Good and bad children in metabolic networks

Nicola Vassena; Nicola Vassena

doi:10.3934/mbe.2020388

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 7621-7644. doi: 10.3934/mbe.2020388

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Research article

Good and bad children in metabolic networks

Nicola Vassena ^,

Department of Mathematics, Free University of Berlin, German

Received: 20 July 2020 Accepted: 14 October 2020 Published: 03 November 2020

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.
- metabolic networks,
- structural analysis,
- Jacobian determinant,
- multistationarity,
- stability change,
- saddle-node bifurcation
Citation: Nicola Vassena. Good and bad children in metabolic networks[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7621-7644. doi: 10.3934/mbe.2020388

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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.

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