Algebraic systems biology

Algebraic systems biology, AIMS South Africa, April 2025

Course schedule

Lecture notes

Part 0. What is algebraic biology?
Part I. Biochemical reactions, gene regulation, and ODE models
1. Chemical reaction networks. 17 pages. Updated Aug 18, 2024.
2. Gene regulation by operons. 15 pages. Updated Sept 5, 2024.
3. Delay differential equation models of gene regulation. 29 pages. Updated Sept 11, 2024.
Part II. Boolean models of molecular networks
1. Basics of Boolean modeling. 17 pages. Updated Sept 16, 2024.
2. Fixed points of Boolean models. 14 pages. Updated Sept 18, 2024
3. Advanced features in Boolean models. 26 pages. Updated Sept 23, 2024.
4. Reduction of Boolean models. 21 pages. Updated Oct 4, 2024.
Part III. Algebraic models and methods
1. Algebraic models and finite dynamical systems. 45 pages. Updated Oct 18, 2024.
2. The network inference problem. 48 pages. Updated Nov 14, 2024.
3. The model space. 28 pages. Updated Oct 30, 2024.
Part IV. Dynamics of Boolean models
1. Random Boolean networks. 33 pages. Updated Nov 18, 2024.
2. Biological feedback. 38 pages. Updated Dec 2, 2024.

Homework

HW 1: pdf | tex. Due Saturday 26 April, 2025.
HW 2: pdf | tex. Due Saturday 3 May, 2025.

Books

Mathematical Concepts and Methods in Modern Biology: using modern discrete models, edited by Raina Robeva and Terrell Hodge. Academic Press, 2013. Freely available with ScienceDirect subscription.
Algebraic and Discrete Mathematical Methods for Modern Biology, edited by Raina Robeva. Academic Press, 2015. Freely available with ScienceDirect subscription.
Algebraic and Combinatorial Computational Biology, edited by Raina Robeva and Matthew Macauley. Academic Press, 2018. Freely available with ScienceDirect, subscription.
Probabilistic Boolean Networks: The Modeling and Control of Gene Regulatory Networks, by Ilya Shmulevich and Edward R. Dougherty. SIAM, 2010.
Biological Feedback, by René Thomas and Thomas D'Ari. CRC Press, 1990 (updated 2006).
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, by David A. Cox, John Little, and Donal O'Shea. Spring, 2015 (4th edition).

Papers

M. Macauley and N. Youngs. The case for algebraic biology: from research to education. Bull. Math. Biol., 82(115), 2020.
M. Macauley and R. Robeva. Algebraic models, pseudomonomials, and inverse problems in algebraic biology. Lett. Biomath. 7(1):81-104, 2020.
B. Sturmfels. Can biology lead to new theorems? Annual report of the Clay Mathematics Institute, 2005, 13-26.
A. Veliz-Cuba and B. Stigler. Boolean models can explain bistability in the lac operon. J. Comput. Biol., 2011, 783-794.

Software

The "Cyclone" program for computing the phase space of an algebraic model: Web version | GitHub.

Macaulay2: free open-source software for computational algebraic geometry and commutative algebra. Can be downloaded or run online.

Analyze Boolean models with the BoolNet library [paper | documentation | vignette] in RStudio.

Singular: free open-source computer algebra system for polynomial computations, with special emphasis on commutative and non-commutative algebra, algebraic geometry, and singularity theory. Can be downloaded or run online.

Sage: free open-source mathematics software. Can be downloaded or run online.

GINsim (Gene Interaction Network simulation), a computer tool for modeling and simulation of Boolean and logical networks.

MaBoSS: Markovian Boolean Stochastic Simulator. C++ software for simulating continuous/discrete time Markov processes on a Boolean network.

Code

Matlab
- Starter code for Robeva/Hodge Chapter 2 Exercise 2.4.
- Starter code for Robeva/Hodge Chapter 2 Exercise 2.5.
- Code for difference equations: a (single species population model), and for cobwebbing. Written by Elizabeth Allman and John Rhodes, authors of Mathematical Models in Biology.
BoolNet
- The 10-variable lac operon model of Stigler/Veliz-Cuba (2011).
- The 10-variable mammalian cell cycle model of Fauré et al. (2006).
Cyclone
- The lac operon toy model with 5 variables, including 2 parameters, from Chapter 1 of Robeva/Hodge (2013).
- Flawed lac operon model with 5 variables from Chapter 1 of Robeva/Hodge (2013).
- The lac operon model with 13 variables (including 3 parameters) of Stigler/Veliz-Cuba (2011).
- The tna operon model with 10 variables (including 3 parameters) of Deal et al (2023).
Macaulay2
- The 9-variable lac operon model from Chapter 1 in Robeva/Hodge (2013).
- The 13-variable lac operon model (10 vars, 3 parameters) of Stigler/Veliz-Cuba (2011).

Important

Fundamental axioms of mathematics, by Federico Ardilla
The Lesson of Grace in Teaching, by Francis Su. Also available as a YouTube lecture.