Thermodynamic Space of Chemical Reaction Networks

Living systems are usually maintained out of equilibrium and exhibit complex dynamical behaviors. The external energy supply often comes from chemical fluxes that can keep some species concentrations constant. Furthermore, the properties of the underlying chemical reaction networks (CRNs) are also instrumental in establishing robust biological functioning. Hence, capturing the emergent complexity of living systems and the role of their non-equilibrium nature is fundamental to uncover constraints and properties of the CRNs underpinning their functions. In particular, while kinetics plays a key role in shaping detailed dynamical phenomena, the range of operations of any CRN must be fundamentally constrained by thermodynamics, as they necessarily operate with a given energy budget. Here, we derive universal thermodynamic upper and lower bounds for the accessible space of species concentrations in a generic CRN. The resulting region determines the "thermodynamic space" of the CRN, a concept we introduce in this work. Moreover, we obtain similar bounds also for the affinities, shedding light on how global thermodynamic properties can limit local non-equilibrium quantities. We illustrate our results in two paradigmatic examples, the Schl\"ogl model for bistability and a minimal self-assembly process, demonstrating how the onset of complex behaviors is intimately tangled with the presence of non-equilibrium driving. In summary, our work unveils the exact form of the accessible space in which a CRN must work as a function of its energy budget, shedding light on the non-equilibrium origin of a variety of phenomena, from amplification to pattern formation. Ultimately, 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 behaviors and design artificial chemical reaction systems.