Exploring Mathematical Modeling in Biology Through Case Studies and Experimental Activities

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