Mathematical Biology (eBook)

Mathematical Biology An Introduction with Maple and Matlab 3
Preface 5
Acknowledgments 8
Contents 9
1 Biology, Mathematics, and a Mathematical BiologyLaboratory 14
1.1 The Natural Linkage Between Mathematics and Biology 14
1.2 The Use of Models in Biology 15
1.3 What Can Be Derived from a Model and How Is It Analyzed? 18
References 21
2 Some Mathematical Tools 22
Introduction 22
2.1 Linear Dependence 23
Exercises 28
2.2 Linear Regression, the Method of Least Squares 30
Exercises 38
2.3 Multiple Regression 41
Exercises 47
2.4 Modeling with Differential Equations 49
Exercises 58
2.5 Modeling with Difference Equations 61
2.6 Matrix Analysis 65
Exercises 69
2.7 Statistical Data 71
Exercises 79
2.8 Probability 83
Exercises 91
References and Suggested Further Reading 94
Part I Cells, Signals, Growth, and Populations 95
3 Reproduction and the Drive for Survival 96
Introduction 96
3.1 The Darwinian Model of Evolution 97
3.2 Cells 100
3.3 Replication of Living Systems 101
3.4 Population Growth and Its Limitations 104
3.5 The Exponential Model for Growth and Decay 106
Exercises/Experiments 115
Questions for Thought and Discussion 116
References and Suggested Further Reading 116
4 Interactions Between Organisms andTheir Environment 117
Introduction 117
4.1 How Population Growth Is Controlled 117
4.2 Community Ecology 127
4.3 Environmentally Limited Population Growth 128
Exercises/Experiments 135
4.4 A Brief Look at Multiple Species Systems 138
Exercises/Experiments 145
Questions for Thought and Discussion 149
References and Suggested Further Reading 149
5 Age-Dependent Population Structures 151
Introduction 151
5.1 Aging and Death 151
5.2 The Age Structure of Populations 156
5.3 Predicting the Age Structure of a Population 160
Exercises/Experiments 169
Questions for Thought and Discussion 170
References and Suggested Further Reading 171
6 Random Movements in Space and Time 172
Introduction 172
6.1 Biological Membranes 173
6.2 The Mathematics of Diffusion 177
Exercises/Experiments 194
6.3 Transplacental Transfer of Oxygen: Biological andBiochemical Considerations 196
6.4 Oxygen Diffusion Across the Placenta: PhysicalConsiderations 198
Exercises/Experiments 206
Questions for Thought and Discussion 206
References and Suggested Further Reading 207
7 Neurophysiology 209
Introduction 209
7.1 Communication Between Parts of an Organism 209
7.2 The Neuron 211
7.3 The Action Potential 214
7.4 Synapses: Interneuronal Connections 218
7.5 A Model for the Conduction of Action Potentials 222
7.6 Fitzhugh–Nagumo Two-Variable Action Potential System 231
Exercises/Experiments 234
Questions for Thought and Discussion 235
References and Suggested Further Reading 235
8 The Biochemistry of Cells 237
Introduction 237
8.1 Atoms and Bonds in Biochemistry 237
8.2 Biopolymers 245
8.3 Molecular Information Transfer 260
8.4 Enzymes and Their Function 267
8.5 Rates of Chemical Reactions 274
Exercises/Experiments 278
8.6 Enzyme Kinetics 279
Exercises/Experiments 284
Questions for Thought and Discussion 287
References and Suggested Further Reading 288
Part II Systems and Diseases 289
9 The Biological Disposition of Drugs and InorganicToxins 290
Introduction 290
9.1 The Biological Importance of Lead 290
9.2 Early Embryogenesis and Organ Formation 291
9.3 Gas Exchange 297
9.4 The Digestive System 299
9.5 The Skin 301
9.6 The Circulatory System 302
9.7 Bones 311
9.8 The Kidneys 312
9.9 Clinical Effects of Lead 314
9.10 A Mathematical Model for Lead in Mammals 315
Exercises/Experiments 319
9.11 Pharmacokinetics 320
Exercises/Experiments 327
Questions for Thought and Discussion 327
References and Suggested Further Reading 327
10 A Biomathematical Approach to HIV and AIDS 329
Introduction 329
10.1 Viruses 329
10.2 The Immune System 334
10.3 HIV and AIDS 343
10.4 An HIV Infection Model 349
Exercises/Experiments 356
10.5 A Model for a Mutating Virus 357
10.6 Predicting the Onset of AIDS 364
Exercises/Experiments 372
Questions for Thought and Discussion 374
References and Suggested Further Reading 375
11 Parasites and Their Diseases 376
Introduction 376
11.1 Protozoan Parasites Cause Important Human Diseases 376
11.2 The Life Cycle of the Malaria Parasite 381
11.3 Host–Parasite Interactions 383
11.4 Mathematical Models for Parasitic Diseases 387
Exercises/Experiments 399
Questions for Thought and Discussion 402
References and Suggested Further Reading 402
12 Cancer: A Disease of the DNA 403
Introduction 403
12.1 Cell Growth and Division Is an Involved and ComplicatedProcess 403
12.2 Types and Stages of Cancer 406
12.3 The Role of Proto-Oncogenes and Tumor Suppressor Genes 406
12.4 A Model for Retinoblastoma 409
12.5 Application to the Retinoblastoma Data 416
12.6 Persistence of Germinal Cases 418
Exercises/Experiments 419
Questions for Thought and Discussion 420
References and Suggested Further Reading 420
Part III Genomics 421
13 Genetics 422
Introduction 422
13.1 Asexual Cell Reproduction: Mitosis 422
13.2 Sexual Reproduction: Meiosis and Fertilization 428
13.3 Classical Genetics 432
13.4 Genetic Drift 441
13.5 A Final Look at Darwinian Evolution 443
13.6 The Hardy–Weinberg Principle 445
13.7 The Fixation of a Beneficial Mutation 451
Exercises/Experiments 456
Questions for Thought and Discussion 462
References and Suggested Further Reading 462
14 Genomics 464
14.1 The Promise of Genomics 464
14.2 Structural Genomics 466
14.3 Comparative Genomics 468
14.4 Genomics in Medicine 470
14.5 Protein Substitution Matrices 474
14.6 BLAST for Protein and DNA Search 478
14.7 The Mathematical Underpinnings of BLAST 487
Exercises/Experiments 497
Questions for Thought and Discussion 498
References and Suggested Further Reading 498
15 Phylogenetics 499
Introduction 499
15.1 Phylogeny 500
15.2 Branch Lengths Estimate the Separation of Species 510
15.3 Introduction to Algebraic Statistics 516
15.4 Algebraic Analysis of Maximum Likelihood 522
15.5 Characterizing Trees by Their Variety, PhylogeneticInvariants 529
15.6 Constructing the Phylogenetic Tree 532
Exercises/Experiments 537
Questions for Thought and Discussion 539
References and Suggested Further Reading 539
Code Index 540
Index 542