Abstract
We present conditions which guarantee a parametrization of the set of positive equilibria of a generalized mass-action system. Our main results state that (1) if the underlying generalized chemical reaction network has an effective deficiency of zero, then the set of positive equilibria coincides with the parametrized set of complex-balanced equilibria and (2) if the network is weakly reversible and has a kinetic deficiency of zero, then the equilibrium set is nonempty and has a positive, typically rational, parametrization. Via the method of network translation, we apply our results to classical mass-action systems studied in the biochemical literature, including the EnvZ–OmpR and shuttled WNT signaling pathways. A parametrization of the set of positive equilibria of a (generalized) mass-action system is often a prerequisite for the study of multistationarity and allows an easy check for the occurrence of absolute concentration robustness, as we demonstrate for the EnvZ–OmpR pathway.
Highlights
Networks of biochemical reactions can be represented as directed graphs where the vertices are combinations of interacting species and the edges are the reactions
A generalized mass-action system for which the underlying network is weakly reversible and has deficiency zero is known to have an equilibrium set with a monomial parametrization (Müller and Regensburger 2012, 2014; Johnston 2014)
The equilibrium set Xk is determined by setting the right-hand sides of the ODEs (12) to zero, whereas the set Zk of complex-balanced equilibria (CBE) is determined by the Laplacian matrix
Summary
Networks of biochemical reactions can be represented as directed graphs where the vertices are combinations of interacting species (so-called complexes) and the edges are the reactions. We develop a method for explicitly constructing positive, typically rational, parametrizations of the set of positive equilibria for a broad class of biochemical reaction networks. A generalized mass-action system for which the underlying network is weakly reversible and has deficiency zero is known to have an equilibrium set with a monomial parametrization (Müller and Regensburger 2012, 2014; Johnston 2014). Ciency of zero, the corresponding generalized mass-action system permits a positive parametrization of the set of positive equilibria This parametrization can be computed by linear algebra techniques and does not require tools from algebraic geometry such as Gröbner bases. 2, we review the relevant terminology regarding generalized chemical reaction networks and introduce several new notions, including effective and phantom edges, parametrized sets of equilibria, condensed networks, and effective deficiency.
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