Exploring Mathematical Modeling in Biology Through Case Studies and Experimental Activities

Table of Contents

Unit 1 Introduction to Modeling using Difference Equations1.1 Discrete-Time Models 1.1.1 Solutions to First-Order Difference Equations1.1.2 Using Linear Regression to Estimate Parameters 1.2 Putting it all together: The Whooping Crane 1.3 CaseStudy1: Island Biogeography1.3.1 Background1.3.2 Model Formulation1.3.3 Rakata Story1.3.4 Modern Approach: Lineage Data1.3.5 Back to MacArthur and Wilson: Effects of Distance and Area1.4 CaseStudy2: Pharmacokinetics Model1.4.1 Background1.4.2 Formulating the model1.4.3 Understanding the Model 1.4.4 Parameter Estimation 1.4.5 Model Evaluation/Analysis1.4.6 Further Exploration 1.5 CaseStudy3: Invasive Plant Species1.5.1 Background1.5.2 Model Formulation 1.5.3 Parameter Estimation1.5.4 Model Predictions1.5.5 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 Analysis1.6.5 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 Background2.2.1 Differential Equation Based Model Formulation 2.2.2 Solutions to Ordinary Differential Equations 2.2.3 Investigating Parameter Space2.2.4 Nonlinear Fitting2.3 Model Selection2.4 Case Study 1: How Leaf Decomposition Rates Vary with Anthropogenic Nitrogen Deposition2.4.1 Background2.4.2 The Data2.4.3 Model Formulation2.4.4 Parameter Estimation 2.4.5 Model Evaluation2.5 Case Study 2: Exploring Models to Describe Tumor Growth Rates 2.5.1 Background2.5.2 The Data 2.5.3 Model Formulation2.5.4 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 Background2.6.2 Analysis of functional response data: determining the parameters 2.6.3 Exploring functional responses as a function of temperature2.7 Wet Lab: Enzyme Kinetics of Catechol Oxidase2.7.1 Overview of Activities 2.7.2 Introduction to Enzyme Catalyzed Reaction Kinetics 2.7.3 Deriving the model2.7.4 Estimating KM and Vmax2.7.5 Our Enzyme: Catechol Oxidase 2.7.6 Experiment: Collecting Initial Rates for the Michaelis-Menten Model2.7.7 Effects of Inhibitors on Enzyme Kinetics2.7.8 Experiment: Measuring the Effects of Two Catechol Oxidase Inhibitors, Phenylthiourea and Benzoic Acid

Unit 3 Differential Equations: Numerical Solutions, Model Calibration, and Sensitivity Analysis3.1 Biological Background 3.2 Mathematical and R Background3.2.1 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 Cells3.3 Case Study: Influenza: Adapting the Classic SIR Model to the 2009 Influenza Pandemic 3.3.1 Background3.3.2 The SIR Model3.3.3 Cumulative Number of Cases 3.3.4 Epidemic Threshold 3.3.5 Public Health Interventions 3.3.6 2009 H1N1 Influenza Pandemic3.4 Case Study 2: Prostate Cancer: optimizing immuno-therapy 3.4.1 Background3.4.2 Model Formulation3.4.3 Model Implementation3.4.4 Parameter Estimation3.4.5 Vaccination Protocols and Model Predictions 3.4.6 Sensitivity Analysis 3.4.7 Simulating Other Treatment Strategies3.5 Case Study 3: Quorum Sensing3.5.1 Introduction 3.5.2 Model Formulation3.5.3 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 Activities3.6.2 Introduction to blood glucose regulation and its importance 3.6.3 Developing a model 3.6.4 Experiment: Measuring Blood Glucose Concentrations Following Glucose Ingestion3.6.5 Analysis3.6.6 Thoughts to Consider for Potential Follow-up Experiments

Unit 4 Technical Notes for Laboratory Activities 4.1 Introduction4.2 Population Growth4.3 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

Authors

Rebecca Sanft Assistant Professor of Mathematics, University of North Carolina Asheville, USA. 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.