Exploring Mathematical Modeling in Biology through Case Studies and Experimental Activities provides supporting materials for courses taken by students majoring in mathematics, computer science or in the life sciences. The book's cases and lab exercises focus on hypothesis testing and model development in the context of real data. The supporting mathematical, coding and biological background permit readers to explore a problem, understand assumptions, and the meaning of their results. The experiential components provide hands-on learning both in the lab and on the computer. As a beginning text in modeling, readers will learn to value the approach and apply competencies in other settings.
Included case studies focus on building a model to solve a particular biological problem from concept and translation into a mathematical form, to validating the parameters, testing the quality of the model and finally interpreting the outcome in biological terms. The book also shows how particular mathematical approaches are adapted to a variety of problems at multiple biological scales. Finally, the labs bring the biological problems and the practical issues of collecting data to actually test the model and/or adapting the mathematics to the data that can be collected.
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Unit 1 Introduction to Modeling using Difference Equations 1.1 Discrete-Time Models 1.1.1 Solutions to First-Order Difference Equations 1.1.2 Using Linear Regression to Estimate Parameters 1.2 Putting it all together: The Whooping Crane 1.3 CaseStudy1: Island Biogeography 1.3.1 Background 1.3.2 Model Formulation 1.3.3 Rakata Story 1.3.4 Modern Approach: Lineage Data 1.3.5 Back to MacArthur and Wilson: Effects of Distance and Area 1.4 CaseStudy2: Pharmacokinetics Model 1.4.1 Background 1.4.2 Formulating the model 1.4.3 Understanding the Model 1.4.4 Parameter Estimation 1.4.5 Model Evaluation/Analysis 1.4.6 Further Exploration 1.5 CaseStudy3: Invasive Plant Species 1.5.1 Background 1.5.2 Model Formulation 1.5.3 Parameter Estimation 1.5.4 Model Predictions 1.5.5 Management Strategies 1.6 Wet Lab: Logistic Growth Model of Bacterial Population Dynamics 1.6.1 Introduction 1.6.2 Modeling populations 1.6.3 The Experiment 1.6.4 Model Calibration and Analysis 1.6.5 Experiment Part2: Effect of changing Media
Unit 2 Differential Equations: Model Formulation, Nonlinear Regression, and Model Selection 2.1 Biological Background 2.2 Mathematical and R Background 2.2.1 Differential Equation Based Model Formulation 2.2.2 Solutions to Ordinary Differential Equations 2.2.3 Investigating Parameter Space 2.2.4 Nonlinear Fitting 2.3 Model Selection 2.4 Case Study 1: How Leaf Decomposition Rates Vary with Anthropogenic Nitrogen Deposition 2.4.1 Background 2.4.2 The Data 2.4.3 Model Formulation 2.4.4 Parameter Estimation 2.4.5 Model Evaluation 2.5 Case Study 2: Exploring Models to Describe Tumor Growth Rates 2.5.1 Background 2.5.2 The Data 2.5.3 Model Formulation 2.5.4 Parameter Estimation 2.5.5 Model Evaluation: Descriptive Power 2.5.6 Model Evaluation: Predictive Power 2.6 Case Study 3: Predator Responses to Prey Density Vary with Temperature 2.6.1 Background 2.6.2 Analysis of functional response data: determining the parameters 2.6.3 Exploring functional responses as a function of temperature 2.7 Wet Lab: Enzyme Kinetics of Catechol Oxidase 2.7.1 Overview of Activities 2.7.2 Introduction to Enzyme Catalyzed Reaction Kinetics 2.7.3 Deriving the model 2.7.4 Estimating KM and Vmax 2.7.5 Our Enzyme: Catechol Oxidase 2.7.6 Experiment: Collecting Initial Rates for the Michaelis-Menten Model 2.7.7 Effects of Inhibitors on Enzyme Kinetics 2.7.8 Experiment: Measuring the Effects of Two Catechol Oxidase Inhibitors, Phenylthiourea and Benzoic Acid
Unit 3 Differential Equations: Numerical Solutions, Model Calibration, and Sensitivity Analysis 3.1 Biological Background 3.2 Mathematical and R Background 3.2.1 Numerical Solutions to Differential Equations 3.2.2 Calibration: Fitting Models to Data 3.2.3 Sensitivity Analysis 3.2.4 Putting it all together: The Dynamics of Ebola Virus Infecting Cells 3.3 Case Study: Influenza: Adapting the Classic SIR Model to the 2009 Influenza Pandemic 3.3.1 Background 3.3.2 The SIR Model 3.3.3 Cumulative Number of Cases 3.3.4 Epidemic Threshold 3.3.5 Public Health Interventions 3.3.6 2009 H1N1 Influenza Pandemic 3.4 Case Study 2: Prostate Cancer: optimizing immuno-therapy 3.4.1 Background 3.4.2 Model Formulation 3.4.3 Model Implementation 3.4.4 Parameter Estimation 3.4.5 Vaccination Protocols and Model Predictions 3.4.6 Sensitivity Analysis 3.4.7 Simulating Other Treatment Strategies 3.5 Case Study 3: Quorum Sensing 3.5.1 Introduction 3.5.2 Model Formulation 3.5.3 Parameter Estimation 3.5.4 Model Simulations 3.5.5 Sensitivity Analysis 3.6 Wet Lab: Hormones and Homeostasis-Keeping Blood Glucose Concentrations Stable 3.6.1 Overview of Activities 3.6.2 Introduction to blood glucose regulation and its importance 3.6.3 Developing a model 3.6.4 Experiment: Measuring Blood Glucose Concentrations Following Glucose Ingestion 3.6.5 Analysis 3.6.6 Thoughts to Consider for Potential Follow-up Experiments
Unit 4 Technical Notes for Laboratory Activities 4.1 Introduction 4.2 Population Growth 4.3 Enzyme Kinetics 4.3.1 Notes on other enzymes or similar experiments 4.4 Instructor Notes for the Blood Glucose Monitoring Lab 4.4.1 Tips for glucose monitoring 4.4.2 Other Lab Activities
Dr. Sanft received her Ph.D. in applied mathematics from the University of Arizona. She then worked at Bryn Mawr College as a Howard Hughes Medical Institute Postdoctoral Fellow. During this time, she developed courses in mathematical modelling and worked with biology faculty to incorporate more quantitative training throughout their curriculum. Following this appointment, Dr. Sanft was an Assistant Professor of Mathematics at St. Olaf College where she led a team of faculty in mathematics and biology to develop a concentration (interdisciplinary minor) in Mathematical Biology. She is passionate about designing interdisciplinary classroom experiences. Her research interests are at the interface of mathematics, biology, and mechanics, with an emphasis on modelling growth in soft tissues.
Anne Walter Professor of Biology, St. Olaf College, USA.
Dr. Walter received her Ph.D. in Physiology and Pharmacology from Duke University. Dr. Walter was a Fellow at the National Institutes of Health where she first worked collaboratively with mathematicians in the Laboratory of Mathematical Biology. Her research areas have included transport physiology, physical properties of biological membranes and their lipids and proteins. During her 30 years of teaching, she has taught courses in renal physiology, comparative animal physiology, cell physiology and neuroscience as well as general education. Dr. Walter has been at the forefront of interdisciplinary course development including an integrated introduction to chemistry and biology program, writing and science literacy, and the mathematics of biology. She also enjoys opportunities to guide students in international study to take interdisciplinary approaches to problems related to human and environmental health in India, evolution, ecology and conservation in Ecuador and water and climate change in Morocco.