# Good and bad children in metabolic networks

## TL;DR

This paper investigates how the Jacobian determinant's sign in metabolic networks influences equilibrium bifurcations, introducing Child Selections to classify reactions affecting stability and bifurcation.

## Contribution

It introduces the concept of Child Selections to analyze Jacobian sign stability in metabolic networks, distinguishing between reactions that preserve or change sign.

## Key findings

- Identifies 'good' Child Selections with constant Jacobian sign.
- Defines 'bad' Child Selections that lead to sign changes and bifurcations.
- Provides a framework for analyzing stability in metabolic networks.

## Abstract

Equilibrium bifurcations arise from sign changes of Jacobian determinants, as parameters are varied. Therefore we address the Jacobian determinant for metabolic networks with general reaction kinetics. Our approach is based on the concept of Child Selections: each (mother) metabolite is mapped, injectively, to one of those (child) reactions that it drives as an input. Our analysis distinguishes reaction network Jacobians with constant sign from the bifurcation case, where that sign depends on specific reaction rates. In particular we distinguish "good" Child Selections, which do not affect the sign, from more interesting and mischievous "bad" children, which gang up towards sign changes, instability, and bifurcation.