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On the sum of chemical reactions

Part of: Applications - Dynamical systems and ergodic theory Physiological, cellular and medical topics Chemistry

Published online by Cambridge University Press:  24 May 2022

LINARD HOESSLY ,
CARSTEN WIUF Open the ORCID record for CARSTEN WIUF [Opens in a new window]  and
PANQIU XIA
LINARD HOESSLY
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark emails: [email protected]; [email protected]; [email protected].
CARSTEN WIUF
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark emails: [email protected]; [email protected]; [email protected].
PANQIU XIA
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark emails: [email protected]; [email protected]; [email protected].
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Abstract

It is standard in chemistry to represent a sequence of reactions by a single overall reaction, often called a complex reaction in contrast to an elementary reaction. Photosynthesis $6 \text{CO}_2+6 \text{H}_2\text{O} \longrightarrow \text{C}_6\text{H}_{12}\text{O}_6 + 6 \text{O}_2$ is an example of such complex reaction. We introduce a mathematical operation that corresponds to summing two chemical reactions. Specifically, we define an associative and non-communicative operation on the product space ${\mathbb{N}}_0^n\times {\mathbb{N}}_0^n$ (representing the reactant and the product of a chemical reaction, respectively). The operation models the overall effect of two reactions happening in succession, one after the other. We study the algebraic properties of the operation and apply the results to stochastic reaction networks (RNs), in particular to reachability of states, and to reduction of RNs.

Keywords

Reaction networkreductionMarkov chaingraph

MSC classification

Primary: 37N25: Dynamical systems in biology
Secondary: 92C42: Systems biology, networks 92E20: Classical flows, reactions, etc.
Type
Papers
Information
European Journal of Applied Mathematics , Volume 34 , Issue 2 , April 2023 , pp. 303 - 325
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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