BiRNe: Symbolic bifurcation analysis of reaction networks with Python
Richard Golnik, Thomas Gatter, Peter F. Stadler, Nicola Vassena
TL;DR
BiRNe tackles the computational bottleneck of bifurcation analysis in reaction networks by using a symbolic Jacobian $G=SR$ under parameter-rich monotone kinetics, avoiding explicit fixed-point parametrization. It provides a principled framework based on Child-Selections and CS-matrices to detect stability changes, multistationarity, autocatalysis, and oscillations, applicable to networks of roughly 15 species. The module automates construction of the CS-based spectral signature, builds a Hasse diagram of CB-summands, and reports minimal unstable motifs that certify bifurcation behavior, demonstrated on a 14- metabolite network with dozens of oscillatory cores. This approach enables rigorous motif-based bifurcation analysis for medium-sized networks and offers an open-source tool that complements purely algebraic or numerical methods, with future work aimed at certified bifurcations and broader oscillatory mechanisms.
Abstract
Computer algebra methods for analyzing reaction networks often rely on the assumption of mass-action kinetics, which transform the governing ODEs into polynomial systems amenable to techniques such as Gröbner basis computation and related algebraic tools. However, these methods face significant computational complexity, limiting their applicability to relatively small networks involving only a handful of species. In contrast, building on recent theoretical advances, we introduce here \textsc{BiRNe} (BIfurcations in Reaction NEtworks) Python module, which relies on a symbolic approach designed to detect bifurcations in larger reaction networks (up to 10-20 species, depending on the network's connectivity) equipped with parameter-rich kinetics. This class includes enzymatic kinetics such as Michaelis--Menten, ligand-binding kinetics like Hill functions, and generalized mass-action kinetics. For a given network, the current algorithm identifies all minimal autocatalytic subnetworks and fully characterizes the presence of bifurcations associated with zero eigenvalues, thus determining whether the network admits multistationarity. It also detects oscillatory bifurcations arising from positive-feedback structures, capturing a significant class of possible oscillations.