Modelling Metabolism with Mathematica

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ISBN: 9780849314681 | 0849314682
Cover: Hardcover
Copyright: 5/14/2003

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With the advent of sophisticated general programming environments like Mathematica, the task of developing new models of metabolism and visualizing their responses has become accessible to students of biochemistry and the life sciences in general. Modelling Metabolism with Mathematica presents the approaches, methods, tools, and algorithms for modelling the chemical-dynamics of metabolic pathways. The authors explain the concepts underpinning the deterministic theory of chemical and enzyme kinetics, present a graded series of computer models of metabolic pathways leading up to that of the human erythrocyte, and document a consistent set of rate equations and associated kinetic parameters.The experimental and theoretical study of metabolism in mammalian cells has a long and fruitful history, but our understanding of cellular metabolism at the molecular level is far from complete. This book enables its readers to formulate their own models of time-dependent metabolic systems and aids them in the quest for the many fundamental and clinically relevant discoveries that remain to be made.

Introduction to Chemical Kinetics and Numerical Integration
Aims and Objectives	p. 1
Complexity	p. 2
Definitions	p. 3
Principle of mass action	p. 3
Equilibrium constant	p. 4
Molecularity	p. 5
Order of a reaction	p. 5
Units of rate constants	p. 6
Extent of reaction	p. 6
Time Courses of Reactions	p. 7
Introduction	p. 7
Half-life	p. 10
Reaction lifetime	p. 11
Coupled first-order reactions	p. 12
Multiple-coupled reactions	p. 16
Non-first-order systems	p. 18
Numerical Integration of Differential Equations	p. 20
General	p. 20
Numerical integration - overview	p. 22
Taylor series solution	p. 25
Runge-Kutta methods - overview	p. 26
Improved Euler method	p. 28
Modified Euler method	p. 29
General Runge-Kutta methods	p. 30
Higher-order Runge-Kutta methods	p. 31
Predictor Corrector Methods	p. 31
The predictor	p. 32
The corrector	p. 33
Choosing the value of h	p. 34
Numerical integration in Mathematica - NDSolve and stiffness	p. 35
Conclusions	p. 35
Exercises	p. 36
References	p. 37
Elements of Enzyme Kinetics
Kinetics of Enzymic Reactions	p. 39
Purpose	p. 39
The Michaelis-Menten equation	p. 39
Graphical evaluation of V[subscript max] and K[subscript m]	p. 41
Lineweaver-Burk plot	p. 41
Eadie-Hofstee plot	p. 42
Hanes-Woolf plot	p. 44
Eisenthal and Cornish-Bowden equation (direct linear plot)	p. 44
Enzyme Inhibition	p. 44
Degree of inhibition	p. 44
Michaelis-Menten equations that include inhibitor effects	p. 45
Enzyme Mechanisms	p. 45
Michaelis-Menten mechanism	p. 45
The steady state	p. 47
Reversible Michaelis-Menten enzyme	p. 48
Regulatory Enzymes	p. 50
Exercises	p. 51
References	p. 51
Basic Procedures for Simulating Metabolic Systems
Introduction	p. 53
Inborn errors of metabolism	p. 54
Constancy of K[subscript eq]	p. 54
Relationships between Unitary Rate Constants and Steady-State Parameters	p. 55
Progress Curve of a Michaelis-Menten reaction	p. 55
Pre-steady-state Michaelis-Menten scheme	p. 57
Enzyme oligomerization and the turnover number	p. 59
Specific examples of enzyme mechanisms	p. 59
Upper Limit of Values for Unitary Rate Constants	p. 63
Diffusion control of reaction rate	p. 63
Realistic Enzyme Models	p. 65
Deriving steady-state rate equations	p. 65
The RateEquation function	p. 67
Calculating a consistent set of unitary rate constants	p. 69
Deriving Expressions for Steady-State Parameters	p. 69
Multiple Equilibria	p. 75
pH Effects on Kinetic Parameters	p. 79
Ionization of the substrate	p. 79
A Simple Model of the Urea Cycle	p. 81
Conclusions	p. 89
Exercises	p. 90
References	p. 93
Advanced Simulation of Metabolic Pathways
Introduction	p. 95
Simulating the Time-Dependent Behavior of Multienzyme Systems	p. 95
Using Matrix Notation in Simulating Metabolic Pathways	p. 97
Generating the Stoichiometry Matrix	p. 102
Determining Steady-State Concentrations	p. 105
Conversation Relations	p. 110
Stability of a Steady State	p. 113
When Cell Volume Changes with Time	p. 118
Decomposition of N and Calculation of the Link Matrix (Optional)	p. 130
Exercises	p. 132
References	p. 132
Metabolic Control Analysis
Introduction	p. 133
Control Coefficients	p. 134
Calculation of Control Coefficients by Numerical Perturbation	p. 140
Elasticity Coefficients	p. 142
Response Coefficients	p. 145
Internal Response Coefficients	p. 149
Conclusions	p. 151
Exercises	p. 152
References	p. 153
Parameter Estimation
Introduction	p. 155
Approaches to Parameter Estimation	p. 155
Least Squares	p. 156
Maximum a Posteriori (MAP)	p. 157
Parameters in Rate Equations	p. 159
Linear least squares	p. 159
Nonlinear estimation	p. 162
Nonlinear least squares	p. 164
Nonlinear MAP	p. 166
Parameters in Systems of Differential Equations	p. 167
Optimal Parameters	p. 171
Variances of Parameters	p. 172
Exercises	p. 173
References	p. 174
Model of Erythrocyte Metabolism
Introduction	p. 175
Models of Erythrocyte Metabolism	p. 175
Stoichiometry of Human Erythrocyte Metabolism	p. 177
In Vivo Steady State of the Erythrocyte	p. 182
Conservation of Mass Relationships	p. 187
Simulating a Time Course	p. 192
Exercises	p. 195
References	p. 196
Metabolic Control Analysis of Human Erythrocyte Metabolism
Introduction	p. 197
Normal In Vivo Steady State	p. 197
Identifying Zero Fluxes	p. 198
Flux Control Coefficients	p. 200
Concentration Control Coefficients	p. 206
Response Coefficients and Partitioned Responses	p. 207
Elasticity Coefficients	p. 211
Internal Response Coefficients	p. 212
Concluding Remarks	p. 213
Exercises	p. 214
References	p. 215
Rate Equation Deriver	p. 217
Metabolic Control Analysis Functions	p. 223
Rate Equations for Enzymes of the Human Erythrocyte	p. 265
Initial Conditions and External Parameters for the Erythrocyte Model	p. 291
Equation List Describing the Erythrocyte Model of Chapters 7 and 8	p. 295
Index	p. 299
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Modelling Metabolism with Mathematica

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