Exploring Mathematical Modeling in Biology
Through Case Studies and Experimental Activities 

Rebecca Sanft and Anne Walter

Exploring Mathematical Modeling in Biology Through Case Studies and Experimental Activities is designed to actively engage students in the process of modeling through a collection of case studies and wet labs connecting mathematical models to real data. The supporting mathematical, coding and biological background helps readers practice and build confidence in asking questions, formulating mathematical models, articulating model assumptions, estimating parameters, analyzing models, and interpreting the results. Through the cases studies and labs, the reader will see the utility of models for understanding complex systems, making predictions, and identifying further questions.

Contents

Unit 1 begins with an introduction to R, followed by an introduction to basic lab skills. The next three units each include a general biological introduction to motivate the modeling, an introduction to the mathematical and computational concepts, embedded R code, and exercises. The background material is followed by three case studies and a laboratory activity. Case study and lab topics are given below.


Bringing mathematics and biology together through modeling
R basics
Prelab lab: practicing the fundamentals

Case Study 1: Island biogeography
Case Study 2: Pharmacokinetics model
Case Study 3: Invasive plant species
Wet lab: Logistic growth model of bacterial population dynamics


Case Study 1: How leaf decomposition rates vary with anthropogenic nitrogen deposition
Case Study 2: Exploring models to describe tumor growth rates
Case Study 3: Predator responses to prey density vary with temperature
Wet lab: Enzyme kinetics of catechol oxidase


Case Study 1: Modeling the 2009 influenza pandemic
Case Study 2: Optimizing immunotherapy in prostate cancer
Case Study 3: Quorum sensing
Wet lab: Hormones and homeostasis--keeping blood glucose concentrations stable

Practical tips for lab exercises
Suggestions of other systems
Suggestions of in silico lab experiences

Data Files and R Code

R code is embedded throughout the book. All R files can be downloaded here (215 KB), which also contains data files used in various exercises and case studies.

R Markdown Templates and Solutions

In R Markdown, students can generate high quality reports that weave together text and code. R Markdown templates and solutions for selected background exercises and case studies are available upon request (contact bsanft@unca.edu). 

Sample Courses

Math 236: Mathematics of Biology

This course introduces students to the essential modeling techniques of formulation, implentation, validation, and analysis. Students engage in these areas by combining experiment, mathematical theory, statistics, and computation to better understand a wide variety of biological systems. Offered annually in the spring semester. Also counts toward neuroscience and mathematical biology concentrations.

Prerequisite: Calculus II (MATH 126 or MATH 128), and Linear Algebra (MATH 220).

Instructors: Co-taught by Anne Walter (Biology) and Sara Clifton (Mathematics)

More information coming soon!

Math 374: Special Topics: Mathematical Models in Biology

This courses introduces students to the essential modeling techniques of formulation, implementation, validation, and analysis. While these tools can be used to study systems in many disciplines, this course focuses on biological systems. Students actively engage in the process of modeling through a collection of case studies connecting mathematical models to real data.

Prerequisite: Calculus I (MATH 191) is a required prerequisite. It is highly encouraged that you also have one of the following: Experimental Design, Analysis and Presentation (BIOL 134), Quantitative Chemistry Laboratory (CHEM 145), Calculus-Based Statistics (STAT 225), an introductory programming course, or another math course beyond Calculus I.

Instructor: Becky Sanft (Mathematics)

Schedule: The daily schedule and due dates can be found here.

Goals of the course: At the end of the semester, students should be able to

  • Graphically display data and model outputs clearly and accurately.
  • Formulate discrete and differential equation models that represent a range of biological problems, including identifying assumptions that are appropriate for the problem to be solved.
  • Understand and implement least squares approach to estimate model parameters.
  • Simulate model outcomes using iterative approaches and built-in differential equation solvers.
  • Consider alternative models and use the Akaike Information Criterion (AIC) for model selection.
  • Perform sensitivity analysis to study how changes in parameters affect the model output.
  • Gain confidence in using R.
  • Understand how to use mathematical tools to understand complex biological phenomena, predict future events, and recommend interventions.
  • Effectively communicate results across disciplines.

Homework: Background exercises will be assigned to help you learn the mathematical and computational methods, and case studies will also be assigned to provide practice in applying these tools. All assignments require the use of R, and solutions must be in R Markdown.

Papers: There are two short papers assigned during the semester. The directions and templates can be downloaded here (28 MB).

Final Project: Over the latter part of the semester, research groups will determine a biological system to model and a question to investigate. Groups will apply course concepts and engage in all aspects of the modeling process. Data can be found in the literature or a group can design an experiment to gather their own data. The project will include two major components: a written report and presentation. Project guidelines can be accessed here .