"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

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.

- 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, 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
- Analysis of Dynamic Algebraic Models (ADAM), a web-based software tool for multi-state discrete models of biological networks.
- TED talk by Stephen Wolfram:
*The theory of everything*. - Cellular Automaton Explorer, a free research, teaching, and exploration tool created by David Bahr.
- NetLogo, a multi-agent programmable modeling environment.
- Sage: free open-source mathematics software. Homepage | Sage Notebook home | Clemson Sage Notebook server
- Sage worksheet:
*lac*operon Boolean network model - In-class worksheet on reverse engineering
- In-class worksheet on RNA folding
- New Book! Algebraic and Discrete Mathematical Methods for Modern Biology, edited by Raina Robeva.

Homework 2: pdf | tex. Topic:

Homework 3: pdf | tex. Topic:

Homework 4: pdf | tex. Topic:

Homework 5: pdf | tex. Topic:

Homework 6: pdf | tex. Topic:

Homework 7: pdf | tex. Topic:

Homework 8: pdf | tex. Topic:

Homework 9: pdf | tex. Topic:

1. Introduction to modeling. 4 pages. Updated Jan 22, 2013.

2. Difference equations. 4 pages. Updated Jan 22, 2013. [typed version, updated Jan 12, 2015]

3. Analyzing nonlinear models . 4 pages. Updated Jan 22, 2013.

4. Models of structured populations. 4 pages. Updated Jan 29, 2013. [typed version, updated Jan 21, 2015]

5. Predator-prey models. 5 pages. Updated Jan 31, 2013. [typed version, updated Jan 28, 2015]

6. Infectious disease modeling. 7 pages. Updated Feb 6, 2013. [typed version, updated Feb 9, 2015]

7. Modeling biochemical reactions. 5 pages. Updated Feb 27, 2013. [typed version, updated Feb 4, 2015]

1. Cellular automata and agent-based models. Updated February 11, 2015.

2. The

3. A Boolean network model of the

4. Using Gröbner bases to find fixed points. 7 pages. Updated Feb 19, 2013.

5. Bi-stability and a differential equation model of the

6. Boolean models of bistable systems. 6 pages. Updated Mar 5, 2013.

7. Overview: reverse engineering of polynomial dynamical systems 11 pages. Updated Apr 6, 2015.

8. Finite dynamical systems and computational algebra preliminaries. 9 pages. Updated Mar 15, 2013.

1. Combinatorial approaches to RNA folding. 16 pages. Updated April 15, 2015.

2. RNA folding via energy minimization. 15 pages. Updated April 15, 2015.

3. RNA folding via formal language theory. 14 pages. Updated April 15, 2015.