Math 4500, Spring 2017

"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 onepop.m the predator-prey model twopop.m, 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
TED talk by Stephen Wolfram: The theory of everything.
text file of Boolean lac operon files in polynomial form, for easy entry into ADAM or TURING.
Sage worksheet: lac operon Boolean network model
Final project ideas.

Software

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 | SageMathCloud
Macaulay2: online software for computational algebraic geometry and commutative algebra. Can be downloaded or run online.
Analysis of Dynamic Algebraic Models (ADAM), a web-based software tool for multi-state discrete models of biological networks.
TURING: Algorithms for Computational with Finite Dynamical Systems. A crowd-sourced platform that is replacing ADAM, currently still in beta.
GINsim (Gene Interaction Network simulation), a computer tool for modeling and simulation of Boolean and logical networks.

Homework

Homework 1: pdf | tex. Topic: Difference and differential equation models. Due Wednesday, January 18.
Homework 2: pdf | tex. Topic: Population models. Due Friday, January 27.
Homework 3: pdf | tex. Topic: Models of structured populations. Due Friday, February 3.
Homework 4: pdf | tex. Topic: Infecious disease models. Due Wednesday, February 8.
Homework 5: pdf | tex. Topic: Biochemical reaction networks. Due Wednesday, February 15.
Homework 6: pdf | tex. Topic: Cellular automata and agent-based models. Due Wednesday, February 22.
Homework 7: pdf | tex. Topic: Gröbner bases and fixed points of Boolean networks. Due Monday, March 6.
Homework 8: pdf | tex. Topic: Bistability, degradation, and time-delays in Boolean networks. Due Wednesday, March 15.
Homework 9: pdf | tex. Topic: Reduction of Boolean networks. Due Thursdsay, March 30.
Homework 10: pdf | tex. Topic: Reverse engineering using computational algebra. Due Wednesday, April 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. Boolean models of the lac operon in E. coli. 43 pages. Updated February 8, 2017.
3. Boolean networks, local models, and finite polynomial dynamical systems. 15 pages. Updated February 27, 2017.
4. Networks in systems biology. 15 pages. Updated February 20, 2017.
5. Bistability in ODE and Boolean network models. 28 pages. Updated Mar 09, 2017.
6. Dilution, degradation, and time delays in Boolean network models. 18 pages. Updated Mar 08, 2017.
7. Reduction of Boolean network models. 18 pages. Updated Feb 18, 2016.
8. Reverse engineering using computational algebra 29 pages. Updated March 4, 2016.

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

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