Biological Clocks, Rhythms, and Oscillations: The Theory of Biological Timekeeping

by Daniel B. Forger

Overview

An introduction to the mathematical, computational, and analytical techniques used for modeling biological rhythms, presenting tools from many disciplines and example applications.

All areas of biology and medicine contain rhythms, and these behaviors are best understood through mathematical tools and techniques. This book offers a survey of mathematical, computational, and analytical techniques used for modeling biological rhythms, gathering these methods for the first time in one volume. Drawing on material from such disciplines as mathematical biology, nonlinear dynamics, physics, statistics, and engineering, it presents practical advice and techniques for studying biological rhythms, with a common language.

The chapters proceed with increasing mathematical abstraction. Part I, on models, highlights the implicit assumptions and common pitfalls of modeling, and is accessible to readers with basic knowledge of differential equations and linear algebra. Part II, on behaviors, focuses on simpler models, describing common properties of biological rhythms that range from the firing properties of squid giant axon to human circadian rhythms. Part III, on mathematical techniques, guides readers who have specific models or goals in mind. Sections on “frontiers” present the latest research; “theory” sections present interesting mathematical results using more accessible approaches than can be found elsewhere. Each chapter offers exercises. Commented MATLAB code is provided to help readers get practical experience.

The book, by an expert in the field, can be used as a textbook for undergraduate courses in mathematical biology or graduate courses in modeling biological rhythms and as a reference for researchers.

Product Details

ISBN-13:	9780262036771
Publisher:	MIT Press
Publication date:	11/16/2017
Series:	The MIT Press
Edition description:	New Edition
Pages:	368
Product dimensions:	7.20(w) x 9.10(h) x 0.60(d)
Age Range:	18 Years

About the Author

Daniel B. Forger is Professor in the Department of Mathematics and in the Department of Computational Medicine and Bioinformatics at the University of Michigan.

Table of Contents

Preface xiii

Notation xvii

1 Basics 1

1.1 Introduction 1

1.1.1 Goals 1

1.1.2 Ten Examples of Biological Rhythms 4

1.1.3 Overview of Basic Questions 9

1.2 Models 11

1.2.1 Fundamentals of Modeling 11

1.2.2 Parsimony? 13

1.2.3 Scaling, Nondimensionalization, and Changing Variables to Change Phase 15

1.3 Period 16

1.4 Phase 16

1.5 Frontiers: The Difficulty of Estimating the Phase and Amplitude of a Clock 17

1.6 Plotting Circular Data 19

1.7 Mathematical Preliminaries, Notations, and Basics 19

1.8 Key Problems in the Autonomous Case 23

1.9 Perturbations, Phase Response Curves, and Synchrony 23

1.10 Key Problems when the External Signal u(t) ≠ 0 24

1.11 Frontiers: Probability Distributions with a Focus on Circular Data 26

1.12 Frontiers: Useful Statistics for Circular Data 30

Exercises 32

Part I Models 35

2 Biophysical Mechanistic Modeling: Choosing the Right Model Equations 37

2.1 Introduction 37

2.2 Biochemical Modeling 38

2.3 Law of Mass Action: When, Why, and How 39

2.4 Frontiers: The Crowded Cellular Environment and Mass Action 41

2.5 Three Mathematical Models of Transcription Regulation 42

2.6 The Goodwin Model 48

2.7 Other Models of Intracellular Processes (e.g., Michaelis-Menten) 49

2.7.1 Nuclear Transport 49

2.7.2 Michaelis-Menten Dynamics 50

2.7.3 Reversible Reactions 52

2.7.4 Delay Equations 53

2.8 Frontiers: Bounding Solutions of Biochemical Models 53

2.9 On Complex Formation 54

2.10 Hodgkin-Huxley and Models of Neuronal Dynamics 55

2.11 Frontiers: Rethinking the Ohm's Law Linear Relationship between Voltage and Current 63

2.12 Ten Common Mistakes to Watch for When Constructing Biochemical and Electrophysiological Models 64

2.13 Interesting Future Work: Are All Cellular Oscillations Intertwined? 65

Code 2.1 Spatial Effects 65

Code 2.2 Biochemical Feedback Loops 67

Code 2.3 The Hodgkin-Huxley Model 68

Exercises 69

3 Functioning in the Changing Cellular Environment 73

3.1 Introduction 73

3.2 Frontiers: Volume Changes 73

3.3 Probabilistic Formulation of Deterministic Equations and Delay Equations 75

3.4 The Discreteness of Chemical Reactions, Gillespie, and All That 78

3.5 Frontiers: Temperature Compensation 84

3.6 Frontiers: Crosstalk between Cellular Systems 92

3.7 Common Mistakes in Modeling 93

Code 3.1 Simulations of the Goodwin Model Using the Gillespie Method 94

Code 3.2 Temperature Compensation Counterexample 96

Code 3.3 A Black-Widow DMA-Diffusing Transcription Factor Model 97

Exercises 98

4 When Do Feedback Loops Oscillate? 101

4.1 Introduction 101

4.2 Introduction to Feedback Loops 102

4.3 General Linear Methodology and Analysis of the Goodwin Model 104

4.4 Frontiers: Futile Cycles Diminish Oscillations, or Why Clocks Like Efficient Complex Formation 109

4.5 Example: Case Study on Familial Advanced Sleep Phase Syndrome 112

4.6 Frontiers: An Additional Fast Positive Feedback Loop 115

4.7 Example: Increasing Activator Concentrations in Circadian Clocks 118

4.8 Bistability and Relaxation Oscillations 119

4.9 Frontiers: Calculating the Period of Relaxation Oscillations 122

4.10 Theory: The Global Secant Condition 124

Code 4.1 Effects of Feedback 126

Code 4.2 Effects of the Hill Coefficient on Rhythms in the Goodwin Model 127

Exercises 128

Part II Behaviors 131

5 System-Level Modeling 133

5.1 Introduction 133

5.2 General Remarks on Bifurcations 134

5.3 SNIC or Type I Oscillators 136

5.4 Examples of Type 1 Oscillators: Simplifications of the Hodgkin-Huxley Model 139

5.5 Hopf or Type 2 Oscillators 142

5.6 Examples of Type 2 Oscillators: The Van der Pol Oscillator and the Resonate-and-Fire Model 145

5.7 Summary of Oscillator Classification 150

5.8 Frontiers: Noise in Type 1 and Type 2 Oscillators 150

5.8.1 Type 1 152

5.8.2 Type 2 154

5.9 Frontiers: Experimentally Testing the Effects of Noise in Squid Giant Axon 156

5.10 Example: The Van der Pol Model and Modeling Human Circadian Rhythms 157

5.11 Example: Refining the Human Circadian Model 159

5.12 Example: A Simple Model of Sleep, Alertness, and Performance 163

5.13 Frontiers: Equivalence of Neuronal and Biochemical Models 165

Code 5.1 Simulation of Type 1 and Type 2 Behavior in the Morris-Lecar Model 167

Exercises 168

6 Phase Response Curves 171

6.1 Introduction and General Properties of Phase Response Curves 171

6.2 Type 1 Response to Brief Stimuli in Phase-Only Oscillators 173

6.3 Perturbations to Type 2 Oscillators 175

6.4 Instantaneous Perturbations to the Radial Isochron Clock 177

6.5 Frontiers: Phase Resetting with Pathological Isochrons 182

6.6 Phase Shifts for Weak Stimuli 183

6.7 Frontiers: Phase Shifting in Models with More Than Two Dimensions 185

6.8 Winfree's Theory of Phase Resetting 186

6.9 Experimental PRCs 187

6.10 Entrainment 188

Code 6.1 Calculating a Predicted Human PRC 190

Code 6.2 Iterating PRCs 191

Exercises 192

7 Eighteen Principles of Synchrony 195

7.1 Basics and Definitions of Synchrony 195

7.1.1 What Are Coupled Oscillators? 196

7.1.2 What Types of Coupling Can Be Found? 197

7.1.3 How to Define Synchrony? 197

7.1.4 How Can We Measure Synchrony? 198

7.2 Synchrony in Pulse-Coupled Oscillators 199

7.3 Heterogeneous Oscillators 199

7.4 Subharmonic and Superharmonic Synchrony 205

7.5 Frontiers: The Counterintuitive Interplay between Noise and Coupling 206

7.6 Nearest-Neighbor Coupling 209

7.7 Frontiers: What Do We Gain by Looking at Limit-Cycle Oscillators? 211

7.8 Coupling Damped Oscillators 213

7.9 Amplitude Death and Beyond 214

7.10 Theory: Proof of Synchrony in Homogeneous Oscillators 214

Code 7.1 Two Coupled Biochemical Feedback Loops 218

Code 7.2 Pulse-Coupled Oscillators 218

Code 7.3 Inhibitory Pulse-Coupled Oscillators 219

Code 7.4 Noisy Coupled Oscillators 220

Code 7.5 Coupled Chain of Oscillators 221

Code 7.6 Amplitude Death 222

Exercises 223

Part III Analysis and Computation 225

8 Statistical and Computational Tools for Model Building: How to Extract Information from Timeseries Data 227

8.1 How to Find Parameters of a Model 227

8.1.1 Two Types of Data 227

8.1.2 Determining the Error of a Model Prediction 228

8.1.3 Measuring Biochemical Rate Constants 229

8.1.4 Frontiers: indirect Ways to Measure Rate Constants with Rhythmic Inputs 232

8.1.5 Finding Parameters to Minimize Error 234

8.1.6 Simulated Annealing 236

8.2 Frontiers: Theoretical Limits on Fitting Timecourse Data 236

8.2.1 How Much Information Is in Timecourse Data? 236

8.2.2 Completely Determinable Dynamics 237

8.2.3 Determining Model Dimension from Timecourse Data 239

8.3 Discrete Models, Noise, and Correlated Error 240

8.3.1 An Introduction to ARMA Models for Error 240

8.3.2 Correlated Errors and Masking 242

8.4 Maximum Likelihood and Least-Squares 243

8.5 The Kalman Filter 244

8.6 Calculating Least-Squares 246

8.7 Frontiers: Using the Kalman Filter for Problems with Correlated Errors 247

8.8 Examples 249

8.9 Theory: The Akaike Information Criterion 252

8.10 A Final Word of Caution about Stationarity 254

Code 8.1 Fitting Protein Data 255

Exercises 258

9 How to Shift an Oscillator Optimally 261

9.1 Asking the Right Biological Questions 261

9.2 Asking the Right Mathematical Questions 265

9.3 Frontiers: A Geometric Interpretation of Optimality 267

9.4 Influence Functions 271

9.5 Frontiers: Two Additional Derivations of the Influence Functions 273

9.6 Adding the Cost to the Hamiltonian 275

9.7 Numerical Methods for Finding Optimal Stimuli 276

9.8 Frontiers: Optimal Stimuli for the Hodgkin-Huxley Equation 277

9.9 Examples: Analysis of Minimal Time Problems 279

9.10 Example: Shifting the Human Circadian Clock 281

Code 9.1 Optimal Stimulus for the Hodgkin-Huxley Equations 282

Code 9.2 An Alternate Method to Calculate Optimal Stimuli for the Hodgkin-Huxley Model 286

Exercises 288

10 Mathematical and Computational Techniques for Multiscale Problems 291

10.1 Simplifying Multiscale Systems 291

10.1.1 The Method of Averaging 293

10.1.2 Example: Applying the Method of Averaging to Human Circadian Rhythms 299

10.1.3 What Is Lost in the Method of Averaging? 299

10.1.4 Frontiers: Second-Order Averaging 300

10.2 Frontiers: Averaging in Systems with More Than Two Variables 303

10.2.1 Increasing the Amplitude of Oscillations in Biochemical Feedback Loops Increases the Oscillation Period 305

10.2.2 A Numerical Method for Simulating Coupled High-Dimensional Clock Models 306

10.3 Frontiers: Piecewise Linear Approximations to Nonlinear Equations 307

10.3.1 Simulational Example with the Goodwin Model 309

10.3.2 Analytical Example with the Van der Pol Model 310

10.4 Frontiers: Poincaré Maps and Model Reduction 310

10.5 Ruling out Limit Cycles 311

10.5.1 Theory: A Sketch of a Proof of the Poincaré-Bendixson Theorem for Biochemical Feedback Loops 313

Code 10.1 Five Simulations of the Goodwin Model 315

Code 10.2 Poincaré Maps of a Detailed Mammalian Model 317

Code 10.3 Chaotic Motions 319

Exercises 320

Glossary 323

Bibliography 329

Index 341

What People are Saying About This

Joseph S. Takahashi

This book is a 'must-read' for those interested in oscillators, modeling, and quantitative biology. Forger lays out his insight and advice on how to tackle complex dynamical systems in biology.

Hiroki R. Ueda

Daniel Forger has written a must-read primer for anyone who studies biological timekeeping—circadian clock, cell cycles, and sleep/wake cycles. Read this book and learn from the very best.

Endorsement

This book is a 'must-read' for those interested in oscillators, modeling, and quantitative biology. Forger lays out his insight and advice on how to tackle complex dynamical systems in biology.

—Joseph S. Takahashi, Howard Hughes Medical Institute, University of Texas Southwestern Medical Center

From the Publisher

Daniel Forger has written a must-read primer for anyone who studies biological timekeeping—circadian clock, cell cycles, and sleep/wake cycles. Read this book and learn from the very best.

—Hiroki R. Ueda, Professor, University of Tokyo, and Group Director, RIKEN

This book is a 'must-read' for those interested in oscillators, modeling, and quantitative biology. Forger lays out his insight and advice on how to tackle complex dynamical systems in biology.

—Joseph S. Takahashi, Howard Hughes Medical Institute, University of Texas Southwestern Medical Center