Lumping of reaction networks: Generic and critical parameters
Abstract
We investigate linear lumping for parameter-dependent mass action reaction networks, distinguishing between generic and critical parameter regimes. For generic parameters -- those ranging in some non-empty open subset of parameter space -- we prove that exact linear lumping yields only "obvious" reductions: elimination of non-reactant species or projections along stoichiometric first integrals. This characterization extends to reaction networks with product-form kinetics, including Michaelis-Menten and Hill-type rate laws. For mass action systems we proceed to develop an algorithmic approach to identify critical parameter sets -- algebraic subvarieties in parameter space where non-trivial lumpings become available. This procedure reduces the determination of lumping maps to a system of finitely many polynomial equations. It also applies to constrained lumping scenarios (which are frequently motivated by chemical considerations). We then review and extend results about proper lumpings. Finally, we discuss lumpings of a self-replicator system, and of a two-pathway enzyme mechanism, to document the viability of our methods in relevant scenarios. Our results clarify the relationship between structural (parameter-independent) and fine-tuned (parameter-dependent) reductions, with implications for approximate lumping when system parameters lie near critical values
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BibTeX
@misc{eilertsen2026lumping,
title = {{Lumping of reaction networks: Generic and critical parameters}},
author = {Justin Eilertsen and Valery G. Romanovski and Santiago Schnell and Sebastian Walcher},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.28895},
eprint = {2606.28895},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.28895},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lumping of reaction networks: Generic and critical parameters
AU - Justin Eilertsen
AU - Valery G. Romanovski
AU - Santiago Schnell
AU - Sebastian Walcher
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2606.28895
UR - https://arxiv.org/abs/2606.28895
AB - We investigate linear lumping for parameter-dependent mass action reaction networks, distinguishing between generic and critical parameter regimes. For generic parameters -- those ranging in some non-empty open subset of parameter space -- we prove that exact linear lumping yields only "obvious" reductions: elimination of non-reactant species or projections along stoichiometric first integrals. This characterization extends to reaction networks with product-form kinetics, including Michaelis-Menten and Hill-type rate laws. For mass action systems we proceed to develop an algorithmic approach to identify critical parameter sets -- algebraic subvarieties in parameter space where non-trivial lumpings become available. This procedure reduces the determination of lumping maps to a system of finitely many polynomial equations. It also applies to constrained lumping scenarios (which are frequently motivated by chemical considerations). We then review and extend results about proper lumpings. Finally, we discuss lumpings of a self-replicator system, and of a two-pathway enzyme mechanism, to document the viability of our methods in relevant scenarios. Our results clarify the relationship between structural (parameter-independent) and fine-tuned (parameter-dependent) reductions, with implications for approximate lumping when system parameters lie near critical values
ER -