Linear elimination in chemical reaction networks

Meritxell Sáez; Elisenda Feliu; Carsten Wiuf

doi:10.1007/978-3-030-00341-8_11

Linear elimination in chemical reaction networks

Meritxell Sáez, Elisenda Feliu, Carsten Wiuf^*

^*Corresponding author for this work

Department of Mathematical Sciences

Abstract

We consider dynamical systems arising in biochemistry and systems biology that model the evolution of the concentrations of biochemical species described by chemical reactions. These systems are typically confined to an invariant linear subspace of ℝⁿ. The steady states of the system are solutions to a system of polynomial equations for which only non-negative solutions are of interest. Here we study the set of non-negative solutions and provide a method for simplification of this polynomial system by means of linear elimination of variables. We take a graphical approach. The interactions among the species are represented by an edge labelled graph. Subgraphs with only certain labels correspond to sets of species concentrations that can be eliminated from the steady state equations using linear algebra. To assess positivity of the eliminated variables in terms of the non-eliminated variables, a multigraph is introduced that encodes the connections between the eliminated species in the reactions. We give graphical conditions on the multigraph that ensure the eliminated variables are expressed as positive functions of the non-eliminated variables. We interpret these conditions in terms of the reaction network. The results are illustrated by examples.

Original language	English
Title of host publication	Recent Advances in Differential Equations and Applications
Editors	Juan Luis García Guirao, José Alberto Murillo Hernández, Francisco Periago Esparza
Number of pages	17
Publisher	Springer
Publication date	2019
Pages	177-193
DOIs	https://doi.org/10.1007/978-3-030-00341-8_11
Publication status	Published - 2019
Event	25th Congress on Differential Equations and Applications / 15th Congress on Applied Mathematics - Cartagena, Spain Duration: 26 Jun 2017 → 30 Jun 2017

Conference

Conference	25th Congress on Differential Equations and Applications / 15th Congress on Applied Mathematics
Country/Territory	Spain
City	Cartagena
Period	26/06/2017 → 30/06/2017

Series	SEMA SIMAI Springer Series
Volume	18
ISSN	2199-3041

Keywords

Conservation law
Elimination
Linear system
Noninteracting
Reaction network
Steady states

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10.1007/978-3-030-00341-8_11

Cite this

Linear elimination in chemical reaction networks. / Sáez, Meritxell; Feliu, Elisenda ; Wiuf, Carsten.
Recent Advances in Differential Equations and Applications. ed. / Juan Luis García Guirao; José Alberto Murillo Hernández; Francisco Periago Esparza. Springer, 2019. p. 177-193 (SEMA SIMAI Springer Series, Vol. 18).

Research output: Chapter in Book/Report/Conference proceeding › Book chapter › Research › peer-review

Sáez, M, Feliu, E & Wiuf, C 2019, Linear elimination in chemical reaction networks. in JLG Guirao, JAM Hernández & FP Esparza (eds), Recent Advances in Differential Equations and Applications. Springer, SEMA SIMAI Springer Series, vol. 18, pp. 177-193, 25th Congress on Differential Equations and Applications / 15th Congress on Applied Mathematics , Cartagena, Spain, 26/06/2017. https://doi.org/10.1007/978-3-030-00341-8_11

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abstract = "We consider dynamical systems arising in biochemistry and systems biology that model the evolution of the concentrations of biochemical species described by chemical reactions. These systems are typically confined to an invariant linear subspace of ℝn. The steady states of the system are solutions to a system of polynomial equations for which only non-negative solutions are of interest. Here we study the set of non-negative solutions and provide a method for simplification of this polynomial system by means of linear elimination of variables. We take a graphical approach. The interactions among the species are represented by an edge labelled graph. Subgraphs with only certain labels correspond to sets of species concentrations that can be eliminated from the steady state equations using linear algebra. To assess positivity of the eliminated variables in terms of the non-eliminated variables, a multigraph is introduced that encodes the connections between the eliminated species in the reactions. We give graphical conditions on the multigraph that ensure the eliminated variables are expressed as positive functions of the non-eliminated variables. We interpret these conditions in terms of the reaction network. The results are illustrated by examples.",

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author = "Meritxell S{\'a}ez and Elisenda Feliu and Carsten Wiuf",

year = "2019",

doi = "10.1007/978-3-030-00341-8_11",

language = "English",

series = "SEMA SIMAI Springer Series",

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AU - Sáez, Meritxell

AU - Feliu, Elisenda

AU - Wiuf, Carsten

PY - 2019

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N2 - We consider dynamical systems arising in biochemistry and systems biology that model the evolution of the concentrations of biochemical species described by chemical reactions. These systems are typically confined to an invariant linear subspace of ℝn. The steady states of the system are solutions to a system of polynomial equations for which only non-negative solutions are of interest. Here we study the set of non-negative solutions and provide a method for simplification of this polynomial system by means of linear elimination of variables. We take a graphical approach. The interactions among the species are represented by an edge labelled graph. Subgraphs with only certain labels correspond to sets of species concentrations that can be eliminated from the steady state equations using linear algebra. To assess positivity of the eliminated variables in terms of the non-eliminated variables, a multigraph is introduced that encodes the connections between the eliminated species in the reactions. We give graphical conditions on the multigraph that ensure the eliminated variables are expressed as positive functions of the non-eliminated variables. We interpret these conditions in terms of the reaction network. The results are illustrated by examples.

AB - We consider dynamical systems arising in biochemistry and systems biology that model the evolution of the concentrations of biochemical species described by chemical reactions. These systems are typically confined to an invariant linear subspace of ℝn. The steady states of the system are solutions to a system of polynomial equations for which only non-negative solutions are of interest. Here we study the set of non-negative solutions and provide a method for simplification of this polynomial system by means of linear elimination of variables. We take a graphical approach. The interactions among the species are represented by an edge labelled graph. Subgraphs with only certain labels correspond to sets of species concentrations that can be eliminated from the steady state equations using linear algebra. To assess positivity of the eliminated variables in terms of the non-eliminated variables, a multigraph is introduced that encodes the connections between the eliminated species in the reactions. We give graphical conditions on the multigraph that ensure the eliminated variables are expressed as positive functions of the non-eliminated variables. We interpret these conditions in terms of the reaction network. The results are illustrated by examples.

KW - Conservation law

KW - Elimination

KW - Linear system

KW - Noninteracting

KW - Reaction network

KW - Steady states

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AN - SCOPUS:85060280858

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BT - Recent Advances in Differential Equations and Applications

A2 - Guirao, Juan Luis García

A2 - Hernández, José Alberto Murillo

A2 - Esparza, Francisco Periago

PB - Springer

T2 - 25th Congress on Differential Equations and Applications / 15th Congress on Applied Mathematics

Y2 - 26 June 2017 through 30 June 2017

ER -

Linear elimination in chemical reaction networks

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