This textbook grew out of a course that the highly respected applied mathematician Lee Segel taught at the Weizmann Institute. This book represents the unique perspective on mathematical biology of Segel and his co-author Leah Edelstein-Keshet (author of the popular SIAM book, Mathematical Models in Biology). It introduces differential equations, biological applications, and simulations, with emphasis on molecular events (biochemistry and enzyme kinetics), excitable systems (neural signals), and small protein and genetic circuits. The exposition combines clear and useful mathematical methods with plenty of applications to illustrate the power of such tools, along with many exercises in reasoning, modelling and simulation. The reader will also find suggestions for further study and appendices containing useful background material. These features make the book ideal for students at the advanced undergraduate or graduate level in both biology and mathematics who wish to experience the application of mathematical techniques to the biological sciences.

List of Figures ix

List of Tables xv

Acknowledgments xvii

Preface xix

1 Introduction 1

1.1 When to model 2

1.2 What is a model 2

1.3 Formulation of a mathematical model 3

1.4 Solving the model equations 5

1.5 Drawing qualitative conclusions 6

1.6 Choosing parameters 7

1.7 Robustness 7

1.8 Analysis of results 9

1.9 Successes and failures of modeling 9

1.10 Final remarks 11

1.11 Other sources of information on mathematical modeling in biology 12

2 Introduction to biochemical kinetics 13

2.1 Transitions between states at the molecular level 13

2.2 Transitions between states at the population level 16

2.3 The law of mass action 23

2.4 Enzyme kinetics: Saturating and cooperative reactions 24

2.5 Simple models for polymer growth dynamics 27

2.6 Discussion 36

Exercises 37

3 Review of linear differential equations 43

3.1 First-order differential equations 44

3.2 Linear second-order equations 50

3.3 Linear second-order equations with constant coefficients 52

3.4 A system of two linear equations 55

3.5 Summary of solutions to differential equations in this chapter 61

Exercises 61

4 Introduction to nondimensionalization and scaling 67

4.1 Simple examples 67

4.2 Rescaling the dimerization model 72

4.3 Other examples 76

Exercises 79

5 Qualitative behavior of simple differential equation models 83

5.1 Revisiting the simple linear ODEs 83

5.2 Stability of steady states 86

5.3 Qualitative analysis of models with bifurcations 88

Exercises 99

6 Developing a model from the ground up: Case study of the spread of an infection 103

6.1 Deriving a model for the spread of an infection 103

6.2 Dimensional analysis applied to the model 105

6.3 Analysis 107

6.4 Interpretation of the results 111

Exercises 111

7 Phase plane analysis 115

7.1 Phase plane trajectories 115

7.2 NulIclines 117

7.3 Steady states 118

7.4 Stability of steady states 120

7.5 Classification of steady state behavior 122

7.6 Qualitative behavior and phase plane analysis 124

7.7 Limit cycles, attractors, and domains of attraction 132

7.8 Bifurcations continued 135

Exercises 138

8 Quasi steady state and enzyme-mediated biochemical kinetics 145

8.1 Warm-up example: Transitions between three states 145

8.2 Enzyme-substrate complex and the quasi steady state approximation 152

8.3 Conditions for validity of the QSS 158

8.4 Overview and discussion of the QSS 165

8.5 Related applications 167

Exercises 168

9 Multiple subunit enzymes and proteins: Cooperativity 173

9.1 Preliminary model for rapid dimerization 173

9.2 Dimer binding that induces conformational change: Model formulation 176

9.3 Consequences of a QSS assumption 178

9.4 Ligand binding to dimer 179

9.5 Results for binding and their interpretation: Cooperativity 183

9.6 Cooperativity in enzyme action 184

9.7 Monod-Wyman-Changeaux (MWC) cooperativity 185

9.8 Discussion 190

Exercises 190

10 Dynamic behavior of neuronal membranes 195

10.1 Introduction 195

10.2 An informal preview of the Hodgkin-Huxley model 198

10.3 Working towards the Hodgkin-Huxley model 202

10.4 The full Hodgkin-Huxley model 214

10.5 Comparison between theory and experiment 216

10.6 Bifurcations in the Hodgkin-Huxley model 219

10.7 Discussion 222

Exercises 222

11 Excitable systems and the FitzHugh-Nagumo equations 227

11.1 A simple excitable system 227

11.2 Phase plane analysis of the model 229

11.3 Piecing together the qualitative behavior 234

11.4 Simulations of the FitzHugh-Nagumo model 236

11.5 Connection to neuronal excitation 241

11.6 Other systems with excitable behavior 246

Exercises 247

12 Biochemical modules 251

12.1 Simple biochemical circuits with useful functions 251

12.2 Genetic switches 257

12.3 Models for the cell division cycle 262

Exercises 277

13 Discrete networks of genes and cells 283

13.1 Some simple automata networks 284

13.2 Boolean algebra 294

13.3 Lysis-lysogeny in bacteriophage 入 299

13.4 Cell cycle, revisited 304

13.5 Discussion 306

Exercises 308

14 For further study 311

14.1 Nondimensionalizing a functional relationship 311

14.2 Scaled dimensionless variables 312

14.3 Mathematical development of the Michaelis-Menten QSS via scaled variables 316

14.4 Cooperativity in the Monod-Wyman-Changeaux theory for binding 318

14.5 Ultrasensitivity in covalent protein modification 320

14.6 Fraction of open channels, Hodgkin-Huxley Model 324

14.7 Asynchronous Boolean networks (kinetic logic) 327

Exercises 333

15 Extended exercises and projects 337

Exercises 337

A The Taylor approximation and Taylor series 355

Exercises 361

B Complex numbers 363

Exercises 365

C A review of basic theory of electricity 367

C.1 Amps, coulombs, and volts 367

C.2 Ohm's law 370

C.3 Capacitance 371

C.4 Circuits 373

C.5 The Nernst equation 375

Exercises 377

D Proofs of Boolean algebra rules 379

Exercises 381

E Appendix: XPP files for models in this book 385

E.1 Biochemical reactions 385

E.2 Linear differential equations 386

E.3 Simple differential equations and bifurcations 387

E.4 Disease dynamics models 389

E.5 Phase plane analysis 389

E.6 Chemical reactions and the QSS 391

E.7 Neuronal excitation and excitable systems 392

E.8 Biochemical modules 394

E.9 Cell division cycle models 396

E.10 Boolean network models 401

E.11 Odell-Oster model of Exercise 15.7 403

Bibliography 405

Index 417

中科院文献情报中心