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This mathematical study develops alternative systems of equations to analyze positive equilibria in mass action networks, which model chemical reaction systems. The authors show that these reformulated equations maintain a one-to-one correspondence with the original equilibria while being computationally simpler in many cases. The approach enables improved characterization of equilibrium properties, including bounds on the number of equilibria, identification of regions where multiple stable states exist, and analysis of system bifurcations, with enhanced results for quadratic networks commonly used in applications.
Why it matters
This work provides mathematicians and computational biologists with more efficient tools for analyzing chemical and biochemical reaction networks, potentially accelerating research in systems biology, pharmacology, and chemical engineering where understanding equilibrium behavior is critical for predicting system dynamics and designing interventions.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: Any mass action network gives rise to a parameterised family of polynomial equations whose positive solutions are the positive equilibria of the network. Here, we consider alternative systems of equations, whose solutions are in smooth, one-to-one correspondence with positive equilibria of the network, and capture degeneracy or nondegeneracy of the corresponding equilibria. The construction leads us to consider partitions of networks in a natural sense, and we explore the implications of choosing different partitions. The alternative systems are in some situations simpler than the original mass action equations, which allows us to rapidly identify various algebraic and geometric properties of the positive equilibrium set. This includes the characterisation of toricity and local toricity, bounds on the number of positive nondegenerate equilibria on stoichiometric classes, semialgebraic descriptions of the parameter regions for multistationarity, and the study of bifurcations. After discussing the construction of the alternative systems, various consequences for particular classes of networks and numerous examples are presented. We also develop additional techniques specifically for quadratic networks, the most common class of networks in applications, and use these techniques to derive strengthened results for quadratic networks.
Source: Positive equilibria in mass action networks: geometry and bounds