Math 4500, Fall 2016

"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.

We will begin with some classical models such as the logistic and predator-prey models for population growth and the SIR model in epidemiology. The second half of the class will be spent learning about a relatively new but widely popular trend of discrete modeling. In particular, the field of mathematical biology has been transformed over the past 15 years by researchers using novel tools from discrete mathematics and computational algebra to tackle old and new problems. These ideas have impacted a wide range of topics such as gene regulatory networks, RNA folding, genomics, infectious disease modeling, phylogenetics, and ecology networks and food-webs. In some cases they have even spawned completely new research areas. This is approach is arguably more accessible and appealing to many scientists and engineers, encouraging cross-disciplinary communication and collaborations.

Resources

Course Syllabus
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
MATLAB files for cobwebbing, a single species population model the predator-prey model, and the SIR model. Written by Elizabeth Allman and John Rhodes, authors of Mathematical Models in Biology
If smallpox strikes Portland C.L. Barrett, S.G. Eubank, J.P. Smith. Scientific American, Vol. 292 (2005), pp. 54-61.
Michaelis-Menten kinetics
2-minute video on gene expression
John Conway's Game of Life segment, from Stephen Hawking's The Meaning of Life.
Game of Life applet
Cellular Automaton Explorer, a free research, teaching, and exploration tool created by David Bahr.
TED talk by Stephen Wolfram: The theory of everything.
NetLogo, a multi-agent programmable modeling environment.
Biological Feedback (book, pdf version), by René Thomas and Thomas D'Ari, 1990 (updated 2006).
Analysis of Dynamic Algebraic Models (ADAM), a web-based software tool for multi-state discrete models of biological networks.
text file of Boolean lac operon files in polynomial form, for easy entry into ADAM.
Sage: free open-source mathematics software. Homepage | SageMathCloud
Sage worksheet: lac operon Boolean network model
The CpG Educate suite
Final project ideas.

Homework

Homework 1: pdf | tex. Topic: Difference and differential equation models. Due Wednesday, August 24.
Homework 2: pdf | tex. Topic: Population models. Due Monday, August 29.
Homework 3: pdf | tex. Topic: Models of structured populations. Due Monday, September 5.
Homework 4: pdf | tex. Topic: Infecious disease models. Due Friday, September 9.
Homework 5: pdf | tex. Topic: Biochemical reaction networks. Due Wednesday, September 14.
Homework 6: pdf | tex. Topic: Cellular automata and agent-based models. Due Wednesday, September 21.
Homework 7: pdf | tex. Topic: Gröbner bases and fixed points of Boolean networks. Due Wednesday, September 28.
Homework 8: pdf | tex. Topic: Bistability, degradation, and time-delays in Boolean networks. Due Wednesday, October 5.
Homework 9: pdf | tex. Topic: Reduction of Boolean networks. Due Wednesday, October 12.
Homework 10: pdf | tex. Topic: Reverse engineering using computational algebra. Due Wednesday, October 19.

Lecture notes

Part I. Differential and difference equations

1. Introduction to modeling. 4 pages (handwritten). Updated Jan 22, 2013.
2. Difference equations. 12 pages. Updated Jan 12, 2015.
3. Analyzing nonlinear models. 4 pages (handwritten). Updated Jan 22, 2013.
4. Models of structured populations. 8 pages. Updated Jan 21, 2015.
5. Predator-prey models. 11 pages. Updated Jan 28, 2015.
6. Infectious disease modeling. 12 pages. Updated Feb 9, 2015.
7. Modeling biochemical reactions. 10 pages. Updated Feb 4, 2015.

Part II. Discrete and agent-based models

1. Cellular automata and agent-based models. 18 pages. Updated February 11, 2015.
2. Boolen network models of gene regulatory networks. 22 pages. Updated February 4, 2016.
3. The lac operon in E. coli. 25 pages. Updated Feb 11, 2016.
4. Bistability and a differential equation model of the lac operon. 23 pages. Updated Feb 12, 2016.
5. Bistability in Boolean network models. 18 pages. Updated Feb 12, 2016.
6. Reduction of Boolean network models. 18 pages. Updated Feb 18, 2016.
7. Reverse engineering using computational algebra 28 pages. Updated March 4, 2016.
8. Finite dynamical systems and computational algebra. 9 pages (handwritten). Updated Mar 15, 2013.
9. Asynchronous Boolean models of signaling networks. 14 pages. Updated Mar 1, 2016.

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

1. CpG islands and hidden Markov models. 12 pages. Updated October 28, 2016.
2. Combinatorial approaches to RNA folding. 16 pages. Updated April 15, 2015.
3. RNA folding via energy minimization. 15 pages. Updated April 15, 2015.
4. RNA folding via formal language theory. 14 pages. Updated April 15, 2015.