Exploring Mathematical Modeling in Biology Through Case Studies and Experimental Activities

Contents
Preface
	Tips for using this text - instructors
	Tips for using this text - students
	Online resources
Acknowledgments
1 Preliminaries: models, R, and lab techniques
	1.1 Bringing mathematics and biology together through modeling
	1.2 R basics
		1.2.1 RStudio layout
		1.2.2 Simple calculations
		1.2.3 Data structures
			Vectors
			Matrices
			Data frames
		1.2.4 Basic plotting
		1.2.5 Reading data from files
		1.2.6 Iteration
		1.2.7 Fitting a linear regression model
	1.3 Prelab lab: practicing the fundamentals
		Introduction
		Materials
		Using the micropipettes
		Serial dilution with dye solutions and water
		Measuring absorbance
		Graphing and data analysis
2 Introduction to modeling using difference equations
	2.1 Discrete-time models
		2.1.1 Solutions to first-order difference equations
		2.1.2 Using linear regression to estimate parameters
	2.2 Putting it all together: the whooping crane
	2.3 Case study 1: Island biogeography
		2.3.1 Background
		2.3.2 Model formulation
		2.3.3 Rakata story
			Data
			Parameter estimation
			Model analysis
		2.3.4 Modern approach: lineage data
			Model
			Parameter estimation
			Model analysis
		2.3.5 Back to MacArthur and Wilson: effects of distance and area
	2.4 Case study 2: Pharmacokinetics model
		2.4.1 Background
			Pharmacokinetics: basic concepts and terminology
			Caffeine
		2.4.2 Formulating the model
		2.4.3 Understanding the model
		2.4.4 Parameter estimation
		2.4.5 Model evaluation/analysis
		2.4.6 Further exploration
	2.5 Case study 3: Invasive plant species
		2.5.1 Background
		2.5.2 Model formulation
		2.5.3 Parameter estimation
		2.5.4 Model predictions
		2.5.5 Management strategies
	2.6 Wet lab: logistic growth model of bacterial population dynamics
		2.6.1 Introduction
		2.6.2 Modeling populations
		2.6.3 The experiment
			Preparation
			Measuring growth rates by optical density
			Measuring growth rate by serial dilution and plate count
		2.6.4 Model calibration and analysis
		2.6.5 Experiment part 2: effect of changing media
3 Differential equations: model formulation, nonlinear regression, and model selection
	3.1 Biological background
	3.2 Mathematical and R background
		3.2.1 Differential equation-based model formulation
			Constant flow rate
			Relative rates
			Mass action
			Feedback
		3.2.2 Solutions to ordinary differential equations
		3.2.3 Investigating parameter space
		3.2.4 Nonlinear fitting
	3.3 Model selection
	3.4 Case study 1: How leaf decomposition rates vary with anthropogenic nitrogen deposition
		3.4.1 Background
		3.4.2 The data
		3.4.3 Model formulation
		3.4.4 Parameter estimation
		3.4.5 Model evaluation
	3.5 Case study 2: Exploring models to describe tumor growth rates
		3.5.1 Background
		3.5.2 The data
		3.5.3 Model formulation
		3.5.4 Parameter estimation
		3.5.5 Model evaluation: descriptive power
		3.5.6 Model evaluation: predictive power
	3.6 Case study 3: Predator responses to prey density vary with temperature
		3.6.1 Background
			Can predator-prey interactions predict invasive behavior?
		3.6.2 Analysis of functional response data: determining the parameters
		3.6.3 Exploring functional responses as a function of temperature
	3.7 Wet lab: enzyme kinetics of catechol oxidase
		3.7.1 Overview of activities
		3.7.2 Introduction to enzyme catalyzed reaction kinetics
		3.7.3 Deriving the model
		3.7.4 Estimating KM and Vmax
		3.7.5 Our enzyme: catechol oxidase
		3.7.6 Experiment: collecting initial rates for the Michaelis-Menten model
			Overview of the procedure
			Materials
			Enzyme preparation
			Running the reactions
			Analysis in R
		3.7.7 Effects of inhibitors on enzyme kinetics
		3.7.8 Experiment: measuring the effects of two catechol oxidase inhibitors, phenylthiourea and benzoic acid
			Analysis in R
4 Differential equations: numerical solutions, model calibration, and sensitivity analysis
	4.1 Biological background
	4.2 Mathematical and R background
		4.2.1 Numerical solutions to differential equations
			Example: logistic growth (one variable)
			Example: tumor growth model (a system of two differential equations)
		4.2.2 Calibration: fitting models to data
			Example: calibrating the logistic growth model
		4.2.3 Sensitivity analysis
			Global sensitivity
			Local sensitivity analysis
			Local sensitivity example: logistic model
		4.2.4 Putting it all together: the dynamics of Ebola virus infecting cells
			Numerical solution for the Ebola model
			Fitting parameters to the Ebola model: calibrating the model
			Local sensitivity analysis: assessing key parameters in the Ebola virus-cell system
	4.3 Case study 1: Modeling the 2009 influenza pandemic
		4.3.1 Background
		4.3.2 The SIR model
			Model simulations
		4.3.3 Cumulative number of cases
		4.3.4 Epidemic threshold
		4.3.5 Public health interventions
		4.3.6 2009 H1N1 influenza pandemic
			Estimating parameters
			Taking action
	4.4 Case study 2: Optimizing immunotherapy in prostate cancer
		4.4.1 Background
		4.4.2 Model formulation
		4.4.3 Model implementation
		4.4.4 Parameter estimation
		4.4.5 Vaccination protocols and model predictions
		4.4.6 Sensitivity analysis
		4.4.7 Simulating other treatment strategies
	4.5 Case study 3: Quorum sensing
		4.5.1 Introduction
		4.5.2 Model formulation
		4.5.3 Parameter estimation
		4.5.4 Model simulations
		4.5.5 Sensitivity analysis
			The temporal response
	4.6 Wet lab: hormones and homeostasis-keeping blood glucose concentrations stable
		4.6.1 Overview of activities
		4.6.2 Introduction to blood glucose regulation and its importance
		4.6.3 Developing a model
		4.6.4 Experiment: measuring blood glucose concentrations following glucose ingestion
			Introduction to the procedure
			Experimental procedure
		4.6.5 Analysis
		4.6.6 Thoughts to consider for potential follow-up experiments
5 Technical notes for laboratory activities
	5.1 Introduction
	5.2 Population growth
		Bacterial growth media and tips
		Optional plate counts and converting optical density to numbers of bacteria
		Alternate organisms
	5.3 Enzyme kinetics
		Tips and solution preparation for catechol oxidase
		Notes on data analysis
			5.3.1 Notes on other enzymes or similar experiments
				Predator-prey or the enzyme game
				Other enzymes
	5.4 Blood glucose monitoring
		5.4.1 Tips for glucose monitoring
			Example of a subject consent form
		5.4.2 Other lab activities
Bibliography
Index