Nonequilibrium Thermodynamics of Chemical Reaction Networks: Wisdom from Stochastic Thermodynamics Riccardo Rao and Massimiliano Esposito Complex Systems and Statistical Mechanics, Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg, Luxembourg (Dated: January 13, 2017. Published in Phys. Rev. X, DOI: 10.1103/PhysRevX.6.041064) We build a rigorous nonequilibrium thermodynamic description for open chemical reaction networks of elementary reactions (CRNs). Their dynamics is described by deterministic rate equations with mass action kinetics. Our most general framework considers open networks driven by time-dependent chemostats. The energy and entropy balances are established and a nonequilibrium Gibbs free energy is introduced. The difference between this latter and its equilibrium form represents the minimal work done by the chemostats to bring the network to its nonequilibrium state. It is minimized in nondriven detailed-balanced networks (i.e. networks which relax to equilibrium states) and has an interesting information-theoretic interpretation. We further show that the entropy production of complex balanced networks (i.e. networks which relax to special kinds of nonequilibrium steady states) splits into two non-negative contributions: one characterizing the dissipation of the nonequilibrium steady state and the other the transients due to relaxation and driving. Our theory lays the path to study time-dependent energy and information transduction in biochemical networks. PACS numbers: 05.70.Ln, 87.16.Yc I. INTRODUCTION so surprising since living systems are the paramount ex- ample of nonequilibrium processes and they are powered Thermodynamics of chemical reactions has a long his- by chemical reactions. The fact that metabolic processes tory. The second half of the 19th century witnessed the involve thousands of coupled reactions also emphasized dawn of the modern studies on thermodynamics of chem- the importance of a network description [25–27]. While ical mixtures. It is indeed at that time that J. W. Gibbs complex dynamical behaviors such as oscillations were introduced the concept of chemical potential and used it analyzed in small CRNs [28, 29], most studies on large bio- to define the thermodynamic potentials of non-interacting chemical networks focused on the steady-state dynamics. mixtures [1]. Several decades later, this enabled T. de Don- Very few studies considered the thermodynamic properties der to approach the study of chemical reacting mixtures of CRNs [30–33]. One of the first nonequilibrium thermo- from a thermodynamic standpoint. He proposed the dynamic description of open biochemical networks was concept of affinity to characterize the chemical force ir- proposed in Ref. [34]. However, it did not take advantage reversibly driving chemical reactions and related it to of Chemical Reaction Network Theory which connects the the thermodynamic properties of mixtures established by network topology to its dynamical behavior and which Gibbs [2]. I. Prigogine, who perpetuated the Brussels was extensively studied by mathematicians during the School founded by de Donder, introduced the assumption seventies [35–37] (this theory was also later extended to of local equilibrium to describe irreversible processes in stochastic dynamics [38–41]). As far as we know, the terms of equilibrium quantities [3, 4]. In doing so, he first and single study that related the nonequilibrium pioneered the connections between thermodynamics and thermodynamics of CRNs to their topology is Ref. [22], kinetics of chemical reacting mixtures [5]. still restricting itself to steady states. During the second half of the 20th century, part of the In this paper, we consider the most general setting attention moved to systems with small particle numbers for the study of CRNs, namely open networks driven which are ill-described by “deterministic” rate equations. by chemostatted concentrations which may change over The Brussels School, as well as other groups, produced time. To the best of our knowledge, this was never con- various studies on the nonequilibrium thermodynamics sidered before. In this way, steady-state properties as of chemical systems [6–11] using a stochastic description well as transient ones are captured. Hence, in the same based on the (Chemical) Master Equation [12, 13]. These way that Stochastic Thermodynamics is built on top of studies played an important role during the first decade stochastic dynamics, we systematically build a nonequi- of the 21st century for the development of Stochastic librium thermodynamic description of CRNs on top of Thermodynamics, a theory that systematically establishes deterministic chemical rate equations. In doing so, we a nonequilibrium thermodynamic description for systems establish the energy and entropy balance and introduce arXiv:1602.07257v3 [cond-mat.stat-mech] 11 Jan 2017 obeying stochastic dynamics [14–17], including chemical the nonequilibrium entropy of the CRN as well as its reaction networks (CRNs) [18–22]. nonequilibrium Gibbs free energy. We show this latter Another significant part of the attention moved to the to bear an information-theoretical interpretation similar thermodynamic description of biochemical reactions in to that of Stochastic Thermodynamics [42–45] and to be terms of deterministic rate equations [23, 24]. This is not related to the dynamical potentials derived by mathemati- 2 cians. We also show the relation between the minimal environment chemical work necessary to manipulate the CRNs far system from equilibrium and the nonequilibrium Gibbs free en- ergy. Our theory embeds both the Prigoginian approach k+1 to thermodynamics of irreversible processes [5] and the Xa 2Xb + Xc thermodynamics of biochemical reactions [23]. Making k-1 full use of the mathematical Chemical Reaction Network Theory, we further analyze the thermodynamic behav- +2 ior of two important classes of CRNs: detailed-balanced k Xc + Xd Xe networks and complex-balanced networks. In absence of k-2 time-dependent driving, the former converges to thermo- dynamic equilibrium by minimizing their nonequilibrium Gibbs free energy. In contrast, the latter converges to a specific class of nonequilibrium steady states and always FIG. 1: Representation of a closed CRN. The chemical allow for an adiabatic–nonadiabatic separation of their species are {Xa, ··· , Xe}. The two reaction pathways are labeled by 1 and 2. The nonzero stoichiometric coefficients entropy production, which is analogous to that found in a b c are −∇+1 = −1, ∇−1 = 2 and ∇−1 = 1 for the first Stochastic Thermodynamics [46–50]. We note that while c d forward reaction and −∇+2 = −1, −∇+2 = −1 and finalizing this paper, a result similar to the latter was e ∇−2 = 1 for the second one. Since the network is closed, independently found in Ref. [51]. no chemical species is exchanged with the environment. Outline and Notation integer-valued stoichiometric matrix The paper is organized as follows. After introducing ∇ ≡ ∇− − ∇+ . (2) the necessary concepts in chemical kinetics and Chemical Reaction Network Theory, sec. II, the nonequilibrium The reason for the choice of the symbol “∇” will become thermodynamic description is established, sec. III. As in clear later. Stochastic Thermodynamics, we build it on top of the Example 1. The stoichiometric matrix of the CRN de- dynamics and formulate the entropy and energy balance, picted in fig.1 is § IIID and § IIIE. Chemical work and nonequilibrium Gibbs free energy are also defined and the information- −1 0 theoretic content of the latter is discussed. The special 2 0 properties of detailed-balance and of complex-balanced ∇ = 1 −1 . (3) networks are considered in sec.V andIV, respectively. 0 −1 Conclusions and perspectives are drawn in sec.VI, while 0 1 some technical derivations are detailed in the appendices. We now proceed by fixing the notation. We consider a system composed of reacting chemical species Xσ, each of which is identified by an index σ ∈ S, where S is the Physical quantities associated to species and reactions set of all indices/species. The species populations change are represented in upper–lower indices vectorial nota- due to elementary reactions, i.e. all reacting species and tion. Upper and lower indexed quantities have the same i reactions must be resolved (none can be hidden), and all physical values, e.g. Z = Zi, ∀ i. We use the Einstein reactions must be reversible, i.e. each forward reaction +ρ summation notation: repeated upper–lower indices im- has a corresponding backward reaction −ρ. Each pair of plies the summation over all the allowed values for those forward–backward reactions is a reaction pathway denoted indices—e.g. σ ∈ S for species and ρ ∈ R for reactions. by ρ ∈ R. The orientation of the set of reaction pathways Given two arbitrary vectorial quantities a = {ai} and R is arbitrary. Hence, a generic CRN is represented as b = {bi}, the following notation will be used +ρ ibi Y ibi X σ k X σ a ≡ a . ∇+ρ Xσ ∇−ρ Xσ . (1) k−ρ i σ σ i The constants k+ρ (k−ρ) are the rate constants of the Finally, given the matrix C, whose elements are {Cj}, the T j forward (backward) reactions. The stoichiometric coeffi- elements of the transposed matrix C are {Ci }. cients −∇σ and ∇σ identify the number of molecules The time derivative of a physical quantity A is denoted +ρ −ρ ¯ of Xσ involved in each forward reaction +ρ (the stoi- by dtA, its steady state value by an overbar, A, and its eq chiometric coefficients of the backward reactions have equilibrium value by Aeq or A . We reserve the overdot, opposite signs). Once stacked into two non-negative ma- A˙, to denote the rate of change of quantities which are σ σ trices ∇+ = {∇+ρ} and ∇− = {∇−ρ}, they define the not exact time derivatives. 3 II. DYNAMICS OF CRNS environment system In this section, we formulate the mathematical descrip- tion of CRNs [52, 53] in a suitable way for a thermo- k+1 dynamic analysis. We introduce closed and open CRNs Ya 2Xb + Xc -1 and show how to drive these latter in a time-dependent k way.