Modelling metabolism with Mathematica : detailed examples including erythrocyte metabolism

Modeling Metabolism with Mathematica......Page 1
Preface......Page 3
Authors......Page 8
Table of Contents......Page 9
1.1  Aims and Objectives......Page 13
Table of Contents......Page 0
1.2  Complexity......Page 14
1.3.1 Principle of mass action......Page 15
1.3.2 Equilibrium constant......Page 16
1.3.4 Order of a reaction......Page 17
1.3.6 Extent of reaction......Page 18
1.4.1 Introduction......Page 19
1.4.2 Half-life......Page 22
1.4.3 Reaction lifetime......Page 23
1.4.4 Coupled first-order reactions......Page 24
1.4.5 Multiple-coupled reactions......Page 28
1.4.6 Non-first-order  systems......Page 30
1.5.1 General......Page 32
1.5.2 Numerical integration Πoverview......Page 34
1.5.3 Taylor series solution......Page 37
1.5.4 Runge-Kutta  methods Πoverview......Page 38
1.5.5 Improved Euler method......Page 40
1.5.6 Modified Euler method......Page 41
1.5.7 General Runge-Kutta methods......Page 42
1.6  Predictor Corrector Methods......Page 43
1.6.1 The predictor......Page 44
1.6.2 The corrector......Page 45
1.6.3 Choosing the value of h......Page 46
1.7  Conclusions......Page 47
1.8.4......Page 48
1.9  References......Page 49
2.1.2 The Michaelis-Menten equation......Page 51
2.1.4 Lineweaver-Burk  plot......Page 53
2.1.5 Eadie-Hofstee plot......Page 54
2.2.1 Degree of inhibition......Page 56
2.3.1 Michaelis-Menten mechanism......Page 57
2.3.2 The steady state......Page 59
2.3.3 Reversible Michaelis-Menten enzyme......Page 60
2.4  Regulatory Enzymes......Page 62
2.6  References......Page 63
3.1  Introduction......Page 65
3.1.2 Constancy of Keq......Page 66
3.2.1 Progress curve of a Michaelis-Menten reaction......Page 67
3.2.2 Pre-steady-state Michaelis-Menten scheme......Page 69
3.2.4 Specific examples of enzyme mechanisms......Page 71
3.3.1 Diffusion control of reaction rate......Page 75
3.4.1 Deriving steady-state rate equations......Page 77
3.4.2 The RateEquation function......Page 79
3.5  Deriving Expressions for Steady-State Parameters......Page 81
3.6  Multiple Equilibria......Page 87
3.7.1 Ionization of the substrate......Page 91
3.8  A Simple Model of the Urea Cycle......Page 93
3.9  Conclusions......Page 101
3.10.3......Page 102
3.10.7......Page 103
3.10.9......Page 104
3.11  References......Page 105
4.2  Simulating the Time-Dependent Behaviour of Multi- e n z y m e   S y s t e m s......Page 107
4.3  Using Matrix Notation in Simulating Metabolic Pathways......Page 109
4.4  Generating the Stoichiometry Matrix......Page 114
4.5  Determining Steady-State Concentrations......Page 117
4.6  Conservation Relations......Page 122
4.7  Stability of a Steady State......Page 125
4.8  When Cell Volume Changes with  Time......Page 130
4.9  Decompostion of N and Calculation of the Link Matrix  ( O p t i o n a l )......Page 142
4.11  References......Page 144
5.1  Introduction......Page 145
5.2  Control Coefficients......Page 146
5.3  Calculation of Control Coefficients by Numerical  P e r t u r b a t i o n......Page 152
5.4  Elasticity Coefficients......Page 154
5.5  Response Coefficients......Page 157
5.6  Internal Response Coefficients......Page 161
5.7  Conclusions......Page 163
5.8.3......Page 164
5.9  References......Page 165
6.2  Approaches to Parameter Estimation......Page 167
6.3  Least Squares......Page 168
6.4  Maximum a Posteriori (MAP)......Page 169
6.5.1  Linear least squares......Page 171
6.5.2 Nonlinear estimation......Page 174
6.5.3 Nonlinear least squares......Page 176
6.5.4 Nonlinear MAP......Page 178
6.6  Parameters in Systems of Differential Equations......Page 179
6.7  Optimal Parameters......Page 183
6.8  Variances of Parameters......Page 184
6.9.3......Page 185
6.10 References......Page 186
7.2  Models of Erythrocyte Metabolism......Page 187
7.3  Stoichiometry of Human Erythrocyte  Metabolism......Page 189
7.4 In Vivo Steady State of the Erythrocyte......Page 194
7.5  Conservation of Mass Relationships......Page 199
7.6  Simulating a Time Course......Page 204
7.7.6......Page 207
7.8  References......Page 208
8.2  Normal In Vivo Steady State......Page 209
8.3  Identifying Zero Fluxes......Page 210
8.4  Flux Control Coefficients......Page 212
8.5  Concentration Control Coefficients......Page 218
8.6  Response Coefficients and Partitioned Responses......Page 219
8.7  Elasticity Coefficients......Page 223
8.8  Internal Response Coefficients......Page 224
8.9  Concluding Remarks......Page 225
8.11 References......Page 226
Appendix 1 - Rate Equation Deriver......Page 229
References......Page 234
A2.1 Metabolic Control Analysis Functions......Page 235
A2.2 Metabolic Control Analysis Program......Page 242
A3.1 Glycolytic Enzymes......Page 277
A3.2 Pentose Phosphate Pathway Enzymes......Page 286
A3.3 2,3-BPG Shunt Enzyme......Page 294
A3.4 Energy Consumption and Oxidative Reactions......Page 295
A3.5 Membrane Transport......Page 296
A3.6 Magnesium-Metabolite Binding......Page 297
A3.8 Oxy-Haemoglobin-Metabolite Binding......Page 299
A3.9 Deoxy-Haemoglobin-Metabolite Binding......Page 301
A4.1.5 Donnan Ratio......Page 303
A4.2 Initial  conditions......Page 304
Appendix 5 - Equation List Describing the Erythrocyte Model of Chapters 7 and 8......Page 307