Sign Vector Conditions in Chemical Reaction Network Theory

A successful completion of the project will extend the applicability of chemical reaction network theory (CRNT) to networks that do not follow mass-action kinetics (MAK). The intended results on dynamical systems arising from networks with generalized mass-action kinetics (GMAK) are not only independent of rate constants, as in classical CRNT, but also robust with respect to kinetic orders, as determined by sign vector conditions. Via dynamical equivalence, the results about networks with GMAK are also significant for networks with MAK to which classical CRNT is not applicable. In terms of generalized polynomial equations, the outcome of the project is relevant for real algebraic geometry and algebraic statistics. In terms of practical applications, it contributes to pharmacokinetics and drug development. For achieving the theoretical goals, methods from dynamical systems, graph theory, polyhedral geometry, and oriented matroids will be combined in a novel way. Throughout the project, efficient algorithms and software for the verification of sign vector conditions will be developed.

In our project, we started a comprehensive analysis of chemical reaction networks with generalized mass-action kinetics and the resulting generalized polynomial dynamical systems. Background: Fundamental cellular functions including signaling, gene regulation, and metabolism involve numerous molecular species interacting via chemical reactions. More than one century of biochemistry and several decades of molecular biology have opened an unprecedented window into the complexity of such chemical reaction networks in living cells. Mathematics has played a pivotal role in coping with the complexity of chemical reaction networks and is a cornerstone of current systems biology. Common modeling frameworks include (deterministic) ordinary differential equations and (stochastic) continuous-time Markov chains. In the deterministic setting, the classical assumption of mass-action kinetics leads to polynomial dynamical systems. All models depend on numerous unknown parameters, the rate constants. Still, there are large classes of networks for which the qualitative dynamics is robust with respect to the model parameters. In particular, for complex-balanced mass-action systems, there exists a unique, (globally) stable positive equilibrium independently of the rate constants. Results: In our project, we extended the applicability of chemical reaction network theory (CRNT) to networks that do not follow mass-action kinetics (MAK). Our results on dynamical systems arising from networks with generalized mass-action kinetics (GMAK) are not only independent of rate constants, as in classical CRNT, but also robust with respect to kinetic orders, as determined by sign vector conditions. Via dynamical equivalence, our results about networks with GMAK are also significant for networks with MAK to which classical CRNT does not apply. Most importantly, we studied existence, uniqueness, and stability of positive complex-balanced equilibria, thereby extending the classical deficiency zero theorem in several ways. Technically, we characterized the bijectivity of generalized polynomial maps as well as the injectivity of classes of maps (for example, monomial, monotonic, or differentiable maps). Moreover, we provided conditions that guarantee the parametrization of all positive equilibria. For achieving our goals, we combined methods from dynamical systems, analysis, graph theory, polyhedral geometry, and oriented matroids in a novel way. Throughout the project, we developed efficient algorithms and software for the computation of sign vectors. In terms of generalized polynomial equations, the outcome of the project is relevant for real (positive) algebraic geometry. In terms of sign vectors, it is relevant for bioinformatics and bioengineering.