Stoichiometric network analysis and graph theoretic methods for studying spatial models of chemical reaction networks
by
Mirela Domijan
Unversity of Warwick

Chemical reaction networks play an important role in understanding the biological processes that take part on a cellular level. However, these networks still continue to be a challenge to model and analyse. If we take a deterministic approach and we assume that the chemicals involved cannot diffuse, the chemical reaction networks are modelled by systems of ODEs. Modelling of chemical interactions can involve many chemicals and complex interactions, resulting in large ODE systems with nonlinear terms. If there is lack of quantitative information about the interactions, there will also be parameter uncertainty in the models. These issues can make it impossible to numerically simulate the system dynamics or to perform bifurcation analysis. In such cases, analysis can be successfully performed via graph theory and stoichiometric network analysis. Analysis via these methods does not depend on the unknown parameters. Instead, they relate system's dynamic properties to network structure and more specifically, to reaction stoichiometry.

Yet, in certain cellular processes reactions may not proceed in "well-mixed" environments and chemicals may diffuse. Then, spatial effects on the chemical dynamics need to be considered and analysis of appropriately-constructed systems of reaction-diffusion PDEs needs to be performed. Diffusion can lead to new behaviors in the PDE systems, such as diffusion-driven (Turing) instability. This instability constitutes the basis of Turing's mechanism for pattern formation. Turing instability has been well described for systems of two chemicals, however, theory is still lacking for large systems. We will present concepts from stoichiometric network analysis and related graph theory that can be applied to analysis of spatial reaction networks. The conditions will address the occurrence of Turing instability. A Turing unstable two-chemical system relies on the activation and inhibition between the two chemicals. The conditions that we will give generalise the conditions of positive/negative feedback cycles for Turing instability.

Date received: May 14, 2008