Sign vector conditions in chemical reaction network theory

Mathematics has played a pivotal role in coping with the complexity of biochemical networks and is a cornerstone of current systems biology. In the proposed project, we study chemical reaction networks with generalized mass-action kinetics (GMAK) and the resulting dynamical systems. In particular, we are interested in positive steady states and their stability. Thereby, we extend the applicability of chemical reaction network theory to networks that do not follow mass-action kinetics (MAK), namely to power-law dynamical systems. The intended results on dynamical systems arising from networks with GMAK are not only independent of rate constants, as in the classical theory, but also robust with respect to perturbations of the kinetic orders, as determined by sign vector conditions. Moreover, via dynamical equivalence, our results are significant for networks with MAK to which the classical theory is not applicable. Technically, we are interested in positive steady states called complex-balanced equilibria and toric steady states, which are given by binomial equations and allow a monomial parametrization. A parametrization of the set of positive steady states is often assumed in the study of multistationarity and absolute concentration robustness. Hence, our intended results can be applied to biochemical networks from the literature (either directly or via dynamical equivalence). In terms of real algebraic geometry, we study positive solutions to systems of generalized polynomial equations and inequalities. Positive solutions to polynomial equations and inequalities are relevant for scientists and engineers in many fields, from algebraic statistics and control theory to economics and robotics. The classical theory assumes fixed integer exponents, but considers perturbations of the coefficients (as in Descartes` rule of signs). As our major innovation, we allow real exponents and consider perturbations of both coefficients and exponents. Our recent results can be seen as first multivariate generalizations of Descartes` rule of signs. We continue to investigate the implications of our results (in chemical reaction network theory) for "generalized" real algebraic geometry.

In this follow-up project, we continued our comprehensive analysis of reaction networks with generalized mass-action kinetics and the resulting dynamical systems. In particular, we studied positive steady states and their stability. Background: Reaction networks are a modeling framework used in several fields of chemistry, in areas of biology such as ecology and epidemiology, and even in economics and engineering. In fact, every polynomial/power-law dynamical system (with integer/real exponents) can be written as a reaction network with (generalized) mass-action kinetics (MAK/GMAK). Typically, such models depend on numerous unknown parameters such as "rate constants". Still, under the classical assumption of MAK, there are large classes of networks for which the qualitative dynamics does not depend on the model parameters. Most importantly, if the network is (weakly) reversible and has "deficiency" zero, then - for all rate constants - there exists a unique, stable, "complex-balanced" positive equilibrium in every forward invariant set. In previous work, we had extended this deficiency zero theorem to GMAK, in particular, we had characterized unique existence of a complex-balanced equilibrium (in every invariant set, for all rate constants) in terms of sign vector conditions (relating stoichiometric coefficients and kinetic orders). Results: In this project, we addressed the stability of complex-balanced equilibria and the non-existence of other steady states. In particular, we provided new sign vector conditions that guarantee the linear stability of all complex-balanced equilibria for all rate constants. Technically, our results are based on a new decomposition of the graph Laplacian (and a decomposition of the state space into "strata", that is, regions with given monomial evaluation order). Alternatively, we used cycle decomposition of the graph. In order to address the unique existence of "toric" steady states (not necessarily complex-balanced equilibria) and the existence of steady states (without uniqueness), we studied positive zeros of parametrized systems of generalized polynomial equations (and even inequalities) in abstract terms (independently of reaction networks). Indeed, we established the groundwork for a novel approach to "positive algebraic geometry". First, we identified the crucial geometric objects of generalized multivariate polynomials, namely the "coefficient polytope" and two linear subspaces representing monomial differences and dependencies. Second, we showed that (parametrized systems of generalized) polynomial equations can be written as binomial (!) equations on the coefficient polytope, depending on monomials in the parameters. Our results allow significant contributions to real fewnomial theory, and they further extend the applicability of reaction network theory (from MAK to GMAK). Finally, we developed algorithms for the computation of elementary vectors and sign vectors. I particular, we can now check the conditions for the existence of a unique complex-balanced equilibrium, as provided by our generalized deficiency zero result.