MthSc 450, Spring 2013

MthSc 450, Spring 2013

"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. Most 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.



Homework 1: pdf | tex. Due Tuesday, January 22nd.
Homework 2: pdf | tex. Due Tuesday, January 29nd.
Homework 3: pdf | tex. Due Tuesday, February 5th.
Homework 4: pdf | tex. Due Tuesday, February 12th.
Homework 5: pdf | tex. Due Tuesday, February 19th.
Homework 6: pdf | tex. Due Tuesday, February 26th.
Homework 7: pdf | tex. Due Tuesday, March 5th.
Homework 8: pdf | tex. Due Thursday, March 14th.

Lecture notes

Part I. Differential and difference equations

1. Introduction to modeling. 4 pages. Updated Jan 22, 2013.
2. Difference equations. 4 pages. Updated Jan 22, 2013.
3. Analyzing nonlinear models . 4 pages. Updated Jan 22, 2013.
4. Models of structured populations. 4 pages. Updated Jan 29, 2013.
5. Nonlinear models of interacting populations. 5 pages. Updated Jan 31, 2013.
6. Infectious disease modeling. 7 pages. Updated Feb 6, 2013.
7. Modeling biochemical reactions. 5 pages. Updated Feb 27, 2013.

Part II. Discrete and agent-based models

1. Boolean and algebraic models of gene regulatory networks.
     A. The lac operon regulatory network in E. coli. 10 pages. Updated Feb 14, 2013.
     B. A Boolean network model of the lac operon. 5 pages. Updated Feb 14, 2013.
     C. Using Gröbner bases to find fixed points. 7 pages. Updated Feb 19, 2013.
     D. Bi-stability and a differential equation model of the lac operon. 8 pages. Updated Mar 1, 2013.
     E. Boolean models of bistable systems. 6 pages. Updated Mar 5, 2013.
     F. Finite dynamical systems and computational algebra preliminaries. 9 pages. Updated Mar 15, 2013.
     G. Reverse engineering of polynomial dynamical systems. 7 pages. Updated Apr 1, 2013.