Nonequilibrium Thermodynamics of Chemical Reaction Networks: Wisdom from Stochastic Thermodynamics

Home , Detailed balance, Entropy production

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 example 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 energy. 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 thermodynamic 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 description 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. We then define conservation laws and cycles and review the dynamical properties of two important classes of k+2 CRNs: detailed-balanced networks and complex-balanced Xc + Xd Ye networks. k-2 We consider a chemical system in which the reacting species {Xσ} are part of a homogeneous and ideal dilute solution: the reactions proceed slowly compared to FIG. 2: Representation of an open CRN. With respect to diffusion and the solvent is much more abundant than the CRN in fig.1, the species Xa and Xe are chemostatted, the reacting species. Temperature T and pressure p are hence represented as Ya and Ye. The green boxes aside kept constant. Since the volume of the solution V is represent the reservoirs of chemostatted species. overwhelmingly dominated by the solvent, it is assumed constant. The species abundances are large enough so that the molecules discreteness can be neglected. Thus, B. Driven CRNs at any time t, the system state is well-described by the molar concentration distribution {Zσ ≡ N σ/V }, where σ In open CRNs, matter is exchanged with the environ- N is the molarity of the species Xσ. ment via reservoirs which control the concentrations of The reaction kinetics is controlled by the reaction rate some specific species, fig.2. These externally controlled ±ρ σ functions J {Z } , which measure the rate of occur- species are said to be chemostatted, while the reservoirs rence of reactions and satisfy the mass action kinetics controlling them are called chemostats. The chemostat- [52, 54, 55] ting procedure may mimic various types of controls by the environment. For instance, a direct control could be ±ρ ±ρ σ ±ρ σ∇±ρ J ≡ J {Z } = k Z σ . (4) implemented via external reactions (not belonging to the CRN) or via abundant species whose concentrations are The net concentration current along a reaction pathway negligibly affected by the CRN reactions within relevant ρ is thus given by time scales. An indirect control may be achieved via semipermeable membranes or by controlled injection of ∇+ρ ∇−ρ J ρ ≡ J +ρ − J −ρ = k+ρZσ σ − k−ρZσ σ . (5) chemicals in continuous stirred-tank reactors. Among the chemical species, the chemostatted ones are Example 2. For the CRN in fig.1 the currents are denoted by the indices σy ∈ Sy, and the internal ones by σx ∈ Sx (S ≡ Sx∪Sy). Also, the part of the stoichiometric J 1 = k+1Za − k−1(Zb)2Zc matrix related to the internal (resp. chemostatted) species σ (6) is denoted by ∇X = ∇σx (resp. ∇Y = ∇ y ). J 2 = k+2ZcZd − k−2Ze . ρ ρ Example 3. When chemostatting the CRN in fig.1 as in fig.2 the stoichiometric matrix (3) splits into

2 0  −1 0 ∇X = 1 −1 , ∇Y = . (8) A. Closed CRNs   0 1 0 −1 A closed CRN does not exchange any chemical species with the environment. Hence, the species concentrations vary solely due to chemical reactions and satisfy the rate In nondriven open CRNs the chemostatted species have equations σy constant concentrations, i.e. {dtZ = 0}. In driven open CRNs the chemostatted concentrations change over σ σ ρ dtZ = ∇ρ J , ∀ σ ∈ S . (7) time according to some time-dependent protocol π(t): {Zσy ≡ Zσy (π(t))}. The changes of the internal species Since rate equations are nonlinear, complex dynamical are solely due to reactions and satisfy the rate equations behaviors may emerge [29]. The fact that the rate equations (7) can be thought of as a continuity equation for σx σx ρ dtZ = ∇ρ J , ∀ σx ∈ Sx . (9) the concentration, where the stoichiometric matrix ∇ (2) σy acts as a discrete differential operator, explains the choice Instead, the changes of chemostatted species {dtZ } are σy ρ of the symbol “∇” for the stoichiometric matrix [56]. not only given by the species formation rates {∇ρ J } 4 but must in addition contain the external currents {Iσy }, environment which quantify the rate at which chemostatted species system enter into the CRN (negative if chemostatted species leave the CRN), k+1 H2O 2H + O σy σy ρ σy -1 dtZ = ∇ρ J + I , ∀ σy ∈ Sy . (10) k This latter equation is not a differential equation since the chemostatted concentrations {Zσy } are not dynam- k+2 O + C CO ical variables. It shows that the external control of the -2 chemostatted concentration is not necessarily direct, via k the chemostatted concentrations, but can also be indi- rectly controlled via the external currents. We note that (10) is the dynamical expression of the decomposition of FIG. 3: Specific implementation of the CRN in fig.2. changes of species populations in internal–external introduced by de Donder (see [57, §§ 4.1 and 15.2]). A steady-state distribution {Z¯σx }, if it exists, must where the first term is nonvanishing for at least one ρ ∈ R. satisfy λb λb σ The broken components {L ≡ `σ Z } are no longer constant over time. On the other hand, the unbroken σx ¯ρ ∇ρ J = 0 , ∀ σx ∈ Sx , (11a) conservation laws are characterized by σy ¯ρ ¯σy ∇ρ J + I = 0 , ∀ σy ∈ Sy , (11b) `λu ∇σx +`λu ∇σy = 0 , ∀ ρ ∈ R , (15) σx ρ σy ρ for given chemostatted concentrations {Zσy }. | {z } = 0 where the first term vanishes for all ρ ∈ R. Therefore, the C. Conservation Laws λu λu σ unbroken components {L ≡ `σ Z } remain constant over time. Without loss of generality, we choose the set In a closed CRN, a conservation law ` = {`σ} is a left {`λ} such that the entries related to the chemostatted null eigenvector of the stoichiometric matrix ∇ [23, 25] species vanish, `λu = 0, ∀λ , σ . σy u y σ `σ ∇ρ = 0 , ∀ ρ ∈ R . (12) Example 4. For the CRN in fig.1, an independent set of conservation laws is Conservation laws identify conserved quantities L ≡ σ `σ Z , called components [23, 25], which satisfy `1 = 2 1 0 0 0 , σ 2 dtL = `σ dtZ = 0 . (13) ` = 0 0 0 1 1 and (16) `3 = 0 1 −1 1 0 . We denote a set of independent conservation laws of 2 the closed network by `λ and the corresponding com- When chemostatting as in fig.2, the first two conservation λ λ σ ponents by L ≡ `σ Z . The choice of this set is not laws break while the last one remains unbroken. We unique, and different choices have different physical mean- also note that this set is chosen so that the unbroken ings. This set is never empty since the total mass is 3 3 conservation law satisfies à = è = 0. When considering always conserved. Physically, conservation laws are often the specific implementation in fig.3 of the CRN in fig.2, related to parts of molecules, called moieties [58], which we see that the first two conservation laws in (16) represent are exchanged between different species and/or subject the conservation of the concentrations of the moiety H to isomerization (see example4). and C, respectively. Instead, the third conservation law σ In an open CRN, since only {Z x } are dynamical vari- in (16) does not have a straightforward interpretation. It ables, the conservation laws become the left null eigenvec- is related to the fact that when the species H or C are tors of the stoichiometric matrix of the internal species produced also O must be produced and vice versa. ∇X. Stated differently, when starting from the closed CRN, the chemostatting procedure may break a subset of the conservation laws of the closed network {`λ} [56]. E.g. when the first chemostat is introduced the total mass D. Detailed-Balanced Networks conservation law is always broken. Within the set `λ , we label the broken ones by λ and the unbroken ones by b A steady state (11) is said to be an equilibrium state, λu. The broken conservation laws are characterized by σ {Zeq}, if it satisfies the detailed balance property [57, § 9.4], `λb ∇σx +`λb ∇σy = 0 , ∀ ρ ∈ R , (14) i.e. all concentration currents (5) vanish, σx ρ σy ρ | {z } ρ ρ σ 6= 0 Jeq ≡ J {Zeq} = 0, ∀ ρ ∈ R . (17) 5

σy ρ For open networks, this means that the external currents, their prior values due to the injection of −∇ρ c molecules eq. (11b), must also vanish {Iσy = 0}. By virtue of mass of X performed by the chemostats. Emergent cycles are eq σy action kinetics, eq. (4), the detailed balance property (17) thus pathways transferring chemicals across chemostats can be rewritten as while leaving the internal state of the CRN unchanged.

+ρ We denote by {cε} a set of linearly independent emergent k ρ σ ∇σ cycles. = Zeq , ∀ ρ ∈ R . (18) k−ρ When chemostatting an initially closed CRN, for each A CRN is said to be detailed-balanced if, for given species which is chemostatted, either a conservation law breaks—as mentioned in §IIC—or an independent emer- kinetics {k±ρ} and chemostatting {Zσy }, its dynamics exhibits an equilibrium steady state (17). For each set gent cycle arises [56]. This follows from the rank nul- lity theorem for the stoichiometric matrices ∇ and ∇X, of unbroken components {Lλu }—which are given by the initial condition and constrain the space where the dynam- which ensures that the number of chemostatted species ics dwells—the equilibrium distribution is globally stable |Sy| equals the number of broken conservation laws |λb| [59]. Equivalently, detailed-balanced networks always re- plus the number of independent emergent cycles |ε|: lax to an equilibrium state, which for a given kinetics and |Sy| = |λb| + |ε|. Importantly, the rise of emergent cycles chemostatting is unique and depends on the unbroken is a topological feature: it depends on the species which components only, see also sec.V. are chemostatted, but not on the chemostatted concentra- Closed CRNs must be detailed-balanced. This state- tions. We also note that emergent cycles are modeled as ment can be seen as the zeroth law for CRNs. Conse- “ﬂux modes” in the context of metabolic networks [60–62]. quently, rather than considering eq. (18) as a property Example 5. To illustrate the concepts of cycles and of the equilibrium distribution, we impose it as a prop- emergent cycles, we use the following CRN [56] erty that the rate constants must satisfy and call it local detailed-balance property. It is a universal property of k+1 Y + X X elementary reactions which holds regardless of the net- 1 a 1 b work state. Indeed, while the equilibrium distribution k− +4 2 depends on the components, the r.h.s. of eq. (18) does k k− not. This point will become explicit after introducing the 4 +2 k− k thermodynamic structure, eq. (88) in sec.V. The local 3 k− X Y + X detailed-balance property will be rewritten in a thermo- c +3 2 a dynamic form in §IIIB, eq. (50). k (21) In open nondriven CRNs, the chemostatting procedure may prevent the system from reaching an equilibrium whose Y1 and Y2 species are chemostatted. The stoichio- state. To express this scenario algebraically we now in- metric matrix decomposes as troduce the concepts of emergent cycle and cycle aﬃnity. ρ −1 1 −1 1  A cycle ˜c = {c˜ } is a right null eigenvector of the −1 0 0 1 ∇X = 1 −1 0 0 ∇Y = . stoichiometric matrix [56], namely   0 1 −1 0 0 0 1 −1 σ ρ ∇ρ c˜ = 0 , ∀ σ ∈ S . (19) (22)

Since ∇ is integer-valued, ˜c can always be rescaled to The set of linearly independent cycles (19) consists of only only contain integer coeﬃcients. In this representation, one cycle, which can be written as its entries denote the number of times each reaction occurs T (negative signs identify reactions occurring in backward c˜ = 1 1 1 1 . (23) direction) along a transformation which overall leaves the σ concentration distributions {Z } unchanged, see example As the CRN is chemostatted one linearly independent 5. We denote by {˜cα} a set of linearly independent cycles. emergent cycle (20) arises An emergent cycle c = {cρ} is deﬁned algebraically as [56] T c = 1 1 −1 −1 . (24)

( σx ρ ∇ c = 0 , ∀ σx ∈ Sx ρ (20) We now see that if each reaction occurs a number of times σy ρ ∇ρ c 6= 0 , for at least one σy ∈ Sy . given by the entry of the cycle (23), the CRN goes back to the initial state, no matter which one it is. On the other In its integer-valued representation, the entries of c de- hand, when the emergent cycle (24) is performed, the note the number of times each reaction occurs along a state of the internal species does not change, while two transformation which overall leaves the concentrations of molecules of Y1 are annihilated and two of Y2 are created. the internal species {Zσx } unchanged while changing the However, since the chemostats restore their initial values, concentrations of the chemostatted species by an amount the overall result of c is to transfer two Y1, transformed σy ρ ∇ρ c . These latter are however immediately restored to in Y2, from the ﬁrst to the second chemostat. 6

The closed version of this CRN has two independent Finally, a tacit assumption in the above discussion is conservation laws, that the network involves a finite number of species and reactions, i.e. the CRN is finite-dimensional. Infinite- 1 ` = 0 1 1 1 1 dimensional CRNs can exhibits long-time behaviors differ- (25) `2 = 1 1 1 0 0 ent from equilibrium even in absence of emergent cycles [63]. the first of which, `1, is broken following the chemostatting of any of the two species Y1 or Y2. The other chemostatted species, instead, gives rise to the emergent cycle (24), E. Complex-Balanced Networks so that the relationship |Sy| = |λb| + |ε| is satisfied. To discuss complex-balanced networks and complex- Any cycle ˜c and emergent cycle c bears a cycle affinity α ε balanced distributions, we first introduce the notion of [56] complex in open CRNs. J A complex is a group of species which combines in a A˜ =c ˜ρ RT ln +ρ , (26) α α J reaction as products or as reactants. Each side of eq. (1) −ρ defines a complex but different reactions might involve J ρ +ρ the same complex. We label complexes by γ ∈ C, where Aε = cε RT ln . (27) J−ρ C is the set of complexes. From the definition of cycle (19) and current (5), and Example 7. Let us consider the following CRN [64] the local detailed balance (18), it follows that the cycle +1 affinities along the cycles (19) vanish, {A˜α = 0}, and that k Xa )−−−−−−* Xb the cycle affinities along the emergent cycles only depend −1 k . (29) on the chemostatted concentrations k+2 k+3 Xa + Xb )−−−−−−* 2 Xb )−−−−−−* Xc σy k−2 k−3 ρ k+ρ −∇ρ Aε = cε RT ln Zσy . (28) k−ρ The set of complexes is C = {Xa, Xb, Xa + Xb, 2Xb, Xc}, Since emergent cycles are pathways connecting different and the complex 2Xb is involved in both the second and chemostats, the emergent affinities quantify the chemical third reaction. forces acting along the cycles. This point will become The notion of complex allows us to decompose the clearer later, when the thermodynamic expressions of the stoichiometric matrix ∇ as emergent cycle affinities {Aε} will be given, eq. (49). σ σ γ A CRN is detailed-balanced if and only if all the emer- ∇ρ = Γγ ∂ρ . (30) gent cycle affinities {Aε} vanish. This condition is equiv- σ alent to the Wegscheider’s condition [59]. This happens We call Γ = {Γγ } the composition matrix [35, 37]. Its σ when the chemostatted concentrations fit an equilibrium entries Γγ are the stoichiometric number of species Xσ distribution. As a special case, unconditionally detailed- in the complex γ. The composition matrix encodes the balanced networks are open CRNs with no emergent cycle. structure of each complex in terms of species, see example γ Therefore, they are detailed-balanced for any choice of the 8. The matrix ∂ = {∂ρ } denotes the incidence matrix of chemostatted concentrations. Consequently, even when a the CRN, whose entries are given by time-dependent driving acts on such a CRN and prevents  it from reaching an equilibrium state, a well-defined equi- 1 if γ is the product complex of +ρ ,  librium state exists at any time: the equilibrium state to γ ∂ρ = −1 if γ is the reactant complex of +ρ , which the CRN would relax if the time-dependent driving 0 otherwise. were stopped. (31) Example 6. Any CRN with one chemostatted species only (|Sy| = 1) is unconditionally detailed-balanced. In- The incidence matrix encodes the structure of the network deed, as mentioned in §IIC, the first chemostatted species at the level of complexes, i.e. how complexes are con- always breaks the mass conservation law, |λb| = 1 and nected by reactions. If we think of complexes as network thus no emergent cycle arises, |ε| = |Sy| − |λb| = 0. nodes, the incidence matrix associates an edge to each The open CRN in fig.2 is an example of unconditionally reaction pathway and the resulting topological structure detailed-balanced network with two chemostatted species, is a reaction graph, e.g. fig.1 and eqs. (21) and (29). The since the chemostatting breaks two conservation laws, stoichiometric matrix instead encodes the structure of the see example4. Indeed, a nonequilibrium steady state network at the level of species. If we think of species as would require a continuous injection of Ya and ejection of the network nodes, the stoichiometric matrix does not Ye (or vice versa). But this would necessary result in a define a graph since reaction connects more than a pair of continuous production of Xb and consumption of Xd which species, in general. The structure originating is rather a is in contradiction with the steady state assumption. hyper-graph [56, 65], or equivalently a Petri net [66, 67]. 7

Example 8. The composition matrix and the incidence A steady-state distribution {Z¯σx } (11) is said to be matrix of the CRN in (29) are complex-balanced if the net current flowing in each group ρ of complexes Cj vanishes, i.e. if the currents {J¯ } satisfy −1 0 0    X 1 0 1 0 0  1 0 0  ∂jJ¯ρ ≡ ∂γ J¯ρ = 0 , ∀j . (37) Γ = 0 1 1 2 0 , ∂ =  0 −1 0  , ρ ρ     γ∈C 0 0 0 0 1  0 1 −1 j 0 0 1 Complex-balance steady states are therefore a subclass of (32) steady states (11a) which include equilibrium ones (17) as a special case where the complexes are ordered as in example7. The corresponding reaction hyper-graph is σ Γ x ∂j J¯ρ . (38) j ρ |{z} k+1 = 0 iff DB | {z } k+3 = 0 iff CB Xa Xb Xc | {z } = 0 for generic SS

k+2 (33) While for generic steady states only the internal species formation rates vanish, for complex-balanced ones the where only the forward reactions are depicted. complex formation rates also vanish. ±ρ In an open CRN, we regroup all complexes γ ∈ C of For a fixed kinetics ({k }) and chemostatting (Sy and the closed CRN which have the same stoichiometry for {Zσy }), a CRN is complex-balanced if its dynamics ex- the internal species (i.e. all complexes with the same hibits a complex-balanced steady state (37) [35, 36]. The X internal part of the composition matrix Γγ regardless complex-balanced distribution (37) depends on the unbro- Y ken components {Lλu }, which can be inferred from the of the chemostatted part Γγ ) in sets denoted by Cj, for j = 1, 2,... . Complexes of the closed network made solely initial conditions, and is always globally stable [68]. Hence, of chemostatted species in the open CRN are all regrouped complex-balanced networks always relax to a—complex- balanced—steady state. Detailed-balanced networks are in the same complex C0. This allows to decompose the internal species stoichiometric matrix as a subclass of complex-balanced networks. Whether or not a CRN is complex-balanced depends on σx σx j ∇ρ = Γj ∂ρ . (34) the network topology (∇), the kinetics ({k±ρ}) and the σy σx σx chemostatting (Sy and {Z }). For any given network where {Γj ≡ Γγ , for γ ∈ Cj} are the entries of the composition matrix corresponding to the internal species, topology and set of chemostatted species Sy, one can ∇±ρ and {∂j ≡ P ∂γ } are the entries of the incidence always find a set of effective rate constants {k±ρZσy σy } ρ γ∈Cj ρ matrix describing the network of regrouped complexes. which makes that CRN complex-balanced [37]. However, This regrouping corresponds to the—equivalent—CRN for some CRNs, this set coincides with the one which only made of internal species with effective rate constant makes the CRN detailed-balanced [69]. A characterization ∇±ρ {k±ρZσy σy } ruling each reaction. of the set of effective rate constants which make a CRN complex balanced is reported in Refs. [37, 69]. Example 9. Let us consider the CRN (29) where the Deficiency-zero CRNs are a class of CRN which is species Xa and Xc are chemostatted. The five complexes complex-balanced irrespective of the effective kinetics of the closed network, see example7, are regrouped as ∇±ρ {k±ρZσy σy } [35–37]. The network deficiency is a topo- C = {X , X }, C = {X , X + X } and C = {2 X }. 0 a c 1 b b a 2 b logical property of the CRN which we briefly discuss in In terms of these groups of complexes, the composition app.D, see Refs. [ 22, 52, 53] for more details. Conse- matrix and incidence matrix are quently, regardless of the way in which a deficiency-zero   −1 0 1 CRN is driven in time, it will always remain complex- X C Γ = 0 1 2 , ∂ =  1 −1 0  , (35) balanced. Throughout the paper, we will refer to these 0 1 −1 CRNs as unconditionally complex-balanced, as in the sem- which corresponds to the effective representation inal work [35]. Example 10. The open CRN (36) has a single steady C0 state Z¯b for any given set of rate constants and chemostat- k +3 k +1 a c k − ted concentrations Z and Z [64], defined by (11a) c 1 Z a 3 Z − k 2 ¯b ¯1 ¯2 ¯3 k− dtZ = J + J − 2J 2Xb Xb k+2Za (36) = k+1Za − k−1Z¯b + k+2ZaZ¯b − k−2(Z¯b)2+ + 2k−3Zc − 2k+3(Z¯b)2 = 0 . 8

(39) III. THERMODYNAMICS OF CHEMICAL NETWORKS If the stronger condition (37) holds

3 1 Using local equilibrium, we here build the connection J¯ − J¯ = 0 , (group C0) , between the dynamics and the nonequilibrium thermo- ¯1 ¯2 J − J = 0 , (group C1) , (40) dynamics for arbitrary CRNs. In the spirit of Stochastic 2 3 J¯ − J¯ = 0 , (group C2) , Thermodynamics, we derive an energy and entropy balance, and express the dissipation of the CRN as the which is equivalent to difference between the chemical work done by the reservoirs on the CRN and its change in nonequilibrium free +1 a −1 ¯b +2 a ¯b −2 ¯b 2 k Z − k Z = k Z Z − k (Z ) energy. We finally discuss the information-theoretical = k+3(Z¯b)2 − k−3Zc , (41) content of the nonequilibrium free energy and its relation to the dynamical potentials used in Chemical Reaction the steady state is complex-balanced. Yet, if the steady- Network Theory. state currents are all independently vanishing, J¯1 = J¯2 = J¯3 = 0 , (42) A. Local Equilibrium i.e., eq. (41) is equal to zero, then the steady state is detailed-balanced. Since we consider homogeneous reaction mixtures in When, for simplicity, all rate constants are taken as ideal dilute solutions, the assumption of local equilibrium 1, the complex-balanced set of quadratic equations (41) [57, § 15.1] [70] means that the equilibration following ¯b a c admits a positive√ solution Z only if Z = 2 − Z (0 < any reaction event is much faster than any reaction time Zc < 2) or Za = Zc. The former case corresponds to scale. Thus, what is assumed is that the nonequilibrium a genuine complex-balanced state, Z¯b = 1 with currents nature of the thermodynamic description is solely due ¯1 ¯2 ¯3 c J = J = J = 1 −√Z , while the second to a detailed- to the reaction mechanisms. If all reactions could be balance state Z¯b = Zc with vanishing currents. When, instantaneously shut down, the state of the whole CRN for example, Za = 1 and Zc = 4, neither of the two would immediately become an equilibrated ideal mixture previous√ conditions holds: the nonequilibrium√ steady state√ of species. As a result, all the intensive thermodynamic is Z¯b = 3 with currents J¯1 = 1 − 3, J¯2 = −3 + 3, variables are well-defined and equal everywhere in the and J¯3 = −1. system. The temperature T is set by the solvent, which acts as a thermal bath, while the pressure p is set by the Example 11. Let us now consider the following open environment the solution is exposed to. As a result, each CRN [22] chemical species is characterized by a chemical potential k+1 k+2 k+3 [23, § 3.1] Ya )−−−−−−* Xb )−−−−−−* Xc + Xd −−−)−−−* Ye (43) k−1 k−2 k−3 ◦ Zσ µσ = µσ + RT ln , ∀ σ ∈ S , (45) where the species Ya and Ye are chemostatted. Out of Ztot the four complexes of the closed network, {Ya, Xb, Xc + ◦ ◦ Xd, Ye}, two are grouped into C0 = {Ya, Ye} and the where R denotes the gas constant and {µσ ≡ µσ(T )} other two remain C1 = {Xb} and C2 = {Xc + Xd}. The are the standard-state chemical potentials, which depend effective representation of this open CRN is on the temperature and on the nature of the solvent. The total concentration of the solution is denoted by Z = P Zσ + Z0, where Z0 is the concentration of the C0 tot σ +3 k +1 solvent. We assume for simplicity that the solvent does k k −1 e Z a not react with the solutes. In case it does, our results 3 Z − still hold provided one treats the solvent as a nondriven k 2 k− chemostatted species, as discussed in app.A. Since the Xc + Xd Xb k+2 (44) solvent is much more abundant than the solutes, the total concentration is almost equal to that of the solvent which This network is deficiency-zero and hence unconditionally is a constant, Ztot ' Z0. Without loss of generality, the 0 complex-balanced [22]. Therefore, given any set of rate constant term −RT ln Ztot ' −RT ln Z in eq. (45) is constants k±1, k±2, and k±3, and the chemostatted con- absorbed in the standard-state chemical potentials. Con- centrations Za and Ze, the steady state of this CRN is sequently, many equations appear with non-matching complex-balanced, i.e. the steady state always satisfies a dimensions. We also emphasize that standard state quan- set of condition like those in eq. (40). Indeed, contrary to tities, denoted with “◦”, are defined as those measured example 10, steady state currents {J¯1, J¯2, J¯3} different in ideal conditions, at standard pressure (p◦ = 100 kPa) from each other cannot exist since they would induce a ◦ 3 and molar concentration (Zσ = 1 mol/dm ), but not at a growth or decrease of some concentrations. standard temperature [71, p. 61]. 9

Due to the assumption of local equilibrium and homo- C. Enthalpies and Entropies of Reaction geneous reaction mixture, the densities of all extensive thermodynamic quantities are well-defined and equal ev- To identify the heat produced by the CRN, we need erywhere in the system. With a slight abuse of notation, to distinguish the enthalpic change produced by each we use the same symbol and name for densities as for their reaction from the entropic one. We consider the decompo- corresponding extensive quantity. E.g. S is the molar sition of the standard state chemical potentials [23, § 3.2]: entropy divided by the volume of the solution, but we denote it as entropy. We apply the same logic to rates of ◦ ◦ ◦ change. E.g. we call entropy production rate the molar µσ = hσ − T sσ . (51) entropy production density rate. ◦ The standard enthalpies of formation {hσ} take into account the enthalpic contributions carried by each species B. Affinities, Emergent Affinities and [23, § 3.2] [72, § 10.4.2]. Enthalpy changes caused by Local Detailed Balance reactions give the enthalpies of reaction [23, § 3.2] [57, § 2.4] The thermodynamic forces driving reactions are given σ ◦ by differences of chemical potential (45) ∆rHρ = ∇ρ hσ , (52)

σ ∆rGρ ≡ ∇ρ µσ , (46) which at constant pressure measure the heat of reaction. This is the content of the Hess’ Law (see e.g. [72, § 10.4.1]). also called Gibbs free energies of reaction [57, § 9.3] ◦ The standard entropies of formation {sσ} take into ac- [23, § 3.2]. Since these must all vanish at equilibrium, count the internal entropic contribution carried by each σ eq ∇ρ µσ = 0, ∀ρ, we have species under standard-state conditions [23, § 3.2]. Using (51), the chemical potentials (45) can be rewritten as σ Zσ ∆rGρ = −RT ∇ρ ln eq . (47) Zσ ◦ ◦ µσ = hσ − T (sσ − R ln Zσ) . (53) | {z } The local detailed-balance (18) allows us to express these ≡ sσ thermodynamic forces in terms of reaction aﬃnities, ◦ The entropies of formation {sσ ≡ sσ − R ln Zσ} account J+ρ Aρ ≡ RT ln = −∆rGρ (48) for the entropic contribution of each species in the CRN J−ρ [23, § 3.2]. Entropy changes along reactions are given by which quantify the kinetic force acting along each reaction ∆ S = ∇σ s , (54) pathway [57, § 4.1.3]. r ρ ρ σ The change of Gibbs free energy along emergent cycles, called entropies of reaction [23, § 3.2].

A = −cρ ∆ G = −cρ ∇σy µ , (49) ε ε r ρ ε ρ σy D. Entropy Balance gives the external thermodynamic forces the network is coupled to, as we shall see in eq. (61), and thus provide 1. Entropy Production Rate a thermodynamic meaning to the cycle affinities (28). Combining the detailed-balance property (18) and the The entropy production rate is a non-negative measure eq equilibrium condition on the affinities Aρ = 0 (46), we of the break of detailed balance in each chemical reaction. can relate the Gibbs free energies of reaction to the rate Its typical form is given by [57, § 9.5] [8] constants +ρ ◦ J+ρ k ∆rGρ T S˙ ≡ RT (J − J ) ln ≥ 0 , (55) = exp − , (50) i +ρ −ρ J k−ρ RT −ρ ◦ σ ◦ because where ∆rGρ ≡ ∇ρ µσ. This relation is the thermodynamic counterpart of the local detailed balance (18). It 1. It is non-negative and vanishes only at equilibrium, plays the same role as in Stochastic Thermodynamics, i.e. when the detailed balance property (17) is namely connecting the thermodynamic description to the satisfied; stochastic dynamics. We emphasize that the local detailed balance property as well as the local equilibrium assump- 2. It vanishes to first order around equilibrium, thus tion by no mean imply that the CRN operates close to allowing for quasi-static reversible transformations. equilibrium. Their importance is to assign well defined Indeed, defining equilibrium potentials to the states of the CRN, which σ σ are then connected by the nonequilibrium mechanisms, Z − Zeq σ σ σ = , | | 1 , ∀ σ ∈ S , (56) i.e. reactions. Zeq 10

we ﬁnd that 3. System Entropy

˙ σ σ0 3 Si = Eσ0 σ + O , (57) The entropy of the ideal dilute solution constituting the CRN is given by (see app.A) σ where E ≡ {E 0 } is a positive semideﬁnite symmet- σ σ S ric matrix. S = Z sσ + RZ + S0 . (63)

Furthermore it can be rewritten in a thermodynamically The total concentration term appealing way using (48): X ZS ≡ Zσ (64)

ρ σ∈S T S˙i = −J ∆rGρ . (58) and the constant S0 together represent the entropic con- It can be further expressed as the sum of two distinct tribution of the solvent. S0 may also account for the contributions [56]: entropy of chemical species not involved in the reactions. We also prove in app.B that the entropy (63) can be ˙ σx σy σy obtained as a large particle limit of the stochastic entropy T Si = −µσx dtZ −µσy (dtZ − I ) . (59) | {z } | {z } of CRNs. ˙ ≡ T Sx ≡ T S˙y S would be an equilibrium entropy if the reactions could be all shut down. But in presence of reactions, it The ﬁrst term is due to changes in the internal species becomes the nonequilibrium entropy of the CRN. Indeed, and thus vanishes at steady state. The second term is using eqs. (53), (58) and (62), we ﬁnd that its change due to the chemostats. It takes into account both the can be expressed as exchange of chemostatted species and the time-dependent σ σ S driving of their concentration. If the system reaches a dtS = sσ dtZ + Z dtsσ + R dtZ σ σ nonequilibrium steady state, the external currents {I¯ y } = sσ dtZ (65) do not vanish and the entropy production reads ρ σy = J ∆rSρ + I sσy

= S˙i + S˙e . ¯˙ ¯σy T Si = I µσy . (60) This relation is the nonequilibrium formulation of the This expression can be rewritten as a bilinear form of Second Law of Thermodynamics for CRNs. It demon- emergent cycle aﬃnities {Aε} (49) and currents along strates that the non-negative entropy production (55) ¯ε ε ¯ρ emergent cycle {J ≡ cρJ } [56] measures the entropy changes in the system plus those in the reservoirs (thermal and chemostats) [57]. ¯ ε T S˙i = J¯ Aε , (61) E. Energy Balance which clearly emphasizes the crucial role of emergent cycles in steady state dissipation. 1. First Law of Thermodynamics

Since the CRN is kept at constant pressure p, its en- 2. Entropy Flow Rate thalpy

σ ◦ The entropy flow rate measures the reversible entropy H = Z hσ + H0 (66) changes in the environment due to exchange processes is equal to the CRN internal energy, up to a constant. with the system [57]. Using the expressions for the en- Indeed, the enthalpy H is a density which, when written in thalpy of reaction (52) and entropy of formation (53), we terms of the internal energy (density) U, reads H = U +p. express the entropy flow rate as Using the rate equations (9) and (10), the enthalpy rate of change can be expressed as the sum of the heat flow T S˙ ≡ J ρ ∆ H +Iσy T s . (62) e r ρ σy rate, defined in eq. (62), and the enthalpy of formation | {z } ≡ Q˙ exchange rate

◦ σ σy ◦ dtH = h dtZ = Q˙ + I h . (67) The first contribution is the heat flow rate (positive if heat σ σy is absorbed by the system). When divided by temperature, Equivalently, it can be rewritten in terms of the entropy it measures minus the entropy changes in the thermal flow rate (62) as [57, § 4.1.2] bath. The second contribution accounts for minus the ˙ σy entropy change in the chemostats. dtH = T Se + I µσy . (68) 11

The last term on the r.h.s. of eq. (68) is the free energy free energy G of the generic distribution {Zσ} defined exchanged with the chemostats. It represents the chemical above is related to Geq (73) by work rate performed by the chemostats on the CRN [21, σ σ 23] G = Geq + RT L {Z }|{Zeq} , (74) where we introduced the relative entropy for non- W˙ ≡ Iσy µ . (69) c σy normalized concentration distributions, also called Shear Either eq. (67) or (68) may be considered as the nonequi- Lyapunov Function, or pseudo-Helmholtz function [35, 73, librium formulation of the First Law of Thermodynamics 74] for CRNs. The former has the advantage to solely focus σ 0σ σ Zσ S 0S on energy exchanges. The latter contains entropic contri- L ({Z }|{Z }) ≡ Z ln − Z − Z ≥ 0 . (75) Z0 butions but is appealing because it involves the chemical σ work (69). This quantity is a natural generalization of the relative entropy, or Kullback–Leibler divergence, used to com- pare two normalized probability distributions [75]. For 2. Nonequilibrium Gibbs Free Energy simplicity, we still refer to it as relative entropy. It quan- tifies the distance between two distributions: it is always We are now in the position to introduce the thermody- positive and vanishes only if the two distributions are σ 0σ namic potential regulating CRNs. The Gibbs free energy identical, {Z } = {Z }. Hence, eq. (74) proves that the of ideal dilute solutions reads nonequilibrium Gibbs free energy of a closed CRN is always greater or equal than its corresponding equilibrium σ S G ≡ H − TS = Z µσ − RTZ + G0 . (70) form, G ≥ Geq. We now proceed to show that the nonequilibrium Gibbs As for entropy, the total concentration term −RTZS and free energy is minimized by the dynamics in closed CRNs, σ σ the constant G0 represent the contribution of the solvent viz. G—or equivalently L {Z }|{Zeq} [59, 76]—acts as (see app.A). Furthermore, in presence of reactions, G a Lyapunov function in closed CRNs. Indeed, the time becomes the nonequilibrium Gibbs free energy of CRNs. derivative of G (70) always reads We will now show that the nonequilibrium Gibbs free σ σ S energy of a closed CRN is always greater or equal than dtG = µσ dtZ + Z dtµσ + R dtZ σ (76) its corresponding equilibrium form. A generic nonequi- = µσ dtZ . librium concentration distribution {Zσ} is characterized by the set of components {Lλ = `λ Zσ}. Let {Zσ } be When using the rate equation for closed CRNs (7) we σ eq ρ σ the corresponding equilibrium distribution defined by the find that dtG = −J ∇ρ µσ. Using eq. (74) together with detailed-balance property (18) and characterized by the eqs. (46) and (58), we get same set of components {Lλ} (a formal expression for d G = RT d L {Zσ}|{Zσ } = −T S˙ ≤ 0 , (77) the equilibrium distribution will be given in eq. (88)). At t t eq i equilibrium, the Gibbs free energy (70) reads which proves the aforementioned result.

σ eq S Geq = Zeq µσ − RTZeq + G0 . (71) 3. Chemical Work As discussed in §IIIB, the equilibrium chemical potentials must satisfy ∇σ µeq = 0. We deduce that µeq must be a ρ σ σ In arbitrary CRNs, the rate of change of nonequilibrium linear combination of the closed system conservation laws Gibbs free energy (76) can be related to the entropy (12) production rate (59) using the rate equations of open eq λ CRN (9) and (10) and the chemical work rate (69), µσ = fλ `σ , (72) T S˙i = W˙ c − dtG ≥ 0 . (78) where {fλ} are real coefficients. Thus, we can write the equilibrium Gibbs free energy as This important results shows that the positivity of the entropy production sets an intrinsic limit on the chemical λ S Geq = fλ L − RTZeq + G0 . (73) work that the chemostats must perform on the CRN to change its concentration distribution. The equality sign In this form, the first term of the Gibbs free energy is achieved for quasi-static transformations (S˙ ' 0). λ i appears as a bilinear form of components {L } and con- If we now integrate eq. (78) along a transformation jugated generalized forces {fλ} [23, § 3.3], which can generated by an arbitrary time dependent protocol π(t), be thought of as chemical potentials of the components. which drives the CRN from an initial concentration dis- From eq. (72) and the properties of components (13), the tribution {Zσ} to a final one {Zσ}, we find σ eq σ eq i f equality Zeq µσ = Z µσ follows. Hence, using the definition of chemical potential (45), the nonequilibrium Gibbs T ∆iS = Wc − ∆G ≥ 0 , (79) 12 where ∆G = Gf − Gi is the difference of nonequilibrium Gibbs free energies between the final and the initial state. Let us also consider the equilibrium state {Zσ } (resp. {Zσ} eqi i {Zσ }) obtained from {Zσ} (resp. {Zσ}) if one closes eqf i f σ the network (i.e. interrupt the chemostatting procedure) {Zf } and let it relax to equilibrium, as illustrated in fig.4. The Gibbs free energy difference between these two equilibrium non-equilibrium driving distributions, ∆Geq = Geqf − Geqi, is related to ∆G via the difference of relative entropies, eq. (74), relaxation* ∆G = ∆Geq + RT ∆L , (80) relaxation* where equilibrium driving

σ σ σ σ σ σ ∆L ≡ L {Zf }|{Zeq } − L {Zi }|{Zeq } . (81) {Z } {Z } f i eqi eqf Thus, the chemical work (79) can be rewritten as FIG. 4: Pictorial representation of the transformation between two nonequilibrium concentration distributions. The Wc − ∆Geq = RT ∆L + T ∆iS, (82) nonequilibrium transformation (blue line) is compared with the equilibrium one (green line). The equilibrium which is a key result of our paper. ∆Geq represents transformation depends on the equilibrium states corre- the reversible work needed to reversibly transform the sponding to the initial and final concentration distribu- CRN from {Zσ } to {Zσ }. Implementing such a re- tions. In § IIIE3, for an arbitrary CRN, these equilibrium eqi eqf versible transformation may be difficult to achieve in states are obtained by first closing the network and then practice. However, it allows us to interpret the difference letting it relax to equilibrium. Instead, in sec.V, for a de- irr tailed balance CRN, the equilibrium states are obtained by Wc ≡ Wc − ∆Geq in eq. (82) as the chemical work dissi- pated during the nonequilibrium transformation, i.e. the simply stopping the time-dependent driving and letting the system spontaneously relax to equilibrium. irreversible chemical work. The positivity of the entropy production implies that Let us first observe that whenever a CRN displays a W irr ≥ RT ∆L . (83) c well-defined steady-state distribution {Z¯σx }, the entropy This relation sets limits on the irreversible chemical work production rate (55) can be formally decomposed as the involved in arbitrary far-from-equilibrium transformations. sum of an adiabatic and a nonadiabatic contribution For transformations connecting two equilibrium distribu- J¯ Z ρ +ρ σx σx irr T S˙i = J RT ln −dtZ RT ln , (84) tions, we get the expected inequality Wc ≥ 0. More ¯ ¯ J−ρ Zσx interestingly, eq. (83) tells us how much chemical work | {z } | {z } ˙ the chemostat need to provide to create a nonequilibrium ≡ T S˙a ≡ T Sna distribution from an equilibrium one. It also tells us how much chemical work can be extracted from a CRN in analogy to what was done in Stochastic Thermody- relaxing to equilibrium. namics [46–50]. As discussed in §IIE, unconditionally The conceptual analogue of (82) in Stochastic Thermo- complex-balanced networks have a unique steady-state ¯σx ¯σx dynamics (where probability distributions replace non- distribution Z ≡ Z (π(t)) , eq. (37), for any value of σ σ normalized concentration distributions) is called the the chemostatted concentrations {Z y ≡ Z y (π(t))} and λ nonequilibrium Landauer’s principle [42, 43] (see also of the fixed unbroken components {L u }. The decomposi- [77–79]). It has been shown to play a crucial role to ana- tion (84) is thus well-defined at any time, for any protocol lyze the thermodynamic cost of information processing π(t). As a central result, we prove in app.C that the (e.g. for Maxwell demons, feedback control or proofread- adiabatic and nonadiabatic contribution are non-negative ing). The inequality (83) is therefore a nonequilibrium for unconditionally complex-balanced networks as well as Landauer’s principle for CRN. for complex-balanced networks without time-dependent driving. The adiabatic entropy production rate encodes the ¯σx IV. THERMODYNAMICS OF dissipation of the steady state Z . It can be rewritten COMPLEX-BALANCED NETWORKS in terms of the steady state Gibbs free energy of reaction {∆rG¯ρ} (48) as In this section, we focus on unconditionally complex- ρ T S˙a = −J ∆rG¯ρ ≥ 0 . (85) balanced networks. We shall see that the thermodynamics of these networks bears remarkable similarities with This inequality highlights the fact that the transient Stochastic Thermodynamics. dynamics—generating the currents {J ρ}—is constrained 13 by the thermodynamics of the complex balanced steady detailed-balanced networks always relax to a unique equi- state, i.e. by {∆rG¯ρ}. librium distribution. Since the equilibrium chemical po- The nonadiabatic entropy production rate characterizes tentials can be expressed as a linear combination of con- the dissipation of the transient dynamics. It can be servation laws, eq. (72), we can express the equilibrium decomposed as distribution as µ◦ − f `λ T S˙ = −RT d L {Zσx }|{Z¯σx } + Zeq = exp − σ λ σ , (88) na t σ RT ¯Sx σx + RT dtZ − Z dtµ¯σx ≥ 0 , (86) | {z } ˙ inverting the expression for the chemical potentials (45). ≡ T Sd Since the independent set of unbroken conservation laws, λu λu {` }, are such that `σ = 0, ∀λu, σy, see §IIC, we have where Z¯Sx = P Z¯σx (see Refs. [46, 48] for the anal- y σx∈Sx that ogous decomposition in the stochastic context). The first term is proportional to the time derivative of the relative eq λb µσ = fλb `σ , ∀ σy ∈ Sy . (89) entropy (75) between the nonequilibrium concentration y y distribution at time t and the corresponding complex- We thus conclude that the |λb| broken generalized forces balanced steady-state distribution. Hence, it describes {fλb } only depend on the chemostatted concentrations σy the dissipation of the relaxation towards the steady state. {Z }. Instead, the remaining |λu| unbroken generalized The second term, T S˙d, is related to the time-dependent forces fλu can be determined by inverting the nonlinear driving performed via the chemostatted species and thus set of equations Lλu = `λu Zσx . They therefore depend σx eq denoted driving entropy production rate [46]. It vanishes on both {Zσy } and {Lλu }. in nondriven networks where we obtain One can easily recover the local detailed-balanced property (50) and (18) using eq. (88). σx σx S˙na = −R dtL {Z }|{Z¯ } ≥ 0 . (87)

This result shows the role of the relative entropy B. Open nondriven networks L {Zσx }|{Z¯σx } as a Lyapunov function in nondriven complex-balanced networks with mass action kinetics. It was known in the mathematical literature [35, 80], but As a consequence of the break of conservation laws, we provide a clear thermodynamic interpretation to this the nonequilibrium Gibbs free energy G (70) is no longer result by demonstrating that it derives from the nonadia- minimized at equilibrium in open detailed-balanced net- batic entropy production rate. works. In analogy to equilibrium thermodynamics [23], We mention that an alternative derivation of the the proper thermodynamic potential is obtained from G adiabatic–nonadiabatic decomposition for nondriven by subtracting the energetic contribution of the broken complex-balanced networks with mass action kinetics was conservation laws. This transformed nonequilibrium Gibbs found in Ref. [51], while we were ﬁnalizing our paper. free energy reads

λb G ≡ G − fλ L b (90) σ λb S V. THERMODYNAMICS OF OPEN = Z µσ − fλb `σ − RTZ + G0 . DETAILED-BALANCED NETWORKS We proceed to show that G is minimized by the dy- We ﬁnish our study by considering detailed-balanced namics in nondriven open detailed-balanced networks. σx networks. We discuss the equilibrium distribution, intro- Let {Z } be a generic concentration distribution in a λu duce a new class of nonequilibrium potentials and derive detailed-balanced network characterized by {L } and σy σx a new work inequality. {Z }, and let {Zeq } be its corresponding equilibrium. Using the relation between equilibrium chemical poten- Let us also emphasize that open detailed-balanced tials and conservation laws (72), the transformed Gibbs CRNs are a special class of open complex-balanced CRNs free energy (90) at equilibrium reads for which the adiabatic entropy production rate vanishes (since the steady state is detailed balanced) and thus the G = f Lλu − RTZS + G . (91) nonadiabatic entropy production characterizes the entire eq λu eq 0 dissipation. Yet, combinig eq. (72) and the properties of unbroken com- σ eq λb ponents, one can readily show that Zeq µσ − fλb `σ = σ eq λb Z µσ − fλb `σ . The relation between the nonequi- A. Equilibrium Distribution librium G and the corresponding equilibrium value thus follows As discussed in §IID, for given kinetics {k±ρ}, σy λu σx σx chemostatting {Z } and unbroken components {L }, G = Geq + RT L {Z }|{Zeq } (92) 14

(we show in app.A the derivation of the latter in presence Example 12. A simple implementation of this scenario of reacting solvent). The non-negativity of the relative is the thermodynamic description of CRNs at constant σx σx entropy for concentration distributions L {Z }|{Zeq } pH [23, Ch. 4] where the chemostatted species becomes + ensures that the nonequilibrium transformed Gibbs free the ion H+ and M H is the total amount of H+ ions energy is always greater or equal to its equilibrium value, in the system. The transformed Gibbs potential thus + G ≥ Geq. Since entropy production and nonadiabatic 0 H become G = G−µH+ M and the transformed chemical entropy production coincide, using eqs. (87) and (92), we potentials can be written in our formalism as µ0 = σx obtain ˆH+ b b µσx −µH+ `b `σ, where `σ is the conservation law broken + σx σx ˙ by chemostatting H . dtG = RT dtL {Z }|{Zeq } = −T Si ≤ 0 , (93) Example 13. For the CRN in fig.2, whose conserva- which demonstrates the role of G as a Lyapunov function laws given in example4, the concentrations of the σ 0σ tion. The relative entropy L ({Z x }|{Z x }) was known exchanged moieties are to be a Lyapunov function for detailed-balanced networks 1 a 1 b [76, 81], but we provided its clear connection to the trans- M = Z + 2 Z formed nonequilibrium Gibbs free energy. To summarize, (97) M 2 = Zd + Ze . instead of minimizing the nonequilibrium Gibbs free energy G (70) as in closed CRNs, the dynamics minimizes For the specific implementation of that CRN, fig.3, the the transformed nonequilibrium Gibbs free energy G in first term (resp. second term) is the total number of moi- open nondriven detailed-balanced CRNs. ety 2H (resp. C) in the system, which can be exchanged with the environment only via the chemostatted species H2O (resp. CO). C. Open driven networks We now turn to the new work relation. From the general work relation (78), using (90) and (95), we find We now consider unconditionally detailed-balanced

CRNs. As discussed in §IID, they are characterized T S˙i = W˙ d − dtG ≥ 0 , (98) σx σx by a unique equilibrium distribution Zeq ≡ Zeq (π(t)) , defined by eq. (18), for any value of the chemostatted where the driving work due to the time-dependent driving concentrations {Zσy = Zσy (π(t))}. of the chemostatted species is obtained using the chemical We start by showing that the external fluxes {Iσy } work rate (69) together with eqs. (95) and (96) can be expressed as influx rate of moieties. Since the ˙ ˙ λb CRN is open and unconditionally detailed balanced, each Wd ≡ Wc − dt fλb L chemostatted species broke a conservation law (no emer- eq σy eq σy = µσ dtM − dt µσ M (99) gent cycle is created, §IID). Therefore, the matrix whose y y eq σy λb = −dtµ M . entries are {` } in eq. (89) is square and also nonsingular σy σy [82]. We can thus invert eq. (89) to get Equivalently, the driving work rate (99) can be defined as the rate of change of the transformed Gibbs free energy f = µeq `ˆσy , (94) λb σy λb (90) due to the time dependent driving only, i.e.

σ where {`ˆ y } denote the entries of the inverse matrix of ˙ ∂G eq ∂G λb Wd ≡ ≡ dtµσ eq . (100) λ y that with entries {` b }. Hence, using the definition of ∂t ∂µσy σy λb λb σ broken component, {L ≡ `σ Z }, we obtain that To relate this alternative definition to eq. (99), all {Zσy } must be expressed in terms of {µeq } using the definition σ σy λb eq ˆ y λb σ fλ L = µ ` ` Z . (95) b σy λb σ of chemical potential (45). | {z } The driving work rate W˙ vanishes in nondriven CRNs, ≡ M σy d where (98) reduces to (93). After demonstrating that From the rate equations for the chemostatted concentra- the entropy production rate is always proportional the tions (10), we find that difference between the chemical work rate and the change of nonequilibrium Gibbs free energy in eq. (79), we showed σy σy dtM = I , ∀ σy ∈ Sy . (96) that, for unconditionally detailed-balanced CRNs, it is also proportional the difference between the driving work We can thus interpret M σy as the concentration of a rate and the change in transformed nonequilibrium Gibbs moiety which is exchanged with the environment only free energy, eq. (98). through the chemostatted species Xσy . Eq. (95) shows We end by formulating a nonequilibrium Landauer’s that the energetic contribution of the broken components principle for the driving work instead the chemical work can be expressed as the Gibbs free energy carried by these done in § IIIE3. We consider a time-dependent trans- specific moieties. formation driving the unconditionally detailed-balanced 15

CRN from {Zσ} to {Zσ}. The distribution {Zσ } (resp. initial and final concentration distributions from their i f eqi {Zσ }) denotes the equilibrium distribution obtained corresponding equilibrium ones, obtained by closing the eqf σ σ network. This result is reminiscent of the nonequilibrium from {Zi } (resp. {Zf }) by stopping the time-dependent driving and letting the system relax towards the equilib- Landauer’s principle derived in Stochastic Thermodynam- rium, fig.4. Note that this reference equilibrium state is ics [43] and which proved very useful to study the energetic different from the one obtained by closing the network in cost of information processing [45]. We also highlighted § IIIE3. Integrating (98) over time and using (92), we the deep relationship between the topology of CRNs, their get dynamics and their thermodynamics. Closed CRNs (resp. nondriven open detailed-balanced networks) always relax Wd − ∆Geq = RT ∆L + T ∆iS, (101) to a unique equilibrium by minimizing their nonequilibrium Gibbs free energy (resp. transformed nonequilibrium where Gibbs free energy). This latter is given, up to a constant, by the relative entropy between the nonequilibrium ∆L ≡ L{Zσx }|{Zσx }−L{Zσx }|{Zσx } . (102) f eq f i eq i and equilibrium concentration distribution. Non-driven complex-balanced networks relax to complex-balanced ∆Geq represents the reversible driving work, and the irre- nonequilibrium steady states by minimizing the relative versible driving work satisfies the inequality entropy between the nonequilibrium and steady state irr concentration distribution. In all these cases, even in Wd ≡ Wd − ∆Geq ≥ RT ∆L . (103) presence of driving, we showed how the rate of change of the relative entropy relates to the CRN dissipation. This central relation sets limits on the irreversible work For complex-balanced networks, we also demonstrated spent to manipulate nonequilibrium distributions. It is a that the entropy production rate can be decomposed, nonequilibrium Landauer’s principle for the driving work as in Stochastic Thermodynamics, in its adiabatic and by the same reasons why inequality (83) is a nonequilib- nonadiabatic contributions quantifying respectively the rium Landauer’s principle for the chemical work. The dissipation of the steady state and of the transient dy- key difference is that the choice of the reference equi- namics. librium state is different in the two cases. The above discussed inequality (103) only holds for unconditionally Our framework could be used to shed new light on a detailed-balanced CRNs while eq. (83) is valid for any broad range of problems. We mention only a few. CRNs. Stochastic thermodynamics has been successfully used to study the thermodynamics cost of information processing in various synthetic and biological systems [44, 83–87]. VI. CONCLUSIONS AND PERSPECTIVES However, most of these are modeled as few state systems or linear networks [8, 9]—e.g. quantum dots [88], Following a strategy reminiscent of Stochastic Ther- molecular motors [89, 90] and single enzyme mechanisms modynamics, we systematically build a nonequilibrium [91, 92]—while biochemical networks involve more com- thermodynamic description for open driven CRNs made plex descriptions. The present work overcomes this limi- of elementary reactions in homogeneous ideal dilute so- tation. It could be used to study biological information- lutions. The dynamics is described by deterministic rate handling processes such as kinetic proofreading [93–99] equations whose kinetics satisfies mass action law. Our or enzyme-assisted copolymerization [92, 100–105] which framework in not restricted to steady states and allows have currently only been studied as single enzyme mecha- to treat transients as well as time-dependent drivings nisms. performed by externally controlled chemostatted concen- Our theory could also be used to study metabolic net- trations. Our theory embeds the nonequilibrium thermo- works. However, these require some care, since complex dynamic framework of irreversible processes established enzymatic reaction mechanisms are involved [106]. Nev- by the Brussels School of Thermodynamics. ertheless our framework provides a basis to build effec- We now summarize our results. Starting from the ex- tive coarse-graining procedures for enzymatic reactions pression for the entropy production rate, we established [107]. For instance, proofreading mechanisms operating a nonequilibrium formulation of the first and second law in metabolic processes could be considered [108]. We of thermodynamics for CRNs. The resulting expression foresee an increasing use of thermodynamics to improve for the system entropy is that of an ideal dilute solution. the modeling of metabolic networks, as recently shown in The clear separation between chemostatted and internal Refs. [30, 32, 33]. species allowed us to identify the chemical work done Since our framework accounts for time-dependent driv- by the chemostats on the CRN and to relate it to the ings and transient dynamics, it could be used to repre- nonequilibrium Gibbs potential. We were also able to sent the transmission of signals through CRNs or their express the minimal chemical work necessary to change response to external modulations in the environment. the nonequilibrium distribution of species in the CRN These features become crucial when considering prob- as a difference of relative entropies for non-normalized lems such as signal transduction and biochemical switches distributions. These latter measure the distance of the [24, 109, 110], biochemical oscillations [28, 111], growth 16 and self-organization in evolving bio-systems [112, 113], on the nature of the solvent. Hence, the chemical poten- or sensory mechanisms [85, 87, 114–117]. Also, since tran- tials of the solutes read sient signals in CRNs can be used for computation [118] ◦ Zσ and have been shown to display Turing-universality [119– µσ ' µσ + RT ln , ∀ σ ∈ S , (A3) 122], one should be able to study the thermodynamic cost Z0 of chemical computing [123]. while that of the solvent Finally, one could use our framework to study any S process that can be described as nucleation or reversible ◦ Z µ0 ' µ0 − RT , (A4) polymerization [124–129] (see also Ref. [130, ch. 5–6]) Z0 since these processes can be described as CRNs [63]. S P σ As closing words, we believe that our results consti- where Z ≡ σ∈S Z . Therefore, the Gibbs free energy tute an important contribution to the theoretical study (A1) reads of CRNs. It does for nonlinear chemical kinetics what G ' Zσ µ + Z0 µ◦ − RTZS , (A5) Stochastic Thermodynamics has done for stochastic dy- σ 0 namics, namely build a systematic nonequilibrium ther- which is eq. (70) in the main text, where G0 is equal to modynamics on top of the dynamics. It also opens many 0 ◦ Z µ0 plus possibly the Gibbs free energy of solutes which new perspectives and builds bridges between approaches do not react. from different communities studying CRNs: mathemati- We now consider the case where the solvent reacts with cians who study CRNs as dynamical systems, physicists the solutes. We assume that both the solutes and the who study them as nonequilibrium complex systems, and solvent react according to the stoichiometric matrix biochemists as well as bioengineers who aim for accurate models of metabolic networks. ∇0  ∇ = ∇X , (A6) ∇Y ACKNOWLEDGMENTS where the first row refers to the solvent, the second The present project was supported by the National block of rows to the internal species and the last one Research Fund, Luxembourg (project FNR/A11/02 and to the chemostatted species. The solvent is treated as a AFR PhD Grant 2014-2, No. 9114110) as well as by the chemostatted species such that dtZ0 = 0. European Research Council (project 681456). In order to recover the expression for the transformed Gibbs free energy (92) in unconditionally detailed balanced networks, we observe that, at equilibrium Appendix A: Thermodynamics of Ideal Dilute σ eq 0 eq Solutions ∇ρ µσ + ∇ρ µ0 = 0 . (A7) Therefore, the equilibrium chemical potentials are a linear We show that the nonequilibrium Gibbs free energy combination of the conservation laws of ∇ (A6) (70) is the Gibbs free energy of an ideal dilute solution [131, Ch. 7] (see also [51]). We also show that in open eq λ µ = fλ ` σ σ (A8) detailed-balanced networks in which the solvent reacts eq λ with the solutes, the expression of the transformed Gibbs µ0 = fλ `0 . free energy (92) is recovered by treating the solvent as a As mentioned in the main text, §IIC, the chemostat- special chemostatted species. ting procedure breaks some conservation laws, which are The Gibbs free energy (density) of an ideal dilute mix- labeled by λ . The unbroken ones are labeled by λ . ture of chemical compounds kept at constant temperature b u The transformed Gibbs free energy is defined as in and pressure reads eq. (90), reported here for convenience σ G = Z µσ + Z0µ0 , (A1) λb G ≡ G − fλb L , (A9) where the labels σ ∈ S refer to the solutes and 0 to the where G reads as in eq. (A1), {Lλb } are the broken com- solvent. The chemical potentials of each species (45) read ponents and {fλb } are here interpreted as the conjugated generalized forces. Adding and subtracting the term σ eq eq Z Z µσ + Z0µ0 from the last equation and using eq. (A8) µ = µ◦ + RT ln σ , ∀ σ ∈ S σ σ Z we obtain tot (A2) Z0 σ eq 0 eq ◦ G = Geq + Z (µσ − µ ) + Z (µ0 − µ ) , (A10) µ0 = µ0 + RT ln . σ 0 Ztot where P σ Since the solution is dilute, Ztot = σ∈S Z + Z0 ' Z0 ◦ λu and the standard state chemical potentials {µσ} depend Geq = fλu L . (A11) 17

σy σy From eqs. (A3) and (A4) and the fact that Z = Zeq Dividing by V and using the relation R = NAkB we ﬁnally eq and Z0 = Z0 we obtain get the macroscopic entropy density (63)

σ ◦ σ S σx eq Sx Sx G'G + Z µ − µ − RT Z − Z hSi/V ' Z sσ − Z R ln Zσ + RZ , (B5) eq σx σx eq Zσx σx Sx Sx ◦ where the (molar) standard entropies of formation sσ = Geq + Z RT ln σx − RT Z − Zeq Zeq reads σx σx ≡ Geq + RT L({Z }|{Zeq }) s◦ = N (˜s◦ + k ln Z ) . (B6) (A12) σ A σ B 0 in agreement with the expression derived in the main text, Mindful of the information-theoretical interpretation eq. (92). of the entropy [133], we note that the uncertainty due to the stochasticity of the state disappears (the first term on the r.h.s. of eq. (B1)). However, the uncertainty Appendix B: Entropy of CRNs due to the indistinguishability of the molecules of the same species—quantified by the configurational entropy (B2)—remains and contributes to the whole deterministic We show how the nonequilibrium entropy (63) can entropy function (63). be obtained as a large particle limit of the stochastic entropy. We point out that while finalizing the paper similar derivations for other thermodynamic quantities Appendix C: Adiabatic–Nonadiabatic have obtained in Refs. [51, 132]. Decomposition In the stochastic description of CRNs, the state is characterized by the population vector n = {nσ}. The probability to find the network is in state n at time t is We prove the positivity of the adiabatic and nonadiabatic entropy production rates (84) using the theory of denoted pt(n). The stochastic entropy of that state reads [21, 107], up to constants, complex-balanced networks, see §IIE. We first rewrite the mass action kinetics currents (5) 0 ρ 0 γ S(n) = −kB ln pt(n) + s(n) . (B1) ρ γ γ σΓσ as [53, 81]: J = Kγ0 ψ , where ψ ≡ Z and K = ρ +ρ −ρ The first term is a Shannon-like contribution while the {Kγ ≡ Kγ − Kγ } is the rate constants matrix whose second term is the configurational entropy entries are defined by X nσ!  s(n) ≡ nσs˜◦ − k ln . (B2) k+ρ if γ is the reactant complex of +ρ , σ B nσ  n0 ρ −ρ σ Kγ = −k if γ is the product complex of +ρ , ◦  s˜σ is the standard entropy of one single Xσ molecule, and 0 otherwise . n0 is the very large number of solvent molecules. (C1) We now assume that the probability becomes very narrow in the large particle limit nσ 1 and behaves as Hence, the definition of complex-balanced network (37) a discrete delta function pt(n) ' δ(n − nˆ(t)). The vector reads nˆ(t) ≡ {nˆσ} denotes the most probable and macroscopic 0 σ σ X γ ¯γ amount of chemical species, such that Z = nˆ /(VNA). Wγ0 ψ = 0 , ∀j . (C2) Hence, the average entropy becomes γ∈Cj

X γ ρ γ hSi = pt(n)S(n) ' s(nˆ) . (B3) where W ≡ ∂ K = {∂ρ Kγ0 } ≡ {Wγ0 } is the so-called Γγ n kinetic matrix [35], and ψ¯γ ≡ Z¯σ σ . When using the Stirling approximation (ln m! ' m ln m − The kinetic matrix W is a Laplacian matrix [76, 81]: m for m 1), we obtain any off-diagonal term is equal to the rate constant of the reaction having γ0 as a reactant and γ as product if the nˆσ X s(nˆ) ' nˆσs˜◦ − nˆσ k ln + k nˆσ reaction exists, and it is zero otherwise. Also, it satisfies σ B n B 0 σ X γ W 0 = 0 , (C3) σ ◦ n0 γ =n ˆ s˜σ + kB ln + γ∈C VNA nˆσ X (B4) − nˆσ k ln + k nˆσ which is a consequence of the fact that the diagonal terms B VN B are equal to minus the sum of the off-diagonal terms along A σ the columns. The detailed balanced property (18) implies ≡ nˆσ (˜s◦ + k ln Z ) + σ B 0 that σ σ X σ − nˆ kB ln Z + kB nˆ . γ eq γ0 eq 0 σ Wγ0 ψγ0 = Wγ ψγ , ∀ γ, γ , (C4) 18

σ eq eq Γγ where ψγ ≡ Zσ . Because of (C3), we further get that In order to prove the non-negativity of the adiabatic term (84), we rewrite it as ¯γ0 ˙ γ γ0 ψγ ψ Sna = −W 0 ψ ln . (C10) −∇σ γ ¯ γ0 ¯ ¯ ρ ψγ ψ ˙ ρ J+ρ ρ γ0 Zσ Sa ≡ J ln = Kγ0 ψ ln eq J¯ Zσ −ρ (C5) ¯ From the log inequality − ln x ≥ 1 − x and from (C4) γ γ0 ψγ = −Wγ0 ψ ln eq . ψγ ¯γ0 ! ˙ γ γ0 ψγ ψ γ ¯γ0 ψγ The detailed balance property is used in the ﬁrst equality, S ≥ W 0 ψ 1 − = −W 0 ψ = 0 . na γ ¯ γ0 γ ¯ and the decomposition of the stoichiometric matrix (30) ψγ ψ ψγ in the second one. Also, the constant RT is taken equal (C11) to one. Using (C3), eq. (C5) can be rewritten as

γ0 The last equality again follows from the assumption of 0 ψ¯ ψ ˙ γ γ γ eq S = −W 0 ψ ln . (C6) complex-balance steady state (C2) as in eq. (C8). a γ eq ¯γ0 ψγ ψ Appendix D: Deficiency of CRNs From the log inequality − ln x ≥ 1 − x and the detailed balance property (C4), we obtain The deficiency of an open CRN is defined as [22]

γ0 ! 0 ψ¯ ψ X C ˙ γ γ γ eq δ = dim ker ∇ − dim ker ∂ ≥ 0 , (D1) S ≥ W 0 ψ 1 − a γ eq γ0 ψγ ψ¯ ¯ γ0 where ∂C = {∂j ≡ P ∂γ }. Other equivalent defini- γ γ0 ψγ ψ γ0 ¯γ ψγ0 ρ γ∈Cj ρ = −W 0 ψ = −W ψ = 0 . X γ eq eq ¯γ0 γ ¯ tions can be found in [52] and [53]. The kernel of ∇ ψγ ψ ψγ0 identifies the set of cycles, eqs. (19) and (20), while the (C7) kernel of the incidence matrix ∂ˆ identifies the set of cycles The last equality follows from the assumption of complex- of the reaction graph. Hence, the deficiency measures the balanced steady state (C2), the properties of the groups difference between the number of cyclic transformations ¯ on chemical species and how many of them can be repre- of complexes {Cj} (§IIE), and the fact that {Zσy = Zσy }. Indeed sented as cycles on the reaction graph. Deficiency-zero networks are defined by δ = 0, i.e. they exhibit a one to σx Γ 0 0 ψ 0 0 Z γ one correspondence between the two. This topological γ ¯γ γ X X γ ¯γ σx Wγ ψ = Wγ ψ property has many dynamical consequences, the most ¯ 0 ¯ ψγ 0 Zσx j γ ∈Cj important of which is that deficiency-zero networks are σx (C8) Γj unconditionally complex-balanced [36, 37]. As shown in X Zσ X 0 = x Wγ ψ¯γ = 0 . ¯ γ Ref. [22], deficiency has also implications on the stochastic Zσx 0 j γ ∈Cj thermodynamic description of networks: the stochastic entropy production of a deficiency-zero network converges Concerning the nonadiabatic term (84), using the rate to the deterministic entropy production in the long-time ¯ equations (9) and the fact that {Zσy = Zσy }, we can limit. Linear networks are the simplest class of deficiency- rewrite it as zero networks. Since only one internal species appears in each complex with a stoichiometric coefficient equal to ˙ σ Zσ γ γ0 ψγ X C S ≡ −d Z ln = −W 0 ψ ln . (C9) one, ∇ ≡ ∂ and thus δ = 0. na t ¯ γ ¯ Zσ ψγ

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