Exploring Mathematical Modeling in Biology Through Case Studies and Experimental Activities

Description

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. Presents a single volume on mathematics and biological examples, with data and wet lab experiences suitable for non-expertsContains three real-world biological case studies and one wet lab for application of the mathematical modelsIncludes R code templates throughout the text, which are also available through an online repository, along with the necessary data files to complete all projects and labs 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