Mathematical modeling of biosensors : an introduction for chemists and mathematicians / by Romas Baronas, Feliksas Ivanauskas, Juozas Kulys.

Format:

Book

Author/Creator:

Baronas, Romas.

Contributor:

Ivanauskas, Feliksas.

Kulys, Juozas.

Series:

Springer series on chemical sensors and biosensors 1612-7617 ; 9.

Springer series on chemical sensors and biosensors ; 9

Language:

English

Subjects (All):

Biosensors--Mathematical models.

Biosensors.

Mathematical models.

Physical Description:

xi, 334 pages : illustrations ; 25 cm.

Place of Publication:

Dordrecht ; London : Springer, 2010.

Summary:

This book presents biosensor development and modeling from both a chemical and mathematical point of view. It contains unique modeling methods for catalytical (amperometric, potentiometer and optical) biosensors. It examines processess that occur in the sensors' layers and at their interface, and it provides analytical and numerical methods to solve enzymatic kinetic and diffusion equations. The action of single enzyme as well as polyenzyme biosensors is studied, and 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 triggerd, and how signals are processed via an chemometric approach and artificial neuronal networks. The results of digital modeling are compared with both proximal analytical solutions and experimental data.

Contents:

Part I Analytical Modeling of Biosensors

Biosensor Action 3

1 Kinetics of Biocatalytical Reactions 3

2 Transducer Function 5

3 Scheme of Biosensor Action 6

Modeling Biosensors at Steady State and Internal Diffusion Limitations 9

1 Biosensors Containing Single Enzyme 9

2 Biosensors Containing Multienzymes 10

2.1 Consecutive Substrates Conversion 10

2.2 Parallel Substrates Conversion 14

2.3 Biosensors Utilizing Cyclic Substrates Conversion 15

3 Biosensors Utilizing Synergistic Substrates Conversion 16

4 Biosensors Based on Chemically Modified Electrodes 18

Modeling Biosensors at Steady State and External Diffusion Limitations 21

1 Biosensor Using Single Enzyme 21

2 Biosensors with Multienzymes 22

3 Biosensor Utilizing Non Michaelis-Menten Enzyme 23

Modeling Biosensors Utilizing Microbial Cells 27

1 Metabolite Biosensor 27

2 BOD Biosensor 30

Modeling Nonstationary State of Biosensors 33

1 Potentiometric Biosensors 33

2 Amperometric Biosensors 34

Part II Numerical Modeling of Biosensors

Mono-layer Mono-Enzyme Models of Biosensors 43

1 Mathematical Model of an Amperometric Biosensor 44

1.1 Governing Equations 44

1.2 Initial and Boundary Conditions 45

1.3 Dimensionless Model 46

2 Characteristics of the Biosensor Response 47

2.1 Biosensor Current 47

2.2 Biosensor Sensitivity 48

2.3 Maximal Gradient of the Current 48

2.4 Response Time 49

3 Finite Difference Solution 50

3.1 Numerical Approximation of Equations 50

3.2 Calculation Procedure 51

3.3 Validation of Numerical Solution 53

3.4 Numerical Errors Analysis 57

4 Pecularities of the Biosenser Response 60

4.1 Effect of the Enzyme Membrane Thickness 60

4.2 Stability of the Response 63

4.3 The Response Versus the Substrate Concentration 64

4.4 The Response Verus the Maximal Enzymatic rate 66

4.5 Choosing the Enzyme Membrane Thickness 68

4.6 Biocensor Resistance 70

4.7 Maximal gradient of the Current 71

5 Flow Injection Analysis 72

5.1 Mathematical Model 73

5.2 Numerical Solution 73

5.3 Biosensor Response 75

5.4 Peculiarities of the Biosensor Response 77

5.5 Sequential Injection analysis 80

6 Biosensors with chemical Amplification 81

6.1 Mathematical Model 82

6.2 Finite difference solution 84

6.3 Concentration profiles 85

6.4 Peculiarities of the Biosensor Response 86

7 Potentiometric Biosensors 91

7.1 Mathematical model 91

7.2 Biosensor response 92

7.3 Finite difference solution 93

7.4 Validation of numerical solution 94

7.5 Simulated Biosensor Response 95

7.6 Peculiarities of the Biosensor Response 98

8 Enzyme Inhibition 103

8.1 Substrate Inhibition 103

8.2 Effect of Substrate Inhibition 105

8.3 Product Inhibition 108

8.4 Effect of Product Inhibition 109

One-Layer Multi-Enzyme Models of Biosensors 113

1 Biosensors Response to Mixture of Compounds 114

1.1 Mathematical Model 114

1.2 Solution of the Problem 116

1.3 Generation of Data Sets 117

1.4 Concluding Remarks 120

2 Biosensors Acting in Trigger Mode 120

2.1 Mathematical Models 121

2.2 Finite Difference Solution 126

2.3 Simulated Response 128

2.4 Peculiarities of the Response 129

2.5 Concluding Remarks 136

Multi-Layer Models of Biosensors 139

1 Multi-Layer Approach 140

1.1 Mathematical Model of Multi-Layer System 141

1.2 Numerical Approximation 143

1.3 Three-Layer Model 145

2 Two-Compartment Model 147

2.1 Mathematical Model 147

2.2 Transient Numerical Solution 150

2.3 Validation of Numerical Solution 152

2.4 Simulated Biosensor Responses 153

2.5 Effect of the Diffusion Layer 155

2.6 The Nernst Diffusion Layer 157

2.7 Dimensionless Model 160

2.8 Impact of the Diffusion Module 162

3 Biosensors with Outer Porous Membrane 163

3.1 Mathematical Model 164

3.2 Numerical Simulation 166

3.3 Effect of the Porous Membrane 168

3.4 Concluding Remarks 171

4 Biosensors with Selective and Outer Perforated Membranes 172

4.1 Mathematical Model 173

4.2 Numerical Simulation 176

5 Biosensors Based on Chemically Modified Electrode 178

5.1 Mathematical Model 178

5.2 Numerical Simulation 182

5.3 Dimensionless Model 184

5.4 Simulated Biosensor Action 186

5.5 Impact of the Diffusion Module 189

5.6 Impact of the substrate concentration 191

5.7 Concluding Remarks 192

6 Optical and Fluorescence Biosensors 193

6.1 Mathematical Model 193

6.2 Numerical Biosensor Action 198

6.3 Impact of the substrate concentration 201

Modelising biosensors of complex geometry 203

1 Biosensor based on heterogeneous microreactor 204

1.1 Structure of Modeling Biosensor 204

1.2 Mathematical Model 206

1.3 Numerical Simulation 211

1.4 Effect of the Tortuosity of the Microreactor matrix 214

1.5 Effect of the Porosity of the Microreactor Matrix 214

1.6 Concluding remarks 216

2 Biosensor Based on Array of Microreactors 217

2.1 Principal Structure of Biosensor 217

2.2 Mathematical model 219

2.3 Numerical simulation 222

2.4 Effect of the electrode coverage with enzyme 225

2.5 Concluding remarks 227

3 Plate-Gap biosensor 228

3.1 Prinicipal structure of biosensor 228

3.2 Mathematical model 229

3.3 Governing equations 230

3.4 Intial conditions 231

3.5 Boundary and matching conditions 231

3.6 Biosensor response 232

3.7 Numerical simulation 233

3.8 Effect of the Gaps Geometry 236

3.9 Concluding Remarks 236

4 Biosensors with selective and perforated membranes 237

4.1 Principal structure of biosensor 238

4.2 Mathematical model 239

4.3 Numerical simulation 242

4.4 Effect of the perforation topology 245

4.5 Concluding remarks 246

Part III Numerical methods for reaction-diffusion equations

The Difference Schemes for the Diffusion Equation 249

1 The Grids 250

1.1 An Equidistant Grid in the Straight 251

1.2 A Non-equidistant Grid in a Straight 251

1.3 The Equidistant Grid in a Plane 252

1.4 The Non-equidistant Grid in a Plane 253

1.5 The Grid in a Multidimensional Case 253

2 The Approximation of the Function Derivatives 254

2.1 The Derivative of the First Order 254

2.2 The Approximation of the Second Order Derivative 256

2.3 The Approximation of the Second Order Derivative on a Non-equidistant Grid 256

3 The Explicit Difference Scheme 258

3.1 The Calculation of a Solution 261

3.2 The Convergence and the Stability 262

The Implicit Difference Scheme 264

4.1 The Convergence and the Stability 266

4.2 The Calculation of a Solution 266

5 The Elimination Method for the System of Linear Equations 268

5.1 Stability of the Elimination Method 271

6 The Crank-Nicolson Difference Scheme 271

6.1 The Calculation of a Solution 273

6.2 The Convergence and the Stability 274

7 The Difference Scheme with the Weights 274

7.1 The Convergence and the Stability 276

8 The Crank-Nicolson Difference Scheme on Non-equidistance Grid 276

8.1 The Calculation of a Solution 278

8.2 The Convergence and the Stability 279

9 The Explicit Difference Scheme in the Cylindrical Coordinates 280

9.1 The Calculation of a Solution 281

9.2 The Convergence and Stability 282

10 The Crank-Nicolson Difference Scheme in the Cylindrical Coordinates 282

10.1 The Calculation of a Solution 283

10.2 The Convergence and Stability 285

11 The Discontinuous Diffusion Coefficient 285

12 The Explicit Difference Scheme 287

12.1 The Calculation of a Solution 288

13 The Crank-Nicolson Difference Scheme 288

13.1 The Calculation of a Solution 289

The Difference Schemes for the Reaction-Diffusion Equations 293

1 The Boundary-Value Problem for the System of Reaction-Diffusion Equations 293

2 The Explicit Difference Scheme 295

2.1 The Calculation of a Solution 296

2.2 The

Convergence and the Stability 297

3 The Non-linear Crank-Nicolson Type Difference Scheme 298

3.1 Calculation of a Solution 299

3.2 The Convergence and the Stability 303

4 The Linear Crank-Nicolson Type Difference Scheme 303

4.1 The Calculation of a Solution 304

4.2 The Convergence and the Stability 307

5 Law of Conservation of Mass 307

6 The Alternating Directions Method 309

6.1 Calculation of a Solution 311

6.2 The Convergence and the Stability 312

7 The Explicit Method for the Multidimensional Problems 312

7.1 The Calculation of a Solution 314

7.2 The Convergence and the Stability 315.

Notes:

Includes bibliographical references and index.

ISBN:

9789048132423

9048132428

OCLC:

428029857