G. Craciun, J. W. Helton and R. J. Williams

Dynamical system models of complex biochemical reaction networks are usually high-dimensional, nonlinear, and contain many unknown parameters. In some cases the reaction network structure dictates that positive equilibria must be unique for all values of the parameters in the model. In other cases multiple equilibria exist if and only if special relationships between these parameters are satisfied. We describe methods based on homotopy invariance of degree which allow us to determine the number of equilibria for complex biochemical reaction networks and how this number depends on parameters in the model.
Published in Mathematical Biosciences, Volume 216, Issue 2, December 2008, pages 140-149. For the published article, click here. For a copy of a preprint, for personal scientific non-commercial use, click here for pdf.
Software Complement: Mathematica Notebooks for Computing Chemical Reaction Network Jacobians
For many of the examples in this paper, the determinant of the Jacobian of the right hand side of the dynamic equations was computed symbolically using Mathematica. The webpage accessed by clicking here contains Mathematica notebooks for many of the examples in this paper, as well as a demonstration notebook that readers may edit to run their own examples. These notebooks were developed by Igor Klep, Karl Fredrickson and J. William Helton.
Last updated December 26, 2008.