Math 4500, Spring 2025

"Mathematics is biology's next microscope, only better; Biology is mathematics' next physics, only better." --Joel E. Cohen

"All models are wrong, but some are useful." --George E. P. Box

About the class

This class will be an introduction to mathematical modeling with a particular focus on mathematical biology. We will sample from a variety of problems and modeling techniques throughout the class. Unlike most undergraduate math classes, the scope of this class will be more about breadth than depth.

The course will be roughly divided into thirds: The first third will cover continuous models such as differential equations and difference equations, covering topics such as logistic, predator-prey, and infectious disease modeling. The second third will cover discrete models such as cellular automata, agent-based models, and Boolean models of gene networks. The last third will cover stochastic models, such as phylogentics, RNA folding, and hidden Markov models in genomics.

Resources

Course Syllabus
Course Schedule
Mathematical Contest in Modeling
RadioLab podcasts: Patient Zero
Webpage on the logistic map with cobwebbing applet.
Simple mathematical models with very complicated dynamics, by Robert May, published in Nature, 1976
Animated gif of cobwebbing in the logistic map. Compare to the bifurication diagram. Both of these from Wikipedia
Blog post by Geoff Boeing about chaos theory and the logistic map.
Webpage for to book Modeling infectious diseases by Matt Keeling and Pejman Rohani, with online resources such as MATLAB files.
Michaelis-Menten kinetics
John Conway's Game of Life segment, from Stephen Hawking's The Meaning of Life.
TED talk by Stephen Wolfram: The theory of everything.
Final project ideas.

Software

NetLogo, a multi-agent programmable modeling environment.
The "Cyclone" program for computing the phase space of an algebraic model: Web version | GitHub.
Macaulay2: online software for computational algebraic geometry and commutative algebra. Can be downloaded or run online.
GINsim (Gene Interaction Network simulation), a computer tool for modeling and simulation of Boolean and logical networks.
Conway's Game of Life applet.

Code

MATLAB files for difference equations: cobwebbing (cobweb.m), a single species population model (onepop.m) the predator-prey model (twopop.m), and the SIR model (sir.m). Written by Elizabeth Allman and John Rhodes, authors of Mathematical Models in Biology.
MATLAB files for epidemic models, from Chapter 2 of Modeling infectious diseases by Matt Keeling and Pejman Rohani, with online resources such as MATLAB files: Programs 2.1, 2.2, 2.5, and 2.6.
BoolNet
- The 10-variable lac operon model of Stigler/Veliz-Cuba (2011).
- The 10-variable mammalian cell cycle model of Fauré et al. (2006).
Cyclone
- The lac operon toy model with 5 variables, including 2 parameters, from Chapter 1 of Robeva/Hodge (2013).
- Flawed lac operon model with 5 variables from Chapter 1 of Robeva/Hodge (2013).
- The lac operon model with 13 variables (including 3 parameters) of Stigler/Veliz-Cuba (2011).
- The tna operon model with 10 variables (including 3 parameters) of Deal et al. (2023).
Macaulay2
- The 9-variable lac operon model from Chapter 1 in Robeva/Hodge (2013).
- The 13-variable lac operon model (10 vars, 3 parameters) of Stigler/Veliz-Cuba (2011).

Homework

HW 1: pdf | tex. Topic: Differential equation models. Due Friday, January 24, 2025.
HW 2: pdf | tex. Topic: Difference equations, the logistic map, population models and linearization. Due Tuesday, February 4.
HW 3: pdf | tex. Topic: Models of structured populations. Due Tuesday, February 11.
HW 4: pdf | tex. Topic: Models of interacting populations. Due Tuesday, February 18.
HW 5: pdf | tex. Topic: Infectious disease modeling. Due Wednesday February 26.
HW 6: pdf | tex. Topic: Chemical reaction networks. Due Friday, March 7.
HW 7: pdf | tex. Topic: Cellular automata and agent-based models. Due Monday, March 31.
HW 8: pdf | tex. Topic: Boolean models of molecular networks. Due Friday, April 18.

Lecture notes

These are the versions from a previous time that I taught this class, and they will likely be improved and updated as we get to them throughout the semester.

Part I. Differential and difference equations

Introduction to modeling. 3 pages (handwritten). Updated Jan 23, 2022.
Difference equations. 12 pages. Updated Jan 20, 2022.
Analyzing nonlinear models. 4 pages (handwritten). Updated Jan 22, 2013. (will replace soon)
Models of structured populations. 8 pages. Updated Jan 21, 2015.
Predator-prey models. 11 pages. Updated Jan 28, 2015.
Infectious disease modeling. 12 pages. Updated Feb 9, 2015.
Modeling biochemical reactions. 17 pages. Updated Feb 20, 2025.
Gene regulation by operons. 16 pages. Updated Feb 20, 2025.
Delay differential equation models of gene regulation. 29 pages. Updated Feb 27, 2025.

Part II. Discrete and agent-based models

Cellular automata and agent-based models. 18 pages. Updated February 11, 2015.
Basics of Boolean modeling. 18 pages. Updated Feb 27, 2025.
Boolean models of molecular networks. 14 pages. Updated Feb 27, 2025.
Advanced features of Boolean models. 26 pages. Updated Mar 25, 2025.
Reduction of Boolean network models. 21 pages. Updated Mar 25, 2025.

Part III. Stochastic models: genetics, nucleic acids and phylogenetics

CpG islands and hidden Markov models. 12 pages. Updated October 28, 2016.
Hidden Markov models and dynamic programming. 12 pages. Updated April 19, 2017.
Combinatorial approaches to RNA folding. 16 pages. Updated April 15, 2016.
RNA folding via energy minimization. 15 pages. Updated April 15, 2016.
RNA folding via formal language theory. 14 pages. Updated April 15, 2016.