Math 4500, Fall 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
The Circada's Love Affair with Prime Numbers, from the New Yorker.
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.
TED talk by Stephen Wolfram: The theory of everything.
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.
Game of Life applet
A cellular automaton applet, and another one.

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.
text file of Boolean lac operon files in polynomial form, for easy entry into ADAM or TURING.
Macaulay2 files for a 10-variable lac operon model and reverse-engineering a 3-node Boolean model

Sage worksheet of the lac operon Boolean model

Homework

HW 1: pdf | tex. Topic: Difference and differential equation models. Due Thursday, August 31.
HW 2: pdf | tex. Topic: Population models and linearization. Due Thursday, September 7.
HW 3: pdf | tex. Topic: Models of structured populations and predator-prey models. Due Thursday, September 14.
HW 4: pdf | tex. Topic: Predator-prey models and infecious disease models. Due Tuesday, September 26.
HW 5: pdf | tex. Topic: Biochemical reaction networks. Due Tuesday, October 3.
HW 6: pdf | tex. Topic: Cellular automata and agent-based models. Due Tuesday, October 10.
HW 7: pdf | tex. Topic: Gröbner bases and fixed points of Boolean networks. Due Thursday, October 19.
HW 8: pdf | tex. Topic: Bistability, degradation, and time-delays in Boolean networks. Due Tuesday, November 1.
HW 9: pdf | tex. Topic: Reduction of Boolean networks. Due Tuesday, November 7.
HW 10: pdf | tex. Topic: Reverse engineering using computational algebra. Due Thursday, November 16.

Lecture notes

Part I. Differential and difference equations

Introduction to modeling. 4 pages (handwritten). Updated Jan 22, 2013.
Difference equations. 12 pages. Updated Jan 12, 2015.
Analyzing nonlinear models. 4 pages (handwritten). Updated Jan 22, 2013.
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. 10 pages. Updated Feb 4, 2015.

Part II. Discrete and agent-based models

Cellular automata and agent-based models. 18 pages. Updated February 11, 2015.
Boolean models of the lac operon in E. coli. 43 pages. Updated February 8, 2017.
Bistability in ODE and Boolean network models. 28 pages. Updated Mar 09, 2017.
Dilution, degradation, and time delays in Boolean network models. 18 pages. Updated Mar 08, 2017.
Reduction of Boolean network models. 18 pages. Updated Oct 31, 2017.
Reverse engineering using computational algebra 29 pages. Updated Nov 10, 2017.

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.