Simple Mathematical Models of Gene Regulatory Dynamics

This is a short and self-contained introduction to the field of mathematical modeling of gene-networks in bacteria. As an entry point to the field, we focus on the analysis of simple gene-network dynamics. The notes commence with an introduction to the deterministic modeling of gene-networks, with extensive reference to applicable results coming from dynamical systems theory. The second part of the notes treats extensively several approaches to the study of gene-network dynamics in the presence of noise―either arising from low numbers of molecules involved, or due to noise external to the regulatory process.  The third and final part of the notes gives a detailed treatment of three well studied and concrete examples of gene-network dynamics by considering the lactose operon, the tryptophan operon, and the lysis-lysogeny switch.  The notes contain an index for easy location of particular topics as well as an extensive bibliography of the current literature. The target audience of these notes are mainly graduates students and young researchers with a solid mathematical background (calculus, ordinary differential equations, and probability theory at a minimum), as well as with basic notions of biochemistry, cell biology, and molecular biology. They are meant to serve as a readable and brief entry point into a field that is currently highly active, and will allow the reader to grasp the current state of research and so prepare them for defining and tackling new research problems.

Michael C. Mackey is a mathematically trained physiologist from McGill University in Montréal who has spent his entire career working in mathematical biology.  He has extensive experience in the modeling and analysis of biologically motivated mathematical models and their use to illuminate biological functioning. 

Moisés Santillán is a Professor of Biomedical Engineering and Physics at the Monterrey Campus of the Center for Research and Advanced Studies of the IPN (Cinvestav), in México. He has BSc, MSc, and PhD degrees in physics, but most of his career has been devoted to tackling biological problems with tools from physics and mathematics. 

Marta Tyran-Kaminska is a faculty member of the Institute of Mathematics at the University of Silesia in Katowice, Poland. Her mathematical research has been in dynamical systems and stochastic processes. 

Eduardo Santillan Zeron is a mathematician working at the Centre for Research and Advanced Studies (Cinvestav) of the IPN in México City, México. His Doctoral Thesis was on the area of general topology and his research interests are: several complex variables, probability and theoretical biology.