1 − 2k3 Ered S2,1 + k4 S1,1 P2,1 − k5 P1,1 S2,1 , ∂t ∂x 2

(12d)

∂P1,1 ∂ 2 P1,1 = DP1,1 + 2k2 Ered S1,1 + k4 S1,1 P2,1 − k5 P1,1 S2,1 , ∂t ∂x 2

(12e)

∂P2,1 ∂ 2 P2,1 = DP2,1 + 2k3 Ered S2,1 − k4 S1,1 P2,1 + k5 P1,1 S2,1 , ∂t ∂x 2

(12f)

∂G1 ∂ 2 G1 = DG1 − k1 Eox G1 , ∂t ∂x 2

(12g)

where x and t stand for space and time, respectively, Ered (x, t) and Eox (x, t) are the concentrations of the oxidized (Eox ) and reduced (Ered ) enzyme, respectively, Si,1 (x, t) and Pi,1 (x, t) are the concentrations of the substrate Sj and the product Pi , G1 (x, t) is the glucose concentration, DEred , DEox , DSi,1 , DPi,1 and DG1 are the corresponding diffusion coefficients in the enzyme layer and a1 = a1 − a0 is the thickness of the enzyme layer, i = 1, 2 [1]. No enzymatic reaction takes place outside the enzyme layer (x > a1 ). The kinetics of the non-enzymatic reactions (10d) and the mass transport by the diffusion in the dialysis membrane (a1 < x < a2 ) and the external diffusion (Nernst) layer (a2 < x < a3 ,) are described by the following system of equations (t > 0): ∂ 2 S1,j ∂S1,j = DS1,j − k4 S1,j P2,j + k5 P1,j S2,j , ∂t ∂x 2

(13a)

∂S2,j ∂ 2 S2,j = DS2,j + k4 S1,j P2,j − k5 P1,j S2,j , ∂t ∂x 2

(13b)

∂P1,j ∂ 2 P1,j = DP1,j + k4 S1,j P2,j − k5 P1,j S2,j , ∂t ∂x 2

(13c)

∂P2,j ∂ 2 P2,j = DP2,j − k4 S1,j P2,j + k5 P1,j S2,j , ∂t ∂x 2

(13d)

∂Gj ∂ 2 Gj = DGj , ∂t ∂x 2

(13e)

3 Modeling Glucose Dehydrogenase-Based Amperometric Biosensor

163

where Si,j (x, t), Pi,j (x, t) and Gj (x, t) are the concentrations of the substrate, product and glucose in the dialysis membrane (j = 2) and the diffusion layer (j = 3), and DSi,j , DPi,j and DGj are the corresponding diffusion coefficients, i = 1, 2 [1].

3.2.2 Initial Conditions The biosensor operation starts (t = 0) when substrates appear on the outer boundary of the diffusion layer (x = a3 ), Si,j = 0,

x ∈ [aj −1 , aj ],

Si,3 = 0,

x ∈ [a2 , a3 ),

(14b)

x = a3 ,

(14c)

Si,3 = Si,0 ,

j = 1, 2,

(14a)

where Si,0 is the concentration of the i-th substrate in the buffer solution, i = 1, 2. The products are assumed to have zero initial concentrations, while the concentration of glucose is assumed to be uniformly distributed in all the layers (t = 0), Pi,j = 0,

Gj = G0 ,

x ∈ [aj −1 , aj ],

j = 1, 2, 3,

i = 1, 2,

(15)

where G0 is the initial concentration of the glucose. Initially, the whole enzyme is in the reduced form (t = 0), Ered + Eox = E0 ,

Eox = 0,

Ered = E0 ,

x ∈ [a0, a1 ],

(16)

where E0 stands for the total concentration of the enzyme [1].

3.2.3 Boundary Conditions On the electrode surface (x = a0 ), the reaction products take part in the electrochemical reactions (11a) and (11b). The rates of these reactions are so high that the concentrations of the products at the electrode surface are permanently reduced to zero, and the substrates are regenerated (t > 0) [1, 16], Pi,1 (a0 , t) = 0, ∂Si,1  ∂Pi,1  DSi,1 = −DPi,1 ,   ∂x x=a0 ∂x x=a0

(17a) i = 1, 2.

(17b)

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Biosensors Utilizing Synergistic Substrates Conversion

Assuming the impenetrable plate surface, the mass flux of the glucose is to be equal to zero at this boundary (t > 0), ∂G1  = 0.  ∂x x=a0

(18)

During the biosensor operation, the enzyme remains locked in the enzyme layer (t > 0), ∂Eox  ∂Eox  ∂Ered  ∂Ered  = = = = 0.     ∂x x=a0 ∂x x=a1 ∂x x=a0 ∂x x=a1

(19)

In the bulk solution (x >= a3 ), the concentrations of the substrates, products and glucose remain constant (t > 0), Si,3 (a3 , t) = Si,0 ,

Pi,3 (a3 , t) = 0,

G3 (a3 , t) = G0 ,

i = 1, 2.

(20)

On the boundary between the adjacent regions having different diffusivities, the following matching conditions are defined (t > 0, x = aj , j = 1, 2, i = 1, 2): ∂Si,j  ∂Si,j +1  = DSi,j+1 , Si,j (aj , t) = Si,j +1 (aj , t),   ∂x x=aj ∂x x=aj ∂Pi,j  ∂Pi,j +1  DPi,j = DPi,j+1 , Pi,j (aj , t) = Pi,j +1 (aj , t),   ∂x x=aj ∂x x=aj ∂Gj  ∂Gj +1  = DGj+1 , Gj (aj , t) = Gj +1 (aj , t). DGj   ∂x x=aj ∂x x=aj DSi,j

(21a) (21b) (21c)

3.2.4 Biosensor Response The measured current is usually accepted as a response of an amperometric biosensor in physical experiments. The output current I (t) depends upon the flux of the ferrocyanide and the reduced mediator at the electrode surface and is expressed explicitly from Faraday’s and Fick’s laws,  ∂P1,1  ∂P2,1  , + ne2 DP2,1 i(t) = AF ne1 DP1,1   ∂x x=a0 ∂x x=a0

(22)

where F is Faraday’s constant, A stands for the geometrical surface of the electrode and ne1 and ne2 are the numbers of electrons involved in the electrochemical reactions (11a) and (11b), respectively.

3 Modeling Glucose Dehydrogenase-Based Amperometric Biosensor

165

It is assumed that the system (12)–(21) approaches a steady state as t → ∞, I = lim i(t), t →∞

(23)

where I is the steady state biosensor current.

3.3 Digital Simulation Due to nonlinearity of the governing equations (12)–(13) of the problem, no exact analytical solutions are possible [9]. Because of this, the initial boundary value problem (12)–(21) was solved numerically by applying the finite difference technique [30]. In the space coordinate x, each layer was divided into the subintervals of equal length: N1 for the enzyme layer, N2 for the dialysis membrane and N3 for the external diffusion layer. A uniform discrete grid was also introduced in the time coordinate t. An explicit finite difference scheme has been built as a result of the difference approximation [9, 10, 30]. Although explicit difference schemes have the stability limitations, these schemes have a convenient algorithm of the calculation and are simple for programming [10]. N1 = N3 = 200 and N2 = 100 was constant in the simulation of all the responses, while the time step size was recalculated for each simulation to make the difference scheme stable. The digital simulator has been implemented in C++ programming language [28]. The steady state biosensor current I was approximated by the output current IR calculated at the moment TR ,     t  di(t)  I ≈ i(TR ), TR = min t :  0 dt i(t) We used ε = 10−3 for the calculations. The values of the model parameters employed in the numerical experiments are summarized in Table 1 [1]. The values k1 and E0 were adjusted to fit the experimental results. It should be emphasized that the actual enzyme concentration E0 on the surface of the bioelectrode cannot be precisely determined due to the possible dilution of the enzyme solution placed on the top of a graphite electrode. Moreover, a part of GDH might become inactive or denatured in contact with graphite and/or the net material [16].

3.4 Model Validation with Experimental Data The numerical solution of the mathematical model (12)–(21) was compared with the published experimental data [16]. The results are depicted in Fig. 3 [1]. The calculated bioelectrode current (curve 2) did not precisely fit the experimental

166

Biosensors Utilizing Synergistic Substrates Conversion

Table 1 Simulation parameters for the GDH amperometric biosensor Parameter d1 d2 d3 k1 k2 k3 k4 k5 E0 S1,0 S2,0 G0 DS1,1 , DS2,1 , DP1,1 , DP2,1 , DG1 DS1,2 , DS2,2 , DP1,2 , DP2,2 , DG2 DS1,3 , DS2,3 , DP1,3 , DP2,3 , DG3 DEred , DEox ne1 , ne2 A

Value 10−4 m 1.87 × 10−5 m 6.42 × 10−5 m 7.9 × 103 M−1 s−1 1.4 × 103 M−1 s−1 4.9 × 107 M−1 s−1 1.4 × 107 M−1 s−1 3.3 × 106 M−1 s−1 1.17 × 10−6 M 7.6 × 10−3 M 1.14 × 10−6 M 10−1 M 3.4 × 10−10 m2 s−1 4.4 × 10−11 m2 s−1 6.3 × 10−10 m2 s−1 3.9 × 10−11 m2 s−1 1 10−5 m2

Reference [16] [16] [16] – [22] [22] [22] [22] – [16] [16] [16] [11] [14] [14] Calculated [16] [16]

100

2

90

3 1

80

i, μA

70 60

4

50 40 30 20 10 0 0

100 200 300 400 500 600 700 800 900 1000 1100 1200

t, s Fig. 3 The change of the experimental (1) and simulated (2–4) biosensor current i(t) at three values of reaction rate constant k1 : 6.9 × 105 (2), 6.9 × 104 (3) and 7.9 × 103 (4) M−1 s−1 . Other parameters are listed in Table 1

data (curve 1) although all the reaction rate and diffusion constants as well as the concentrations were derived from the experimental set-up [16]. Firstly,

3 Modeling Glucose Dehydrogenase-Based Amperometric Biosensor

167

a disagreement is noticeable due to a difference in ferricyanide responses: the bioelectrode current of 8 µA was obtained in the experiment, while in the simulated case it is only 5 µA. This could be attributed to a mediator adsorption on the graphite electrode that is coming from previous measurements (it was noticed that the mediator can be penetrated into graphite matrix more than 1 mm depth). By further analyzing the calculated current shape and comparing it with the experimental one, it is clear that they differ mainly due to the low saturation rate in the modeled system indicating too high enzyme turnover rate. In the GDH bioelectrode, this corresponds to the too fast GDH reaction with glucose that generates the active form of enzyme-reduced GDH. Therefore, k1 was reduced tenfold (Fig. 3, curve 3). However, this proved unsuccessful. A nearly perfect match was observed with the ca. 100-fold smaller k1 value (7.9 × 103 M−1 s−1 , Fig. 3, curve 4). The responses differ by approximately the same amount (not larger than 3µA) in all time domains. These differences arose in the initial stage of the biosensor operation—the ferricyanide current. In [36], a kred was determined as 6.9 × 105 M−1 s−1 . Due to the high concentration of glucose used (10−1 M ) and the reaction inhibition by glucose (Ki = 4.11 × 10−4 M ) [35], kred was recalculated as 2.83 × 103 M−1 s−1 : 0.69(0.69 × 105 + 49 × 1.14) kred (kred S + kox M) = (kred S(S/Ki + 1) + kox M) (0.69 × 105 × (105 /411 + 1) + 49 × 1.14) = 2.83 × 103 M−1 s−1 . The recalculated value is in a good agreement with k1 = 7.9 × 103 M−1 s−1 , which in our model gave the best fit to the experimental data. Therefore, the digital simulation of the synergistic model developed here proves that the glucose dehydrogenase bioelectrode utilizes synergistic substrate conversion. The dependency of the steady state response on the concentration of the mediator can be considered as the calibration curve for the analysed GDH biosensor [2, 32, 37]. A longer linear part of the calibration curve as well as larger values of the response corresponds to a configuration less prone to errors occurring during the biosensor action. The dependency was investigated at three different concentrations E0 of enzyme in the physical set-up: 4.7 × 10−6 M (Fig. 4, curve 1), 9 × 10−7 M (curve 2) and 5 × 10−7 M (curve 3). When simulating the corresponding responses, the enzyme concentration was decreased to match the physical experiment (Fig. 4) [1]. The dependence of the bioelectrode steady state current on the total mediator concentration (Fig. 4) matches well the experimentally obtained data showing that the mathematical model (12)–(21) correctly represents the GDH biosensor. The observed saturation at different enzyme concentrations occurs due to the limiting amount of GDH. The GDH concentration was adjusted to fit the experimental data set. For the largest concentration of enzyme used in the experiment (E0 = 4.7 × 10−6 M , curve 1), the concentration E0 used in the simulation had to be decreased approximately 4 times down to E0 = 1.17 × 10−6 M (curve 4). For the second (curve 2) and third (curve 3) concentrations of enzyme used in the experiments, the

168

Biosensors Utilizing Synergistic Substrates Conversion

80 70 60

I, μA

50 40

1 2 3 4 5 6

30 20 10 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

s2,0, μM Fig. 4 The bioelectrode steady state current I vs. the mediator concentration S2,0 at different concentrations (E0 ) of GDH in physical experiments: 4.7 × 10−6 (1), 9 × 10−7 (2) and 5 × 10−7 (3) M and in the simulation: 1.17 × 10−6 (4), 3.9 × 10−7 (5) and 1.75 × 10−7 (6) M . Other parameters are listed in Table 1

enzyme concentration in the simulation had to be decreased approximately 2.3 and 3 times, respectively. The decreased activity of the immobilized enzyme compared to that in solution can be associated with denaturation of some enzyme, particularly in the layer closest to the graphite electrode [1].

3.5 Biosensor Sensitivity To define the optimal configuration of the analysed biosensor, the dependence of the steady state current on the concentrations of the mediator (S2,0 ) and the glucose (G0 ) was analysed (Fig. 5a). The ratio of the steady state current I (S2,0 , G0 ) simulated as the response to the glucose of the concentration GO with the presence of the mediator (S2,0 > 0) to the corresponding steady state current measured without mediator (S2,0 = 0) can be considered as the gain GI of the response [4], GI (S2,0 , G0 ) =

I (S2,0 , G0 ) . I (0, G0 )

(25)

The dependencies of the gain GI and the steady state current I on the concentration of oxidized mediator (S2,0 ) and glucose (G0 ) are displayed in Fig. 5 [1]. As one can see from Fig. 5a, the steady state response is an increasing function of the glucose and mediator concentrations. By increasing the glucose concentration

3 Modeling Glucose Dehydrogenase-Based Amperometric Biosensor

169

16

-4

10

14 12

GI

I, μA

10 -5

10

8 6

10

10

S2,0 , μM

4

0.1

b)

10

G0 ,

mM

100

mM

100

10

G0 ,

1

0

1

1

a)

S2,0 , μM

2

1

-6

10

0.1

Fig. 5 The dependency of the biosensor steady state current I (a) and gain GI (b) on the concentrations of the mediator (S2,0 ) and glucose (G0 ). The parameters are as defined in Table 1

100 times, the response increases approximately 5 times (from I ≈ 1, 5 µA to I ≈ 7, 5 µA) for the smallest concentration of the mediator (S2,0 = 0.1 µM ) and approximately 40 times (from I ≈ 2 µA to I ≈ 80 µA) for the largest concentration of the mediator (S2,0 = 100 µM ). The influence of mediator concentration on the biosensor response is shown in Fig. 5b, where the largest gain of GI ≈ 17 times is reached at S2,0 = 100 µM and G0 = 100 mM . The sensitivity is a characteristic indicating how properly the biosensor responds to the concentration changes and is defined as the gradient of the steady state current with respect to the concentration (in this case, G0 ) to be determined [2, 29, 32, 37]. Since the concentrations of the mediator and glucose, as well as the values of the steady state current, vary in a few orders of magnitude, we use the dimensionless sensitivity, BG (S2,0 , G0 ) =

∂I (S2,0 , G0 ) G0 . × ∂G0 I (S2,0 , G0 )

(26)

The increase in sensitivity due the mediator concentration is defined as follows: GB (S2,0 , G0 ) =

BG (S2,0 , G0 ) . BG (0, G0 )

(27)

The largest biosensor sensitivity to glucose (BG ≈ 1) is achieved at the smallest analysed concentrations of glucose (G0 = 1mM ) and the largest concentrations of mediator (S2,0 = 10 µM , Fig. 6a). No gain in glucose sensitivity due to the mediator is noticeable in all the domains of S2,0 values for the lowest analysed concentration of glucose (G0 = 1 mM ). However, at the largest concentration of glucose (G0 = 100 mM ), the increase in sensitivity changes from GB ≈ 2 at S2,0 =

170

Biosensors Utilizing Synergistic Substrates Conversion

1.0

20

0. 8 15 0.4 10

0

100

a)

0.1

S2,0 , μ

G, 0 m M 10

0

10

G, 0 m M

1

M

S2, , μ

1

1

1

0.0

10

5

b)

100

0.2

10

M

GB

BG

0.6

0.1

Fig. 6 The biosensor sensitivity BG (a) and the gain GB of the sensitivity (b) vs. the concentrations of the mediator (S2,0 ) and the glucose (G0 ). The parameters are as defined in Table 1

0.1 µM to GB ≈ 22 at S2,0 = 10 µM. One can see in Fig. 6 that the use of the mediator acting in the synergy with ferricyanide increased the biosensor sensitivity from practically inapplicable (BG < 0.2) to reasonable (BG > 0.4) for relatively large concentrations of the mediator (S2,0 > 1 µM ) and glucose (G0 > 10 mM ). The biosensor sensitivity BS to the concentration S2,0 of the mediator S2 was defined similarly to the sensitivity BG and also investigated numerically, BS (S2,0 , G0 ) =

∂I (S2,0 , G0 ) S2,0 . × ∂S2,0 I (S2,0 , G0 )

(28)

The sensitivity BS is a non-monotonous function of S2,0 (see Fig. 7), which does not reach the maximum theoretically possible sensitivity of BS = 1. The nonmonotony is mostly noticeable for the largest concentrations of glucose (G0 = 100 mM ) reaching the maximum value of BS ≈ 0.7 at S2,0 ≈ 1 µM . A larger than 0.4 sensitivity is obtained only when G0 > 30 mM and 0.2 µM < S2,0 < 6 µM [1].

4 Modeling Laccase-Based Amperometric Biosensor Recently, the laccase-based bioelectrode utilizing synergistic N-substituted phenothiazine and phenoxazine oxidation in the presence of hexacyanoferrate (II) was built and investigated [21]. The synergistic process was analytically investigated assuming the steady state conditions, ignoring the mass transport and applying many other simplifications. The action of bioelectrodes includes not only the biocatalytical conversion but also the mass transport of substrates as well as products [25, 31,

4 Modeling Laccase-Based Amperometric Biosensor

171

0.7 0.6 0.5

0.3 0.2

BS

0.4

0.1 01.000

10

0.1

1

G

0,m

M ,μ S 2,0

1

M

10

Fig. 7 The biosensor sensitivity BS vs. the concentrations of the glucose (G0 ) and oxidized mediator (S2,0 ). The parameters are as defined in Table 1

37, 41]. The diffusion limitations cause bioelectrode sensitivity changes. For the accurate prediction of the bioelectrode response, the mass transport by diffusion has to be considered together with the biocatalytical conversion [3, 6, 24]. The modeling of these processes by analytical solution of system of differential equations is practically impossible [8, 26]. In this section, a laccase-based biosensor is modeled and investigated with a special emphasis to the influence of the species concentrations on the synergy of the simultaneous substrates conversion, and the effect of the chemical amplification was investigated for biosensors used for determination of heterocyclic compounds [13, 21, 38].

4.1 Reaction Scheme The laccase biosensor model involves the following regions: an enzyme-loaded nylon net (mesh = 160, thickness of the thread 100 µm), which is separated from the buffer solution by means of the dialysis membrane, an outer diffusion limiting region and a convective region where the concentrations of the substrates as well as the products are maintained constant [13, 21]. The schematic view of the modeled biosensor is presented in Fig. 8. The scheme of the laccase action contains the stadium of oxidized laccase interaction with two substrates as well as a cross reaction of the oxidized mediator

172

Biosensors Utilizing Synergistic Substrates Conversion

Enzyme layer

Fe(CN)64e-

Dialysis Diffusion layer membrane

Fe(CN)64-

laccaseox

Fe(CN)64Mred

H2O Buffer solution

Elec trode

Fe(CN)63-

Fe(CN)63-

Mred

Mred e-

O2

Mox

d2

d1

a1

a0

H-

Mox

laccasered

Fe(CN)63Mox

d3 a2

a3

x

Fig. 8 The schematic view of the laccase-based amperometric biosensor

and ferrocyanide [13, 21]. The laccase is activated with oxygen, k1

Laccase(red) + O2 + 4H+ −→ Laccase(ox) + 2H2 O, k2

3− Laccase(ox) + 4Fe(CN)4− 6 −→ Laccase(red) + 4Fe(CN)6 , k3

Laccase(ox) + 4Mred −→ Laccase(red) + 4Mox , k4

3− Mox + Fe(CN)4− 6 −→ Mred + Fe(CN)6 ,

(29a) (29b) (29c) (29d)

where Laccase(red) and Laccase(ox) are the reduced and the oxidized forms of laccase, O2 —oxygen, H2 O—water, H+ stands for the hydrogen ion, Fe(CN)4− 6 is the hexacyanoferrate (II) (ferrocyanide), Fe(CN)3− is the hexacyanoferrate 6 (III) (ferricyanide), Mred and Mox stand for the reduced and oxidized mediators, respectively, and k1 , k2 , k3 and k4 are the reaction rate constants. On the electrode surface, the ferricyanide is reduced to ferrocyanide, whereas the oxidized mediator is converted to its reduced form, 4− − Fe(CN)3− 6 + e −→ Fe(CN)6 ,

(30a)

Mox + e− −→ Mred .

(30b)

4 Modeling Laccase-Based Amperometric Biosensor

173

In terms of substrates and products, the reaction scheme (29a)–(30b) can be written as follows: k1

Ered + O + 4H+ −→ Eox + 2H2 O, k2

Eox + 4S1 −→ Ered + 4P1 , k3

Eox + 4S2 −→ Ered + 4P2 , k4

(31) (32) (33)

P2 + S1 −→ S2 + P1 ,

(34)

P1 + e− −→ S1 ,

(35)

P2 + e− −→ S2 ,

(36)

where Ered and Eox correspond to the reduced and the oxidized laccase enzyme, respectively, S1 and S2 are the substrates and P1 and P2 stand for the products of the reactions. S2 and P2 are called the reduced and the oxidized mediators, respectively. The products P1 and P2 are the electrochemically active substances [13, 21].

4.2 Mathematical Modeling Due to the relatively small volume of the nylon net in comparison with the volume of the enzyme, the enzyme-loaded mesh can be assumed as a homogenized enzyme layer [39]. Like in modeling the GDH biosensor (Sect. 3), the modeling laccase biosensor also comprises three compartments, the enzyme layer, the dialysis membrane and the outer diffusion layer [13]. 4.2.1 Governing Equations The mass transport and the reaction kinetics in the homogenized enzyme layer are described by a system of the reaction–diffusion equations (a0 < x < a1 , t > 0), ∂ 2 Ered ∂Ered = DEred − k1 Ered,1 O1 + 4k2 Eox S1,1 + 4k3Eox S2,1 , ∂t ∂x 2

(37a)

∂Eox ∂ 2 Eox = DEox + k1 Ered O1 − 4k2Eox,1S1,1 − 4k3 Eox S2,1 , ∂t ∂x 2

(37b)

∂S1,1 ∂ 2 S1,1 = DS1,1 − 4k2 Eox S1,1 − k4 P2,1 S1,1 , ∂t ∂x 2

(37c)

∂S2,1 ∂ 2 S2,1 = DS2,1 − 4k3 Eox S2,1 + k4 P2,1 S1,1 , ∂t ∂x 2

(37d)

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Biosensors Utilizing Synergistic Substrates Conversion

∂P1,1 ∂ 2 P1,1 = DP1,1 + 4k2Eox S1,1 + k4 P2,1 S1,1 , ∂t ∂x 2

(37e)

∂P2,1 ∂ 2 P2,1 + 4k3Eox S2,1 − k4 P2,1 S1,1 , = DP2,1 ∂t ∂x 2

(37f)

∂O1 ∂ 2 O1 = DO1 − k1 Ered O1 , ∂t ∂x 2

(37g)

where Ered , Eox , Si,1 , Pi,1 and O1 stand for the concentrations of the reduced and the oxidized forms of the enzyme, i-th substrate, i-th product and oxygen, respectively, a1 is the thickness of the enzyme layer and DEred , DEox , DSi,1 , DPi,1 and DO1 are the diffusion coefficients, i = 1, 2 [13, 21]. In the dialysis membrane as well as in the outer diffusion layer, no enzymatic reaction occurs. Hence, only the mass transport by diffusion and the electrochemical reaction (35) are modeled (aj −1 < x < aj , t > 0), ∂ 2 S1,j ∂S1,j = DS1,j − k4 P2,j S1,j , ∂t ∂x 2

(38a)

∂S2,j ∂ 2 S2,j = DS2,j + k4 P2,j S1,j , ∂t ∂x 2

(38b)

∂P1,j ∂ 2 P1,j = DP1,j + k4 P2,j S1,j , ∂t ∂x 2

(38c)

∂P2,j ∂ 2 P2,j = DP2,j − k4 P2,j S1,j , ∂t ∂x 2

(38d)

∂Oj ∂ 2 Oj = DOj , ∂t ∂x 2

(38e)

where Si,j , Pi,j and Oj stand for the concentrations of the i-th substrate, i-th product and oxygen in the dialysis membrane (j = 2) and the diffusion layer (j = 3), respectively, a1 is the thickness of the enzyme layer, a2 − a1 and a3 − a2 are the thicknesses of the membrane and the diffusion layer, respectively, and DSi,j , DPi,j and DOj are the diffusion coefficients, i = 1, 2, j = 2, 3.

4.3 Initial and Boundary Conditions Initially, the oxygen is assumed to be of uniform concentration, while the products are assumed of zero concentration. The biosensor operation starts when substrates appear on the boundary of the diffusion layer (t = 0), Si,j = 0,

x ∈ [aj −1 , aj ],

Si,3 = 0,

x ∈ [a2 , a3 ),

j = 1, 2,

(39a) (39b)

4 Modeling Laccase-Based Amperometric Biosensor

Si,3 = Si,0 , Pi,j = 0,

175

x = a3 , Oj = O0 ,

(39c) x ∈ [aj −1 , aj ],

j = 1, 2, 3,

(39d)

where O0 is the initial concentration of the oxygen, and Si,0 is the concentration of the i-th substrate in the buffer solution, a0 = 0, i = 1, 2. The whole enzyme is initially in the reduced form (t = 0), Ered + Eox = E0 ,

Ered = E0 ,

Eox = 0,

x ∈ [a0, a1 ],

(40)

where E0 stands for the initial as well as the total concentration of the enzyme. On the impenetrable electrode surface (x = a0 ), the reaction products are consumed and the substrates are regenerated by reactions (35) and (36), while the non-leakage condition is applied to the electro-inactive oxygen (t > 0), Pi,1 (a0 , t) = 0, ∂Si,1  ∂Pi,1  DSi,1 = −DPi,1 ,   ∂x x=a0 ∂x x=a0 ∂O1  DO1 = 0.  ∂x x=a0

(41a) i = 1, 2,

(41b) (41c)

During the biosensor operation, the enzyme remains locked in the enzyme layer (t > 0), ∂Eox  ∂Eox  ∂Ered  ∂Ered  = = = = 0.     ∂x x=a0 ∂x x=a1 ∂x x=a0 ∂x x=a1

(42)

In the bulk solution, the concentrations of the substances remain constant (t > 0, x = a3 ), S1,3 = S1,0 ,

S2,3 = S2,0 ,

P1,3 = 0,

P2,3 = 0,

O3 = O0 .

(43)

On the boundary between the three regions having different diffusivities, we define the matching conditions (t > 0, i = 1, 2, j = 1, 2), ∂Si,j  ∂Si,j +1  = DSi,j+1 , Si,j = Si,j +1 , x = aj ,   ∂x x=aj ∂x x=aj ∂Pi,j  ∂Pi,j +1  = DPi,j+1 , Pi,j = Pi,j +1 , x = aj , DPi,j   ∂x x=aj ∂x x=aj ∂Oj  ∂Oj +1  DOj = DOj+1 , Oj = Oj +1 , x = aj .   ∂x x=aj ∂x x=aj DSi,j

(44a) (44b) (44c)

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Biosensors Utilizing Synergistic Substrates Conversion

4.3.1 Response of the Laccase Biosensor The current i(t) depends upon the fluxes of the hexacyanoferrate (III) and mediator at the electrode surface,  ∂P1,1  ∂P2,1   i(t) = ne FA DP1,1 , + DP2,1   ∂x x=0 ∂x x=0

(45)

where ne is the number of electrons involved in the electrochemical reaction, F is Faraday’s constant and A stands for the geometrical surface of the electrode. The system (37)–(44c) approaches a steady state as t → ∞, I = lim i(t),

(46)

t →∞

where I is the steady state current. The geometry (parameters d1 , d2 , d3 and A) of the laccase biosensor being modeled was the same as in modeling the GDH biosensor (Sect. 3, Table 1). Values of the other model parameters used in the numerical experiments are summarized in Table 2 [13, 21].

4.4 Digital Simulation Due to nonlinearity of the governing equations (37)–(44c) of the problem, no exact analytical solutions are possible [9]. The number and structure of model equations are practically the same as for the GDH biosensor (Sect. 3). Because of this, the initial boundary value problem (37)– (44c) can be solved numerically by applying the same technique as for the GDH Table 2 Simulation parameters for the laccase biosensor

Parameter k1 k2 k3 k4 E0 S1,0 S2,0 00 DS1,1 , DS2,1 , DP1,1 , DP2,1 DS1,2 , DS2,2 , DP1,2 , DP2,2 DS1,3 , DS2,3 , DP1,3 , DP2,3 DO1 , DO2 , DO3 DEred , DEox

Value 2.44 µM−1 s−1 0.26 µM−1 s−1 18 µM−1 s−1 330 µM−1 s−1 2.76 µM 28 µM 11 µM 253 µM 3.4 × 10−10 m2 s−1 4.4 × 10−11 m2 s−1 6.3 × 10−10 m2 s−1 2.01 × 10−9 m2 s−1 3.6 × 10−11 m2 s−1

Reference [34] [21] [21] [22] [21] [21] [21] [21] [11] [14] [14] [7] [33]

4 Modeling Laccase-Based Amperometric Biosensor

177

biosensor (see Sect. 3.3) [13]. The same numerical technique can also be used for the glucose oxidase-based biosensor described in Sect. 2 as well as in [18]. The values and references of the model parameters employed in the numerical experiments with laccase-based biosensor are summarized in Table 2 [13, 21].

4.5 Limits of the Synergistic Effect In order to estimate the limits of synergistic effect, the difference IS between the overall steady state current I and the steady state current I0 is obtained with no mediator [13, 21], IS = I − I0 .

(47)

IS is the current calculated by withdrawing, from the overall bioelectrode current, the bioelectrode current in the presence of ferrocyanide at zero mediator concentration. The difference IS between the steady state currents I and I0 can be called as the synergistic current [13, 21]. Figure 9 shows the dependence of the synergistic current IS on the enzyme concentration E0 . The source currents IR and I0R are also depicted in Fig. 9, where one can see the non-monotony of the synergistic current IS [13]. Figure 9 shows that the synergistic current reaches a maximum at E0 = 5 µM. Below that value, a small amount of product P2 is produced for the synergetic reaction to occur. Above 1.8 1.6

I, I0, IS, μA

1.4 1.2 1.0

I I0 IS

0.8 0.6 0.4 0.2 0.0 0.1

1

E0, μM

10

Fig. 9 The steady state bioelectrode currents I and I0 and the synergistic current IS vs. the enzyme concentration E0 . The other parameters are listed in Table 2

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Biosensors Utilizing Synergistic Substrates Conversion

2.0 1.8

I, I0, IS, μA

1.6 1.4 1.2

I I0 IS

1.0 0.8 0.6 0.4 0.2 0.0 1

10

100

S1,0, μM Fig. 10 The dependencies of the steady state bioelectrode currents I and I0 and the synergistic current IS on the substrate concentration S1,0 at the values of the model parameters given in Table 2

the peak value, most of S1 is consumed in reaction (29b) hindering the synergetic reaction. Figure 10 shows the dependence of the synergistic current IS on the concentration S1,0 of the ferrocyanide (substrate S1 ). This dependence can be explained similarly as the dependence of IS on the enzyme concentration E0 . At very high concentrations S1,0 of the substrate S1 , the enzyme is mostly involved in reaction (29b), and this slows down the production of the product P2 which is necessary for the synergetic reaction [13].

5 Concluding Remarks The mathematical model (12)–(21) of the glucose dehydrogenase biosensor utilizing the synergistic scheme of substrates conversion can be successfully used to investigate the peculiarities of the biosensor response and sensitivity at steady as well as transition states. Simulated responses to the ferricyanide and mediator matched relatively well to the experimental results, when considering the diminished reaction rate between GDH and glucose. This reaction rate can be reduced almost 100 times due to high concentration of glucose use and enzyme inhibition as described in [35]. In this case, the simulated biosensor response to the mediator matches the experimental one, with the main difference being influenced by a mediator adsorption on the graphite electrode (Fig. 3).

References

179

The simulated bioelectrode responses match well the experimental ones when the enzyme concentration in simulations is a few times (reaching maximum at four times) less than that used in the physical experiments (Fig. 4). This can be attributed to the dilution of the enzyme when setting up the bioelectrode device [1]. The mediator concentration S2,0 significantly influences the biosensor response, as the mediator and ferricyanide act in synergy (Fig. 5a). Digital simulations showed that the gain in the steady state response due to the presence of mediator can be as large as GI ≈ 16 times (Fig. 5b) and the sensitivity to glucose as large as GB ≈ 22 times (Fig. 6b). The sensitivity of GDH biosensor to the mediator is lower at all analysed concentrations of glucose and mediator comparing with the corresponding sensitivity to glucose and reached the maximum of only BS ≈ 0.7 when S2,0 ≈ 1 µM and G0 = 100 mM (Fig. 7) [1]. The synergistic current IS is a non-monotonous function of the enzyme concentration E0 (Fig. 9) as well as the substrate concentration S1,0 (Fig. 10). The limits of synergistic effect were estimated with the maximum synergistic current IS obtained at E0 =5 µM and S1,0 =12 µM. Taken together, these results confirm the synergistic effect in the laccase biosensor [13, 21]. The synergistic effect of the laccase-based biosensors can be increased by selecting an appropriate concentration of the enzyme as well as the values of some other model parameters. The computational simulation of the biosensor response can be used as a tool in the design of novel highly sensitive laccase-based biosensors [13, 21].

References 1. Ašeris V, Gaidamauskait˙e E, Kulys J, Baronas R (2014) Modelling the biosensor utilising parallel substrates conversion. Electrochim Acta 146:752–758 2. Banica FG (2012) Chemical sensors and biosensors: fundamentals and applications. Wiley, Chichester 3. Baronas R, Ivanauskas F, Kulys J (2004) The effect of diffusion limitations on the response of amperometric biosensors with substrate cyclic conversion. J Math Chem 35(3):199–213 4. Baronas R, Kulys J, Ivanauskas F (2004) Modelling amperometric enzyme electrode with substrate cyclic conversion. Biosens Bioelectron 19(8):915–922 5. Baronas R, Žilinskas A, Litvinas L (2016) Optimal design of amperometric biosensors applying multi-objective optimization and decision visualization. Electrochim Acta 211:586– 594 6. 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 7. Bird RB, Stewart WE, Lightfoot EN (2006) Transport phenomena, 2nd edn. Wiley, New York 8. Britz D (2005) Digital simulation in electrochemistry, 3rd edn. Springer, Berlin 9. Britz D, Strutwolf J (2016) Digital simulation in electrochemistry. Monographs in electrochemistry. Springer, Cham (2016) 10. Britz D, Baronas R, Gaidamauskait˙e E, Ivanauskas F (2009) Further comparisons of finite difference schemes for computational modelling of biosensors. Nonlinear Anal Model Control 14(4):419–433

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11. Chang HC, Wu CC, Ding SJ, Lin IS, Sun IW (2005) Measurement of diffusion and partition coefficients of ferrocyanide in protein-immobilized membranes. Anal Chim Acta 532(2):209– 214 12. Durand F, Limoges B, Mano N, Mavre F, Miranda-Castro R, Saveant J (2011) Effect of substrate inhibition and cooperativity on the electrochemical responses of glucose dehydrogenase. Kinetic characterization of wild and mutant types. J Am Chem Soc 133:12801–12809 13. Gaidamauskait˙e E, Baronas R, Kulys J (2011) Modelling synergistic action of laccase-based biosensor utilizing simultaneous substrates conversion. J Math Chem 49(8):1573–158 14. Gough DA, Leypoldt JK (1979) Membrane-covered, rotated disk electrode. Anal Chem 51(3):439–444 15. Ivanec-Goranina R, Kulys J, Bachmatova I, Marcinkeviˇcien˙e L, Meškys R (2015) Laccasecatalyzed bisphenol a oxidation in the presence of 10-propyl sulfonic acid phenoxazine. J Environ Sci 30:135–139 16. Kulys J, Bratkovskaja I (2012) Glucose dehydrogenase based bioelectrode utilizing a synergistic scheme of substrate conversion. J Electroan 24(2):273–277 17. Kulys J, Dapkunas Z (2007) The effectiveness of synergistic enzymatic reaction with limited mediator stability. Nonlinear Anal Model Control 12(4):495–501 18. Kulys J, Tetianec L (2005) Synergistic substrates determination with biosensors. Biosens Bioelectron 21(1):152–158 19. Kulys J, Dapkunas Z, Stupak R (2009) Intensification of biocatalytical processes by synergistic substrate conversion. Fungal peroxidase catalyzed n-hydroxy derivative oxidation in presence of 10-propyl sulfonic acid phenoxazine. Appl Bioch Biotech 158(2):445–456 20. Kulys J, Vidziunaite R (1990) Amperometric enzyme electrodes with chemically amplified response. In Wise D (ed) Bioinstrumentation. Butterworths, Boston, pp 1263–1283 21. Kulys J, Vidziunaite R (2009) Laccase based synergistic electrocatalytical system. J Electroanal 21(20):2228–2233 22. Kulys J, Tetianec, L, Bratkovskaja I (2010) Pyrroloquinoline quinone-dependent carbohydrate dehydrogenase: activity enhancement and the role of artificial electron acceptors. Biotechnol J 5(8):822–828 23. Laurynenas A, Kulys J (2015) An exhaustive search approach for chemical kinetics experimental data fitting, rate constants optimization and confidence interval estimation. Nonlinear Anal Model Control 20(1):145–157 24. Lyons MEG (2006) Modelling the transport and kinetics of electroenzymes at the electrode/solution interface. Sensors 6(12):1765–1790 25. Lyons MEG, Murphy J, Rebouillat S (2000) Theoretical analysis of time dependent diffusion, reaction and electromigration in membranes. J Solid State Electrochem 4(8):458–472 26. Meena A, Eswari A, Rajendran L (2010) Mathematical modelling of enzyme kinetics reaction mechanisms and analytical solutions of non-linear reaction equations. J Math Chem 48(2):179– 186 27. Nernst W. (1904) Theorie der Reaktionsgeschwindigkeit in heterogenen Systemen. Z Phys Chem 47(1), 52–55 28. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, Cambridge 29. Sadana A, Sadana N (2011) Handbook of biosensors and biosensor kinetics. Elsevier, Amsterdam (2011) 30. Samarskii A (2001) The theory of difference schemes. Marcel Dekker, New York 31. Scheller F, Pfeiffer D (1978) Enzymelektroden. Z Chem 18(2):50–57 32. Scheller FW, Schubert F (1992) Biosensors. Elsevier, Amsterdam 33. Shleev S, Christenson A, Serezhenkov V, Burbaev D, Yaropolov A, Gorton, L, Ruzgas T (2005) Electrochemical redox transformations of T1 and T2 copper sites in native trametes hirsuta laccase at gold electrode. Biochem J 385(3):745–754 34. Tetianec L, Kulys J (2009) Kinetics of N-substituted phenothiazines and N-substituted phenoxazines oxidation catalyzed by fungal laccases. Cent Eur J Biol 4(1):62–67

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Biosensors Acting in Injection Mode

Contents 1 2

3 4

5

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Injection Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Numerical Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Dynamics of the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Peculiarities of the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequential Injection Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Diffusion Limitations on the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Two-Compartment Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Dynamics of the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Impact on the Apparent Michaelis Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

184 184 185 186 188 190 194 195 196 198 199 202

Abstract This chapter numerically investigates the sensitivity of an amperometric biosensor acting in the flow injection mode when the biosensor contacts an analyte for a short time. The analytical system is modeled by non-stationary reaction– diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetics of an enzymatic reaction. At first, the biosensor action is modeled by a mono-layer mono-enzyme model assuming no external diffusion limitation. Then, the model is extended to a two-compartment model by adding an outer diffusion layer. The biosensor operation is analysed with a special emphasis to the conditions at which the biosensor sensitivity can be increased and the calibration curve can be prolonged by changing the injection duration, the permeability of the external diffusion layer, the thickness of the enzyme layer and the catalytic activity of the enzyme. The apparent Michaelis constant is used as a main characteristic of the sensitivity and the calibration curve of the biosensor. The numerical simulation was carried out using the finite difference technique. Keywords Flow injection mode biosensor · Mathematical model · Finite difference technique · Non-stationary process

© Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_6

183

184

Biosensors Acting in Injection Mode

1 Introduction The biosensors are combined with flow injection analysis (FIA) for on-line monitoring of raw materials, product quality and the manufacturing process [25, 26, 34, 38, 42]. In the FIA, a biosensor contacts with the substrate for a short time (seconds to tens of seconds), whereas in the batch analysis the biosensor remains immersed in the substrate solution for a long time [23]. Compared to the batch systems, the FIA systems present the advantages of the reduction in analysis time allowing a high sample throughput and the possibility to work with small volumes of the substrate [13, 33, 40]. The growing interest of using biosensors as quantitative detectors in FIA offers convenience arising from automation and higher precision [12, 15]. The FIA arrangement also presents a wide response range as well as high sensitivity and selectivity [29, 31]. Actual biosensors acting in the FIA mode have already been modeled using an empirical approach based on neural networks [14, 17] and a deterministic approach that takes into account the reaction–diffusion equations [17, 30]. The biosensors have usually been modeled by ignoring the external diffusion [5, 22, 43]. However, in practical biosensing systems, the mass transport outside the enzyme region is of crucial importance, and it has to be taken into consideration when modeling the biosensor action [19, 20, 24, 39]. Theoretical investigation of the FIA biosensing systems presented a higher quality of the concentrations prediction than the corresponding batch systems [5, 7]. This chapter presents results of the biosensor modeling at mixed enzyme kinetics and with external/internal diffusion limitations in the FIA [2, 4, 5]. The biosensing systems are modeled by non-stationary reaction–diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetics of an enzymatic reaction [6, 11]. Firstly, the biosensor action is modeled by one-layer model assuming no external diffusion limitation. Then, the model is extended to a two-compartment model by adding an outer diffusion layer. The biosensor operation is analysed with a special emphasis to the conditions at which the biosensor sensitivity can be increased and the calibration curve can be prolonged by changing the injection duration, the permeability of the external diffusion layer, the thickness of the enzyme layer and the catalytic activity of the enzyme. The apparent Michaelis constant is used as one of the main characteristics of the sensitivity and the calibration curve of the biosensor [18, 20, 28, 37, 41]. The numerical simulation was carried out using the finite difference technique [6, 11].

2 Flow Injection Analysis The mono-layer mono-enzyme model introduced in Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors” describes the operation of the biosensors acting in the batch mode where the concentration of the substrate

2 Flow Injection Analysis

185

as well as the product over the enzyme surface (bulk solution/membrane interface) remains constant, while the biosensor comes into contact with the substrate. In the flow injection mode, the biosensor contacts the substrate only for a short term (seconds to tens of seconds) [22, 35]. When the analyte disappears, the buffer solution swills the enzyme surface reducing the substrate concentration at this surface to zero.

2.1 Mathematical Model The governing equations and the initial conditions for the injection mode are the same as for the batch mode, ∂S ∂ 2S Vmax S = DS 2 − , ∂t KM + S ∂x ∂P ∂ 2P Vmax S = DP , + 2 ∂t ∂x KM + S

(1) x ∈ (0, d),

S(x, 0) = 0,

P (x, 0) = 0,

S(d, 0) = S0

P (d, 0) = P0 ,

t > 0,

x ∈ [0, d) ,

(2)

where x and t stand for space and time, respectively, d is the thickness of the enzyme layer (membrane), S(x, t) and P (x, t) are the concentrations of the substrate and the product, respectively, S0 and P0 are the concentrations of the substrate and the product in the bulk solution, DS and DP are the diffusion coefficients, Vmax is the maximal enzymatic rate and KM is the Michaelis constant [9, 27]. The boundary conditions applied for the electrode surface are also the same as for the batch mode, DS

∂S  = 0,  ∂x x=0

P (0, t) = 0,

t > 0.

(3)

In the FIA mode of the biosensor operation, the substrate appears in the bulk solution only for a short time period called the injection time. Later, the substrate disappears from the bulk solution, and the concentrations of the substrate as well as the product reduce to zero,  S(d, t) =

P (d, t) =

S0 , t ≤ TF , 0,

t > TF ,

 P0 ,

t ≤ TF ,

0,

t > TF ,

(4)

(5)

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Biosensors Acting in Injection Mode

where TF is the injection time, i.e. the time when the substrate disappears from the bulk solution/membrane interface. Usually, the zero concentration of the reaction product is also accepted for the initial phase of the operation, i.e. P0 = 0. The governing equations (1) together with the initial conditions (2) and the boundary conditions (3)–(5) together form the mathematical model of the amperometric biosensor acting in the injection mode [2, 4, 5]. Let us notice that the boundary conditions (4) and (5) for the injection analysis generalize the conditions used in the batch analysis. Assuming that the injection time equals the biosensor operation time (TF = T ), the model of the biosensor acting in the injection mode can also be used for the biosensors acting in the batch mode.

2.2 Numerical Solution Solving the initial boundary value problem of the injection analysis for the specific boundary conditions (4) and (5) the substrate and the product concentration can be approximated as follows: j

j

SN = S0 , j

SN = 0,

PN = P0 , j

PN = 0,

j = 1, . . . , MF ,

(6)

j = MF + 1, . . . , M,

(7)

  j : tj ≤ TF .

(8)

where MF =

max

j =1,...,M

Figure 1 shows the concentration profiles calculated by using the numerical simulation at a relatively low concentration of the substrate (S0 = 0.001KM ) and the following values of other parameters: DS = DP = 300 µm2/s, Vmax = 100 µM/s,

KM = 100 µM,

d = 100 µm,

ne = 2,

P0 = 0, TF = 10 s.

(9)

When comparing these concentration profiles with the corresponding profiles obtained for the batch mode (see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”), one can see notably different shapes of the corresponding curves. However, until the moment TF , i.e. while the enzyme surface contacts the analyte, the concentration profiles are practically identical for both modes of the analysis, i.e. the batch and the injection. The boundary conditions (4) and (5) are discontinuous. Discontinuity in the boundary conditions increases the error of the numerical solution [1, 36]. In order

S, μM

2 Flow Injection Analysis

187

0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00

1

2

4

0

10

20

30

40

50

3

60

70

80

90

100

90

100

x, μm

a) 0.5

1 0.4

2

P, μM

3 0.3

4

0.2

0.1

0.0 0

b)

10

20

30

40

50

60

70

80

x, μm

Fig. 1 The concentration profiles of the substrate (a) and the product (b) at S0 = 0.001KM and the following values of time t: 10.5 (1), 11 (2), 12 (3) and 13 s (4). Values of the other parameters are as defined in (9)

to circumvent the discontinuity, the following boundary conditions are also used instead of (4) and (5): ⎧ ⎪ t ≤ TF , ⎪ ⎨S0 , S(d, t) = ϕ(S0 , t, ε), TF < t ≤ TF + ε, (10) ⎪ ⎪ ⎩0, t > T + ε, F

188

Biosensors Acting in Injection Mode

Fig. 2 The profiles of the substrate concentration S0 in the case of the discontinuous (a) boundary conditions (4) and (5) as well as the continuous (b) conditions (10) and (11)

S0

S0

0

a)

P (d, t) =

⎧ ⎪ ⎪ ⎨P0 , ϕ(P0 , t, ε), ⎪ ⎪ ⎩0,

TF

t

0

b)

TF TF+ε

t

t ≤ TF , TF < t ≤ TF + ε,

(11)

t > TF + ε,

where ϕ(C, t, ε) is a continuous function such as ϕ(C, TF , ε) = C, ϕ(C, TF + ε, ε) = 0 and ϕ(C, t, ε). ϕ(C, t, ε) monotonously decreases when t changes form TF to TF + ε. Several different expressions of ϕ as well as small values of ε were used in the numerical simulation. The following linear function is one of the simplest functions to be used in (10) and (11): ϕ(C, t, ε) =

C (TF − t + ε) , ε

TF ≤ t ≤ TF + ε

(12)

Figure 2 shows the profiles of the substrate concentration S0 in the case of the discontinuous boundary conditions (4) and (5) as well as the continuous boundary conditions (10) and (11) assuming the ϕ definition (12). Accepting a relatively small value of ε, e.g. ε = 0.01TF , no notable difference between the numerical solutions was observed.

2.3 Dynamics of the Biosensor Response Figure 3 shows the dynamics of the density i of the output current of the biosensor acting in the injection mode,  ∂P  . i(t) = ne F DP ∂x x=0

(13)

One can see that the current of the biosensor acting in the injection mode is a nonmonotonous function of time t. Because of the substrate remaining in the enzyme membrane, the mass diffusion as well as the enzyme reaction remains for some time even after the disconnection of the biosensor and the substrate.

2 Flow Injection Analysis

189

1.2 1.1

2

1.0

1

i, nA/mm2

0.9 0.8 0.7 0.6 0.5

3

0.4

4

0.3 0.2 0.1 0.0 0

2

4

6

8

10 12 14 16 18 20 22 24 26 28 30

t, s Fig. 3 The dynamics of the density i of the biosensor current at four values of the membrane thickness d: 0.01 (1), 0.015 (2), 0.1 (3) and 0.15 (4) mm. The other parameters are the same as in Fig. 1

Figure 3 shows that the shapes of two curves (1 and 2) corresponding to the thin enzyme membranes (0.01 and 0.015 mm) noticeably differ from the other two curves (3 and 4). The enzyme kinetics determines mainly the response of the two thin biosensors, while the response of the two thicker biosensors is controlled by the diffusion. In the case of the thin membranes, the diffusion module σ 2 equals 0.33 and 0.75, respectively, i.e. both of them are less than unity, σ2 =

d 2 Vmax . DS KM

(14)

σ 2 for the other two biosensors is a hundred times greater, i.e. significantly greater than unity. The small diffusion module shows a very fast mass transport by diffusion. Because of this, the reaction–diffusion process in the thin biosensors approaches the steady state before the time TF , and the current falls to zero very shortly. In the case of the thick enzyme layer (σ 2 > 1), the diffusion is relatively slow, and it determines the smooth curve of the current. The system (1)–(5), describing the biosensor action in the injection analysis, approaches to a steady state as t → ∞. Because of the zero concentration of the surrounding substrate at t > TF , the density I of the steady state current falls to zero, I = lim i(t) = 0. t →∞

(15)

190

Biosensors Acting in Injection Mode

Because of this, the steady state current is not practically useful in the analytical systems. Since the current density i(t) of the biosensor acting in the injection mode is a non-monotonous function, the maximal current is one of the mostly used characteristics for this kind of biosensors, Imax = max {i(t)} , t >0

(16)

where Imax is the maximal density of the biosensor current. The corresponding time Tmax of the maximal biosensor current is used instead of the steady state one, Tmax = {t : i(t) = Imax } .

(17)

In the numerical simulation, the calculation can be stopped when the current reaches the maximum. The minimal time at which the current starts to decrease may be accepted as the time of the maximal current,   Tmax ≈ min tj : ij > ij +1 , j =1,...

(18)

where tj = j τ , ij = i(tj ), τ is the step of the discrete grid in the direction of time. Using the maximal current as the main characteristic of the biosensor acting in the injection mode leads to the following definition of the dimensionless sensitivity BS : BS (S0 ) =

dImax (S0 ) S0 × , Imax (S0 ) dS0

(19)

where Imax (S0 ) is the density of the maximal biosensor current calculated at the substrate concentration S0 .

2.4 Peculiarities of the Biosensor Response Using the computer simulation, the dependence of the biosensor response on the substrate concentration and the injection duration was investigated.

2.4.1 Maximal Current versus Substrate Concentration Figure 4 shows the density Imax of the maximal current versus the substrate concentration S0 at different values of the maximal enzymatic rate Vmax . As it is possible to notice, the shape of curves in the injection analysis is very similar to that

2 Flow Injection Analysis

191

101

Imax, μA/mm2

100

4 5

1 2 3

10-1 10-2 10-3 10-4

KM 10

5KM

-5

10-6

10-5

10-4

10-3

10-2

10-1

S0 , M Fig. 4 The density Imax of the maximal biosensor current versus the substrate concentration S0 at five maximal enzymatic rates Vmax : 0.1 (1), 1 (2), 10 (3), 100 (4) and 1000 (5) µM/s, and the membrane thickness d = 0.15 mm. The other parameters are the same as in Fig. 1

in the batch analysis (see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”). The Michaelis constant KM has been defined as the substrate concentration at which the reaction rate is the half of its maximal value [9, 27]. In the batch analysis, the effect of halving the maximal current is valid when the biosensor response is under enzyme kinetics control, i.e. when the diffusion module σ is less than unity. In such a case, the Michaelis constant KM is approximately equal to the apparent app Michaelis constant KM , which is used as one of the main characteristics of the sensitivity and the calibration curve of biosensors [18, 20, 28, 37, 41], app

KM =

  1 S0∗ : Imax (S0∗ ) = 0.5 lim Imax (S0 ) , S0 →∞ S0

app Kˆ M =

app

KM , KM

(20)

where Imax (S0 ) is the maximal density of the biosensor current calculated at the app substrate concentration S0 , and Kˆ M is the dimensionless apparent Michaelis app app constant. For the biosensor of a concrete configuration, KM as well as Kˆ M can be rather easily calculated by multiple simulations of the maximal response changing the substrate concentration S0 . The effect of halving was verified if it is valid in the injection analysis as in the batch one. As one can see in Fig. 4, the maximum of the density of the maximal current is approximately equal to 1.44 mA/mm2 at the maximal enzymatic rate of 0.1 µM/s. The half of the maximum, 0.72 mA/mm2 , of the maximal current is reached at S0 ≈ app app 0.5 mM/s = 5KM , which is notably greater than KM , i.e. KM ≈ 5KM and Kˆ M = 5.

192

Biosensors Acting in Injection Mode

For the tenfold greater value of Vmax , 1 µM/s, the half of the maximum of the maximal current reaches at a slightly greater value of the substrate concentration, app S0 ≈ 5.2KM , Kˆ M = 5.2. In both these cases of Vmax , 0.1 and 1 µM/s, the diffusion 2 module σ is less than unity, and σ 2 equals 0.075 and 0.75, respectively. So, the app apparent Michaelis constant KM is greater than the theoretical Michaelis constant KM even when the biosensor response is about to be under the enzyme kinetics app control. Figure 4 shows that KM increases with increasing the diffusion module [2, 5]. The Michaelis constant KM has been defined as the substrate concentration at which the reaction rate is the half of its maximal value [9, 27]. The higher substrate concentration, at which the half of the maximum of the maximal current is reached, app (greater KM ) corresponds to a longer linearly increasing part of the maximal biosensor current Imax as a function of S0 . Consequently, the biosensors with a app greater KM are sensitive to a wider range of the substrate concentrations than app those with a lower KM . Below, this is discussed in detail. 2.4.2 Response Time Versus Substrate Concentration Figure 5 shows the evolution of the time Tmax of the maximal current vs. the substrate concentration S0 in the injection analysis. When comparing curves in Fig. 5 with the corresponding curves of the half of the steady state time obtained in the batch analysis (see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”), it is possible to notice a considerable difference in the shape. In the injection analysis (Fig. 5), Tmax is a monotonous increasing

45

Tmax , s

40 35

1 2 3 4 5

30 25 20 15 10-6

10-5

10-4

10-3

10-2

10-1

S0 , M Fig. 5 The time Tmax of the maximal biosensor current versus the substrate concentration S0 . The parameters and the notations are the same as in Fig. 4

2 Flow Injection Analysis

193

function of S0 at all values of Vmax , while in the batch analysis, T0.5 is a monotonous decreasing or even a non-monotone function of S0 . Figure 5 shows a considerable increase of the time Tmax of the maximal response with the increase of the substrate concentration S0 only at high values of S0 . This can be explained by a sufficient supply of the substrate after the time TF when the substrate disappears from the surrounding analyte. The more substrate penetrates into the enzyme layer until the time TF , the longer enzyme reaction occurs. At low concentrations of the substrate, the time of the maximal current increases very weakly.

2.4.3 Sensitivity Versus Substrate Concentration As has been mentioned in Sect. 2.4.1, the biosensors acting in the injection mode have the higher half of the maximum of the maximal current than the corresponding biosensors acting in the batch mode. The half of the maximum of the maximal current influences directly the biosensor sensitivity. Biosensors with the higher half of the maximum of the maximal current are sensitive to a wider range of the substrate concentration than those with a lower half of the maximum of the maximal current. Figure 6 shows the dependence of the biosensor sensitivity for a wide range of the substrate concentrations calculated for several values of the enzyme activity Vmax . At low substrate concentrations, the amperometric biosensors are very sensitive in both modes of the analysis: the batch and the injection. At higher concentrations

1.0 0.9 0.8 0.7

BS

0.6 0.5 0.4 0.3

1 2 3 4 5

0.2 0.1 0.0 10-6

10-5

10-4

10-3

10-2

10-1

S0, M Fig. 6 The normalized biosensor sensitivity BS versus the substrate concentration S0 . The parameters and the notations are the same as in Fig. 4

194

Biosensors Acting in Injection Mode

1.0 0.9

1 2 3 4 5

0.8 0.7

BS

0.6 0.5 0.4 0.3 0.2 0.1 0.0 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

TF , s Fig. 7 The normalized biosensor sensitivity BS versus the injection duration TF at concentration S0 = 1 mM = 10KM of the substrate. The other parameters and the notations are the same as in Fig. 4

of S0 , the sensitivity of the biosensor acting in the injection analysis is noticeably higher than the sensitivity of the corresponding biosensor acting in the batch mode (see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”). To find out if the injection duration TF really influences the sensitivity of biosensors, the normalized sensitivity BS is calculated for different values of the duration TF and the enzymatic activity Vmax . The results of calculation are depicted in Fig. 7. Figure 7 shows a notable effect of the injection duration TF on the biosensor sensitivity for low maximal enzymatic activities Vmax . As has been discussed above (see Fig. 3), the thickness of the enzyme membrane also effects the behaviour of the response of the biosensor acting in the injection mode. Additional calculations showed that the injection duration TF effects the biosensor sensitivity when the diffusion module σ 2 is less than unity, i.e. when the biosensor response is under the enzyme kinetics control.

3 Sequential Injection Analysis The injection analysis can be generalized to the sequential injection analysis [35], where two surrounding environments of the biosensor action, i.e. the substrate solution of the concentration S0 and the buffer solution containing no substrate, alternate sequentially. Figure 8 shows the profile of the concentration S0 of the

4 Effect of Diffusion Limitations on the Biosensor Response Fig. 8 The profile of the substrate concentration S0 in the case of the sequential injection

195

S0

t

0

TF

t

3TF+2T0

substrate surrounding the biosensor in the sequential injection analysis. The product concentration out of the enzyme layer equals zero at all values of t > 0. Assuming the time intervals of the injection and the swilling of the uniform duration, the boundary conditions corresponding to the profile of the substrate concentration depicted in Fig. 8 are defined as follows:

S(d, t) =

 S0 , 0,

(TF + T0 )k ≤ t ≤ (TF + T0 )k + TF , (TF + T0 )k + TF < t < (TF + T0 )(k + 1),

,

k = 1, . . . , K, (21)

P (d, t) =

 P0 , (TF + T0 )k ≤ t ≤ (TF + T0 )k + TF , 0,

(TF + T0 )k + TF < t < (TF + T0 )(k + 1),

,

k = 1, . . . , K, (22)

where TF is the duration of the injection, T0 is the duration of the swilling and K is the number of injections. The steady state current is practically meaningless for the sequential injection analysis. One can see in Fig. 9 that the biosensor current oscillates. The number of peaks corresponds to the number of injections. If the next injection starts when the current is above zero, then the next peak may be higher than the previous one. For example, the first local maximum of curve 4 in Fig. 9 equals 0.244 nA/mm2 (at t = 13.9 s), the second one equals 0.269 (t = 33.6 s) and the third equals 0.271 (at t = 53.6 s). Starting from the second peak, the local maximum of the current as well as the time period between two peaks practically remains unchanged. As one can see in Fig. 9, i(t) is practically the periodical function with the period of TF + T0 .

4 Effect of Diffusion Limitations on the Biosensor Response For investigating the effect of the external diffusion limitation on the biosensor response, the biosensing system was modeled by two- and three-compartment models [2, 4]. In this section, an amperometric biosensor acting in the flow injection

196

Biosensors Acting in Injection Mode 1.2 1.1

1

1.0

i, nA/mm2

0.9 0.8

2

0.7 0.6

3

0.5 0.4

4

0.3 0.2 0.1 0.0 0

10

20

30

40

50

60

t, s Fig. 9 The dynamics of the biosensor current at four values of the thickness d, 0.01 (1), 0.015 (2), 0.1 (3) and 0.15 (4) mm, the injection duration TF = 10 s and the swilling duration TF = 10 s. The other parameters are the same as in Fig. 1

mode is modeled by a two-compartment model, and the biosensor operation is analysed with a special emphasis to the conditions at which the biosensor sensitivity can be increased and the calibration curve can be prolonged by changing the injection duration, the permeability of the external diffusion layer, the thickness of the enzyme layer and the catalytic activity of the enzyme.

4.1 Two-Compartment Model The two-compartment mathematical model involves three regions: the enzyme layer where enzymatic reaction as well as the mass transport by diffusion takes place, a diffusion limiting region where only the diffusion takes place and a convective region [2, 39]. The dynamics of the concentrations of the substrate as well as the product in the enzyme layer are described by the reaction–diffusion system, ∂ 2 Se ∂Se Vmax Se = DSe − , ∂t ∂x 2 KM + Se ∂Pe ∂ 2 Pe Vmax Se = DPe + , 2 ∂t ∂x KM + Se

(23a) x ∈ (0, d),

t > 0,

(23b)

4 Effect of Diffusion Limitations on the Biosensor Response

197

where x and t stand for space and time, d is the thickness of the enzyme layer, Se and Pe are the concentrations of the substrate and the product in the enzyme layer, DSe and DPe are the diffusion coefficients, Vmax is the maximal enzymatic rate and KM is the Michaelis constant [8, 21, 39]. In the outer layer, only the mass transport by diffusion of both species takes place, ∂ 2 Sb ∂Sb = DSb , ∂t ∂x 2 ∂Pb ∂ 2 Pb = DPb , ∂t ∂x 2

(24a) x ∈ (d, d + δ),

t > 0,

(24b)

where Sb and Pb are the substrate and the product concentrations in the outer layer, DSb and DPb are the diffusion coefficients and δ is the thickness of the diffusion layer. The biosensor operation starts when the substrate appears in the bulk solution (t = 0), Pe (x, 0) = 0,

Se (x, 0) = 0,

x ∈ [0, d],

Pb (x, 0) = 0, x ∈ [d, d + δ],  0, x ∈ [d, d + δ), Sb (x, 0) = S0 , x = d + δ,

(25a) (25b) (25c)

where S0 is the substrate concentration in the bulk solution. The most boundary conditions (t > 0) are typical for two-compartment model of an amperometric biosensor (see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”),   ∂Se  ∂Sb  = DSb , Se (d, t) = Sb (d, t), DSe ∂x x=d ∂x x=d   ∂Pe  ∂Pb  DPe = DPb , Pe (d, t) = Pb (d, t), ∂x x=d ∂x x=d  ∂Se  Pe (0, t) = 0, = 0, ∂x x=0  S0 , t ≤ TF , Pb (d + δ, t) = 0, Sb (d + δ, t) = 0, t > TF , where TF is the injection time.

(26a) (26b) (26c) (26d)

198

Biosensors Acting in Injection Mode

4.2 Dynamics of the Biosensor Response When solving the mathematical model, an implicit finite difference scheme was built on a uniform discrete grid [2, 10, 11, 39]. The computational model was developed in the C language [32]. The mathematical model and the numerical solution were validated using a known analytical solution [3, 39]. Assuming TF → ∞, the mathematical model (23)–(26) approaches the two-compartment model of the amperometric biosensor acting in the batch mode [3, 39]. A number of experiments were carried out, while the values of some parameters were kept constant [2, 16], DSe = DPe = 300 µm2/s,

DS2 = 2DS1 ,

KM = 100 µM,

d = 200 µm.

ne = 1,

DP2 = 2DP1 ,

(27)

Figure 10 shows the evolution of the density i(t) of the biosensor current,  ∂Pe  . i(t) = ne F DPe ∂x x=0

(28)

The biosensor action was simulated at a moderate concentration S0 of the substrate (S0 = KM ) and different values of the other model parameters, the dimensionless 5

1

i, nA/mm2

4

3

2 3 5

2

6

4

1

8 7

0 0

10

20

30

40

50

60

70

80

90

t, s Fig. 10 The dynamics of the biosensor current at different values of the diffusion module σ 2 : 1 (5–8), 2 (1–4), the Biot number β: 1 (3, 4, 7, 8), 2 (1, 2, 5, 6) and the injection time TF : 3 (2, 4, 6, 8), 6 s (1, 3, 5, 7)

4 Effect of Diffusion Limitations on the Biosensor Response

199

diffusion module σ 2 (1 and 2), the injection time TF (3 and 6 s) and the dimensionless Biot number β (1 and 2), σ2 =

d 2 Vmax , DSe KM

β=

d/DSe DSb d . = δ/DSb DSe δ

(29)

Assuming (27), two values (1 and 2) of σ 2 were obtained at the following values of the maximal enzymatic rate Vmax : 0.75 and 1.5 µM, respectively. Accordingly, β = 2 corresponds to the thickness δ of the external diffusion layer equal to the thickness d of the enzyme layer, while β = 1 at δ = (DSb /DSe )d = (DPb /DPe )d = 2d = 400 µm. One can see in Fig. 10 the non-monotonic behaviour of the biosensor current. In all the cases, the current increases during the injection period (t ≤ TF ). However, the current increases still some time after the substrate disappears from the bulk solution (t ≥ TF ). The time moment of the maximum current as well as the maximal current itself depends on all the three model parameters: σ 2 , β and TF . Figure 10 shows that the density Imax of the maximal current (as defined in (16)) increases almost two times when the injection time TF doubles. However, the influence of doubling the time TF on the time of the maximal current is rather slight. When comparing curves 1 (TF = 6 s) and 2 (TF = 3 s), one can see that the time of the maximal response increases little, i.e. from 13.9 s to 16 s, then Imax increases almost 2 times from 2.3 to 4.4 nA/mm2 at σ 2 = 2, β = 2. Figure 10 also shows that the biosensor response significantly depends on the Biot number β. A decrease in β noticeable prolongs the response. As one can see in Fig. 10, the maximal current decreases when the thickness of the external diffusion layer increases, i.e. β decreases. FIA biosensing systems have already been investigated by using mathematical models at zero thickness (β → ∞) of the external diffusion layer (see Sect. 2). Figure 10 visually substantiates the importance of the external diffusion layer.

4.3 Impact on the Apparent Michaelis Constant Using the numerical simulation, the biosensor operation was analysed with a special emphasis to the conditions at which the biosensor sensitivity can be increased and the calibration curve can be prolonged by changing the injection duration, the biosensor geometry and the catalytic activity of the enzyme. In order to investigate the influence of the model parameters on the dimensionless app apparent Michaelis constant Kˆ M , the simulation was performed at wide ranges of the values of the diffusion module σ 2 , the Biot number β and the injection time TF . app The constant Kˆ M , introduced in (20), expresses the relative prolongation (in times) of the calibration curve in comparison with the theoretical Michaelis constant KM .

200

Biosensors Acting in Injection Mode 106

1 2 3 4 5 6

105

103

^

KMapp

104

102 101 100 10-1

100

101

102

Fig. 11 The dependence of the dimensionless apparent Michaelis constant Kˆ M on the Biot number β at different values of the diffusion module σ 2 : 0.1 (1, 2), 1 (3, 4), 10 (5, 6) and the injection time TF : 1 (1, 3, 5), 10 s (2, 4, 6) app

Figure 11 shows the dependence of the dimensionless apparent Michaelis app app constant Kˆ M on the Biot number β. The constant Kˆ M was calculated at three values of the diffusion module σ 2 : 0.1 (curves 1 and 2), 1 (3, 4) and 10 (5, 6) and two practically extreme values of the injection time TF : 1 (1, 3, 5) and 10 s (2, 4, 6). At concrete values of σ 2 and TF , the calculations were performed by changing the thickness δ of the diffusion layer from 40 µm (δ = 0.2d) to 4 mm (δ = 20d) and keeping constant the thickness d = 200 µm of the enzyme layer. One can see in Fig. 11 that at relatively large values of the Biot number (β > 10), app app the dimensionless apparent Michaelis constant Kˆ M (as well as dimensional KM ) is almost insensitive to changes in β. However, when β < 1, a decrease in β affects app a drastic increase of Kˆ M . By increasing the thickness δ of the external diffusion layer as well as decreasing the diffusivity DSb in this layer, i.e. by decreasing β, the calibration curve of the biosensor can be prolonged by a few orders of magnitude. The diffusivity of species in the diffusion layer is usually relative to the permeability of the diffusion layer. The Biot number β might also be decreased by decreasing the permeability of the external diffusion layer. In the case of the batch analysis, an advantageous effect of the external diffusion on the length of the calibration curve of amperometric biosensors is rather well known [18, 20, 37, 41]. Figure 11 shows that, due to the FIA, the linear part of the calibration curve becomes even longer. This figure also shows a weak dependence app of Kˆ M on the diffusion module σ 2 when σ 2 ≤ 1. To properly investigate the impact of the injection time TF on the length of app the linear part of the calibration curve, the apparent Michaelis constant Kˆ M was

4 Effect of Diffusion Limitations on the Biosensor Response

201

107 10

7 8 9

4 5 6

1 2 3

6

104

^

KMapp

105

103 102 101 100 0

1

2

3

4

5

6

7

8

9

10

TF , s app Fig. 12 The dimensionless apparent Michaelis constant Kˆ M vs. the injection time TF , σ 2 : 0.1 (1, 4, 7), 1 (2, 5, 8), 10 (3, 6, 9), β: 0.1 (1–3), 10 (4–6), 100 (7–9)

app calculated by changing TF from 1 up to 10 s. Values of Kˆ M were calculated at 2 three values of the diffusion module σ (0.1, 1 and 10) and three values of the Biot number β (0.1, 10 and 100). The calculation results are depicted in Fig. 12. app As one can see in Fig. 12, Kˆ M exponentially increases with a decrease in the injection time TF . The calibration curve of the biosensor can be prolonged by a few orders of magnitude only by decreasing the injection time TF . The impact of TF is practically invariant to the Biot number β and the diffusion module σ 2 . The exponential increase is specifically characteristic at low values of TF (TF < 3 s). One can also see in Fig. 12 that there is no noticeable difference between curves 4, 5, 7 and 8. Two other curves 6 and 9 only slightly differ from each other. app So, at relatively high values of the Biot number (β ≥ 10), Kˆ M is only slightly sensitive to changes in β. This effect was even more easily shown in Fig. 11. Figure app 12 additionally shows that Kˆ M increases more at greater values of the diffusion 2 module σ rather than at lower ones. Finally, the impact of the diffusion module σ 2 on the apparent Michaelis constant app was evaluated. The result is presented in Fig. 13. The constant Kˆ M was calculated at three values of the diffusion Biot number β: 0.1 (curves 1 and 4), 10 (2, 5) and 100 (3, 6) and two values of the injection time TF : 1 (1–3) and 10 s (4–6). At concrete values of β and TF , the calculations were performed by changing the maximal enzymatic rate Vmax from 75 nM/s (σ 2 = 0.1) to 7.5 µM/s (σ 2 = 10) and keeping other parameters constant. app As one can see in Fig. 13, Kˆ M is a monotonous increasing function of σ 2 . When the enzyme kinetics predominates in the response (σ 2 < 1) of a FIA biosensing app system with a relatively large Biot number (β ≥ 10), the Kˆ M is approximately

202

Biosensors Acting in Injection Mode 106 5

10

1 2 3

4 5 6

103

^

KMapp

104

102 101 100 10-1

100

101

2 app Fig. 13 The apparent Michaelis constant Kˆ M vs. the diffusion module σ 2 , β: 0.1 (1, 4), 10 (2, 5), 100 (3, 6), TF : 1 (1–3), 10 s (4–6)

a constant function (curves 2, 3, 5 and 6). When the biosensor response is under app diffusion control (σ 2 > 1), Kˆ M exponentially increases with an increase in the 2 diffusion module σ . These features were particularly noticed in Figs. 11 and 12. In real applications of biosensors, the diffusion module σ 2 can be modified by changing the enzyme activity (Vmax ) as well as the thickness d of the enzyme layer. The maximal enzymatic rate Vmax is actually a product of two parameters: the catalytic constant k2 and the total concentration Et of the enzyme [18, 37]. It is usually impossible to modify the k2 part. The maximal rate Vmax might be modified by changing the enzyme concentration Et in the enzyme layer. Vmax is relative to the total enzyme used in a biosensor. In the batch analysis (TF → ∞), when the enzyme kinetics distinctly predominates in the biosensor response (σ 2  1 and β → ∞), the apparent Michaelis app constant KM approaches the theoretical Michaelis constant KM , i.e. KM,app ≈ app app KM , Kˆ M ≈ 1 [18, 20, 28, 37, 41]. As one can see in Figs. 11, 12, and 13, Kˆ M 2 is rather near to 1 also in the case of the FIA biosensing systems when σ < 1, TF = 10 and β = 100.

5 Concluding Remarks The mathematical model (23)–(26) of the flow injection analysis system based on an amperometric biosensor can be successfully used to investigate the kinetic peculiarities of the biosensor response. The two-compartment model (23)–(26)

References

203

approaches the one-layer model (1)–(5) assuming zero thickness δ of the external diffusion layer (the Biot number β approaches infinity). By increasing the thickness δ of the external diffusion layer or by decreasing the permeability of the external diffusion layer (by decreasing the Biot number β), the calibration curve of the biosensor can be prolonged by a few orders of magnitude. At relatively large values of the Biot number (β > 10), the apparent Michaelis constant app KM is almost insensitive to changes in β (Fig. 11). app The apparent Michaelis constant KM exponentially increases with a decrease in the injection time TF . The calibration curve of the biosensor can be prolonged by a few orders of magnitude only by decreasing the injection time TF . The impact of TF is practically invariant to the Biot number β and the diffusion module σ 2 . The exponential increase is specifically characteristic at low values of TF (TF < 3 s) (Fig. 12). app The KM is a monotonous increasing function of the diffusion module σ 2 . When the enzyme kinetics distinctly predominates in the response (σ 2 < 1 and β ≥ 10), app app the KM is approximately a constant function, while at σ 2 > 1, KM exponentially 2 increases with an increase in σ (Fig. 13).

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13. Cervini P, Éder Tadeu, Cavalheiro G (2008) Determination of paracetamol at a graphitepolyurethane composite electrode as an amperometric flow detector. J Braz Chem Soc 19(5):836–841 14. Ferreira LS, Souza MBD, Trierweiler JO, Hitzmann B, Folly R (2003) Analysis of experimental biosensor/FIA lactose measurements. J Braz Chem Soc 20(1):7–13 15. Frangu A, Pravcová K, Šilarová P, Arbneshi T, Sýs M (2019) Flow injection tyrosinase biosensor for direct determination of acetaminophen in human urine. Anal Bioanal Chem 411(11):2415–2424 16. Gough DA, Leypoldt JK (1979) Membrane-covered, rotated disk electrode. Anal Chem 51(3):439–444 17. 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 18. Gutfreund H (1995) Kinetics for the life sciences. Cambridge University Press, Cambridge 19. Ivanauskas F, Baronas R (2008) Modeling an amperometric biosensor acting in a flowing liquid. Int J Numer Meth Fluids 56(8):1313–1319 20. Ivanauskas F, Kaunietis I, Laurinaviˇcius V, Razumien˙e J, Šimkus R (2008) Apparent Michaelis constant of the enzyme modified porous electrode. J Math Chem 43(4):1516–1526 21. Kulys J (1981) The development of new analytical systems based on biocatalysts. Anal Lett 14(6):377–397 22. Lammertyn J, Verboven P, Veraverbeke E, Vermeir S, Irudayaraj J, Nicolai B (2006) Analysis of fluid flow and reaction kinetics in a flow injection analysis biosensor. Sens Actuator B Chem 114(2):728–736 23. Lüdi H, Garn M, Bataillard P, Widmer H (1990) Flow injection analysis and biosensors: applications for biotechnology and environmental control. J Biotechnol 14(1):71–79 24. Lyons MEG (2006) Modelling the transport and kinetics of electroenzymes at the electrode/solution interface. Sensors 6(12):1765–1790 25. MacLaurin P, Andrew K, Worsfold P (1995) Flow injection analysis. In: McLennan F, Kowalski B (eds) Process analytical chemistry. Springer, Dordrecht, pp 159–182 26. Mello LD, Kubota LT (2002) Review of the use of biosensors as analytical tools in the food and drink industries. Food Chem 77(2):237–256 27. Michaelis L, Menten ML (1913) Die Kinetik der Invertinwirkung. Biochem Z 49:333–369 28. 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 29. Oliveira MC, Lanza MR, Tanaka AA, Sotomayor MDPT (2010) Flow injection analysis of paracetamol using a biomimetic sensor as a sensitive and selective amperometric detector. Anal Methods 2(5):507–512 30. Osipov AP, Zaitseva NV, Egorov AM (1996) Chemiluminescent immunoenzyme biosensor with a thin-layer flow-through cell. Application for study of a real-time bimolecular antigenantibody interaction. Biosens Bioelectron 11(9):881–887 31. Piano M, Serban S, Pittson R, Drago G, Hart J (2010) Amperometric lactate biosensor for flow injection analysis based on a screen-printed carbon electrode containing Meldola’s BlueReinecke salt, coated with lactate dehydrogenase and NAD+. Talanta 82(1):34–37 32. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, Cambridge 33. Prieto-Simón B, Campás M, Andreescu S, Marty JL (2006) Trends in flow-based biosensing systems for pesticide assessment. Sensors 6(10):1161–1186 34. Nenkova R, Atanasova DI, Godjevargova T (2010) Flow injection analysis for amperometric detection of glucose with immobilized enzyme reactor. Biotechnol Biotechnol Equ 24(3):1986–1992 35. Ruzicka J, Hansen E (1988) Flow injection analysis. Wiley, New York 36. Samarskii A (2001) The theory of difference schemes. Marcel Dekker, New York 37. Scheller FW, Schubert F (1992) Biosensors. Elsevier, Amsterdam

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Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

Contents 1 2 3

4

5

6

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Biosensors Utilizing First Order Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Biosensors Utilizing Michaelis–Menten Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Solving the Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Dimensionless Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Simulated Biosensor Action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Impact of the Diffusion Module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Impact of the Substrate Concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Multi-Layer CME-Based Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Three-Layer Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Effective Diffusion Coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Concentration Profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of the Electrocatalysis Using Michaelis–Menten and Second-Order Reaction Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Characteristics of the Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Numerical Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Effect of the Combination of Two Types of Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Impact on the Apparent Michaelis Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

208 209 210 211 215 218 220 223 225 227 227 229 230 231 232 234 235 235 237 238

Abstract The chemically modified enzyme electrodes (CMEEs) and biomimetic catalysts (BC) based electrodes (BCEs) exhibit numerous remarkable properties, such as high sensitivity, specificity and stability. In this chapter mathematical models of several types of these biosensors are considered at stationary and transient conditions. Firstly, the action of CMEEs produced by modifying carbon electrodes with redox active component (mediator) and an enzyme is considered at stationary conditions. Then, the two compartment modeling is applied to the biosensor utilizing an ordered ping-pong scheme of the enzyme catalysed substrate conversion in the presence of the mediator. After that, an approach where two diffusion layers are modeled by one layer by introducing an effective diffusion © Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_7

207

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coefficient is discussed and applied to a model comprising three layers. The final study is dedicated to analysis of biosensors based on biomimetic catalysts utilizing a combination of two kinds of redox interaction—a simple chemical second-order reaction and Michaelis-type redox reaction scheme. By applying these two types of reactions the influence of the physical and the kinetic parameters on the biosensor response is investigated. Keywords Biosensor steady-state/non-stationary response · Chemically modified enzyme electrode · Biomimetic catalysts electrode

1 Introduction The chemical modification of electrodes aims to give certain chemical and physical properties that did not naturally exist in the materials used as electrical conductors. Atoms, molecules or particles are attached to the surface of different materials to modify their electronic and structural properties, leading to changing their functionality [1]. A large part of commercially available and disposable biosensors is prepared by screen-printing technology [25, 53, 60]. They usually contain chemically modified (CM) graphite together with an enzyme [1, 22, 59]. At a CM electrode (CME) electrocatalysis is accomplished by an immobilized redox substance acting as an electron transfer mediator between the graphite electrode and the reaction agent [27, 42, 43, 45, 57]. The immobilization of an enzyme on the surface of CM produces chemically modified enzyme electrode (CMEE) (biosensor). The CMEEs exhibit numerous remarkable properties, such as high sensitivity and stability [1, 46]. To improve the efficiency of the biosensors design and to optimize their configuration the mathematical modeling has rather been widely used [3, 6, 18, 20, 21, 62]. Mathematical modeling has also been successfully applied to specific biosensors based on the CME [13, 16, 17, 28, 32, 33, 44, 47, 63]. ˇ A mathematical model for a generic CMEE (biosensor) was proposed by Cenas and Kulys [16] and later improved [11]. The model was formulated in the onedimensional space and composed of two layers with different properties: an enzyme layer and an outer diffusion layer. The enzyme layer was attached to the CM electrode and the analysed buffer solution was considered as being well mixed. Practical biosensors often contain an additional outer membrane—a thin layer of polyvinyl alcohol, polyurethane, cellulose, latex or other material [9, 23]. Outer diffusion limiting membranes are used to prevent the enzyme from dissolution, to improve the biosensors’ stability and to prolong the linear part of its calibration curve. The outer membrane can be modeled as a diffusive layer placed between the enzyme and the Nernst diffusion layers [8]. In this chapter mathematical models of several types of amperometric biosensors based on CME are considered at stationary and transient conditions. Firstly, CMEE produced by modifying carbon electrodes with redox active component (mediator) and an enzyme is investigated at stationary conditions [27]. Then, the

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209

two compartment modeling is applied to a biosensor utilizing an ordered pingpong scheme of the enzyme catalysed substrate conversion in the presence of the mediator [8, 11]. After that, an approach where two diffusion layers are modeled by one layer by introducing an effective diffusion coefficient is discussed and the approach is applied to a model comprising three layers: an enzyme layer, a dialysis membrane and an outer diffusion layer [8]. The final study is devoted to biosensors utilizing a combination of two kinds of redox interaction—a simple chemical second-order reaction and Michaelis-type redox reaction [50, 51].

2 Modeling Biosensors Utilizing First Order Kinetics Chemically modified electrodes (CME) can be produced by modifying carbon electrodes with redox active component (mediator) which reacts with immobilized enzyme [27]. For modification of electrodes adsorption or covalent immobilization of the mediator is used. The peculiarities of CME-based biosensors modeling arise due to the mediator location on the electrode that produces special boundary conditions. It was assumed that the concentration of the mediator (Ms ) on the electrode is constant due to the desorption of the mediator. The enzyme membrane with thickness d is fixed on CME (Fig. 1). At the steady state and in the absence of a substrate the concentration of the mediator decreases uniformly in the membrane. In the presence of the substrate a part of the mediator is reduced (oxidized) and the reaction product is oxidized (reduced) on the electrode. For calculations the concentration of the mediator was assumed to be much less than KM and the enzymatic reaction was described as a

1.0 0.9

M/MS , P/MS

0.8

1 2

0.7 0.6

3

0.5 0.4

2

0.3 0.2

3

0.1 0.0 0.0

1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

x/d Fig. 1 Concentration profiles of the mediator (solid lines) and the product (dashed lines) in the enzyme membrane. αd = 0 (1), αd = 3 (2), αd = 6 (3)

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Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

bimolecular process of the mediator interaction with an enzyme. The constant of reaction rate is kb : V (M) = kb ME0 .

(1)

The steady state diffusion and enzymatic equation (2) was solved with the boundary conditions M = Ms , P = 0 at x = 0 and M = 0, P = 0 at x ≥ d, De

d2 M − k b E0 M = 0 , dx 2

d2 P De 2 + kb E0 M = 0 , dx

(2)

where M and P are the concentrations of the mediator and the product in the enzyme membrane of thickness d, respectively. The mediator (M) and the product (P ) concentrations in the enzyme membrane were calculated form the following equations: M exp(αd) sinh(αx) , = exp(αx) − Ms sinh(αd) P x exp(αd) sinh(αx) − exp(αx) − + 1 , = Ms sinh(αd) d

(3)

where α 2 = kb E0 /De . The density I of the stationary biosensor current is I = ne F De

  1 dP  Ms . α coth(αd) − = n F D e e dx x=0 d

(4)

When αd ≈ 1, the response is I = 0.33ne F dkox E0 Ms .

(5)

For the first time the model was used to describe the action of glucose biosensors based on organic metal or on carbon electrodes modified with organic metal components [16]. Later the biosensors of this type have found one of the largest practical applications [36, 60].

3 Modeling Biosensors Utilizing Michaelis–Menten Kinetics This section presents a model allowing an effective computer simulation of the amperometric biosensors based on the CME [11]. Using this model the influence

3 Modeling Biosensors Utilizing Michaelis–Menten Kinetics

211

of the physical and the kinetic parameters on the biosensor response is investigated. An ordered ping-pong scheme of the enzyme catalysed substrate conversion in the presence of the mediator is considered. The CMEE is considered as an electrode containing a relatively thin layer of the low soluble mediator and covered with an enzyme membrane. The developed model approaches the two-compartment model. In order to define the main governing parameters of the mathematical model the corresponding dimensionless model is derived. By changing the input parameters the output results are numerically analysed at transition and steady state conditions [11].

3.1 Mathematical Model An ordered ping-pong scheme of the enzyme (E) catalysed substrate (S) conversion in the presence of the mediator (M) is considered, k1

k2

Eox + S  ES → Ered + P1 , k−1

k3

Ered + M → Eox + P,

(6a)

(6b)

where Eox , Ered and ES are oxidized enzyme, reduced enzyme and the enzyme substrate, respectively, P and P1 are the reaction products [11]. The reaction takes place on a chemically modified electrode (CME). The CMEE is considered as an electrode containing a relatively thin layer of the low soluble mediator and covered with an enzyme membrane. The developed model approaches the two-compartment model and involves three regions: the enzyme layer where the enzymatic reaction as well as the mass transport by diffusion take place, the diffusion limiting region where only the mass transport by diffusion takes place and the convective region where the analyte concentration is maintained constant.

3.1.1 Governing Equations Assuming the quasi-steady state approximation, the concentration of the intermediate complex (ES) does not change and is usually neglected when simulating the biochemical behaviour of the biosensors [24, 54, 60]. Additionally assuming the symmetrical geometry of the electrode and the homogeneous distribution of the immobilized enzyme in the enzyme layer of a uniform thickness, the mass transport and the kinetics in the enzyme layer can be expressed by the system of reaction–

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diffusion equations (t > 0) [11], ∂ 2 Se ∂Se = DSe − V (Me , Se ), ∂t ∂x 2

(7a)

∂Me ∂ 2 Me = DMe − V (Me , Se ), ∂t ∂x 2

(7b)

∂Pe ∂ 2 Pe = DPe + V (Me , Se ), ∂t ∂x 2

0 < x < d,

(7c)

where x stands for space, t stands for time, Se (x, t), Me (x, t), Pe (x, t) are the concentrations of the substrate, the mediator and the reaction product, respectively, d is thickness of the enzyme layer, DSe , DMe and DPe are the diffusion coefficients for the substrate, mediator and reaction product, respectively, and V (Me , Se ) is the quasi-steady state enzyme reaction rate of the ordered ping-pong scheme (6). According to the scheme 1 Et 1 1 = + + , V kcat kred Se kox Me

(8)

where Et is the total concentration of the enzyme, kcat is the catalytic constant of the ES conversion, kcat = k2 , kox is a constant of the enzyme interaction with the mediator, kox = k3 , kred is an apparent bimolecular constant of the enzyme and the substrate interaction, kred = k1 k2 /(k−1 + k2 ). The total sum Et of the concentrations of all the enzyme forms is assumed to be constant in the entire enzyme layer, Et = Eox + Ered + Es , where Eox , Ered , Es are the concentrations of Eox , Ered , ES, respectively. From (8) the following nonlinear expression of the reaction rate is obtained: V (Me , Se ) =

Et kcat kred kox Me Se . kred kox Me Se + kcat kox Me + kcat kred Se

(9)

Outside the enzyme layer only the mass transport by diffusion takes place. The outer mass transport obeys a finite diffusion regime, ∂Sb ∂ 2 Sb = DSb , ∂t ∂x 2

(10a)

∂Mb ∂ 2 Mb = DMb , ∂t ∂x 2

(10b)

∂Pb ∂ 2 Pb = DPb , ∂t ∂x 2

d < x < d + δ,

(10c)

where Sb (x, t), Mb (x, t) and Pb (x, t) are the concentrations of the substrate, the mediator and the product, respectively, in the diffusion layer, δ is the thickness of the diffusion layer, DSb , DMb , DPb are the diffusion coefficients [11].

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213

According to the Nernst approach the layer (d < x < d + δ) remains unchanged with time. Away from it the solution is in motion and uniform in concentration [2, 19, 31, 34, 48, 61].

3.1.2 Initial Conditions Let x = 0 represent the surface of the CMEE, while x = d—the boundary between the enzyme membrane and the buffer solution. The biosensor response starts when some substrate appears in the bulk solution. This occurs in the initial conditions (t = 0) [11], Se (x, 0) = 0, 0 ≤ x ≤ d,  M0 , x = 0, Me (x, 0) = 0, 0 < x ≤ d, Pe (x, 0) = 0, 0 ≤ x ≤ d,  0, d ≤ x < d + δ, Sb (x, 0) = S0 , x = d + δ,

(11a) (11b) (11c) (11d)

Mb (x, 0) = 0,

d ≤ x ≤ d + δ,

(11e)

Pb (x, 0) = 0,

d ≤ x ≤ d + δ,

(11f)

where M0 is the concentration of the mediator at the boundary between the electrode and the enzyme layer, S0 is the concentration of the substrate in the bulk solution.

3.1.3 Boundary and Matching Conditions On the boundary between two regions having different diffusivities, the matching conditions are defined (t > 0) [11], ∂Se  ∂Sb  = DSb , Se (d, t) = Sb (d, t),   ∂x x=d ∂x x=d ∂Me  ∂Mb  DMe = DMb , Me (d, t) = Mb (d, t),   ∂x x=d ∂x x=d ∂Pe  ∂Pb  DPe = DPb , Pe (d, t) = Pb (d, t).   ∂x x=d ∂x x=d DSe

(12a) (12b) (12c)

These conditions mean that the fluxes of the substrate, the mediator and the product through the stagnant external layer are equal to the corresponding fluxes

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Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

entering the surface of the enzyme membrane. The partition of the substrate, mediator and product in the membrane versus the bulk is assumed to be equal. In the bulk solution the concentrations of the substrate, the mediator and the product remain constant (t > 0) [11], Sb (d + δ, t) = S0 ,

(13a)

Mb (d + δ, t) = 0,

(13b)

Pb (d + δ, t) = 0.

(13c)

The concentration Pe of the reaction product at the electrode surface (x = 0) is being permanently reduced to zero due to the electrode polarization. Following the scheme (6), the substrate is an electro-inactive substance. The concentration of the mediator covering the electrode surface is kept constant. This is described by the following boundary conditions (t > 0): DSe

∂Se  = 0,  ∂x x=0

(14a)

Me (0, t) = M0 ,

(14b)

Pe (0, t) = 0.

(14c)

The constant concentration M0 of the mediator on the electrode can be achieved by permanent dissolution of adsorbed mediator. The direct measurements show that M0 can be as low as 10−6 M [29].

3.1.4 Biosensor Response The measured current is accepted as a response of an amperometric biosensor in physical experiments. The anodic current is directly proportional to the flux of the reaction product at the electrode surface [24, 54, 60], i.e. on the border x = 0. Since the total current is also directly proportional to the area of the electrode surface, the total current is normalized with the area of that surface. The density i(t) of the biosensor current at time t can be obtained explicitly from the Faraday and the Fick laws, ∂Pe  (15) i(t) = ne F DPe  , ∂x x=0 where ne is the number of electrons involved in a charge transfer at the electrode surface, and F is the Faraday constant.

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215

The system (7), (10)–(14) approaches a steady state as t → ∞, I = lim i(t),

(16)

t →∞

where I is assumed as the density of the steady state biosensor current.

3.2 Solving the Problem The concentrations S, M and P of the substrate, the mediator and the reaction product, respectively, can be defined for the entire interval x ∈ [0, d + δ] as follows (t ≥ 0) [11]: U (x, t) =

 Ue (x, t), Ub (x, t),

x ∈ [0, d], x ∈ (d, d + δ],

U = S, M, P .

(17)

The concentration functions (S, M and P ) are continuous in the entire interval x ∈ [0, d + δ]. The initial boundary value problem (7), (10)–(14) can be solved numerically by applying the finite difference technique [15, 52]. Since the governing equations, the initial and the boundary conditions are of the same type as in a two-compartment model and as in modeling a biosensor with porous membrane discussed in previous sections, the model equations can be approximated to the difference equations by using the same technique. The adequacy of the mathematical model of the biosensor can be evaluated using a known analytical solution of the two-compartment model of amperometric biosensors (see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”). As one can see from the reaction rate v introduced by (9) the kinetics of the biochemical reaction significantly depends on the ratio of the substrate and the mediator concentrations. Let us introduce the dimensionless ratio  of the substrate (S0 ) to the mediator (M0 ) concentrations combining them with the rates of the corresponding reactions (6), =

S0 kred . M0 kox

(18)

At relatively low concentrations of the substrate when   1 (S0 kred  M0 kox ), the reaction rate V (M, S) introduced by (9) reduces as follows: V (M, S) ≈

Et kcat kred S Et kcat S . = kred S + kcat kcat /kred + S

(19)

216

Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

Consequently, in this case the mediator concentration does not affect the product concentration, and the governing equation (7b) can be neglected when simulating the biosensor response. Assuming (19), the governing equations (7a), (7c), (10a), (10c), together with the initial conditions (11a), (11c), (11d), (11f) and the boundary conditions (12a), (12c), (13a), (13c), (14a), (14c) form an initial boundary value problem which can be solved analytically in cases when the reaction function (19) approaches a linear function [55]. At so low concentrations of the substrate as S0  kcat /kred , the reaction rate V (M, S) reduces further to Et kred S. Assuming V (M, S) ≈ Et kred S, the density I of the steady state current can be calculated as follows [10, 55] (also see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”): I = ne F S0

σred sinh(σred ) + βP cosh(σred ) − βP βS DSe × × , d σred sinh(σred ) + βS cosh(σred ) βP + 1

(20)

where 2 σred = kred q,

q=

Et d 2 , DSe

βS =

DSb d , DSe δ

βP =

DPb d . DPe δ

(21)

The dimensionless parameters βS and βP are the Biot numbers for the substrate and product, respectively [9, 34]. Large values of Biot number means that the internal diffusion is very slow compared with the external diffusion. This case when a biosensor operates under internal diffusion control is rather typical. The case of relatively low Biot number-values implies much slower diffusion within the outer membrane compared with the diffusion within the enzyme layer, which leads to the external diffusion control case [9]. 2 is known as the diffusion module or the DamköhThe dimensionless factor σred ler number [5]. The diffusion module compares the rate (Et kred ) of the enzyme reaction with the diffusion rate (DSe /d 2 ). At relatively low concentrations of the mediator when   1 (M0 kox  S0 kred ), the reaction rate V (M, S) reduces to V (M, S) ≈

Et kcat kox M . kox M + kcat

(22)

In this case the substrate concentration may be neglected when simulating the biosensor response. Assuming (22), the governing equations (7b), (7c), (10b), (10c), together with the initial conditions (11b), (11c), (11e), (11f) and the boundary conditions (12b), (12c), (13b), (13c), (14b), (14c) form an initial boundary value problem which can be solved analytically in cases when the reaction function (22) approaches a linear function [8, 16]. At concentrations of the mediator as low as M0  kcat /kox , V (M, S) ≈ Et kox M and the density I of the steady state current can be calculated

3 Modeling Biosensors Utilizing Michaelis–Menten Kinetics

217

as follows [8]: I = ne F DPe 

M0 d

2 sinh(σ ) − β β sinh(σ ) + β β σ cosh(σ ) (βP + 1)σox ox M P ox M P ox ox βM (1 + βP )(sinh(σox ) + σox cosh(σox )) sinh2 (σox ) − cosh2 (σox ) + cosh(σox ) (βM − βP )σox × + , βM (1 + βP ) sinh(σox ) + σox cosh(σox )

×

(23) where 2 = kox q = σox

Et kox d 2 kox 2 = σ . DSe kred red

(24)

A zero thickness of the external diffusion layer, δ = 0, the density I of the steady state current can be calculated as follows [16]: I = ne F DPe M0

1 (σox coth(σox ) − 1) . d

(25)

This formula is an analogue of (4). The number q introduced by (21) incorporates the diffusion rate (DSe /d 2 ) and the total concentration Et of the enzyme. q includes all the parts of the diffusion module except the constant kred of the enzyme–substrate interaction and the constant kox of the enzyme-mediator interaction. Assuming constant values of kred as well as of kox , the number q can be used as a reduced diffusion module instead of two module 2 and σ 2 . σred ox It is rather well known that an ordinary enzyme electrode acts under diffusion limitation when the diffusion module is much greater than unity [9, 13, 55]. If the diffusion module is significantly less than unity, then the enzyme kinetics predominates in the biosensor response. In the case of CM electrode, the kinetics of the enzymatic reaction was expressed by two rates: kred and kox . These two rates of the reactions (6) lead to two 2 and σ 2 . Assuming k diffusion module: σred red < kox and taking into consideration ox definitions (21) and (24), one can state that the biosensor acts under limitation of 2  1). If q  1/k the enzyme-mediator interaction when q  1/kox (σox red ( 2  1), then the response is under control of the mass transport by diffusion. σred At intermediate values of q (1/kox < q < 1/kred ) the biosensor acts under mixed limitation of the diffusion and the enzyme–substrate interaction.

218

Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

3.3 Dimensionless Model In order to define the main governing parameters of the mathematical model the following dimensionless parameters are introduced: tDS x δ , tˆ = 2 e , δˆ = , d d d kred S kox M kox P Sˆ = , Mˆ = , Pˆ = , kcat kcat kcat xˆ =

kred S0 Sˆ0 = , kcat

(26)

kox M0 Mˆ 0 = , kcat

where S, P and M are the concentrations introduced by (17), xˆ is the dimensionless distance from the electrode surface, tˆ stands for the dimensionless time, δˆ is ˆ M, ˆ Pˆ , Sˆ0 , Mˆ 0 are the the dimensionless thickness of the diffusion layer and S, dimensionless concentrations. The dimensionless thickness of enzyme membrane equals one. The governing equations (7) in dimensionless coordinates are expressed as follows: ∂ Sˆ Mˆ Sˆ ∂ 2 Sˆ 2 − σred = , 2 ∂ xˆ ∂ tˆ Mˆ Sˆ + Mˆ + Sˆ

(27a)

Mˆ Sˆ DMe ∂ 2 Mˆ ∂ Mˆ 2 − σ = , ox DSe ∂ xˆ 2 ∂ tˆ Mˆ Sˆ + Mˆ + Sˆ Mˆ Sˆ DPe ∂ 2 Pˆ ∂ Pˆ 2 , + σox = 2 ˆ ˆ ˆ DSe ∂ xˆ ∂t M S + Mˆ + Sˆ

(27b) 0 < xˆ < 1,

tˆ > 0.

(27c)

The governing equations (10) yield to the following equations: DSb ∂ 2 Sˆ ∂ Sˆ = , DSe ∂ xˆ 2 ∂ tˆ

(28a)

∂ Mˆ DMb ∂ 2 Mˆ , = DSe ∂ xˆ 2 ∂ tˆ

(28b)

∂ Pˆ DPb ∂ 2 Pˆ , = DSe ∂ xˆ 2 ∂ tˆ

ˆ 1 < xˆ < 1 + δ,

tˆ > 0.

(28c)

The initial conditions (11) transform to the following conditions:  ˆ 0, 0 ≤ xˆ < 1 + δ, ˆ x, S( ˆ 0) = ˆ Sˆ0 , xˆ = 1 + δ,

(29a)

3 Modeling Biosensors Utilizing Michaelis–Menten Kinetics

ˆ x, M( ˆ 0) =

 Mˆ 0 , 0,

Pˆ (x, ˆ 0) = 0,

219

xˆ = 0, ˆ 0 < xˆ ≤ 1 + δ,

ˆ 1 ≤ xˆ ≤ 1 + δ.

(29b) (29c)

The matching (12) and the boundary (13)–(14) conditions are rewritten as follows (tˆ > 0): ∂ Sˆ  DSb ∂ Sˆ  =   , ˆ ˆ ∂ xˆ x=1 DSe ∂ xˆ x=1  DMb ∂ Mˆ  ∂ Mˆ  =   , ˆ ˆ ∂ xˆ x=1 DMe ∂ xˆ x=1 ∂ Pˆ  DPb ∂ Pˆ  =   , ˆ ˆ ∂ xˆ x=1 DPe ∂ xˆ x=1

(30b)

ˆ + δ, ˆ tˆ) = Sˆ0 , S(1

(31a)

(30a)

(30c)

ˆ + δ, ˆ tˆ) = 0, M(1

(31b)

ˆ tˆ) = 0, Pˆ (1 + δ,

(31c)

∂ Sˆ  = 0,  ˆ ∂ xˆ x=0 ˆ M(0, tˆ) = Mˆ 0 ,

(32b)

Pˆ (0, tˆ) = 0.

(32c)

(32a)

The dimensionless current (flux) iˆ and the corresponding dimensionless stationary current Iˆ are defined as follows: ˆ i(t)kox d ˆ tˆ) = ∂ P  = , i( ˆ ∂ xˆ x=0 ne F DPe kcat

ˆ tˆ). Iˆ = lim i( tˆ→∞

(33)

Assuming the same diffusion coefficients for all three species, only the following dimensionless parameters remain in the dimensionless mathematical model (27)– ˆ (32): δ—the thickness of the diffusion layer, Sˆ0 —the substrate concentration in the 2 and bulk solution, Mˆ 0 —the mediator concentration at the electrode surface, σox 2 σred —the diffusion module and Drel —the ratio of the external diffusivity to the internal diffusivity, Drel = DSb /DSe = DMb /DMe = DPb /DPe . In all the calculations

220

Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

Drel was equal to 2. As has been mentioned above, it is reasonable to use the reduced diffusion module q instead of two the modules: σox and σred .

3.4 Simulated Biosensor Action Figures 2 and 3 show the profiles of concentrations of the substrate, the mediator and the product in the enzyme membrane (x ∈ (0, d), xˆ ∈ (0, 1)) as well as ˆ at d = 100 µm, in the external diffusion layer (x ∈ (d, d + δ), xˆ ∈ (1, 1 + δ)) δ = 300 µm [11]. The dynamics of the biosensor current is presented in Fig. 3. The biosensor action was simulated for two concentrations (0.01 and 1 M) of the substrate (S0 ) as well as for two concentrations (10−5 and 10−3 M) of the mediator (M0 ). The corresponding dimensionless concentrations of the substrate (Sˆ0 ) as well as of the mediator (Mˆ 0 ) are 0.1 and 10. The values of all the other parameters were constant in the numerical simulation, DSe = DMe = DPe = 300 µm2/s, DSb = 2DSe ,

DMb = 2DMe ,

kcat = 103 s−1 , Et = 3 µM,

DPb = 2DPe ,

kred = 104 M−1 /s,

kox = 107 M−1 /s,

(34)

ne = 1.

In Figs. 2 and 3, the concentration profiles were normalized as follows: ˆ Sˆ0 = S/S0 , SN = S/ ˆ Mˆ 0 = M/M0 , MN = M/

(35)

PN = Pˆ /Mˆ 0 = P /M0 . In Figs. 2 and 3, the concentration profiles were plotted at the time when the process reaches steady state and the time Tˆ0.5 when 50% of the steady state current has been reached [11]. As one can see in Fig. 2, there is rather a long shoulder in the profile of the mediator concentration at 1.3 < xˆ < 2.3. The shoulder appears in the case of a relatively high concentration M0 of the mediator and a low concentration S0 of the substrate. At these conditions (M0  S0 ,   1), the rate of the enzymatic reaction depends practically only on the substrate concentration as defined in (19). In the beginning of the biosensor action, there is no substrate in the enzyme membrane, and the mediator diffuses fast from the electrode surface along the enzyme membrane and even to the bulk solution. The enzymatic reaction starts only when some substrate touches the enzyme. Due to a relatively high concentration of the mediator, the reaction progresses rapidly and the concentration of the mediator inside the enzyme near the border also reduces rapidly. Consequently, for a short time the

3 Modeling Biosensors Utilizing Michaelis–Menten Kinetics

221

1.0 0.9

SN , M N , PN

0.8 0.7

SN

0.6 0.5 0.4 0.3

PN

0.2 0.1

MN

0.0 0

1

2

3

4

x Fig. 2 The profiles of the normalized concentrations of the substrate (SN ), the mediator (MN ) and the product (PN ) in the enzyme layer xˆ ∈ (0, 1) and in the diffusion layer xˆ ∈ (1, 4) at an approximate steady state dimensionless time tˆ = 5.73 (solid lines) and at the dimensionless halftime Tˆ0.5 = 1.86 (dashed lines), Sˆ0 = 0.1, Mˆ 0 = 10. The values of the other parameters are defined in (34)

1.0 0.9

SN , M N , PN

0.8 0.7 0.6

SN

0.5 0.4

MN

0.3 0.2

PN

0.1 0.0 0

1

2

3

4

x Fig. 3 The profiles of the normalized concentrations of the substrate (SN ), the mediator (MN ) and the product (PN ) in the enzyme and the diffusion layers at an approximate steady state dimensionless time tˆ = 1.2 (solid lines) and the dimensionless half-time Tˆ0.5 = 0.435 (dashed lines), Sˆ0 = 10, Mˆ 0 = 0.1. The other parameters are the same as in Fig. 2

222

Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

mediator concentration inside the enzyme becomes slightly lower than outside the membrane. No similar effect can be noticed in Fig. 3 which shows the concentration profiles of the opposite case of (S0  M0 ,   1). Additional numerical experiments approved that a shoulder in the profile of the mediator concentration appears only in cases when   1 [11]. Figure 4 shows the dynamics of the current calculated at different concentrations of the substrate and mediator [11]. Particularly, the dynamics of the response at Sˆ0 = 0.1 and Mˆ 0 = 10, at which Fig. 2 shows the profiles of the concentrations of the species. The profiles of the concentrations depicted in Fig. 3 correspond to curve 3 in Fig. 4. One can see in Fig. 4 that the biosensor current is affected by both concentrations: S0 and M0 . The current grows notably faster at higher concentration S0 (curves 3 and 4) of the substrate rather than at lower one (curves 1 and 2). The effect of concentration M0 of the mediator on the biosensor response becomes notable with some delay. The mediator diffuses very quickly from the CME into the enzyme layer in a sufficient for the reaction amount, while the substrate has to diffuse across the Nernst diffusion and enzyme layers. Therefore, at the very beginning of the response, the biosensor acts under the limitation of the substrate diffusion [11].

102 10

4

2

1

3

100

1

î

10-1 10-2 10-3 10-4 10-5 0

1

2

3

4

^

t

ˆ tˆ) at two concentrations of the Fig. 4 The dynamics of the dimensionless biosensor current i( substrate Sˆ0 : 0.1 (1, 2), 10 (3, 4) and two concentrations of the mediator Mˆ 0 : 0.1 (1, 3), 10 (2, 4). The other parameters are the same as in Fig. 2

3 Modeling Biosensors Utilizing Michaelis–Menten Kinetics

223

3.5 Impact of the Diffusion Module 2 and σ 2 . The dimensionless model (27)–(32) contains two diffusion modules: σred ox 2 2 The reduced diffusion module q is a common part of σred and σox (see (20) and (24)). At constant rates kred and kox of the reactions (6), it is reasonable to use the 2 and σ 2 . To investigate reduced diffusion module q instead of the two modules: σred ox the effect of the diffusion module q on the biosensor response, the biosensor action was simulated at different concentrations of the substrate and the mediator changing the enzyme layer thickness [11]. Figure 5 shows the dependence of the steady state dimensionless current IS on the module q, while Fig. 6 shows the corresponding dependence of the sensitivity BS . The calculations were made at three concentrations of the substrate (S0 ) and three concentrations of the mediator (M0 ) changing exponentially the thickness of the enzyme layer from 0.3 µm up to 1.5 mm. Values of all other parameters were assumed constant as defined in (34) [11]. Let us notice that accepting these values 2 becomes equal to unity when q = 10−7 Ms, and σ 2 = 1 at q of the parameters, σox red = 10−4 Ms. 2 < 1, the As one can see in Fig. 5 at small values of the diffusion module, σox ˆ dimensionless current I is approximately a linearly increasing function of q as well 2  1, Iˆ becomes a non-monotonous function as of d 2 . At large values of q, σred of q (curves 1 and 5). To see the behaviour of the biosensor response versus the diffusion module the results of calculations were re-plotted in Fig. 7 in terms of the dimensional density I of the steady state current.

106

Î

105

1 2 3 4 5

104 103 102 ox=

10

red=

1

1

1

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

q, Ms Fig. 5 The steady state dimensionless current Iˆ versus the reduced diffusion module q at different concentrations of the substrate and the mediator, Sˆ0 : 0.1 (1), 1 (2, 4, 5), 10 (3), Mˆ 0 : 0.1 (4), 1 (1–3), 10 (5). The other parameters are the same as in Fig. 2

224

Chemically Modified Enzyme and Biomimetic Catalysts Electrodes 1.0

1 2 3 4 5

0.9 0.8 0.7

BS

0.6 0.5 0.4 0.3 0.2 0.1 0.0 10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

q, Ms Fig. 6 The biosensor sensitivity BS versus the reduced diffusion module q. The parameters and notation are the same as in Fig. 5

I , nA/mm2

104

103

1 2 3 4 5

102

ox=

101 10-9

10-8

red=

1

10-7

10-6

10-5

1

10-4

10-3

10-2

q, Ms Fig. 7 The density I of the steady state biosensor current versus the reduced diffusion module q. All the parameters and the notation are the same as in Fig. 5

3 Modeling Biosensors Utilizing Michaelis–Menten Kinetics

225

Figure 7 shows clearly the non-monotony of the density I of the steady state 2 current versus the module q. As one can see in Fig. 7, increasing q from 1/kox (σox 2 = 1) up to 1/kred (σred = 1), the density I of the steady state current changes only 2 > 1, the density I of the steady state current slightly. At greater values of q, σred monotonously decreases [11]. The complex effect of the diffusion module on the biosensor response can also be seen in Fig. 6. In cases when CMEE acts under the limitation of the 2 < 1, q < 10−7 Ms), the biosensor sensitivity enzyme-mediator interaction (σox BS practically does not depend on the diffusion module. It means that at these conditions the biosensor sensitivity is very resistant to changes in the thickness d of the enzyme layer as well as in the total concentration Et of the enzyme. This resistance notably decreases at higher values of the diffusion module. The sensitivity BS changes even non-monotonously when q increases from 10−7 to 10−4 Ms, i.e. 2 > 1 and σ 2 < 1. In cases when the CMEE acts under the control of the when σox red 2 > 1, q > 10−4 Ms), the biosensor sensitivity slightly increases mass transport (σred with the increase in q. The diffusion module especially affects the sensitivity in cases of low substrate concentration (curve 1) and of high concentration of the mediator (curve 5). This can also be observed in Fig. 7. Figures 5, 6 and 7 show the linearity of the biosensor response in cases when the CMEE acts under the limitation of the enzyme-mediator interaction (σox < 1, q < 10−7 Ms).

3.6 Impact of the Substrate Concentration The dependence of the biosensor response on the dimensionless ratio  of the substrate and mediator concentrations is depicted in Figs. 8 and 9. The biosensor responses were simulated at different values of the diffusion module by changing the substrate concentration S0 in the bulk solution and keeping the mediator concentration M0 constant [11]. One can see in Fig. 8 a linear range of the calibration curve up to  ≈ 0.1 (Sˆ0 ≈ 0.1, S0 ≈ 10 mM). The dependence of the steady state current on the ratio  is noticeably affected by the diffusion module. The current is directly proportional 2 as well as to σ 2 . At low values of the diffusion module, the tenfold increase to σred ox 2 also increases the steady state dimensionless current Iˆ approximately tenfold in σred 2 ≥ 1 (curves 4–6), the effect of the diffusion modules (curves 1–3). However, at σred 2 increases on the density I of the steady state current notably decreases. When σred from 10 (curve 5) to 100 (curve 6), the current Iˆ increases only about 2–3 times. Figure 9 shows that the biosensor sensitivity notably decreases with an increase in the ratio  of the substrate and mediator concentrations at all values of the diffusion module. In general, the fact that the effect of the sensitivity decreases increasing the substrate concentration is rather well known [24, 54, 60]. As usual, the biosensors are highly sensitive at very low concentrations of the substrate

226

Chemically Modified Enzyme and Biomimetic Catalysts Electrodes 102 101 100

Î

10-1 10-2 10

4 5 6

1 2 3

-3

10-4 10-3

10-2

10-1

100

101

Σ Fig. 8 The steady state dimensionless current Iˆ versus the ratio  of the substrate and the mediator 2 : 10−4 (1), 10−3 (2), 10−2 (3), 0.1 concentrations at different values of the diffusion module σred 2 = 103 σ 2 , the other ˆ (4), 1 (5), 10 (6), keeping constant concentration M0 = 1 of the mediator. σox red parameters are the same as in Fig. 2

1.0 0.9 0.8 0.7

BS

0.6 0.5 0.4 0.3 0.2

1 2 3 4 5 6

0.1 0.0 10-3

10-2

10-1

100

101

Σ Fig. 9 The biosensor sensitivity BS versus the ratio  of the substrate and the mediator concentrations at different values of the diffusion module. The parameters and notation are the same as in Fig. 8

4 Modeling Multi-Layer CME-Based Biosensor

227

( ≤ 10−2 ) and they are of very low sensitivity at high concentrations of the substrate ( > 1). This effect can also be noticed in Fig. 8. One can see no notable difference between the shapes of curves 1 and 2 in 2 ≤ 1 (σ 2 −3 Figs. 8 and 9. So, in case when σox red ≤ 10 , curves 1 and 2) the diffusion module practically has no influence on the biosensor sensitivity. When 2 > 1 and σ 2 ≤ 1 (curves 3–5) the sensitivity B decreases with increase in the σox S red diffusion module. The diffusion module especially affects the biosensor sensitivity at moderate concentrations of the substrate (0.01 ≤  ≤ 1). When the response is 2 > 1, curve 6) the sensitivity slightly increases [11]. under the diffusion control (σred This was clearly shown in Fig. 6.

4 Modeling Multi-Layer CME-Based Biosensor Practical biosensors often contain an additional outer membrane—a thin layer of polyvinyl alcohol, polyurethane, cellulose, latex or other material [9, 23]. Dialysis membranes are often used to cover the enzyme layer in order to prevent it from dissolution. Outer diffusion limiting membranes are used to prevent the enzyme from dissolution, to improve the biosensors’ stability and to prolong the linear part of its calibration curve [4, 30, 58]. This section discusses the modeling of CMEbased biosensors containing outer diffusion limiting membranes [7, 8]. Taking into consideration additional layers leads to introducing new parameters and increasing the model complexity [12, 35, 56] (see Chapter “Biosensors with Porous and Perforated Membranes”). An approach where two diffusion layers are modeled by one layer by introducing an effective diffusion coefficient is discussed. The approach is then applied to a mathematical model comprising three layers: an enzyme layer, a dialysis membrane and an outer diffusion layer [8].

4.1 Three-Layer Model The model is constructed of three layers with different properties [7, 8]. The enzyme layer (a0 < x < a1 ) is placed on the CME and touches it at x = a0 = 0. The dialysis (diffusion limiting) membrane (a1 < x < a2 ) covers the enzyme and protects it from dissolution to the analysed solution (x > a2 ). Applying the Nernst approach leads to a layer of limited diffusion (a2 < x < a3 ) [19, 31, 48]. The mathematical model of the CME-based biosensor is formulated as an extension of the model (7)– (16) [8].

228

Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

The mass transport and the reaction kinetics in the enzyme layer is expressed by the system of reaction–diffusion equations (t > 0) [8, 11]: ∂ 2 S1 ∂S1 = DS1 − V (M1 , S1 ), ∂t ∂x 2

(36a)

∂M1 ∂ 2 M1 = DM1 − V (M1 , S1 ), ∂t ∂x 2

(36b)

∂P1 ∂ 2 P1 = DP1 + V (M1 , S1 ), ∂t ∂x 2

a0 < x < a1 ,

(36c)

where S1 (x, t), M1 (x, t), P1 (x, t) are the concentrations of the substrate, the mediator and the reaction product, respectively, and the reaction rate V is as defined in (9). Only the mass transport by diffusion is modeled in the dialysis membrane and the outer diffusion layer. Governing equations for the concentrations of the species in the dialysis (l = 2) and the diffusion (l = 3) layers are described by the following equations: ∂ 2 Ul ∂Ul = DUl , ∂t ∂x 2

U = S, M, P ,

l = 2, 3,

(37)

where Ul (x, t) is the molar concentration of species U in layer l, and DUl is the corresponding diffusion coefficient [8]. For simplicity, the diffusion coefficients are assumed to be equal for all species in each layer, DU1 = D1 ,

DU2 = D2 ,

DU3 = D3 ,

U = S, M, P .

(38)

At the beginning of the biosensor operation (t = 0) the concentrations of all the species are equal to zero, except the substrate concentration at the external boundary of the diffusion layer (x = a3 ) and the mediator concentration at the electrode surface (x = a0 ), Ul (x, 0) = 0,

al−1 ≤ x ≤ al ,

M1 (a0 , 0) = M0 ,

U = S, M, P ,

l = 1, 2, 3,

S3 (a3 , 0) = S0 .

(39a) (39b)

During the biosensor operation (t > 0), at the surface of the electrode, the reaction product is consumed due to the CME polarization, the mediator concentration is maintained constant at its initial value, and the substrate concentration is modeled by the non-leakage condition,  ∂ 2 S1  D1 = 0, ∂x 2 x=a0

M1 (a0 , t) = M0 ,

P1 (a0 , t) = 0.

(40)

4 Modeling Multi-Layer CME-Based Biosensor

229

The concentrations in the buffer solution are assumed to be constant, S3 (a3 , t) = S0 ,

M3 (a3 , t) = 0,

P3 (a3 , t) = 0.

(41)

The matching conditions define the materials diffuse through the layers, Dl

  ∂Ul  ∂Ul+1  = D , l+1 ∂x x=al ∂x x=al Ul (al , t) = Ul+1 (al , t),

U = S, M, P , U = S, M, P ,

l = 1, 2, l = 1, 2.

(42) (43)

The output current of the biosensor is proportional to the flux of reaction product at the electrode surface and the system approaches steady state, i(t) = ne F D1

 ∂P1  , ∂x x=a0

I = lim i(t). t →∞

(44)

4.2 Effective Diffusion Coefficient In order to minimize number of the model parameters, we investigate a possibility of merging the dialysis (a1 < x < a2 ) and the diffusion (a2 < x < a3 ) layers of the biosensor. Two mathematical models are further analysed: three-layer (3L) mathematical model (36)–(44) and the corresponding two-layer (2L) mathematical model, where the outer layers are merged by using the effective diffusion coefficient. In order for this merger to be accurate, the requirement is set that at the steady state the fluxes of 2L and 3L models of any concentration at boundary x = a1 must be equal, D1

  ∂U1∗  ∂U1  = D , 23 ∂x x=a1 ∂x x=a1

U = S, M, P ,

(45)

where D23 is the effective diffusion coefficient in the merged layer, U1 —the concentration of the substance U (U = M, P, S) at the boundary x = a1 in 3L model, U1∗ is the concentration of the substance U in the merged layer of the 2L mathematical model at the same boundary [8]. By rewriting the matching conditions (43) at layer x = a2 for the steady state, the concentration U2 of the substance U at this boundary can be obtained for the 3L mathematical model as follows: U2 =

D3 d2 U3 + D2 d3 U2 , D2 d3 + d2 D3

U = S, M, P ,

(46)

230

Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

where U3 is the concentration of the corresponding substance at the boundary x = a3 (set by the (41) boundary condition), d2 = a2 − a1 and d3 = a3 − a2 are the thicknesses of the dialysis and the diffusion layers, respectively. By inserting this expression into the (45) equation rewritten for the steady state, the effective diffusion coefficient D23 for the merged dialysis and diffusion layer is derived to be D23 =

D2 D3 , νD3 + (1 − ν)D2

(47)

where ν is the relative thickness of the dialysis membrane in the dialysis-diffusion layer system, ν = d2 /(d2 + d3 ). Using the effective diffusion coefficient D23 , the 3L model (36)–(44) reduces to 2L model: the enzyme layer (thickness and diffusion coefficient of this layer remain the same as in 3L model) and the merged layer of the thickness d23 = d2 + d3 . Similarly, the effective diffusion coefficient was also applied to model the biosensors with a perforated membrane, when applying it to reduce a two-dimensional model to a one-dimensional one [49]. Extensive simulations showed that the relative error of the 2L model steady state response was less than 1% for all the analysed parameter values, therefore the transient concentrations of product and reactants were compared [8].

4.3 Concentration Profiles To investigate the precision of the two layers merging, the concentration profiles were analysed [8]. Due to large differences in absolute values of the substance concentrations, the concentrations were normalized as defined in (35). The simulations were carried out for both mathematical models, 3L and 2L. The normalized concentration profiles are depicted in Fig. 10. For the 3L model, the thickness of the enzyme (d1 = a1 ) and diffusion (d3 ) layers were d1 = d3 = 100 µm and the thickness of the dialysis membrane was d2 = 50 µm. For the 2L model, the thickness of the merged diffusion layer was d23 = d2 + d3 = 150 µm, while the effective diffusion coefficient was calculated by using (47), D23 = 150 µm2/s. The approximate steady state was achieved at t = 550 s when using 3L model and at t = 686 s when using 2L model. The corresponding half-times of the steady state are 33 s and 42 s. So, the processes simulated by 2L model proceed slightly slower than the processes simulated by the corresponding more precise 3L model. At the half-time of the steady state response (Fig. 10a), i.e. at the time moment of occurrence of a half of steady state biosensor current, differences between concentration profiles calculated by different models are noticeable in all three layers. However, in the enzyme membrane (a0 < x < a1 ), the differences in the concentration profiles are only slight.

a)

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

SN , MN , PN

SN , MN , PN

5 Analysis of the Electrocatalysis Using Michaelis–Menten and Second-Order. . .

SN 3L 2L

MN a2

a1 0

50

PN

100

150

x, μm

200

250

b)

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

231

SN

PN

MN a1 0

3L 2L

50

a2

100

150

x, μm

200

250

Fig. 10 The profiles of the normalized concentrations of the substrate SN , mediator MN and product PN simulated by the three-layer (3L) and the two-layer (2L) models at the half-time of the steady state (a) and at the steady state (b), S0 = 0.1 M, M0 = 10−4 M, D1 = 300 µm2 /s, D2 = 0.2D1 , D3 = 2D1 , and the other parameters are defined in (34)

At the steady state conditions of the biosensor action, no differences between the concentration profiles obtained by using 3L model and those obtained by using 2L model are noticeable in the enzyme membrane (a0 < x < a1 , Fig. 10b). The perfect matching of the curves in the enzyme layer confirms that the relative difference between the steady state responses is less than 1%, despite the variance of the product and substrate concentration profiles in the dialysis (a1 < x < a2 ) and diffusion (a2 < x < a3 ) layers (Fig. 10b).

5 Analysis of the Electrocatalysis Using Michaelis–Menten and Second-Order Reaction Schemes Chemically modified electrodes prepared by using biomimetic catalysts (BC) and other biomimic materials (BM) can demonstrate high electrocatalytical activity and serve as selective recognition elements for biosensors building [26]. If these BM’s show high conversion rate of substrate and specificity, the selectivity of biosensors approaches the selectivity of enzyme based biosensors. A basic question concerning electrocatalysis at chemically modified electrodes relates to the location of reaction sites. During electrocatalytical redox transformation of solution species, a balance of two partial processes establishes: the flow of species from the bulk of solution to the modifier-solution boundary along with the possible diffusion of species within the modifier layer appears to be compensated by charge carrier (electrons or holes) diffusion in the opposite direction, i.e. from the background electrode through the modifier layer to the reaction zone [50, 51]. At a relatively fast diffusion of charge carriers within the modifier layer, the reaction site appears to be located at a modifier-solution boundary. Adversely, at a relatively slow diffusion of charge carriers, the reaction zone is located within this layer, and shifts within it towards the background electrode with decreasing mobility of

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Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

charge carriers [50, 51]. Experimentally, both cases were observed with the use of in situ Raman spectroelectrochemistry. Electrooxidation of hydroquinone and ascorbic acid at electrode, modified with poly(toluidine blue), was found to occur at a modifier-solution interface (“metal-like electrocatalysis”) [39], whereas same or related processes proceeded within the modifier layer, as it was shown for electrodes, modified with polyaniline [37, 38], or poly(neutral red) [40], or Prussian blue [41] (“redox electrocatalysis”). The dependence of the location of reaction zone on various parameters has been studied [47]. As a result, the deviation from the linearity for the dependence of electric current on the concentration of solution species was shown. The dependence of the nonlinearity on the mobility of charge carriers was also shown [50]. At a fast diffusion of charge carriers, the reaction proceeds at an outer interface, and the current-concentration dependence appears to be linear, whereas lowering of charge carrier mobility results both in the shift of reaction zone into the bulk of a modifier layer, and the deviation of the said dependence from linearity. On the other hand, a nonlinear, hyperbolic dependence of current on concentration is well known for biosensors—bioelectrocatalytic devices, based on enzyme-containing layer placed at an electrode surface. For these devices, however, the nonlinearity is determined by the hyperbolic nature of Michaelis–Menten type enzyme kinetics, as distinct from simple electrocatalytic systems, where a simple bimolecular reaction of solution species with “active centres” of a modifier occurs [47, 50]. This section presents a model of the CME utilizing electrocatalytical substrate conversion. The model involves a combination of two kinds of redox interactions in the modified layer, i.e. a simple chemical second-order reaction and the Michaelis– Menten-type redox process [51]. Depending on relative increments from these two kinetic models, either linear or hyperbolic dependencies of electric current on substrate (reactant) concentration are obtained.

5.1 Mathematical Model A flat surface of electrode is assumed to be covered with a uniform layer of a conducting polymer of a thickness d [47, 50]. The electrode is immersed into a solution containing reactant. No concentration gradient is assumed to be either for reactant or for reaction product outside of a polymer layer in the course of electrocatalytic process [51]. By applying a suitable electrode potential, an electrochemical conversion of a reactant into a product proceeds. This conversion means either anodic oxidation (i.e. withdrawing of electrons from reactant) or cathodic reduction (i.e. addition of electrons to reactant) [51]. It was supposed that the charge transfer process follows two mechanisms. One of them relates to a simple redox reaction with active centres (charge carriers) in

5 Analysis of the Electrocatalysis Using Michaelis–Menten and Second-Order. . .

233

polymer film according to relation k

R + n −→ P,

(48)

where R and P are the reactant and the reaction product, respectively, n is a charge carrier, i.e. an electron for cathodic reduction, or a hole for anodic oxidation processes, and k is a second-order rate constant of the chemical reaction [47, 50, 51]. The second mechanism for the conversion of R to P proceeds following the Michaelis–Menten type mechanism, which involves the formation of a virtual complex of R (R ∗ n) with active centres carrying electric charges, and the next following split of this complex leading to reaction product P [51], k1 R + n FGGGGGB GGGGG R ∗ n, k−1

(49a)

kcat R ∗ n GGGGGGGA P.

(49b)

A combination of both possible mechanisms was modeled by the dimensionless coefficient α corresponding to the partial amount of the second-order reaction and Michaelis–Menten process. It varied between 0 and 1 [51]. For α = 1, the reaction follows a simple chemical reaction (48) without any increment from Michaelis– Menten-like mechanism. In contrary, the reaction follows the Michaelis–Menten type scheme(49) without any increment from a simple chemical reaction for α = 0. Combining the one-dimensional-in-space diffusion with BC-catalysed reaction (49) and the redox reaction (48) leads to the following rate equations for R, P and n (0 < x < d, t > 0): ∂R ∂ 2R kcat Rn = DR 2 − αkRn − (1 − α) , ∂t ∂x KM + R

(50a)

∂P kcat Rn ∂ 2P + αkRn + (1 − α) = DP , ∂t KM + R ∂x 2

(50b)

∂ 2n ∂n kcat Rn = Dn 2 − αkRn − (1 − α) , ∂t ∂x KM + R

(50c)

where x and t stand for space and time, respectively, R(x, t), P (x, t) and n(x, t) are the concentrations of the reactant R, the reaction product P, and the charge carrier n, respectively, DR , DP and Dn are the diffusion coefficients, and KM is the Michaelis constant, KM = (k−1 + kcat )/k1 [51]. Let x = 0 represents the electrode/polymer film boundary while x = d is the thickness of a polymer layer. The electrocatalytic processes start when the reactant appears over the surface of the polymer layer. This

234

Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

is used in the initial conditions (t = 0) [51], R(x, 0) =

 R0 ,

x = d,

0,

0 ≤ x < d,

P (x, 0) = 0,

0 ≤ x ≤ d,

n(x, 0) = n0 ,

0 ≤ x ≤ d,

(51a) (51b) (51c)

where R0 is the reactant concentration in buffer solution, and n0 is the initial concentration of charge carriers. The boundary conditions are as follows (t > 0) [51]: ∂R  = 0,  ∂x x=0 ∂R  DP = 0,  ∂x x=0 DR

n(0, t) = n0 ,

Dn

R(d, t) = R0 ,

(52a)

P (d, t) = 0,

(52b)

∂n  = 0.  ∂x x=d

(52c)

5.2 Characteristics of the Response The density i(t) of the output current at time t can be obtained explicitly from the Faraday and the Fick laws, i(t) = ne F Dn

∂n   , ∂x x=0

(53)

where ne is the number of electrons involved in a charge transfer, and F is the Faraday constant. It was assumed that the system (50)–(52) approaches a steady state as t → ∞, I = lim i(t), t →∞

(54)

where I is assumed as the density of the steady state biosensor current. app The apparent Michaelis constant KM is often accepted as a characteristic of the sensitivity as well as of the calibration curve for the biosensors [24, 54]. The greater app value of KM corresponds to a wider range of the linear part of the calibration curve. app KM is assumed to be the concentration of the reactant (substrate) at which the response reaches a half of the maximal current Imax when the reactant concentration

5 Analysis of the Electrocatalysis Using Michaelis–Menten and Second-Order. . .

235

is extrapolated to infinity keeping the other model parameters constant,   app KM = R0∗ : I (R0∗ ) = 0.5Imax ,

Imax = lim I (R0 ), R0 →∞

(55)

where I (R0 ) is the density of the steady state current calculated at the concentration R0 of the reactant. The apparent Michaelis constant is also known as the half maximal effective concentration of the substrate to be determined [14].

5.3 Numerical Simulation The problem (50)–(52) was solved numerically using the finite difference technique [15, 50, 51]. For simplicity, the diffusion coefficients for reactant R, product P and charge carriers n were chosen to be equal, D = DR = DP = Dn = 10−9 m2 /s [47, 50]. In a physical sense, the electrode matter (conducting polymer) was considered as a semiconductor [51]. The only value for n0 = 4 M was used in numerical experiments [50]. The redox reaction rate constant k was varied within the limits of two orders of magnitude between 10 and 103 M−1 s−1 , whereas the reactant concentration R0 was varied within the limits of 1–10 mM at intervals of 1 mM. Additionally, three values for catalytic constant kcat of 1, 10 and 100 s−1 , and, for simplicity, the only value for KM of 5 mM were taken for calculations [51]. The thickness d of the polymer layer was kept constant at 1 µm throughout all the numerical experiments [47, 50, 51].

5.4 Effect of the Combination of Two Types of Kinetics Figure 11 presents the dependence of the steady state current I on the reactant concentration R0 at a moderate value of the second-order reaction rate constant k = 103 M−1 s−1 and different values of catalytic rate constant kcat [51]. As one can see in Fig. 11a, at the lowest value of kcat (0.1 s−1 ), no noticeable response is observed for “pure” Michaelis–Menten type reaction (α = 0). An increase in α results in a progressive increase of the steady state current retaining a linear shape [51]. An increase of kcat by an order of magnitude (up to 1 s−1 , Fig. 11b) causes an increase of the slope for the current-concentration profile at α = 0, whereas the increment from Michaelis–Menten scheme diminishes with increasing α and appears to be negligible for α approaching unity. It is seen from Fig. 11b that the current-concentration dependence bears a hyperbolic character for α = 0, and turns gradually into a linear dependence at increasing α up to α = 1 [51].

236

Chemically Modified Enzyme and Biomimetic Catalysts Electrodes 2.0

2.0

1 2 3 4 5 6

1.0

0.5

0.0 0

1

2

3

4

5

6

R0, mM

7

8

9

10

0

b)

2.0

1

2

3

4

5

6

7

8

9

10

7

8

9

10

R0, mM 6.0

1 2 3 4 5 6

1.0

1 2 3 4 5 6

5.0

I, mA/mm2

1.5

I, mA/mm2

1.0

0.5

0.0

a)

1 2 3 4 5 6

1.5

I, mA/mm2

I, mA/mm2

1.5

4.0 3.0 2.0

0.5 1.0 0.0

c)

0.0 0

1

2

3

4

5

6

R0, mM

7

8

9

10

d)

0

1

2

3

4

5

6

R0, mM

Fig. 11 The steady state current I versus the concentration R0 of reactant at different values of α (as indicated) and four values of catalytic rate constant kcat : 0.1 (a), 1 (b), 10 (c) and 100 (d) s−1 ; k = 103 M−1 s−1 , Dn = 10−9 m2 s−1 [51]

A progressive increase of kcat leads to further increase of the slope for profiles, and a phenomenon of “compensation” could be observed at a definite ratio of kcat and k, as shown in Fig. 11c. Here, for kcat = 10 s−1 , close related slopes are observed for α varying between 0 and 1, i.e. independent on a relative increment of either chemical, or Michaelis–Menten reaction. The only difference in this case is that the dependence obtained possesses either a linear or a hyperbolic shape for chemical (α = 1) or Michaelis–Menten (α = 0) reaction scheme, respectively [51]. As it could be expected, a further increase of kcat causes an increase of a relative increment from the Michaelis–Menten reaction, exceeding that from a simple chemical interaction. For kcat exceeding 10 s−1 , a decrease of current response is observed for α increasing from 0 to 1, as opposed to kcat < 10 s−1 (Fig. 11d). Again, a hyperbolic dependence of the output current on the concentration R0 is obtained for α = 0, i.e. without any increment from the simple second-order chemical reaction [51]. Additional calculations showed that the slope of curves obtained and thus the sensitivity of response to concentration increases at increasing α for the ratio of k/kcat > 100 M, or decreases for k/kcat < 100 M [51]. This decrease appears well expressed for the lowest values of k/kcat = 1 or even 0.1 M. Also, the evolution of hyperbolic to linear shape for these dependencies with increasing α is well seen for all combinations of k and kcat [51].

5 Analysis of the Electrocatalysis Using Michaelis–Menten and Second-Order. . .

237

5.5 Impact on the Apparent Michaelis Constant app

It is well known that the apparent Michaelis constant KM is subjected to changes depending on various factors [14, 24], mainly on the diffusion conditions within a modifier layer on the electrode surface [8, 10, 11]. For α approaching unity, the only way for conversion of reactant to product app is the direct chemical interaction according to (48). In this case, KM should be indefinitely high so that diffusion of reactant within the modifier layer proceeds fast as compared to the chemical reaction. However, even for this “pure” chemical interaction, a hyperbolic Michaelis–Menten-like dependencies can be observed, caused by the relatively slow diffusion of reactant [50, 51]. app Figure 12 discloses the dependence of the apparent KM on kinetic characterapp istics. For α = 0 (i.e. Michaelis–Menten-like scheme), KM depends greatly on app −1 kcat . For the lowest value of kcat of 1 s , KM equals 6.5 mM and does not differ greatly from that taken into calculations, KM = 5 mM. For greater values of kcat app (10 and 100 s−1 ), KM increases up to 20.2 and 24.6 mM, respectively (Fig. 12a). It could be concluded that an increase of the rate of catalytic transformation of reactant app according to reaction (49a) leads to increase of KM . Indeed, an increase of kcat under a constant diffusion coefficient D for reactant means a progressive decrease of a relative diffusion rate as compared to the rate of chemical transformation of reactant [50, 51]. app It is seen from Fig. 12, that an increase of α causes different changes of KM , −1 −1 greatly influenced by kcat . For k = 10 M s (Fig. 12a) and the lowest kcat of 1 s−1 , app a slight increase of KM from 6.5 to 10.4 mM is observed for α = 0.8, whereas a app sharp decrease of KM from 20.2 to 8.9 mM proceeds by changing of α from 0 to app 0.8 for kcat of 10 s−1 , and negligible changes of KM from 24.6 to 23.7 mM occur when changing α from 0 to 0.8 for the largest kcat = 100 s−1 [51].

25

25

20

KMapp, mM

30

KMapp, mM

30

20

1 2 3

15

15

10

a)

5 0.0

1 2 3

10

0.2

0.4

0.6

0.8

b)

5 0.0

0.2

app

0.4

0.6

0.8

Fig. 12 Dependence of the apparent Michaelis constant KM on parameter α at two values of the second-order reaction rate constant k: 10 (a) and 100 (b) M−1 s−1 and three values of the catalytic rate constant kcat : 1 (a), 10 (b) and 100 s−1 [51], Dn = 10−9 m2 s−1

238

Chemically Modified Enzyme and Biomimetic Catalysts Electrodes

These tendencies possess complicated character and could not be predicted app without calculations [51]. Noteworthy changes in the dependence of KM on α occur by increasing k, i.e. by increasing the relative rate of the direct chemical interaction relative to Michaelis–Menten-type one. As an example, Fig. 12b shows the corresponding dependencies for k = 100 M−1 s−1 . Here, the tendency for an app increase of KM for lowest kcat with increasing α appears much stronger as for app a lower value of k. Also, the decrease of KM for the intermediate kcat of 10 s−1 appears not as steep as for lower k. For the greatest kcat of 100 s−1 , no noticeable app change of KM occurs with increasing α. In all cases (except for a combination app of lowest kcat and highest k), however, KM values do not approach indefinite high values at α approaching 1, as it could be expected for “pure” chemical interaction according to (48), and for very fast diffusion of reactant. This means that, even at a relatively high increment from “pure” chemical interaction (and low increment from Michaelis–Menten kinetics, accordingly), the dependence of current on concentration bears hyperbolic (nonlinear) character [51].

6 Concluding Remarks The mathematical model (7), (10)–(14) of an amperometric biosensor based on a chemically modified electrode can be successfully used to investigate the kinetic peculiarities of the biosensor response. The corresponding dimensionless mathematical model (27)–(32) can be used as a framework for numerical investigation of the impact of model parameters on the biosensor action and to optimize the biosensor configuration. The biosensor current grows notably faster at higher substrate concentrations in the bulk solution than at lower ones (Fig. 4). At the very beginning of the response, the biosensor acts under the limitation of the substrate diffusion from the bulk solution to the electrode. The value of the diffusion module substantially determines the behaviour of the response and sensitivity of the biosensor. The steady state biosensor current is a non2 ≤ 1, monotonous function of the diffusion module (Fig. 7). In all cases when σox the diffusion module practically has no influence on the biosensor sensitivity. When 2 > 1 and σ 2 ≤ 1, the sensitivity changes non-monotonously with the diffusion σox red 2 module. When the response is fully under the diffusion control (σred > 1), the sensitivity slightly increases with increase in the diffusion module (Figs. 6 and 9). The two-layered mathematical model can be used to simulate the steady state response of the biosensor comprising three layers, an enzyme layer, a dialysis membrane and an outer diffusion layer, by using the effective diffusion coefficient to merge two layers, the dialysis membrane and the outer diffusion layer. However, the modeling error must be taken into consideration when analysing the transient biosensor response (Fig. 10). The modeling of electrocatalytical conversion of substrates on CME showed that two increments from chemical kinetics, one related to a simple second-order

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239

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Biosensors with Porous and Perforated Membranes

Contents 1 2

3

4

5

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biosensors with Outer Porous Membrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Numerical Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Effect of the Porous Membrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biosensors with Selective and Outer Perforated Membranes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Numerical Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Effect of the Selective Membrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Dimensional Modeling of Biosensors with Selective and Perforated Membranes. . . 4.1 Principal Structure of Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Effect of the Perforation Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

244 245 245 248 250 254 254 258 259 261 262 262 266 269 271

Abstract The additional application of catalytically inactive membranes can solve some drawbacks of biosensors, such as a relatively short linear range of the calibration graph, an instability and a low specificity. The selective membranes are usually used to increase the biosensors specificity. In this chapter, amperometric biosensors with inert and selective membranes are mathematically modeled by nonlinear reaction–diffusion equations containing a nonlinear term related to the Michaelis– Menten kinetics of an enzymatic reaction. At first, a biosensor, containing enzymatic and outer inert membranes, is mathematically and numerically modeled by a threecompartment model in one-dimensional space at transient conditions. Then, the model is extended to cover a transducer with an additional selective membrane permeable for the product of the enzymatic reaction, and the output results are numerically analysed with a special emphasis on the influence of the selective membrane to the biosensor response. And finally, the biosensor with selective and outer perforated membranes is modeled in two-dimensions. The biosensor response is analysed with a special focus on the geometry of the membrane perforation © Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_8

243

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Keywords Amperometric biosensor · Inert/selective membranes · Nonlinear reaction–diffusion equation

1 Introduction A practical biosensor contains a multi-layer composed of enzyme and catalytically inactive membranes [2, 3, 14, 15, 23]. The application of an additional catalytically inactive membrane can completely or partially solve some drawbacks, like a relatively short linear range of the calibration curve, an instability and a low specificity caused by interfering compounds to the biosensor transducer [22, 35]. Moreover, components of solution, such as high substrate concentration, temperature, pH and inhibitors may influence the catalytic properties of the enzyme [5, 32, 40]. Polyvinyl alcohol, cellulose acetate, polyurethane, latex and a number of other membranes have been used to protect the surface of the electrodes from electrochemically active compounds, like uric acid, ascorbic acid, free amino acids, paracetamol and a number of other electrochemically active compounds of natural and artificial nature [1, 5, 24, 44, 49]. The outer porous membrane can also create a diffusion limitation to the substrate, i.e. to lower the substrate concentration in the enzymatic layer and thereby prolong the calibration curve of the biosensor [30, 31, 33, 34, 39, 48]. The outer membrane can be prepared from a dialysis membrane or, in modern biosensors, from a perforated nuclepore type membranes [1]. The electrode acting as a transducer of the biosensor is rather often covered by a selective membrane, following a layer of immobilized enzyme and an outer membrane [5, 19, 32, 40]. The selective membrane is used to increase the biosensors selectivity [17, 28, 38, 42]. The selective membrane is permeable for the selected species only, usually, for the product of the biochemical reaction. The modeling of biosensors with outer perforated membrane has been performed by Schulmeister and Pfeiffer [46]. Their model did not take into account the geometry of the holes in the membranes. It included the diffusion coefficients having a limited physical sense, and as the authors acknowledged “its quantitative value is limited” [46]. However, this assumption allowed formulating the mathematical model of the biosensor operation in a one-dimensional space. The perforation of the outer membrane has been often modeled as the permeability (porosity or diffusivity) of the porous (dialysis) membrane [10, 26, 43]. An inert membrane is usually modeled by a diffusion layer [14, 29, 31, 39, 45, 47]. Possible degradation of substrate and/or product was also considered when modeling biosensors containing an inert membrane [33, 34]. In this chapter, firstly, an amperometric biosensor, containing enzymatic and outer inert membranes, is mathematically and numerically modeled by a threecompartment model in one-dimensional space at transient conditions. Then, the model is extended to cover an additional selective membrane permeable for the product of the enzymatic reaction, and the output results are numerically analysed with a special emphasis on the influence of the selective membrane to the biosensor response. And finally, the biosensor with selective and outer perforated membranes

2 Biosensors with Outer Porous Membrane

245

is modeled in two-dimensions. The biosensor response is analysed with a special focus on the geometry of the membrane perforation [12, 13, 27].

2 Biosensors with Outer Porous Membrane Here an amperometric biosensor is considered as a flat electrode deposited with a mono-layer of enzyme and covered with an inert porous membrane [10, 12, 26, 43]. Figure 1 shows the principal structure of the biosensor, where de is the thickness of the enzyme layer immobilized onto the surface of the electrode, dp is the thickness of the porous membrane and δ is the thickness of the external diffusion layer. In the enzyme region the mass transport by diffusion and the enzyme-catalysed reaction are considered, E + S  ES → E + P.

(1)

In this scheme the substrate (S) combines reversibly with an enzyme (E) to form a complex (ES), which then dissociates into a product (P) and the enzyme is regenerated. Assuming the quasi-steady state approximation, the concentration of the intermediate complex (ES) does not change and may be neglected when simulating the biochemical behaviour of analytical systems [43, 50].

2.1 Mathematical Model

Porous membrane

dp

Diffusion layer

Enzyme

de

Fig. 1 A principal structure of a biosensor with an outer porous membrane. The figure is not to scale

δ

The model to be considered consists of two main regions: the enzyme layer where the enzyme reaction (1) as well as the mass transport by diffusion takes place and an outer porous membrane where only the mass transport by diffusion takes place. Assuming relatively thin layers of the enzyme and the porous membrane, an additional diffusion limiting region where only the mass transport by diffusion takes place, has to be considered even in modeling of a well-stirred buffer solution. If the bulk solution is slightly stirred or not stirred at all, then the layer of external diffusion especially has to be taken into account. The analyte concentration is maintained constant farther from the diffusion layer.

Electrode

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2.1.1 Governing Equations Assuming the homogeneous distribution of the immobilized enzyme in the enzyme membrane of the uniform thickness, the dynamics of the concentrations of the substrate as well as product in the enzyme layer can be described by a system of the reaction–diffusion equations [8, 10, 45], ∂Se ∂ 2 Se Vmax Se = DSe − , 2 ∂t KM + Se ∂x ∂ 2 Pe ∂Pe Vmax Se = DPe + , 2 ∂t ∂x KM + Se

(2) x ∈ (0, de ),

t > 0,

where x and t stand for space and time, respectively, Se (x, t) and Pe (x, t) are the concentrations of the substrate and reaction product, respectively, de is the thickness of the enzyme membrane, DSe and DPe are the diffusion coefficients, Vmax is the maximal enzymatic rate and KM is the Michaelis constant. Assuming the porous membrane as a periodic media, the homogenization process can be applied to the membrane domain [4, 6, 21, 51]. After this, the porous membrane is modeled as a diffusion layer with an effective (averaging) diffusion coefficient, ∂Sp ∂ 2 Sp = DSp , ∂t ∂x 2 ∂Pp ∂ 2 Pp = DPp , ∂t ∂x 2

(3) x ∈ (de , de + dp ),

t > 0,

where dp is the thickness of the porous membrane, Sp (x, t) and Pp (x, t) are the concentrations of the substrate and the reaction product, respectively, DSp , DPp are the effective diffusion coefficients of the species in the membrane. In the homogeneous external region, only the mass transport by diffusion of the substrate as well as of the product takes place, ∂ 2 Sb ∂Sb = DSb , ∂t ∂x 2 ∂Pb ∂ 2 Pb = DPb , ∂t ∂x 2

(4) x ∈ (de + dp , de + dp + δ),

t > 0,

where δ is the thickness of the external diffusion layer, Sb (x, t) and Pb (x, t) are the concentrations of the substrate and reaction product, respectively, DSb , DPb are the diffusion coefficients of the species in the bulk solution.

2 Biosensors with Outer Porous Membrane

247

2.1.2 Initial and Boundary Conditions Let x = 0 represent the electrode surface. The simulation of the biosensor operation starts when some substrate appears in the bulk solution (t = 0), Se (x, 0) = 0,

Pe (x, 0) = 0,

x ∈ [0, de ],

Sp (x, 0) = 0,

Pp (x, 0) = 0,

x ∈ [de , de + dp ],

Sb (x, 0) = 0,

Pb (x, 0) = 0,

x ∈ [de + dp , de + dp + δ),

Sb (de + dp + δ, 0) = S0 ,

(5)

Pb (de + dp + δ, 0) = 0,

where S0 is the concentration of the substrate in the bulk solution. Due to the electrode polarization, the substrate is assumed as an electro-inactive substance, while the product is an electro-active substance. The electrode potential is chosen to keep zero concentration of the product at the electrode surface (t > 0), Pe (0, t) = 0,

DSe

∂Se  = 0.  ∂x x=0

(6)

On the boundary between the enzyme layer and the porous membrane as well as between the porous membrane and the external diffusion layer the matching conditions are defined (t > 0), ∂Sp  ∂Se  = DSp , Se (de , t) = Sp (de , t),   ∂x x=de ∂x x=de ∂Pp  ∂Pe  DPe = DPp , Pe (de , t) = Pp (de , t),   ∂x x=de ∂x x=de

DSe

DSp

(7)

∂Sp  ∂Sb  = DSb ,   ∂x x=de +dp ∂x x=de +dp

Sp (de + dp , t) = Sb (de + dp , t), ∂Pp  ∂Pb  DPp = DPb ,   ∂x x=de +dp ∂x x=de +dp

(8)

Pp (de + dp , t) = Pb (de + dp , t). The external diffusion layer of thickness δ remains unchanged with time. Away from it the concentration of the substrate and the product remains constant (t > 0), Sb (de + dp + δ, t) = S0 ,

Pb (de + dp + δ, t) = 0.

(9)

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Biosensors with Porous and Perforated Membranes

2.1.3 Biosensor Response The system (2)–(9) approaches a steady state as t → ∞. The density i(t) of the biosensor current at time t as well as the steady state current I is defined as in the two-compartment model, i(t) = ne F DPe

∂Pe   , ∂x x=0

I = lim i(t), t →∞

(10)

where ne is the number of electrons involved in the charge transfer at the electrode surface, F is the Faraday constant.

2.2 Numerical Simulation At the transient conditions the initial boundary value problem (2)–(9) can be solved numerically using the finite difference technique. Since the governing equations, initial as well as boundary conditions are of the same type as in a two-compartment model discussed in Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”, the model equations can be approximated with the difference equations by using the same technique. For δ → 0, the solution of the model approaches the solution of the corresponding two-compartment discussed in Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”. Similarly, vanishing the thickness of the porous membrane, dp → 0, the solution of three-compartment model (2)– (9) also approaches the solution of two-compartment model with the governing equations (2) and (4). These features can be applied to validation of the numerical solution of the three-compartment model by using analytical solutions presented in the previous section. As in a general multi-layer model as well as in a two-compartment model, the concentrations of both species, S and P , can be defined continuously in the entire domain x ∈ [0, de + dp + δ] as follows (t ≥ 0):

S(x, t) =

⎧ ⎪ ⎪Se (x, t), ⎨ Sp (x, t), ⎪ ⎪ ⎩S (x, t),

x ∈ [0, de ], x ∈ (de , de + dp ],

x ∈ (de + dp , de + dp + δ], b ⎧ ⎪ ⎪ ⎨Pe (x, t), x ∈ [0, d], P (x, t) = Pp (x, t), x ∈ (de , de + dp ], ⎪ ⎪ ⎩P (x, t), x ∈ (d + d , d + d + δ]. b e p e p

(11)

2 Biosensors with Outer Porous Membrane

249

Figure 2 shows the simulated concentration profiles of the substrate S and the product P calculated from the three-compartment model (2)–(9) at the following

1.0

6

0.9

5

0.8 0.7

4

S / S0

0.6 0.5 0.4

3

0.3 0.2

2

0.1

1

0.0 0

1

2

3

4

5

6

7

8

5

6

7

8

x, μm

a) 0.0012

6

0.0011 0.0010

5

0.0009 0.0008

4

P / S0

0.0007 0.0006 0.0005

3

0.0004

2

0.0003 0.0002

1

0.0001 0.0000 0

b)

1

2

3

4

x, μm

Fig. 2 Concentration profiles of the substrate (a) and the product (b) obtained at t = 0.05(1), 0.1(2), 0.2(3), 0.4(4), 0.8(5) and 4.14 (6, steady state time) s. The profiles are normalized with the bulk concentration S0 . Dot lines show the boundaries between the enzyme layer and the porous membrane, dashed lines show the boundaries between the porous membrane and the external diffusion layer. Values of the model parameters are defined in (12)

250

Biosensors with Porous and Perforated Membranes

values of the model parameters: DSe = DPe = De = 300 μm2 /s,

DSp = DPp = Dp = 30 μm2 /s,

DSb = DPb = Db = 600 μm2 /s,

de = 4 μm,

KM = 100 μM,

Vmax = 10 μM/s.

S0 = 100 μM,

dp = δ = 2 μm,

(12)

In Fig. 2, the concentration profiles are depicted at the steady state (curves 6) and the intermediate (curves 1–5) conditions. Approximate steady state conditions were achieved at time T = 4.14 s. The half of the steady state was achieved at 0.19 s (T0.5 = 0.19 s). Curves 3 correspond approximately to the half-time T0.5 of the steady state. One can see in Fig. 2 linear curves 6 at x ∈ (de , de + dp + δ). At the steady state conditions, the concentrations approach straight lines because of the linearity of the governing equations (3) and (4).

2.3 Effect of the Porous Membrane Using numerical simulation, the influence of the permeability and of the thickness of the outer porous membrane on the biosensor response has been investigated [10]. In terms of the mathematical model (2)–(9), the permeability as well as the porosity is expressed by the diffusion coefficients DSp and DPp .

2.3.1 Effect of the Membrane Permeability When calculating the concentrations shown in Fig. 2, the diffusion coefficients DSp and DPp for the porous membrane were tenfold less than the corresponding coefficients for the enzyme, Dp = DSp = DPp = DSe /10 = DPe /10 = De /10. To investigate the dependence of the biosensor response on the diffusivity (permeability) of the outer porous membrane, the biosensor response was simulated at constant thickness dp = 2 μm of the outer membrane changing the diffusion coefficient Dp (Dp = DSp = DPp ) from 600 to 6 μm2 /s, i.e. from Db to Db /100, where Db = DSb = DPb . The simulated biosensor current was normalized with respect to the stationary current calculated at theoretically maximal value Dp of the diffusivity of the outer membrane to be analysed, IDp =

I (Dp ) , I (Db )

(13)

where I (Dp ) is the density of the steady state current calculated at the diffusivity Dp of the outer porous membrane. The maximal limiting value Dp = Db of diffusivity corresponds to a limiting case when the diffusivity in the membrane

2 Biosensors with Outer Porous Membrane

251

ID

p

matches the diffusivity in the bulk. Practically, the diffusivity Dp = Db of the outer membrane corresponds to the extreme case when the porous membrane becomes just a diffusion layer. Calculations were done at three maximal enzymatic rates Vmax : 10, 100, 1000 μM/s, combined with the three values of the substrate concentration S0 in bulk: 10, 100, 1000 μM/s. When normalizing these values of S0 with respect to the Michaelis constant K M = 100 μM/s the following dimensionless concentrations of the substrate are obtained: 0.1, 1 and 10. Figure 3a represents the

1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

1 2 3 4 5 10

a)

6 7 8 9

100 2

Dp, μm /s 1.0 0.9 0.8

BS

0.7 0.6 0.5 0.4

1 2 3

0.3 0.2

4 5 6

7 8 9

0.1

b)

10

100 2

Dp, μm /s

Fig. 3 The normalized steady state current IDp (a) and the sensitivity BS (b) versus the diffusivity Dp of the outer membrane at three maximal enzymatic rates Vmax : 10 (1, 4, 7), 100 (2, 5, 8), 1000 (3, 6, 9) μM/s, and three substrate concentrations S0 : 10 (1–3), 100 (4–6), 1000 (7–9) μM. Values of the other parameters are defined in (12)

252

Biosensors with Porous and Perforated Membranes

dependence of the normalized steady state current IDp on the diffusivity Dp , while Fig. 3a represents the corresponding dependence of the biosensor sensitivity BS . As one can see in Fig. 3a the biosensor current notably depends on the diffusivity of the porous membrane. A decrease in the diffusivity corresponds to a decrease in the permeability and, usually, to a decrease in the porosity. The effect of the diffusivity Dp entirely depends on the maximal enzymatic rate Vmax and the substrate concentration S0 . At moderate values of the enzyme activity Vmax (10 and 100 μM/s) and relatively high values of the concentration S0 (100 and 1000 μM, curves 4, 7 and 8), the stationary current monotonously increases when Dp changes 100 fold from 600 down to 6 μm2 /s. Due to the increase in the diffusivity Dp , the stationary current can increase half as much again. However, at a high enzyme activity Vmax (1 mM/s) and low concentration S0 (1 mM, curve 3), the stationary current monotonously decreases when Dp changes from 600 to 6 μm2 /s. In all other cases of Vmax and S0 , IDp is a non-monotonous function at Dp ∈ [6, 600] (μm2 /s). Since at the theoretically minimal limiting value of Dp = 0 no current upstarts, the normalized current IDp is a non-monotonous function of Dp changing Dp from the maximal limiting value (Db ) to the minimal one (0). Figure 3b shows a well known feature of the biosensors that the biosensor sensitivity is higher at lower substrate concentrations rather than at higher ones. However, in cases of a high enough enzymatic activity Vmax , the sensitivity BS can be notably increased by decreasing the diffusivity Dp of the outer membrane even at a relatively high substrate concentration S0 (curves 5, 6, 9 in Fig. 3b). This is a very important advantage of the outer membranes in extending the region of the biosensor application [30, 31, 44].

2.3.2 Effect of the Membrane Thickness To investigate the effect of the thickness dp of the outer porous membrane on the biosensor response the steady state current changing the thickness dp has to be calculated at different values of the maximal enzymatic rate Vmax and the substrate concentration S0 . To evaluate the effect of the membrane thickness on the biosensor response the biosensor stationary current is normalized with respect to the steady state current of the corresponding biosensor having no outer membrane, Idp =

I (dp ) , I (0)

(14)

where I (dp ) is the density of the steady state current calculated at the thickness dp of the outer porous membrane. Figure 4a shows the dependence of the steady state biosensor response on the thickness dp of the outer membrane at different values of the maximal enzymatic rate Vmax , the substrate concentration S0 keeping the membrane diffusivity Dp = 0.1 De = 30 μm2 /s and the other parameters constant as defined in (12). Figure 4b

Id

p

2 Biosensors with Outer Porous Membrane 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

1 2 3

0

a)

253

1

2

3

4 5 6

4

5

7 8 9

6

7

8

dp, μm 1.0 0.9 0.8

BS

0.7 0.6 0.5

4 5 6

1 2 3

0.4 0.3 0.2

7 8 9

0.1 0

b)

1

2

3

4

5

6

7

8

dp, μm

Fig. 4 The normalized steady state current Idp (a) and the sensitivity BS (b) versus the thickness dp of the outer membrane. The parameters and notations are the same as in Fig. 3

shows the effect of the membrane thickness on the biosensor sensitivity at the same values of the parameters. One can see in Fig. 4a that the shape of the normalized steady state current Idp (as well as of the non-normalized one I ) is very sensitive to changes of the maximal enzymatic rate Vmax and the substrate concentration S0 . Idp is a monotonous increasing function of the outer membrane thickness dp at moderate values of the enzyme activity Vmax (10 and 100 μM/s) and relatively high values of the concentration S0 (100 and 1000 μM, curves 4, 7 and 8) when dp

254

Biosensors with Porous and Perforated Membranes

changes from 0 to 8 μm. Idp is a monotonous decreasing function of the thickness dp at a high value of Vmax (1 mM/s) and a low value of S0 (10 μM) (curve 3). Similar behaviour of the biosensor response was observed when analysing the effect of the thickness of diffusion layer for the amperometric biosensors when applying a two-compartment model [8] (see also Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”). Although the shapes of curves in Fig. 4 notably differ from those in Fig. 3, the effect of the diffusivity Dp of the porous membrane is very similar to that of the membrane thickness dp . A decrease in diffusivity Dp influences the steady state current as well as the sensitivity similarly to an increase in thickness Dp of the membrane. To make the similarities more apparent, the normalized steady state current Idp and the sensitivity BS (Fig. 3) were replotted in Fig. 5 as functions of the inverse thickness 1/dp of the outer membrane. One can see that the shapes of curves in Fig. 5 are approximately the same as in Fig. 3. When preparing highly sensitive biosensors, the increase of the outer membrane can be effectively combined with the decrease of the membrane diffusivity. The membrane diffusivity usually decreases with the decrease in the porosity.

3 Biosensors with Selective and Outer Perforated Membranes In this section, the model, presented in the previous section, is extended to cover the transducer with an additional selective membrane permeable for the product of the enzymatic reaction [12, 13, 27]. This section concentrates on the modeling of the selective membrane. By changing the input parameters the output results are numerically analysed with a special emphasis on the influence of the selective membrane to the response of the biosensors. Figure 6 shows the principal structure of a biosensor to be considered, where ds is the thickness of the selective membrane, de is the thickness of the enzyme layer, dp is the thickness of the perforated membrane and δ is the thickness of the external diffusion layer.

3.1 Mathematical Model The thicknesses of the selective membrane as well as of the perforated membrane of the biosensor are assumed to be much less than its length and width. In the biosensor, the selective and the perforated membranes are of the uniform thickness. Figure 6 shows the profile of a one-dimensional unit of the biosensor, where ds = b1 − b0 stands for the thickness of the selective membrane, de = b2 − b1 is the thickness of

Id

p

3 Biosensors with Selective and Outer Perforated Membranes 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1

255

1 2 3 4 5 1

a)

6 7 8 9

10 -1

100

-1

dp , μm 1.0 0.9 0.8 0.7

BS

0.6 0.5 0.4

1 2 3

0.3 0.2

4 5 6

7 8 9

0.1 0.1

b)

1

10

100

dp-1, μm-1

Fig. 5 The normalized steady state current Idp (a) and the sensitivity BS (b) versus the inverse thickness dp−1 of the outer membrane. The parameters and notations are the same as in Fig. 3

the enzyme layer, dp = b3 − b2 is the thickness of the perforated membrane, and δ = b4 − b3 is the thickness of the external diffusion (Nernst) layer [12, 13, 27].

3.1.1 Governing Equations The dynamics of the concentrations of the substrate as well as the product in the enzyme layer can be described by a one-dimensional-in-space system of the

256

Biosensors with Porous and Perforated Membranes

Fig. 6 The principal structure of the biosensor with selective and outer perforated membranes. The figure is not to scale

x δ

b4 Diffusion layer

Enzyme Selective membrane

Electrode

ds de

Perforated membrane

dp

b3

b2 b1 b0

reaction–diffusion equations [12, 46]. In the selective membrane only the mass transport by diffusion of the reaction product takes place, ∂P1 ∂ 2 P1 = DP1 , ∂t ∂x 2

x ∈ (b0 , b1 ),

t > 0,

(15)

where x and t stand for space and time, respectively, P1 (x, t) is the concentration of the reaction product in the selective membrane, ds = b1 − b0 is the thickness of the enzyme membrane, DP1 is the diffusion coefficient. In the enzyme layer the mass transport by diffusion and the enzyme-catalysed reaction are considered, ∂S2 Vmax S2 ∂ 2 S2 − , = DS2 2 ∂t KM + S2 ∂x ∂P2 ∂ 2 P2 Vmax S2 = DP2 + , 2 ∂t ∂x KM + S2

(16) x ∈ (b1 , b2 ),

t > 0,

where S2 (x, t) and P2 (x, t) are the concentrations of the substrate and reaction product, respectively, de = b2 − b1 is the thickness of the enzyme membrane, Vmax is the maximal enzymatic rate and KM is the Michaelis constant. Assuming the perforated membrane as a periodic media, a homogenization process can be applied to the membrane domain [4, 6, 21]. After this, the perforated membrane is modeled as a diffusion layer with effective diffusion coefficients. So, the dynamics of concentrations in the perforated and the external diffusion layer are described by the systems of the diffusion equations, ∂Sj ∂ 2 Sj = DSj , ∂t ∂x 2 ∂Pj ∂ 2 Pj = DPj , ∂t ∂x 2

(17) x ∈ (bj −1 , bj ),

t > 0,

j = 3, 4,

3 Biosensors with Selective and Outer Perforated Membranes

257

where Sj (x, t) and Pj (x, t) are the concentrations of the substrate and the reaction product, respectively, dp = b3 − b2 is the thickness of the perforated membrane, δ = b4 − b3 is the thickness of the external diffusion layer, j = 3, 4.

3.1.2 Initial and Boundary Conditions Let x = 0 represent the electrode surface. The simulation of the biosensor operation starts when some substrate appears in the bulk solution (t = 0), Pi (x, 0) = 0,

x ∈ [bi−1 , bi ],

Sj (x, 0) = 0,

x ∈ [bj −1 , bj ],

S4 (x, 0) = 0,

x ∈ [b3, b4 ),

i = 1, 2, 3, 4, j = 2, 3,

(18)

S4 (b4 , 0) = S0 ,

where S0 is the concentration of the substrate in the bulk solution. Due to the amperometry, the electrode potential is chosen to keep zero concentration of the product at the electrode surface (t > 0), P1 (b0 , t) = 0.

(19)

On the boundary between the selective membrane and the enzyme layer the matching conditions for the product and the non-leakage condition for the substrate are defined (t > 0), ∂P1  ∂P2  = DP2 ,   ∂x x=b1 ∂x x=b1 ∂S2  DS2 = 0.  ∂x x=b1 DP1

P1 (b1 , t) = P2 (b1 , t), (20)

On the other boundaries between the adjacent layers the matching conditions are applied (t > 0), ∂Sj  ∂Sj +1  = DSj+1 , Sj (bj , t) = Sj +1 (bj , t),   ∂x x=bj ∂x x=bj ∂Pj  ∂Pj +1  = DPj+1 , Pj (bj , t) = Pj +1 (bj , t), DPj   ∂x x=bj ∂x x=bj

DSj

(21)

j = 2, 3. The external diffusion layer of thickness δ remains unchanged with time. Away from it the concentrations of the substrate and the product remain constant (t > 0), S4 (b4 , t) = S0 ,

P4 (b4 , t) = 0.

(22)

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Biosensors with Porous and Perforated Membranes

3.1.3 Biosensor Response The system (15)–(22) approaches a steady state as t → ∞. The biosensor current is proportional to the gradient of the concentration P1 of the reaction product at the electrode surface. The density i(t) of the current at time t as well as the steady state current I is defined as usual, i(t) = ne F DP1

∂P1  ,  ∂x x=b0

I = lim i(t), t →∞

(23)

where ne is the number of the electrons involved in a charge transfer at the electrode surface, F is the Faraday constant.

3.2 Numerical Simulation The initial boundary value problem (15)–(22) can be solved numerically using the finite difference technique [18, 41]. Since the governing equations, the initial and the boundary conditions are of the same type as in the modeling of a biosensor with the porous membrane discussed in the previous Sect. 2, the model equations can be approximated with the difference equations by using the same technique. For ds → 0, the solution of the model approaches the solution of the corresponding three-layer model of the biosensor with the outer porous membrane discussed in Sect. 2. This can be applied to validate the numerical solution of the four-compartment model (15)–(22). As for the model of the biosensor with an outer porous membrane, the concentrations of both species, S and P , can be defined continuously in the entire domain x ∈ [b0 , b4 ] as follows (t ≥ 0): ⎧ ⎪ ⎪ ⎨S2 (x, t), x ∈ [b1 , b2 ], S(x, t) = S3 (x, t), x ∈ (b2 , b3 ], (24) ⎪ ⎪ ⎩S (x, t), x ∈ (b , b ], 4

⎧ ⎪ P1 (x, t), ⎪ ⎪ ⎪ ⎨P (x, t), 2 P (x, t) = ⎪P3 (x, t), ⎪ ⎪ ⎪ ⎩ P4 (x, t),

3

4

x ∈ [b0, b1 ], x ∈ [b1, b2 ], x ∈ (b2 , b3 ],

(25)

x ∈ (b3 , b4 ].

Both concentration functions (S and P ) are now continuous in the entire domain x ∈ [b0 , b4 ]. In the simulation of the biosensor action, the diffusion coefficient for the selective membrane was accepted 100 fold less and the diffusion coefficient for the perforated

3 Biosensors with Selective and Outer Perforated Membranes

259

membrane—tenfold less than the diffusion coefficient for the enzyme layer. Figure 7 shows the simulated concentration profiles of the substrate S and the product P calculated from the four-compartment model (15)–(22) at the following values of the model parameters: DP1 = 3 μm2 /s,

DS2 = DP2 = 300 μm2 /s,

DS3 = DP3 = 30 μm2 /s,

DS4 = DP4 = 600 μm2 /s,

ds = b1 − b0 = b1 = 2 μm, dp = b3 − b2 = 8 μm, KM = 100 μM,

de = b2 − b1 = 4 μm,

(26)

δ = b4 − b3 = 2 μm,

S0 = 100 μM,

Vmax = 1 μM/s.

Figure 7 shows the concentration profiles at the steady state (curves 5) and the intermediate (curves 1–4) conditions. An approximate steady state condition was achieved at time T = 10.04 s. The half of the steady state was achieved at 2.66 s (T0.5 = 2.66 s). Curves 3 correspond to the half-time T0.5 of the steady state. The other curves show the concentrations at the intermediate values of time: T0.1 = 1.1(1), T0.25 = 1.67(2), T0.75 = 4.12(4) s.

3.3 Effect of the Selective Membrane To investigate the dependence of the biosensor response on the thickness ds of the selective membrane, the biosensor response was calculated by changing the thickness ds from 0.4 up to 8 μm keeping constant thicknesses de , dp and δ of all other layers as defined in (26) [12]. The calculated steady state current was normalized with respect to the minimal value ds,min of the thickness of the selective membrane to be analysed, Is =

I (ds ) , I (ds,min )

ds ≥ ds,min ≥ 0,

(27)

where I (ds ) is density of steady state current as defined in (23) and Is is the normalized steady state current, both calculated at the thickness ds of the selective membrane. The results of the calculations are depicted in Fig. 8. One can see in Fig. 8, that the effect of the thickness ds is practically independent from the maximal enzymatic rate Vmax as well as from the substrate concentration S0 . The steady state current is a monotonously decreasing function of ds at values of Vmax and S0 differing in orders of magnitude [12]. The normalized currents were well fitted with the following exponential function (ds in μm): Is (ds ) = 0.14 + 1.05 × exp(−0.61ds ).

(28)

260

Biosensors with Porous and Perforated Membranes 1.0

5 0.9

4

S / S0

0.8 0.7

3

0.6 0.5

2 0.4

1

0.3 0

a)

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16

x, μm 0.004

5 0.003

P / S0

4 0.002

3

0.001

2 1

0.000 0

b)

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16

x, μm

Fig. 7 Concentration profiles of the substrate (a) and the product (b) obtained at t = 1.1(1), 1.67(2), 2.66(3), 4.12(4) and 10.04 (5) s. The profiles are normalized with the bulk concentration S0 . Dash-dot lines show the boundaries between the selective membrane and the enzyme layer (x = 2 μm), dot lines—the boundaries between the enzyme layer and the perforated membrane (x = 6 μm) and dashed lines—the boundaries between the perforated membrane and the external diffusion layer (x = 14 μm). The values of the model parameters are defined in (26)

4 Two-Dimensional Modeling of Biosensors with Selective and Perforated. . .

261

1.0 0.9

0.7 0.6

Is

6 7 8 9 10

1 2 3 4 5

0.8

0.5 0.4 0.3 0.2 0.1 0.0 0

1

2

3

4

5

6

7

8

ds, μm Fig. 8 The normalized steady state current Is versus the thickness ds of the selective membrane at three maximal enzymatic rates Vmax : 1(1, 4, 7), 10(2, 5, 8), 100(3, 6, 9) μM/s, and three substrate concentrations S0 : 10(1–3), 100(4–6), 1000(7–9) μM. Curve 10 shows exponential fit (28). Values of all other parameters are as defined in (26)

4 Two-Dimensional Modeling of Biosensors with Selective and Perforated Membranes The initial boundary value problem (2)–(9) has been formulated in one-dimensional domain by assuming a porous membrane as a periodic media and applying the homogenization process [12]. The accuracy of one-dimensional model depends on the geometry of the membrane perforation as well as on the level of filling the holes with the enzyme [37]. The relative error of the one-dimensional modeling decreases with a decrease in the filling level, while the size of the holes has inverse influence to the modeling accuracy. When the holes of the perforated membrane are relatively very small, the two-dimensional modeling should be applied for an accurate prediction of the biosensor response [37]. This section focuses on modeling the amperometric biosensor with selective and outer perforated membranes where the topology of the perforation is taken into consideration. The mathematical model of the biosensor is formulated in twodimensional domain. The perforated membrane is analysed with a special emphasis to the geometry of the membrane perforation [12, 13, 27].

262

Biosensors with Porous and Perforated Membranes

Perforated membrane

Holes

Enzyme Selective membrane Electrode Fig. 9 The principal structure of the biosensor. The figure is not to scale

4.1 Principal Structure of Biosensor The thickness of the selective membrane as well as of the perforated membrane of the biosensor are assumed to be much less than its length and width. Both membranes are assumed to be of uniform thickness. The holes in the perforated membrane are modeled by right cylinders. The holes are of uniform diameter and spacing, forming a hexagonal pattern. Figure 9 presents the biosensor schematically [12]. Due to the uniform distribution of the holes, the entire biosensor may be divided into equal hexagonal prisms with regular hexagonal bases. For simplicity, it is reasonable to consider a circle whose area equals that of the hexagon and to regard one of the cylinders as a unit cell of the biosensor. Figure 10a shows the profile of the biosensor schematically represented in Fig. 9. Due to the symmetry of the unit cell, only a half of the transverse section of the unit cell can be considered. The unit cell to be considered is presented in Fig. 10b. A very similar approach has been taken in modeling of the analytical system based on the array of enzyme microreactors [9, 11] (also see Chapter “Biosensors Based on Microreactors”) and of the partially blocked electrodes [7, 20, 25]. In Fig. 10b, a2 is the radius of the base of the unit cell, a1 is the radius of the holes, b1 stands for the thickness of the selective membrane, b4 − b2 is the thickness of the perforated membrane, b5 − b4 is the thickness of the external diffusion layer. The holes can be fully or partially filled with an enzyme.

4.2 Mathematical Model The mathematical model of the enzyme electrode with the selective and the perforated membranes (Figs. 9 and 10) can be formulated in a two-dimensional

4 Two-Dimensional Modeling of Biosensors with Selective and Perforated. . .

263

z b5

Analyte

Ω3

b4

b3

Insulator

b2

Enzyme Selective membrane

a)

Ω2

b1

b)

Ω1

0

a1

a2

r

Fig. 10 The profile (a) and the unit cell (b) at y-plane. z = 0 corresponds to the electrode surface, a1 is the radius of holes, a2 is the half distance between the centres of adjacent holes. b1 , b2 −b1 and b3 − b2 are the thicknesses of the selective membrane, the basic enzyme layer and the perforated membrane, respectively

domain 1 ∪ 2 ∪ 3 in the cylindrical coordinates [12], 1 = (0, a2 ) × (0, b1 ), 2 = ((0, a2 ) × (b1 , b2 )) ∪ ((0, a1 ) × [b2 , b3 )) ,

(29)

3 = ((0, a1 ) × (b3 , b4 ]) ∪ ((0, a2 ) × (b4 , b5 )) . Let i denote the closed region corresponding to the open region i , Si (r, z, t) be the concentration of the substrate in the region i , Pi (r, z, t)—the concentration of the reaction product in i , (r, z) ∈ i , t ≥ 0, i = 1, 2, 3.

4.2.1 Governing Equations In the selective membrane (region 1 ) only the mass transport by diffusion of the reaction product takes place, ∂P1 = DP1 P1 , ∂t

(r, z) ∈ 1 ,

(30)

where  is the Laplace operator in the cylindrical coordinates. In the enzyme region 2 the mass transport by diffusion and the enzymecatalysed reaction are considered, E

S −→ P.

(31)

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Biosensors with Porous and Perforated Membranes

Coupling the enzyme-catalysed reaction with the two-dimensional-in-space mass transport by diffusion leads to the system of the reaction–diffusion equations (t > 0), Vmax S2 ∂S2 = DS2 S2 − , ∂t KM + S2 ∂P2 Vmax S2 = DP2 P2 + , ∂t KM + S2

(32) (r, z) ∈ 2 ,

where Vmax is the maximal enzymatic rate, and KM is the Michaelis constant. The region 3 stands for the external diffusion, ∂S3 = DS3 S3 , ∂t ∂P3 = DP3 P3 , ∂t

(33) (r, z) ∈ 3 .

4.2.2 Initial Conditions Let  be the upper boundary of 3 ,  = [0, a2 ] × {b5 }.

(34)

The enzyme electrode operation starts when the substrate of the concentration S0 appears in the bulk solution, i.e. on the boundary . This is defined in the initial conditions (t = 0) S2 (r, z, 0) = 0,

(r, z) ∈ 2 ,

S3 (r, z, 0) = 0,

(r, z) ∈ 3 \ ,

S3 (r, z, 0) = S0 ,

(r, z) ∈ ,

Pi (r, z, 0) = 0,

(r, z) ∈ i ,

(35)

i = 1, 2, 3.

4.2.3 Boundary and Matching Conditions Assuming the amperometry and the electric activity of the product leads to the condition P1 (r, 0, t) = 0,

r ∈ [0, a2 ].

(36)

On the boundary between the selective membrane and the enzyme layer the matching conditions for the product and the non-leakage condition for the substrate

4 Two-Dimensional Modeling of Biosensors with Selective and Perforated. . .

265

are defined (t > 0), ∂P1  ∂P2  = DP2 , P1 (r, b1 , t) = P2 (r, b1 , t),   ∂z z=b1 ∂z z=b1 ∂S2  = 0, r ∈ [0, a2 ]. DS2  ∂z z=b1

DP1

(37)

On the boundary between adjacent regions 2 and 3 the matching conditions are defined (t > 0, r ∈ [0, a1 ]), ∂S2  ∂S3  = DS3 , S2 (r, b3 , t) = S3 (r, b3 , t),   ∂z z=b3 ∂z z=b3 ∂P2  ∂P3  DP2 = DP3 , P2 (r, b3 , t) = P3 (r, b3 , t).   ∂z z=b3 ∂z z=b3

DS2

(38)

Assuming well-stirred and being in the powerful motion bulk solution, the upper layer of the constant thickness b5 − b4 can be treated as the Nernst diffusion layer [18, 45]. Away from this layer, the solution is in motion and is of the uniform concentration. This is described in the following boundary condition (t > 0): S3 (r, z, t) = S0 ,

P3 (r, z, t) = 0,

(r, z) ∈ .

(39)

Non-leakage boundary conditions are used for the boundaries of the symmetry of the unit cell and for the insulator boundary, DP1

∂P1  ∂P1  = DP1 = 0,   ∂r r=0 ∂r r=a2

z ∈ [0, b1 ],

(40)

DS2

∂S2  ∂P2  = DP2 = 0,   ∂r r=0 ∂r r=0

z ∈ [b1 , b3 ],

(41)

DS2

∂S2  ∂P2  = DP2 = 0,   ∂r r=a2 ∂r r=a2

z ∈ [b1 , b2 ],

(42)

DS2

∂S2  ∂P2  = DP2 = 0,   ∂r r=a1 ∂r r=a1

z ∈ (b2 , b3 ],

(43)

DS3

∂S3  ∂P3  = DP3 = 0,   ∂r r=0 ∂r r=0

z ∈ [b3 , b5 ],

(44)

DS3

∂S3  ∂P3  = DP3 = 0,   ∂r r=a1 ∂r r=a1

z ∈ [b3 , b4 ),

(45)

DS3

∂S3  ∂P3  = DP3 = 0,   ∂r r=a2 ∂r r=a2

z ∈ [b4 , b5 ],

(46)

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Biosensors with Porous and Perforated Membranes

DS2

∂S2  ∂P2  = DP2 = 0,   ∂z z=b2 ∂z z=b2

r ∈ (a1 , a2 ],

(47)

DS3

∂S3  ∂P3  = DP3 = 0,   ∂z z=b4 ∂z z=b4

r ∈ (a1 , a2 ].

(48)

4.2.4 Biosensor Response The density i(t) of the anodic current at time t can be obtained explicitly from the Faraday and the Fick laws,  ! 2π ! a2 1 ∂P1  i(t) = ne F DP1 2 rdrdϕ ∂z z=0 πa2 0 0  ! a2 2 ∂P1  = ne F DP1 2 r dr, a2 0 ∂z z=0

(49)

where ϕ is the third cylindrical coordinate, ne is the number of electrons involved in a charge transfer at the electrode surface, and F is the Faraday constant. The system approaches the steady state when t → ∞, I = lim i(t), t →∞

(50)

where I is the density of the steady state current.

4.3 Numerical Simulation Because of the nonlinearity of the governing equations and a rather complex geometry of the domain, the initial boundary value problem was solved numerically using the finite difference technique [16, 18, 41]. To find a numerical solution of the problem a non-uniform discrete grid in all the directions, r, z and t, was introduced [12, 13, 27]. Using the alternating direction method an implicit linear finite difference scheme has been built as a result of the difference approximation [41]. The resulting system of the linear algebraic equations was solved rather efficiently because of the tridiagonality of the matrix of the system. Due to the high gradients of the concentrations of the species, an accurate and stable numerical solution was only achieved at a very small step size in z direction at the boundaries z = 0 and z = b5 . Because of the concavity of angles at points (a1 , b2 ) and (a1 , b4 ) it was also necessary to use a very small step size at the boundaries r = a1 , z = b2 and z = b4 in both space directions: r and z.

4 Two-Dimensional Modeling of Biosensors with Selective and Perforated. . .

267

Due to the matching conditions between the adjacent regions with very different diffusivities, a small step size was also used at the boundaries z = b1 and z = b3 . In the direction r, an exponentially increasing step size was used to both sides from a1 : to a2 and down to 0. In the direction z, an exponentially increasing step size was used from 0 to b1 /2, from b5 down to (b4 + b5 )/2, from bj down to (bj + bj −1 )/2 and from bj to (bj + bj +1 )/2, j = 1, 2, 3, 4, where b0 = 0. The step size in the time direction was restricted due to the nonlinear reaction term in (32), the boundary conditions and the domain geometry. Since the biosensor action obeys the steady state assumption when t → ∞, it was reasonable to apply an increasing step size in the time direction. The final step size in time was in a few orders of magnitude higher than the first one. The numerical simulator has been programmed in JAVA language [36]. The upper layer of the thickness δ = b5 − b4 was assumed to be the Nernst diffusion layer. Assuming intensively stirred the buffer solution, the thickness δ of 2 μm was used in the numerical simulation of the biosensor action changing some other parameters. The following values of the model parameters were constant in all the numerical experiments discussed below: DP1 = 1 μm2 /s,

DS2 = DP2 = 300 μm2 /s,

DS3 = DP3 = 2DS2 = 2DP2 , KM = 100 μM,

ne = 2,

Vmax = 100 μM/s,

(51) S0 = 100 μM.

Some results of the simulation are depicted in Figs. 11 and 12 [12].

4.0

2

3.5 3.0

i, nA/mm2

4 6

5 1 7

2.5 2.0

3

1.5 1.0 0.5 0.0 0

5

10

15

20

t, s Fig. 11 The dynamics of the density i of the biosensor current, a1 : 0.1 (1, 4, 5–7), 0.2 (2), 0.4 (3), a2 : 1 (1–3, 6, 7), 2 (4) , 4(5), b3 : b2 (6), (b2 + b4 )/2 (1–5), b4 (7), b1 = 2, b2 = b1 + 2, b4 = b2 + 10, b5 = b4 + 2 (μm). The values of all other parameters are defined in (51)

268

Biosensors with Porous and Perforated Membranes

S, μM 16

100

14

80

z, μm

12 10

60

8 40

6 4

20

2 0

0 -1

a)

-0.5

0

0.5

1

r, μm P, μM 16

35

14

30

z, μm

12

25

10

20

8

15

6

10

4

5

2 0

b)

0 -1

-0.5

0

0.5

1

r, μm

Fig. 12 The steady state concentrations of the substrate (a) and the product (b) in the selective membrane, enzyme region and buffer solution at t = 26 s, a 1 = 0.1, a 2 = 1, b1 = 2, b2 = b1 + 2, b3 = b2 + 5, b4 = b3 + 5, b5 = b4 + 2 (μm). The values of the model parameters are defined in (51)

One can see in Fig. 11 that the biosensor current is very sensitive to changes in the radius of the holes and in the degree of the holes filling with the enzyme. The density of the steady state current changes non-monotonously from 2.9 (curve 1) to 1.9 (curve 3) nA/mm2 when the holes radius a1 changes from 0.1 to 0.4 μm (curves 1–3). On the other hand, three curves (1, 4, 5) show the current dynamics at the same relative radius of the holes when the ratio a1 /a2 equals 0.1. Although, in these three cases the radius a2 differs in several times (1, 2 and 4 μm, respectively), the

4 Two-Dimensional Modeling of Biosensors with Selective and Perforated. . .

269

density of the steady state current only differs in about 10% (changes from 2.9 to 3.3 nA/mm2 ). The next section discusses the effect of the relative radius of the holes on the biosensor response in detail. Figure 12 shows the steady state concentrations of the substrate (S) and the product (P ) in the enzyme and the diffusion regions at time t = 26 s. The concentrations were introduced as follows (t ≥ 0): ⎧ ⎪ (r, z) ∈ 1 , ⎪ ⎨0, S(r, z, t) = S2 (r, z, t), (r, z) ∈ 2 , ⎪ ⎪ ⎩S (r, z, t), (r, z) ∈  \  , 3 3 2 (52) ⎧ ⎪ P (r, z, t), (r, z) ∈  , ⎪ 1 ⎨ 1 P (r, z, t) = P2 (r, z, t), (r, z) ∈ 2 \ 1 , ⎪ ⎪ ⎩P (r, z, t), (r, z) ∈  \  . 3

3

2

At any time, t ≥ 0, both concentration functions, S(r, z, t) and P (r, z, t) are continuous at all (r, z) ∈ 1 ∪ 2 ∪ 3 . To have a more comprehensive view, the mirror-image along the z-axis is also shown in Fig. 12.

4.4 Effect of the Perforation Topology Using computer simulation the dependence of the response of the enzyme electrode on the geometry of the membrane perforation has been investigated [12]. For this purpose the density I of the steady state current was calculated at different values of the radius a2 of the entire unit and several degrees of the holes filling with the enzyme. In the calculations the radius a1 of holes was changed in a wide range. To compare the response of the biosensor (a1 < a2 ) with the response of the corresponding flat biosensor (a = b), the steady state current was expressed as a function of the degree α of the membrane openness and was normalized with the steady state current of the corresponding flat biosensor having no perforated membrane, Iα =

I (α) , I (1)

α=

a12 a22

,

0 < α ≤ 1,

(53)

where I (α) is the density of the steady state current calculated at the concrete value of α. The degree α expresses the level of the membrane perforation. The theoretically extreme case of α = 0 corresponds to the fully impermeable membrane having no holes. In the opposite case of α = 1 (a1 = a2 ), the perforated membrane becomes so opened that it disappears at all.

270

Biosensors with Porous and Perforated Membranes

Figure 13 shows the effect of the openness α on the steady state current and on the biosensor sensitivity. One can see in Fig. 13a that the steady state current is a non-monotonous function of the degree α of the perforated membrane openness. The steady state current of the biosensor having a perforated membrane onto the top of the enzyme layer can generate the steady state current which is significantly higher than the current if the enzyme electrode is without the perforated membrane

18 16 14 12 10

I

1 2 3 4 5

8 6 4 2 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

a) 1.0 0.9

1 2 3 4 5

BS

0.8 0.7 0.6 0.5 0.4 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b) Fig. 13 The normalized current Iα (a) and the sensitivity BS (b) versus the degree α of the perforated membrane openness, a2 : 1 (1, 4, 5), 2 (2) , 4 (3), b3 : b2 (4), (b2 + b4 )/2 (1–3), b4 (5) (μm). The values of all other parameters are the same as in Fig. 11

5 Concluding Remarks

271

(a1 = a2 , α = 1). This effect is especially meaningful at small values of the degree α (α ≈ 0.05). Figure 13 also shows that the biosensor response is practically invariant to the absolute value of the holes radius a1 . No notable difference is observed between the cases when the values of the radius a1 differ but the relative radius a1 /a2 remains stagnant (curves 1-3). Such difference was already noticed in Fig. 11. According to Fig. 13, the difference is observed only in the cases of very small values of α. Figure 13b shows that the biosensor sensitivity BS can be notably increased by decreasing the holes radius a1 [12]. The membrane containing the holes of a relatively small radius creates the diffusion limitation to the substrate, i.e. lowers the substrate concentration in the enzymatic layer and thereby prolongs the calibration curve of the biosensor [29–31, 43, 47, 50].

5 Concluding Remarks Assuming the porous membrane to be of a uniform thickness and as a periodic media, the membrane can be modeled as a diffusion layer with an effective diffusion coefficient depending on the porosity of the membrane [4, 51]. The outer porous membrane creates a diffusion limitation to the substrate, i.e. lowers the substrate concentration in the enzymatic layer and thereby prolongs the calibration curve of the biosensor (Figs. 3, 4 and 5) [30, 31, 44, 48]. The diffusivity of the porous membrane influences the steady state biosensor current as well as the sensitivity very similarly to the increase in the membrane thickness (Figs. 3 and 5) [12]. The modeling of biosensor with additionally added selective membrane revealed that the steady state current is almost exponentially decreasing function of the thickness ds of the selective membrane at a wide range of enzymatic activities Vmax as well as of substrate concentrations S0 (Fig. 8). The thinner the selective membrane is, the higher biosensor current is achieved [12]. The numerical simulation of the response of the amperometric biosensor with the selective and the perforated membranes shows that the steady state current is a nonmonotonous function of the relative radius of the holes of the perforated membrane (Fig. 11 and 13a) [12]. The biosensor with the perforated membrane can produce the steady state current which is notably higher than the current of the corresponding biosensor without the membrane (Fig. 13a). The sensitivity of the biosensors can be significantly increased by choosing an appropriate topology of the membrane perforation (Fig. 13b) [12].

272

Biosensors with Porous and Perforated Membranes

References 1. Antiochia R, Lavagnini I, Magno F (2004) Amperometric mediated carbon nanotube paste biosensor for fructose determination. Anal Lett 37(8):1657–1669 2. Bacon NC, Hall E (1999) A sandwich enzyme electrode giving electrochemical scavenging of interferents. Electroanal 11(10–11):749–755 3. Baeumner A, Jones C, Wong C, Price A (2004) A generic sandwich-type biosensor with nanomolar detection limits. Anal Bioanal Chem 378(6):1587–1593 4. Bakhvalov N, Panasenko G (1989) Homogenization: averaging processes in periodic media. Kluwer Academic Publishers, Dordrecht 5. Banica FG (2012) Chemical sensors and biosensors: fundamentals and applications. John Wiley & Sons, Chichester 6. Baronas R, Ivanauskas F, Kulys J (1999) Modeling a biosensor based on the heterogeneous microreactor. J Math Chem 25(3–4):245–252 7. Baronas R, Ivanauskas F, Survila A (2000) Simulation of electrochemical behavior of partially blocked electrodes under linear potential sweep conditions. J Math Chem 27(4):267–278 8. Baronas R, Ivanauskas F, Kulys J (2003) Computer simulation of the response of amperometric biosensors in stirred and non stirred solution. Nonlinear Anal Modell Control 8(1):3–18 9. Baronas R, Ivanauskas F, Kulys J, Sapagovas M (2004) Computational modeling of a sensor based on an array of enzyme microreactors. Nonlinear Anal Model Control 9(1):203–218 10. Baronas R, Ivanauskas F, Kaunietis I (2006) Laurinavicius V mathematical modeling of plategap biosensors with an outer porous membrane. Sensors 6(7):727–745 11. Baronas R, Ivanauskas F, Kulys J (2006) Mathematical modeling of biosensors based on an array of enzyme microreactors. Sensors 6(4):453–465 12. Baronas R, Kulys J, Ivanauskas F (2006) Computational modeling of biosensors with perforated and selective membranes. J Math Chem 39(2):345–362 13. Baronas R, Ivanauskas F, Kulys J (2007) Computational modeling of amperometric enzyme electrodes with selective and perforated membranes. In: Computation in modern science and engineering: proceedings of the international conference on computational methods in science and engineering 2007 (ICCMSE 2007). Parts A and B, vol 2. AIP Press, College Park, pp 457– 460 14. Bartlett PN (2008) Bioelectrochemistry: fundamentals, experimental techniques and applications. John Wiley & Sons, Chichester 15. Bergel A, Comtat M (1984) Theoretical evaluation of transient responses of an amperometricenzyme electrode. Anal Chem 56(14):2904–2909 16. Bieniasz L, Britz D (2004) Recent developments in digital simulation of electroanalytical experiments. Pol J Chem 78(9):1195–1219 17. Bindra D, Zhang Y, Wilson G, Sternberg R, Thévenot D, Moatti D, Reach G (1991) Design and in vitro studies of a needle-type glucose sensor for subcutaneous monitoring. Anal Chem 63(17):1692–1696 18. Britz D (2005) Digital simulation in electrochemistry, 3rd edn. Springer, Berlin 19. Conway GE, Lambertson RH, Schwarzmann MA, Pannell MJ, Kerins HW, Rubenstein KJ, Dattelbaum JD, Leopold MC (2016) Layer-by-layer design and optimization of xerogel-based amperometric first generation biosensors for uric acid. J Electroanal Chem 775:135–145 20. Deslous C, Gabrielli C, Keddam M, Khalil A, Rosset R, Trobollet B, Zidoune M (1997) Impedance techniques at partially blocked electrodes by scale deposition. Electrochim Acta 42(8):1219–1233 21. Dohnal M (1992) Qualitative partial differential equations and their realistic applications. Comput Ind 20(2):209–217 22. Fraser D (1997) Biosensors in the body: continuous in vivo monitoring. John Wiley & Sons, Chichester 23. Frew JE, Hill HAO (1987) Electrochemical biosensors. Anal Chem 59(15):933A–944A

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24. Grieshaber D, MacKenzie R, Vörös J, Reimhult E (2008) Electrochemical biosensors – sensor principles and architectures. Sensors 8(3):1400–1458 25. Gueshi T, Tokuda K, Matsuda H (1978) Voltammetry at partially covered electrodes: Part I. Chronopotentiometry and chronoamperometry at model electrodes. J Electroanal Chem 89(2):247–260 26. Ivanauskas F, Baronas R (2008) Numerical simulation of a plate-gap biosensor with an outer porous membrane. Simul Model Pract Theory 16(8):962–970 27. Ivanauskas F, Baronas R, Kulys J (2005) Mathematical modeling of biosensors with perforated and selective membranes. Rakenteiden Mekaniikka J Struct Mech 38(3):63–66 28. Kasche V, Kapune A, Schwegler H (1999) Potentiometric determination of NAD(P)H coenzymes using a nitrite-selective membrane electrode and a nitrate reductase enzyme and the application to glucose assay in human serum. Anal Sci 15(2):135–139 29. Kulys J, Bratkovskaja I, Ašeris V, Baronas R (2013) Electrochemical peroxidase-catalase Clark-type biosensor: computed and experimental response. Electroanal 25(6):1491–1496 30. Laurinavicius V, Kulys J, Gureviciene V, Simonavicius K (1989) Flow through and cateter biosensors with an extended concentration range. Biomed Biochem Acta 48(11–12):905–909 31. Lyons M, Murphy J, Rebouillat S (2000) Theoretical analysis of time dependent diffusion, reaction and electromigration in membranes. J Solid State Electrochem 4(8):458–472 32. Malhotra BD, Pandey CM (2017) Biosensors: fundamentals and applications. Smithers Rapra, Shawbury 33. Meskauskas T, Ivanauskas F, Laurinavicius V (2013) Degradation of substrate and/or product: mathematical modeling of biosensor action. J Math Chem 51(9):2491–2502 34. Meskauskas T, Ivanauskas F, Laurinavicius V (2013) Numerical modeling of multilayer biosensor with degrading substrate and product. In: AlBegain K, AlDabass D, Orsoni A, Cant R, Zobel R (eds.) 8th EUROSIM congress on modelling and simulation (EUROSIM), Cardiff, pp 24–29 35. Meyerhoff M, Duan C, Meusel M (1995) Novel nonseparation sandwich-type electrochemical enzyme immunoassay system for detecting marker proteins in undiluted blood. Clin Chem 41(9) 36. Moreira JE, Midkiff SP, Gupta M, Artigas PV, Snir M, Lawrence RD (2000) Java programming for high-performance numerical computing. IBM Syst J 39(1):21–56 37. Petrauskas K, Baronas R (2009) Computational modelling of biosensors with an outer perforated membrane. Nonlinear Anal Model Control 14(1):85–102 38. Pfeiffer D, Scheller F, Setz K, Schubert F (1993) Amperometric enzyme electrodes for lactate and glucose determinations in highly diluted and undiluted media. Anal Chim Acta 281(3) 39. Puida M, Ivanauskas F, Laurinaviˇcius V (2010) Mathematical modeling of the action of biosensor possessing variable parameters. J Math Chem 47(1):191–200 40. Sadana A, Sadana N (2011) Handbook of Biosensors and Biosensor Kinetics. Elsevier, Amsterdam 41. Samarskii A (2001) The theory of difference schemes. Marcel Dekker, New York 42. Scheller F, Pfeiffer D (1978) Enzymelektroden. Z Chem 18(2):50–57 43. Scheller FW, Schubert F (1992) Biosensors. Elsevier Science, Amsterdam 44. Schöning M (2005) “Playing around” with field-effect sensors on the basis of EIS structures, LAPS and ISFETS. Sensors 5(3):126–138 45. Schulmeister T (1990) Mathematical modelling of the dynamic behaviour of amperometric enzyme electrodes. Sel Electrode Rev 12(2):203–260 46. Schulmeister T, Pfeiffer D (1193) Mathematical modelling of amperometric enzyme electrodes with perforated membranes. Biosens Bioelectron 8(2):75–79 47. Šimeleviˇcius D, Petrauskas K, Baronas R, Razumien˙e J (2014) Computational modeling of mediator oxidation by oxygen in an amperometric glucose biosensor. Sensors 14(2):2578– 2594 48. Takoh K, Ishibashi T, Matsue T, Nishizawa M (2005) Localized chemical stimulation of cellular micropatterns using a porous membrane-based culture substrate. Sens Actuators B Chem 108(1–2):683–687

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49. Treloar P, Christie I, Vadgama P (1995) Engineering the right membranes for electrodes at the biological interface; solvent cast and electropolymerised. Biosens Bioelectron 10(1–2):195– 201 50. Turner APF, Karube I, Wilson GS (eds.) (1990) Biosensors: fundamentals and applications. Oxford University Press, Oxford 51. Whitaker S (1999) The method of volume averaging. Theory and applications of transport in porous media. Kluwer, Boston

Biosensors Utilizing Non-Michaelis–Menten Kinetics

Contents 1 2 3

4

5

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steady State Modeling of Substrate Inhibition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transient Modeling of Substrate and Product Inhibition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Characteristics of the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Numerical Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Dimensionless Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Effect of Substrate Inhibition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Effect of Product Inhibition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transient Modeling of Allostery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Transient Kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Effect of Cooperativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

276 278 281 281 284 285 287 289 292 293 293 296 297 298

Abstract The action of biosensors utilizing non-Michaelis–Menten kinetics is modeled at mixed enzyme kinetics and diffusion limitation in the cases of substrate and reaction product inhibition as well as of allostery at steady state and transient conditions. Computational modeling of the substrate inhibition at steady state shows multi-steady state concentrations of the substrate at the surface of the enzyme layer (membrane) when the diffusion module is much larger than one and the substrate bulk concentration is much higher than Michaelis–Menten constant. The multi-steady state concentration generates multi-response of the biosensor. At transient conditions, analytical systems are modeled by a two-compartment model comprising a mono-enzyme layer and an external Nernst diffusion layer. The complex enzyme kinetics produces different calibration curves for the response at the transition and the steady state. The cooperative phenomena of allosteric enzymes are modeled by applying the substrate uptake and the Hill equations. The positive cooperativity leads to a steady state current less than that when the biosensor action obeys the Michaelis–Menten kinetics, while negative cooperativity leads to increasing the biosensor response. The substrate concentration, at which © Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_9

275

276

Biosensors Utilizing Non-Michaelis–Menten Kinetics

the saturation curves of allosteric biosensors intersect, increases with increasing the external diffusion limitation. Keywords Biosensor modeling · Non Michaelis–Menten kinetics · Enzyme inhibition · Allostery · Multi-steady state · External diffusion

1 Introduction Usually biosensors operate following the Michaelis–Menten kinetics scheme [36, 51, 53, 60], k2 k1 E + S FGGGGGB GGGGG ESGGGA E + P, k−1

(1)

where E is an enzyme, S is a substrate, ES is an enzyme–substrate complex, P is a reaction product and k1 , k−1 and k2 are the rate constants. Often the kinetics of an enzyme action is much more complicated in comparison with the simplest scheme of the enzyme action (1). An inhibition, an activation, an allostery and other types of non-Michaelis–Menten kinetics are known for the diversity of enzymes [12, 14, 18, 24, 26, 33]. A number of substances may cause a reduction in the rate of an enzymecatalysed reaction. Enzyme inhibitors are molecules that interact with enzymes and decrease their activity [16, 17]. The inhibition is a process when a substance (inhibitor) diminishes the rate of a biochemical reaction [26, 33]. In enzymecatalysed reactions, the inhibitor frequently acts by binding to the enzyme. In addition to the scheme (1), the interaction of the enzyme–substrate complex (ES) with other substrate molecule (S) following the generation of a non-active complex (ESS) (the substrate inhibition) may produce one of the simplest non-Michaelis– Menten scheme of the enzyme action, k3 ES + S FGGGGGB GGGGG ESS. k−3

(2)

The products of many enzyme-catalysed reactions behave as inhibitors when they are presented in the reaction mixture. From this point of view they are called as product inhibitors and the phenomenon is known as the product inhibition. The inhibition is caused by the structural similarity of the product to the substrate. The product inhibition causes a loss in the productivity of the enzyme process at high degrees of substrate conversion. The product inhibition can be described by adding the relationship of the interaction of the product (P) with the enzyme (E) to the Michaelis–Menten scheme. Due to the interaction the enzyme-product complex (EP) is produced. The complex

2 Steady State Modeling of Substrate Inhibition

277

dissociates into the product (P), and the enzyme (E) is regenerated, k4 E + P FGGGGGB GGGGG EP. k−4

(3)

In the model mechanism (1), one substrate molecule combines with one enzyme molecule in the same binding site. There are many enzymes which have more than one binding site for substrate molecules [11, 24, 43, 59]. The indirect interaction between distinct and specific binding sites is called allostery, or an allosteric effect, if an enzyme with several binding sites is such that the binding of one substrate molecule at one site affects the activity of binding other substrate molecules at another site. The enzymes that display that effect are known as allosteric enzymes. If a substrate that binds at one site increases the binding activity at another site, then the substrate is called an activator, and it is an inhibitor if it decreases the activity [7, 15, 24, 28, 43, 53, 59]. In a simple case where a dimeric enzyme has two enzyme binding sites, in addition to the scheme (1), the complex ES can also combine with another substrate molecule S to form a dual bound enzyme–substrate complex ESS. This ESS complex breaks down to form the product P and the single bound complex ES. A reaction mechanism for this model includes not only the scheme (1), but also the following scheme: k5 k3 ES + S FGGGGGB GGGGG ESSGGGA ES + P, k−3

(4)

where k3 , k−3 and k5 are the rate constants [24, 25, 43, 59]. Mathematical modeling has proven to be a useful tool to study effect of enzyme inhibition and allostery [23, 27, 30, 35, 41, 42, 56–58, 64]. Very different approaches have been applied for the modeling [19, 38–40, 67]. Actual biosensors with the substrate as well as the product inhibition have been already modeled at various, often steady state, conditions [1, 25, 29, 37, 43, 47, 49]. The amperometric biosensors utilizing the enzyme with only the substrate inhibition has been also modeled at the external diffusion limitation and the steady state [31] as well as the transition conditions [2, 32, 61, 62]. This chapter presents some results of stationary and non-stationary biosensor modeling at mixed enzyme kinetics, the external and the internal diffusion limitations with both kinds of the inhibition, the substrate and the product, and with an allosteric enzyme [31, 32, 50, 61–63].

278

Biosensors Utilizing Non-Michaelis–Menten Kinetics

2 Steady State Modeling of Substrate Inhibition In this section, the action of a biosensor utilizing the substrate inhibition is modeled at mixed enzyme kinetics and external diffusion limitation [31]. The reaction network includes reactions (1) and (2). The dependence of the steady √ state “initial” rate on the substrate concentration shows the maximum at S = KM KI in contrast to the typical Michaelis–Menten kinetics, V (S) = −

dS Vmax S = , dt KM + S + S 2 /KI

(5)

where KI is the constant of ESS dissociation. The biosensor is considered as an infinite nontransparent plate covered by a thin (molecular) layer of an enzyme. The substrate flux to the layer is perpendicular to the surface of the transducer. The thickness of the stagnant (the Nernst) layer covering the enzyme layer is δ. At the steady state conditions the concentration of the substrate on the transducer can be calculated from the equality of the substrate flux and the enzyme conversion on the surface [31], D

Vmax Ss S0 − Ss = . δ KM + Ss + Ss2 /KI

(6)

At the steady state conditions the biosensor response I can be expressed as follows [31]: I = ne F

KM

Vmax Ss . + Ss + Ss2 /KI

(7)

Assuming for simplicity KI = KM and using dimensionless parameters, cb = S0 /KM and cs = Ss /KM , the expression (6) simplifies cb − cs =

ρcs , 1 + cs + cs2

(8)

where ρ is the diffusion module, ρ = Vmax δ/(KM D) [31]. Equation (8) can be solved using the Cardano formula or graphically. For the calculations the enzyme concentration 10−11 mol/cm2 that corresponds to monolayer of the enzyme molecules adsorbed on the geometrically flat surface was used. The values of the other parameters used for the calculations were: the catalytic constant (k2 = 103 1/s) corresponds to the moderately active enzyme, the Michaelis constant (10−5 mol/cm3 ) is typical for many enzymes, the diffusion coefficient of the substrate (10−6 cm2 /s), the thickness of the stagnant layer (0.03 cm). For these parameters the calculated dimensionless parameter ρ equals 30 [31].

2 Steady State Modeling of Substrate Inhibition

279

Y y, = 30 y, = 3

Functions Y, y

0.3

0.2

0.1

0.0 0

1

2

3

4

5

6

7

8

9

10

11

Normalized surface concentration cs Fig. 1 Graphical surface concentration calculation. The parameters of calculations are presented in the text

The solution of (8) at 0 < cb < 9.714 gave a single value of surface concentration that was less in comparison with cb . At cb = 9.714 two values of cs = 0.589 and cs = 4.063 were calculated. At 9.714 < cb < 11.091 three values of cs were found. At cb = 11.091 two values of cs (1.201 and 7.688) were found again. At cb > 11.091 a single value was calculated. To verify the correctness of the calculations a graphical solution of (8) was found. A function Y = cs /(1 + cs + cs2 ) was plotted to show the enzymatic rate, and the function y = (cb − cs )/ρ at fixed cb concentration—to show the diffusion rate. The crossing of these functions gave cs . In Fig. 1, three approximate values of cs = 0.64, cs = 2.8 and cs = 5.5 gave the crossing of the functions at cb = 10 and ρ = 30. These values fitted the calculated cs (0.636, 2.833 and 5.529) [31]. The dependence of the substrate surface concentration on the bulk concentration is shown in Fig. 2. The presented results demonstrate that a surface concentration is less than a bulk concentration. At critical bulk concentrations 9.714 and 11.091, two steady state concentrations are available whereas at the intermediate three concentrations are possible. However, an intermediate concentration is not stable, since any perturbation of parameters produces a low or high concentration [31]. The biosensor response (7) is related to the surface concentration of the substrate. The generation of different surface concentrations produces multi-response of biosensor (Fig. 3). The multi-response can be achieved at 11.091 ≥ cb ≥ 9.714 [31]. The modeling shows that surface multi-concentration is possible at large diffusion parameter (ρ). The decrease of the ρ value up to 3 generates just 1 concentration cs = 9.7 at cb = 10 (Fig. 3) [31]. It is worth noticing that many interfaces used for the enzyme adsorption have a surface being much more than geometrical. Therefore,

280

Biosensors Utilizing Non-Michaelis–Menten Kinetics 11 10

Normalized surface concentration cs

9 8 7 6 5 4 3 2 1 0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

Normalized bulk concentration cb Fig. 2 The dependence of normalized surface concentration cs on normalized bulk concentration cb at ρ = 30. Vertical dot lines mark the zone of multi-steady state concentrations

I, μA/cm2 ( = 30)

350

= 30

300

= 0.3

3

250 2

200 150

1

100

I, μA/cm2 ( = 0.3)

4

400

50 0 0

1

2

3

4

5

6

7

8

9

10

11

12

0 13

Normalized bulk concentration cb Fig. 3 The dependence of the biosensor response on normalized bulk concentration cb of the substrate. The data calculated with ρ = 30 belongs to the left hand y axis, with ρ = 0.3—to the right hand y axis

3 Transient Modeling of Substrate and Product Inhibition

281

the enzyme concentration and consequently ρ may increase in many orders of magnitude. At ρ = 0.3 the biosensor acts in a kinetic regime. The surface concentration of the substrate is little less than the bulk concentration. The response of the biosensor drops down almost 10–100 times, and the decrease of the response in the concentration range 1–12 is associated with the enzyme activity decrease (Fig. 3). From the calculations it follows that all parameters which change the diffusion module (ρ) perturb the multi-steady state zone. The thickness δ of the stagnant layer is the most difficult controllable parameter. Using a rotating disk electrode or the precious flow rate may help to control the thickness of this layer. Multi-steady state surface concentration may have far-reaching consequences for the stability of the biosensors response. It can generate oscillations of the concentration and the response of the biosensor if the negligible perturbation of an enzyme activity or mass transport occurs [31].

3 Transient Modeling of Substrate and Product Inhibition In this section the response of an amperometric biosensor is considered at mixed enzyme kinetics and diffusion limitations in the case of the substrate and the product inhibition when the reaction network includes reactions (1)–(3) [32, 61–63]. The amperometric biosensor is considered as an electrode and a relatively thin layer of an enzyme (enzyme membrane) applied onto the electrode surface. The model involves three regions: the enzyme layer where the enzymatic reaction as well as the mass transport by diffusion takes place, a diffusion limiting region where only the mass transport by diffusion takes place and a convective region where the analyte concentration is maintained constant [32, 61–63].

3.1 Mathematical Model Assuming the symmetrical geometry of the electrode and a homogeneous distribution of the immobilized enzyme in the enzyme membrane, the mathematical model of the biosensor action was defined in a one-dimensional-in-space domain [32, 54, 61–63, 66].

3.1.1 Governing Equations The governing equations for a chemical reaction network can be formulated by the law of mass action [3, 6, 9, 24]. Coupling the enzyme-catalysed reactions (1)–(3) in the enzyme layer with the one-dimensional-in-space diffusion, described by Fick

282

Biosensors Utilizing Non-Michaelis–Menten Kinetics

law leads to the following equations of the reaction–diffusion type (t > 0): ∂ 2 Se ∂Se = DSe − k1 ESe + k−1 Es − k3 Es Se + k−3 Ess , ∂t ∂x 2

(9a)

∂ 2 Pe + k2 Es − k4 EPe + k−4 Ep , ∂x 2

(9b)

= −k1 ESe + k−1 Es + k2 Es − k4 EPe + k−4 Ep ,

(9c)

= k1 ESe − k−1 Es − k2 Es − k3 Es Se + k−3 Ess ,

(9d)

= k3 Es Se − k−3 Ess ,

(9e)

∂Pe ∂t ∂E ∂t ∂Es ∂t ∂Ess ∂t ∂Ep ∂t

= DPe

= k4 EPe − k−4 Ep ,

0 0.5KM ) the current becomes a non-monotone function of time. In the case of S0 = KM , the maximal biosensor current is about 22% greater than the steady state current. Additional calculations show that the appearance of the maximal response value at high substrate concentrations is associated with the substrate inhibition [31,

65 60 55 50 45 40 35 30 25 20 15 10 5 0 0.0

0.5

0.6

0.4

0.8

1.0 0.3 0.2 0.1

0.5

1.0

1.5

2.0

2.5

3.0

t, s Fig. 4 The evolution of the density i of the biosensor current at different values of the substrate concentration S0 simulated at Vmax = 300 µM/s, Ks = 0.1KM , Kp = 0.1KM . The numbers on the curves show values of S0 /KM . The other parameters are as defined in (24)

3 Transient Modeling of Substrate and Product Inhibition

287

32, 62, 63]. No non-monotony in the behaviour of the response of amperometric biosensors is usually observed in the absence of the inhibition [14, 24, 29, 32, 54]. The appearance of the maximal response at the non-steady conditions is associated with the change of the substrate concentration in the enzyme layer that agrees with the dependence of the enzyme rate on the substrate concentration. The steady state response is determined by the stationary substrate concentration and the enzymatic rate. This response is less than the maximal value due to the inhibition of the enzyme activity with the substrate [31, 32, 62]. From the curves depicted in Fig. 4 one can also observe another important relationship: at low substrate concentrations (S0 ≤ 0.5KM ) the density I of the steady state current increases with increasing the substrate concentration S0 , while at higher concentrations I decreases with increasing S0 . This effect is more thoroughly discussed below.

3.4 Dimensionless Model In order to identify the main governing parameters of the mathematical model, the dimensionless mathematical model was derived [62]. For simplicity, the concentrations S and P of the substrate and the product were defined for the entire domain x ∈ [0, d + δ] as follows (t ≥ 0):  S=  P =

Se , Sb ,

0≤x ≤d, d < x ≤ d +δ,

(25a)

Pe , Pb ,

0≤x ≤d, d < x ≤d +δ.

(25b)

Both concentration functions (S and P ) are continuous in the entire domain x ∈ [0, d + δ]. The following dimensionless parameters were introduced: xˆ =

x , d

Ks Kˆ s = , KM DSb , Dˆ Sb = DSe

tˆ =

tDSe , d2

Kˆ p =

δˆ =

Kp , KM

DPe Dˆ Pe = , DSe

δ , d

S Sˆ = , KM

iˆ =

id , ne F DPe KM

DPb Dˆ Pb = , DSe

P Pˆ = , KM

σ2 =

(26) Vmax d 2 , DSe KM

where xˆ is the dimensionless distance from the electrode surface, tˆ stands for the dimensionless time, δˆ is the dimensionless thickness of the diffusion layer, Sˆ and Pˆ are the dimensionless (normalized) concentrations, Kˆ s and Kˆ p are the dimensionless (normalized) inhibition rates, and Dˆ Sb , Dˆ Pe , Dˆ Pb are the dimensionless

288

Biosensors Utilizing Non-Michaelis–Menten Kinetics

diffusion coefficients. σ 2 is the dimensionless diffusion module [36, 43, 51, 53, 54]. The dimensionless thickness of enzyme layer equals one. The governing equations (18) in dimensionless coordinates were expressed as follows (tˆ > 0): ∂ Sˆ ∂ 2 Sˆ ˆ Pˆ ), − Vˆ (S, = ∂ xˆ 2 ∂ tˆ ∂ Pˆ ∂ 2 Pˆ ˆ Pˆ ), = Dˆ Pe 2 + Vˆ (S, ∂ xˆ ∂ tˆ

(27a) 0 < xˆ < 1,

(27b)

ˆ Pˆ ) is the dimensionless quasi-steady state reaction rate, where Vˆ (S, ˆ Pˆ ) = σ 2 Vˆ (S,

Sˆ 1 + Pˆ /Kˆ p + Sˆ + Sˆ 2 /Kˆ s

.

(28)

The governing equations (10) take the following form (tˆ > 0): ∂ Sˆ ∂ 2 Sˆ = Dˆ Sb 2 , ∂ xˆ ∂ tˆ ∂ Pˆ ∂ 2 Pˆ = Dˆ Pb 2 , ∂ xˆ ∂ tˆ

(29a) ˆ 1 < xˆ < 1 + δ.

(29b)

The initial (11), boundary (12)–(13) and matching (14) conditions take the following form: ˆ x, S( ˆ 0) = 0,

Pˆ (x, ˆ 0) = 0,

ˆ 0 ≤ xˆ < 1 + δ,

ˆ 0) = 0, ˆ + δ, ˆ 0) = Sˆ0 , Pˆ (1 + δ, S(1  ∂ Sˆ  ˆ + δ, ˆ tˆ) = S0 , Pˆ (0, tˆ) = 0, Pˆ (1 + δ, ˆ tˆ) = 0, = 0, S(1 ∂ xˆ x=0 ˆ     ˆ ˆ ˆ ∂ Sˆ  ˆ Sb ∂ S  , Dˆ Pe ∂ P  ˆ Pb ∂ P  . = D = D ∂ xˆ x=1 ∂ xˆ x=1 ∂ xˆ x=1 ∂ xˆ x=1 ˆ ˆ ˆ ˆ

(30a) (30b) (30c) (30d)

Assuming the same diffusivities for both species, the substrate and the product, ˆ the dimensionless model (27)–(30) contains only six following parameters: δ— ˆ the thickness of the diffusion layer, S0 —the substrate concentration in the bulk, Kˆ s —the substrate inhibition constant, Kˆ p —the product inhibition constant, σ 2 — the diffusion module, and Dˆ Sb = DPb /DPe = Dˆ Pb /Dˆ Pe —the ratio of the diffusivity in the diffusion layer to the diffusivity in the enzyme layer [62].

3 Transient Modeling of Substrate and Product Inhibition

289

3.5 Effect of Substrate Inhibition The dependence of the maximal as well as the steady state biosensor current on the substrate concentration was investigated at different normalized rates of the substrate inhibition Kˆ s : 0.01, 0.1 and 1 [62]. To see the effect of only the substrate inhibition, the response was also simulated in the absence of the inhibition (Kˆ s → ∞). These responses were simulated at mixed enzyme kinetics and external diffusion limitations (σ = 1) as the responses depicted in Fig. 4 but assuming no product inhibition (Kˆ p → ∞). In the case of only substrate inhibition, the reaction scheme (1)–(3) reduces to (1)–(2). Figure 5 shows calculated the maximal Iˆmax and the steady state Iˆ dimensionless biosensor currents versus the dimensionless substrate concentration Sˆ0 , while Fig. 6 presents the dimensionless biosensor sensitivities BS and Bmax calculated at the same values of the model parameters [62]. One can see in Fig. 5 very sharp response changes which occur at moderate substrate concentrations Sˆ0 , when due to the substrate inhibition the steady state and the maximal currents sharply diverge. At relatively low concentrations of the substrate (Sˆ < 0.1), the maximal and the steady state currents are approximately the same, while at higher concentrations the substrate inhibition leads to diverge the output currents Iˆ and Iˆmax [62]. The effect of the divergence of the maximal and the steady state biosensor currents has been explained by a multi-concentration generation [32, 62]. This was

10-2

Î, Îmax

10-3

10-4

1 2 3 4 5

10-5

10-6 0.01

0.1

6 7 1

10

100

0

Fig. 5 The dimensionless steady state Iˆ (1, 2, 4, 6) and maximal Iˆmax (3, 5, 7) currents vs. the normalized substrate concentration Sˆ0 at different rates of the substrate inhibition Kˆ s : ∞ (no inhibition) (1), 1 (2, 3), 0.1 (4, 5) and 0.01 (6, 7), assuming no product inhibition (Kˆ p → ∞). All the other parameters are the same as in Fig. 4

290

Biosensors Utilizing Non-Michaelis–Menten Kinetics

1.0 0.8 0.6

BS , Bmax

0.4 0.2 0.0

-0.2 -0.4 -0.6 -0.8 -1.0 0.01

1 2 3 4 5 6 7 0.1

1

10

100

0

Fig. 6 The dimensionless biosensor sensitivities BS (1, 2, 4, 6) and Bmax (3, 5, 7) vs. the normalized substrate concentration Sˆ 0 . All the other parameters and notations are the same as in Fig. 5

also confirmed in Sect. 2 by the analytical solution of a simplified model with the external diffusion limitation at the steady state conditions [31]. As can see in Fig. 6, the sensitivity of the biosensors with the substrate inhibition can be even negative. A negative biosensor sensitivity means that the maximal (in the case of Bmax ) or the steady state (in the case of BS ) current decreases with an increase in the substrate concentration S0 (Sˆ0 ). In the case of the notable substrate inhibition, the sensitivity BS decreases monotonously from 1 down to −1, while the sensitivity Bmax is a non-monotone function of the substrate concentration. Bmax (curves 3, 5 and 7) finally approaches zero like BS in the case of no inhibition (curve 1). Because of this, the biosensors acting under the substrate inhibition and measuring only the maximal current are practically inapplicable to the prediction of the substrate concentrations higher than about 10KM (Sˆ0 > 10) [62]. In the case of no product inhibition (Kˆ p → ∞) and at low substrate concentrations (Sˆ0  1) the dimensionless reaction rate Vˆ , introduced in (28), reduces to ˆ + Sˆ 2 /Kˆ s ). Furthermore, additionally assuming a relatively low substrate σ 2 S/(1 inhibition (Sˆ02 < Kˆ s ), the rate Vˆ reduces to a linear function of the substrate ˆ i.e., to the rate when no enzyme inhibition occurs. At practical concentration σ 2 S, values on the substrate inhibition rate (Kˆ s ≥ 0.01), this is easily shown in Figs. 5 and 6 [62]. In the opposite case of relatively high substrate concentrations (Sˆ0  1), the ˆ Kˆ s ) = σ 2 Kˆ s /(Kˆ s + S). ˆ Furtherdimensionless reaction rate Vˆ reduces to σ 2 /(1 + S/ more, additionally assuming a relatively high substrate inhibition (Kˆ s  Sˆ0 ), the

3 Transient Modeling of Substrate and Product Inhibition

291

rate Vˆ reduces to an inversely proportional function of the substrate concentration ˆ At such values of the model parameters, a linearly decreasing dependence σ 2 Kˆ s /S. of the biosensor response was observed as shown in Fig. 5 [62]. The biosensors acting under the substrate inhibition and supporting both, the steady state and the maximal output currents, could be applied to predict the substrate concentration in the range where Bs is noticeably negative. Such an intelligent biosensor should support two linear calibration curves, one (increasing) for relatively low substrate concentrations when Iˆmax ≈ Iˆ and then the response is directly proportional to the substrate concentration, and the other calibration curve (decreasing) for high concentrations when Iˆmax > Iˆ. Two linear calibration curves are clearly shown in Fig. 5 for high substrate inhibition (Ks = 0.001, curve 6) [62]. The simulation results depicted in Figs. 4, 5 and 6 were obtained at mixed enzyme kinetics and external diffusion limitation (σ 2 = 1). To investigate the effect of the substrate inhibition the biosensor responses were also simulated for very different values of the diffusion module σ 2 when the limitation of the biosensor action changes from the enzyme kinetics limitation (σ 2  1) to the internal diffusion limitation (σ 2  1). Figure 7 shows dependence of the steady state (I ) and the maximal (Imax ) output currents on the diffusion module σ at a moderate substrate concentration (Sˆ0 = 1, S0 = KM ). The other parameters were the same as in simulations shown in Fig. 5 [62]. One can see in Fig. 7 noticeable effect of the substrate inhibition when the biosensor acts under enzyme kinetics control (σ 2 < 1). The effect decreases with further increase of the diffusion module σ 2 . No noticeable effect on either steady 10-2

Î, Îmax

10-3

1 2 3 4 5 6 7

10-4

10-5

10-6 0.01

0.1

1

10

100

2

Fig. 7 The steady state Iˆ (1, 2, 4, 6) and the maximal Iˆmax (3, 5, 7) biosensor currents vs. the diffusion module σ 2 at the dimensionless substrate concentration Sˆ0 = 1. The other parameters and notations are the same as in Fig. 5

292

Biosensors Utilizing Non-Michaelis–Menten Kinetics

state or maximal current was observed when the biosensor is significantly under diffusion control (σ 2  1). On the other hand, the substrate inhibition can extend more than tenfold a linear part of the calibration curve. However, the inhibition leads to a noticeable decrease in the output current. The disadvantage of a decrease in the response can be reduced by using the maximal current instead of steady state one [62]. The complex nature of biosensors utilizing the substrate inhibition involves consideration of the simultaneous optimization of several conflicting objectives, which means that if it is desired to improve one of them, it must allow others to get worse [5, 45, 51] (also see Chapter “Application of Mathematical Modeling to Optimal Design of Biosensors”).

3.6 Effect of Product Inhibition The dependence of the biosensor response on the substrate concentration was investigated at different normalized rates of the product inhibition [62]. The response was simulated at mixed enzyme kinetics and external diffusion limitation (σ = 1) as responses depicted in Figs. 4 and 5, assuming no substrate inhibition (Kˆ s → ∞). Figure 8 presents the calculated dimensionless steady state current Iˆ versus the dimensionless substrate concentration Sˆ0 at four rates (Kˆ p ) of the product inhibition: ∞ (no inhibition), 0.1, 0.01 and 0.001. In all these cases the transient output current monotonously increased with time (Iˆ ≈ Iˆmax ) [62]. 10-2

10-3

Î

1 2 3 4

10-4

10-5 0.01

0.1

1

10

100

0

Fig. 8 The dimensionless steady state current Iˆ vs. the dimensionless substrate concentration Sˆ0 at four rates of the product inhibition Kp : ∞ (no inhibition) (1), 0.1 (2), 0.01 (3) and 0.001 (4), assuming no substrate inhibition Ks → ∞. All the other parameters are the same as in Fig. 4

4 Transient Modeling of Allostery

293

One can see in Fig. 8 noticeably different effect of the product inhibition on the biosensor response from that of the substrate inhibition (Fig. 5). The product inhibition leads to a decrease in the output current, but this decrease is significant only in the case of moderate concentrations of the substrate. On the other hand, due to the product inhibition, a linear part of the calibration curve becomes not so straight but it is noticeably longer than in the case of no inhibition (curve 1) [62]. In the case of only the product inhibition and low substrate concentrations (Sˆ0  ˆ ˆ Kˆ p + 1), the dimensionless reaction rate Vˆ reduces to σ 2 S/(1 + Pˆ /Kˆ p ) = σ 2 Kˆ p S/( Pˆ ). Furthermore, additionally assuming a relatively high inhibition rate (Kˆ p  Sˆ0 ), ˆ i.e. to the the rate Vˆ reduces to a linear function of the substrate concentration σ 2 S, ˆ rate when no enzyme inhibition occurs. As one can see in Fig. 8, at S0 = 0.01 the dimensionless steady state current I is approximately the same for all Kˆ p ≥ Sˆ0 [62]. In the case of high substrate concentrations (Sˆ0  1), the dimensionless ˆ reaction rate Vˆ reduces to σ 2 S/(1 + Pˆ /Kˆ p ) = σ 2 Kˆ p /(Kˆ p + Pˆ ). At especially high concentrations of substrate (Sˆ0 → ∞), the rate Vˆ reduces to a zero order reaction rate σ 2 (see Fig. 8). The product inhibition prolongs the calibration curve though the curve is not strictly linear [62]. Additional simulation of the biosensor response showed that the impact of the product inhibition is notably smaller than the impact of the substrate inhibition [62, 63]. In both opposite cases of the enzyme activity, low and high, no notable effect of the product inhibition on the steady state current was observed at very low as well as at very high substrate concentrations. Due to the product inhibition the steady state current decreases notably only at intermediate concentrations of the substrate [62].

4 Transient Modeling of Allostery In this section the response of an amperometric biosensor is considered at mixed enzyme kinetics and diffusion limitations in the case of the enzyme allostery when the reaction network includes reactions (1) and (4) [6, 15, 21, 24, 28, 43, 50]. The amperometric biosensor is of the same principal structure as the biosensor discussed in the previous section: an electrode and a relatively thin layer of an enzyme (enzyme membrane) applied onto the electrode surface [50].

4.1 Mathematical Model Coupling the enzyme-catalysed reactions (1) and (4) in the layer of an allosteric enzyme with the one-dimensional-in-space diffusion, described by Fick law leads

294

Biosensors Utilizing Non-Michaelis–Menten Kinetics

to the following equations of the reaction–diffusion type (t > 0): ∂ 2 Se ∂Se = DSe − k1 ESe + k−1 Es − k3 Es Se + k−3 Ess , ∂t ∂x 2 ∂Pe ∂t ∂E ∂t ∂Es ∂t ∂Ess ∂t

(31a)

∂ 2 Pe + k2 Es + k5 Ess , ∂x 2

(31b)

= −k1 ESe + k−1 Es + k2 Es ,

(31c)

= k1 ESe − k−1 Es − k2 Es − k3 Es Se + k−3 Ess + k5 Ess ,

(31d)

= k3 Es Se − k−3 Ess − k5 Ess ,

(31e)

= DPe

0 0), ∂ 2 Sˆe ∂ Sˆe = − Vˆa (Sˆe ), ∂ xˆ 2 ∂ tˆ ∂ Pˆe ∂ 2 Pˆe = Dˆ Pe + Vˆa (Sˆe ), ∂ xˆ 2 ∂ tˆ

(34a) 0 < xˆ < 1,

(34b)

4 Transient Modeling of Allostery

295

where Vˆa (Sˆe ) is the dimensionless quasi-steady state reaction rate, ˆ = σ2 Vˆa (S)

Kˆ a Sˆ + k Sˆ 2 , Kˆ a + Kˆ a Sˆ + Sˆ 2

Ka Kˆ a = , KM

k=

k5 , k2

(35)

where σ 2 is the diffusion module introduced in (26). In the case of k2 → 0 and small substrate concentrations (S0  KM , Sˆ02  Kˆ a ), the reaction rate Va (dimensionless rate Vˆa ) becomes proportional to S 2 (Sˆ 2 ), ˆ ≈ σ 2 k Sˆ 2 /Kˆ a [43]. Va (S) ≈ E0 k5 S 2 /(KM Ka ) = Vmax kS 2 /(KM Ka ), Vˆa (S) The reactions (1) and (4) involve a very simple case of allosteric enzymes when a dimeric enzyme has only two substrate binding sites [24, 25, 43, 59]. Tetrameric enzymes have four substrate binding sites, while some other enzymes have even more binding sites [24, 25, 43, 59]. In such cases the models become very complex and require many constants [11, 21]. Because of this, the general equations that describe cooperative kinetics have been developed [43, 59, 65]. The most popular of these is the Hill equation, an empirical expression that has a form similar to the Michaelis–Menten equation, Va (S) =

Vmax S h , h + Sh K0.5

ˆ = σ2 Vˆa (S)

Sˆ h 1 + Sˆ h

,

Sˆ =

S , K0.5

(36)

where K0.5 represents midpoint of the substrate versus the velocity curve and is not related to the substrate dissociation constant, h is the Hill coefficient [27, 43, 59, 65]. The constant K0.5 is analogous to the Michaelis constant KM . If the value of the Hill coefficient is set to 1 (h = 1), K0.5 will be the same as the KM . The Hill coefficient is a purely empirical measure of the degree of cooperativity [10, 11, 43, 59, 65]. For example, the Hill coefficient for tetrameric haemoglobin is typically around 2.8, although the number of substrate binding sites on the haemoglobin molecule is 4 [7, 27]. The Hill coefficient provides a measure of the cooperativity of substrate binding to the enzyme. If h > 1, the reaction is thought to exhibit positive cooperativity with respect to substrate binding to the enzyme. A value of larger than 1 (h > 1) also suggests that there are more than one substrate binding sites in the enzyme under study. Positive cooperativity refers to the fact that the binding of one substrate facilitates the binding of another substrate to the enzyme. If h = 1, the Hill equation reduces to the Michaelis–Menten equation, and there is no cooperativity with respect to substrate binding to the enzyme. If h < 1, the reaction exhibits negative cooperativity with respect to substrate binding [10, 15, 33, 59].

296

Biosensors Utilizing Non-Michaelis–Menten Kinetics

4.2 Transient Kinetics The governing equations (32) together with the appropriate initial and boundary conditions form together a two-compartment mathematical model as an initial boundary value problem. The action of biosensors utilizing the allostery of an enzyme was simulated applying the Hill equation (36) as an expression of the quasisteady reaction rate. The problem was solved numerically by applying the finite difference technique very similarly to that discussed above [50]. Figure 9 shows the evolution of the density i(t) of biosensor current as defined in (20). The biosensor response was simulated at mean concentration S0 = K0.5 = KM of the substrate using different values of the Hill coefficient h. The maximal enzymatic rate Vmax = 300 µM/s was chosen so that the diffusion module σ 2 equals one, i.e. the biosensor acts at mixed enzyme kinetics and diffusion limitations. One can see in Fig. 9 that increasing the Hill coefficient prolongs the biosensor response time and decreases the steady state current. In the case of positive cooperativity (h > 1), the steady state current is less than that when the biosensor action obeys the Michaelis–Menten kinetics (h = 1). While the negative cooperativity (h < 1) leads to the steady state current greater than that at Michaelis–Menten conditions [10]. 180 160

0.5

1

i, nA/mm2

140

2

120 100

3

80

5

60

10

40 20 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

t, s Fig. 9 The evolution of the density of the biosensor current i at different values of the Hill coefficient h simulated at S0 = K0.5 = KM and Vmax = 300 µM/s. The numbers on the curves show values of h. The other parameters are as defined in (24)

4 Transient Modeling of Allostery

297

4.3 Effect of Cooperativity Cooperative binding is one of the most interesting and complex phenomena involved in control and regulation of biological processes [10]. To investigate the effect of cooperativity (the Hill coefficient) on the biosensor response, the biosensor response was simulated numerically for very different values of the substrate concentration S0 keeping all other parameter-values constant. Figure 10 shows calculated the dimensionless steady state biosensor current Iˆ versus the dimensionless substrate concentration Sˆ0 . The currents were simulated by the same values of the Hill coefficient h as those presented in Fig. 9. ˆ At so low concentrations of the substrate as Sˆ0  1, the reaction term Vˆa (S) defined in (36) reduces to σ 2 Sˆ h , which is an exponentially decreasing function of h. When the substrate concentration is very high in comparison with the constant K0.5 ˆ reduces to σ 2 , i.e. it becomes invariant to the Hill (Sˆ0  1), the reaction term Vˆa (S) coefficient h. These effects of the Hill coefficient h on the biosensor response can be also noticed in Fig. 10. Figure 10 also shows that all saturation curves intersect at a dimensionless substrate concentration Sˆ0 slightly greater than one, Sˆ0 ≈ 1.7. When the internal and especially the external mass transport by diffusion is ignored, the saturation curves intersect at the dimensionless substrate concentration equal to one [7, 10, 18]. Additional numerical simulations showed that the substrate concentration, at which the saturation curves intersect, increases with increasing external diffusion limitation, e.g., when increasing the thickness δ of the external diffusion layer.

10-2 10-3

1 2 3 4 5 6

Î

10-4 10-5 10-6 10-7 10-8 0.01

0.1

1

10

100

0

Fig. 10 The dimensionless steady state current Iˆ vs. the dimensionless substrate concentration Sˆ0 at six values of the Hill coefficient h: 0.5 (1), 1 (2), 2 (3), 3 (4), 5 (5), 10 (6), assuming K0.5 = KM . All the other parameters are the same as in Fig. 9

298

Biosensors Utilizing Non-Michaelis–Menten Kinetics

5 Concluding Remarks The modeling of biosensors utilizing non-Michaelis–Menten kinetics reveals the complex behaviour of the biosensor response. At low substrate concentration, the kinetics of biosensor with enzyme inhibition looks like a simple substrate diffusion. At the substrate concentration comparable to the Michaelis constant KM , the substrate inhibition leads to the non-monotone response change. The numerical simulations show that the substrate inhibition produces different calibration graphs for the biosensor response at the transition and the steady state. At low substrate concentrations the steady state current as well as the maximal current can be equally used to predict the substrate concentration independent of the rate of the substrate as well as the product inhibition. In the case of the substrate inhibition, knowing both biosensor currents, the steady state and the maximal, can be applied to significantly prolong the biosensor calibration curve. The effect of the product inhibition on the biosensor response and sensitivity is noticeably less in comparison to that of the substrate inhibition. In the case of positive cooperativity (the Hill coefficient greater than 1), the steady state current is less than that when the biosensor action obeys the Michaelis– Menten kinetics. The negative cooperativity (the Hill coefficient less than 1) leads to the steady state current greater than that at Michaelis–Menten conditions. The substrate concentration, at which the saturation curves intersect, increases with increasing external diffusion limitation.

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Biosensors Based on Microreactors

Contents 1 2

3

4

5

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biosensor Based on Heterogeneous Microreactor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Structure of Modeling Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Numerical Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Effect of the Tortuosity of the Microreactor Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Effect of the Porosity of the Microreactor Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biosensor Based on Array of Microreactors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Principal Structure of Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Numerical Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Effect of the Electrode Coverage with Enzyme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plate–Gap Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Principal Structure of Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Effect of the Gaps Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

304 305 305 307 313 314 316 319 319 321 324 326 330 330 331 338 338

Abstract The reduction of the device size, reagents and sample consumption is among the most important advantages of miniaturization of analytical systems. An integration of the systems with enzymatic microreactors proved to be a very suitable approach to the biosensor miniaturization. In this chapter, three types of amperometric biosensors are mathematically and numerically modeled in a twodimensional space at transient conditions. The biosensing systems are modeled by reaction–diffusion equations containing a nonlinear term related to the Michaelis– Menten kinetics of an enzymatic reaction. A biosensor based on a carbon paste electrode encrusted with a single microreactor is modeled by a two-compartment model. The constructed biosensor explores an idea to separate the enzyme and the electron transfer components in a microreactor, the silica particle, and use the wellestablished carbon paste electrode. Then, a biosensing system based on an array of enzyme microreactors immobilized on a single electrode is modeled. Carbon paste porous electrodes are also modeled and investigated by applying a plate–gap model.

© Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_10

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Using the numerical simulation, the influence of the geometry of the microreactors as well as of the diffusion region on the output current and sensitivity is investigated. Keywords Amperometric biosensor · Micro-reactors array · Two-dimensional space · Steady state/transient conditions · Numerical modeling

1 Introduction A miniaturization of analytical systems is regarded as a trend in the development of the modern biosensors [44, 66, 76, 79, 80, 84]. The reduction of the device size, reagents and sample consumption is among the most important advantages. An integration of analytical systems with enzymatic microreactors proved to be a very suitable approach to the biosensor miniaturization [25, 57, 68, 69, 82, 85, 96]. In amperometric enzyme-based biosensing, microreactors have also been introduced as beneficial tools for increasing the sensitivity of the biosensors [56, 79, 83, 86, 95]. A microreactor is a miniaturized reaction unit that in a sensing system can act as a concentrator for the measurable products such as hydrogen peroxide [81]. A remarkable variety of the carbon paste electrodes (CPEs) belonging to a special group of heterogeneous electrodes has been used for the construction of biosensors [24, 29, 35, 58, 64]. A carbon paste (CP) consists of a mixture of graphite powder and an organic binder, e.g. mineral oil, which is immiscible with water. The electrodes prepared from the CP show exceptionally low background current, a wide operating potential window, a convenient modification, renewability, miniaturization and low cost [45, 90]. Because of these fascinating properties, the CPEs are currently in extensive use in electroanalysis [20, 59, 87]. A number of biosensors based on the CPE have been constructed for determination of glucose [2, 3, 33, 40, 42, 46, 47, 49, 52, 54, 60–62, 67, 71, 73, 89, 93]. A working electrode is often considered as a complex device consisting of the conducting electrode (metal, carbon or carbon paste) coated with a biochemical film [24, 27, 32, 74]. Such definitions suggest a planar structure of a working electrode widely investigated in mathematical models of biosensors [31, 75]. These approaches to the electrodes omit a specific, but widely used in practice, class of non-planar biosensors that are based on a bulk modification of entire electrode material, e.g. enzyme modified porous carbon electrodes [29]. The dimensionality of such electrodes is seldom taken into account [1, 7, 12, 37, 39, 77]. When immobilized enzyme is attached to an impermeable solid support, the substrate is carried to the active sites of catalyst through the external diffusion layer. However, the enzyme is rather often entrapped within a porous ceramic or silica particles [36, 45, 56]. In such cases, the substrate must also diffuse through the porous media to reach the enzyme [43]. Thus, intraparticle diffusion resistance as well as external mass transfer resistance must be considered [7, 15, 16]. When modeling microbioreactors where the intraparticle and external diffusion resistances are considered, multi-compartment models are required to achieve a sufficient accuracy of the model [13, 15, 75, 85]. Nevertheless, mono-compartment

2 Biosensor Based on Heterogeneous Microreactor

305

models, in which the internal mass transport by diffusion and substrate conversion is considered, are still used in different applications due to the model simplicity [4]. Furthermore, the substrate conversion is often studied only in the case where the enzyme kinetics approaches either first or zero order kinetics [43, 70]. In this chapter, three types of amperometric biosensors are mathematically and numerically modeled in a two-dimensional space at transient conditions. The biosensing systems are modeled by reaction–diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetics of an enzymatic reaction [14, 19]. Firstly, a biosensor based on a carbon paste electrode encrusted with a single microreactor (MR) is modeled by a two-compartment model [6, 7]. The constructed biosensor explores an idea to separate the enzyme (glucose oxidase) and the electron transfer components in a microreactor, the silica particle, and use the wellestablished carbon paste electrode [45]. Then, a biosensing system based on an array of enzyme microreactors immobilized on a single electrode is modeled [11, 13]. The enzyme microreactors are modeled by identical particles (right cylinders) and by strips (right longitudinal quadrangular prisms) distributed uniformly on the electrode surface. And finally, carbon paste porous electrodes are modeled and investigated by applying a plate–gap model [12, 37–39]. The ultimate goal of the modeling of plate– gap biosensors was verification of the proposed model by comparison of theoretical and experimental responses for biosensors based on the PQQ-dependent glucose dehydrogenases [39, 51]. Using the numerical simulation, the influence of the geometry of the microreactors as well as of the diffusion region on the biosensor current and sensitivity is investigated.

2 Biosensor Based on Heterogeneous Microreactor This section considers an amperometric biosensor based on a carbon paste electrode encrusted with a single microreactor (MR). The constructed biosensor explores an idea to separate the enzyme and the electron transfer components in the microreactor, the silica particle, and use the well-established carbon paste electrode [45]. The microreactor contained an enzyme (glucose oxidase), a mediator and an electron acceptor together with a polymer. A biosensor was assembled by encrusting the CPE with a single MR. The biosensor showed a linear dynamic response up to 50 mM of glucose, a high stability and selectivity [45].

2.1 Structure of Modeling Biosensor The MR was prepared by loading CPC-silica carrier (CPC) with the glucose oxidase (GO), mediator (M) and acceptor (A). The volume of the MR was about 0.16 mm3 .

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Biosensors Based on Microreactors Matrix of CPC-silica carrier (CPC)

Enzyme (GO) loaded within CPC Pores in CPC Graphite

Paraffin oil

Fig. 1 The principal structure of a biosensor based on CPE and microreactor. The figure is not to scale. The diameter of the microreactor is 0.67 mm and the carbon paste electrode is 2.3 mm

The MR was placed onto a freshly cut CPE and pressed into the carbon paste [45]. The principal structure of the biosensor is shown in Fig. 1. The biosensor operation starts after placing the buffer solution (50 mm3 ) of glucose onto the surface of the electrode. During the biosensor action, the glucose (substrate) diffuses into the microreactor (MR), where the glucose reacts with the GO and the reduced GO is formed. Then, the reduced GO acts with the oxidized mediator (M+) followed by the mediator reoxidation with the acceptor (A) [45], β-D-Glucose + GOox −→ δ-Glucolactone + GOred ,

(1a)

GOred + 2 M+ −→ GOox + 2 M + 2 H+ ,

(1b)

M + A −→ M+ + A− ,

(1c)

where GOox and GOred are oxidized and reduced glucose oxidase, respectively. Due to the large excess of the acceptor used and the fast mediator reoxidation, the simplified scheme of the biosensor action includes the glucose oxidase-catalysed glucose oxidation as well as the reaction of the reduced glucose oxidase with the mediator, the concentration of which does not change due to larger excess of the acceptor, β-D-Glucose + 2 A −→ δ-Glucolactone + 2 A− .

(2)

The reduced acceptor (A− ) diffuses out from the MR and is oxidized on the carbon paste electrode (CPE) producing the current, A− −→ A + e.

(3)

2 Biosensor Based on Heterogeneous Microreactor

307

Following this scheme, the biosensor current is a function of the concentration gradient of the acceptor on the CPE. In mathematical modeling of the biosensor, it was assumed that the GO, the mediator and the acceptor are homogeneously distributed within the MR. The modeling of the heterogeneous enzymatic process is associated with the solving diffusion equations containing a nonlinear term related to the enzymatic reaction. In the simplest case, this term is expressed by the Michaelis–Menten function. In terms of the substrate (S) and the reaction product (P), the simplified scheme of the biosensor action is expressed as GO

S −→ P,

(4)

where the substrate (S) binds to the enzyme (GO) and is converted to the product (P).

2.2 Mathematical Model The biosensor action includes a heterogeneous enzymatic process and the diffusion [7, 15, 16]. The mathematical model involves a system of nonlinear differential equations with the inclusion of the enzymatic reaction and the diffusion of glucose (substrate) and acceptor (product). The model to be considered consists of three regions: the enzyme (GO) region where the enzyme reaction as well as the mass transport by diffusion takes place, the impermeable CPC-silica carrier (CPC) and the diffusion limiting region where only the mass transport by diffusion takes place. Let b be the open region of the bulk solution containing some substrate and mr the open region of the entire microreactor. Since the microreactor was constructed from the CPC and was loaded with GO, the region mr of the entire MR consists of two subregions: cpc —the CPC and go —the GO, mr = cpc ∪ go (see Fig. 1). ¯ denote a closure of the corresponding open region  and  denote a boundary Let  ¯ =  ∪ ,  =  ¯ \ . of the corresponding domain , 

2.2.1 Governing Equations The biosensor operation includes the heterogeneous enzymatic process (reaction) and the diffusion. The stimulus of the reaction is the MR, but the reaction performs only in the domain go of the MR which was filled with the GO. Assuming the homogeneous distribution of the GO in the CPC pores, the mass transport and the reaction kinetics in the enzyme region go can be described by the

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following system of the reaction–diffusion equations: Vmax Sgo ∂Sgo = DSgo Sgo − , ∂t KM + Sgo Vmax Sgo ∂Pgo = DPgo Pgo + , ∂t KM + Sgo

(5) x ∈ go ,

t > 0,

where the vector x stands for space, t is time,  is the Laplace operator, Sgo (x, t) and Pgo (x, t) are the concentrations of the substrate and the reaction product, respectively, in the region loaded with the GO, DSgo and DPgo are the diffusion coefficients, Vmax is the maximal enzymatic rate and KM is the Michaelis constant. However, due to the technology of the MR preparation, the number of cells loaded with the GO is very large, and the geometrical shape of the cells cannot be precisely defined. Nevertheless, it was ascertained that the average size of a cell was much less than the size of the entire MR. Because of the inaccurate geometry of the domain go , there is no hope of solving equations (5) neither analytically nor numerically. The governing equations (5) were reduced by applying the homogenization process [5]. Let V () denote the volume of the domain  and γ the volume fraction of the GO in the MR, i.e. γ = V (go ) / V (mr ). The fraction γ can also be regarded as the porosity of the MR. It was easy to calculate the porosity γ experimentally. Assuming the MR as a periodic media, the governing equations (5) were reformulated for the entire MR [5, 92], ∂Smr Vmax Smr = DSmr Smr − γ , ∂t KM + Smr Vmax Smr ∂Pmr , = DPmr Pmr + γ ∂t KM + Smr

(6) x ∈ mr ,

t > 0,

where Smr (x, t) and Pmr (x, t) are the averaged concentrations of the substrate and the reaction product in the MR, respectively, DSmr and DPmr are the effective diffusion coefficients averaged for the entire MR and γ is the porosity of the MR. Outside the MR, only the mass transport by diffusion of the substrate and the product takes place. Due to restricted amount of the buffer solution placed onto the surface of the electrode, the external mass transport obeys a finite diffusion regime ∂Sb = DSb Sb , ∂t ∂Pb = DPb Pb , ∂t

(7) x ∈ b ,

t > 0,

where Sb (x, t) and Pb (x, t) are the concentrations of the substrate and the reaction product in the buffer solution, respectively.

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309

Fig. 2 The profile of the modeling homogenized biosensor based on the CPE and the MR. The figure is not to scale

Due to the symmetry of the homogenized biosensor, only a half of the crosssection of the biosensor can be considered. Figure 2 shows the profile of the homogenized biosensor in the Descartes as well as in the spherical coordinates. In Fig. 2, mr denotes the MR, b represents the buffer solution, r1 is the radius of the homogenized MR, r2 is the radius of the entire surface of the CPE and r3 is the radius of the biosensor (outer boundary of the bulk solution), mr = (0, r1 ) × (0, π),

b = (r1 , r3 ) × (0, π/2),

mr = {r1 } × [0, π/2],

b = {r3 } × [0, π/2],

cpe,mr = {r1 } × [π/2, π],

cpe,b = [r1 , r2 ] × {π/2},

(8)

p = [r2 , r3 ] × {π/2}, where mr corresponds to the interface between the MR and the bulk solution, cpe,mr is the interface between the CPE and the MR, cpe,b is the interface between the CPE and the bulk, cpe,mr ∪ cpe,b corresponds to the entire surface of the CPE, p corresponds to the surface of the plate and b is the outer boundary of the bulk solution. In the governing equations (6) and (7) adjusted for the two-dimensional domains (8), the substrate and the product concentrations are functions of r, θ and t, Smr (r, θ, t), Pmr (r, θ, t), Sb (r, θ, t) and Pb (r, θ, t). Let us remind that the Laplace operator  in two dimensions in spherical coordinates r and θ is described as follows [21]: U =

1 ∂ r 2 ∂r

  ∂ ∂U 1 ∂U r2 + 2 sin θ . ∂r r sin θ ∂θ ∂θ

(9)

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Biosensors Based on Microreactors

2.2.2 Effective Diffusion Coefficient The values of the effective diffusion coefficients DSmr and DPmr in the MR depend on the corresponding diffusion coefficients in the GO, the porosity γ of the MR and the geometry of pores [5, 92]. The diffusion coefficients DSmr and DPmr of the substrate and the product in the MR are lower than those in the loaded GO [15, 16]. In the case when the model material is a two-phase composite, the effective diffusion coefficient Deff in a periodic media usually satisfies the following condition [5]: D1 + D2 2D1 D2 , ≤ Deff ≤ D1 + D2 2

(10)

where Di is the diffusion coefficient of the species in the phase i, i = 1, 2. When one of the aggregates is impermeable (D2 = 0), this estimation reduces to 0 ≤ Deff ≤

D1 . 2

(11)

There are several models to evaluate the effective diffusion coefficient more accurately for a porous material. The coefficient Deff is considered as a function of D1 , D2 and the porosity γ [23, 34] and is estimated by D1 D2 ≤ Deff ≤ γ D1 + (1 − γ )D2 . γ D2 + (1 − γ )D1

(12)

When the constituent is impermeable (D2 = 0), Deff is estimated by 0 ≤ Deff ≤ γ D1 .

(13)

Assuming a regular spherical matrix configuration of a porous material (D2 = 0) [28, 41, 94], the value of Deff is given by Deff = D1

2D1 γ + D2 (3 − 2γ ) . D1 (3 − γ ) + D2 γ

(14)

When D2 = 0, Eq. (14) reduces to the following estimation of Deff : Deff = D1 γ

2 . 3−γ

(15)

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311

In a more common case of an irregular matrix of a porous material, the estimation (15) is replaced by Deff = D1 γβ,

(16)

where β is the tortuosity of the pore space [23, 28]. Although the geometry of pores in the MR was really irregular (Fig. 1), assuming zero diffusivities of the substrate and the product in the CPC and the regular matrix of the CPC, from (15), an estimation for DSmr and DPmr is obtained, DSmr = DSgo γβ, β=

DPmr = DPgo γβ, (17)

2 , 3−γ

where β corresponds to the tortuosity (irregularity) of the CPC-matrix. A very similar approach to the effective diffusion coefficient was applied in the modeling of glucose diffusion through an isolated pancreatic islet of the Langerhans [17].

2.2.3 Initial and Boundary Conditions The biosensor operation starts after placing the buffer solution of the substrate (glucose) onto the surface of the electrode and the MR. Let x = 0 represent the electrode surface. The simulation of the biosensor operation starts when some substrate appears in the bulk solution (t = 0), Smr (r, θ, 0) = 0,

¯ mr \ mr , (r, θ ) ∈ 

Smr (r, θ, 0) = S0 ,

(r, θ ) ∈ mr ,

Sb (r, θ, 0) = S0 ,

¯ b, (r, θ ) ∈ 

Pmr (r, θ, 0) = 0,

¯ mr , (r, θ ) ∈ 

Pb (r, θ, 0) = 0,

¯ b, (r, θ ) ∈ 

(18)

where S0 is the concentration of the substrate in the bulk solution. Assuming the amperometry, the substrate is an electro-inactive substance, while the product is an electro-active substance. The electrode potential is chosen to keep

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zero concentration of the product at the electrode surface (t > 0), Pmr (r, θ, t) = 0, ∂Smr  DSmr = 0,  ∂r cpe,mr Pb (r, θ, t) = 0, ∂Sb  DSb = 0.  ∂θ cpe,b

(r, θ ) ∈ cpe,mr ,

(r, θ ) ∈ cpe,b ,

(19)

Due to the symmetry of the biosensor, the non-leakage boundary conditions on the borders θ = 0 and θ = π are defined, ∂Smr   ∂θ θ=0 ∂Pmr  DPmr  ∂θ θ=0 ∂Sb  DSb  ∂θ θ=0 ∂Pb  DPb  ∂θ θ=0 DSmr

∂Smr  = 0, r ∈ [0, r1 ],  ∂θ θ=π ∂Pmr  = DPmr = 0, r ∈ [0, r1 ],  ∂θ θ=π = DSmr

= 0,

r ∈ [r1 , r3 ],

= 0,

r ∈ [r1 , r3 ].

(20)

Under the circumstances where both external and internal diffusion gradients are found, the flux of the substrate and the product through the stagnant layer must equal the flux entering the surface of the MR (t > 0),   ∂Smr  ∂Sb    = DSb = Sb  ,   , Smr  mr mr ∂r mr ∂r mr   ∂Pmr  ∂Pb    = DPb = Pb  . DPmr   , Pmr  mr mr ∂r mr ∂r mr DSmr

(21)

The external diffusion layer of the thickness r3 −r1 remains unchanged with time. Due to a limited volume of the bulk solution, at the boundary of the bulk solution, the non-leakage condition for both species, the substrate and the product, is used (t > 0), ∂Sb   = 0, ∂r b ∂Pb  DPb  = 0. ∂r b DSb

(22)

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2.2.4 Biosensor Response The total current i(t) at time t of the amperometric biosensor is proportional to the gradient of the product concentration at the electrode surface cpe,mr ∪ cpe,b , ⎛ !2π !π ∂Pmr  ⎜ r 2 sinθ dθ i(t) = ne F ⎝ DPmr  ∂r r=r1 1 0

π/2

!r2 +

DPb r1

∂Pb  r dr dϕ  ∂θ θ=π/2

⎛

⎜ = 2πne F ⎝r12 DPmr

!π

⎞ !r2   ∂Pmr  ∂Pb  ⎟ sinθ dθ + DPb r dr ⎠ ,   ∂r r=r1 ∂θ θ=π/2 r1

π/2

(23) where ne is the number of electrons, and F is the Faraday constant. The reaction–diffusion process (6), (7), (18)–(22) approaches a steady state as t → ∞. I = lim i(t),

(24)

t →∞

where I is the density of the stationary current.

2.3 Numerical Simulation The initial boundary value problem (6), (7), (18)–(22) can be solved numerically using the finite difference technique [6, 7, 18, 72]. Although the mathematical model has been formulated in a two-dimensional space, the governing, the initial and the boundary conditions are of the same type as in a one-dimensional space and can be approximated very similarly. The following values of the model parameters were employed in the modeling of a practical biosensor [45, 55, 91]: DSb = DPb = 673 µm2/s, r1 = 0.34 mm, γ = 0.5,

DSgo = DPgo = 0.5DSb = 0.5DPb ,

r2 = 1.15 mm,

r3 = 2.29 mm,

(25)

β = 0.8,

KM = 83 mM,

Vmax = 44 mM/s,

S0 = 25 mM,

ne = 2.

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Figure 3 shows the concentrations of the substrate (S) and the product (P ) in the enzyme and diffusion regions at the steady state conditions (at time t = 334 s). The concentrations S and P for the entire domain mr ∪b can be introduced as follows (t ≥ 0): S(r, θ, t) =

P (r, θ, t) =

 Smr (r, θ, t),

(r, θ ) ∈ mr ,

(r, θ ) ∈ b \ mr , Sb (r, θ, t),  Pmr (r, θ, t), (r, θ ) ∈ mr , Pb (r, θ, t),

(26)

(r, θ ) ∈ b \ mr .

Although the dynamics of the concentrations was simulated in the spherical coordinates, Fig. 3 presents the profiles in the Descartes coordinates. The process was simulated in a domain shown in Fig. 2. To have a more comprehensive view, the mirror-image along the z-axis is also shown in Fig. 3, where the point (0, 0) corresponds to the centre of the MR. Only a part of the entire domain is presented in the figure because a very small variation of the concentrations appeared far from the centre of the MR. Since the porosity γ and the tortuosity β are undersell grounded parameters of the model, and the geometry of the CPC-matrix of the MR cannot be defined precisely (Fig. 1), it is important to evaluate the sensitivity of the biosensor response to the changes in these parameters. Figure 4 shows the dynamics of the biosensor current calculated at different values of the porosity γ and the tortuosity β. One can see in Fig. 4 that both parameters, γ and β, defining the internal structure of the CPC-matrix of the MR noticeably effect the stationary current as well as the time moment of occurrence of the stationary current. Below, the effect of the porosity γ as well as of the tortuosity β on the biosensor response is investigated in detail.

2.4 Effect of the Tortuosity of the Microreactor Matrix Since the geometry of the CPC-matrix of the MR cannot be defined precisely (Fig. 1), it is important to know how the tortuosity β effects the biosensor response. The dependence of the steady state biosensor current on the tortuosity β of the CPCmatrix is investigated at different concentrations of the substrate. Because the steady state current is very sensitive to the concentration S0 of the substrate, the current is normalized to evaluate the effect of the tortuosity β on the biosensor response. The normalized steady state biosensor current Iβ is expressed by the steady state current at the tortuosity β divided by the steady state current assuming a unitary tortuosity

2 Biosensor Based on Heterogeneous Microreactor

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S, mM 0.8

25

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20

z, mm

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0 -0.2

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18 16

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0.2

8

0

6 4

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2

-0.4

b)

0 -1

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1

x, mm

Fig. 3 The stationary concentrations of the substrate (a) and the product (b) in the MR and the surrounding region, obtained at t = 334 s. The values of the model parameters are defined in (25)

β of the CPC-matrix, Iβ (S0 , β) =

I (S0 , β) , I (S0 , 1)

0 < β ≤ 1,

(27)

where I (S0 , β) is the density of the steady state current, as defined by (24), calculated at a given substrate concentration S0 and the tortuosity β of the CPCmatrix of the MR. The upper limiting value of the tortuosity (β = 1) corresponds to an unrealistic CPC-matrix containing no curvature. The results of calculations are depicted in Fig. 5. The results were obtained at four values of the substrate

316

Biosensors Based on Microreactors 6.0 5.5 5.0

4

4.5

3

i, μA

4.0

2

3.5 3.0

1

2.5 2.0 1.5 1.0 0.5 0.0 0

50

100

150

200

250

300

t, s Fig. 4 The dynamics of the biosensor current i at two values of the porosity γ : 0.5 (1, 2), 0.8 (3, 4) and two values of tortuosity β: 0.5 (1, 3), 0.8 (2, 4). The other parameters are the same as in this figure

concentration S0 changing the tortuosity β in a wide range and keeping constant values of all other parameters as defined in (25). As one can see in Fig. 5a, the stationary current (Iβ as well as I ) increases with an increase in the tortuosity β. The effect of the tortuosity β on the steady state current practically does not depend on the substrate concentration S0 . All calculated values of Iβ were fitted with a positive allometric function Iβ (β) = β 0.35 .

(28)

The allometric function (28) is presented in Fig. 5a by a solid curve. Figure 5b shows a relatively stable sensitivity BS of the biosensor at high values of the tortuosity, β > 0.5. At lower values of the tortuosity, β < 0.5, the sensitivity notably decreases with a decrease in the tortuosity β. As usual, the biosensor sensitivity is higher at a low substrate concentration S0 rather than at a higher one.

2.5 Effect of the Porosity of the Microreactor Matrix The porosity γ of the MR influences the effective diffusion coefficients and the rate of the enzymatic reaction (see (6) and (17)). To investigate the dependence of the steady state biosensor current on the porosity γ of the MR, the biosensor response was simulated at different values of γ and of the substrate concentration S0 . The

2 Biosensor Based on Heterogeneous Microreactor

317

1.0 0.9 0.8

I

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1 2 3 4

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0.80 0.0

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0.6

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b) Fig. 5 The normalized steady state current Iβ (a) and the sensitivity BS versus the tortuosity β at four substrate concentrations (S0 ): 25 (1), 50 (2), 100 (3) and 200 (4) mM. The values of the other parameters were defined in (25)

normalized stationary current Iγ was expressed as the steady state current calculated at the porosity γ and divided by the stationary current assuming a unitary porosity,

Iγ (S0 , γ ) =

I (S0 , γ ) , I (S0 , 1)

0 < γ ≤ 1,

(29)

where I (S0 , γ ) is the stationary current as defined by (24) calculated at given substrate concentration S0 and the porosity γ of the MR. The upper limiting value

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Biosensors Based on Microreactors

1.0 0.9 0.8 0.7

I

0.6

1 2 3 4

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b) Fig. 6 The normalized steady state current Iγ (a) and the sensitivity BS versus the microreactor porosity γ . The parameters and the notations are the same as in Fig. 5

of the porosity (γ = 1) corresponds to an unrealistic case of the MR containing no CPC-silica carrier, while the lower limiting value (γ = 0) corresponds to another unrealistic case of the MR constructed exclusively from the CPC-silica carrier. The calculated stationary currents as well as the sensitivities at four values of the substrate concentration S0 are depicted in Fig. 6. As one can see in Fig. 6, the effect of the porosity γ on the stationary current (Iβ and I ) as well as on the biosensor sensitivity BS is very similar to that of the tortuosity β shown in Fig. 5. The calculated values of the normalized stationary

3 Biosensor Based on Array of Microreactors

319

current Iγ were fitted with the following allometric function (solid line in Fig. 6a): Iγ (γ ) = β 0.63 .

(30)

3 Biosensor Based on Array of Microreactors This section discusses mathematical modeling of a biosensor system based on an array of enzyme microreactors immobilized on a single electrode [11, 13]. The model involves three regions: an array of microreactors where the enzymatic reaction as well as the mass transport by diffusion takes place, a diffusion limiting region where only the diffusion takes place and a convective region where the analyte concentration is maintained constant. The enzyme microreactors were modeled by identical particles (right cylinders) and by strips (right longitudinal quadrangular prisms) distributed uniformly on the electrode surface. Using the numerical simulation, the influence of the geometry of the microreactors as well as of the diffusion region on the biosensor response has been investigated.

3.1 Principal Structure of Biosensor Two shapes of the enzyme microreactors immobilized on a single electrode are investigated. In the case of the first kind of the biosensor geometry, the microreactors were modeled by identical enzyme-filled right cylinders. Figure 7 shows a biosensor system, where the enzyme cylinders of radius a and height c are arranged in a rigid hexagonal array. The distance between the centres of two adjacent cylinders equals 2b. The mass transport during the biosensor action obeys a finite diffusion regime. The principal structure of the electrode and the profile of the biosensor at the perpendicular plane are depicted in Fig. 8, where d stands for the thickness of the external diffusion layer. Assuming the uniform distribution of the enzyme microreactors on the electrode surface, the biosensor may be divided into equal hexagonal prisms with regular Fig. 7 The principal structure of an array of the enzyme microreactors immobilized on a single electrode. Microreactors are modeled by cylinders. The figure is not to scale

c enzyme a

2b

electrode

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Biosensors Based on Microreactors

d

2b 2a c

a)

b)

Fig. 8 The principal structure of the enzyme electrode (a) and the profile at the perpendicular plane (b)

z d c

a

b

x

Fig. 9 The modeling domain as the profile of a unit cell perpendicular to the electrode plane

c 2b

2a

enzyme

electrode

Fig. 10 The principal structure of an array of enzyme microreactors immobilized on a single electrode. The microreactors are modeled by strips. The figure is not to scale

hexagonal bases. For simplicity, it is reasonable to consider a circle of radius b whose area equals that of the hexagon and regard one of the cylinders as a unit cell. Due to the symmetry of the unit cell, only a half of the transverse section of the unit cell can be considered. The profile of the biosensor perpendicular to the electrode plane is depicted in Fig. 9. A very similar approach has been used in modeling of partially blocked electrodes [8, 22, 30] and in modeling of surface roughness of the enzyme membrane [10]. In the second case of the biosensor geometry, the microreactors were modeled by identical strips distributed uniformly on the electrode surface. Figure 10 shows the principal structure of a biosensor, where enzyme microreactors are right quadrangular prisms of base 2a by c distributed uniformly, so that the distance between adjacent prisms equals 2(b − a). Due to the uniform distribution of the enzyme strips, it is reasonable to consider only a unit consisting of a single strip together with region between two adjacent strips. Because of the symmetry and the

3 Biosensor Based on Array of Microreactors

321

relatively great length of the unit cell, only the transverse section of a half of the unit can be considered. Figure 9 also represents the profile of that kind of microreactors.

3.2 Mathematical Model Consider a scheme where a substrate (S) is enzymatically converted to a product (P), E

S −→ P.

(31)

Two different kinds of the geometry of the enzyme microreactors, the cylinders and the strips, have been discussed. Although the profile at y-plane (Fig. 8) is the same for both kinds of the microreactors, the corresponding mathematical models have to be formulated differently. In the case of the cylinders (Fig. 7), a twodimensional-in-space (2D) model in the cylindrical coordinates is formulated, while in the second case (Fig. 10), a 2D model is formulated in the Cartesian ones. In the profile (Fig. 9), parameter b stands for the half-width (radius) of the entire unit cell, while a is the half-width (radius) of the enzyme microreactor, and c is the height of the microreactor. The fourth parameter d is the thickness of the diffusion layer. The diffusion region surrounding the microreactors is assumed as the Nernst diffusion layer [21, 53, 65, 88]. Away from it, the solution is in motion and uniform in concentration. The thickness d of the diffusion layer remains unchanged with time. Let  and 0 be open regions corresponding to the entire domain to be considered and to the enzyme region, respectively,  0 the boundary between the buffer solution and the enzyme region and  N the upper boundary of the entire cell,  = (0, b) × (0, d), 0 = (0, a) × (0, c),  0 = {a} × [0, c] ∪ [0, a] × {c},

(32)

 N = [0, b] × {d}. Let  and 0 denote the corresponding closed regions.

3.2.1 Governing Equations In the enzyme region, coupling the enzyme-catalysed reaction with the twodimensional-in-space mass transport by diffusion described by Fick’s law leads to

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the system of the reaction–diffusion equations (t > 0), ∂Se Vmax Se , = DSe Se − ∂t KM + Se Vmax Se ∂Pe = DPe Pe + , ∂t KM + Se

(33) (x, z) ∈ 0 ,

where z stands for the distance from the electrode, x is the distance from the z-axis, t is time,  is the Laplace operator, Se (x, z, t) and Pe (x, z, t) are the concentrations of the substrate and the reaction product, respectively, DSe and DPe are the diffusion coefficients, Vmax is the maximal enzymatic rate and KM is the Michaelis constant. The expression of the Laplace operator depends on the system of coordinates. Let us recall that the Laplace operator  in two dimensions in cylindrical coordinates x and z is defined as follows [21]:  1 ∂ ∂U ∂ 2U U = . (34) x + x ∂x ∂x ∂z2 The operator  in the Descartes coordinates x and z is defined by U =

∂ 2U ∂ 2U + . ∂x 2 ∂z2

(35)

Outside the enzyme region, only the mass transport by diffusion of the substrate and the product takes place, ∂Sb = DSb Sb , ∂t ∂Pb = DPb Pb , ∂t

(36) (x, z) ∈  \ 0 ,

where Sb (x, z, t) and Pb (x, z, t) are the concentrations of the substrate and the reaction product, respectively, in the buffer solution.

3.2.2 Initial and Boundary Conditions In the domain presented in Fig. 9, z = 0 represents the electrode surface. The simulation of the biosensor operation starts when some substrate appears in the bulk solution (t = 0), Se (x, z, 0) = 0,

Pe (x, z, 0) = 0,

(x, z) ∈ 0 ,

Sb (x, z, 0) = 0,

Pb (x, z, 0) = 0,

(x, z) ∈  \ (0 ∪  N ),

Sb (x, z, 0) = S0 ,

Pb (x, z, 0) = 0,

(x, z) ∈  N ,

where S0 is the concentration of the substrate to be analysed.

(37)

3 Biosensor Based on Array of Microreactors

323

The following boundary conditions express the symmetry of the biosensor:   ∂Se  ∂Pe  = D = 0, z ∈ [0, c], Pe ∂x x=0 ∂x x=0   ∂Sb  ∂Pb  DSb = DPb = 0, z ∈ [c, d], ∂x x=0 ∂x x=0   ∂Sb  ∂Pb  DSb = DPb = 0, z ∈ [0, d]. ∂x x=b ∂x x=b

DSe

(38)

Assuming the amperometry, the product is an electro-active substance. The electrode potential is chosen to keep zero concentration of the product at the electrode surface. The substrate does not react at the electrode surface. These characteristics are expressed in the boundary conditions (t > 0) given by  ∂Se  DSe = 0, ∂z z=0  ∂Sb  DSb = 0, ∂z z=0

Pe (x, 0, t) = 0, Pb (x, 0, t) = 0,

x ∈ [0, a], (39) x ∈ [a, b].

According to the Nernst approach, away from the diffusion layer, the buffer solution is uniform in concentration,  Sb  = S0 , N

 Pb  = 0. N

(40)

On the boundary  0 , the matching conditions are defined (t > 0)     ∂Se  ∂Sb  DSe = DSb , Se  = Sb  ,   0 0 ∂n  0 ∂n  0     ∂Pe  ∂Pb  DPe = DPb , Pe  = Pb  , 0 0 ∂n  0 ∂n  0

(41)

where n stands for the normal direction. In a very special case when a = b, the model (33), (36), (37)–(41) describes an operation of a mono-layer enzyme biosensor discussed in Chapter “Introduction to Modeling of Biosensors”.

3.2.3 Biosensor Response The generated current of the amperometric biosensor is proportional to the gradient of the product concentration at the electrode surface, i.e. on the border z = 0. Due to the direct proportionality, the total current is normalized with the area of the

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Biosensors Based on Microreactors

electrode surface. In the case of the modeling enzyme reactors by right cylinders, the current density i(t) has to be calculated as follows:   ! b ∂Pe  ∂Pb  DPe x dx + DPb x dx dϕ ∂z z=0 ∂z z=0 0 0 a    ! a ! b 2ne F ∂Pe  ∂Pb  = DPe  x dx + DPb  x dx , b2 0 ∂z z=0 a ∂z z=0

ne F i(t) = πb2

!

2π

!

a

(42)

where ne is the number of electrons involved in a charge transfer, F is the Faraday constant and ϕ is the third cylindrical coordinate. In the case of the Cartesian coordinates, the density i(t) of the biosensor current is expressed as follows: ne F i(t) = b

!

a 0

  ! b ∂Pe  ∂Pb  DPe dx + D dx . P b ∂z z=0 ∂z z=0 a

(43)

The density I of the steady state biosensor current is calculated alike for both kinds of the reactor geometry, I = lim i(t) . t →∞

(44)

3.3 Numerical Simulation The initial boundary value problem (33), (36), (37)–(41) can be successfully solved using the finite difference technique [11, 13]. In the case of a = b, this model describes the operation of a mono-layer enzyme biosensor. This feature of the model can be applied for validation of the mathematical and the numerical models. Accepting a = b, the solution of the problem (33), (36), (37)–(41) must coincide with the corresponding analytical solution known for the two-compartment model of the amperometric biosensor discussed in Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”. Being invariant to both considered systems (the cylindrical and the Descartes) of coordinates is another useful feature of mathematical model (33), (36), (37)– (41), which could be used when evaluating the correctness of the corresponding numerical model. The upper layer of the thickness δ = d − c from the enzyme region was assumed as the Nernst diffusion layer. The thickness δ of the Nernst layer depends upon the nature and stirring of the buffer solution (see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”). In practice, the zero thickness of the Nernst layer cannot be achieved. In the case when the buffer solution to be analysed is stirred by magnetic stirrer, the thickness δ may be minimized up to

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325

20 µm by increasing stirring intensity [48, 88]. That thickness of the Nernst layer, δ = d − c = 20 µm, was used in the simulation of the biosensor action changing some other parameters, DSe = DPe = 300 µm2/s, a = 40 µm, KM = 100 µM,

DSb = DPb = 2DSe = 2DPe ,

b = 100 µm,

c = 100 µm,

Vmax = 100 µM/s,

d = 120 µm,

S0 = 100 µM,

(45)

ne = 2.

Some results of the numerical simulations are depicted in Figs. 11 and 12. The enzyme microreactors were modeled by cylinders (Figs. 11a and 12) and by strips (Fig. 11b). One can see in Fig. 11 that the radius (half-width) a of the enzyme microreactors significantly affects the density i of the biosensor current as well as the response time. The biosensor current was simulated at the following five values of a: 20 µm (curve 1 in Fig. 11), 40 (2), 60 (3), 80 (4) and 100 (5). The case of a = 100 µm = b corresponds to a flat biosensor with continuous enzyme membrane of thickness c. The calculated biosensor current notably differs for different shapes of the microreactors unchanging values of all the model parameters. For example, at a = 20 µm = 0.2b (curves 1 in Fig. 11), the density of the steady state current (I = 7.1 nA/mm2 at t = 41.1 s, Fig. 11a) in the case of the cylinders is about 4.7 times less than the corresponding current calculated in the case of strips (I = 33.3 nA/mm2 at t = 38.3 s, Fig. 11b). Figure 11 also shows a non-monotonic dependence of the stationary current on the base area of microreactors. The next section discusses this effect in detail. Figure 12 shows the concentrations of substrate (S) and the product (P ) in the enzyme and diffusion regions at time t = 34.8 s when the steady state was reached. The enzyme reactors were modeled by right cylinders. The concentration S of the substrate and the concentration P of the reaction product for the entire domain  were introduced as follows (t ≥ 0): S(x, z, t) =

P (x, z, t) =

 Se (x, z, t),

(x, z) ∈ 0 ,

Sb (x, z, t), (x, z) ∈  \ 0 ,  Pe (x, z, t), (x, z) ∈ 0 , Pb (x, z, t),

(46)

(x, z) ∈  \ 0 .

At any time, t ≥ 0, both concentration functions, S(x, z, t) and P (x, z, t), are continuous at all (x, z) ∈ . Although the dynamics of the concentrations was simulated in the spherical coordinates, Fig. 12 presents the profiles in the Descartes coordinates. To have a more comprehensive view, the mirror-image along the z-axis is also shown in Fig. 12, where the point (0, 0) corresponds to the centre of the base of an enzyme cylinder.

Biosensors Based on Microreactors

i, nA/mm2

326 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

i, nA/mm2

a)

b)

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1

0

10

20

30

t, s

70 65 60 55 50 45 40 35 30 25 20 15 10 5 0

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1

0

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20

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Fig. 11 The dynamics of the biosensor current at different values of the half-width a of the microreactors in the cases of cylinders (a) and strips (b), a = 0.2b (1), 0.4b (2), 0.6b (3), 0.8b (4) and b (5). The values of all the other parameters are defined in (45)

3.4 Effect of the Electrode Coverage with Enzyme Figure 11 shows a non-monotonic dependence of the stationary current on the radius (half-width) of the base of microreactors. To investigate that effect in detail, the density I of the steady state current has to be calculated at different values of the radius (half-width) b of the entire unit changing the radius (half-width) a of the microreactor. To compare the response of the biosensor based on an array of the

3 Biosensor Based on Array of Microreactors

327

S , μM 100

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-100

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Fig. 12 The stationary concentrations of the substrate (a) and the product (b) in the enzyme and the surrounding regions, obtained at t = 34.8 s. The values of the model parameters are defined in (45)

microreactors (a < b) with the response of the corresponding flat (membrane) biosensor (a = b), the normalized steady state current is introduced as a function of the degree θ of the electrode surface coverage, Iβ (b, θ ) =

I (b, θ ) , I (b, 1)

0 < θ ≤ 1,

(47)

where Iβ (b, θ ) is the density of the steady state biosensor current calculated as defined in (44) at the radius (half-width) b and the degree θ of the electrode coverage, 0 < θ ≤ 1. The dimensionless degree θ was expressed as the area of

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Biosensors Based on Microreactors

I

1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0

a)

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0.7

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b) Fig. 13 The normalized steady state current Iθ (a) and the sensitivity BS (b) versus the electrode coverage θ, b = 50 (1, 5), 100 (2, 6), 200 (3, 7), 400 (4, 8) µm, c = 100 (1–4), 10 (5–8) µm. The values of the other parameters are defined in (45)

the bases of all enzyme microreactors divided by the area of the whole electrode surface. The case of θ = 1 corresponds to a flat biosensor, i.e. an electrode fully covered by an enzyme mono-layer. In the case when microreactors are modeled by cylinders (Fig. 7), θ = a 2 /b2 , and θ = a/b in the case of strips (Fig. 10) Figure 13 shows the normalized steady state current Iθ (Fig. 13a) and the biosensor sensitivity BS (Fig. 13b) versus the degree θ of the electrode coverage at four values of b. The values of all other parameters were the same as defined in (45). Since in the case of strips, the behaviour of the stationary current versus the degree

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329

θ is very similar to that in the case of cylinders, Fig. 13 shows the calculation results obtained only for the microreactors modeled by right cylinders. The calculations have been performed at four values of b (50 (curves 1 and 5), 100 (2, 6), 200 (3, 7), 400 (4, 8) µm) and two values of c (100 (1–4), 10 (5–8) µm) changing the radius a from 0.05b up to b. The electrode coverage θ changes from 0.0025 to 1 when a increases from 0.05b to b. As one can see in Fig. 13a, in the case of c = 100 µm (curves 1–4), the steady state current is a non-monotonous function of the degree θ of the electrode coverage. At 10 times smaller value of c (curves 5–8), the normalized current Iθ is approximately a linearly increasing function of θ . Additional calculations confirmed this property also for the Descartes coordinates [13]. In the case of c = 100 µm and b = 0.5c (curve 1 in Fig. 13a), the relative difference between the steady state current at θ = 0.5 and another one at θ = 1 exceeds 38%. Let us notice that the volume of enzyme microreactors is directly proportional to the degree θ , when the height c of the microreactors is kept constant. Although the biosensor based on an array of microreactors (θ < 1) is of less enzyme volume than the corresponding flat biosensor (θ = 1), the array biosensor can generate even higher steady state current than the flat one. The variation of the half-width b of the entire unit keeping θ and c constant does not change the volume of the microreactors. Since IN varies with b (Fig. 13a), the biosensor response depends on not only the volume but also the shape of the enzyme microreactors. The smaller value of b corresponds to the denser distribution of the enzyme microreactors on the electrode surface. According to Fig. 13a, the denser the microreactors are distributed, the higher steady state current is generated. Assuming the continuous enzyme layer (a = b, θ = 1) of the thickness c = 100 µm, the diffusion module σ 2 at values (45) is approximately equal to 33.3, i.e. the biosensor response is under diffusion control. While at ten times smaller value (0.01 mm) of c, σ 2 is approximately equal to 0.33, and consequently the enzyme kinetics controls the biosensor response. The steady state biosensor current is a notably non-monotonous function of the degree θ of the electrode coverage when σ 2 > 1, i.e. when the biosensor response is significantly under diffusion control. Additional calculations confirmed this property. When comparing curves 1–4 with the corresponding curves 5–8, one can see that the biosensor sensitivity is notable higher when the response is under diffusion control (curves 1–4) rather than under the enzyme kinetics (curves 5–8). Figure 13b shows that a moderate decrease in the electrode coverage θ practically does not change the biosensor sensitivity BS . In the case of c = 100 µm, the sensitivity BS decreases from only 1 down to 0.98 changing the coverage θ from 1 down to 0.5 (curves 1–4), i.e. reducing the volume of the enzyme two times. Although at c = 10 µm, the overall sensitivity is significantly less than at c = 100 µm, the shape of curves BS is very similar in both these cases. Selecting the geometry of the microreactors, one can minimize the volume of an enzyme practically without losing the sensitivity.

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4 Plate–Gap Biosensor This section considers computational modeling of an enzyme modified porous screen printed electrode. The porous electrode-based biosensors seem particularly promising for the detection of glucose, galactose, ethanol, phenol and some other substrates [26, 50, 78]. These biosensors have theoretically been investigated by applying a plate–gap model [38, 39]. The ultimate goal of the modeling of the plate– gap biosensors was verification of the proposed model by comparison of theoretical and experimental responses for biosensors based on the PQQ-dependent glucose dehydrogenases [39, 51].

4.1 Principal Structure of Biosensor The pores of the carbon paste electrode are assumed to be the enzyme deposited. The electrode is covered with an inert porous membrane [50, 51]. The enzyme activity is gradually dispersed in the volume of porous electrode, and the distances between the enzymatic reaction sites and the conducting walls of the porous electrode are as short as an average radius of pores. According to this physical model, the enzyme activity is uniformly dispersed in the gap between two parallel conducting plates. The modeled physical system, in general, mimics the main features of the porous electrode. Firstly, the uniform dispersion of the enzyme activity is affirmed according to the definition of the modeled physical system. Secondly, the gap width dependent characteristic distances between the enzymatic reaction sites and the conducting plates of the modeled system can be admitted to be similar to the average radius of pores in the porous electrode. In addition, the substrate or product molecules in the modeled plate–gap electrode may diffuse distantly in the directions, which are parallel to the surface of the electrode, i.e. as it is in the threedimensional network of a porous electrode. Figure 14 shows the principal structure of a biosensor, where enzyme-filled gaps are modeled by right quadrangular prisms distributed uniformly. The thickness of the outer membrane as well as the depth of the gaps of the electrode is assumed to be much less than its length. The porous membrane is assumed to be of a uniform thickness. Because of the symmetry and the relatively great length of the gaps, only the transverse section of the biosensor can be considered. Figure 15a shows the profile of a biosensor where the rectangular section of the enzyme-filled gaps is 2a1 by c, the distance between adjacent prisms equals 2(a2 − a1 ), a1 is the half-width of the gaps, c is the gap depth and d is the thickness of the outer porous membrane. Due to the uniform distribution of the gaps, it stands to reason to consider only a unit consisting of a single gap together with the region between two adjacent gaps. Figure 15b shows the profile of a unit cell to be considered in mathematical modeling of the biochemical behaviour of the plate–gap biosensor represented schematically in Fig. 14.

4 Plate–Gap Biosensor

331 Membrane

Enzyme

Electrode

Fig. 14 The principal structure of a plate–gap biosensor. The figure is not to scale

y b3

d

Porous membrane

Ω3

b2

Ω2

c

Enzyme

Electrode

Enzyme

Electrode

Enzyme

b1

Ω1

0

a)

2a1

2a2

a1

a2

x

b)

Fig. 15 The profile (a) and the unit cell (b) of a plate–gap biosensor

In Fig. 15b, 1 represents the enzyme-filled gaps, 2 corresponds to the porous membrane, 3 stands for the external diffusion layer, c = b1 is the depth of the gaps, d = b2 − b1 is the thickness of the porous membrane and δ = b3 − b2 is the thickness of the external diffusion layer. A very similar approach has been used in modeling of partially blocked electrodes [8, 22].

4.2 Mathematical Model The mathematical model of a plate–gap biosensor with an outer porous membrane (Fig. 15) can be formulated in a two-dimensional domain consisting mainly of three regions: the enzyme region 1 , the region 2 corresponding to the porous membrane and the region 3 of the external diffusion, 1 = (0, a1 ) × (0, b1 ), 2 = (0, a2 ) × (b1 , b2 ), 3 = (0, a2 ) × (b2 , b3 ). Let i denote the closed region corresponding to i , i = 1, 2, 3.

(48)

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Biosensors Based on Microreactors

In the enzyme region 1 , the mass transport by diffusion and the enzymecatalysed reaction are considered, E

S −→ P,

(49)

where a substrate (S) is enzymatically converted to a product (P), Assuming the outer porous membrane as the periodic media, the homogenization process has been applied to the domain 2 corresponding of the membrane [5]. After this, the porous membrane was modeled as a diffusion layer with an effective diffusion coefficient. Thus, in the region 2 of the outer membrane, only the mass transport by diffusion takes place. In the homogeneous external region 3 also, only the mass transport by diffusion of the substrate as well as of the product takes place. According to the Nernst approach, the thickness δ = b3 − b2 of the diffusion layer was assumed to be constant during the biosensor action.

4.2.1 Governing Equations The action of the biosensor, presented schematically in Figs. 14 and 15, can be described by the following system of reaction–diffusion equations (t > 0) [12, 37, 75]: Vmax S1 ∂S1 = DS1 S1 − , ∂t KM + S1 Vmax S1 ∂P1 = DP1 P1 + , ∂t KM + S1 ∂Sj = DSj Sj , ∂t ∂Pj = DPj Pj , ∂t

(50) (x, y) ∈ 1 ,

(51) (x, y) ∈ j ,

j = 2, 3,

where  is the Laplacian, Si (x, y, t) is the concentration of the substrate in the region i , Pi (x, y, t) is the concentration of the reaction product in i , i = 1, 2, 3, Vmax is the maximal enzymatic rate and KM is the Michaelis constant.

4.2.2 Initial Conditions Let 1 be the electrode surface and 2 the porous membrane/bulk solution boundary, 1 = ([0, a1 ] × {0}) ∪ ({a1} × [0, b1 ]) ∪ ([a1 , a2 ] × {b1 }) , 2 = [0, a2 ] × {b3 }.

(52)

4 Plate–Gap Biosensor

333

The biosensor operation starts when the substrate of concentration S0 appears in the bulk solution. This is used in the initial conditions (t = 0) Sk (x, y, 0) = 0, (x, y) ∈ k ,

k = 1, 2,

S3 (x, y, 0) = 0, (x, y) ∈ 3 \ 2 ,

(53)

S3 (x, y, 0) = S0 , (x, y) ∈ 2 , Pi (x, y, 0) = 0, (x, y) ∈ i ,

i = 1, 2, 3.

4.2.3 Boundary and Matching Conditions Assuming b0 = 0, the following boundary conditions express the symmetry of the biosensor (t > 0): DPi

∂Pi  ∂Si  = DSi = 0,   ∂x x=0 ∂x x=0

DPj

∂Pj  ∂Sj  = DSj = 0,   ∂x x=a2 ∂x x=a2

y ∈ [bi−1 , bi ], y ∈ [bj −1 , bj ],

i = 1, 2, 3,

(54)

j = 2, 3.

(55)

The substrate is an electro-inactive substance, while the product is an electroactive substance. The electrode potential is chosen to keep zero concentration of the product at the electrode surface (t > 0), DSk

∂Sk   = 0, ∂n 1

Pk = 0,

(x, y) ∈ 1 ,

k = 1, 2,

(56)

where n stands for the normal direction. Assuming the bulk solution to be well-stirred and in a powerful motion, the diffusion layer (b2 < y < b3 ) may be treated as the Nernst diffusion layer [21, 53, 65, 88]. According to the Nernst approach, a layer of thickness δ = b3 − b2 remains unchanged with time. Away from it, the bulk solution is in motion and is uniform in concentration (t > 0), S3 (x, b3, t) = S0 , P3 (x, b3 , t) = 0,

(57)

x ∈ [0, a2].

On the boundary between adjacent regions k and k+1 , the matching conditions are defined (t > 0), DSk

∂Sk  ∂Sk+1  = DSk+1 ,   ∂y y=bk ∂y y=bk

Sk (x, bk , t) = Sk+1 (x, bk , t),

(x, y) ∈ k ∩ k+1 ,

(58) k = 1, 2.

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Biosensors Based on Microreactors

DPk

∂Pk  ∂Pk+1  = DPk+1 ,   ∂y y=bk ∂y y=bk

Pk (x, bk , t) = Pk+1 (x, bk , t),

(x, y) ∈ k ∩ k+1 ,

(59) k = 1, 2.

The governing equations (50) and (51) together with the initial (53), the boundary (54)–(57) and the matching (59) conditions form together a boundary value problem.

4.2.4 Biosensor Response The measured current is accepted as a response of a biosensor in an actual experiment. The current depends upon the flux of the reaction product at the electrode surface, i.e. on the border 1 . The density i(t) of the current at time t can be obtained explicitly from the Faraday and the Fick laws i(t) =

  ! a1 ! b1 ∂P1  ∂P1  ne F  DP1 dx + D dy P1 a2 ∂y y=0 ∂x x=a1 0 0  ! a2  ∂P2  dx , + DP2  a1 ∂y y=b1

(60)

where ne is the number of electrons involved in a charge transfer and F is the Faraday constant. The system (50)–(59) approaches a steady state when t → ∞, I = lim i(t), t →∞

(61)

where I is the density of the steady state current of the plate–gap biosensor.

4.2.5 Numerical Simulation The finite difference technique can be successfully applied to solve numerically the initial boundary value problem (50), (51), (53)–(59) [12, 37–39]. To find an efficient numerical solution of the problem, a bilinear discrete grid in all directions, x, y and t, was introduced [9, 12]. Using alternating direction method, a semi-implicit linear finite difference scheme has been built as a result of the difference approximation [72]. The resulting system of linear algebraic equations was solved rather efficiently because of the tridiagonality of the matrix of the system. Due to high gradients of the concentrations of both species, the substrate and the product, an accurate and stable numerical solution was achieved only at a very small step size in y direction at the boundaries y = 0 and y = b3 . Because of the

4 Plate–Gap Biosensor

335

concavity of an angle at point (a1 , b1 ), it was necessary to use a very small step size in both space directions, x and y, also at the boundaries x = a1 , y = b1 . Due to the matching conditions between the adjacent regions with different diffusivities, a small step size near the boundary y = b2 was also used. In the direction x, an exponentially increasing step size was used to both sides from a1 to a2 and down to 0. In the direction y, an exponentially increasing step size was used from 0 to b1 /2, from b3 down to (b2 + b3 )/2, from bj down to (bj + bj −1 )/2 and from bj to (bj + bj +1 )/2, j = 1, 2, where b0 = 0. The step size in the direction of time was restricted due to the nonlinear reaction term in (50), boundary conditions and the geometry of the domain. In order to achieve an accurate and stable solution of the problem, at the beginning of the reaction–diffusion process, the restrictive condition was required. Since the biosensor action obeys the steady state assumption when t → ∞, it was reasonable to apply an increasing step size in the time direction. The final step size was in a few orders of magnitude higher than the first one. The numerical simulator has been programmed in JAVA language [63]. Assuming a1 = a2  b1 and the zero thickness of either the porous membrane (d = 0, b2 = b1 ) or the external diffusion layer (δ = 0, b3 = b3 ), the mathematical model (50), (51), (53)–(59) approaches the two-compartment model [75] (see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”). At relatively low (S0  KM ) as well as at very high (S0  KM ) concentrations of the substrate, the two-compartment mathematical model can be solved analytically [75]. The adequacy of the mathematical model (50), (51), (53)–(59) of the plate–gap biosensor as well as of the numerical solution of the problem was evaluated using well-known analytical solutions for the two-compartment model [75]. Accepting a1 = a2 = 20b1, the density of the steady state biosensor current was calculated at different values of the model parameters: the maximal enzymatic rate Vmax , the substrate concentration S0 (S0  KM as well as S0  KM ), the gap depth c = b1 , the thickness d of the outer membrane (accepting δ = 0) and the thickness δ of the external diffusion layer (accepting d = 0). In all these cases, the relative difference between the numerical and analytical solutions was less than 1%. The upper layer of the thickness δ = b3 − b2 was assumed to be the Nernst diffusion layer. Assuming the intensively stirred buffer solution, the thickness δ of 2 µm was used in the numerical simulation of the biosensor action changing some other parameters. The following values of the model parameters were constant in all the numerical experiments discussed below: DS1 = DP1 = 300 µm2/s, KM = 100 µM,

DS2 = DP2 = 2DS1 = 2DP1 ,

Vmax = 100 µM/s,

S0 = 100 µM,

ne = 2.

(62)

Some results of the numerical simulation are depicted in Figs. 16 and 17. One can see in Fig. 16 that the biosensor current is very sensitive to changes in the depth c = b1 and in the relative width of the gaps. The threefold increase in gap depth c (from 2 to 6 µm) increases the steady state current about 2.7 times

336

Biosensors Based on Microreactors 22

6

20 18

5

i, nA/mm2

16

4 3

14 12 10

2

8 6

1

4 2 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

t, s Fig. 16 The dynamics of the density i of the biosensor current, a1 : 0.5 (1, 3, 4), 1 (2, 6), 2 (5) a2 : 1.5 (1, 3, 4, 6) , 3.0 (2, 5) b1 : 2 (1), 4 (2, 3, 5, 6), 6 (4), b2 = b1 + 2, b3 = b2 + 2 (µm). The values of all other parameters are defined in (62)

(from 6.1 to 16.4 nA/mm2, curves 1 and 4). At a2 = 1.5 µm, the increase in b1 from 2 up to 6 µm increases the total area of the electrode surface about 2.14 times, (1.5 + 6)/(1.5 + 2) ≈ 2.14. As it is possible to notice in Fig. 16, the two pairs of curves (2, 3) and (5, 6) are very close to each other. When comparing the parameters of the simulation corresponding to curves 2 and 3, one can see the significant (2 times) difference in absolute width a1 . However, the relative width a1 /a2 of the gaps is the same. A very similar situation is observed when comparing another pair (5 and 6) of curves. Probably, the biosensor response depends mainly on the relative width a1 /a2 of the gaps, and the biosensor current is practically invariant to the absolute width a1 . The next section discusses this effect in greater detail. Figure 17 shows the concentrations of the substrate (S) and the product (P ) in the enzyme and the diffusion regions at time t = 3.2 s when the steady state was reached. The concentration S of the substrate and the concentration P of the reaction product in the entire modeling domain were introduced as follows (t ≥ 0): ⎧ ⎪ ⎪ ⎨S1 (x, y, t), (x, y) ∈ 1 , S(x, y, t) = S2 (x, y, t), (x, y) ∈ 2 \ 1 , ⎪ ⎪ ⎩S (x, y, t), (x, y) ∈  \  , 3 3 2 (63) ⎧ ⎪ P (x, y, t), (x, y) ∈  , ⎪ 1 1 ⎨ P (x, y, t) = P2 (x, y, t), (x, y) ∈ 2 \ 1 , ⎪ ⎪ ⎩P (x, y, t), (x, y) ∈  \  . 3

3

2

4 Plate–Gap Biosensor

337

S , μM 100

8 7

80

y, μm

6 5

60

4 40

3 2

20

1 0

0 -2

a)

-1.5

-1

-0.5

0

0.5

1

1.5

2

x , μm

y, μm

P , μM

b)

8

0.08

7

0.07

6

0.06

5

0.05

4

0.04

3

0.03

2

0.02

1

0.01

0

0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x , μm

Fig. 17 The steady state concentrations of the substrate (a) and the product (b) in the gaps, the outer membrane and the surrounding regions obtained at t = 3.2 s, a1 = 1, a2 = a1 + 1, b1 = 2, b2 = b1 + 2, b3 = b2 + 2 (µm). The values of the model parameters are defined in (62)

At any time t ≥ 0, both concentration functions, S(x, y, t) and P (x, y, t), are continuous at all (x, z) ∈ 1 ∪ 2 ∪ 3 . To have a more comprehensive view, the mirror-image along the y-axis is also shown in Fig. 17, where the point (0, 0) corresponds to the centre of the base of the enzyme-filled gap.

338

Biosensors Based on Microreactors

4.3 Effect of the Gaps Geometry To investigate the effect of the geometry of gaps on the biosensor response, the density of the steady state current has to be calculated at different values of the gap depth c and the half-width a2 of the cell changing the half-width a1 of the gaps. To compare the response of a plate–gap biosensor with the response of the corresponding flat (membrane) biosensor, the normalized dimensionless steady state current was defined as a function of the degree ρ of the rimosity of the plate–gap electrode, Iρ (ρ) =

I (ρ) , IF

ρ=

a1 , a2

0 < ρ < 1,

(64)

where Iρ (ρ) is the density of the steady state biosensor current calculated as defined in (61) at the relative width a1 /a2 of gaps of the electrode. The dimensionless degree ρ was expressed as the total area of the bases of all gaps divided by the base area of the whole electrode. IF corresponds to the density of the steady state current of the flat biosensor having the enzyme mono-layer immobilized on a flat electrode. In general, Iρ (1) = IF , and however, Iρ (1) → IF when a1 = a2 → ∞. Figure 18 shows the dependence of the steady state current Iρ and of the sensitivity BS of the plate–gap biosensor on the rimosity ρ (relative width) of the plate–gap electrode at different values of the gap depth c and the half-width a2 of the cell. Figure 18a shows that the steady state current of the plate–gap biosensor decreases with the decrease in the rimosity ρ of the electrode. The model parameter a2 stands for the density of the gaps distribution. As one can see in Fig. 18, there is no notable difference between the curves 1–3 obtained at a constant gap depth c = b1 and different values of a2 . The biosensor response is practically invariant to the topology of the electrode gaps. It is possible to notice in Fig. 18b that the narrowing of the gaps slightly decreases the biosensor sensitivity BS and thereby truncates the calibration curve of the plate–gap biosensor. This is definitely a disadvantage of the plate–gap biosensors. However, significantly less amount of the enzyme used in a plate–gap biosensor in comparison with the corresponding flat biosensor redresses this disadvantage with a vengeance. The increase in the rimosity ρ increases linearly the total volume of the enzyme used in the plate–gap biosensors.

5 Concluding Remarks When modeling an amperometric biosensor based on a carbon paste electrode encrusted with a single MR prepared from the CPC-silica carrier and loaded with a glucose oxidase, the homogenization process can be applied to model the MR as

5 Concluding Remarks

339

1.2 1.1 1.0 0.9 0.8

I

0.7 0.6

1 2 3 4 5

0.5 0.4 0.3 0.2 0.1 0.0 0.0

a)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.60

1 2 3 4 5

BS

0.55

0.50

0.45 0.0

0.1

b) Fig. 18 The normalized steady state current Iρ (a) and the sensitivity BS (b) versus the rimosity θ of the plate–gap electrode, a2 : 2 (1), 4 (2, 4, 5), 6 (3) µm, b1 : 2 (4), 4 (1–3), 6 (5) µm. The other parameters and the notations are the same as in Fig. 16

a homogeneous media [5, 92]. Assuming the microreactor as a porous media, the MR can be modeled as a reaction–diffusion system (6) with an effective diffusion coefficient directly proportional to the porosity and tortuosity of the CPC-silica carrier. Due to the homogenization process, the rate of the enzymatic reaction has to be corrected with the porosity of the MR. The stationary biosensor current increases in an allometric fashion with an increase in the porosity γ as well as the tortuosity β (Figs. 5a and 6a). The biosensor

340

Biosensors Based on Microreactors

sensitivity increases with an increase in the porosity and the tortuosity (Figs. 5b and 6b). In the case when the biosensor response is under the diffusion control, a biosensor based on an array of microreactors is able to generate a greater steady state current than a corresponding flat mono-layer biosensor, the thickness of which being the same as the height of the microreactors (Fig. 13a). The denser the microreactors are distributed on the electrode surface, the higher the generated steady state current. This feature of array biosensors can be applied to design novel highly sensitive biosensors when the minimization of the enzyme volume is of crucial importance. Selecting the geometry of microreactors, one can minimize the volume of enzyme without losing the sensitivity (Fig. 13b). Reducing the relative width of gaps in plate–gap biosensors reduces the effective steady state current (Fig. 18a) and can slightly decrease the biosensor sensitivity BS (Fig. 18b). However, the sensitivity decrease of biosensor is compensated by significant enzyme amount decrease.

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Modeling Carbon Nanotube Based Biosensors

Contents 1 2

3

4

5

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carbon Nanotube Based Mediated Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Principal Structure of the Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Numerical Simulation and Model Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Effect of the Structural Anisotropy of the CNT Mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Effect of the Partition Coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Dimensional Modeling of CNT Based Mediated Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Numerical Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Impact of the Perforation Level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Impact of the Tortuosity in the Perforated Membrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carbon Nanotube Based Unmediated Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Principal Structure of the Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Experimental Model Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Impact of Enzyme Concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Impact of Electrochemical Reaction Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

346 347 347 349 355 356 358 359 359 360 361 362 363 364 365 368 369 371 372 373

Abstract This chapter presents two-dimensional and one-dimensional-in-space mathematical models of mediated and unmediated (mediatorless) amperometric biosensors based on an enzyme-loaded carbon nanotube (CNT) layer deposited on the perforated membrane. The models are based on nonlinear reaction–diffusion equations and involve four regions: the enzyme and the CNT regions where enzymatic reactions as well as the mass transport by diffusion take place, a diffusion limiting layer where only the mass transport by diffusion takes place and a convective region where the analyte concentration is maintained constant. By changing input parameters the output results are numerically analysed with an emphasis to the influence of the geometry and the catalytic activity of the biosensors to their response and sensitivity. The mediatorless transfer of the electrons in the © Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_11

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region of enzyme-loaded carbon nanotubes is especially investigated. The numerical simulation at transient conditions was carried out using the finite difference technique. The mathematical models and the numerical solutions were validated by experimental data. The obtained agreement between the simulation results and the experimental data was admissible at different concentrations of the substrate. Keywords Mediated/mediatorless amperometric biosensor · Carbon nanotube · Enzymatic reaction · Nonlinear reaction–diffusion equation · Glucose dehydrogenase · Laccase

1 Introduction Since carbon nanotubes (CNT) were discovered [25], they were used in various applications because of their unique structural and electronic properties. The CNT based biosensors are recognized to be a next generation building block for ultrasensitive and ultra-fast biosensing systems [56]. Multi (MWCNT) as well as single walled carbon nanotubes (SWCNT) emerged as a promise component for the developing novel biosensors [1, 3, 24, 26, 53, 59]. Mediated biosensors as a kind of amperometric biosensors require the participation of redox molecules in signal transduction. The mediated biosensors are often constructed with the enzymes that can donate electrons to electrochemically active artificial electron acceptors [14]. A more advanced method is to have no mediator and to ensure the direct electron transfer between the enzyme and the electrode [28]. The mediatorless biosensors are so called third generation biosensors and are successors for the mediated biosensors [16, 17, 28]. The mediatorless biosensors are advantageous over the mediator based biosensors because of simplicity in their application and modeling. The biosensors with the direct electron transfer have higher selectivity and are less prone to the interfering reactions [18, 22, 35, 41, 42, 46]. One of the main obstacles in the development of the mediatorless biosensors is that only few enzymes can support the direct electron transfer [58]. To solve the problems currently limiting the practical use of CNT based biosensors, the mass transport and the enzyme kinetics within the biosensors have to be comprehensively investigated [31]. The two-compartment modeling has been successfully applied by Lyons [31, 32] to the transport and the kinetics of the substrate and the mediator within chemically modified electrodes comprising redox enzymes immobilized in CNT meshes dispersed on support electrode surfaces. The one-dimensional-in-space boundary value problem was solved analytically assuming the steady state conditions [31, 32]. Practical biosensors are usually covered by outer porous or perforated membranes [1, 3, 24, 26, 43, 53, 59] that increase their stability and prolong their calibration curve [6, 21, 48, 50]. This chapter presents mathematical models of mediated and unmediated amperometric biosensors based on an enzyme-loaded CNT layer deposited on the perforated membrane [8, 9, 43, 45, 46]. Both models are based on nonlinear

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reaction–diffusion equations and involve the following regions (compartments): the enzyme and the CNT layers where enzymatic reactions as well as the mass transport by diffusion take place, a diffusion limiting region where only the mass transport by diffusion takes place and a convective region where the analyte concentration is maintained constant [8, 9, 39]. The biosensor models are defined in two-dimensional and in one-dimensional space domains. The accuracy of the biosensor response simulated by using onedimensional model is evaluated by the response simulated by the corresponding two-dimensional model [8, 9, 39]. The numerical simulation at transition conditions was carried out using the finite difference technique [7, 12, 36]. The mathematical models and the numerical solutions were validated by experimental data [8, 9, 43]. By changing input parameters the output results were numerically analysed with a special emphasis to the influence of the geometry and the catalytic activity of the biosensor on the output current and biosensor sensitivity [8, 9].

2 Carbon Nanotube Based Mediated Biosensor This section presents a two-dimensional-in-space mathematical model of an amperometric mediated biosensor based on an enzyme-loaded carbon nanotubes layer deposited on a perforated membrane [8, 43, 45]. The developed model is based on nonlinear non-stationary reaction–diffusion equations. By changing input parameters the output results are numerically analysed with a special emphasis to the influence of the geometry and the catalytic activity of the biosensor to its response. The mathematical model and the numerical solution were validated by experimental data [8, 43, 45].

2.1 Principal Structure of the Biosensor Figure 1 shows a schematic representation of the modeling biosensor based on an enzyme-loaded CNT composite electrode and a perforated membrane [8, 43]. The biosensor was built by assembling several layers of different materials and sizes [43, 44]. These layers were deposited on the insulating film shown in Fig. 1 as the bottom layer. When preparing the biosensor, SWCNT layer was deposited on a polycarbonate perforated membrane and loaded with an enzyme. The enzyme was additionally constrained in a relatively thin layer between the insulating film and the CNT layer. The single walled carbon nanotubes form a composite electrode of the biosensor. Due to the technology of the biosensor preparation, some CNT were sinked into the holes of the perforated membrane. Because of this the structural alignment of nanotubes in holes was notably different from that in the enzyme layer [43, 44].

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Perforated membrane

Enzyme-loaded CNT Enzyme layer Insulating film Fig. 1 Principal structure of the biosensor based on an enzyme-loaded CNT composite electrode and a perforated membrane [8]. The figure is not to scale

In the enzyme-loaded region, the enzyme (E) catalyses the substrate (S) conversion to the product (P) in the presence of the mediator (M), k1

Eox + S −→ Ered + P, k2

Ered + Mox −→ Eox + Mred ,

(1) (2)

where Eox and Ered stand for the oxidized and reduced forms of the enzyme, respectively, while Mox and Mred represent the oxidized and reduced forms of the mediator, respectively. In the CNT region, the reduced mediator Mred is electrochemically re-oxidized and electrons are released forming the output current, Mred −→ Mox + ne e− ,

(3)

where ne is the number of electrons released in the mediator re-oxidation. As it is common for the amperometric biosensors, the electrochemical reaction (3) is assumed to be very fast [21, 48, 50]. The holes in the perforated membrane were modeled by right cylinders of uniform diameter and spacing, forming a regular hexagonal pattern [8]. For simplicity, it was reasonable to consider a circle whose area equals to that of the hexagon and to regard one of the cylinders as a unit cell of the biosensor. Due to the symmetry of the unit cell, only a half of the transverse section of the unit cell was considered in cylindrical coordinates [6]. Very similar approach has been used in modeling of partially blocked electrodes [4, 15]. The profile of the unit cell of the biosensor is shown in Fig. 2 [8]. In this figure, r1 stands for the radius of the holes in the perforated membrane, r2 is the radius of the unit cell of the biosensor, regions 1 and 2 represent the enzyme and the enzyme-loaded CNT layers, respectively, z1 is the thickness of the enzyme layer, z2 − z1 is the thickness of the CNT layer, z3 − z2 —the thickness of the perforated

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349 z

Fig. 2 Profile of the unit cell of the biosensor [8]. The figure is not to scale

z4 4 z3 3

z2

2

z1 0

r1

1

r r2

membrane, 3 represents a hole in the perforated membrane, 4 stands for the external diffusion layer of the thickness z4 − z3 . The analysed substrate gets into the biosensor from the buffer solution through the holes of the perforated membrane. The external diffusion layer 4 can be treated as the Nernst diffusion layer [29]. According to the Nernst approach the layer of the thickness z4 − z3 remains unchanged with time. It is also assumed that away from it the solution is uniform in the concentration.

2.2 Mathematical Model Let j be the open region presented in Fig. 2, j = 1, 2, 3, 4, 1 = (0, r2 ) × (0, z1 ),

2 = (0, r2 ) × (z1 , z2 ),

3 = (0, r1 ) × (z2 , z3 ), 4 = (0, r2 ) × (z3 , z4 ).

(4)

Additionally, let j be the closed region corresponding to the open region j , j = 1, 2, 3, 4. For further convenience, we introduce the boundaries of the regions: 0 = [0, r2 ] × {0}, j = j ∩ j +1 ,

j = 1, 2, 3,

(5)

4 = [0, r2 ] × {z4 }, where 0 , 4 stand for the lower and the upper boundaries of the unit cell of the biosensor, and j denotes the boundary between adjacent regions j and j +1 , j = 1, 2, 3.

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2.2.1 Governing Equations The biosensor operation was described by nonlinear reaction–diffusion equations [7, 8, 12]. The mathematical model was formulated in a two-dimensional domain in cylindrical coordinates (see Fig. 2). The direction-dependent diffusion coefficients for each species were applied to directions r and z due to anisotropic properties of CNT and pores [19, 54, 55]. To simplify the expression of the governing equations of the model, an operator  was introduced to describe the diffusion term in cylindrical coordinates r and z, (U ) = DU,r

1 ∂ r ∂r

 ∂U ∂ 2U r + DU,z 2 , ∂r ∂z

(6)

where U is the concentration of a particular species U , DU,r and DU,z are the effective diffusion coefficients of the species U in the directions r and z, respectively [8]. The enzyme in the biosensor is immobilized and therefore is not affected by the diffusion. In the enzyme-loaded regions 1 and 2 , the dynamics of the enzyme concentration is affected only by the enzymatic reactions (1) and (2). The dynamics of the enzyme concentration is described by the following reaction equations: ∂Eox,j = −k1 Eox,j Sj + k2 Mox,j Ered,j , ∂t ∂Ered,j = k1 Eox,j Sj − k2 Mox,j Ered,j , (r, z) ∈ j , j = 1, 2, t > 0, ∂t

(7)

where t stands for time, Uj = Uj (r, z, t), U ∈ {Eox , Ered , S, Mox } is the concentration of the corresponding species in the area j , coefficients k1 and k2 are the rates of the reactions (1) and (2), respectively [8]. The mass transport by diffusion of the substrate takes place in all the regions: the enzyme layer (1 ), the enzyme-loaded CNT layer (2 ), the holes of the perforated membrane (3 ) and the Nernst diffusion layer (4 ). In both layers containing the enzyme, the substrate also participates in the reaction (1). The dynamics of the substrate concentration Sj in the region j is described as follows (j = 1, 2, 3, 4):  ∂Sj (Sj ) − k1 Eox,j Sj , = ∂t (Sj ),

j = 1, 2, j = 3, 4,

, (r, z) ∈ j , t > 0.

(8)

The mass transport by diffusion of the oxidized mediator Mox takes place inside the entire biosensor as well as in the outer diffusion layer. In both layers containing the enzyme, the mediator is involved in the enzymatic reaction (2). In the CNT regions, the oxidized mediator is regenerated by the electrochemical reaction (3). The reaction (3) is assumed to be very fast as it is common for amperometric biosensors [21, 48, 50]. The whole mediator in its reduced form

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(Mred ) is immediately re-oxidized. Therefore, whole Mox consumed in the reaction (2) is immediately regenerated in the reaction (3). The immediate regeneration of Mox leads to skipping the reaction terms in the equation describing the dynamics of the concentration Mox,2 in the region 2 . Consequently, the dynamics of the Mox in the region j is described by the following equations (t > 0) [8]:  ∂Mox,j (Mox,j ) − k2 Mox,j Ered,j , j = 1, = , (r, z) ∈ j . ∂t (Mox,j ), j = 2, 3, 4,

(9)

In the enzyme layer (region 1 ), the reduced mediator Mred is only produced by the reaction (2). The immediate re-oxidation of the reduced mediator in the reaction (3) leads to zero concentration of Mred in the enzyme-loaded CNT layer (region 2 ). Therefore, the dynamics of the concentration Mred,1 = Mred,1 (r, z, t) of the Mred was described only in the region 1 [8], ∂Mred,1 = (Mred,1 ) + k2 Mox,1 Ered,1, ∂t

(r, z) ∈ 1 , t > 0.

(10)

2.2.2 Boundary Conditions Since in physical experiments the buffer solution was well-stirred [43], the constant concentrations of the species were assumed above the Nernst diffusion layer [8, 29], S4 (r, z4 , t) = S0 ,

Mox,4(r, z4 , t) = M0 ,

r ∈ [0, r2 ], t > 0,

(11)

where S0 and M0 are the concentrations of the substrate and the mediator in the buffer solution. The non-leakage boundary conditions were used to represent the symmetry of the unit cell of the biosensor. These conditions were also applied to the surface of the perforated membrane when t > 0,  ∂Mred,1  = 0, ∂r 1

  ∂Sj  ∂Mox,j  = = 0, ∂n j ∂n j

j = 1, 2, 3, 4,

(12)

where n stands for the normal direction, and j = j \ j \ j −1 \ j . The boundary 0 represents the surface of the insulating film where non-leakage conditions were also applied to all the diffusive species,    ∂Mox,1  ∂Mred,1  ∂S1  = = = 0, ∂z 0 ∂z 0 ∂z 0

t > 0.

(13)

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On the boundary between adjacent regions having different diffusivities, the matching conditions (t > 0) were defined, DSj ,z

    ∂Sj  ∂Sj +1  = D , Sj  = Sj +1  , Sj+1 ,z   j j ∂z j ∂z j DMox,j ,z

j = 1, 2, 3,

  ∂Mox,j  ∂Mox,j +1  = DMox,j+1 ,z  , ∂z j ∂z j     Mox,j  = Mox,j +1  , j = 2, 3. j

(14)

(15)

j

On the boundary 1 , where the enzyme-loaded CNT layer (2 ) touches the entirely enzyme-loaded layer (1), the concentration of Mox is influenced by the reaction (3). The mediator re-oxidation reaction (3) was assumed to be so fast, that whole diffusive Mred touching the border 1 is immediately re-oxidized, i.e. Mred is converted to Mox (t > 0),  Mred,1  = 0, 1

DMox,2 ,z

   ∂Mox,2  ∂Mox,1  ∂Mred,1  = DMox,1 ,z + DMred,1 ,z , ∂z 1 ∂z 1 ∂z 1   Mox,1 = Mox,2 . 1

(16)

(17)

1

2.2.3 Initial Conditions The biosensor operation starts when the substrate S and the mediator Mox are infused into the buffer solution (t = 0) [8], Sj = 0,

Mox,j = 0,

(r, z) ∈ j \ 4 , j = 1, 2, 3, 4,

 S4 |4 = S0 , Mox,4 = M0 . 4

(18) (19)

Initially, the whole enzyme was in the oxidized form (t = 0), Eox,1 = E0 , (r, z) ∈ 1 , Ered,j = 0, (r, z) ∈ j , j = 1, 2, Mred,1 = 0, (r, z) ∈ 1 , where E0 stands for the total concentration of the enzyme [8, 43].

(20)

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It was also assumed that the CNT mesh homogeneously saturates with the enzyme at a concentration that is different in comparison to the enzyme layer, Eox,2 = ηE0 ,

(r, z) ∈ 2 , t = 0,

(21)

where η is the formal partition coefficient [20, 27, 52] assumed as the ratio the enzyme initial concentration in the enzyme layer 1 to that in the CNT layer 2 , practically, 0 ≤ η < 1 [8, 43].

2.2.4 Biosensor Response The response of the biosensor is generated due the electrochemical reaction (3). The electrons released in this reaction form a current that is amplified and presented to the end-user. The biosensor current is usually proportional to the area of the electrode surface [10, 34]. Therefore, when analysing unit cells of the amperometric biosensor, the current is often normalized with the area of the base of the unit cell [5, 6]. The electrochemical reaction (3) takes place in the layer of the enzyme-loaded CNTs. As the reaction was assumed to be very fast, its rate fully depends on the concentration of Mred , which is generated in the reaction (2). Consequently, in the region 2 the rate of the electrochemical reaction (3) is equal to the rate of the enzymatic reaction (2). Therefore, the current density i2 (t) generated in the enzyme-loaded CNT layer at time t was defined as ! ! ! ne F k2 2π z2 r2 i2 (t) = Ered,2 Mox,2r dr dz dϕ, (22) πr22 0 z1 0 where F is the Faraday constant [8]. Mred is also generated in the enzyme layer (1). Differently from the region 2 , here the mediator is not re-oxidized. The Mred produced in 1 is consumed on the boundary 1 . The density i1 of the current induced by the Mred in this layer can be explicitly obtained from the Faraday and Fick laws [8],  ! ! ne F DMred,1 ,z 2π r2 ∂Mred,1   r dr dϕ. i1 (t) = − (23) ∂z 1 πr22 0 0 The total current density i(t) is a sum of densities defined in (22) and (23), i(t) = i2 (t) + i1 (t).

(24)

For biosensors it is common to consider the current generated by the biosensor at the steady state conditions [48, 49, 51]. The steady state current density for the biosensor being considered is defined as usually, I = lim i(t). t →∞

(25)

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2.2.5 Effective Diffusion Coefficients For the simplicity, the species S, Mox and Mred were assumed to have the same diffusion properties, i.e. the diffusion coefficients of all these species were assumed to be the same for a certain medium [8, 39]. The enzyme (1 ) and the Nernst (4) layers were assumed to be homogeneous, DU1 ,r = DU1 ,z = De ,

U1 ∈ {S1 , Mox,1, Mred,1 },

DU4 ,r = DU4 ,z = Db ,

U4 ∈ {S4 , Mox,4 },

(26)

where Db stands for the diffusion coefficient of the substrate and the mediator in the buffer solution as well as in the Nernst diffusion layer, and De is the corresponding diffusion coefficient in the enzyme. These diffusion coefficients are applied to all the space directions. The enzyme-loaded CNT composite layer and the perforated membrane both are non-homogeneous mediums. Assuming these layers as periodic media, the volume averaging approach can be applied to estimating the effective (averaged) diffusion coefficients for these layers [2, 54, 57]. The enzyme-loaded CNT layer (2 ) can be considered as a three-compartment domain: the enzyme, the carbon nanotubes and the buffer solution. The holes of the perforated membrane (3 ) consist of two compartments: the carbon nanotubes and the buffer solution. The effective diffusion coefficients for these two consisting compartments can be expressed in terms of the volume fraction, the tortuosity and the diffusion coefficients [2, 8, 11, 54, 57], DU2 ,ζ = θ2,ζ (v2 DU + η2 De + (1 − v2 − η2 )Db ), U2 ∈ {S2 , Mox,2 }, DU3 ,ζ = θ3,ζ (v3 DU + (1 − v3 )Db ),

U3 ∈ {S3 , Mox,3}, ζ ∈ {r, z},

(27)

where vj and η2 stand for the volume fractions of the CNTs and the enzyme, respectively, 0 < vj , η2 < 1, DU is the diffusion coefficient of the species U in the carbon nanotubes, and θj,r , θj,z are the tortuosities of the medium in r and z directions, respectively, j = 2, 3 [8, 9]. Values of the tortuosities θj,r and θj,z depend on the structural anisotropy (arrangement, alignment) of the CNTs, particularly on the nanotubes orientation [31, 54]. A near-unity tortuosity corresponds to a fully vertical alignment of the CNTs (Fig. 1). Assuming relatively small values of the diffusion coefficient DU as well as of the volume fractions v2 , v3 [37], the volume fraction η2 approaches to η, and the expression (27) of the effective diffusion coefficients reduces to DU2 ,ζ = θ2,ζ (ηDe + (1 − η)Db ), DU3 ,ζ = θ3,ζ Db ,

U2 ∈ {S2 , Mox,2 },

U3 ∈ {S3 , Mox,3 },

ζ ∈ {r, z},

where η is the formal partition coefficient introduced in (21) [8, 38, 39].

(28)

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2.3 Numerical Simulation and Model Validation The mathematical model (7)–(25) of the biosensor has been defined as a twodimensional-in-space initial boundary value problem based on a system of partial differential equations containing nonlinear terms representing biochemical reactions. Similar systems can be analytically solved only in very special cases [33, 49]. Therefore, the problem (7)–(25) was solved numerically by applying the method of finite differences together with the method of alternating directions [8, 12, 47]. The domain of the model was discretized using a uniform grid individually for each region j , j = 1, 2, 3, 4. Each region was divided into 50 equal parts in the space directions r and z. A constant step τ = 10−4 s was used in the time direction [8]. The approximation was done applying semi-implicit finite difference schemes. The diffusion terms were approximated by the backward differences, while the reaction terms were approximated by the forward differences. The approximation resulted by a system of linear algebraic equations with a tri-diagonal matrix. Systems having tri-diagonal matrices are solved very effectively and therefore are advantageous for solving similar problems [12, 47]. The computer simulation of this biosensor was carried out using a software developed by the authors in C++ programming language [8, 40]. In the simulation, the biosensor response time was assumed as the time when the change of the biosensor current over time remains very small during a relatively long term [8]. When analysing an influence of the geometry on the biosensor response, it is more convenient to operate with thicknesses of the layers instead of points zj ,  dj =

j = 1,

z1 ,

(29)

zj − zj −1 , j = 2, 3, 4,

where dj is the thickness of j -th layer, j = 1, 2, 3, 4. The following parameter values were used as a basic configuration of the biosensor [8, 30, 43]: r1 = 2 × 10−7 m,

r2 = 8 × 10−7 m,

d2 = 4 × 10−7 m,

d3 = 10−5 m,

E0 = 4.55 × 10−5 M,

d4 = 1.5 × 10−4 m,

De = 3 × 10−10 m2 s−1 ,

k1 = 6.9 × 105 M−1 s−1 , η = 0.5,

d1 = 10−7 m,

Db = 2De ,

k2 = 6.9 × 107 M−1 s−1 ,

θ2,r = θ3,r = θr = 0.125,

ne = 2,

θ2,z = θ3,z = θz = 0.25.

(30)

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2.5

1 2 3

i, A/m2

2.0

4 5 6

1.5

1.0

0.5

0.0 0

10

20

30

40

50

60

t, s Fig. 3 The dynamics of the simulated (1–3) and experimental (4–6) biosensor current i(t) at different concentrations of the mediator and the substrate: M0 = 0.005, S0 = 1.99 (1, 4), M0 = 0.05, S0 = 4.98 (2, 5), M0 = 0.2, S0 = 9.9 mM (3, 6) [8]. Values of other parameters are as defined in (30)

The mathematical model (7)–(25) was validated by comparing the biosensor response obtained by using computer-aided experiments with the results of the corresponding physical experiments [43]. The experimental data along with the corresponding simulated data are depicted in Fig. 3 [8]. As one can see in Fig. 3, the simulation results are similar to the experimental data. In the cases presented in the figure the relative error of the simulation is less than 10% when the sensor approaches to the steady state. This error was considered as small enough, referring to the fact that the results of repeated physical experiments can also vary about 10% [43]. The behaviour of the biosensor response was investigated by using computer simulation. In order to investigate the influence of the model parameters on the response behaviour, the simulation was performed at wide ranges of the parameter values [8].

2.4 Effect of the Structural Anisotropy of the CNT Mesh The mesh of carbon nanotubes influences the diffusivity of the species inside the composite medium containing the nanotubes. An experimental measurement of the diffusion coefficients is rather complicated [37]. The tortuosity is a characteristic of the CNT electrode mesh substantially influencing the effective diffusion coefficients (28). In terms of the biosensor

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3.0 2.5

I, A/m2

2.0 1.5

1 2 3

1.0 0.5 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

z

Fig. 4 The density I of the stationary current versus the height tortuosity θz calculated at S0 = 10 mM, M0 = 0.05 mM and two values of the radial tortuosity θr : 0.01 (1), 0.1 (2) and 0.5 (3) [8]. Values of other parameters are as defined in (30)

structure, the tortuosity mainly depends on the structural anisotropy (arrangement) of the nanotubes inside the composite electrode. Two model parameters, θr and θz , stand for the directional anisotropy of the CNTs. Numerical experiments were employed in order to investigate the impact of the directional anisotropy on the biosensor response. Figure 4 shows the dependence of the density of the steady state current on the height tortuosity θz [8]. As one can see in Fig. 4, the impact of the diffusivity along the axis r on the response of the biosensor is relatively low. Fifty times increase of θr increases the response current less than 3% for all employed values of θz . The impact of θz , in comparison to θr , is much higher. In general, the directional anisotropy of the CNTs can result in a noticeable change of the steady state biosensor current. The sensitivity of the response to changes in θz was explained by a relatively great thickness of the perforated membrane. The membrane thickness d3 is almost in two orders of magnitude greater than the radius r1 of the membrane holes. An insensibility of the biosensor response to changes in the tortuosity θr in the radial direction can be applied to further reduction of the mathematical model (7)– (25) to the corresponding one-dimensional-in-space mathematical model for the response simulation of biosensors having the structure as presented in Fig. 1. For much more efficient 1-D modeling, the perforated membrane can be approximated as proposed in [38, 39].

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2.5 Effect of the Partition Coefficient During the biosensor preparation, some enzyme penetrates into the mesh of the carbon nanotubes. The concentration of the enzyme filling the prepared CNT mesh cannot be precisely determined because of the enzyme permanent inactivation and some other factors. Numerical simulation was employed in order to investigate the effect of the concentration of the enzyme in the CNT layer (region 2 ). The concentration of the enzyme in this region was expressed as ηE0 , and therefore the biosensor response was simulated at different values of the enzyme filling degree (partition coefficient) η. Figure 5 shows the simulated stationary biosensor current versus the substrate concentration S0 [8]. The initial concentration of the enzyme in the CNT layer has relatively low impact on the biosensor output at low concentrations of the substrate. At higher concentrations (S0 > 10 mM), the influence of η on the response notably increases. At relatively low substrate concentrations the overall kinetics of the enzymatic reaction is usually of the first order while at high concentrations the kinetics is of the zero order [48, 49, 51]. So, the effect of η on the biosensor response is notably greater when the reaction kinetics is of the zero order rather than the kinetics is of the first order. A greater η corresponds to the higher enzyme concentration in the CNT layer. A higher enzyme concentration relates to a higher substrate concentration S0 at which the reaction kinetics switches from the first to the zero order. The Michaelis constant well characterizes the length of the biosensor calibration curve [48, 51]. The apparent Michaelis constant KMapp is known to be the substrate concentration 102

I, A/m2

101

100

1 2 3

10-1

1 10-2 10-1

100

2

101

3 102

103

S0, mM Fig. 5 The density I of the stationary current versus the substrate concentration S0 calculated at the mediator concentration M0 = 0.05 mM and three values of the filling degree η: 0 (1), 0.5 (2) and 1 (3), keeping all other parameters constant as defined in (30) [8]

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resulting in a half-maximum of the biosensor activity [21, 50],   KMapp = S : I (S) = 0.5 lim I (S0 ) , S0 →∞

(31)

where I (S) is the density of the stationary biosensor current calculated at the substrate concentration S, i.e. S0 = S. The dashed and dot lines and arrows in Fig. 5 schematically show the corresponding values of KMapp . As one can see, the apparent Michaelis constant KMapp is about 3.5 times higher for the biosensor with the highly enzyme-loaded CNT layer (dot– dashed curve 3) than for the biosensor with no enzyme-loaded CNT layer (dashed curve 1).

3 One-Dimensional Modeling of CNT Based Mediated Biosensor In this section, a corresponding one-dimensional-in-space model of the biosensor based on a CNT enzyme-loaded electrode deposited on a perforated membrane is considered [39]. The accuracy of the biosensor response simulated by using onedimensional model is evaluated by the response simulated by the corresponding two-dimensional model [8].

3.1 Mathematical Model Principal structure of the biosensor being modeled was shown in Fig. 1, and the reaction network was defined in (1)–(3) [8, 43–45]. Applying the homogenization process to the perforated membrane allows to reformulate the mathematical model (7)–(25) of the biosensor in a one-dimensional domain, on a line perpendicular to the surface of the biosensor [2, 54]. The surface of the insulating film was assumed to be zero point (x = 0) in the space interval. The model of the biosensor involves four layers (j ) with different properties, j = (xj −1 , xj ), 0 = {0},

x0 = 0,

j = {xj },

xj = xj −1 + dj , j = 1, 2, 3, 4,

(32)

where d1 is the thickness of the enzyme layer, d2 —the thickness of the enzymeloaded CNT mesh, d3 —the thickness of the perforated membrane and d4 stands for the thickness of the Nernst diffusion layer forming on the surface of the perforated membrane [39].

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Modeling Carbon Nanotube Based Biosensors

The governing equations of the model are analogous to (7)–(10) with differently defined operator  which here reduces to the one-dimensional diffusion, (U ) = DU

∂ 2U , ∂x 2

(33)

where U is the concentration of a particular species U , DU is the effective diffusion coefficient. Here, the concentration U is a function of one space coordinate x and time t. The enzyme (1 ) and the Nernst (4) layers were assumed to be homogeneous, DU1 = De ,

U1 ∈ {S1 , Mox,1, Mred,1 },

DU4 = Db ,

U4 ∈ {S4 , Mox,4 }.

(34)

Specifying expressions (27) and (28) lead to the following expressions of the effective diffusion coefficients: DU2 = θ2 (ηDe + (1 − η)Db ) ,

U2 ∈ {S2 , Mox,2 },

DU3 = θ3 ρDb ,

U3 ∈ {S3 , Mox,3 },

(35)

where ρ stands for the perforation level of the membrane as the ratio of the volume occupied by the holes to the overall volume of the membrane, θ2 and θ3 are the tortuosities defining the structural properties of the corresponding media [9, 39]. The boundary conditions involve equations (11), (13)–(17) and the initial conditions consist of Eqs. (18)–(21), all of them mapped to one-dimensional space. The total current density i(t) corresponding to (24) in one-dimensional domain was expressed as follows: ⎛ i(t) = ne F ⎝k2

!x2 Ered,2Mox,2 dx − DMred,1 x1

⎞  ∂Mred,1  ⎠ . ∂x 1

(36)

The density I of the steady state current was expressed exactly as in the corresponding two-dimensional model (25) [9, 39].

3.2 Numerical Simulation The mathematical model was defined as an initial boundary value problem involving nonlinear reaction terms. Due to the nonlinearity, the problem was solved numerically by applying the finite differences technique [47]. The model domain was discretized by applying the regular mesh to each space region j (j = 1, 2, 3, 4), and a constant step was used to discretize the time. The model equations were

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361

approximated by semi-implicit finite difference schemes. The diffusion terms were approximated by the backward differences, while the reaction terms were approximated by the forward differences [13, 23]. The resulting system of linear tri-diagonal algebraic equations was solved effectively. The computer simulation was carried out using a software developed in C++ programming language [39, 40]. The biosensor response was simulated at values most of which (d1 , . . . , d4 , De , Dn , k1 , k2 , ne are η) were the same as in (30), and values of only the following parameters were specific in one-dimensional modeling: θ2 = 1/3,

θ3 = 0.5,

ρ = 0.0625.

(37)

In the numerical simulations, the steady state was assumed at the time, when the increase of the output current becomes small enough [9]. An individual numerical simulation runs approximately 5 min using the one-dimensional model and approximately 25 h using the corresponding twodimensional model at the basic configuration of the biosensor. The simulations were performed on a computer with Intel® CoreTM i5-540M (2.53 GHz) CPUs, each simulation running on one CPU only [39]. The two-dimensional model takes into consideration the geometry of the perforated membrane, while the perforation topology is approximated by introducing the effective diffusion coefficients in the one-dimensional model. Therefore, the onedimensional model can be considered as an approximation of the corresponding two-dimensional one. The two-dimensional model was reduced by introducing two perforation parameters: the perforation ratio ρ and the tortuosity θ3 . In order to investigate the conditions under which the one-dimensional mathematical model can be applied to an accurate simulation of the biosensor response, a number of simulations were performed and the relative modeling error ν of the one-dimensional model was calculated, assuming the two-dimensional model as a precise one, ν=

I1D − I2D , I2D

(38)

where I2D is the density of the steady state current obtained by using the twodimensional model (7)–(25) [8], and I1D is the output current obtained by using the corresponding one-dimensional model discussed in this section. A similar approach to a comparison of one and two-dimensional models was also used in [38, 39].

3.3 Impact of the Perforation Level The dimensionless perforation level ρ depends on the topology and the form of the holes in the membrane. In the case of two-dimensional modeling, the holes were modeled by right cylinders of uniform diameter and spacing, forming a

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Modeling Carbon Nanotube Based Biosensors

0.10 0.08 0.06 0.04 0.02 0.00

1 2 3

-0.02 -0.04 -0.06 0.001

0.01

0.1

1

Fig. 6 The dependence of the one-dimensional modeling error ν on the membrane perforation level ρ at the following concentrations of the mediator and the substrate: M0 = 0.2, S0 = 9.9 (1), M0 = 0.05, S0 = 4.98 (2), M0 = 0.005, S0 = 1.99 mM (3) [39]. Values of other parameters are defined in (30) and (37)

regular hexagonal pattern. The corresponding one-dimensional modeling treats the membrane as a homogeneous medium with the corresponding effective (averaged) diffusion coefficients. In order to investigate the impact of the perforation geometry on the modeling error ν, the numerical simulations were performed using the one and two-dimensional models at various levels of the membrane perforation. The simulations of the biosensor response were repeated for the same concentrations of the substrate and the mediator as in the physical experiments depicted in Fig. 3. Figure 6 shows the calculated values of the one-dimensional modeling error ν. As one can see in Fig. 6, the absolute value |ν| of the modeling error is less than 10% when the perforation level ρ is greater than 0.002. The real biosensor was modeled at the perforation level ρ = 0.0625 (see Fig. 3). At this value of the level ρ the modeling error ν is approximately equal to 0.075. Similar modeling errors are usually admissible for an investigation of the peculiarities of the biosensor response [38]. When the perforation level ρ decreases below 0.002, the error escalates.

3.4 Impact of the Tortuosity in the Perforated Membrane The tortuosity θ3 introduced for the perforated membrane varies due to the structural properties of the carbon nanotubes sinked into the holes. The tortuosity impacts the effective diffusion coefficient of the substances in the perforated membrane. In order

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0.14 0.12 0.10 0.08

1 2 3

0.06 0.04 0.02 0.00 -0.02 -0.04 0.001

0.01

0.1

1

3

Fig. 7 The modeling error ν versus the membrane tortuosity θ3 calculated at the perforation level ρ = 0.0625 [39]. The other parameters and the notation are the same as in Fig. 6

to investigate the impact of the tortuosity on the accuracy of the one-dimensional model, multiple numerical experiments were performed using both models, and the relative error ν of the one-dimensional modeling was calculated. The simulation results are shown in Fig. 7. As one can see in Fig. 7, the error ν of the one-dimensional modeling decreases with decreasing the tortuosity θ3 . The maximum error reaches at the theoretical maximum tortuosity θ3 = 1. Although the error ν is quite high (ν ≈ 0.12) at θ3 = 1, it is still at the level allowing one to use the one-dimensional model for approximate estimations of the biosensor behaviour.

4 Carbon Nanotube Based Unmediated Biosensor This section presents a one-dimensional-in-space mathematical model of carbon nanotubes based mediatorless biosensor [9]. The biosensor response and sensitivity are analysed by changing the model parameters with a special emphasis on the mediatorless transfer of the electrons in the layer of the enzyme-loaded carbon nanotubes [9, 44–46]. The numerical solution of the mathematical model is compared with experimental data [9, 45, 46].

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Modeling Carbon Nanotube Based Biosensors

4.1 Principal Structure of the Biosensor The biosensor being modeled has the layered structure and is composed of different materials and sizes [9, 44–46]. The active surface of the biosensor is built by binding the mesh of SWCNTs to a perforated membrane. Some of the nanotubes are sinked into the holes of the membrane during the preparation procedure. CNT’s were preliminary oxidized using laccase from Basidiomycete Lac. After this procedure the layer of CNT was thoroughly washed with distilled water up to total clearing of laccase and covered by the layer of the enzyme. The changeable enzyme layer of proposed CNT based biosensor was designed by immobilization of soluble type of pyrroloquinoline quinone dependent glucose dehydrogenase from Acinetobacter calcoaceticus L.M.D. 79.41 to the semi-permeable membrane of terylene [9, 44– 46]. All electrochemical experiments were performed using a conventional threeelectrode system containing a planar CNT electrode as a working electrode, a platinum wire as a counter electrode and an Ag/AgCl in saturated KCl as a reference electrode. 0.05 M acetate buffer (pH 6.0) containing 1 mM of Ca2+ was used as a default buffer. The steady state current of the biosensors was recorded at 0.4 V. Principal structure of this biosensor is practically identical to that shown in Fig. 1, but the reaction network is different [9, 44– 46]. Enzymatic reaction was employed in the biosensor to selectively detect the substrate (S) in the target analyte. The enzymatic reaction takes place in the regions of the biosensor filled with the enzyme, k1

Eox + S −→ Ered + P,

(39)

where k1 is an enzymatic reaction rate constant. In the reaction, the substrate S reacts with the oxidized form of the enzyme (Eox ) and reduces it (Ered ) producing the product P. The latter is considered as not impacting the processes in the biosensor and therefore is omitted in the model. The output current of the biosensor is generated due to the direct enzyme oxidation taking place in the layer of the carbon nanotubes, k2

Ered −→ Eox + ne e− ,

(40)

where k2 is a constant of the electrochemical reaction rate and ne is the number of electrons released in the enzyme re-oxidation. The released electrons are collected by the CNT electrode [9, 44–46].

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4.2 Mathematical Model The mathematical model for the biosensor was formulated as a system of nonlinear reaction–diffusion equations with the corresponding initial and boundary conditions [9]. The model was described in the one-dimensional space by applying the homogenization process to the perforated membrane and the mesh of carbon nanotubes [2, 54]. A line perpendicular to the biosensor surface was considered as the domain of the model with the zero point at the surface of the terylene membrane. The model involves four layers (1 , . . . , 4 ) as defined in (32).

4.2.1 Governing Equations The enzyme in the oxidized form participates in the enzymatic reaction (39) in both enzyme-loaded regions, 1 and 2 . Additionally, the enzyme that is properly conjoined with the active sides of the CNTs is also involved in the electrochemical reaction (40) and is re-oxidized. Due to the procedure of the mediatorless biosensor preparation, it was assumed that only a part of the whole enzyme has direct contact with the electro-active sides of the CNTs, and only that part of the enzyme is involved in the electrochemical reaction (40) [43, 44, 46]. The dynamics of the concentration of the oxidized enzyme was mathematically described as follows (t > 0) [9]: ∂Eox,j = −k1 Eox,j Sj , ∂t ∂Ee,ox = −k1 Ee,ox S2 + k2 Ee,red , ∂t

x ∈ j , j = 1, 2, (41) x ∈ 2 ,

where x stands for the space, t is time, Sj = Sj (x, t) is the substrate concentration in the j − th layer j , Eox,j = Eox,j (x, t) is the concentration of the oxidized enzyme that is not involved in the reaction (40). Ee,ox = Ee,ox (x, t) and Ee,red = Ee,red (x, t) are the concentrations of the enzyme involved in the reaction (40) in the oxidized and the reduced forms, respectively [9]. The dynamics of the concentration of the reduced enzyme is practically opposite to that of the oxidized enzyme. The reduced enzyme Ered is produced in the enzymatic reaction (39) in both the enzyme-loaded layers. The electrochemically active part of the reduced enzyme (Ee,red ) is additionally re-oxidized in the electrochemical reaction (40) (t > 0), ∂Ered,j = k1 Eox,j Sj , ∂t ∂Ee,red = k1 Ee,ox S2 − k2 Ee,red , ∂t

x ∈ j , j = 1, 2, (42) x ∈ 2 ,

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where Ered,j = Ered,j (x, t) is the concentration of the reduced enzyme that is not involved in the reaction (40) [9]. The molecules of both forms of the enzyme, Eox and Ered , were considered as immobilized, and therefore there was no diffusion terms in the corresponding equations [9]. The substrate S diffuses from the bulk through the holes of the perforated membrane to inner layers of the biosensor. The substrate also participates in the enzymatic reaction (39) in the enzyme-loaded layers (1 and 2 ). The dynamics of the substrate concentration was described by the following reaction–diffusion equations (t > 0) [9]: ∂ 2 S1 ∂S1 = DS1 − k1 Eox,1S1 , ∂t ∂x 2

x ∈ 1 ,

∂S2 ∂ 2 S2 = DS2 − k1 Eox,2S2 − k1 Ee,ox S2 , ∂t ∂x 2 ∂Sj ∂ 2 Sj = DSj , ∂t ∂x 2

x ∈ j ,

x ∈ 2 ,

(43)

j = 3, 4,

where DS1 and DS4 are the constant diffusion coefficients of the substrate S in the enzyme and the diffusion layers, respectively. The coefficients DS2 and DS3 stand for the effective diffusivity of the substrate in the CNT layer and the perforated membrane, respectively. The effective diffusion coefficients DS2 and DS3 can by calculated by (35) [9].

4.2.2 Boundary Conditions Assuming well-stirred buffer solution leads to the constant thickness of the diffusion layer as well as the constant concentration above that layer, S4 |4 = S0 ,

t > 0,

(44)

where S0 is the concentration of the substrate in the bulk solution. The terylene membrane is placed between the enzyme and the insulating layer of the biosensor and plays a role of an insulating film immobilizing the enzyme. Assuming low volume of the membrane and the insulating layer behind it, the nonleakage condition was used for the substrate on the surface of the terylene membrane (0 ),  ∂S1  = 0, t > 0. (45) ∂x 0

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On the boundaries between adjacent regions (j and j +1 , j = 1, 2, 3), the merge conditions were defined for the substrate (t > 0) [9]: DSj

  ∂Sj  ∂Sj +1  = D , Si+1 ∂x j ∂x j

  Sj  = Sj +1  , j

j

j = 1, 2, 3.

(46)

The diffusion layer 4 (x3 < x < x3 + d4 ) was treated as the Nernst diffusion layer [9]. According to the Nernst approach a layer of thickness d4 remains unchanged with time [12].

4.2.3 Initial Conditions The modeled experiment starts (t = 0) when the substrate is poured into the buffer solution. At this time, the substrate is absent in the entire biosensor and the diffusion layer, except only the outer boundary 4 , where the substrate concentration is considered equal to that in the bulk solution, Sj = 0,

x ∈ j \ 4 ,

S4 |4 = S0 ,

j = 1, 2, 3, 4, t = 0,

(47)

where j is the closed region corresponding to the open region j . At the beginning of the experiment all the enzyme was assumed to be in the oxidized form. The initial concentrations of the oxidized (Eox,1) and the reduced (Ered,1) enzyme in the enzyme layer 1 are defined as follows: Eox,1 = E0 ,

Ered,1 = 0,

x ∈ 1 ,

t = 0,

(48)

where E0 is the enzyme concentration in the layer filled with the enzyme only (1 ) [9]. Due to the procedure of the mediatorless biosensor preparation, the concentration of the enzyme in the CNT layer is assumed to be lower than in the enzyme layer [43, 46]. Moreover, only a part of the enzyme is properly conjoined with the CNTs to be electrochemically active. Assuming a uniform distribution of the enzyme in the CNT layer leads to following conditions (t = 0): Eox,2 = (1 − α)ηE0 , Ee,ox = αηE0 , Ered,2 = Ee,red = 0,

x ∈ 2 ,

(49)

where η (0 ≤ η ≤ 1) is the ratio of the enzyme concentration in the CNTs to that in the enzyme layer 1 , and α (0 ≤ α ≤ 1) is the volume fraction of the electrochemically active enzyme in the CNT layer [9].

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4.2.4 Biosensor Response and Sensitivity The output current of the biosensor is generated due to the enzyme re-oxidation in the electrochemical reaction (40) taking place only in the enzyme-loaded CNT electrode (region 2 ). It was assumed that only the enzyme molecules properly attached to the CNTs (Ee,red ) can be re-oxidized in the reaction (40). The density i(t) of the output current is proportional to the rate of the electrochemical reaction (40), ! i(t) = ne F k2

x2

Ee,red dx,

(50)

x1

where F is the Faraday constant [9]. The density I of the steady state current is expressed exactly as for the corresponding mediated biosensor, i.e. by (25). The sensitivity is one of the most important characteristic of the biosensor operation [48, 51]. The normalized sensitivity to the substrate concentration BS was measured when investigating the impact of the model parameters on the behaviour of the biosensor, BS (S0 ) =

d I (S0 ) S0 . × d S0 I (S0 )

(51)

4.3 Numerical Simulation The formulated mathematical model (41)–(50) was defined as an initial boundary value problem involving nonlinear reaction terms. The structure of the mathematical model is very similar to that of the mediated biosensor discussed above in this chapter. Because of this the numerical solution is also very similar to that discussed above. The basic configuration of the biosensor was accepted almost the same as for the corresponding mediated biosensor as defined in (30) and (37) [9], d1 = 10−7 m,

d2 = 4 × 10−7 m,

De = 3 × 10−10 m2 s−1 , η = 0.5,

θ2 = 1/3,

d3 = 10−5 m,

Dn = 2De , θ3 = 0.5,

d4 = 3 × 10−4 m,

k1 = 6.9 × 105 M−1 s−1 ,

ρ = 0.0625,

ne = 2. (52)

4 Carbon Nanotube Based Unmediated Biosensor

369

4.4 Experimental Model Validation The mathematical model (41)–(50) and the corresponding numerical solution were validated by experimental data. The experimental data along with the corresponding simulated responses of the biosensor are shown in Figs. 8 and 9 [9]. The concentration of the enzyme involved in the electrochemical reaction (40) is complicated to measure experimentally. The volume fraction α of the electrochemically active enzyme highly depends on the properties of carbon nanotubes and the technology of the electrode preparation [44]. The rate constant k2 of the electrochemical reaction (40) is also very specific for the developed CNT electrode. Multiple numerical simulations were performed in order to estimate the values of the α- and k2 -parameters. These two parameters were fitted by minimizing the relative error between the simulated and experimental responses. A satisfactory match was obtained at α = 0.005 and k2 = 550 s−1 , when only 0.5% of the enzyme is assumed as participating in the electrochemical reaction (40) in the CNT layer. The steady state responses of the simulated as well as physical experiments are shown in Fig. 8. As it can be observed in Fig. 8, the responses obtained in numerical experiments using α = 0.005 are close to the experimental data. At that value of α, the relative error of the simulated steady state responses is less than 10%, except two concentrations S0 of the substrate: S0 = 0.199 and 0.299 mM. At low substrate concentrations the relative error reaches almost 20%. Taking into account relatively low biosensor currents and possible measuring errors in the physical experiments,

6

I, mA/m2

5 4

1 2 3 4

3 2 1 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

S0, mM Fig. 8 The density I of the steady state current obtained experimentally (1) and numerically (2, 3, 4) [9]. The simulation was performed at k2 = 550 s−1 , E0 = 45.5 µM and three values of the ratio α : 0.0075 (2), 0.005 (3) and 0.0033 (4). The other parameters are as defined in (52)

370

Modeling Carbon Nanotube Based Biosensors

the error of the simulated responses can be considered admissible. The fitted value 550 s−1 of the electrochemical reaction rate constant k2 suitably matches with reported values of apparent heterogeneous electron transfer rate constants [17]. Figure 8 also shows how sensitive is the biosensor response to the α-parameter. The steady state current directly depends on the α. Increasing the volume fraction α of the electrochemically active enzyme increases the biosensor current and prolongs the linear part of the biosensor calibration curve. The proposed model was also validated at the transient conditions. The simulation was carried out using the same biosensor configuration as in the simulations shown in Fig. 8. The dynamics of the simulated density i(t) of the biosensor current along with the corresponding experimental data are shown in Fig. 9. The scattering of the experimental data shown in Fig. 9 are caused by the measurement errors. Due to relatively low output currents, the errors are relatively high. Taking into account the measurement errors, the simulated responses can be assumed adequate to that observed in the physical experiments. When modeling the corresponding mediated biosensor, no similar parameter was used [8]. The response of the mediated biosensor based on an enzymeloaded CNTs was modeled assuming whole enzyme involved in the corresponding mediated electrochemical reaction. Multiple numerical simulations of the response of the mediatorless biosensor showed that involving the whole enzyme into the electrochemical reaction (40) leads to the steady state currents being in about two order greater than the experimental ones. Because of this, the volume fraction α of

4.0

1 2 3 4

3.5

i, mA/m2

3.0 2.5

5 6 7 8

2.0 1.5 1.0 0.5 0.0 0

20

40

60

80

100

120

140

160

180

t, s Fig. 9 The dynamics of the density i(t) of the output current obtained numerically (1, 2, 3 and 4) and experimentally (5, 6, 7 and 8) at four substrate concentrations (S0 ): 0.099 (1, 5), 0.99 (2, 6), 1.96 (3, 7) and 2.92 mM (4, 8) [9]. The simulation was performed at α = 0.005, while the other parameters are the same as in Fig. (8)

4 Carbon Nanotube Based Unmediated Biosensor

371

the electrochemically active enzyme is of crucial importance when modeling the mediatorless biosensor based on enzyme-loaded CNT electrode.

4.5 Impact of Enzyme Concentration The proper conjunction of the enzyme with the active sites of the CNTs is essential for the mediatorless biosensor operation. The proposed model of the biosensor assumes the proportional dependence between the total concentration of the enzyme and the concentration of the enzyme capable to participate in the electrochemical reaction (40). To investigate the impact of the enzyme concentration E0 on the behaviour of the biosensor action, the biosensor responses were simulated in wide ranges of the enzyme (E0 ) as well as the substrate (S0 ) concentrations [9]. The biosensor current varies in orders of magnitude with changing the concentrations of both these species, therefore the dimensionless sensitivity BS was considered instead of the density I of the steady state current. The simulation results are provided in Fig. 10 [9]. As one can see in Fig. 10, an increase in the enzyme concentration E0 noticeably prolongs the linear part of the biosensor calibration curve, ensuring qualitative determination of higher substrate concentrations. The linear part of the biosensor calibration curve corresponds to BS ≈ 1. At the high enzyme concentrations (E0 > 40 mM) the length of the linear part of the calibration curve is practically pro1.0 0.9 0.8 0.7

BS

0.6 0.5 0.4 0.3 0.2

1 2 3 4 5 6

0.1 0.0 10-3

10-2

10-1

100

101

102

103

S0, mM Fig. 10 The dimensionless biosensor sensitivity BS versus the substrate concentration S0 at k2 = 550 s−1 , α = 0.005 and six enzyme concentrations E0 : 0.0455 (1), 0.455 (2), 4.55 (3), 45.5 (4), 455 (5) and 4550 mM (6) [9]. Other parameters are as defined in (52)

372

Modeling Carbon Nanotube Based Biosensors

portional to the enzyme concentration E0 , while at the low enzyme concentrations (E0 < 4 mM) the sensitivity of the biosensor loosely depends on the concentration E0 . The especially drastic change in the sensitivity appears at the moderate values of E0 . Figure 10 shows that a tenfold increase of the enzyme concentration from 45.5 (curve 4) to 455 mM (curve 5) leads to an approximately 50 times longer linear part of the biosensor calibration curve. The noticeable change in the behaviour of the biosensor sensitivity (Fig. 10) at the moderate enzyme concentrations could be explained by the transition from the kinetic-limited to the diffusion-controlled mode of the biosensor action [7, 10, 49]. The diffusion module σ 2 essentially compares the rate of the enzyme reaction (k1 αE0 ) with the mass transport through the enzyme-loaded layer (DS2 /d22 ), σ 2 = k1 αE0 d22 /DS2 . The biosensor response is known to be under diffusion control when the diffusion module σ 2  1. In the very opposite case, when σ 2  1, the enzyme kinetics predominates in the response. The diffusion module σ 2 approaches one at the concentration αE0 of the electrochemically active enzyme approximately equal to 0.75 mM. This value of αE0 favourably matches with the enzyme concentrations at which the behaviour of the biosensor sensitivity distinctly changes (see Fig. 10).

4.6 Impact of Electrochemical Reaction Rate Mathematical models for amperometric biosensors are usually formulated assuming electrochemical reactions to be extremely fast when all the available reagent is immediately consumed [7, 49]. In modeling the mediatorless biosensor, it was assumed that only a small fraction of the enzyme participates in the electrochemical reaction taking place in the CNT electrode [9]. Because of the relatively spare sites, where the direct electron transfer is possible, the electrochemical reaction (40) cannot be assumed very fast. The appropriate value 550 s−1 of the k2 -parameter was fitted by minimizing the relative error between the simulated and experimental responses. In order to show the impact of the rate constant k2 of the electrochemical reaction (40) on the biosensor response, the response was numerically simulated and the corresponding dimensionless biosensor sensitivity BS was calculated at different values of the k2 -parameter by changing the substrate concentrations S0 in a wide range. The simulation results are depicted in Fig. 11 [9]. As one can see in Fig. 11 an increase in the electrochemical reaction rate constant k2 proportionally shifts the curve representing the sensitivity BS to the right. Thus an increase in the coefficient k2 proportionally prolongs the linear part of the biosensor calibration curve. The shape of the curve remains unchanged for all k2 used in the investigation.

5 Concluding Remarks

373

1.0 0.9

1 2 3 4

0.8 0.7

BS

0.6 0.5 0.4 0.3 0.2 0.1 0.0 10-3

10-2

10-1

100

101

102

103

S0, mM Fig. 11 The dimensionless biosensor sensitivity BS versus the substrate concentration S0 at α = 0.005, E0 = 0.0455 mM and four values of the electrochemical reaction rate constant k2 : 5.5 (1), 55 (2), 550 (3) and 5500 s−1 (4) [9]. Other parameters are as defined in (52)

5 Concluding Remarks The sensitivity of biosensors based on the enzyme-loaded CNT electrode can be noticeably increased and the linear part of the calibration curve can be prolonged by selecting an appropriate geometry of the biosensor (Figs. 5 and 8) [8, 9]. The biosensor sensitivity can be also significantly increased by increasing the enzyme concentration. The linear part of the calibration curve is longer when the biosensor acts in the diffusion limiting mode rather than in the enzyme reactioncontrolled mode (Fig. 10) [9]. One-dimensional-in-space mathematical modeling can be successfully used to simulate the response of the biosensor based on the carbon nanotubes deposited on the perforated membrane at different configurations of the biosensor operation (Figs. 6 and 7). However, the accuracy of the simulated response using onedimensional model considerably depends on the geometry of the outer perforated membrane [39]. The numerical simulation showed that only a fraction of the enzyme participates in the direct electron transfer at the nanoscale (Fig. 8) [9, 46]. The overall rate of the electrochemical reaction (40) for the mediatorless biosensor was determined to be relatively low. Therefore, the electrochemical reaction rate constant k2 should be used when modeling the mediatorless biosensor action.

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Modeling Biosensors Utilizing Microbial Cells

Contents 1 2 3 4

5

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metabolite Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BOD Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Bacterial Self-Organization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 3D Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Dimensionless Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 3D Simulation of Population Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Population Dynamics Near the Top Surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Population Dynamics Near the Contact Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Population Dynamics Near the Lateral Surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

378 379 382 384 384 387 389 391 394 397 400

Abstract This chapter deals with the field of biosensors in which the biological component consists of microbial cells. Whole-cell biosensors provide an opportunity to elicit functional information for different applications including drug discovery, cell biology, toxicology and ecology. Metabolite and biochemical oxygen demand (BOD) biosensors as special cases of biosensors based on microorganisms are mathematically considered at steady state and transient conditions. This chapter also considers the bacterial self-organization in the fluid cultures of luminous E. coli in a small rounded container as detected by bioluminescence imaging. Assuming that the luminescence in experiments is proportional to the cell density, the threedimensional pattern formation in a bacterial colony is modeled by the nonlinear reaction–diffusion-chemotaxis equations in which the reaction term for the cells is a logistic (autocatalytic) growth function. The numerical simulation showed that the developed model captures fairly well the sophisticated patterns observed in the experiments. Since the simulation based on three-dimensional model is very time-consuming, the reducing spatial dimensionality for simulating one and twodimensional spatiotemporal patterns is investigated. The patterns simulated by the models of different dimensionality are compared with each other and with the experimental patterns © Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_12

377

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Modeling Biosensors Utilizing Microbial Cells

Keywords Microbial cell based biosensor · Metabolite · Biochemical oxygen demand · Bacterial self-organization · Chemotaxis · Steady-state/transient conditions · Numerical simulation

1 Introduction Enzymes are most widely used biological recognition elements because they are highly selective and sensitive [15]. However, relatively low stability and the need for costly and tedious purification is often a limitation of constructing enzyme-based biosensors [52]. Intact cells of microorganisms contain all enzymatic systems necessary for their vital functions. They transform chemical and other forms of energy with high efficiency, possess the necessary stability and regeneration. In the cells, consecutive selective transformations of the substrate occur under the action of poly enzyme complexes and systems in which coenzymes are involved. The application of whole cells for biosensors building are an alternative to enzyme-based biosensors since they offer the benefits of low cost and improved stability [8, 39]. A microbial biosensor is an analytical device which integrates microorganism(s) with a transducer to generate a measurable signal proportional to the analyte concentration [15, 41, 52]. Whole-cell biosensors of key metabolites are ideal for the monitoring of carbon flow in important metabolic pathways, thus guiding metabolic engineering for microbial improvement [33]. The special case of biosensors based on microorganisms is the so-called BOD (biochemical oxygen demand) biosensor [30]. The BOD biosensor determinates organic compounds that can be oxidized by a microbial cell. The ability of microorganisms to consume oxygen in the presence of organic compounds leads to the development of BOD determination methods. These biosensors are used for water quality management, ecology and environmental science [8, 41]. In this chapter, a BOD flow-through electrode based on the yeast cells Hansenula anomala is mathematically investigated at steady state conditions [30]. The microbial cells can be genetically modified that makes it possible to obtain recombinant organisms with determined biocatalytic properties [8, 41]. The specificity, selectivity and stability of the cell-based analytical systems can be improved by genetic engineering. Genetically engineered cell-based sensing systems can elicit a response in the presence of an analyte by coupling the sensing element to a reporter gene through gene fusion, which upon expression produces a readily measurable signal [14]. Lux-gene engineered bacteria are among the biological cells successfully used to develop whole cell-based biosensors [14]. These biosensors work properly when suspensions of bacteria are diluted and well stirred. Dense and static suspensions of bacteria may exhibit fluctuating/oscillatory signals [45, 47]. Bacterial growth and movement in confined suspensions often result in the emergence of millimeter-scale patterns, mainly near the contact lines and surfaces [7, 9, 11, 45, 54]. The interaction of several active processes in the living suspensions leads to very complex dynamic systems [12, 48].

2 Metabolite Biosensor

379

Chemotaxis, as the direct movement of cells along the gradient of certain chemicals in their environment, is one of the main phenomena determining the pattern formation [19]. A vast amount of research, both experimental and theoretical, has been devoted to understanding the chemotaxis, and the analysis of pattern formation in chemotactic systems is in full swing [12, 21]. Since the pioneering work by Keller and Segel [27] mathematical modeling plays a crucial role in understanding the mechanism of chemotaxis [20]. Quite a number of mathematical models based on reaction-advection-diffusion equations have been developed for modeling the pattern formation in bacterial colonies [12, 17, 34, 40, 42, 53, 56]. However, the system introduced by Keller and Segel remains among the most widely used [18, 20, 22, 31, 37, 38]. A special emphasis in this chapter was placed to the bacterial self-organization in the fluid cultures of luminous E. coli in a small rounded container as detected by bioluminescence imaging [45, 47]. Assuming that the luminescence in experiments is proportional to the cell density, the three-dimensional (3D) pattern formation in a luminous E. coli colony is modeled by the nonlinear reaction–diffusion-chemotaxis equations in which the reaction term for the cells is a logistic (autocatalytic) growth function [3, 4, 6, 32, 46, 48–50]. The numerical simulation at the transient conditions was carried out using the finite difference technique [10, 35, 43]. The numerical simulation showed that the developed model captures fairly well the sophisticated patterns observed in the experiments. Since the simulation based on 3D model is very time-consuming, therefore reducing spatial dimensionality in a model for simulating 1D and 2D spatiotemporal patterns is investigated [6]. The patterns simulated by the models of different dimensionality are compared with each other and with the experimental patterns. Due to the accumulation of luminous cells near the top three-phase contact line the experimental patterns of the bioluminescence were qualitatively simulated using 1D and 2D models by adjusting values of the diffusion coefficient and/or chemotactic sensitivity.

2 Metabolite Biosensor The biocatalytical system of microbial cells can be used as biocatalysers for the biosensor preparation [28, 30]. They can show very high specificity for some substrates. For example, yeast cells Hansenula anomala grown in lactate reach breeding media induce cytochrome b2 , and show high specificity to L-lactate [29]. The scheme of substrates distribution in a microbial biosensor is depicted in Fig. 1. The peculiarity of the modeling of the microbial biosensors is a slow substrate and product transport through the microbial cell wall. If the substrate transport is slower than the diffusion through the bulk solution and the semipermeable

380 Fig. 1 The scheme of the substrates distribution in a microbial biosensor. Transducer (1), biocatalytical membrane with microbial cell (2), semipermeable membrane (3), bulk solution (4), radius of cell (R), thickness of cell wall (l)

Modeling Biosensors Utilizing Microbial Cells 1 Pe

Sc R

2

l Pc

S0

3 S0

4

membrane the substrate and the product concentration change in the cell can be written: dSc = k(S0 − Sc ) − Vc , dt dPc = Vc − k  Pc , dt

(1)

where t is time, Sc and Pc are the concentrations of the substrate and the product in the cell, respectively, k and k  are constants of substrate transport into the cell and the product transport from the cell, respectively, and Vc is the rate of the enzymatic process in the cell. The constants k and k  are related to permeability (h) of the cell wall that can be expressed as hs , l hp surc , k = volc k=

(2)

where surc and volc are the surface area and the volume of the microbial cell, respectively. For permeability calculations the equilibrium distribution of the substrate and the product between the cell wall and the solution is used, hs =

Ds Sm,eq , l S0,eq

hp =

Dp Pm,eq . l Pc,eq

(3)

2 Metabolite Biosensor

381

If the kinetics of the substrate conversion in the cell obeys the Michaelis–Menten scheme (see Chapter “Introduction to Modeling of Biosensors”) the substrate concentration change in the cell can be rewritten dSc Vmax,c Sc = k(S0 − Sc ) − . dt KM,c + Sc

(4)

This equation can be integrated with the initial conditions Sc = 0 at t = 0. A more simple solution can be produced at Sc  KM,c : Sc =

− exp



−t kKM,c −t Vmax,c +ln(kS0 )KM,c KM,c V k + Kmax,c M,c



+ kS0 .

(5)

Inserting Sc solution into (1) gives a very bulky expression of Pc [30]. Therefore, it is omitted. To simplify further analysis the biosensor action was analysed at the steady state conditions when dSc /dt and dPc /dt equals zero. In this case Pc =

ρ  S0 , 1+ρ

(6)

where dimensionless parameters ρ and ρ  represent the ratios of the enzymatic reaction in the cell and the compounds transfer, ρ=

Vmax,c , KM,c k

Vmax,c ρ = . KM,c k 

(7)



The concentration Pc of the product produced in the cell is the basis for the calculation of the product concentration Pe at the transducer since tp Pe dPe = k  Pc − , dt δ

(8)

where tp and δ are the transfer coefficient and the thickness of the semipermeable membrane. At the steady state conditions, Pe is equal to Pe =

δ Vmax,c S0 . KM,c (1 + ρ)tp

(9)

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The response E of the biosensor measured as an electrode potential in the case of the potentiometric transducer is E = E0 +

RT RT ln Pe = ln S0 + C, ne F ne F

(10)

where E0 is a characteristic constant for the ion-selective electrode, R is the universal gas constant, T is the absolute temperature (K) and C is a constant. In the case of the amperometric transducer the expression of the response is more complex due to the product consumption at the electrode. This can be accounted for by adding a transfer rate (tp /δ) and a constant of the electrochemical conversion (kel ). Therefore, the response of the biosensor can be calculated as I = ne F

ρ kel k  kel k  S0 . Pc = ne F tp /δ + kel tp /δ + kel 1 + ρ

(11)

3 BOD Biosensor The BOD (biochemical oxygen demand) biosensor action was modeled assuming that the rate of oxygen consumption by the cells in the layer near the transducer (Clark type oxygen electrode) is determined by the substrate concentration [30]. If the rate of the substrate conversion in the cells is limited by the transport through the cell wall, the substrate concentration in the layer near the electrode surface is determined by the diffusion from the external solution and the transport into the cell. At a steady state Vs S Ds S0 − S = , d d Ks + S

(12)

where Ds is the diffusion coefficient of the substrate through the semipermeable membrane, d is the thickness of the layer, S0 and S are the concentrations of the substrate in the external solution and in the layer near the oxygen electrode, respectively, Vs is the rate of transport through the cell membrane at saturating concentrations of the substrate and Ks is the saturation constant [30]. The solution of the equation gives the expression for the relative concentration of the substrate in the layer near the oxygen electrode: 1 − (Ks /S0 )(1 + ρ) S + = S0 2



1 − (Ks /S0 )(1 + ρ) 2

2

Ks + S0

1/2 ,

(13)

where the dimensionless module ρ = (Vs /Ks )(d 2 /Ds ) quantitatively characterizes the diffusion limitations [30].

3 BOD Biosensor

383

In the case of two extreme concentrations of the substrate the solution of (6) is simpler: S0 − S = ρKs

at S0  Ks +

Vs d 2 Ds

(14)

and S=

S0 1+ρ

at S0  Ks +

Vs d 2 . Ds

(15)

To calculate the stationary concentration O2 of the oxygen it was assumed that the substrate flow through the membrane is equal to the oxygen flow:  1 DO2  Ds O2 0 − O2 , (S0 − S) = d n d

(16)

where DO2 is the diffusion coefficient of oxygen, n is the stoichiometric coefficient of the substrate conversion [30]. The decrease in the oxygen electrode current is proportional to the oxygen concentration change: I0 − I Ds O2 0 = O2 0 − O2 = n (S0 − S) . I0 DO2

(17)

At saturating substrate concentration Vs d 2 I0 − I O2 0 = n . I0 DO2

(18)

At a low concentration of S0 Ds I0 − I ρ S0 . O2 0 = n I0 DO2 1 + ρ

(19)

Thus, at a high substrate concentration the maximal decrease in the current is determined by the rate of transport into the cell and diffusion ratio. At low concentrations, the response of the sensor is directly proportional to S0 . The sensitivity varies with the value of the module. When ρ  l the sensor operates under diffusive conditions and the sensitivity is maximal. At ρ  l the action of the sensor is limited by the kinetic parameters and the sensitivity is determined by the rate of the transport of the substances into the cell, Vs d 2 I0 − I O2 0 = n S0 . I0 DO2 Ks

(20)

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The examination of the calibration curves for the microbial biosensor showed that a 50% consumption of oxygen took place at substrate (L-lactate) concentration much higher than Ks indicating an important role of diffusion [28].

4 Modeling Bacterial Self-Organization This section considers the spatiotemporal pattern formation in the fluid cultures of luminous E. coli placed in a rounded glass container [3–6, 32, 46, 48–50]. The spatiotemporal patterns in the fluid cultures of E. coli have been observed by employing lux-gene engineered cells and a bioluminescence imaging technique [45, 47]. Assuming the direct proportionality between the bioluminescence and the number of active cells the bacterial self-organization is modeled by the dynamics of the density of bioluminescent cells [46].

4.1 3D Mathematical Model The container is modeled by a right circular cylinder. Figure 2 shows the principal structure of the rounded container, where r and h are the base radius and the height of the cylinder, respectively. For simplicity, it was assumed that the fluid fills the container [6]. According to the Keller and Segel approach, the dynamics of the bacteria, chemoattractant and nutrient (stimulant) are modeled mathematically and give rise to a system of nonlinear partial differential equations [11, 27]. Assuming that the liquid medium contains sufficient nutrient for the cells (or organism), two governing equations are mostly used to describe the dynamics of the cell density and concentration of the chemical signal (chemoattractant) under quasi-steady state assumption [21, 38, 55, 57]. Fig. 2 Principal structure of the rounded container containing bacterial population [6]

z r

h

ρ

4 Modeling Bacterial Self-Organization

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4.1.1 Governing Equations Translating the main biological processes into a mathematical model leads to a system of three conservation equations [11, 55, 57], ∂n = Dn n − ∇ (fs (n, c)n∇c) + fg (n, s), ∂t ∂c = Dc c + gp (n, c)n − gd (n, c)c, ∂t ∂s = Ds s − h(n, s), x ∈  ⊂ Rn , t > 0, ∂t

(21)

where  is the Laplace operator, x and t stand for space and time, n(x, t) denotes the cell density, c(x, t) is the chemoattractant concentration, s(x, t) is the concentration of a nutrient (succinate, stimulant), Dn , Dc and Ds are the diffusion coefficients usually assumed to be constant, fg (n, s) stands for cell growth and death, fs (n, c) denotes the chemotactic sensitivity, gp and gd stand for the production and degradation of the chemoattractant, respectively, and h(n, s) stands for the nutrient consumption. The function fs (n, c) controls the chemotactic response of the cells to the chemoattractant. The form of fs (n, c) depends on the sensitivity of cells at different concentrations of the attractant [37]. It was shown that the sensitivity of cells to attractant can be successfully assumed to be independent of the chemoattractant concentration when modeling the bacterial self-organization in a rounded container along the contact line as detected by bioluminescence imaging [3], i.e. fs (n, c) can be constant, fs (n, c) = k1 . The cell growth is usually assumed to be logistic, i.e. fg (n, c) = k2 n(1 − n/k), where k2 is the growth rate of the cell population, and k is the cell density under steady state conditions or the carrying capacity [18, 20, 21, 38, 40]. Although k2 and k are often assumed to be constant, in a more general case k is assumed to be a linear function of the nutrient concentration, i.e. k = k3 s [6]. The linear dependence between carrying capacity and limited resources was also explored, for example, in certain population growth models [23, 26]. A number of chemoattractant production functions have been used in chemotactic models [20]. Usually, a saturating function of the cell density is used indicating that, as the cell density increases, the chemoattractant production decreases. The Michaelis–Menten function is widely used to express the chemoattractant production, gp (n, c) = k4 /(k5 + n) [27, 34, 37]. The degradation or consumption of the chemoattractant is typically linear, gd (n, c) = k6 , where k6 is a constant [20]. Consumption h(n, s) of the nutrient was assumed to be directly proportional to the population density, h(n, s) = k7 n. Similar approach was used in investigating the rate of dissolved oxygen consumption by different viable cells in a bioreactor [25], as well as in certain population growth models [23, 26].

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Modeling Biosensors Utilizing Microbial Cells

Inserting the concrete expressions of fs , fg , gp , gd and h to system (21) leads to the following governing equations of the population kinetics model:  n ∂n = Dn n − ∇ (k1 n∇c) + k2 n 1 − , ∂t k3 s ∂c k4 n = Dc c + − k6 c, ∂t k5 + n ∂s = Ds s − k7 n, ∂t

x ∈ ,

(22) t > 0,

where k1 is the chemotactic sensitivity, k2 is the growth rate of the cell population, k3 stands for the cell density under steady state conditions, k4 and k5 stand for saturating chemoattractant production, k6 and k7 are the consumption rates of the chemoattractant and the nutrient, respectively, the other notations are the same as in model (21). All the parameters are assumed to be constant and positive [6, 20]. Assuming the rounded container as a right circular cylinder, the mathematical model of the bacterial self-organization in the container (domain ) can be defined in cylindrical coordinates, x = (ρ, ϕ, z),  = (0, r) × (0, 2π) × (0, h),  ∂ 2F 1 ∂ ∂F 1 ∂ 2F + , F = ρ + 2 ρ ∂ρ ∂ρ ρ ∂ϕ 2 ∂z2

(23)

where r and h are the base radius and the height of the cylinder  as shown in Fig. 2.

4.1.2 Initial and Boundary Conditions We assume a possibly non-uniform initial (at t = 0) distribution of cells, chemoattractant and nutrient, n(ρ, ϕ, z, 0) = n0x (ρ, ϕ, z),

c(ρ, ϕ, z, 0) = c0x (ρ, ϕ, z),

s(ρ, ϕ, z, 0) = s0x (ρ, ϕ, z),

(ρ, ϕ, z) ∈ [0, r] × [0, 2π) × [0, h],

(24)

where n0x (ρ, ϕ, z), c0x (ρ, ϕ, z) and s0x (ρ, ϕ, z) stand for the initial (t = 0) cell density, chemoattractant and nutrient concentrations, respectively [6]. The no-leak boundary conditions (t > 0) are applied on the base of the glass vessel, Dn

∂n  ∂c  ∂s  = Dc  = Ds  = 0,  ∂z z=0 ∂z z=0 ∂z z=0

(ρ, ϕ) ∈ [0, r] × [0, 2π).

(25)

4 Modeling Bacterial Self-Organization

387

At the top surface the fluid contacts with the atmosphere containing a nutrient, e.g. oxygen. We assume a constant concentration of nutrient at that surface, while no-leak conditions for the cells as well as for the chemoattractant, Dn

∂n  ∂c  = Dc  = 0,  ∂z z=h ∂z z=h

s(ρ, ϕ, h, t) = s0 ,

(26)

(ρ, ϕ) ∈ [0, r] × [0, 2π).

Due to the continuity in the azimuth direction of the vessel, the periodicity conditions are used in ϕ direction (t > 0), n(ρ, 0, z, t) = n(ρ, 2π, z, t), c(ρ, 0, z, t) = c(ρ, 2π, z, t), s(ρ, 0, z, t) = s(ρ, 2π, z, t),

∂n  ∂n  = Dn  ,  ∂ϕ ϕ=0 ∂ϕ ϕ=2π ∂c  ∂c  Dc  = Dc  , ∂ϕ ϕ=0 ∂ϕ ϕ=2π ∂s  ∂s  = Ds  , Ds  ∂ϕ ϕ=0 ∂ϕ ϕ=2π Dn

(27)

(ρ, z) ∈ [0, r] × [0, h]. The rotational symmetry and non-permeability of the lateral surface of the tube leads to the following boundary conditions: ∂n  ∂n  ∂c  ∂c  = Dn  = 0, Dc  = Dc  = 0,  ∂ρ ρ=0 ∂ρ ρ=r ∂ρ ρ=0 ∂ρ ρ=r ∂s  ∂s  = Ds  = 0, (ϕ, z) ∈ [0, 2π) × [0, h]. Ds  ∂ρ ρ=0 ∂ρ ρ=r

Dn

(28)

4.2 Dimensionless Model In order to define the main governing parameters of the mathematical model (22), (24)–(28), a dimensionless mathematical model has to be derived [20, 36, 37]. A dimensionless model can be derived by setting u=

n , n0

t ∗ = k6 t, χ=

k5 k6 c k3 s Dn Ds , w= , Du = , Dw = , k4 n0 n0 Dc Dc   k k6 6 ρ∗ = ρ, ϕ ∗ = ϕ, z∗ = z, Dc Dc

v=

k1 k4 n0 , k5 k6 Dc

α=

k2 , k6

β=

n0 , k5

γ =

k7 k3 , k6

(29)

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Modeling Biosensors Utilizing Microbial Cells

where n0 is the cell density under steady state conditions. When modeling the carrying capacity by a linear function of the nutrient concentration (k3 s), the cell density under steady state conditions is directly proportional to the concentration s0 of the nutrient near the top surface, n0 = k3 s0 [6]. Dropping the asterisks, the dimensionless governing equations then become  ∂u u = Du u − χ∇ (u∇v) + αu 1 − , ∂t w ∂v u = v + − v, ∂t 1 + βu ∂w = Dw w − γ u, ∂t

(ρ, ϕ, z) ∈ (0, R) × (0, 2π) × (0, H ),

(30) t > 0,

where u is the dimensionless cell density, v is the dimensionless chemoattractant concentration, w is the dimensionless concentration of the nutrient, α is the dimensionless growth rate of the cell population, β stands for saturating of the signal production, γ is dimensionless consumption rate √ of the nutrient,√R and H are the relative radius and height of the cylinder, R = r k6 /Dc , H = h k6 /Dc . The initial conditions (24) take the following dimensionless form: u(ρ, ϕ, z, 0) = u0x (ρ, ϕ, z), w(ρ, ϕ, z, 0) = w0x (ρ, ϕ, z),

v(ρ, ϕ, z, 0) = v0x (ρ, ϕ, z), (ρ, ϕ, z) ∈ [0, R] × [0, 2π) × [0, H ],

(31)

where u0x (ρ, ϕ, z) = n0x (ρ, ϕ, z)/n0 , v0x (ρ, ϕ, z) = k5 k6 c0x (ρ, ϕ, z)/(k4 n0 ) and w0x (ρ, ϕ, z) = k3 s0x (ρ, ϕ, z)/n0 . The boundary conditions (25)–(28) transform to the following dimensionless equations (t > 0): Du

∂u  ∂w  ∂u  ∂v  ∂v  = = Dw = Du  = = 0,     ∂z z=0 ∂z z=0 ∂z z=0 ∂z z=H ∂z z=H

w(ρ, ϕ, H, t) = w0 ,

(32)

(ρ, ϕ) ∈ [0, R] × [0, 2π),

∂u  ∂u  = Du  ,  ∂ϕ ϕ=0 ∂ϕ ϕ=2π ∂v  ∂v  v(ρ, 0, z, t) = v(ρ, 2π, z, t), = ,   ∂ϕ ϕ=0 ∂ϕ ϕ=2π ∂w  ∂w  = Dw , w(ρ, 0, z, t) = w(ρ, 2π, z, t), Dw   ∂ϕ ϕ=0 ∂ϕ ϕ=2π

u(ρ, 0, z, t) = u(ρ, 2π, z, t),

(ρ, z) ∈ [0, R] × [0, H ],

Du

(33)

4 Modeling Bacterial Self-Organization

389

∂u  ∂u  ∂v  ∂v  = Du  = 0, = = 0,    ∂ρ ρ=0 ∂ρ ρ=R ∂ρ ρ=0 ∂ρ ρ=R ∂w  ∂w  = Dw = 0, (ϕ, z) ∈ [0, 2π) × [0, H ]. Dw   ∂ρ ρ=0 ∂ρ ρ=R

Du

(34)

4.3 3D Simulation of Population Dynamics The above described mathematical model has been defined as an initial boundary value problem based on a system of nonlinear partial differential equations. Because of the nonlinearity of the problem, no exact analytical solutions could be derived in the general case [36, 42, 57]. Hence the numerical simulation of the bacterial selforganization was used [6]. The functions used in the initial and boundary conditions were defined as follows [6]: u0x (ρ, ϕ, z) = 1 + 0.2 sin w0x (ρ, ϕ, z) = 1,

 ϕ ∗ 11  ρ 2

R

,

v0x (ρ, ϕ, z) = 0,

(35)

(ρ, ϕ, z) ∈ [0, R] × [0, 2π) × [0, H ].

The numerical simulation was carried out using the finite difference technique [43]. To find a numerical solution of the problem a uniform discrete grid 40 × 224 × 80 was introduced in space directions and the constant dimensionless step size 5 × 10−5 was also used in the time direction. These values were used for all four cases analysed accordingly for the existing dimensions. The simulations have been checked with a variety of space and time discretizations, and verified that the obtained patterns shown below are almost independent of the space and time steps. An explicit finite difference scheme has been built as a result of the difference approximation [43]. The digital simulator has been programmed by the authors in Free Pascal language [6]. Figure 3 shows a visualization of two arbitrary frames of calculated cell density u at the following values of the dimensionless parameters [6]: Du = 0.1, χ = 8.3,

Dw = 0.2, α = 1,

R = 5,

β = 0.73,

H = 10, γ = 0.025.

(36)

One can see in Fig. 3 that the cell density u inside the rounded container forms foam-like structures similar to the experimentally observed structures [46–48]. In the initial stage of the population evolution the cells are non-uniformly distributed in entire container, while later the population concentrates near the top surface were the oxygen concentrates [6]. The bioluminescence along the circular three-phase contact line, which can be considered as quasi-one-dimensional, was experimentally measured using the

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Modeling Biosensors Utilizing Microbial Cells

Fig. 3 Visualization of arbitrary frames at u concentrations obtained by 3D simulation at two time moments: 65 (a) and 329 (b). The snapshot (c) of the experimental culture is shown for comparison [6]

Fig. 4 Spatiotemporal plots of experimental bioluminescence near the three-phase contact line of microtiter plate (a) and of the dimensionless cell density u1D ( = 0.075) simulated by 3D model at values of the parameters defined in (36) (b)

image processing software [46, 48–50]. An example of a space-time plot of the bioluminescence intensity in a microtiter plate is shown in Fig. 4a. The volume of the experimental sample was 0.25 ml and the length of the contact line—19 mm (621 pixels). The patterns of bioluminescence observed during the long run experiments can be characterized by the formation of meandering traveling waves and relative stability of the corresponding wave number (number of waves along the contact line). The characteristic wave speed of the bioluminescent waves was about 7 μm/s [46, 48]. Qualitatively, the observed space-time plot exhibits the so-called merging and emerging dynamics in chemotaxis [20].

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The spatiotemporal pattern formation along the three-phase contact line can be studied by the numerical simulation based on the 3D model. To simulate the spatiotemporal patterns of the quasi-one-dimensional cell density u1D in a vessel near the three-phase contact line, the density u of cells was integrated over the whole depth and a thin ring of the thickness  ( = 0.075 = 0.015R) close to the lateral surface and then averaged, 1 u1D (ϕ, t) = H

!

H 0



2 R 2 − (R − )2

!



R

u(ρ, ϕ, z, t) ρdρ dz, R−

ϕ ∈ [0, 2π],

(37)

t ≥ 0.

The pattern simulated at values of the model parameters defined in (36) is depicted in Fig. 4b. Values (36) of the model parameters were determined experimentally by changing input parameters and aiming to achieve a meandering wave pattern comparable to the one shown in Figs. 3 and 4 [6]. Taking into account the transformation (29) of variables, one can determine values of the dimensional parameters. The dimensional characteristic speed of the bioluminescence waves and some other parameters compare favourably with the corresponding values observed experimentally [46, 48, 49].

4.4 Population Dynamics Near the Top Surface The bacterial self-organization near the inner top surface of a rounded container can be modeled by applying the common 3D mathematical model (22), (24)–(28) as well as the corresponding dimensionless model (30)–(34) [6]. However, transient computational simulations based on 3D mathematical models are extremely time and resource consuming. The dimension reduction is a widely used approach for increasing efficiency of the numerical simulation [53]. When modeling the bacterial self-organization near the top surface of a right circular container, the mathematical model can be defined in the polar coordinates on a 2D domain—a circle [4, 6]. Due to a constant concentration of the nutrient near the top surface (s(ρ, ϕ, h, t) = s0 and w(ρ, ϕ, H, t) = w0 ), the dynamics of the nutrient (oxygen) concentration can be ignored. Due to modeling the carrying capacity by a linear function of the nutrient concentration (k3 s in (22)) and the assumption n0 = k3 s0 , the term k2 n(1−n/(k3s)) of the logistic cell growth reduces to k2 n(1 − n/n0 ), while the corresponding dimensionless term αu(1 − u/w) approaches αu(1 − u).

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The dynamics of the bacterial population near the top surface of a right circular container can be described by the following governing equations formulated in polar coordinates: ∂u = Du u − χ∇ (u∇v) + αu (1 − u) , ∂t ∂v u = v + − v, (ρ, ϕ) ∈ (0, R) × (0, 2π), ∂t 1 + βu

(38) t > 0,

where  is the Laplace operator in the polar coordinates ρ and ϕ, u and v are functions of two parameters ρ and ϕ. The initial conditions (31) take the following form: u(ρ, ϕ, 0) = u0x (ρ, ϕ),

v(ρ, ϕ, 0) = v0x (ρ, ϕ), (ρ, ϕ) ∈ [0, R] × [0, 2π).

(39)

The boundary conditions (32)–(34) reduce to the following equations (t > 0): u(ρ, 0, t) = u(ρ, 2π, t), v(ρ, 0, t) = v(ρ, 2π, t), ∂v  ∂v  ∂u  ∂u  Du  = Du  , = ,   ∂ϕ ϕ=0 ∂ϕ ϕ=2π ∂ϕ ϕ=0 ∂ϕ ϕ=2π Du

∂u  ∂u  = Du  = 0,  ∂ρ ρ=0 ∂ρ ρ=R

ρ ∈ [0, R],

∂v  ∂v  = = 0,   ∂ρ ρ=0 ∂ρ ρ=R

(40)

(41)

ϕ ∈ [0, 2π). To simulate the spatiotemporal patterns of the quasi-one-dimensional cell density u1D in a vessel near the three-phase contact line by applying the 2D polar model (38)–(41), the density u of cells was integrated over the thin ring close to the outer boundary of the thickness  ( = 0.075) and then averaged [6], u1D (ϕ, t) =

2 2 R − (R − )2

!

R

u(ρ, ϕ, t) ρdρ, R−

ϕ ∈ [0, 2π],

(42)

t ≥ 0.

The mathematical and the corresponding numerical models were validated by top view bioluminescence patterns experimentally observed in a small circular container made of glass [47]. Figure 5 shows typical top view bioluminescence images of experimental bacterial cultures illustrating an accumulation of luminous bacteria near the contact line [47]. Figure 6 shows the simulated patterns of the dimensionless cell density u and the chemoattractant concentration v at different moments starting from t = 0 and

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393

Fig. 5 Top view bioluminescence images of the glass vessel filled with the bacterial culture. The images were recorded at different time moments: 10 (a), 30 (b), 50 (c), 70 (d) and 90 (e) min [47]

Fig. 6 Patterns of the cell density u and the chemoattractant concentration v at the top surface of the cylinder simulated by 2D model (38)–(41) at different dimensionless time t moments: 0 (a), 26 (b), 41 (c), 81 (d), 193 (e), 236 (f), 301 (g), 401 (h), 415 (i); Du = 0.31, χ = 10.5, β = 0.95, and the other parameters are as in (36) [4]

ending with t = 415 (non-dimensional time units) [4]. The corresponding spacetime plots of the quasi-one-dimensional cell density u1D and the chemoattractant concentration v1D near the contact line (at ring (r, ϕ) ∈ [0.925, 1] × [0, 2π] assuming  = 0.075) are depicted in Fig. 7a and b. The evolution of the corresponding values (uavg and vavg ) averaged on circumference of the ring near the contact line are depicted in Fig. 7c, 1 uavg (t) = π(R 2 − (R − )2 ) 1 vavg (t) = 2 π(R − (R − )2 )

!

2π

!

R

u(ρ, ϕ, t)ρ dρ dϕ, !

0 2π

!

R−

(43)

R

v(ρ, ϕ, t)ρ dρ dϕ. 0

R−

As one can see in Figs. 6 and 7, regular oscillations as well as chaotic fluctuations similar to experimental ones were computationally simulated [4]. When comparing the spatiotemporal patterns of the quasi-one-dimensional cell density u1D shown in Fig. 7a with those obtained by applying the corresponding one-dimensional model [3, 46], one can see that the 2D computational model more accurately

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Fig. 7 Space-time plots of the cell density u1D (a) and the chemoattractant concentration v1D (b) along the contact line and the dynamics of the corresponding averaged values uavg and vavg (c) obtained by 2D model (38)–(41) at the same parameter values as in Fig. 6 [4]

simulates the pattern formation than the corresponding one-dimensional model. However, when comparing the simulated two-dimensional distributions (Fig. 6) of the bacterial population with the top view bioluminescence images of the bacterial cultures (Fig. 5), one can see that a more precise 3D model is required to take into consideration the distribution on cells over the depth of the vessel.

4.5 Population Dynamics Near the Contact Line The dynamics of the bacterial population near the three-phase contact line can be modeled by applying the common 3D mathematical model (30)–(34) (accepting ρ = R, z = H ) as well as by 2D mathematical model (38)–(41) (accepting ρ = R) [4, 6]. However, when considering the population dynamics near the three-phase contact line, the radial transport of cells and chemoattractant can be ignored because

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of the zero flux conditions at the three-phase contact line (ρ = R in model (30)– (34)), and the spatial dimension of these models could be reduced to one. The dynamics of the bacterial population near the circumference of the top surface of right circular cylinder can be approximated by the following governing equations formulated in one polar coordinate ϕ, 1 ∂ 2u 1 ∂ ∂u = Du 2 2 − χ 2 ∂t R ∂ϕ R ∂ϕ



1 ∂ 2v u ∂v = 2 2+ − v, ∂t R ∂ϕ 1 + βu

∂v u ∂ϕ

+ αu (1 − u) , (44)

ϕ ∈ (0, 2π),

t > 0,

where u and v are functions of one parameter ϕ [4, 6]. The initial conditions (39) reduce to the following equations: u(ϕ, 0) = u0x (ϕ),

v(ϕ, 0) = v0x (ϕ),

ϕ ∈ [0, 2π).

(45)

The boundary conditions take the following form (t > 0): u(0, t) = u(2π, t), v(0, t) = v(2π, t), ∂u  ∂u  ∂v  ∂v  Du  = Du  , = .   ∂ϕ ϕ=0 ∂ϕ ϕ=2π ∂ϕ ϕ=0 ∂ϕ ϕ=2π

(46)

When modeling the bacterial self-organization in a quasi-one dimensional ring, using the longitudinal analysis is often more reasonable than the azimuth one [3, 46, 47]. The 1D mathematical model (44)–(46) can be reformulated by replacing the azimuth parameter ϕ with the longitudinal parameter x by applying x = ϕR, ∂ 2u ∂ ∂u = Du 2 − χ ∂t ∂x ∂x

 ∂v u + αu (1 − u) , ∂x

∂ 2v u ∂v = 2+ − v, ∂t ∂x 1 + βu

x ∈ (0, L),

(47)

t > 0,

where u and v are functions of one parameter x. L is the dimensionless length √ of the contact line, i.e. the circumference of the vessel (a circle), L = 2πR = 2πr k6 /Dc , where r is dimensional radius of the base of the cylinder as shown in Fig. 2. The initial (45) and the boundary (46) conditions take the following form: u(x, 0) = u0x (x),

v(x, 0) = v0x (x),

x ∈ [0, L),

u(0, t) = u(L, t), v(0, t) = v(L, t), ∂u  ∂u  ∂v  ∂v  Du  = Du  , = ,   ∂x x=0 ∂x x=L ∂x x=0 ∂x x=L

(48)

(49) t > 0.

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Fig. 8 Spatiotemporal plots of the dimensionless cell density u simulated by using 1D model at values of the parameters defined in (36) (a), with changed dimensionless diffusion coefficient Du = 0.2 (b) and with changed two parameters, Du = 0.15, and χ = 6.8 (c) [6]

The mathematical model (47)–(49) has been successfully used to study the bacterial self-organization of luminous E. coli along the contact line of the circular container as detected by bioluminescence imaging [3, 13, 46]. Here this model was derived as a very special case of the common 3D mathematical model (30)–(34). Since the dimensionless diffusion coefficient Du and the chemotactic sensitivity χ are among the main governing parameters of the reaction–diffusion-chemotaxis model, the sensitivity of the spatiotemporal pattern formation to the model parameters Du and χ has to be studied [6, 12, 20, 21]. Figure 8 shows patterns simulated at three values of the diffusion coefficient Du (0.1, 0.2 and 0.3) and two values of the chemotactic sensitivity χ (6.8 and 8.3). When increasing Du -value, the spatiotemporal patterns seem to become thicker and a bit less prone to merge which makes them more similar to Fig. 4b and to patterns observed experimentally [46– 48]. Similarly, when decreasing χ parameter value, patterns (Fig. 8 [6]) also tend to become thicker and in a similar way become more familiar to Fig. 4b. The patterns shown in Figs. 4b and 8a have been produced at the same values (36) of the model parameters but by using different models, 3D and 1D, respectively. The difference in these spatiotemporal patterns could be explained by assumptions used for reduction of the dimensionality, a relatively large domain size and the sensitivity to the initial conditions [6]. Distinct initial conditions lead not only to distinct pattern types, but certain initial conditions may evolve even to a steady state, while others lead to periodic patterns of varying period [21]. As the domain size increases, more and more modes become unstable [38], and this corresponds to an increase in the dimension of the attractor, and supports higher levels of complexity [1].

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In three dimensions, for small growth rate α and constant w, the governing equations (30) may actually generate even unbounded solutions [58]. Using 3D model also permits producing very specific patterns, such as P-surface, which never exist in lower dimensions [24, 44]. Important differences between the physics of the 2D and 3D systems were also found when investigating the activity-induced phase separation in concentrated suspensions of active particles, for instance, the shape of the phase diagram and the region within which phase separation was observed were significantly different [51]. Figures 4b, 7 and 8 as well as additional simulations showed that the modeling error raised when reducing the model dimensionality can be at least partially compensated by adjusting values of Du and/or χ [6]. The compensation mechanism for 1D chemotaxis model, when an increase in one parameter can be compensated by decreasing or increasing another one, has been investigated in detail [3, 6, 18, 38]. Particularly, cell growth could compensate any loss of cells from one aggregate to a neighbour [38].

4.6 Population Dynamics Near the Lateral Surface The bacterial self-organization in a circular glass test-tube near the inner lateral surface of the vessel has been mathematically described in a 2D domain and numerically investigated [48]. Simulated populations of luminous E. coli formed bamboo foam-like structures similar to the experimentally observed structures near the inner lateral surface of the vessel [48]. The dynamics of the bacterial population near the lateral surface can be modeled also by applying the common 3D mathematical model (30)–(34) (accepting ρ = R) [6]. The radial transport of cells u, chemoattractant v and nutrient w can be ignored because of the zero flux condition for u, v and w at the lateral surface (ρ = R in model (30)–(34)). The dynamics of the bacterial population near the lateral surface of a right circular cylinder can be approximately described by Eqs. (30)–(33) by replacing parameter ρ with a constant R and assuming functions u, v and w as of only two parameters ϕ and z. Since the lateral surface of a right circular cylinder is a rectangle, the corresponding mathematical model could be defined in the Cartesian coordinates system. The mathematical model (30)–(33) defined in a 2D domain [0, 2π) × [0, H ] can be transformed into a model defined in Cartesian system in a domain [0, L] ×[0, H ] by replacing the azimuth parameter ϕ with the longitudinal parameter x, x = Rϕ and keeping the height parameter h unchanged [6].

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The bacterial self-organization near the lateral surface can be defined in Cartesian system in the same manner as (30) only adjusting the domain and the Laplace operator, ∂ 2u ∂ 2u + 2 − ∂x 2 ∂z     ∂v ∂ ∂v u ∂ u + u + αu 1 − , −χ ∂x ∂x ∂z ∂z w  ∂ 2v ∂ 2v u = 2+ 2 + −v , ∂x ∂z 1 + βu  2 ∂ w ∂ 2w =Dw + − γ u, (x, z) ∈ (0, L) × (0, H ), ∂x 2 ∂z2

∂u =Du ∂t

∂v ∂t ∂w ∂t



(50)

t > 0,

where u, v and w are functions of two parameters x and z, L is the dimensionless circumference of √ the cylinder base, √and H is dimensionless height of the cylinder, L = 2πR = 2πr k6 /Dc , H = h k6 /Dc [6, 48]. The initial conditions (31) take the following form (t = 0): u(x, z, 0) = u0x (x, z), w(x, z, 0) = w0x (x, z),

v(x, z, 0) = v0x (x, z), (x, z) ∈ [0, L] × [0, H ].

(51)

The boundary conditions (32) and (33) transform to the following conditions (t > 0): ∂u  = 0,  ∂z z=0

∂v  = 0,  ∂z z=0

Dw

∂u  = 0,  ∂z z=H

∂v  = 0,  ∂z z=H

w(x, H, t) = w0 ,

Du Du

∂w  = 0,  ∂z z=0

∂u  ∂u  = ,   ∂x x=0 ∂x x=L ∂v  ∂v  v(0, z, t) = v(L, z, t), = ,   ∂x x=0 ∂x x=L ∂w  ∂w  w(0, z, t) = w(L, z, t), = ,   ∂x x=0 ∂x x=L

x ∈ [0, L], x ∈ [0, L],

(52)

(53)

u(0, z, t) = u(L, z, t),

(54) z ∈ [0, H ].

The 2D mathematical model (50)–(54) was applied to study the bacterial selforganization of luminous E. coli near the lateral surface of the circular test-tube as detected by bioluminescence imaging [48]. The simulation result obtained at the cell diffusivity Du = 0.04, the oxygen diffusivity Dw = 0.12, the oxygen consumption rate γ = 0.048 and dimensionless time t = 240 is depicted in Fig. 9 [48]. To see

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399

Fig. 9 Snapshots of cell density u (a), the chemoattractant v (c) and the oxygen w (d) concentrations on the inner lateral surface of the vessel as well as the profile of the vertical distribution ux of cells (b) at t = 240; Du = 0.04, Dw = 0.12, γ = 0.048 while the other parameters are as in (36) [48]

the vertical profile of bioluminescence ux (Fig. 9c), the cell density u was integrated over the circumference of the tube, 1 ux (z, t) = L

!

L

u(x, z, t) dx,

z ∈ [0, H ],

t ≥ 0.

(55)

0

One can see in Fig. 9 that the cell density u on the inner lateral surface of the vessel forms a bamboo foam-like structure similar to the experimentally observed structures [48]. Structured assembly of active bacteria can be viewed as parallel venation patterns in the bacterial culture. The simulated vertical distribution of active cells (Fig. 9b) clearly shows two peaks: one near the top surface and another one noticeably below that surface. Similar peaks were noticed in physical experiments with E. coli (Fig. 3c) [48] and with other bacteria [2, 16]. To simulate spatiotemporal patterns of the quasi-one-dimensional cell density u1D near the three-phase contact line by applying the 2D Cartesian model (50)– (54), the density u of cells was integrated over the detectable layer of height H0 of the culture and then averaged [48], u1D (x, t) =

1 H0

!

H H −H0

u(x, z, t) dz,

x ∈ [0, L],

t ≥ 0.

(56)

Figure 10 shows three almost identical patterns simulated at different values of the height H0 . Only the intensity of the bioluminescence is different. At the greater depth, the higher intensity is observed. The patterns presented in Fig. 10 compare well with those seen in the experiment, Fig. 3 [48].

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Fig. 10 Space-time plots of the cell density u1D along the three-phase contact line simulated by using model (50)–(54) and (56) at three heights of the detectable bioluminescence H0 : H , 0.5H and 0.1H . The other parameters are the same as in Fig. 9 [48]

The 2D Cartesian model based on governing equations (50) generalizes the 1D model based on (47) in the same way as 3D model (30) generalizes the 2D polar model (38), as both these more common models can be produced by introducing the depth dimension and the additional equation for the oxygen concentration w. On the other hand, when reducing the mathematical model dimensionality the radial mass transport was assumed to be negligible due to the zero flux boundary conditions at the lateral surface (ρ = R) [6].

5 Concluding Remarks The modeling of three (A, B, C) principal possibilities of the application of the cell enzymatic reactions for the development of biosensors are performed in this chapter. A. The induction in cells of certain enzymatic system and it conjugation with an electrochemical process by using electron transfer mediators produced metabolite biosensor. In this case the simulation of biosensor action was considered as a microreactor. The selectivity of these biosensors is determined by the specificity of the enzymatic reaction and the permeability of cell membranes. B. If a metabolic rate of entire microbial cell is analysed in relation to oxygen consumption, the biochemical oxygen demand (BOD) biosensors have been simulated. C. The modeling demonstrate that microbial population as a whole can produce specific macrostructures or generate specific metabolites that amount can change in a time and a space. These changes can be used for the development of principally novel biosensing methods and building original biosensors. In the case of the third type (C) biosensors Keller and Segel approach was successfully used to describe the formation of bioluminescence patterns representing the self-organization of the bacteria in a rounded container (Fig. 3). The 3D mathematical model (22)–(28) and the corresponding dimensionless model

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(30)–(34) of the bacterial self-organization in a rounded container as detected by bioluminescence imaging can be successfully used to simulate structures similar to the experimentally observed structures (Fig. 3) as well as to study the pattern formation in a colony of luminous E. coli. Although the rounded container is best represented by the 3D model, due to the accumulation of luminous cells near the three-phase contact line, the experimental spatiotemporal patterns of the bioluminescence can be qualitatively simulated also by using 1D and 2D models (Figs. 7, 8 and 10). Nevertheless, important differences in the shape of the patterns are observed between the 1D, 2D and 3D cases when the same values of the model parameters are applied in the simulations (Figs. 4b and 8). Similar spatiotemporal patterns of the bioluminescence can be simulated using mathematical models of different dimensionality by adjusting values of the model parameters, particularly of the diffusion coefficient and/or chemotactic sensitivity (Figs. 4b, 7a and 8).

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Application of Mathematical Modeling to Optimal Design of Biosensors

Contents 1 2

3

4

5

6

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization of Bi-Layer Biosensors: Trade-off Between Sensitivity and Enzyme Amount. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Biosensor Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Bi-Objective Optimization Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Results of Computational Experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applying Multi-Objective Optimization and Decision Visualization. . . . . . . . . . . . . . . . . . . . . . 3.1 Modeling Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Biosensor Characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Optimal Design of the Biosensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Visualization of the Optimization Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization of the Analytes Determination with Biased Biosensor Response. . . . . . . . . . . 4.1 Biosensors Array for Long-Term Glucose Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Multianalyte Determination with Biased Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . Neural Networks for an Analysis of the Biosensor Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Generation of Data Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Prediction of Concentrations Using Neural Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Input Data Compression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Locally Weighted Neural Network Setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Biosensor Calibration Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

406 407 408 409 411 412 416 416 417 418 422 424 425 426 434 435 435 435 436 439 440

Abstract This chapter presents a method combining mathematical modeling, chemometrics, multi-objective optimization and multi-dimensional visualization intended for the design and optimization of biosensors. An approach for optimizing the biosensor parameters is based on the availability of mathematical model of the catalytic biosensor. A multi-objective visualization of trade-off solutions and Pareto optimal decisions is applied for the selection of the most favourable decision by a human expert when designing the biosensor. The proposed method is applied to a bi-layer mono-enzyme biosensor utilizing the Michaelis–Menten kinetics as well as to the glucose dehydrogenase-based amperometric biosensor utilizing © Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1_13

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the synergistic substrates conversion. The following objectives are optimized: the biosensor sensitivity, the apparent Michaelis constant, the output current and the enzyme amount. This chapter also presents a semi-global mathematical model for the multivariate calibration of signals of amperometric biosensors. Artificial neural networks are applied to an analysis of the biosensor response to multi-component mixtures. The output signal is analysed with respect to the concentration of each component of the mixture. The locally weighted regression is applied for improving the quality of the concentration prediction. Keywords Biosensors optimization · Chemometrics · Multivariate calibration · Multi-objective Optimization · Multi-dimensional visualization · Artificial neural network · Pareto optimal decision

1 Introduction With the aid of computer tools, the efforts for the design and optimization of biosensors can be remarkably reduced [26]. Moreover, these tools often allow to observe processes inside the devices, which are not accessible by the measurement technology. An optimization in biosensor engineering is often concentrated on a unique objective [50, 65, 74]. The complex nature of practical biosensors involves consideration of the simultaneous optimization of several objectives (multiobjective optimization). These objectives are usually conflicting, which means that if it is desired to improve one of them, it must allow others to get worse [65, 70]. The multi-objective optimization of biochemical processes and systems has been successfully performed in different applications, particularly, for the technological improvement of biochemical systems [76, 79, 93], for increasing the productivity and yield of a multi-enzymatic system [2], for the optimal design of a pressure swing adsorption system [92]. The importance of the multi-objective optimization in chemical and biochemical engineering permanently increases due to the development of new methods sustained by increased computational resources [65, 67, 76]. Multi-objective optimization tools provide a mechanism to obtain a certain number of trade-off solutions, largely known as Pareto optimal solutions. Establishing an efficient approach to find a set of solutions with good trade-off among different objectives has a great practical significance, as these allow engineers to gain insight into the key characteristics of potentially good designs before moving on to more detailed simulations and pilot plant tests. Trade-off curves as a visualization of tradeoff solutions are widely used for learning and making decisions when designing products [55]. Optimization methods have been successfully applied also to quantification of mixtures of multiple substrates [14, 15, 87]. Methods, such as artificial neural networks (ANN) [41, 63], also become powerful tools for experimental data analysis to improve sensitivity and selectivity of sensor systems [21, 60, 84].

2 Optimization of Bi-Layer Biosensors: Trade-off Between Sensitivity and. . .

407

The computer based design of industrial analytical systems is still a challenging task due to not only multiple often conflicting objectives, but also to a combination of factors with complex nonlinear mathematical models [26, 50, 76, 92, 93]. The action of catalytic biosensors is associated with the substrates diffusion from a bulk solution into a biocatalytic membrane and an enzyme-catalysed substrates conversion to products [20, 73]. The action is mathematically described by nonlinear partial differential equations of the reaction–diffusion type [22, 73]. This chapter demonstrates the potential of mathematical modeling, chemometrics, multi-objective optimization and multi-dimensional visualization for the design and optimization of biosensors [16, 17, 46, 53]. An approach for optimizing the biosensor parameters with the prior knowledge of mathematical model of the biosensor is presented. The approach is demonstrated by applying it to a biosensor utilizing a synergistic scheme, in which an enzyme catalyses parallel conversion of substrates followed by chemical cross reaction of the products [17, 45, 47, 53]. The synergistic schemes of the substrates conversion are of particular interest due to applying them to producing highly sensitive bioelectrodes [45, 69] and powerful biofuel cells [68, 81]. This chapter also discusses the optimization of the multianalyte determination with biased biosensor response and the improvement of the long-term response stability of biosensors [14, 15, 46, 87]. A semi-global mathematical model for the multivariate calibration of signals of biosensors is presented and analysed [12, 13, 52]. Artificial neural networks are also applied to the analysis of the biosensor response to multi-component mixtures. The output signal is analysed with respect to the concentration of each component of the mixture. The calibration data is allocated into two different data sets: a learning set, with which the neural network as global model was trained, and a test set, which was used to improve the results of the calibration by the locally weighted linear regression method [4, 71]. The application of locally weighted regression significantly improved the quality of the prediction of the concentrations.

2 Optimization of Bi-Layer Biosensors: Trade-off Between Sensitivity and Enzyme Amount This section introduces to the determination of the biosensor parameters using an optimization approach. The approach is demonstrated on a simple bi-layer mono-enzyme biosensor utilizing the Michaelis–Menten kinetics. The parameters are determined with respect to the maximization of the biosensor sensitivity and simultaneously minimization of the enzyme amount [16]. The amperometric biosensor is treated as a flat electrode covered with a relatively thin layer of an enzyme (biocatalytic membrane) applied onto the electrode surface by using a dialysis membrane.

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Application of Mathematical Modeling to Optimal Design of Biosensors

2.1 Mathematical Model Consider a mono-enzyme biosensor utilizing the Michaelis–Menten kinetics [72, 78], k+1 k3 E + S FGGGGGGGB GGGGGGG ESGGGA E + P, k−1

(1)

where E denotes the enzyme, S is the substrate to be determined, ES is the enzyme– substrate complex, P is the reaction product and k+1 , k−1 , k3 are the kinetic constants. The model of the amperometric biosensor involves three regions: the enzyme layer, where the biochemical reaction (1) as well as the mass transport by diffusion takes place, the dialysis membrane (diffusion layer), where only the mass transport by diffusion of the substrate as well as product takes place, and a convective region, where the concentrations of the substrate and product remain constant. Usually, the steady state current is used as a response of commercial amperometric biosensors. Further, for the quantitative analysis, the measurement of the steady state response is one of the easiest electrochemical methods [6, 72]. A steady state electrochemical signal is reached when the rate of the reaction product formation equals the rate at which the product diffuses out of enzyme membrane. Assuming a symmetrical geometry of the electrode and a homogeneous distribution of the immobilized enzyme in the enzyme layer of a uniform thickness, the mathematical model of the biosensor response can be defined in a one-dimensionalin-space domain [18, 19, 73].

2.1.1 Governing Equations The mass transport and the kinetics of the enzyme-catalysed reaction (1) in the enzyme layer under steady state conditions can be described by the following system of stationary reaction–diffusion equations [16, 18, 19, 73]: DSe

d2 S Vmax S , = 2 dx KM + S

Vmax S d2 P DPe 2 = − , dx KM + S

(2) 0 < x < d,

where x stands for space, S(x) and P (x) are the molar concentrations of the substrate S and the product P in the enzyme layer, respectively, V is the maximal enzymatic rate, KM is the Michaelis constant, d is the thickness of enzyme layer, DSe and DPe are the diffusion coefficients, Vmax = k3 E0 , KM = (k−1 + k3 )/k+1 and E0 is the total concentration of the enzyme.

2 Optimization of Bi-Layer Biosensors: Trade-off Between Sensitivity and. . .

409

In the dialysis membrane, only the mass transport by diffusion of the substrate and product takes place during the biosensor operation. Away from the biosensor, the solution is assumed to be in motion and uniform in the concentration, S(d + δ) = S0 ,

P (d + δ) = P0 ,

(3)

where δ is the thickness of the external diffusion layer and S0 and P0 are the concentrations of the substrate and product in the buffer solution. Usually, the zero concentration of the reaction product in the bulk is assumed, P0 = 0, while the concentration S0 is to be quantitatively determined from the biosensor response. In the case of the amperometric biosensors, due to the electrode polarization the concentration of the reaction product at the electrode surface (x = 0) is being permanently reduced to zero [56]. The substrate does not react at the electrode surface, and therefore, the non-leakage (zero flux) boundary condition is applied to the substrate,  dS  DSe  = 0, P (0) = 0. (4) dx x=0 At the steady state, fluxes of the substrate and product through the boundary of the dialysis membrane/bulk solution (x = d + δ) are equal to the corresponding fluxes through the boundary of the biocatalytic/dialysis membranes (x = d) [73],  S0 − S(d) dS  = DSe  DSm , δ dx x=d  P0 − P (d) dP  DPm = DPe , δ dx x=d

(5)

where DSm and DPm are the diffusion coefficients of the species in the dialysis membrane.

2.2 Biosensor Characteristics The density I of the steady state biosensor current is directly proportional to the flux of the reaction product at the electrode surface and can be expressed explicitly from the Faraday and the Fick laws [39, 56], I = ne F DPe

 dP  , dx x=0

(6)

where ne is the number of electrons involved in a charge transfer at the electrode surface and F = 96,486 C/mol is the Faraday constant.

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Application of Mathematical Modeling to Optimal Design of Biosensors

The sensitivity is one of the most important characteristics of the biosensors [6, 72, 78]. The biosensor sensitivity can be expressed as the gradient of the steady state current with respect to the substrate concentration. Since the biosensor current as well as the substrate concentration varies even in orders of magnitude, another useful parameter to consider is a dimensionless sensitivity. The dimensionless sensitivity BS for the substrate concentration S0 is given by BS (S0 ) =

I (S0 + S0 ) − I (S0 ) dI (S0 ) S0 S0 ≈ , × × dS0 I (S0 ) S0 I (S0 )

(7)

where I (S0 ) is the steady state current obtained at the concentration S0 of the substrate. Despite the concentration S0 , the steady state current I (S0 ) also depends on the maximal enzymatic rate Vmax and thicknesses d and δ of the enzyme and diffusion layers, respectively. Aiming to increase the biosensor sensitivity, the biosensor parameters can be optimized at a certain concentration of the substrate. Since the Michaelis KM constant is the concentration of the substrate at which half of the maximum velocity of an enzyme-catalysed reaction is achieved, the concentration KM of the substrate is widely used to evaluate the general sensitivity of biosensors. KM is also used as a measure of the enzyme affinity for substrate, the higher the value of KM , the lower is the affinity [6, 38]. Thus, the dimensionless biosensor sensitivity to be optimized at a specific substrate concentration S0 = KM can be considered as a three-variable function [16], f1 (Vmax , d, δ) = BS (KM ).

(8)

In some applications of biosensors, enzymes are archival and only available in every limited quantity or are the products of combinatorial synthesis procedures, and thus they are only produced in microgram to milligram quantities [31, 56]. In such applications, the minimization of the enzyme amount is of crucial importance. In the case of enzyme mono-layer two-compartment model of biosensors, the enzyme volume equals the product of the enzyme concentration E0 (E0 = Vmax /k3 ) and the thickness d of the enzyme layer. Without loss of generality, the mathematical model (2)–(5) involves the concentration E0 implicitly as a parameter of Vmax (Vmax = E0 k3 ) [38, 39]. Since the enzyme concentration E0 can be freely selected and the maximal enzymatic rate Vmax is directly proportional to the enzyme concentration, the rate Vmax can be considered as a free variable. Therefore, an enzyme amount factor, as the objective to be minimized, can be expressed as follows [16]: f2 (Vmax , d) = Vmax × d.

(9)

2 Optimization of Bi-Layer Biosensors: Trade-off Between Sensitivity and. . .

411

2.2.1 Numerical Simulation Due to the nonlinearity of the governing equations (2), the boundary value problem (2)–(5) was solved numerically by applying the finite difference technique [22, 23]. The mathematical as well as the corresponding computational models of the biosensor were validated using known analytical and numerical solutions for two-compartment model of mono-enzyme single substrate amperometric biosensors [73]. Those analytical solutions were derived only for relatively low as well as high concentrations of the substrate (see Chapter “Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors”).

2.3 Bi-Objective Optimization Problem Assuming fixed substrate concentration S0 , the maximal enzymatic rate V , and thicknesses d and δ of the enzyme and dialysis (diffusion) layers can be determined aiming to maximize the dimensionless biosensor sensitivity BS and simultaneously minimize the enzyme amount (amount factor). This leads to a solution of the multiobjective optimization problem by maximizing the function f1 of the biosensor sensitivity [16], max f1 (Vmax , d, δ),

Vmax ,d,δ

(10)

and minimizing the function f2 of the enzyme amount, min f2 (Vmax , d),

Vmax ,d

(11)

assuming that the maximal enzymatic rate Vmax varies from the lower bound Vmax∗ ∗ , and the thicknesses of the enzyme layer (d) and the dialysis to the upper bound Vmax membrane (δ) are defined as d ∈ [d∗ , d ∗ ] and δ ∈ [δ∗ , δ ∗ ], respectively. In terms of multi-objective optimization, p = (Vmax , d, δ) is called the decision vector, which is taken from a search space [16], ∗ ] × [d∗ , d ∗ ] × [δ∗ , δ ∗ ]. D = [Vmax∗ , Vmax

(12)

The corresponding vector f (p) = (f1 (p), f2 (p)) representing values of the objective functions f1 (p) and f2 (p) obtained using decision vector p is called objective vector. The joint optimization of f1 (·) and f2 (·) is a contradictory task. Depending on specific circumstances, a certain trade-off between these objectives is accepted. To aid a rational decision, the set of compromising decision vectors all of which are optimal in some sense will be constructed. But theretofore we will recall relevant definitions used in multi-objective optimization theory.

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Application of Mathematical Modeling to Optimal Design of Biosensors

In general, a decision vector is called Pareto optimal if the value of any of objective functions cannot be improved without deterioration of value of any other  , d  , and δ  can be considered as Pareto objective, i.e. the values of parameters Vmax optimal if there are no other values in the search space D, which would increase the sensitivity of the biosensor without increment of enzyme amount or would reduce the enzyme amount without reduction of the sensitivity. The set of Pareto optimal decision vectors is called Pareto set, and the corresponding set of objective vectors is called Pareto front. To derive an analytical expression for the Pareto front is a hard task even in the cases where the objective functions are defined by the analytical formulas. Therefore, a discrete representation is usually computed to aid a decision maker in the selection of a proper trade-off between contradictive objectives [33]. To substantiate a choice of a suitable algorithm for computation of a discrete representation of the Pareto front, analytical properties of the objective functions would be very helpful. However, in the problem considered, objective functions are available only as computer algorithms, and their analytical investigation is difficult. In such circumstances, it seems reasonable to choose the algorithm that simply selects Pareto optimal decision vectors from the set of those computed at vertices of a quadratic lattice (in logarithmic scale). Such an algorithm is approximately optimal in worst case setting [85].

2.4 Results of Computational Experiments A number of numerical simulations of the biosensor response have been performed in order to determine a discrete approximation of the Pareto front of the multiobjective optimization problem (10)–(11). The numerical simulator of the biosensor response has been implemented by C++ programming language [66]. The following constant values of the parameters of the mathematical model (2)– (5) have been used [16]: DSe = DPe = 3 × 10−6 cm2 /s, DSm = DPm = 5 × 10−7 cm2 /s,

KM = 10−4 M.

(13)

Since the biosensor sensitivity increases with reduction of the concentration S0 , when values of other parameters (Vmax , d, δ) are fixed, the constant value S0 = KM of the substrate concentration was used. The remaining decision variables (Vmax , d, δ) were varied in a wide range within their lower and upper bounds, as typical for practical biosensors [6, 36, 37], Vmax ∈ [10−13, 10−2 ] (M/s),

d, δ ∈ [0.01, 1.0] (mm),

(14)

2 Optimization of Bi-Layer Biosensors: Trade-off Between Sensitivity and. . .

413

assuming that the difference of thicknesses of the layers would be no larger than a level of magnitude. The boundary values of the decision variables have been chosen with respect to the real-life experiments. Polyvinyl alcohol, polyurethane, cellulose, latex or other membranes are often used to cover the enzyme layer in order to prevent it from dissolution and make biosensors more stable [6, 37]. The thickness of most of them varies from several micrometres up to a millimetre, and the thickness of enzyme membranes in practical biosensors varies similarly [6, 24, 37]. The maximal enzymatic rate can vary in many orders of magnitude [6, 39].

2.4.1 Dependence of the Enzyme Amount on the Maximal Enzymatic Rate The dependence of the biosensor dimensionless sensitivity f1 = BS (KM ) on the maximal enzymatic rate Vmax at different thicknesses of the enzyme layer (d) and the dialysis membrane (δ) is illustrated in Fig. 1 [16], where the horizontal axis corresponds to the maximal enzymatic rate V , the vertical axis to the biosensor sensitivity f1 , and different curves to the different combinations (d, δ) of thicknesses of both layers. Computations have been performed for Vmax = 10−k mM/s, k = 2, . . . , 13, except the cases (d, δ) = (0.1, 0.1) and (d, δ) = (0.01, 0.1), which have been investigated using intermediate values of Vmax as these combinations of parameters appeared to be the most promising in the sense of Pareto optimality [16].

1.0 0.9 0.8

f1

(d; ) (1.0; 1.0) (0.1; 1.0) (1.0; 0.1) (0.1; 0.1) (0.01; 0.1) (0.1; 0.01)

0.7 0.6 0.5 10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Vmax, M/s Fig. 1 The dimensionless biosensor sensitivity f1 = BS (KM ) versus the maximal enzymatic rate Vmax at different thicknesses d and δ (mm) of the enzyme and dialysis layers [16]

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Application of Mathematical Modeling to Optimal Design of Biosensors

1.0 0.9

(d; ) (1.0; 1.0) (0.1; 1.0) (1.0; 0.1) (0.1; 0.1) (0.01; 0.1) (0.1; 0.01)

f1

0.8 0.7 0.6 0.5 10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

f2, mm M/s Fig. 2 The dimensionless biosensor sensitivity f1 versus the enzyme amount factor f2 at different thicknesses d and δ (mm) of the enzyme and dialysis membranes [16]

One can see in Fig. 1 that a higher value of the maximal enzymatic rate leads to the higher biosensor sensitivity. However, the refractive point from which the increment of the maximal enzymatic rate is not useful in a sense of the sensitivity can be indicated as well as its dependence on the thicknesses of the catalytic and dialysis membranes. It is also clear from the figure that the best sensitivity is achieved with the thickest layers (curve denoted by rectangles), and thinning of enzyme or dialysis layers leads to the lower sensitivity of the biosensor (see curves denoted by circles and triangles, respectively). The same tendency can also be envisaged for the layers, thinner by an order of magnitude. Similarly, Fig. 2 shows the dependence of the biosensor sensitivity f1 on the enzyme amount factor f2 , considering different thicknesses of the enzyme and dialysis layers [16]. Since the enzyme amount factor f2 was expressed as the product of the maximal enzymatic rate Vmax and the thickness d (f2 = Vmax d), the latter dependency corresponds to the same curves as presented in Fig. 1 but shifted on the vertical axis. One can see from Fig. 2 that usage of the thickest dialysis layer (δ) leads to the largest sensitivity. If we are interested in lower enzyme amount rather than biosensor sensitivity, then it is useful to use the thickest enzyme layer (d = 1.0 mm)—the sensitivity is higher for relatively small enzyme volumes. If we are more interested in the greater sensitivity rather than saving the enzyme amount, then it is useful to use a thinner enzyme layer (d = 0.1 mm) as it produces slightly greater sensitivity when the enzyme amount (volume) is larger.

f2, mm M/s

2 Optimization of Bi-Layer Biosensors: Trade-off Between Sensitivity and. . .

10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 10-13 10-14 10-15 10-16

415

Pareto (d; ) (1.0, 1.0) (0.1; 1.0) (0.01; 0.1)

(d; ) (0.1; 0.1) (0.01; 0.1) (0.1; 0.01)

(d; ) (1.0; 1.0) (0.1; 1.0) (1.0; 0.1) 0.5

0.6

0.7

0.8

0.9

1.0

f1 Fig. 3 The interdependence of the dimensionless biosensor sensitivity f1 and the enzyme amount factor f2 with the distinguished Pareto front (solid symbols) at different thicknesses d and δ (mm) of the enzyme and dialysis membranes [16]

2.4.2 The Discrete Approximation of Pareto Front The discrete approximation of the Pareto front with the context of all other decision vectors is illustrated in Fig. 3, where the horizontal axis stands for the sensitivity of the biosensor, the vertical axis for the enzyme amount factor and different marks correspond to different thicknesses (d and δ) of the enzyme and dialysis layers, correspondingly. Pareto optimal decision vectors are denoted by solid (filled) symbols. One can see from the figure that the Pareto front mainly consists of the decision vectors referring to the thickest dialysis layer, δ = 1.0 mm. As it was shown in Fig. 1, the reduction of the value of δ always leads to the lower sensitivity of the biosensor without any impact to the enzyme amount (volume). More interesting is the thickness d of the enzyme layer as it has a direct impact on the enzyme volume. One can see from Fig. 3 that it is better to use d = 0.1 mm if the larger sensitivity has priority against the saving of the enzyme, and d = 1.0 mm—if we have a limit for the enzyme volume. We can also distinguish the Pareto optimal decision vector (Vmax = 10−13 mM/s, d = 0.01 mm, δ = 0.1 mm), referring to the lowest enzyme volume. However, this decision vector is not reasonable as it leads to the lowest sensitivity of the biosensor. In general, the most interesting Pareto optimal decision vectors are illustrated between the dashed lines as they provide a reasonable trade-off between parameters—the dimensionless sensitivity of the biosensor can be significantly increased (from 0.7 to almost 1) without significant (relative) increment of the

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Application of Mathematical Modeling to Optimal Design of Biosensors

enzyme volume. Therefore, the decision vectors indicated by the circle should be considered as the most relevant.

3 Applying Multi-Objective Optimization and Decision Visualization This section presents a method combining mathematical modeling, multi-objective optimization and multi-dimensional visualization intended for the design and optimization of biosensors [17, 53]. An approach for optimizing the biosensor parameters is based on the availability of mathematical model of a catalytic biosensor (bioelectrode). A multi-objective visualization of trade-off solutions and Pareto optimal decisions is applied for the selection of the most favourable decision by a human expert when designing the biosensor [17, 53]. The proposed method is applied to the industrially relevant optimization of a glucose dehydrogenase-based amperometric biosensor utilizing the synergistic substrates conversion [5, 17, 45].

3.1 Modeling Biosensor The glucose dehydrogenase (GDH)-based amperometric biosensor is a particular case of biosensors utilizing the synergistic substrates conversion used for glucose measurement in blood [45]. The GDH biosensor being modeled is assumed to be composed of a graphite electrode covered with an enzyme (GDH) layer [45]. The enzyme layer is separated from the bulk solution by means of the inert dialysis membrane. 3.1.1 Reaction Network The reaction scheme of the GDH-based bioelectrocatalytical system involves the oxidised GDH (GDHox ) reaction (15a) with glucose followed by the reduced GDH (GDHred ) oxidation (15b) with ferricyanide (Fox ) and (15c) with the oxidized mediator (Mox ) as well as a cross reaction (15d) of the ferricyanide and the reduced mediator (Mred ) resulting in ferrocyanide (Fred ) and oxidised mediator [5, 45], k1

GDHox + glucose −→ GDHred + δ-glucolactone, k2

GDHred + 2Fox −→ GDHox + 2Fred , k3

GDHred + 2Mox −→ GDHox + 2Mred , k4

Fox + Mred ←→ Fred + Mox , k5

(15a) (15b) (15c) (15d)

3 Applying Multi-Objective Optimization and Decision Visualization

417

where the reaction rate constants k1 , k2 and k3 correspond to the respective biocatalytical process and k4 and k5 belong to the electron exchange reactions. On the electrode surface, the electrons are released and the biocatalytical current is produced during the oxidation of the ferrocyanide and the reduced mediator, Fred −→ Fox + e− ,

(16a)

Mred −→ Mox + e− .

(16b)

Optimization-based design methods require mathematical models of the analytical system [16, 17, 61]. A mathematical model of the GDH-based amperometric biosensor has been described in detail in Chapter “Biosensors Utilizing Synergistic Substrates Conversion”. The governing equations of the model are typical for a class of biosensors used for the synergistic substrates determination [35, 47, 48, 75].

3.2 Biosensor Characteristics There are several important characteristics of the biosensor response [77]. The measured current is usually accepted as a response of an amperometric biosensor in physical experiments. In the case of the GDH-based amperometric biosensor, the output current i(t) depends upon the flux of the ferrocyanide and the reduced mediator at the electrode surface and is expressed explicitly from Faraday and Fick laws [73],  ∂P1  ∂P2  i(t) = AF DP1 + DP2 ,   ∂z z=0 ∂z z=0

(17)

where z and t stand for space and time, respectively, P1 (z, t) and P2 (z, t) are the concentrations of the reaction products (ferrocyanide and reduced mediator), DP1 and DP2 are the corresponding diffusion coefficients in the enzyme layer, A is area of the electrode surface and F is the Faraday constant. Assuming zero concentration of the ferricyanide (substrate S1 ) in the buffer solution, the output current is a non-monotone function of time t, and the maximal current Imax is accepted as the biosensor response [5, 17], Imax = max{i(t)}. t >0

(18)

The time interval TR from the beginning of the biosensor operation up to the moment of the maximal current is called the biosensor response time, Imax = I (TR ). The biosensor response time is influenced by the kinetic parameters, diffusion barriers, enzyme loading, enzyme activity and even by enzyme immobilization procedure [6, 30].

418

Application of Mathematical Modeling to Optimal Design of Biosensors app

The apparent Michaelis constant KM characterises the sensitivity as well app as the calibration curve for the amperometric biosensors [39, 77]. KM is the concentration of the analyte to be determined at which the biosensor response reaches a half of the maximal response when the concentration of analyte is extrapolated to infinity keeping the other model parameters constant,  app KM

=

G∗0

:

Imax (G∗0 )

 = 0.5 lim Imax (G0 ) , G0 →∞

(19)

where G0 is the concentration of the glucose (analyte) in the buffer solution.

3.3 Optimal Design of the Biosensor 3.3.1 Biosensor Parameters to be Optimized The designing of a biosensor, alike designing in general, is reducible to a multiobjective optimization where minimum or maximum values of numerous parameters are desirable, e.g. the biosensor sensitivity, the response time, material costs and some others. In this work, three objectives are considered: the apparent Michaelis app constant KM , the maximal current Imax and the enzyme amount A d1 E0 , where E0 is the total concentration of the enzyme. For the electrochemical biosensors, an important parameter is the upper limit of app the linear concentration range [6, 37]. The greater value of KM corresponds to a wider range of the linear part of the calibration curve [39, 77]. This limit is directly related to the biocatalytic properties of the biochemical receptor. The linear part of the calibration curve can be substantially extended by using a diffusion barrier (an outer membrane) to the analyte [72]. However, an additional diffusion barrier increases the response time [9]. A reasonably short response time is of crucial importance in many applications of biosensors [72]. A response time below 1 min is usually accepted as excellent, and a response time of several minutes is still satisfactory [6]. The amperometric biosensors utilizing the synergistic substrate conversion provide an admissible response time [5, 35, 45, 47, 48, 75]. Since the response time TR of the considered biosensor was short enough (TR < 207 s) for all feasible values of the model parameters, the response time was not accepted as the subject of the optimization. In some applications of biosensors, enzymes are archival and only available in every limited quantity or are the products of combinatorial synthesis procedures, and thus they are only produced in microgram to milligram quantities [6, 70]. In such applications, the amount of the enzyme used in the biosensor has to be minimized. In terms of the mathematical model [16, 17], the total quantity of the enzyme (GDH) is expressed as the product of the initial (total) concentration E0 of GDH and volume A d1 of the enzyme layer, i.e. total quantity of GDH equals A d1 E0 .

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419

The limit of detection of sensors is also determined by the signal-to-noise ratio [70]. The trend and noise or background current typically bring about a level of uncertainty in an analytical system [3]. In miniaturized sensors, the signalto-noise ratio is low because of their small sensing area, and the problems may become prominent. Substrate cyclic and other techniques are applied to amplify the electrochemical signal of biosensors [72]. The signal amplification enhances the signal-to-noise ratio of the biosensor, resulting in the detection limit for the target analyte. Therefore, it is reasonable to maximize the biosensor current Imax .

3.3.2 Multi-Objective Optimization Problem The complex nature of biosensors involves consideration of the simultaneous optimization of several objectives some of which are conflicting [70]. Therefore, the goal of design is to find a representative set of Pareto optimal solutions, quantify the trade-offs and find a solution that satisfies the subjective preferences of a human decision maker [27, 59]. The considered optimal design problem mathematically is stated as a three-objective optimization problem with the objective function (x) = app (ϕ1 (x), ϕ2 (x), ϕ3 (x))T , where ϕ1 (x) is KM , ϕ2 (x) is Imax and ϕ3 (x) is A d1 E0 . The decision variables for the optimal biosensor design problem are given in Table 1. The minimum as well as maximum values of the decision parameters should be expertly estimated. Values of some of them depend on the technological possibilities, e.g. on the thicknesses of the commercially available dialysis membranes or of the thread thicknesses of nylon nets to be used for preparation of the enzyme layer [6, 45, 48, 72]. A zero value could be used as the minimum level of a decision variable in unclear cases, as in the case of the oxidized mediator concentration S2,0 . First two objectives should be maximized, and the last one should be minimized. However, to facilitate the analysis, it is convenient to reformulate the optimization problem into a problem with all objectives aimed at, e.g. minimization. To equalize the ranges of objectives, their minimum and maximum ϕi− , ϕi+ , i = 1, 2, 3, are  T Table 1 Decision variables x = d1 , d2 , E0 , S1,0 , S2,0 for the biosensor design problem 1 2 3 4 5

Variable

Description

Range

Units

d1 d2 E0 S1,0 S2,0

Enzyme layer thickness Dialysis membrane thickness Enzyme concentration Ferricyanide concentration Oxidized mediator concentration

[2 × 10−6 , 10−4 ] [10−6 , 2 × 10−5 ] [5 × 10−8 , 5 × 10−5 ] [10−3 , 10−2 ] [0, 10−5 ]

m m M M M

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Application of Mathematical Modeling to Optimal Design of Biosensors

computed (using the multi-start with Hooke–Jeeves algorithm [44]) and the ranges are normalized to [0, 1]: fi (x) = f3 (x) =

ϕi+ − ϕi (x) ϕi+ − ϕi−

,

ϕ3 (x) − ϕ3− ϕ3+ − ϕ3−

i = 1, 2,

(20)

(21)

,

ϕi+ = max ϕi (x),

ϕi− = min ϕi (x),

x = (x1 , . . . , x5 )T ,

A = {x : 0 ≤ xj ≤ 1, j = 1, . . . , 5},

x∈A

x∈A

i = 1, . . . , 3,

(22) (23)

where xj , j = 1, 2, . . . , 5, denote the optimization variables (decision parameters) re-scaled to the unit interval [17, 53, 93]. Thus, the considered optimal design problem is reduced to the following: FP = min F (x), x∈A

F (x) = (f1 (x), f2 (x), f3 (x))T .

(24)

The optimization result FP is the Pareto front of the formulated three-objective optimization problem. For the convenience of the readers, who are not experts in multi-objective optimization, let us recall some important facts of the latter; for the comprehensive presentation, we refer to [27, 59]. The solution of a multi-objective minimization problem (Pareto front) is the set of Pareto optimal objective vectors, i.e. those vectors of all components of which cannot be decreased by the variations of x ∈ A. The Pareto front of bi-objective problems has very clear geometric interpretation: it is the south-west border of the feasible objective region. Similarly, in the threeobjective case, the Pareto front is surface limiting the feasible objective region from the side of smallest values of objectives. The vector of variables corresponding to a Pareto optimal solution is named the Pareto optimal decision. Besides computing an approximation of FP , we are also interested in the representation of the set of Pareto optimal decisions XP = {x : F (x) ∈ FP }.

(25)

3.3.3 Solution of the Multi-Objective Optimization Problem The stated multi-objective optimization problem (23)–(24) is difficult since the objectives should be considered expensive black-box functions. Such a characterization is implied by the definition of the objectives as functionals of the solutions of the governing equations of the mathematical [5, 17], which can be solved only numerically, consuming relatively much computer time.

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421

To aid the optimal design of a biosensor, the designer should be provided as detailed as possible representation of the Pareto front and of the set of Pareto optimal decisions. The latter is important to enable the designer to perceive the properties of the set of Pareto optimal decisions XP that not always can be guessed analysing the Pareto front FP . In selecting an appropriate algorithm to represent FP and XP , the crucial difficulty is the characterization of the problem as an expensive black-box problem. Moreover, the numerical experiments showed the non-convexity of at least one objective function [17]. The classical methods [59], which are efficient for smooth convex problems, as well as their adaptive versions [62], are not suitable here because of the non-convexity and of possible non-smoothness of the objective functions implied by the numerical errors of the solution of the mathematical model. The application of various metaheuristic methods is limited because of expensiveness of the objectives [27]; a vector value of the objective function on average takes almost 6 min using a relatively new personal computer with Intel Core i7-4770 3.5 GHz processor. The most suitable for the problems with the characteristic of interest would be an algorithm based on a statistical model of the objectives [86]; however, at present the corresponding software is available only for bi-objective problems. Among the other available alternatives, the most promising method for the considered problem is the so-called Chebyshev scalarization method [59]. By the latter, the minimization problem (24) is reduced to the following parametric single objective problem: f (x) = max wi fi (x), 1≤i≤3

w = (w1 , w2 , w3 ) , 0 ≤ wi ≤ 1, T

x(w) = arg min f (x), x∈A

3 

wi = 1,

(26)

(27)

i=1

where the minimizer x(w) is a Pareto optimal decision of the original multiobjective problem. All the Pareto optimal decisions can be found by the solution of (27) with an appropriate vector of weights w. To represent the whole sets FP and XP , the minimization problem (27) should be solved repeatedly with different vectors of weights. The scalarized function f (x) can be minimized by using a combination of a randomized selection of a starting point with the Hooke–Jeeves optimization algorithm [44], which has been successfully applied to similar problems [90]. Let us note that some solutions of (27) can be weak Pareto optimal but they can be easily filtered out from the results. The choice of weights aiming at the uniform distribution of solutions in FP is complicated. The desirably distributed solutions can be computed by a two or even more step procedure [17]. At the first step, the optimization problem is solved with the uniformly distributed weights. At each subsequent step, the weights are adjusted in accordance with the optimization results obtained in the previous step [17].

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Although the empirical investigation of the optimization problems in question shows that the approximate Pareto solutions well represent FP , the alternative representation of FP by the exhaustive uniform search can also be applied [17].

3.4 Visualization of the Optimization Results By the analysis of a visual representation of the Pareto front FP (W), an appropriate Pareto solution can be selected as well as the decision x(w) that corresponds to the selected Pareto solution. However, such a choice is not always satisfactory since it does not pay respect to such properties of the corresponding decision as, e.g. the location of the selected decision vector in the feasible region A. The analysis of the location of the set of efficient points in A can be especially valuable in cases of structural properties of the considered set important for the decision making [90, 92]. For example, some subsets of A might not be forbidden but may be unfavourable, and that property may not be easy to introduce into a mathematical model. The analysis of the properties of the set of Pareto optimal decisions can enable the discovery of latent variables, the relation between which essentially defines the Pareto front. An approximate analytic description of the set of Pareto optimal decisions would be desirable, but even much simpler problems of nonlinear least squares are hardly solvable [89]. A suitable method for the discovery of a structure in multi-dimensional data is the visualization [29]. A graphical drawing of the Pareto front is the standard presentation of results of a bi-objective optimization problem [27, 59]. Threedimensional computer graphics also enables a visualization of Pareto fronts of three-objective optimization problems, although the analysis of an image can be considerably more complicated than in the bi-objective case. However, to visualize a set of Pareto optimal decisions, which is a subset of the feasible region in a multidimensional space, special methods of visualization of multi-dimensional data are needed. A suitable method here is the multi-dimensional scaling [25]. The approximation FP (W) of the Pareto front FP computed by the Chebyshev algorithm consists of N = 136 three-dimensional vectors [17]. The corresponding set XP (W) of five-dimensional points x(wi ) ∈ A, i = 1, . . . , N, is an approximation of XP . To get an idea of the location of XP (W) in the five-dimensional unit cube, a multi-dimensional scaling based algorithm has been applied to the two-dimensional visualization of a set of five-dimensional points consisting of x(wi ), i = 1, . . . , N, and the cube vertices. To visualize such sets of multidimensional data, the MDS algorithm, utilizing a special method for solving an auxiliary global optimization procedure, has been proposed [88]. However, intending to encourage the wider application of the proposed methodology for the analysis of sets of Pareto optimal decisions, a procedure based on the wellknown so-called SMACOF algorithm was applied [51]. The earlier versions of the procedure have been successfully applied in different applications [90, 92].

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3 Applying Multi-Objective Optimization and Decision Visualization

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Fig. 4 Pareto optimal solutions (d) and their two-dimensional (a–c) projections in original dimensions [17]

For the selection of the most favourable parameters of the biosensor, the projections of trade-off (Pareto optimal) solutions could be analysed in the original dimensions. Figure 4 shows the three-dimensional Pareto optimal solutions and their two-dimensional projections, where the solutions correspond to three objectives: the app apparent Michaelis constant KM , the maximal output current Imax and the enzyme amount A d1 E0 . app One can see in Fig. 4b that the high apparent Michaelis constant (KM ≈ 100 mM) is reached at the maximal amount of the enzyme (A d1 E0 = 49 pmol) as well as at minimal its amount (A d1 E0 = 5 fmol). On the other hand, Fig. 4a shows that the output current is maximized (Imax = 196 μA) only at maximal amount of the enzyme, while at the minimal amount the current is extremely reduced (Imax = 600 nA). As one can see in Fig. 4c, the Pareto front includes high output app currents together with rather high values of KM . Some intermediate trade-off solutions can be picked by biosensor designers from the three-dimensional Pareto front depicted in Fig. 4d. The most attractive compromise decision seems to be where the Pareto front app shown in Fig. 4d makes a bend, i.e. where KM ∈ [60, 65] (mM), Imax ∈ [60, 80]

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Table 2 Attractive optimal solutions and the corresponding values of the decision variables in the original dimensions [17]

app

KM mM 62.1 60.6 62.1 60.6 62.1 62.1 62.1 63.6

Imax μA 62.7 74.6 75.5 75.9 64.4 76.9 70.1 62.3

A d1 E0 pmol 1.98 2.62 2.48 2.48 1.88 2.54 2.16 1.77

d1 μm 24.2 21.3 15.5 14.0 15.9 14.1 14.1 15.7

d2 μm 19.5 19.5 19.5 18.3 16.0 19.5 18.5 18.1

E0 μM 8.2 12.3 16.0 17.8 11.8 18.0 15.4 11.3

S1,0 mM 9.41 9.41 9.41 9.91 9.41 9.41 9.65 9.60

S2,0 μM 9.98 9.98 9.98 9.58 9.98 9.98 9.82 9.98

(μA) and Ad1 E0 ∈ [1.5, 3] (pmol). Those optimal solutions together with the corresponding values of the decision variables in the original dimensions are presented in Table 2. When comparing the optimal solutions (Table 2) with parameters of the experimental GDH biosensor [45], one can see that the thickness 18.7 μm of the experimental dialysis membrane is very similar to the calculated optimal thicknesses (d2 ). Although the thickness (d1 = 100 μm) of the experimental enzyme membrane notably differs from the optimal decisions shown in Table 2, the total amount of GDH is rather similar: A d1 E0 varies from 0.5 up to 4.7 pmol for the experimental biosensor and varies from 1.77 to 2.62 pmol for the optimized biosensor. The optimization procedure as well as physical experiments showed that increasing concentrations of S1,0 and S2,0 increases the biosensor sensitivity and thus the apparent app Michaelis constant KM increases. The configurations of the GHD biosensor used in experiments are rather similar to the optimal decisions; however, the number of potentially good configurations could be reduced and the configurations could be purposefully improved by applying the multi-objective optimization and decision visualization.

4 Optimization of the Analytes Determination with Biased Biosensor Response The problem of the determination of analyte concentration becomes more complex if the biosensors response is perturbed by noise, e.g. white noise, sinusoidal power electrical noise, or if the biosensor response is biased, e.g. by temperature change or long-term storage [3, 46, 58]. This section analyses an optimization of the multianalyte determination with biased biosensor response and an improvement of the long-term response stability of biosensors [14, 15, 46, 87].

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425

4.1 Biosensors Array for Long-Term Glucose Measurement A chemometric method was applied to carbon-paste biosensors for the improvement of the long-term stability of the biosensor response [46]. The biosensor action follows the following scheme: the glucose oxidase (GO) adsorbed on graphite is reduced by glucose (S); the reduced mediator dissolved in paraffin (Mpar ) diffuses into the buffer solution at the interface is oxidized electrochemically and interacts with the reduced glucose oxidase, GOox + S −→ GOred + P,

(28)

Mpar −→ Msol − e− −→ M+ sol ,

(29)

GOred + M+ sol −→ GOox + Msol .

(30)

The long-term responses of the biosensor containing different samples of enzyme indicated a decrease in response [46]. The decrease depended on the amount of enzyme as well as on the type of enzyme utilized in the carbon paste. When biosensor contained lyophilized glucose oxidase and a low enzyme activity, the response decreased from 230 to 52 μC over a period of 13 weeks. During the same period of time, the response of the biosensor containing a high amount of enzyme only changed from 364 to 261 μC. The kinetics of the decrease in the response of the biosensor with low enzyme activity fit a bi-exponential decay curve with halftimes of inactivation of 0.8 and 9 weeks and partial magnitudes of 60 and 40%, respectively. Introducing trehalose into the carbon paste significantly decreased the sensor inactivation rate. At low enzyme activity, the response decreased by 34%, whereas at high activity it changed only 5% during both 13 and 12.1 weeks. The kinetics of the changes of the sensor with low activity was approximated by a bi-exponential decay curve with half-times of 1 and 25 weeks and partial magnitudes of 35 and 65%, respectively [46]. The biosensor contained two components the loss of which might account for a decrease in response: the mediator and the enzyme. It was shown that the mediator concentration change is not a dominant factor in the long-term decrease in sensor response. The characteristics of enzyme inactivation in carbon paste were obtained from studies of glucose oxidase inactivation in paraffin oil [46]. The model of the limitation of the external diffusion of biosensor action was used to explain the change in biosensor response and to predict the response not perturbed by enzyme inactivation. At low enzyme activity, the biosensor acts in the kinetic regime and the decrease in response follows the change in enzyme activity. Therefore, the half-time of the decrease in response corresponds to the enzyme lifetime. At high enzyme activity and at the beginning of inactivation, the biosensor acts in the conditions that limit external diffusion and the enzyme inactivation has

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little influence on the decrease in response. The calculated output currents rather convincingly approximate the experimental data of both types of sensors [46]. To compensate for the decrease in the biosensor response, a correction factor CF was introduced that corresponds to the relation between the responses of noninactivated and inactivated biosensors. CF is different for the two biosensors and is a function of the ratio (R) between the two sensor responses. Even if a precise model of the limitation of external diffusion is established, the functional dependence CF = f (R) cannot be solved analytically, and therefore, the response was predicted numerically [46]. In the case of the biosensor array, where the two channels were filled with carbon pastes containing different amounts of enzyme, a more general chemometric approach was used [46]. The ratio R was derived directly from the experimental data, and the dependence of the correction factor on R was evaluated statistically using a third-order polynomial approximation. The predicted response of the biosensor containing paraffin oil as well as silicon oil decreased by only 0.11 and 0.14% per week. The relative standard errors of the predicted response were 54 and 24% during the periods of 124 and 166 days, respectively. For an evaluation of the chemometric approach, it is important to stress two points. One point concerns the substrate concentration. It is easy to show that in external (as well as in internal) diffusion limiting conditions, the correction factor CF is not very sensitive to the substrate concentration when the substrate concentration is less than apparent Michaelis–Menten constant and the diffusion module is larger than 1. Temperature influences the response as well as the rate of enzyme inactivation. The most important effect is response compensation. From analysis of the models of the biosensor action at external or internal diffusion limiting conditions, it follows that the temperature coefficient can vary from 2% as in the case of mass-transport limitation[8] to about 10% as in the case of a kinetic regime [49]. The ratio R contains information about the temperature factor, but this functional dependence should be evaluated in independent measurements [46]. This approach may not only provide new tools for the development of long-term stable biosensors but also promote new opportunities to construct biosensors for the detection of inhibitors and for biosensors with other novel properties [46].

4.2 Multianalyte Determination with Biased Biosensor Response An optimization-based approach was applied for quantification of mixtures by a catalytic biosensor assuming noise-corrupted measurements and temperature variance-affected reaction rate [14, 15, 87].

4 Optimization of the Analytes Determination with Biased Biosensor Response

427

4.2.1 Modeling Biosensor The amperometric biosensor was treated as an electrode and a relatively thin layer of an enzyme (enzyme membrane) applied onto the probe surface. In the enzyme layer, the mass transport by diffusion and the following mono-enzyme catalysed multi-substrate conversion take place [64, 82]: k1k

k2k

E + Sk  ESk −→ E + Pk , k−1k

k = 1, . . . , K,

(31)

where E denotes the enzyme, Sk is the substrate, ESk stands for the enzyme and substrate complex, Pk is the reaction product, kinetic constants k1k , k−1k and k2k correspond to the respective reactions: the enzyme–substrate interaction, the reverse enzyme–substrate decomposition and the product formation, and K is the number of substrates to be analysed. Applying the quasi-steady state approach (QSSA) to the enzyme-catalysed reactions (31) and then coupling these reactions with the one-dimensional-in-space diffusion, described by Fick’s law, lead to the following system of 2K equations of the reaction–diffusion type (t > 0, 0 < x < d): ∂Sk Vmaxk Sk ∂ 2 Sk  , = DSk − 2 ∂t ∂x KMk 1 + K j =1 Sj /KMj ∂ 2 Pk Vmaxk Sk ∂Pk   , k = 1, . . . , K, = DPk + 2 ∂t ∂x KMk 1 + K j =1 Sj /KMj

(32)

where d is the thickness of the enzyme layer, Sk and Pk are the concentrations of the substrate and the product, respectively, Vmaxk = k2k E0 , KMk = (k−1k + k2k )/k1k and E0 is the total concentration of the enzyme. The biosensor current density i(t) at time t was expressed explicitly from the Faraday and the Fick laws [39], i(t) =

K  k=1

nk F DPk

 ∂Pk  , ∂x x=0

(33)

where nk is the number of electrons involved in a charge transfer at the electrode surface in the corresponding reaction and F is the Faraday constant. The detail formulation of the mathematical model is discussed in Chapter “Biosensors Utilizing Consecutive and Parallel Substrates Conversion” as well as in [14, 15, 52].

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4.2.2 Optimization Problem for Multianalyte Determination We are interested in estimating the concentrations of substrates using the measurements of the biosensor response. The physicochemical processes are supposed to be corresponding to the biosensor model described mathematically. That problem is an inverse problem with respect to the problem of computation of the biosensor output signal where the concentrations of the substrates are given. The method developed is oriented to process experimental data. However, pseudo-experimental simulated data have been used for testing with reference to the proved adequacy of the mathematical model. The method was tested for a particular case of two substrates (dual biosensor, K = 2) [14, 15]. Let W = (w1 , . . . , wn ) be a sequence of measurements where the output of a biosensor (the faradaic current) has been measured at the time moments t1 , . . . , tn . The output of the biosensor simulated according to the mathematical model is denoted by y(t, c), where t is time, and c = (c1 , . . . , cK )T denotes a vector of concentrations of the substrates S1 , . . . , SK , respectively. The output values computed at the time moments t1 , . . . , tn constitute a vector denoted by (y(t1 , c), . . . , y(tn , c))T . The concentrations of the substrates are supposed to be evaluated by tuning the theoretical output of the biosensor to the corresponding measurements. The least squares approach is usually applied to tune the output data, computed according to a theoretical model, to the experimental measurements. This approach is especially appropriate for statistical data where measurements are corrupted by random noise. The least squares approach when applied to the solution of the stated above problem reduces to the following minimization problem: c˜ = arg min c∈C

n 

(y(ti , c) − wi )2 ,

(34)

i=1

where C denotes the feasible region for the values of concentrations of the substrates [14]. The minimization problem (34) is relatively easy to solve in a case when y(t, c) is an affine function of x. However, in the considered case here, y(t, c) is a solution of a system of partial differential equations, and the desirable property of y(t, c) cannot be rigorously proved. In the case of nonlinearity of y(t, c) with respect to c, the optimization problem (34) is usually difficult to solve because of the multimodality of the objective function. Although various global optimization methods theoretically can be applied to the solution of the considered problem, practically the solution time frequently is unacceptably long even for model functions defined by simple analytical formulas [89, 91]. In the considered case, the additional difficulty of the problem (34) is long computing time of y(t, c) [14]. In real applications, the reaction rate can be affected by the temperature change that causes a trend of the output of the considered sensor [7, 83]. Let us describe the trend of the output as the multiplication of the initial output y(ti , c) by a factor

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429

R(ti ), and assume that at the time moment t0 = 0 the initial conditions are valid implying the equality R(t0 ) = R(0) = 1. The trend factor R(t) can be expressed using the Arrhenius equation  that defines the dependency of the reaction rate on temperature R(t) = A exp −Ea /(R¯ T ) , where A is the pre-exponential factor, Ea is the activation energy, R¯ = 1.98 cal/(K mol)—the gas constant and T is the absolute temperature [32, 74, 80]. In calculations, it was assumed that temperature T is changing lineally with time, T (t) = T0 + at, where T0 equals 298 K, and a (K/s) is the coefficient of proportionality. Calculating the pre-exponential factor A from the initial condition R(0) = 1, (A = exp (Ea /(R¯ T0 ))) leads to the following expression for the factor R(t) of the exponential trend:  R(t) = exp

at Ea . × T0 (T0 + at) R¯

(35)

Assuming the sensor output influenced by the Arrhenius trend, the concentration evaluation problem should be reformulated as follows: (c, ˜ a) ˜ = arg

min

n 

{c∈C, a− ≤a≤a+ }

(y(ti , c) R(ti ) − wi )2 ,

(36)

i=1

where the optimization variable a is the unknown parameter a in (35). In some cases, a linear trend is also of interest [80, 83]. In the case of the linear signal trend, the optimization problem related to the evaluation of concentrations can be described as follows: (c, ˜ α) ˜ = arg

min

{c∈C, α− ≤α≤α+ }

n 

(y(ti , c) + αti − wi )2 ,

(37)

i=1

where the optimization variable α is interpreted as the rate of a linear trend and the feasible regions C and [α− , α+ ] are defined at due place. In the all stated optimization problems, data can be corrupted by noise.

4.2.3 Solution of the Optimization Problem The optimization problems stated are difficult because of the following reasons [14]: • Neither convexity nor unimodality of the objective function is mathematically provable, and therefore, applicability of local optimization algorithms here can be questionable. • Long computing time is needed to compute a single value of the objective function.

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• Analytical gradients of the objective function are not available, and their numerical estimates can be strongly corrupted because of errors in numerical solution of the equations that describe the mathematical model of the considered biosensor. On the other hand, the considered optimization problem is of low dimensionality, and it possibly can be solved by brute force methods spending (allowably) much of computing time. However, some heuristically established properties of the considered optimization problem induce an idea on how to solve that optimization problem more efficiently. The function y(t, c) is not an affine function with respect to concentrations c to be determined. However, practical biosensors are usually designed in such a way that the steady state current y(t∞ , c) would be linear in some intervals for the concentrations of the substrates to be quantitatively determined [6, 72, 78]. Therefore, it can be expected that transient responses y(t, c) deviate not too much from a linear function of concentrations c, implying unimodality of the stated above optimization problems in an appropriate feasible region for c. To reduce the computation time of objective functions required in the optimization process, the values of a surrogate function y(t, ˜ c) have been computed instead of the original values of the transient biosensor responses y(t, c) simulated numerically or measured experimentally. To ensure the sufficient precision of such a replacement, the function y(t, ˜ c) is defined as a set of k-dimensional splines constructed for every ti using the values of y(t, c) obtained at various concentrations c1 , . . . , cK of the substrates, i = 1, . . . , n. Since at relatively low concentrations of the substrates (S0i  KMi , i = 1, . . . , k), the nonlinear governing equations describing the biosensor action become linear, and therefore, quantitative analysis of such mixtures becomes rather easy [28, 34, 42], the proposed method was evaluated only for moderate and relatively high concentrations of the substrates. For a modeling dual biosensor (k = 2), the reasonable intervals for concentrations of both substrates are 3.2 ≤ c1 , c2 ≤ 12.8, where c1 and c2 are assumed to be the dimensionless (normalized) concentrations, ci = S0i /KMi , i = 1, 2. To derive the approximation y(t, ˜ c), the entire quadratic domain [3.2, 12.8]2 of the substrate concentrations was discretized with the uniform mesh step size of 0.3. To have detailed enough of the transient responses, the simulated output current was recorded every second, so that tn = 300 s, ti+1 − ti = 1 s, i = 1, . . . , n − 1 and n = 300 [14]. Two-dimensional splines were then fitted to values of y(t, c) obtained at the grid nodes. To check the precision of the surrogate model of the two-dimensional bilinear spline type, the differences of the values of y(t, c) and y(t, ˜ c) at the randomly generated points have been computed. The points have been generated with uniform distribution over 322 = 1024 squares obtained in the discretization of the concentrations domain. The maximum of relative error was 3.48 × 10−5 [14]. A standard gradient descent method is appropriate for the minimization of y(t, ˜ c), especially because of the availability of the analytical formula for gradient

4 Optimization of the Analytes Determination with Biased Biosensor Response

Preprocessing Analytical system model y(t, c)

Biochemical information

Evaluation Surrogate model ỹ(t, c)

Experimental responses y(t, c)

431

Output

Optimization

c1 ... cK

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Fig. 5 The general schema of the proposed method of the evaluation of concentrations [14]

of y(t, ˜ c). Moreover, the objective functions defined in the previous section appear unimodal in the region of interest although that property is not provable theoretically. Nevertheless, the computation of minimum point is not trivial because of the large condition number of Hessian at minimum points of the functions, which is equal to 0.43 × 103 and 0.47 × 105, correspondingly [14]. The proposed method for the evaluation of the concentrations was implemented in MATLAB using the local minimization subroutine f mincon [14]. A special MATLAB function has been written to implement formulae defining gradients of the objective functions (34)–(37). The stopping condition was defined by the tolerance of function values equal to 10−10 , and the tolerance of optimization variables was equal to 10−5 . Taking into account the above-mentioned large condition number of Hessian at minimum points of the objective functions, the local descent was repeated three times using randomly generated starting points. The general schema of the proposed method of the evaluation of concentrations is presented in Fig. 5. When applying the proposed method, the biochemical parameters of the considered biosensor are used to specify the mathematical model (32)–(33), which in turn is used to compute the values of y(t, c). The preprocessing is a onetime procedure for the considered biosensor, and the results of the preprocessing are saved and used by the subroutine for the computation of values and gradients of y(t, ˜ c) during the optimization process [14].

4.2.4 Application of the Model The workability of the proposed method was validated by the investigation aimed at the evaluation of concentrations of two substrates in a mixture (K = 2). Then the influence of the linear and exponential trends as well as of the white noise to the precision of the concentrations estimation was investigated [14]. Each component of the mixture was characterized by the individual maximal enzymatic rate differing in an order of magnitude, while values of the Michaelis constant were assumed to be the same (KM1 = KM2 ). Without reducing the

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generality, it was assumed that the maximal enzymatic rate for the second substrate is notably greater than for the first one, Vmax2 = 10 Vmax1 . Similar approach was used in a quantitative analysis of mixtures using artificial neural networks [12, 13]. Since the analysis quality may depend on whether the biosensor response is under the diffusion or the enzyme kinetics control, the maximal enzymatic rates were chosen so that the biosensor would operate under different conditions, i.e. at different values of the diffusion modules σ12 and σ22 , keeping σ22 = 10 σ12 , σk2 =

Vmaxk d 2 , DSk KMk

k = 1, . . . , K.

(38)

4.2.5 Impact of Signal Linear Trend In order to investigate the influence of the signal linear trend to the precision of the evaluation of the concentrations, a linear trend has been added to the simulated biosensor responses. Three different values of the trend parameter α (see (37)) have been chosen so that the biosensor current increases by 1%, 3% and 5% in tn seconds of the biosensor operation, i.e. at the final moment of the biosensor operation. The following three configurations of the biosensor operation were investigated: (a) The biosensor response to both substrates is controlled by the enzyme kinetics (σ12 ≈ 0.067, σ22 ≈ 0.67). (b) The response is under mixed control (controlled by the enzyme kinetics for the first component and by mass transport for the second one, σ12 ≈ 0.67, σ22 ≈ 6.7). (c) The mass transport controls the biosensor response to both components (σ12 ≈ 6.7, σ22 ≈ 67) [14]. Results of the investigation showed that the linear trend has significant influence to the error of the evaluation of the concentrations—the relative error of the estimated concentrations increased up to 33% for the concentration c1 of the first substrate and up to 15% for the concentration c2 of the second substrate, with the largest linear trend rate investigated. The more precise estimation of the concentration of the second substrate than that of the first substrate was explained by the difference in the diffusion module. In all three configurations of the biosensor operation, the diffusion module σ22 for the second substrate was tenfold greater than the diffusion module σ12 corresponding to the first substrate, σ22 = 10 σ12 . The numerical experiments showed that the concentration estimation is more accurate for a component corresponding to a greater diffusion module than for another component corresponding to a lower diffusion module. This approves and generalizes a known property that the concentration of a mixture component can be estimated more accurately when the biosensor response is under diffusion control rather than the response is controlled by the enzyme kinetics [12, 13]. Due to a notable influence of the trend rate α to the precision of the concentration estimation, the value of the trend parameter α should be evaluated. In order to do so, the measure of the difference defined in (37) has been used, thus expanding

4 Optimization of the Analytes Determination with Biased Biosensor Response

433

the optimization problem to three variables: two values of concentrations of the substrates and the trend parameter α. Results of the investigation showed that the evaluation of the value of the trend parameter α gives significant advantage to the precision of the evaluation of the concentrations—the maximum discrepancy was less than 0.006%. Thus, we can conclude that prediction of a value of the linear trend rate α helps to evaluate concentrations of substrates with the similar precision as they have been evaluated when biosensor response was unaffected by any trend [14].

4.2.6 Impact of Signal Exponential Trend Further, the influence of the exponential trend (defined by (35)) was investigated [14]. Since values of the activation energy parameter Ea and the coefficient a of the proportionality are not known precisely, three combinations of theses parameters were used in the calculations. The value of the proportionality coefficient a has been supposed to be unknown and the subject to evaluate considering it as the third variable of the problem. A measure of the difference (36) was used. Results of the investigation showed that the concentrations of both substrates can be evaluated precisely independent of the rate of the exponential trend. The maximal relative errors of the estimation of the concentrations c1 and c2 were around 1.5 × 10−4 and 9 × 10−5 correspondingly. The most intractable situation occurred when values of the activation energy and the coefficient of the proportionality are the largest. On the other hand, the response of the biosensor increased by approximately 45% using these values of the parameters. Although the relative errors of the concentration estimations were very small, the error for the second substrate (c2 ) was notably less than that for the first one (c1 ). This result confirms that the concentration evaluation is more accurate for a substrate corresponding to a greater diffusion module than that for another one corresponding to a lower diffusion module [14].

4.2.7 Impact of White Noise The results of the experiments discussed above showed that the concentrations of the substrates can be precisely estimated from the transient biosensor response, i.e. from the values of the biosensor current measured at time moments t1 , . . . , tn . However, the latter experiments were performed under ideal conditions—it was supposed that the transient biosensor response W = (w1 , . . . , wn ) was precise and noise-free. Measurements in real experiments are usually not so precise [3, 58, 83]. In order to get closer to the real-life experiments, we added a white noise to the modeled biosensor response. Two different types of white noise were investigated: multiplicative white Gaussian noise (MWGN) and additive white Gaussian noise (AWGN) [14].

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The noisy signal W N = (wiN , . . . , wnN ) was modeled as follows: wiN = wi + ξ ∼ N(0, σ0 wi ),

(39)

in the case of MWGN, and wiN = wi + ξ ∼ N(0, σ0 wn ),

(40)

for AWGN. The parameter σ0 in the Gaussian (normal) distribution N stands for the level of the noise. If σ0 -value is chosen to be, for instance, equal to 0.05, the standard deviation of MWGN equals 5% of the biosensor current at any time moment ti (i = 1, . . . , n), or 5% of the steady state biosensor current at the final time moment tn in the case of AWGN. The influence of the standard deviation of the noise to the precision of the evaluation of the substrate concentrations was investigated. Different values of parameter σ0 were chosen to be 0.05, 0.1, 0.15 and 0.20, so that standard deviation of the noise was 5%, 10%, 15% and 20%. The impact of both types of white noise on the biosensor response with different exponential trends was investigated, and the relative errors of the evaluations of the substrates concentrations were measured [14]. Numerical experiments showed significant influence of the noise on the precision of the evaluation of the concentrations. However, the influence of the parameters of the exponential trend to the relative errors of the estimation was only slight. The relative error for the first substrate increased from around 0.05 to 0.25, while the standard deviation of MWGN increases from 5% to 20%. A more significant noise is AWGN—the relative error of the estimated concentration of the first substrate increases from around 0.1 to almost 0.45, using the same values of the standard deviation of the noise. Although the concentration of the second substrate has been evaluated more precisely, the influence of the noise has the same tendency—AWGN has significantly larger influence to the precision of the evaluation than MWGN.

5 Neural Networks for an Analysis of the Biosensor Response This section presents a semi-global mathematical model for an analysis of a signal of amperometric biosensors [12, 13, 52]. Artificial neural networks are applied to an analysis of the biosensor response to multi-component mixtures [1, 54]. The signal is analysed with respect to the concentration of each component of the mixture. The paradigm of locally weighted linear regression is used for retraining the neural networks [13].

5 Neural Networks for an Analysis of the Biosensor Response

435

5.1 Generation of Data Sets The overall biosensor response to a mixture can be represented as the total sum of individual responses to each constituent substrate Sk (k = 1, . . ., K). In the mathematical model, each component Sk of the mixture can be characterized by the individual maximal enzymatic rate Vmaxk [12, 13]. See Chapter “Biosensors Utilizing Consecutive and Parallel Substrates Conversion” for detail description of the mathematical model of an amperometric biosensor utilizing multi-substrate conversion to multi-product. and the procedure of pseudo-experimental data generation.

5.2 Prediction of Concentrations Using Neural Networks Let c = (c1 , . . . , cK ) be a vector of concentrations of K components S1 , S2 , . . . , SK of a mixture and z = z(c) = (z1 (c), . . . , zN (c)) = (ic (t1 ), . . . , ic (tN )) be a vector of the biosensor current densities at times t1 , . . . , tN obtained computationally for the concentrations c. Thus, z defines a response of the biosensor to the mixture of K components of the concentrations c. Note that z implicitly depends also on the diffusion module σk2 , k = 1, . . ., K, defined in (38) and, hence, on the membrane thickness d. Our goal is to define a nonlinear map N, such that N(z) = c. The widely used class of feedforward neural networks (FNN) can be chosen to approximate the map N [41]. While an artificial neural network (ANN) provides a nonlinear approach that needs no a priori knowledge of functional dependencies, it requires training [63]. Training is based upon cumulative experimental data [94]. An accurate and reliable calibration of the system as well as a proper test of the methods of chemometrics requires a lot of experimental data. A mathematical model of a biosensor can be used to synthesize experimental biosensor responses to mixtures [10, 12] (also see Chapter “Biosensors Utilizing Consecutive and Parallel Substrates Conversion”). Assuming good enough adequacy of a mathematical model to the physical phenomena, data synthesized using computer simulation can be employed instead of experimental data. A computer simulation of the physical experiment is usually more affordable and faster than actual experimentation [12, 13, 52].

5.3 Input Data Compression Data collected in complicated processes contains a lot of redundant information, since the variables are collinear [57]. Preprocessing methods can be applied in such situations to enhance the relevant information to make the resulting models

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Application of Mathematical Modeling to Optimal Design of Biosensors

simpler and easier to interpret. Different approaches can be applied to reduce the dimensionality of the vector of input data, e.g. the canonical correlation analysis (CCA) or the principal component analysis (PCA) [43, 57]. With CCA, the correlation coefficients are calculated for every point of the input vector and each component of the mixture. Basing oneself on the calculated correlation coefficients, only a few points of z with the highest values of the coefficients are accepted as inputs to a neural network. Therefore, every vector z of the original data is replaced with a resulting vector x, having the dimensionality J , J ≤ N, x = (x1 , . . . , xJ ). Each element of the resulting vector x also belongs to z. However, most elements of z are usually missing in x. The CCA resulting vector x can be then passed to a neural network. With the more advanced PCA, the so-called principal components are extracted, which are statistically independent from each other and which are therefore orthogonal relative to one another, yet are still capable of adequately reconstructing the original data [43, 57]. Using PCA, the input data z are expressed as a linear combination of a few basic vectors, where the basic vectors capture as much variation of the original data as possible. The purpose of PCA is to find basic vectors w1 , . . . , wJ , J ≤ N, for i = 1, . . . , J , satisfying the following conditions: 1. E{wTi z}2 are maximized under the constraints, 2. wTi wj = δij , for j = 1, . . . , i, where E stands for an expectation operator, N is the number of measurements of biosensor current during the biosensor operation and the superscript T indicates transposition. The vectors w1 ,. . . ,wJ satisfying those conditions can be found as dominant eigenvectors of the data covariance matrix  = E{zzT }. Therefore, each vector z of the original data can be represented by its principal component vector x having dimensionality J , J ≤ N. In the cases of large dimensionality of input vector z, the dimensionality of the resulting vector x is usually significantly less than the dimensionality of z, J  N. There exist some rules of thumb on how many dimensions to use, such as keeping all dimensions whose contributions to the total variation exceed 80%. The forecasting quality of concentrations is usually considerably higher when the calibration data is processed using the more advanced PCA rather than CCA [13, 43]. Because of this, the PCA resulting vector x is passed to a neural network.

5.4 Locally Weighted Neural Network Setup Let cx = (c1 , . . . , cK ) be a vector of concentrations of K components of a mixture and x = (x1 , . . . , xJ ) be the data vector of the biosensor signal after the preprocessing. cx is the target concentrations for the input data x.

5 Neural Networks for an Analysis of the Biosensor Response Fig. 6 Schematic diagram of three-layer feedforward artificial neural network, where x1 , . . . , xJ are the values of the biosensor current and c1 , . . . , cK are the determined concentrations of mixture components

437

x1

c1

x2

c2

cK

xJ Input layer

Output layer Hidden layer

The kth component of the quested nonlinear mapping from x to c can be expressed by an artificial neural network(ANN) Nk (x) = β0k +

p 

βsk ϕ(< α s , x > +α0s ),

k = 1, . . . , K,

(41)

s=1

where Nk (x) is the output of the kth output node expressing the concentration of the kth component Sk of the mixture, p is the number of nodes in the hidden layer, β k = (β0k , β1k , . . . , βpk ), α0s , α s are the parameters and ϕ is the nonlinear activation function. The sigmoid (logistic) function was employed as the activation function ϕ, ϕ(u) = 1/(1 + exp(−u)). The number p of nodes in the hidden layer was chosen according to a rule of thumb. Having L observations (elements) in the learning set, the degrees of freedom in the neural network should not exceed 0.1 × L, i.e. (p + 1) × K + p(J + 1) < 0.1 × L. Figure 6 shows the overall architecture of the network used for the analysis of the biosensor response. The data analysis showed that one hidden layer is enough to achieve sufficiently good results in concentration estimation [12]. The neural network is usually trained by a supervised batch learning procedure that requires a set of examples for which the desired network response is known. An advanced variant of the back-propagation algorithm called Levenberg–Marquardt was used to optimize the process of learning [63]. After the first phase of learning ∗ , α ∗ of α , α , s = 1, . . ., p, were kept fixed. In the second the estimated values α0s 0s s s phase, we used the locally weighted regression method to re-estimate the value of β k [4, 71]. Let 1 be the learning set and 2 be the test set of examples. 1 is used in the first phase of learning, while data of 2 is employed for retraining the neural network in the second phase. ∗ , α = α ∗ , s = 1, . . ., p, the neural network For the fixed coefficients α0s = α0s s s can be expressed as a linear regression β Tk (x), where  T ∗ ∗ (x) = 1, ϕ(< α ∗1 , x > +α01 ), . . . , ϕ(< α ∗p , x > +α0p ) .

(42)

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Application of Mathematical Modeling to Optimal Design of Biosensors

Given a test vector xq ∈ 2 and a weight function w(x, xq ), the locally weighted loss functions are defined as follows: 

 2 w2 (x, xq ) ck (x) − β Tk (x) ,

k = 1, . . ., K.

(43)

x∈1

Let 1 = (x1 , . . . , xL1 ). Then the parameter vectors β ∗k = β ∗k (xq ) minimizing loss functions (43) are given by ( )−1 β ∗k = (W )T (W ) (W )T W C,

(44)

where  is a L1 × (p + 1) (design) matrix with the elements ls = (s (xl )), l = 1, . . ., L1 , s = 0, . . ., p; W is a L1 ×L1 diagonal matrix of the local weights with the    T diagonal elements wll = w(xl , xq ) , l = 1, . . ., L1 ; C = ck (x1 ), . . . , ck (xL1 ) . In (44), we use a pseudo-inverse. Here for brevity, we omit the subscript k. From (41), (42) and (44), we obtain the final estimate of Nk (xq ), Nk∗ (xq )

=

∗ β0k (xq ) +

p 

∗ ∗ βsk (xq )ϕ(< α ∗s , xq > +α0s ), k = 1, . . . , K.

(45)

s=1

Locally weighted learning systems require a measure of relevance. The major assumption that locally weighted learning rests upon is that the relevance can be measured using the distance between data points. Nearby training points are more relevant. A weighting or kernel function is used to calculate the weight for a given distance between the two points. A typical weighting function is Gaussian   w(xn , xq ) = exp −xn − xq 2 /(2h2 ) ,

(46)

where h is the bandwidth or the kernel width. It determines the range over which the generalization is performed. The forecasting quality Qk of concentration ck of each component Sk , k = 1, . . . , K, is estimated by the percentage of true interval predictions [12], Qk =

L2   1  I nd Nk∗ (xi ) ∈ y × I nd (ck (xi ) = y) × 100%, L2

(47)

i=1

  where the indicator function I nd Nk∗ (xi ) ∈ y equals unity when the kth output of the network training for xi as a query point) belongs to   (after locally weighted the interval y − δ1,y , y + δ2,y of the target concentration ck (xi ) = y, and zero otherwise. L2 is the number of observations in the test set 2 . When using locally weighted retraining, it is necessary to retrain the network each time a new point is evaluated. This operation is quite time-consuming. Another

5 Neural Networks for an Analysis of the Biosensor Response

439

disadvantage is the need to keep the original training set. However, the improvement in accuracy in concentration prediction offsets these disadvantages in different applications [40, 56].

5.5 Biosensor Calibration Results The modeling biosensor was calibrated for mixtures of four (K = 4) components. Each component of eight (M = 8) different concentrations was employed in the calibration to have the biosensor response to a wide range of substrate concentrations [10, 12, 13]. The total set of full factorial of M K = 84 = 4096 responses was split randomly into 1 (learning) and 2 (test) sets having approximately the same number of responses. Two thousand response curves were chosen independently as the 2 set. The remaining 2096 samples were accepted as the 1 set, L = 2096 [12, 13]. The following M concentrations for each of the K components S1 , . . . , SK of the mixture were used:   ck ∈ sk,m : sk,m = γm μM, m = 1, . . . , M , k = 1, . . . , K, K = 4, M = 8; γ1 = 1, γ2 = 2, γ3 = 4, γ4 = 8, γ5 = 12, γ6 = 16, γ7 = 32, γ8 = 64. The biosensor response depends upon the concentration ck of the component Sk (k = 1, . . ., K) and the membrane thickness d. The response of the first biosensor, having a membrane of thickness d = 0.2 mm, was under dual control: by diffusion (for three components S1 , S2 , S3 ) and by enzyme kinetics (for component S4 ). Another thickness d = 0.4 mm was chosen, so that the response was controlled by the diffusion for all four components. The biosensor responses were simulated until a steady state was reached. During the computer simulation, values z = (z1 , . . ., zN ) of the density of the biosensor current were stored every second of the biosensor operation, i.e. the current density at time ti = i s was accepted as the zi , i = 1, . . . , N. The moment of the occurrence of the steady state depends on the thickness of the enzyme membrane [11, 72, 78]. The application of PCA to the 1 sets resulted in 6 principal components for both thicknesses on the enzyme membrane, d = 0.2 mm and d = 0.4 mm, N = 300. Due to the PCA, the neural networks having J = 6 nodes in the input layer were employed. Since the mixtures to be analysed consist of four components, the networks have K = 4 nodes in the output layer. An additional analysis showed that one hidden layer was enough to achieve sufficiently good results in concentration estimation [12]. A single hidden layer feedforward neural network with sigmoid activation in the hidden layer has universal approximation capabilities [41, 63]. Twelve nodes in a

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Application of Mathematical Modeling to Optimal Design of Biosensors

Table 3 The accuracy intervals y for prediction of the concentration y using forecasting quality (47)

Table 4 The forecasting quality Qk of the concentration prediction in bath analysis at the test set and membrane thickness of 0.2 mm

y 1 2 4 8 Component, k 1 2 3 4

y [0, 1.5) [1.5, 2.9] [3.1, 5) [7, 9) Standard ANN 100 100 96 57.4

y 12 16 32 64

y [11, 13) [15, 17) [31, 33) [63, 65)

Locally weighted ANN 100 100 99.55 76.75

single hidden layer were used for the membrane thickness d of 0.2 mm. In the case of d = 0.4 mm, eight nodes in a single hidden layer were enough. The forecasting quality Qk of concentration y = sk,m of the component Sk of a mixture was estimated using quality measure (47) with the tolerance intervals y presented in Table 3. The quality of prediction depends on whether the biosensor response is under diffusion or enzyme kinetics control. When biosensor response was under enzyme kinetics control, a prediction quality of 76.75 % was achieved. In all other cases, the forecasting quality was over 99 %. Let us remind that the biosensor response was under diffusion control when predicting only the component S4 with the biosensor having an enzyme membrane thickness of 0.2 mm. The prediction quality of 76.75% is fairly low (Table 4). However, this value is significantly higher than the corresponding value of the forecasting quality (57.4%) achieved using a standard ANN [12].

6 Concluding Remarks The designing of a biosensor is reducible to a multi-objective optimization where minimum or maximum values of numerous parameters are desirable. The complex nature of practical biosensors involves consideration of the simultaneous optimization of several objectives, some of whose are conflicting. The stated multi-objective optimization problem is difficult to solve since the objectives are numerical solutions of the nonlinear mathematical model. Chebyshev scalarization method can be efficiency applied to find trade-off solutions (Pareto optimal decisions). The multi-dimensional scaling is a suitable method for visualization of the Pareto optimal decisions that are a subset of the feasible regions in a multi-dimensional space. The presented method of the biosensor design integrating the multi-objective optimization with the visualization is valuable in exploration of the relation between

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the Pareto optimal decision and solution spaces aiming at search for an appropriate trade-off between the conflicting objectives. An application of the method to the optimization of a glucose dehydrogenase-based amperometric biosensor utilizing the synergistic substrates conversion showed that the advantage of the method is attained by the combination of the advantages of mathematical methods in generating a set of admissible decisions and human heuristics in analysing visual information. The optimization-based method of the quantitative analysis of the biosensor response is appropriate for the evaluation of the concentrations of the substrates by single enzyme amperometric biosensors utilizing the Michaelis–Menten kinetics (31). A computer simulation based on the mathematical model can be used to generate pseudo-experimental biosensor responses to mixtures of substrates. The generated data can be used to validate a method of the quantitative analysis as well as to calibrate the analytical system. The influence of a signal trend (linear as well as exponential) as of a background current to the precision of the concentrations estimation can be compensated by means of including the corresponding term (ati ) into the model of measurements (34). The additive white Gaussian noise has much more significant impact on the precision of the evaluation of the substrate concentrations compared with the multiplicative white Gaussian noise with the same standard deviation. Artificial neural networks can be successfully used to discriminate components of mixtures and to estimate the concentration of each component from the biosensor response data. The prediction quality depends on whether the biosensor response is under diffusion or enzyme kinetics control. The concentration of a component is predicted more accurately when the biosensor response is under diffusion control, i.e. when the diffusion module is greater than one. Because of this, the enzyme membrane thickness as one of the factors determining the diffusion module is of crucial importance for the detection limit of the biosensor. Prediction quality can be significantly increased by the application of the locally weighted linear regression for retraining the neural networks (Table 4).

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Index

Symbols 1D model, 395–397, 400, 401 2D Cartesian model, 400 2D model, 393, 394, 398, 401 2D polar model, 392, 400 2L model, 229 3D model, 397, 400, 401 3L model, 229

A Absorbance, 42 Acetaminophen, 86, 97, 99, 108 Acinetobacter calcoaceticus, 156 Activation, 276 Activation energy, 429 Activation function, 437 Activator, 277 Active sites of the CNTs, 371 Additive white Gaussian noise (AWGN), 433, 434 Adenosine triphosphate (ATP), 86, 95 Alcohol dehydrogenase, 125 Allometric function, 316, 319 Allosteric effect, 277 Allosteric enzyme, 277, 293, 295 Allostery, 276, 277, 293 Amperometric biosensor, 2, 9, 10, 281, 293, 427 Amperometric electrode, 5 Amperometric transducer, 382 Amperometry, 2, 9, 52, 60 Amplification, 121, 122, 125, 131–134, 145, 148–150, 152

Amplification cycle, 122 Analytical device, 2 Analytical solution, 19 Anodic current, 9, 111, 266 Apparent bimolecular constant, 212 Apparent Michaelis constant, 14, 183, 184, 191, 192, 199–203, 418, 423, 424 Array of enzyme microreactors, 303, 305, 319 Array of microreactors, 319, 329 Arrhenius equation, 429 Arrhenius trend, 429 Artificial neural networks (ANN), 113, 406, 407, 432, 435, 437, 440, 441 Averaged concentration, 308 Averaged diffusion coefficient, 362 Averaging diffusion coefficient, 246 Azimuth parameter, 395

B Backward difference, 355, 361 Bacterial culture, 393, 399 Bacterial population, 392, 394, 395, 397 Bacterial self-organization, 384, 389, 391, 395–398 Bamboo foam-like structures, 397 Bandwidth width, 438 Basic vectors, 436 Basidiomycete Lac, 364 Batch analysis, 184, 186, 191–193, 200, 202 BC-catalyzed reaction, 233 BCE, 207 Beer–Lambert law, 42 Biased response, 407, 424

© Springer Nature Switzerland AG 2021 R. Baronas et al., Mathematical Modeling of Biosensors, Springer Series on Chemical Sensors and Biosensors 9, https://doi.org/10.1007/978-3-030-65505-1

447

448 Bienzyme biocatalytic membrane, 90 Bienzyme biosensor, 86, 90, 95, 96 Bienzyme electrode, 86, 88–90, 96 Bilinear spline, 430 Bimolecular electron exchange, 157 Binding site, 277, 295 Bi-objective problem, 420, 421 Biocatalyst, 3, 6 Biocatalytical current, 160 Biocatalytic membrane, 3, 5, 6 Biocatalytic process, 5 Biochemical oxygen demand, 377, 382 Biochemical oxygen demand biosensor, 378 Biochemical recognition, 5 Biofuel cells, 156 Biological catalyst, 3 Bioluminescence images, 392–394 Bioluminescence imaging, 396 Bioluminescence patterns, 392 Bioluminescence waves, 391 Bioluminescent cells, 384 Biomimetic catalysts (BC), 207, 231 Biomimic materials (BM), 231 Biosensor, 2, 3 Biosensor action, 3, 6 Biosensor array, 426 Biosensor based on microorganisms, 378 Biosensor calibration, 112, 439 Biosensor calibration curve, 358, 371, 372 Biosensor characteristics, 409, 417 Biosensor current, 9, 53, 58, 67, 69, 74, 75, 95, 104, 111, 210, 214, 215, 220, 222, 224, 234, 238, 248, 250, 252, 258, 268, 271, 313, 324 Biosensor current density, 9 Biosensor potential, 40 Biosensor response, 6, 9, 41, 42, 65, 79, 90, 91, 95, 102, 105, 159, 164, 169, 178, 179, 208, 211, 214, 216, 217, 222, 223, 225, 238, 243, 245, 248, 250, 252, 258, 266, 271, 279, 313, 323, 334, 347, 353, 356, 359, 361, 363, 368, 417 Biosensor response time, 355, 417 Biosensor selectivity, 5, 244 Biosensor sensitivity, 13, 24, 60, 121, 123, 131, 132, 156, 159, 168–171, 285, 347, 368, 372, 412, 418 Biot number, 73, 75, 77, 79–81, 198, 199, 201, 203, 216 Black-box functions, 420 Black-box problem, 421 BOD biosensor, 378, 382

Index Boundary conditions, 6, 11, 39, 52, 59, 61, 67, 72, 89, 95, 96, 102, 163, 174, 185–188, 195, 197, 209, 210, 214–216, 219, 234, 247, 257, 258, 264, 265, 267, 282, 294, 312, 323, 333, 351, 360, 366, 387–389, 392, 395, 398 Butler–Volmer, 5

C C+ language, 285 C+ programming language, 355, 361 Calibration, 14 Calibration curve, 50, 81, 106, 227, 293, 298, 418 Calibration data, 407 Canonical correlation analysis (CCA), 436 Carbon nanotube, 345 Carbon paste, 304 Carbon-paste biosensor, 425 Carbon paste electrode (CPE), 304–306, 309, 338 Carbon paste porous electrode, 305 Cardano formula, 278 Carrying capacity, 385, 391 Cartesian coordinates, 321, 324 Cartesian system, 398 Catalase, 86, 97, 105, 106, 116 Catalase-peroxidase, 116 Catalase-peroxidase biosensor, 106, 108 Catalytic activity, 3 Catalytic biosensor, 2 Catalytic constant, 212, 235, 239 Cathodic current, 9 CCE, 122, 143, 146, 152 CCE mode, 138, 140, 141, 144, 145, 147, 150, 152 CCE scheme, 135, 138 CEC, 122, 139, 143, 146, 152 CEC mode, 137–141, 144, 146, 150, 151 CEC scheme, 135, 136, 138, 142 Cell death, 385 Cell density, 386, 388, 393, 394, 396, 399 Cell growth, 385 CE scheme, 139, 140 Chaotic fluctuations, 393 Chemical amplification, 125, 126, 148, 157, 171 Chemical conversion, 122, 156 Chemically modified (CM), 208 Chemically modified electrode (CME), 208–211, 222, 232, 238

Index Chemically modified enzyme electrode (CMEE), 207, 208, 211, 213, 225 Chemoattractant, 384, 385 Chemoattractant concentration, 393, 394, 399 Chemoattractant production, 385 Chemometric approach, 426 Chemometric method , 425 Chemometrics, 112, 435 Chemotactic response, 385 Chemotactic sensitivity, 379, 385, 386, 401 Chemotaxis, 379, 390 Clark-type electrode (CE), 86, 108 Clark type oxygen electrode, 382 CNT composite electrode, 347 CNT composite layer , 354 CNT electrode, 364, 372 CNT enzyme-loaded electrode, 359 CNT mesh, 356, 358 Coenzymes, 378 Co-immobilized enzymes, 86 Competition pathway, 86 Competitive inhibitor, 114 Composite electrode, 347, 357 Compound, 5 Concentration profiles, 55, 67, 230 Conflicting objectives, 292 Consecutive conversion, 91 Consecutive substrates conversion, 86, 87, 90, 91, 122 Consumption-regeneration cycle, 122 Contradictive objectives, 412 Convective region, 161, 171 Cooperative binding, 297 Cooperative kinetics, 295 Cooperativity, 295, 297 Correction factor, 426 Correlation coefficients analysis, 114 Cottrell equation, 4 CPC, 305, 311 CPC-matrix, 314 Crank-Nicolson, 103 Creatine, 92 Creatininase, 92, 93 Creatinine, 92 Cross reaction, 159, 161, 171 Current density, 89, 91, 324, 334 Cyclic coenzyme conversion, 125 Cyclic mediator conversion, 5 Cyclic substrates conversion, 122, 123, 125, 148 Cylindrical coordinates, 321, 324

449 D Damköhler number, 12, 63, 216 Data covariance matrix, 436 Data pre-processing, 114 Data synthesized, 113 Decision parameters, 420 Decision variables, 412, 413, 419, 424 Decision vector, 411 Decision visualization, 416, 424 Dehydrogenase-based bioelectrode, 159 Descartes coordinates, 314, 322, 329 D-glucose oxidase-hexokinase biosensor, 95 D-glucose oxidase-peroxidase biosensor, 88 Dialysis membrane, 108, 155, 159, 161, 165, 171, 174, 227, 244 Difference equations, 127 Diffusion coefficient, 52, 59, 77, 282 Diffusion control, 75, 432 Diffusion layer, 50, 61, 62, 69, 74, 87, 183, 184, 197, 199, 200, 203, 244, 246, 247, 254, 256, 282, 321, 332 Diffusion limitation, 50, 217, 244, 271, 382, 426 Diffusion-limiting membrane, 227 Diffusion limiting region, 161, 171 Diffusion module, 12, 14, 41, 63, 71, 73, 79, 89, 91–94, 134, 149, 158, 189, 191, 192, 194, 198–203, 216, 217, 219, 223, 225, 227, 238, 288, 295, 296, 372, 426, 432 Diffusion rate, 216, 217, 237 Digital simulation, 155, 165, 167, 176 Dimensionless apparent Michaelis constant, 14, 191, 199–201 Dimensionless cell density, 388 Dimensionless chemoattractant concentration, 388 Dimensionless coefficient, 233 Dimensionless concentration, 11, 71, 218, 287 Dimensionless consumption rate, 388 Dimensionless coordinate, 12, 218 Dimensionless current, 12, 72, 219, 223, 225, 226 Dimensionless diffusion coefficient, 288 Dimensionless distance, 218, 287 Dimensionless governing equations, 388 Dimensionless growth rate, 388 Dimensionless inhibition rate, 287 Dimensionless maximal gradient, 13 Dimensionless model, 11, 70, 211, 218, 219, 223, 287, 387

450 Dimensionless nutrient concentration, 388 Dimensionless parameters, 11, 71, 218, 278, 287, 389 Dimensionless reaction rate, 106 Dimensionless sensitivity, 13, 60 Dimensionless thickness, 218, 287 Dimensionless time, 71, 218, 221, 287 Dimeric enzyme, 277, 295 Direct electron transfer, 372 Direct enzyme oxidation, 364 Directional anisotropy, 357 Discrete grid, 54, 127, 140 Dual biosensor, 430 Dual catalase-peroxidase bioelectrode, 86, 97 Dual enzyme-substrate complex, 277 Dynamics of the biosensor current, 356

E Effective diffusion coefficient, 62, 229, 230, 246, 271, 308, 310, 311, 332, 339, 350, 354, 360–362, 366 Effective diffusion layer, 4 Effective rate, 104 Effective thickness, 42 Efficient points, 422 Eigenvectors, 436 Electro-active substance, 11, 247 Electrocatalysis, 208, 231 Electrochemical amplification, 139 Electrochemical biosensor, 2 Electrochemical conversion, 122, 126, 127, 135, 382 Electrochemically active enzyme, 369–371 Electrochemical reaction rate, 364, 372 Electrochemical reaction rate constant, 370 Electrochemical reactions, 5, 163, 164, 285, 350, 353, 372 Electrode coverage, 327, 329 Electrode potential, 382 Electrode stability, 90 Electro-inactive substance, 247 Electronic signal, 5 Enzymatic amplification, 122, 139 Enzymatic reaction, 350, 364 Enzymatic trigger reaction, 122 Enzyme activity, 3, 28, 425 Enzyme allostery, 277, 293 Enzyme amount factor, 410, 414 Enzyme-catalyzed reactions, 58, 59, 109 Enzyme electrode, 10 Enzyme filling degree, 358 Enzyme inactivation, 425 Enzyme inhibition, 277, 298

Index Enzyme inhibitor, 276 Enzyme kinetics control, 14, 75, 432 Enzyme layer, 51, 59, 281, 293, 294 Enzyme life-time, 425 Enzyme-loaded carbon nanotubes (CNT), 353, 363 Enzyme-loaded CNT layer, 350, 351, 354 Enzyme-loaded CNT mesh, 359 Enzyme-loaded mesh, 161, 173 Enzyme-loaded nylon net, 161, 171 Enzyme membrane, 6, 7, 58, 281, 293, 427 Enzyme membrane thickness, 75 Enzyme microreactor, 320, 325 Enzyme-product complex, 276 Enzyme reaction, 2, 3, 6 Enzyme re-oxidation, 364 Enzymes, 2–4 Enzyme-substrate complex, 4, 276 Excitation light, 42 Exhaustive search, 422 Experimental bioluminescence, 390 Experimental data, 165, 167, 346, 347, 356, 363, 369, 370 Experimental model validation, 369 Experimental sample, 390 Explicit finite difference scheme, 22 Exponential trend, 431, 433 External diffusion, 183, 184, 195, 196, 200, 425, 426 External diffusion layer, 7, 59–61, 69, 71, 74, 77, 245, 246, 249, 254, 256, 257, 260, 262, 297, 304, 312, 319, 331, 335 External diffusion limitation, 7, 64, 81, 298 Extinction coefficient, 42 F Faradaic current, 5 Faraday constant, 60, 89, 91, 285, 353, 368, 417 Faraday law, 60, 284, 417 Feasible objective region, 420 Feasible region, 422, 428, 429, 440 Feedforward artificial neural network, 437 Feedforward neural network (FNN), 435, 439 Ferricyanide, 155, 159, 161, 167, 170, 172, 178 Ferrocyanide, 86, 88, 155, 159, 160, 164, 172, 178, 417 Fick law, 59, 60, 282, 293, 417 Filling degree, 358 Finite difference scheme, 103, 285 Finite difference technique, 51, 65, 183, 184, 248, 258, 266, 285, 296, 360, 379, 389

Index Finite diffusion regime, 282 First-order kinetics, 104 First order reaction, 7 First order reaction constant, 7 First order reaction rate, 62, 86 First type transducer, 5, 6, 10 Flow injection analysis (FIA), 184, 185, 202 Fluctuating signal, 378 Fluorescence, 42 Fluorescent, 42 Foam-like structures, 389 Forecasting quality, 438, 440 Forward difference, 355, 361 Four-compartment model, 258, 259 Four substrate binding sites, 295

G Gain of the sensitivity, 131, 139 Gaussian, 438 Gaussian distribution, 434 GDH action, 159 GDH amperometric biosensor, 160, 166 GDH-based amperometric biosensor, 417, 441 GDH biosensor, 159, 167, 173, 176, 178, 179, 416, 424 GDH reaction, 159, 167 Generation of pseudo-experimental data, 112 Glucose, 418 Glucose biosensor, 9, 88 Glucose dehydrogenase (GDH), 122, 156, 159, 160, 364, 416 Glucose electrode, 90 Glucose oxidase (GO), 9, 86, 88, 90, 95, 157, 158, 177, 305–308, 338, 425 Glucose oxidation, 86, 95 Governing equation, 141, 211, 216, 218, 263, 308, 321, 332 Governing equations, 12, 59, 71, 100, 161, 165, 173, 176, 246, 248, 250, 255, 258, 266, 281, 307, 309, 334, 350, 360, 365, 395, 397 GOx, 9 Graphite electrode, 156, 159, 165, 167, 168, 178 Green function, 91, 96 Growth rate, 385, 386, 397

H Haemoglobin, 295 Half maximal effective concentration, 14 Half-time of steady state, 15

451 Hansenula anomala, 378, 379 Height tortuosity, 357 Hessian, 431 Heterogeneous microreactor, 305 Hexacyanoferrate(II), 155–157, 170 Hexacyanoferrate(III), 156–158, 176 Hexokinase, 86, 95, 96 Hexokinase reaction, 95 Hidden layer, 439 Hill coefficient, 295–298 Hill equation, 295, 296 Homogenization process, 161, 246, 256, 261, 359, 365 Homogenized biosensor, 309 Homogenized MR, 309 Hooke–Jeeves algorithm, 420 Hooke–Jeeves optimization algorithm, 421 Hydrogen peroxide, 9, 88, 92, 97, 100

I Immobilized enzyme, 3, 58 Impermeable membrane, 50 Implicit finite difference scheme, 66, 140 Inactivated biosensor, 426 Indicator electrode, 2 Inert dialysis membrane, 416 Inert membrane, 61, 243, 244 Inert porous membrane, 245 Inhibition, 276, 287 Inhibition constant, 286 Inhibitor, 276, 277 Inite difference scheme, 266 Initial boundary value problem, 20, 54, 62, 63, 91, 124, 285, 296, 313, 324, 334, 355, 360, 368 Initial concentration, 4, 89 Initial conditions, 6, 10, 59, 66, 72, 96, 101, 174, 213, 216, 218, 234, 247, 257, 264, 282, 294, 311, 322, 333, 352, 360, 367, 381, 388, 389, 392, 395, 396, 398 Insulating film, 347, 359 Intermediate complex, 211 Internal diffusion, 184 Internal diffusion limitation, 7, 81, 86, 91 Ion-selective electrodes, 2, 5, 7, 39, 382 K Keller and Segel approach, 384, 400 Keller and Segel model, 379 Kernel function, 438 Kernel width, 438

452 L Laccase, 126, 131, 364 Laccase-based bioelectrode, 155, 170 Laccase-based biosensor, 170 Laccase biosensor, 171, 173, 176, 179 Laccase-catalysed bisphenol A, 156 Laplace operator, 308, 309, 322, 385, 392, 398 Laplacian, 332 Least squares approach, 428 Levenberg–Marquardt algorithm, 437 L’Hopital rule, 91 Light absorbance, 41 Linear combination, 436 Linear regression, 437 Linear trend, 431, 432 Locally weighted learning, 438 Locally weighted linear regression, 407, 434, 441 Locally weighted loss function, 438 Locally weighted neural network, 434 Locally weighted regression, 407, 437 Locally weighted retraining, 438 Local weights, 438 Logistic cell growth, 391 Logistic function, 437 Longitudinal parameter, 395 Long-term response, 425 Loss function, 438 Luminous E. coli, 384, 396 Lux-gene engineered bacteria, 378

M Main governing parameters, 287 Mass conservation relation, 52 Mass transport, 184, 189, 197 Mass transport by diffusion, 59, 350 Matching conditions, 53, 67, 72, 102, 213, 219, 247, 257, 264, 265, 294, 323, 333, 352 Mathematical model, 10, 244, 250, 261, 262 MATLAB, 431 Maximal current, 14, 190–193, 199, 285 Maximal density of the biosensor current, 190, 191, 285 Maximal enzymatic activity, 79 Maximal enzymatic rate, 4, 10, 32, 52, 59, 75, 110, 126, 129, 133, 134, 136, 144–152, 185, 190, 191, 197, 199, 201, 202, 246, 252, 264, 284, 296, 308, 322, 332, 435 Maximal enzyme rate, 64 Maximal gradient, 13, 24 MDS algorithm, 422

Index Mediated biosensor, 347 Mediator, 207–217, 219–223, 225, 226, 228, 305–307, 350 Mediatorless biosensor, 363, 365, 371 Mediatorless electron transfer, 363 Mediator re-oxidation, 306, 348 Membrane perforation, 243, 261, 269 Membrane permeability, 250 Membrane thickness, 252, 271 Membrane thickness optimization, 28 Metabolite biosensor, 379, 400 Method of alternating directions, 355 Michaelis constant, 4, 7, 10, 14, 15, 31, 52, 59, 63, 110, 126, 136, 152, 185, 191, 197, 246, 256, 264, 278, 284, 294, 298, 308, 322, 332 Michaelis–Menten equation, 295 Michaelis–Menten function, 18, 385 Michaelis–Menten kinetics, 3, 183, 184, 243, 276, 278, 284, 296, 298, 303, 305 Michaelis–Menten scheme, 381 Microbial biosensor, 378, 379, 384 Microbial cell, 378, 379 Microbial cell wall, 379 Microreactor (MR), 303, 305–312, 314–316, 318, 326, 338, 339 Microtiter plate, 390 Minimization problem, 421, 428 Mixture, 116 Mixture of compounds, 87, 109 Mixture of substrates, 109 Model equations, 115 Modeling substrates interaction, 114 Modelling error, 362, 397 Model parameters, 391 Model validation, 165, 355 Mono-enzyme electrode, 91, 97 Mono-layer biosensor, 58 Mono-layer enzyme biosensor, 324 Mono-layer enzyme electrode, 86 Multianalyte determination, 407, 424 Multi-compartment model, 62 Multi-component analysis, 113 Multi-component mixtures, 407, 434 Multi-dimensional data, 422 Multi-dimensional scaling, 422, 440 Multi-enzyme, 109 Multi-enzyme biosensor, 86, 109 Multi-enzyme system, 50 Multi-fold complex, 115 Multi-layer approach, 50, 51, 55 Multi-layer biosensor, 50, 58 Multi-layer enzyme membrane, 244 Multi-layer model, 50, 51, 66

Index Multi-layer system, 51 Multi-mbjective optimization problem, 420 Multi-objective optimization, 406, 411, 416, 418, 420, 424, 440 Multi-objective optimization problem, 412 Multi-objective problem, 421 Multiplicative white Gaussian noise (MWGN), 433, 434 Multi-response, 279 Multi-substrate, 114 Multi-substrate conversion, 114, 427, 435 Multivariate calibration, 407 Multi walled carbon nanotubes (MWCNT), 346 N Nanotubes orientation, 354 Negative biosensor sensitivity, 290 Negative cooperativity, 295, 296, 298 Nernst approach, 77, 213, 227, 283, 323, 332, 333, 349 Nernst diffusion layer, 50, 77, 79, 81, 265, 267, 321, 324, 333, 335, 349–351, 354, 359, 367 Nernst equation, 40 Nernstian boundary condition, 6 Nernst layer, 162, 278 Neural networks, 184, 436 Noise-corrupted measurement, 426 No-leak boundary conditions, 386 Non-Clark-type electrode (NCE), 108 Non-leakage boundary condition, 11, 39, 61, 257, 264, 265, 312 Non Michaelis-Menten kinetics, 276, 298 Non-stirred buffer solution, 61 Nontransparent plate, 278 Normal distribution, 434 Normalized biosensor sensitivity, 133 Normalized steady state current, 259, 327, 339 N-substituted phenothiazines, 155, 156, 170 N-substituted phenoxazines, 156 Nuclepore type membranes, 244 Numerical approximation, 54 Numerical simulation, 103, 116, 248, 250, 258, 266, 267, 285, 355, 358, 361, 368 Numerical solution, 54, 103, 116 Nutrient, 384, 385 Nutrient concentration, 385, 388, 391 Nutrient consumption, 385 O Objective functions, 411, 412, 421, 428–430 Objective vectors, 411, 412

453 One-compartment model, 76 One-dimensional-in-space diffusion, 51, 59, 233 One-dimensional model, 361 One-layer model, 50, 61, 81 Optical biosensor, 41 Optical fibber, 5 Optical signal, 41 Optical system, 5 Optical transducer, 6, 7 Optimal decisions, 424 Optimal design, 416, 421 Optimal design problem, 419, 420 Optimal solutions, 424 Optimization-based approach, 426 Optimization problem, 419, 421, 428 Optimization process, 430 Optimization variables, 420 Oscillation, 281 Oscillatory signal, 378 Other membrane, 244 Outer membrane, 50, 61, 62, 227, 244, 245, 250, 252, 254, 332, 337 Outer perforated membrane, 244 Output current, 368 Overall biosensor response, 111 Oxgen concentration, 399 Oxidized compound, 5 Oxidized enzyme, 365 Oxidized GDH, 161 Oxidized mediator, 159, 168, 171, 172, 350 Oxygen, 86, 100, 387 Oxygen bioelectrode, 108 Oxygen consumption rate, 398 Oxygen diffusivity, 398 Oxygen electrode, 86, 382 Oxygen electrode current, 383 P Parallel substrate conversion, 95–97, 100 Pareto front, 412, 415, 420–423 Pareto optimal, 412 Pareto optimal decisions, 415, 420–422, 440, 441 Pareto optimal decision vectors, 412, 415 Pareto optimal solutions, 406, 419, 423 Pareto set, 412 Pareto solution, 422 Partially blocked electrode, 348 Partial stationary current, 15 Partition coefficient, 353, 354, 358 Perforated membrane, 254, 256, 259, 261, 262, 271, 347–349, 354, 357, 359, 361, 362, 364, 365

454 Perforated membrane openness, 269, 270 Perforation geometry, 362 Perforation level, 360–362 Perforation topology, 269, 361 Periodicity conditions, 387 Periodic media, 161, 246, 256, 261, 271, 310, 332, 354 Permeability, 244, 250, 252, 380 Peroxidase, 86, 88, 97, 105, 106, 109, 116 Peroxidase activity, 90 Perturbed response, 425 Phenol sensitive biosensor, 126 Ping-pong scheme, 159, 207, 209, 211, 212 Plate-gap biosensor, 305, 330, 331, 334, 335, 338 Plate-gap electrode, 330, 338 Plate-gap model, 303, 305, 330 Polar coordinates, 392, 395 Polycarbonate perforated membrane, 347 Polyenzyme system, 87 Polymer membrane, 61 Polynomial approximation, 426 Population density, 385 Population kinetics model, 386 Porosity, 250, 308, 310, 314, 316, 318, 340 Porous electrode, 330 Porous membrane, 244–252, 258, 261, 271, 330–332, 335 Positive cooperativity, 295, 296, 298 Potentiometric biosensor, 2, 61 Potentiometric biosensor response, 40 Potentiometric transducer, 382 Potentiometry, 2 PQQ-dependent glucose dehydrogenases, 305, 330 Prediction quality, 440 Pre-exponential factor, 429 Principal component analysis (PCA), 114, 436, 439 Principal component vector, 436 Product degradation, 62 Product fluorescence, 42 Product inhibition, 276, 281, 292, 293, 298 Product inhibition constant, 288 Product inhibition rate, 284 Pseudo-experimental data, 428 Pyrocatechol, 126 Pyrroloquinoline quinone, 364

Q Quantum yield, 42 Quasi-one-dimensional cell density, 392 Quasi-one dimensional ring, 395

Index Quasi-steady state (QSS), 4 Quasi-steady state approach (QSSA), 4, 10, 115, 283, 284, 294, 427 Quasi-steady state approximation, 51, 58, 98, 283 Quasi-steady state conditions, 99 Quasi-steady state reaction rate, 284, 288, 294, 296

R Radial tortuosity, 357 Randomized selection, 421 Random noise, 428 Reaction-diffusion, 59, 282 Reaction-diffusion-chemotaxis, 379, 396 Reaction-diffusion equations, 162, 173, 243, 246, 256, 264, 294, 350, 366 Reaction-diffusion system, 196 Reaction network, 416 Reaction rate, 106 Reaction scheme, 97, 109, 160, 171, 173 Redox reaction, 233 Reduced compound, 5 Reduced diffusion module, 217, 220, 223, 224 Reduced enzyme, 365 Reduced GDH, 159, 161, 167 Reduced mediator, 157, 159, 160, 164, 351 Redundant information, 114 Reference electrode, 2 Regular hexagonal pattern, 348, 362 Regular oscillations, 393 Relative fluorescence units (RFU), 43 Relative modelling error, 361 Relative sensitivity, 158 Response amplification, 139, 145, 146, 148 Response compensation, 426 Response gain, 134, 145 Response stability, 28, 407 Response time, 14, 15, 129, 150–152 Response time prolongation, 150, 151 Resultant relative output signal, 15 Retraining the neural networks, 434 Rotating disk electrode, 281 S Sandwich-like biosensor, 50 Sandwich-like enzyme membrane, 52 Sarcosine, 92 Saturating chemoattractant production, 386 Saturating substrate concentration, 383 Saturation constant, 382

Index Saturation curve, 14, 297, 298 Search space, 411 Second substrate, 105 Second type transducer, 5, 6, 39 Selective layer, 50 Selective membrane, 50, 243, 244, 254, 256–259, 261–264, 268 Semi-implicit finite difference scheme, 23, 355, 361 Semi permeable membrane, 364, 380, 382 Sequence pathway, 86 Sequential injection analysis, 194, 195 Sigmoid activation, 439 Sigmoid function, 437 Signal production, 388 Signal-to-noise ratio, 419 Silica carrier, 305 Single objective problem, 421 Single walled carbon nanotubes (SWCNT), 346, 347 SMACOF algorithm, 422 Smart biosensor, 112 Solution for first order kinetics, 15, 62 Solution for Michaelis–Menten kinetics, 20 Solution for zero order kinetics, 18, 63 Solution of the problem, 111 Space-time plot, 390 Spatiotemporal pattern formation, 391 Spatiotemporal patterns, 377, 379, 384, 391–393, 396, 399, 401 Spatiotemporal plots, 390 Spherical coordinates, 309, 314, 325 Spline, 430 Stationary concentration, 19 Stationary condition, 7 Stationary current, 20 Statistical model, 421 Steady state, 10 Steady state biosensor current, 10, 14, 53, 60, 103, 305, 314, 324, 325, 329, 336, 339 Steady state biosensor response, 103, 105 Steady state concentration, 19 Steady state conditions, 7, 92, 381, 388 Steady state current, 20, 63, 190, 195, 216, 217, 220, 223, 225, 235, 239, 248, 250, 252, 254, 258, 266, 268–271, 315, 325, 329, 334, 335, 338, 340, 360 Steady state fluorescence, 43 Steady state relative biosensor current, 116 Steady state relative current, 107 Steady state response, 41, 69, 107, 108 Stimulant, 384, 385

455 Stoichiometric coefficient, 383 Structural anisotropy, 354, 356 Substrate degradation, 62 Substrate binding, 295 Substrate conversion, 88, 383 Substrate cyclic conversion, 121, 122, 127, 132, 148, 151 Substrate dissociation constant, 295 Substrate inhibition, 278, 281, 286, 289, 290, 293, 298 Substrate inhibition constant, 288 Substrate inhibition rate, 284 Succinate, 385 Surface multi-concentration, 279 Synergistic effect, 159, 177, 179 Synergistic reactions, 5, 156 Synergistic substrates conversion, 416, 441 Synergistic substrates determination, 417 Synergy, 155, 156, 170, 171, 179

T Tchebyshev algorithm, 422 Tchebyshev scalarization method, 421, 440 Terylene, 364 Terylene membrane, 366 Tetrameric enzyme, 295 Three-compartment model, 243, 248, 249 Three-layer biosensor, 57 Three-layer mathematical model, 229 Three-layer model, 55, 227, 258 Three-objective optimization, 419 Three-objective optimization problem, 420, 422 Three-phase contact line, 389, 390, 392 Time of maximal current, 190 Tortuosity, 311, 314, 316, 339, 354, 360–362 Total current density, 353, 360 Total enzyme concentration, 294 Trade-off, 406, 411, 412, 415, 419, 423, 441 Trade-off curves, 406 Trade-off solutions, 423, 440 Training points, 438 Transducer, 5, 6, 39, 244, 254 Transducer selectivity, 5 Transient numerical solution, 65 Transient state, 90 Trehalose, 425 Tri-diagonal matrix, 355 Trienzyme biosensor, 86, 91, 94 Trigger mode, 135, 148 Trigger reaction, 122 Trigger scheme, 122 Triggering, 122, 145, 148, 151, 152

456 Two calibration curves, 291 Two compartment, 100 Two compartment mathematical model, 107, 108 Two-compartment model, 50, 61, 62, 67, 70, 79, 81, 100, 104, 183, 184, 196, 198, 202, 211, 215, 248, 254, 303, 324 Two-dimensional model, 361 Two-dimensional visualization, 422 Two-enzyme biosensor, 86 Two enzyme electrode, 107 Two enzymes, 100 Two-layer mathematical model, 229 Two-phase composite, 310 Two substrate binding sites, 295 U Uniform discrete grid, 103, 116

Index V Viscosity, 77 Visualization of Pareto fronts, 422 Volume averaging approach, 354 Volume fraction, 354

W Weighting function, 438 White noise, 431, 433

X Xenobiotics conversion, 156

Z Zero flux boundary condition, 61 Zero order reaction rate, 63