Invitation to Biomathematics (eBook)

Front Cover 1
An Invitation to Biomathematics 4
Copyright Page 5
Contents 6
Preface 8
Chapter 1: Processes that Change with Time: Introduction to Dynamical Systems 14
I. Using Data to Formulate a Model 15
II. Discrete Versus Continuous Models 19
III. A Continuous Population Growth Model 21
IV. The Logistic Model 25
V. An Alternative Derivation of the Logistic Model 29
VI. Long-Term Behavior and Equilibrium States 31
VII. Analyzing Equilibrium States 32
VIII. The Verhulst Model for Discrete Population Growth 41
IX. A Population Growth Model with Delay 46
X. Modeling Physiological Mechanisms of Drug Elimination 49
XI. Using Computer Software for Solving the Models 57
XII. Some BERKELEY MADONNA Specifics 58
XIII. Suggested Biology Laboratory Exercises for Chapter 1 63
References 63
Internet Resources 63
Further Reading 64
Chapter 2: Complex Dynamics Emerging from Interacting Dynamical Systems 66
I. Introduction to Infectious Disease 67
II. The Spread of an Epidemic 71
III. Phase Plane Analysis 83
IV. Stability of Equilibrium Points 86
V. Epidemic Models with Delay and Models with Intermediate Groups 92
VI. Predator-Prey Interactions 94
VII. A Model of Competitive Interaction 104
VIII. Appendix: Validation of a Mathematical Claim 107
References 110
Further Reading 111
Chapter 3: Mathematics in Genetics 112
I. Introduction 112
II. Chromosomes and the Physical Basis of Heredity 115
III. Hardy-Weinberg Law of Genetic Equilibrium 120
IV. The Effect of a Maladaptive or Lethal Gene 125
V. More Complex Hereditary Patterns 131
VI. Quantitative Traits 133
D. Genes, Environments, and Variation in a Population 140
References 141
Further Reading 141
Chapte 4: Quantitative Genetics and Statistics 142
I. Probability Background 143
II. Relation of Probability Distributions to Statistical Testing 147
III. Statistical Testing 150
References 162
Internet Resources 162
Further Reading 162
Chapter 5: Risk Analysis of Blood Glucose Data 164
I. Historical Overview 165
II. Clinical Blood Glucose Optimization Problem of Diabetes 166
III. Quantifying Characteristics of Diabetes 167
IV. Self-Monitoring of Blood Glucose 170
V. Symmetrization of the Blood Glucose Measurement Scale 172
VI. The Blood Glucose Risk Function 176
VII. The Low and High Blood Glucose Risk Indices 178
VIII. Model Validation Strategies 179
IX. Validation of the Blood Glucose Risk Indices 180
X. More Complex Models 188
References 192
Further Reading 193
Chapter 6: Predicting Septicemia in Neonates 194
I. Premature Births, Low Birth Weights, and Health Risks 194
II. Sepsis: Medical Overview, Clinical Diagnosis, and Diagnostic Challenges 195
III. Heart Rate and Heart Rate Variability 199
IV. Quantifying Heart Rate and Heart Rate Variability 201
V. Time-independent Measures: Interbeat Interval Distribution, Standard Deviation, and Skewness 204
VI. Time-independent Measures: Sample Asymmetry of a Random Variable 207
VII. Data Validating the Utility of Sample Asymmetry Analysis of Heart Rate Variability 212
VIII. Validating the Properties of Sample Asymmetry Through Computer Simulation 215
IX. Time-dependent Measures: Sample Entropy 216
X. Combining Various Measures of Heart Rate Variability Abnormality 221
References 222
Internet Resources 223
Further Reading 223
Chapter 7: Cooperative Binding: How Your Blood Transports Oxygen 224
I. Introduction 225
II. Binding Reactions 226
III. Mathematical Models of Hemoglobin-Oxygen Binding 230
IV. Deriving Fractional Saturation Functions with Binding Polynomials 239
V. Appendix: Justifying Equation (7-20) 243
References 244
Further Reading 244
Chapter 8: Ligand Binding, Data Fitting, and Least-Squares Estimates of Model Parameters 246
I. Data-Fitting Terms, Definitions, and Examples 248
II. A Ligand-Binding Example 253
III. A Primer for Solving Nonlinear Equations 256
IV. Weighted Least-Squares Criterion and the Gauss-Newton Methods for Weighted Least Squares 265
V. Objectives of the Data-Fitting Procedured 269
VI. Appendix: Basic Matrix Arithmetic 275
References 278
Chapter 9: Endocrinology and Hormone Pulsatility 280
I. Introduction 282
II. Experimental Design, Data Collection, and Errors of Measurement 288
III. Classical Methods for Analyzing Hormone Concentration Time Series 292
IV. Deconvolution Methods 301
References 311
Internet Resources 312
Further-Reading 312
Chapter 10: Endocrine Network Modeling: Feedback Loops and Hormone Oscillations 314
I. Introduction 315
II. Symbolic Scheme Representations of Theoretical Models and Modeling Goals 317
III. Evolution and Control of Hormone Concentration 321
IV. Oscillations Driven by a Single-System Feedback Loop 332
V. Networks with Multiple Feedback Loops 348
References 351
Chapter 11: Detecting Rhythms in Confounded Data 354
I. Biological Clocks 355
II. Example: Simulation of the Effects of SCN Ablation and Transplantation on Free-Running Locomotor Activity 365
III. Fundamentals of Rhythmic Data and Time Series 368
IV. Data Preprocessing Strategies 372
V. Methods for Rhythm Analysis and Analytical Strategies 378
VI. Preprocessing Before Analysis 386
VII. Example Analysis: Simulation of the Effects of SCN Ablation and Transplantation on Free-Running Locomotor Activity 389
References 399
Internet Resources 400
Further Readings 400
Chapter 12: Using Microarrays to Study Gene Expression Patterns 402
I. Fabricating and Using Microarrays 404
II. Analysis of Microarray Data 410
VI. Microarrays and Circadian Rhythms 420
References 429
Internet Resources 430
Further Readings 431
Solutions 432
Index 460