Math 4500, Fall 2017
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
predatorprey 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 foodwebs. 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 crossdisciplinary 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 predatorprey model twopop.m, and the SIR model. Written by
Elizabeth Allman and John Rhodes, authors of Mathematical Models in Biology
 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. 5461.
 MichaelisMenten kinetics
 2minute 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
 CA applet
Another CA applet
Software
 Cellular Automaton Explorer, a free research, teaching, and exploration tool created by David Bahr.
 NetLogo, a multiagent programmable modeling
environment.
 Sage: free opensource 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 webbased software tool for
multistate discrete models of biological networks.
 TURING:
Algorithms for Computational with Finite Dynamical Systems. A crowdsourced 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
 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 predatorprey models. Due Thursday, September 14.
 HW 4: pdf 
tex. Topic:
Predatorprey 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 agentbased models. Due Tuesday, October 10.
HW 7: pdf 
tex. Topic: Gröbner bases and fixed points of Boolean networks. Due Thursday, October 19.
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.
 Predatorprey 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 agentbased models
 Cellular automata and agentbased 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.
Part III. Stochastic models: genetics, nucleic acids and phylogenetics