# Uncovering bifurcation behaviors of biochemical reaction systems from network topology

## TL;DR

This paper introduces a method to study how biochemical networks produce different cell states by analyzing their structure, applied to macrophage polarization.

## Contribution

A new method called Structural Bifurcation Analysis (SBA) is introduced to study bifurcation behaviors from network topology alone.

## Key findings

- SBA identifies substructures and parameters responsible for bifurcation behaviors in biochemical networks.
- The method was applied to macrophage polarization, revealing how network structure constrains polarization patterns.
- Gene deletions like SOCS3 are predicted to cause intermediate polarization states, offering testable hypotheses.

## Abstract

The regulation of biological functions is achieved through the modulation of biochemical reaction network dynamics. The diversity of cell states and the transitions between them have been interpreted as bifurcations in these dynamics. However, due to the complexity of networks and limited knowledge of reaction kinetics, bifurcation behaviors in biological systems remain largely underexplored. To address this, we developed a mathematical method, Structural Bifurcation Analysis (SBA), which decomposes the system into substructures and determines important aspects of bifurcation behaviors—such as substructures responsible for bifurcation conditions, bifurcation-inducing parameters, and bifurcating variables—solely from network topology. We establish a direct relationship between SBA and classical bifurcation analysis, enabling the study of systems even in the presence of conserved quantities. Additionally, we provide a step-by-step bifurcation analysis for general use. We applied our method to the macrophage M1/M2 polarization system. Our analysis reveals that the network structure strongly constrains possible patterns of polarization. We also clarify the dependency of the M1/M2 balance on gene expression levels and predict the emergence of intermediate polarization patterns under gene deletions, including SOCS3, which are experimentally testable.