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  • Open Access

Thermodynamic Space of Chemical Reaction Networks

Shiling Liang (梁师翎)1,2,3,4,*, Paolo De Los Rios1,5, and Daniel Maria Busiello3,6,†

  • *Contact author: shiling.liang@outlook.com
  • †Contact author: danielmaria.busiello@unipd.it
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Phys. Rev. X 16, 041026 – Published 26 October, 2026

DOI: https://doi.org/10.1103/g875-ldn5

Abstract

Living systems operate out of equilibrium, continuously consuming energy to sustain organized, functional states. Their emergent behavior usually relies on a set of interconnected chemical reaction networks (CRNs) driven by external fluxes that keep some species at fixed concentrations. Hence, uncovering the principles governing the functioning of these CRNs is crucial to understand how living systems generate and regulate complexity. While kinetics plays a key role in shaping detailed dynamical phenomena, the range of operations of a CRN is fundamentally constrained by thermodynamics. Here, we introduce and analytically derive the “thermodynamic space” of a CRN, i.e., the range of accessible stationary concentrations that can be realized under a given energetic budget. We establish analogous bounds for reaction affinities, shedding light on how global thermodynamic properties, such as the total nonequilibrium driving, can limit local nonequilibrium quantities, and further showing that these bounds can be used to test candidate network topologies and expose hidden multimolecular pathways. We illustrate our results in various paradigmatic examples, demonstrating how the onset of complex behaviors is intimately tangled with the presence of nonequilibrium conditions. By providing a general tool for analyzing CRNs, the presented framework constitutes a stepping stone to deepen our ability to predict complex out-of-equilibrium phenomena and design artificial chemical systems, starting from the sole knowledge of the underlying thermodynamic properties.

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Popular Summary

Chemical reaction systems can use a continuous energy supply to sustain organized states and functions away from equilibrium, enabling many processes essential to life. Yet the thermodynamic limits on what such chemical systems can achieve remain poorly understood. We address this question by combining reaction network structure with a global free-energy budget to bound species concentrations and the driving forces of individual reactions at steady state. These bounds reveal how energy available to the network as a whole constrains the local chemical conditions underlying its emergent behavior, without requiring detailed knowledge of reaction rates. This connection between energy and function provides a general basis for understanding the capabilities of chemical systems and guiding their design.

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Phys. Rev. X 16, 041026– Published 26 October, 2026
  • Received 1 July 2025
  • Accepted 18 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

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