Mathematical Modeling of Biosensors (eBook)

Preface 7
Acknowledgements 9
Contents 10
Introduction 16
Part I Analytical Modeling of Biosensors 19
Biosensor Action 20
1 Kinetics of Biocatalytical Reactions 20
2 Transducer Function 22
3 Scheme of Biosensor Action 23
Modeling Biosensors at Steady State and Internal Diffusion Limitations 26
1 Biosensors Containing Single Enzyme 26
2 Biosensors Containing Multienzymes 27
2.1 Consecutive Substrates Conversion 27
2.2 Parallel Substrates Conversion 31
2.3 Biosensors Utilizing Cyclic Substrates Conversion 32
3 Biosensors Utilizing Synergistic Substrates Conversion 33
4 Biosensors Based on Chemically Modified Electrodes 35
Modeling Biosensors at Steady State and External Diffusion Limitations 38
1 Biosensor Using Single Enzyme 38
2 Biosensors with Multienzymes 39
3 Biosensor Utilizing Non Michaelis–Menten Enzyme 40
Modeling Biosensors Utilizing Microbial Cells 44
1 Metabolite Biosensor 44
2 BOD Biosensor 47
Modeling Nonstationary State of Biosensors 49
1 Potentiometric Biosensors 49
2 Amperometric Biosensors 50
Part II Numerical Modeling of Biosensors 56
Mono-Layer Mono-Enzyme Models of Biosensors 57
1 Mathematical Model of an Amperometric Biosensor 58
1.1 Governing Equations 58
1.2 Initial and Boundary Conditions 59
1.3 Dimensionless Model 60
2 Characteristics of the Biosensor Response 61
2.1 Biosensor Current 61
2.2 Biosensor Sensitivity 62
2.3 Maximal Gradient of the Current 62
2.4 Response Time 63
3 Finite Difference Solution 64
3.1 Numerical Approximation of Equations 64
3.1.1 Explicit Scheme 64
3.1.2 Implicit Scheme 65
3.1.3 Approximation of Initial and Boundary Conditions 65
3.2 Calculation Procedure 65
3.3 Validation of Numerical Solution 67
3.3.1 First-Order Reaction Rate 67
3.3.2 Zero-Order Reaction Rate 70
3.4 Numerical Error Analysis 71
4 Peculiarities of the Biosensor Response 74
4.1 Effect of the Enzyme Membrane Thickness 74
4.2 Stability of the Response 77
4.3 The Response Versus the Substrate Concentration 78
4.4 The Response Versus the Maximal Enzymatic Rate 80
4.5 Choosing the Enzyme Membrane Thickness 82
4.6 Biosensor Resistance 84
4.7 Maximal Gradient of the Current 85
5 Flow Injection Analysis 86
5.1 Mathematical Model 87
5.2 Numerical Solution 87
5.3 Biosensor Response 89
5.4 Peculiarities of the Biosensor Response 91
5.4.1 Maximal Current Versus Substrate Concentration 91
5.4.2 Response Time Versus Substrate Concentration 92
5.4.3 Sensitivity Versus Substrate Concentration 93
5.5 Sequential Injection Analysis 94
6 Biosensors with Chemical Amplification 95
6.1 Mathematical Model 96
6.2 Finite Difference Solution 98
6.3 Concentration Profiles 99
6.4 Peculiarities of the Biosensor Response 100
6.4.1 Dependence of Response on the Substrate Concentration 100
6.4.2 Sensitivity Versus Substrate Concentration 102
6.4.3 Effect of the Enzyme Membrane Thickness 103
6.4.4 Effect of the Reaction Rate 104
7 Potentiometric Biosensors 105
7.1 Mathematical Model 105
7.2 Biosensor Response 106
7.3 Finite Difference Solution 107
7.4 Validation of Numerical Solution 108
7.5 Simulated Biosensor Response 109
7.6 Peculiarities of the Biosensor Response 112
7.6.1 Effect of the Enzyme Membrane Thickness 112
7.6.2 Stability of the Response 114
7.6.3 Effect of the Reaction Rate 116
7.6.4 Dependence of the Response on the Diffusion Module 116
8 Enzyme Inhibition 117
8.1 Substrate Inhibition 117
8.1.1 Mathematical Model 118
8.1.2 Solution of the Problem 119
8.2 Effect of Substrate Inhibition 119
8.3 Product Inhibition 122
8.3.1 Mathematical Model 122
8.3.2 Solution of the Problem 123
8.4 Effect of Product Inhibition 123
One-Layer Multi-Enzyme Models of Biosensors 126
1 Biosensors Response to Mixture of Compounds 127
1.1 Mathematical Model 127
1.2 Solution of the Problem 129
1.3 Generation of Data Sets 130
1.4 Concluding Remarks 133
2 Biosensors Acting in Trigger Mode 133
2.1 Mathematical Models 134
2.1.1 Modeling Biosensor Acting in CEC Mode 134
2.1.2 Modeling Biosensor Acting in CCE Mode 136
2.1.3 Modeling Biosensor Acting in CE Mode 137
2.1.4 Enzymatic Amplification 138
2.2 Finite Difference Solution 139
2.2.1 CEC Mode 139
2.2.2 CCE Mode 140
2.3 Simulated Response 141
2.4 Peculiarities of the Response 142
2.4.1 Dependence of the Steady State Current on the Reactions Rates 142
2.4.2 Effect the Reaction Rates on the Amplification 144
2.4.3 The Amplification Versus the Substrate Concentration 145
2.4.4 Effect of the Enzyme Membrane Thickness on the Amplification 146
2.4.5 Effect of the Membrane Thickness on the Response Time 148
2.5 Concluding Remarks 149
Multi-Layer Models of Biosensors 151
1 Multi-Layer Approach 152
1.1 Mathematical Model of Multi-Layer System 153
1.2 Numerical Approximation 155
1.3 Three-Layer Model 157
2 Two-Compartment Model 159
2.1 Mathematical Model 159
2.2 Transient Numerical Solution 162
2.3 Validation of Numerical Solution 164
2.4 Simulated Biosensor Responses 165
2.5 Effect of the Diffusion Layer 167
2.6 The Nernst Diffusion Layer 169
2.7 Dimensionless Model 172
2.8 Impact of the Diffusion Module 174
3 Biosensors with Outer Porous Membrane 175
3.1 Mathematical Model 176
3.1.1 Governing Equations 176
3.1.2 Initial and Boundary Conditions 177
3.1.3 Biosensor Response 178
3.2 Numerical Simulation 178
3.3 Effect of the Porous Membrane 180
3.3.1 Effect of the Membrane Permeability 180
3.3.2 Effect of the Membrane Thickness 182
3.4 Concluding Remarks 183
4 Biosensors with Selective and Outer Perforated Membranes 184
4.1 Mathematical Model 185
4.1.1 Governing Equations 186
4.1.2 Initial and Boundary Conditions 187
4.1.3 Biosensor Response 188
4.2 Numerical Simulation 188
5 Biosensors Based on Chemically Modified Electrode 190
5.1 Mathematical Model 190
5.1.1 Governing Equations 191
5.1.2 Initial Conditions 192
5.1.3 Boundary and Matching Conditions 192
5.1.4 Biosensor Response 193
5.2 Numerical Simulation 194
5.3 Dimensionless Model 196
5.4 Simulated Biosensor Action 198
5.5 Impact of the Diffusion Module 201
5.6 Impact of the Substrate Concentration 203
5.7 Concluding Remarks 204
6 Optical and Fluorescence Biosensors 205
6.1 Mathematical Model 205
6.1.1 Governing Equations 206
6.1.2 Initial and Boundary Conditions 207
6.1.3 Response of Optical Biosensor 208
6.1.4 Response of Fluorescence Biosensor 208
6.2 Numerical Simulation 209
6.3 Simulated Biosensor Action 210
6.4 Impact of the Substrate Concentration 213
Modeling Biosensors of Complex Geometry 215
1 Biosensor Based on Heterogeneous Microreactor 216
1.1 Structure of Modeling Biosensor 216
1.2 Mathematical Model 218
1.2.1 Governing Equations 218
1.2.2 Effective Diffusion Coefficient 220
1.2.3 Initial and Boundary Conditions 221
1.2.4 Biosensor Response 223
1.3 Numerical Simulation 223
1.4 Effect of the Tortuosity of the Microreactor Matrix 226
1.5 Effect of the Porosity of the Microreactor Matrix 226
1.6 Concluding Remarks 228
2 Biosensor Based on Array of Microreactors 229
2.1 Principal Structure of Biosensor 229
2.2 Mathematical Model 231
2.2.1 Governing Equations 231
2.2.2 Initial and Boundary Conditions 232
2.2.3 Biosensor Response 233
2.3 Numerical Simulation 234
2.4 Effect of the Electrode Coverage with Enzyme 237
2.5 Concluding Remarks 239
3 Plate-Gap Biosensor 240
3.1 Principal Structure of Biosensor 240
3.2 Mathematical Model 241
3.3 Governing Equations 242
3.4 Initial Conditions 243
3.5 Boundary and Matching Conditions 243
3.6 Biosensor Response 244
3.7 Numerical Simulation 245
3.8 Effect of the Gaps Geometry 248
3.9 Concluding Remarks 248
4 Biosensors with Selective and Perforated Membranes 249
4.1 Principal Structure of Biosensor 250
4.2 Mathematical Model 251
4.2.1 Governing Equations 251
4.2.2 Initial Conditions 252
4.2.3 Boundary and Matching Conditions 253
4.2.4 Biosensor Response 254
4.3 Numerical Simulation 254
4.4 Effect of the Perforation Topology 257
4.5 Concluding Remarks 258
Part III Numerical Methods for Reaction–Diffusion Equations 259
The Difference Schemes for the Diffusion Equation 260
1 The Grids 261
1.1 An Equidistant Grid in the Straight 262
1.2 A Non-equidistant Grid in a Straight 262
1.3 The Equidistant Grid in a Plane 263
1.4 The Non-equidistant Grid in a Plane 264
1.5 The Grid in a Multidimensional Case 264
2 The Approximation of the Function Derivatives 265
2.1 The Derivative of the First Order 265
2.2 The Approximation of the Second Order Derivative 267
2.3 The Approximation of the Second Order Derivative on a Non-equidistant Grid 267
3 The Explicit Difference Scheme 269
3.1 The Calculation of a Solution 272
3.2 The Convergence and the Stability 273
4 The Implicit Difference Scheme 275
4.1 The Convergence and the Stability 277
4.2 The Calculation of a Solution 277
5 The Elimination Method for the System of Linear Equations 279
5.1 Stability of the Elimination Method 282
6 The Crank–Nicolson Difference Scheme 282
6.1 The Calculation of a Solution 284
6.2 The Convergence and the Stability 285
7 The Difference Scheme with the Weights 285
7.1 The Convergence and the Stability 287
8 The Crank–Nicolson Difference Schemeon Non-equidistance Grid 287
8.1 The Calculation of a Solution 289
8.2 The Convergence and the Stability 290
9 The Explicit Difference Schemein the Cylindrical Coordinates 291
9.1 The Calculation of a Solution 292
9.2 The Convergence and Stability 293
10 The Crank–Nicolson Difference Scheme in the Cylindrical Coordinates 293
10.1 The Calculation of a Solution 294
10.2 The Convergence and Stability 296
11 The Discontinuous Diffusion Coefficient 296
12 The Explicit Difference Scheme 298
12.1 The Calculation of a Solution 299
13 The Crank–Nicolson Difference Scheme 299
13.1 The Calculation of a Solution 300
The Difference Schemes for the Reaction–Diffusion Equations 303
1 The Boundary-Value Problem for the System of Reaction–Diffusion Equations 303
2 The Explicit Difference Scheme 305
2.1 The Calculation of a Solution 306
2.2 The Convergence and the Stability 307
3 The Non-linear Crank–Nicolson Type Difference Scheme 308
3.1 Calculation of a Solution 309
3.2 The Convergence and the Stability 313
4 The Linear Crank–Nicolson Type Difference Scheme 313
4.1 The Calculation of a Solution 314
4.2 The Convergence and the Stability 317
5 Law of Conservation of Mass 317
6 The Alternating Directions Method 319
6.1 Calculation of a Solution 321
6.2 The Convergence and the Stability 322
7 The Explicit Method for the Multidimensional Problems 322
7.1 The Calculation of a Solution 324
7.2 The Convergence and the Stability 325
References 326
Index 337
About Authors 341