Dynamic Models in Biology

List of Figures ixList of Tables xivPreface xviChapter 1: What Are Dynamic Models? 11.1 Descriptive versus Mechanistic Models 21.2 Chinook Salmon 41.3 Bathtub Models 61.4 Many Bathtubs: Compartment Models 71.4.1 Enzyme Kinetics 81.4.2 The Modeling Process 111.4.3 Pharmacokinetic Models 131.5 Physics Models: Running and Hopping 161.6 Optimization Models 201.7 Why Bother? 211.8 Theoretical versus Practical Models 241.9 What's Next? 261.10 References 28Chapter 2: Matrix Models and Structured Population Dynamics 312.1 The Population Balance Law 322.2 Age-Structured Models 332.2.1 The Leslie Matrix 342.2.2 Warning: Prebreeding versus Postbreeding Models 372.3 Matrix Models Based on Stage Classes 382.4 Matrices and Matrix Operations 422.4.1 Review of Matrix Operations 432.4.2 Solution of the Matrix Model 442.5 Eigenvalues and a Second Solution of the Model 442.5.1 Left Eigenvectors 482.6 Some Applications of Matrix Models 492.6.1 Why Do We Age? 492.6.2 Elasticity Analysis and Conservation Biology 522.6.3 How Much Should We Trust These Models? 582.7 Generalizing the Matrix Model 592.7.1 Stochastic Matrix Models 592.7.2 Density-Dependent Matrix Models 612.7.3 Continuous Size Distributions 632.8 Summary and Conclusions 662.9 Appendix 672.9.1 Existence and Number of Eigenvalues 672.9.2 Reproductive Value 672.10 References 68Chapter 3: Membrane Channels and Action Potentials 713.1 Membrane Currents 723.1.1 Channel Gating and Conformational States 743.2 Markov Chains 773.2.1 Coin Tossing 783.2.2 Markov Chains 823.2.3 The Neuromuscular Junction 863.3 Voltage-Gated Channels 903.4 Membranes as Electrical Circuits 923.4.1 Reversal Potential 943.4.2 Action Potentials 953.5 Summary 1033.6 Appendix: The Central Limit Theorem 1043.7 References 106Chapter 4: Cellular Dynamics: Pathways of Gene Expression 1074.1 Biological Background 1084.2 A Gene Network That Acts as a Clock 1104.2.1 Formulating a Model 1114.2.2 Model Predictions 1134.3 Networks That Act as a Switch 1194.4 Systems Biology 1254.4.1 Complex versus Simple Models 1294.5 Summary 1314.6 References 132Chapter 5: Dynamical Systems 1355.1 Geometry of a Single Differential Equation 1365.2 Mathematical Foundations: A Fundamental Theorem 1385.3 Linearization and Linear Systems 1415.3.1 Equilibrium Points 1415.3.2 Linearization at Equilibria 1425.3.3 Solving Linear Systems of Differential Equations 1445.3.4 Invariant Manifolds 1495.3.5 Periodic Orbits 1505.4 Phase Planes 1515.5 An Example: The Morris-Lecar Model 1545.6 Bifurcations 1605.7 Numerical Methods 1755.8 Summary 1815.9 References 181Chapter 6: Differential Equation Models for Infectious Disease 1836.1 Sir Ronald Ross and the Epidemic Curve 1836.2 Rescaling the Model 1876.3 Endemic Diseases and Oscillations 1916.3.1 Analysis of the SIR Model with Births 1936.3.2 Summing Up 1976.4 Gonorrhea Dynamics and Control 2006.4.1 A Simple Model and a Paradox 2006.4.2 The Core Group 2016.4.3 Implications for Control 2036.5 Drug Resistance 2066.6 Within-Host Dynamics of HIV 2096.7 Conclusions 2136.8 References 214Chapter 7: Spatial Patterns in Biology 2177.1 Reaction-Diffusion Models 2187.2 The Turing Mechanism 2237.3 Pattern Selection: Steady Patterns 2267.4 Moving Patterns: Chemical Waves and Heartbeats 2327.5 References 241Chapter 8: Agent-Based and Other Computational Models for Complex Systems 2438.1 Individual-Based Models in Ecology 2458.1.1 Size-Dependent Predation 2458.1.2 Swarm 247 8.1.3 Individual-Based Modeling of Extinction Risk 2488.2 Artificial Life 2528.2.1 Tierra 2538.2.2 Microbes in Tierra 2558.2.3 Avida 2578.3 The Immune System and the Flu 2598.4 What Can We Learn from Agent-Based Models? 2608.5 Sensitivity Analysis 2618.5.1 Correlation Methods 2648.5.2 Variance Decomposition 2668.6 Simplifying Computational Models 2698.6.1 Separation of Time Scales 2698.6.2 Simplifying Spatial Models 2728.6.3 Improving the Mean Field Approximation 2768.7 Conclusions 2778.8 Appendix: Derivation of Pair Approximation 2788.9 References 279Chapter 9: Building Dynamic Models 2839.1 Setting the Objective 2849.2 Building an Initial Model 2859.2.1 Conceptual Model and Diagram 2869.3 Developing Equations for Process Rates 2919.3.1 Linear Rates: When and Why? 2919.3.2 Nonlinear Rates from "First Principles" 2939.3.3 Nonlinear Rates from Data: Fitting Parametric Models 2949.3.4 Nonlinear Rates from Data: Selecting a Parametric Model 2989.4 Nonlinear Rates from Data: Nonparametric Models 3029.4.1 Multivariate Rate Equations 3049.5 Stochastic Models 3069.5.1 Individual-Level Stochasticity 3069.5.2 Parameter Drift and Exogenous Shocks 3099.6 Fitting Rate Equations by Calibration 3119.7 Three Commandments for Modelers 3149.8 Evaluating a Model 3159.8.1 Comparing Models 3179.9 References 320Index 323