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Dynamic Models in Biology

Stephen P. Ellner and John Guckenheimer
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

What is remarkable about Dynamic Models in Biology is that it truly speaks to students of the biological sciences. It puts biology first, and then tries to explain how mathematical tools can explain biological phenomena. Nothing else I’ve seen does this anywhere near as well. The authors (a biologist and a mathematician) have combined their experience and talents to produce an excellent textbook.

There are several identifiable elements that combine to create a success. The authors’ use of case studies focuses on doing a few well-chosen examples in depth. There is no attempt to be comprehensive. The applications range across the areas of biology where dynamic models have made compelling contributions. The authors introduce just enough mathematics to understand the modeling results — and no more. This keeps biology in the forefront and can perhaps inspire students to follow up with more in-depth investigations on their own or through a related mathematics course. The authors spend a significant amount of time on the modeling process itself. In part this is because the complexity of biological problems requires a more subtle approach to the concept of modeling. Questions like ‘What is a model?’, ‘What data are available to estimate model parameters or to validate a model?’, and ‘How much simplification is acceptable to keep a model tractable?’ can be much harder to answer than, for example, in the modeling of a mechanical system.

Computation is well-integrated throughout. The authors rely on high level languages such as MATLAB or R for calculations or to implement the models. (MATLAB is probably well-known to readers, and R is an object-oriented scripting language appropriate for simulating dynamical models.) The authors provide a website with supplementary materials including lab manuals for both MATLAB and R as well as MATLAB files and R scripts. Computation and simulation are viewed simply as tools that are available to explore a model and identify biologically significant results, especially when the phenomena are so complex that extensive computation or simulation are the only means of getting predictions from the model.

The authors explore dynamic models exclusively, and focus more on deterministic than stochastic systems. It would be wonderful to see a companion volume on genomics, bioinformatics, and structural biology, but the authors’ interests and experience are on the dynamics side.

There are four primary case studies. They are: matrix models and structured population dynamics, membrane channels and action potentials, cellular dynamics, and infectious disease. The models that the authors choose to investigate are either time-tested and more traditional (the population and infectious disease models, for example) or models of more prominent current interest, For example, the case study on cellular dynamics includes a discussion of one recently constructed gene network that acts like a clock and another that acts like a switch.

A chapter on spatial models in biology explores reaction-diffusion models as well as stationary patterns (e.g., animal coats and insect markings) or moving patterns (e.g., chemical waves or heartbeats). Another chapter takes a quick look at other computational approaches including agent-based models and artificial life models such as Tierra.

The authors do not discuss prerequisites, but the book presumes some level of familiarity with differential calculus and very basic differential equations. Although its primary audience consists of biology students, mathematics students would benefit considerably from exposure to deeper and more tangible applications than they would typically see in mathematics courses.

Bill Satzer ([email protected]) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

List of Figures ix
List of Tables xiv
Preface xvi

Chapter 1: What Are Dynamic Models? 1
1.1 Descriptive versus Mechanistic Models 2
1.2 Chinook Salmon 4
1.3 Bathtub Models 6
1.4 Many Bathtubs: Compartment Models 7
1.4.1 Enzyme Kinetics 8
1.4.2 The Modeling Process 11
1.4.3 Pharmacokinetic Models 13
1.5 Physics Models: Running and Hopping 16
1.6 Optimization Models 20
1.7 Why Bother? 21
1.8 Theoretical versus Practical Models 24
1.9 What's Next? 26
1.10 References 28

Chapter 2: Matrix Models and Structured Population Dynamics 31
2.1 The Population Balance Law 32
2.2 Age-Structured Models 33
2.2.1 The Leslie Matrix 34
2.2.2 Warning: Prebreeding versus Postbreeding Models 37
2.3 Matrix Models Based on Stage Classes 38
2.4 Matrices and Matrix Operations 42
2.4.1 Review of Matrix Operations 43
2.4.2 Solution of the Matrix Model 44
2.5 Eigenvalues and a Second Solution of the Model 44
2.5.1 Left Eigenvectors 48
2.6 Some Applications of Matrix Models 49
2.6.1 Why Do We Age? 49
2.6.2 Elasticity Analysis and Conservation Biology 52
2.6.3 How Much Should We Trust These Models? 58
2.7 Generalizing the Matrix Model 59
2.7.1 Stochastic Matrix Models 59
2.7.2 Density-Dependent Matrix Models 61
2.7.3 Continuous Size Distributions 63
2.8 Summary and Conclusions 66
2.9 Appendix 67
2.9.1 Existence and Number of Eigenvalues 67
2.9.2 Reproductive Value 67
2.10 References 68

Chapter 3: Membrane Channels and Action Potentials 71
3.1 Membrane Currents 72
3.1.1 Channel Gating and Conformational States 74
3.2 Markov Chains 77
3.2.1 Coin Tossing 78
3.2.2 Markov Chains 82
3.2.3 The Neuromuscular Junction 86
3.3 Voltage-Gated Channels 90
3.4 Membranes as Electrical Circuits 92
3.4.1 Reversal Potential 94
3.4.2 Action Potentials 95
3.5 Summary 103
3.6 Appendix: The Central Limit Theorem 104
3.7 References 106

Chapter 4: Cellular Dynamics: Pathways of Gene Expression 107
4.1 Biological Background 108
4.2 A Gene Network That Acts as a Clock 110
4.2.1 Formulating a Model 111
4.2.2 Model Predictions 113
4.3 Networks That Act as a Switch 119
4.4 Systems Biology 125
4.4.1 Complex versus Simple Models 129
4.5 Summary 131
4.6 References 132

Chapter 5: Dynamical Systems 135
5.1 Geometry of a Single Differential Equation 136
5.2 Mathematical Foundations: A Fundamental Theorem 138
5.3 Linearization and Linear Systems 141
5.3.1 Equilibrium Points 141
5.3.2 Linearization at Equilibria 142
5.3.3 Solving Linear Systems of Differential Equations 144
5.3.4 Invariant Manifolds 149
5.3.5 Periodic Orbits 150
5.4 Phase Planes 151
5.5 An Example: The Morris-Lecar Model 154
5.6 Bifurcations 160
5.7 Numerical Methods 175
5.8 Summary 181
5.9 References 181

Chapter 6: Differential Equation Models for Infectious Disease 183
6.1 Sir Ronald Ross and the Epidemic Curve 183
6.2 Rescaling the Model 187
6.3 Endemic Diseases and Oscillations 191
6.3.1 Analysis of the SIR Model with Births 193
6.3.2 Summing Up 197
6.4 Gonorrhea Dynamics and Control 200
6.4.1 A Simple Model and a Paradox 200
6.4.2 The Core Group 201
6.4.3 Implications for Control 203
6.5 Drug Resistance 206
6.6 Within-Host Dynamics of HIV 209
6.7 Conclusions 213
6.8 References 214

Chapter 7: Spatial Patterns in Biology 217
7.1 Reaction-Diffusion Models 218
7.2 The Turing Mechanism 223
7.3 Pattern Selection: Steady Patterns 226
7.4 Moving Patterns: Chemical Waves and Heartbeats 232
7.5 References 241

Chapter 8: Agent-Based and Other Computational Models for Complex Systems 243
8.1 Individual-Based Models in Ecology 245
8.1.1 Size-Dependent Predation 245
8.1.2 Swarm 247 8.1.3 Individual-Based Modeling of Extinction Risk 248
8.2 Artificial Life 252
8.2.1 Tierra 253
8.2.2 Microbes in Tierra 255
8.2.3 Avida 257
8.3 The Immune System and the Flu 259
8.4 What Can We Learn from Agent-Based Models? 260
8.5 Sensitivity Analysis 261
8.5.1 Correlation Methods 264
8.5.2 Variance Decomposition 266
8.6 Simplifying Computational Models 269
8.6.1 Separation of Time Scales 269
8.6.2 Simplifying Spatial Models 272
8.6.3 Improving the Mean Field Approximation 276
8.7 Conclusions 277
8.8 Appendix: Derivation of Pair Approximation 278
8.9 References 279

Chapter 9: Building Dynamic Models 283
9.1 Setting the Objective 284
9.2 Building an Initial Model 285
9.2.1 Conceptual Model and Diagram 286
9.3 Developing Equations for Process Rates 291
9.3.1 Linear Rates: When and Why? 291
9.3.2 Nonlinear Rates from "First Principles" 293
9.3.3 Nonlinear Rates from Data: Fitting Parametric Models 294
9.3.4 Nonlinear Rates from Data: Selecting a Parametric Model 298
9.4 Nonlinear Rates from Data: Nonparametric Models 302
9.4.1 Multivariate Rate Equations 304
9.5 Stochastic Models 306
9.5.1 Individual-Level Stochasticity 306
9.5.2 Parameter Drift and Exogenous Shocks 309
9.6 Fitting Rate Equations by Calibration 311
9.7 Three Commandments for Modelers 314
9.8 Evaluating a Model 315
9.8.1 Comparing Models 317
9.9 References 320

Index 323