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Computational Cell Biology - Christopher Fall, Eric Marland, John Wagner, John Tyson

Contents


I Introductory Course 1
1 Dynamic Phenomena in Cells 3
1.1 Scope of Cellular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Computational Modeling in Biology . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Cartoons, Mechanisms, and Models . . . . . . . . . . . . . . . . 8
1.2.2 The Role of Computation . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 The Role of Mathematics . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 A Simple Molecular Switch . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Solving and Analyzing Differential Equations . . . . . . . . . . . . . . . 13
1.4.1 Numerical Integration of Differential Equations . . . . . . . . . 15
1.4.2 Introduction to Numerical Packages . . . . . . . . . . . . . . . . 18
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Voltage Gated Ionic Currents 21
2.1 Basis of the Ionic Battery . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 The Nernst Potential: Charge Balances Concentration . . . . . 24
2.1.2 The Resting Membrane Potential . . . . . . . . . . . . . . . . . . 26
2.2 The Membrane Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Equations for Membrane Electrical Behavior . . . . . . . . . . . 28
2.3 Activation and Inactivation Gates . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Models of Voltage–Dependent Gating . . . . . . . . . . . . . . . . 29

2.3.2 TheVoltageClamp . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Interacting Ion Channels: The Morris–Lecar Model . . . . . . . . . . . 34
2.4.1 Phase Plane Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.3 Why Do Oscillations Occur? . . . . . . . . . . . . . . . . . . . . . 40
2.4.4 Excitability and Action Potentials . . . . . . . . . . . . . . . . . . 43
2.4.5 Type I andType II Spiking . . . . . . . . . . . . . . . . . . . . . . 44
2.5 The Hodgkin–Huxley Model . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6 FitzHugh–Nagumo Class Models . . . . . . . . . . . . . . . . . . . . . . 47
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Transporters and Pumps 53
3.1 Passive Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Transporter Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.1 Algebraic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.2 Diagrammatic Method . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.3 Rate of the GLUT Transporter . . . . . . . . . . . . . . . . . . . . 62
3.3 The Na+/Glucose Cotransporter . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 SERCA Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5 Transport Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Fast and Slow Time Scales 77
4.1 The Rapid Equilibrium Approximation . . . . . . . . . . . . . . . . . . . 78
4.2 Asymptotic Analysis of Time Scales . . . . . . . . . . . . . . . . . . . . . 82
4.3 Glucose–Dependent Insulin Secretion . . . . . . . . . . . . . . . . . . . 83
4.4 Ligand Gated Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5 The Neuromuscular Junction . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.6 The Inositol Trisphosphate (IP3) receptor . . . . . . . . . . . . . . . . . 91
4.7 Michaelis–Menten Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Whole–Cell Models 101
5.1 Models of ER and PM Calcium Handling . . . . . . . . . . . . . . . . . 102
5.1.1 Flux Balance Equations with Rapid Buffering . . . . . . . . . . 103
5.1.2 Expressions for the Fluxes . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Calcium Oscillations in the Bullfrog Sympathetic Ganglion Neuron . 107
5.2.1 Ryanodine Receptor Kinetics: The Keizer–Levine Model . . . . 108
5.2.2 Bullfrog Sympathetic Ganglion Neuron Closed–Cell Model . . 111
5.2.3 Bullfrog Sympathetic Ganglion Neuron Open–Cell Model . . . 113
5.3 The Pituitary Gonadotroph . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3.1 The ER Oscillator in a Closed Cell . . . . . . . . . . . . . . . . . . 116

A.2 A Brief Review of Power Series . . . . . . . . . . . . . . . . . . . . . . . . 380
A.3 Linear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
A.3.1 Solution of Systems of Linear ODEs . . . . . . . . . . . . . . . . 383
A.3.2 Numerical Solutions of ODEs . . . . . . . . . . . . . . . . . . . . 385
A.3.3 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . 386
A.4 Phase Plane Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
A.4.1 Stability of Linear Steady States . . . . . . . . . . . . . . . . . . . 390
A.4.2 Stability of a Nonlinear Steady States . . . . . . . . . . . . . . . 392
A.5 Bifurcation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
A.5.1 Bifurcation at a Zero Eigenvalue . . . . . . . . . . . . . . . . . . 396
A.5.2 Bifurcation at a Pair of Imaginary Eigenvalues . . . . . . . . . . 398
A.6 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
A.6.1 Regular Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . 401
A.6.2 Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
A.6.3 Singular Perturbation Theory . . . . . . . . . . . . . . . . . . . . 405
A.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
B Solving and Analyzing Dynamical Systems Using XPPAUT 410
B.1 Basics of Solving Ordinary Differential Equations . . . . . . . . . . . . 411
B.1.1 Creating the ODE File . . . . . . . . . . . . . . . . . . . . . . . . . 411
B.1.2 Running the Program . . . . . . . . . . . . . . . . . . . . . . . . . 412
B.1.3 The Main Window . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
B.1.4 Solving the Equations, Graphing, and Plotting. . . . . . . . . . . 414
B.1.5 Saving and Printing Plots . . . . . . . . . . . . . . . . . . . . . . . 416
B.1.6 Changing Parameters and Initial Data . . . . . . . . . . . . . . . 418
B.1.7 Looking at the Numbers: The Data Viewer . . . . . . . . . . . . 419
B.1.8 Saving and Restoring the State of Simulations . . . . . . . . . . 420
B.1.9 Important Numerical Parameters . . . . . . . . . . . . . . . . . . 421
B.1.10 Command Summary: The Basics . . . . . . . . . . . . . . . . . . 422
B.2 Phase Planes and Nonlinear Equations . . . . . . . . . . . . . . . . . . . 422
B.2.1 Direction Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
B.2.2 Nullclines and Fixed Points . . . . . . . . . . . . . . . . . . . . . . 423
B.2.3 Command Summary: Phase Planes and Fixed Points . . . . . . 426
B.3 Bifurcation and Continuation . . . . . . . . . . . . . . . . . . . . . . . . 427
B.3.1 General Steps for Bifurcation Analysis . . . . . . . . . . . . . . . 427
B.3.2 Hopf Bifurcation in the FitzHugh–Nagumo Equations . . . . . 428
B.3.3 Hints for Computing Complete Bifurcation Diagrams . . . . . . 430
B.4 Partial Differential Equations: The Method of Lines . . . . . . . . . . . 432
B.5 Stochastic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
B.5.1 A Simple Brownian Ratchet . . . . . . . . . . . . . . . . . . . . . 434
B.5.2 A Sodium Channel Model . . . . . . . . . . . . . . . . . . . . . . . 434
B.5.3 A Flashing Ratchet . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

5.3.2 Open–Cell Model with Constant Calcium Influx . . . . . . . . . 122
5.3.3 The Plasma Membrane Oscillator . . . . . . . . . . . . . . . . . . 124
5.3.4 Bursting Driven by the ER in the Full Model . . . . . . . . . . . 126
5.4 The Pancreatic Beta Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4.1 Chay–Keizer Model . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4.2 Chay–Keizer with an ER . . . . . . . . . . . . . . . . . . . . . . . 133
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6 Intercellular Communication 140
6.1 Electrical Coupling and Gap Junctions . . . . . . . . . . . . . . . . . . . 141
6.1.1 Synchronization of Two Oscillators . . . . . . . . . . . . . . . . . 142
6.1.2 Asynchrony Between Oscillators . . . . . . . . . . . . . . . . . . . 143
6.1.3 Cell Ensembles, Electrical Coupling Length Scale . . . . . . . . 144
6.2 Synaptic Transmission Between Neurons . . . . . . . . . . . . . . . . . 146
6.2.1 Kinetics of Postsynaptic Current . . . . . . . . . . . . . . . . . . 147
6.2.2 Synapses: Excitatory and Inhibitory; Fast and Slow . . . . . . . 148
6.3 When Synapses Might (or Might Not) Synchronize Active Cells . . . . 150
6.4 Neural Circuits as Computational Devices . . . . . . . . . . . . . . . . . 153
6.5 Large–Scale Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
II Advanced Material 169
7 Spatial Modeling 171
7.1 One-Dimensional Formulation . . . . . . . . . . . . . . . . . . . . . . . . 173
7.1.1 Conservation in One Dimension . . . . . . . . . . . . . . . . . . . 173
7.1.2 Fick’sLawofDiffusion . . . . . . . . . . . . . . . . . . . . . . . . 175
7.1.3 Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.1.4 Flux of Ions in a Field . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.1.5 The Cable Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.1.6 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . 178
7.2 Important Examples with Analytic Solutions . . . . . . . . . . . . . . . 179
7.2.1 Diffusion Through a Membrane . . . . . . . . . . . . . . . . . . . 179
7.2.2 Ion Flux Through a Channel . . . . . . . . . . . . . . . . . . . . . 180
7.2.3 Voltage Clamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.2.4 Diffusion in a Long Dendrite . . . . . . . . . . . . . . . . . . . . . 181
7.2.5 Diffusion into aCapillary . . . . . . . . . . . . . . . . . . . . . . . 183
7.3 Numerical Solution of the Diffusion Equation . . . . . . . . . . . . . . 184
7.4 Multidimensional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.4.1 Conservation Law in Multiple Dimensions . . . . . . . . . . . . 186
7.4.2 Fick’s Law in Multiple Dimensions . . . . . . . . . . . . . . . . . 187
7.4.3 Advection in Multiple Dimensions . . . . . . . . . . . . . . . . . 188

7.4.4 Boundary and Initial Conditions for Multiple Dimensions . . . 188
7.4.5 Diffusion in Multiple Dimensions: Symmetry . . . . . . . . . . . 188
7.5 Traveling Waves in Nonlinear Reaction–Diffusion Equations . . . . . . 189
7.5.1 Traveling Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . 190
7.5.2 Traveling Wave in the Fitzhugh–Nagumo Equations . . . . . . . 192
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8 Modeling Intracellular Calcium Waves and Sparks 198
8.1 Microfluorometric Measurements . . . . . . . . . . . . . . . . . . . . . . 198
8.2 A Model of the Fertilization Calcium Wave . . . . . . . . . . . . . . . . 200
8.3 Including Calcium Buffers in Spatial Models . . . . . . . . . . . . . . . 202
8.4 The Effective Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . 203
8.5 Simulation of a Fertilization Calcium Wave . . . . . . . . . . . . . . . . 204
8.6 Simulation of a Traveling Front . . . . . . . . . . . . . . . . . . . . . . . 204
8.7 Calcium Waves in the Immature Xenopus Oocycte . . . . . . . . . . . . 208
8.8 Simulation of a Traveling Pulse . . . . . . . . . . . . . . . . . . . . . . . 208
8.9 Simulation of a Kinematic Wave . . . . . . . . . . . . . . . . . . . . . . . 210
8.10 Spark-Mediated Calcium Waves . . . . . . . . . . . . . . . . . . . . . . . 213
8.11 The Fire–Diffuse–Fire Model . . . . . . . . . . . . . . . . . . . . . . . . . 214
8.12 Modeling Localized Calcium Elevations . . . . . . . . . . . . . . . . . . 220
8.13 Steady-State Localized Calcium Elevations . . . . . . . . . . . . . . . . 222
8.13.1 The Steady–State Excess Buffer Approximation (EBA) . . . . . 224
8.13.2 The Steady–State Rapid Buffer Approximation (RBA) . . . . . . 225
8.13.3 Complementarity of the Steady-State EBA and RBA . . . . . . . 226
8.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
9 Biochemical Oscillations 230
9.1 Biochemical Kinetics and Feedback . . . . . . . . . . . . . . . . . . . . . 232
9.2 Regulatory Enzymes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
9.3 Two-Component Oscillators Based on Autocatalysis . . . . . . . . . . . 239
9.3.1 Substrate–Depletion Oscillator . . . . . . . . . . . . . . . . . . . . 240
9.3.2 Activator–Inhibitor Oscillator . . . . . . . . . . . . . . . . . . . . 242
9.4 Three-Component Networks Without Autocatalysis . . . . . . . . . . . 243
9.4.1 Positive Feedback Loop and the Routh–Hurwitz Theorem . . . 244
9.4.2 Negative Feedback Oscillations . . . . . . . . . . . . . . . . . . . 244
9.4.3 The Goodwin Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 244
9.5 Time-Delayed Negative Feedback . . . . . . . . . . . . . . . . . . . . . . 247
9.5.1 Distributed Time Lag and the Linear Chain Trick . . . . . . . . 248
9.5.2 Discrete Time Lag . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
9.6 Circadian Rhythms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

10 Cell Cycle Controls 261
10.1 Physiology of the Cell Cycle in Eukaryotes . . . . . . . . . . . . . . . . . 261
10.2 Molecular Mechanisms of Cell Cycle Control . . . . . . . . . . . . . . . 263
10.3 A Toy Model of Start and Finish . . . . . . . . . . . . . . . . . . . . . . . 265
10.3.1 Hysteresis in the Interactions Between Cdk and APC . . . . . . 266
10.3.2 Activation of the APC at Anaphase . . . . . . . . . . . . . . . . . 267
10.4 A Serious Model of the Budding Yeast Cell Cycle . . . . . . . . . . . . . 269
10.5 Cell Cycle Controls in Fission Yeast . . . . . . . . . . . . . . . . . . . . . 273
10.6 Checkpoints and Surveillance Mechanisms . . . . . . . . . . . . . . . . 276
10.7 Division Controls in Egg Cells . . . . . . . . . . . . . . . . . . . . . . . . 276
10.8 Growth and Division Controls in Metazoans . . . . . . . . . . . . . . . . 278
10.9 Spontaneous Limit Cycle or Hysteresis Loop? . . . . . . . . . . . . . . . 279
10.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
11 Modeling the Stochastic Gating of Ion Channels 285
11.1 Single–Channel Gating and a Two-State Model . . . . . . . . . . . . . . 285
11.1.1 Modeling Channel Gating as a Markov Process . . . . . . . . . . 286
11.1.2 The Transition Probability Matrix . . . . . . . . . . . . . . . . . . 288
11.1.3 Dwell Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
11.1.4 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . 290
11.1.5 Simulating Multiple Independent Channels . . . . . . . . . . . . 291
11.1.6 Gillespie’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
11.2 An Ensemble of Two-State Ion Channels . . . . . . . . . . . . . . . . . . 293
11.2.1 Probability of Finding N Channels in the Open State . . . . . . 293
11.2.2 The Average Number of Open Channels . . . . . . . . . . . . . . 296
11.2.3 The Variance of the Number of Open Channels . . . . . . . . . . 297
11.3 Fluctuations in Macroscopic Currents . . . . . . . . . . . . . . . . . . . 298
11.4 Modeling Fluctuations in Macroscopic Currents with Stochastic
ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
11.4.1 Langevin Equation for an Ensemble of Two-State Channels . . 304
11.4.2 Fokker–Planck Equation for an Ensemble of Two-State
Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
11.5 Membrane Voltage Fluctuations . . . . . . . . . . . . . . . . . . . . . . . 307
11.5.1 Membrane Voltage Fluctuations with an Ensemble of
Two-State Channels . . . . . . . . . . . . . . . . . . . . . . . . . . 309
11.6 Stochasticity and Discreteness in an Excitable Membrane Model . . . 311
11.6.1 Phenomena Induced by Stochasticity and Discreteness . . . . . 312
11.6.2 The Ensemble Density Approach Applied to the Stochastic
Morris–Lecar Model . . . . . . . . . . . . . . . . . . . . . . . . . . 313
11.6.3 Langevin Formulation for the Stochastic Morris–Lecar Model . 314
11.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

12 Molecular Motors: Theory 320
12.1 Molecular Motions as Stochastic Processes . . . . . . . . . . . . . . . . 323
12.1.1 Protein Motion as a Simple Random Walk . . . . . . . . . . . . 323
12.1.2 Polymer Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
12.1.3 Sample Paths of Polymer Growth . . . . . . . . . . . . . . . . . . 327
12.1.4 The Statistical Behavior of Polymer Growth . . . . . . . . . . . 329
12.2 Modeling Molecular Motions . . . . . . . . . . . . . . . . . . . . . . . . . 330
12.2.1 The Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . 330
12.2.2 Numerical Simulation of the Langevin Equation . . . . . . . . . 332
12.2.3 The Smoluchowski Model . . . . . . . . . . . . . . . . . . . . . . 333
12.2.4 First Passage Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
12.3 Modeling Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . 335
12.4 A Mechanochemical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 338
12.5 Numerical Simulation of Protein Motion . . . . . . . . . . . . . . . . . . 339
12.5.1 Numerical Algorithm that Preserves Detailed Balance . . . . . 340
12.5.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 341
12.5.3 Numerical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 342
12.5.4 Implicit Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 344
12.6 Derivations and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . 345
12.6.1 The Drag Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 345
12.6.2 The Equipartition Theorem . . . . . . . . . . . . . . . . . . . . . 345
12.6.3 A Numerical Method for the Langevin Equation . . . . . . . . . 346
12.6.4 Some Connections with Thermodynamics . . . . . . . . . . . . . 347
12.6.5 Jumping Beans and Entropy . . . . . . . . . . . . . . . . . . . . . 349
12.6.6 Jump Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
12.6.7 Jump Rates at an Absorbing Boundary . . . . . . . . . . . . . . . 351
12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
13 Molecular Motors: Examples 354
13.1 Switching in the Bacterial Flagellar Motor . . . . . . . . . . . . . . . . . 354
13.2 A Motor Driven by a “Flashing Potential” . . . . . . . . . . . . . . . . . 359
13.3 The Polymerization Ratchet . . . . . . . . . . . . . . . . . . . . . . . . . 362
13.4 Simplified Model of the F0Motor . . . . . . . . . . . . . . . . . . . . . . 364
13.4.1 The Average Velocity of the Motor in the Limit of Fast
Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
13.4.2 Brownian Ratchet vs. Power Stroke . . . . . . . . . . . . . . . . . 369
13.4.3 The Average Velocity of the Motor When Chemical Reactions
Are as Fast as Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 369
13.5 Other Motor Proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
A Qualitative Analysis of Differential Equations 378
A.1 Matrix and Vector Manipulation . . . . . . . . . . . . . . . . . . . . . . . 379
C Numerical Algorithms 439

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