Circuit Theory for Chemical Reaction Networks

We lay the foundation of a circuit theory for chemical reaction networks. Chemical reactions are grouped into chemical modules solely characterized by their current-concentration characteristic, as electrical devices by their current-voltage (I-V) curve in electronic circuit theory. This, combined with the chemical analog of Kirchhoff's current and voltage laws, provides a powerful tool to predict reaction currents and dissipation across complex chemical networks. The theory can serve to build accurate reduced models of complex networks as well as to design networks performing desired tasks.


I. INTRODUCTION
Chemical reaction networks (CRNs) are ubiquitous in nature and can easily reach high levels of complexity.Combustion [1], atmospheric chemistry [2,3], geochemistry [4], biochemistry [5], biogeochemistry [6,7], ecology [8], provide some examples.e complexity of many of these networks arises from their large size and complex topology (encoded in the stoichiometric matrix), from the non-linearity of chemical kinetics, and from the fact that they do not operate in closed vessels.ey continuously exchange energy and matter with their surroundings thus maintaining chemical reactions out of equilibrium [9,10].
eir detailed characterization would require knowing the currents through all the reactions which, for elementary reactions satisfying massaction kinetics, implies the knowledge of the reaction rate of every reaction and of the concentrations of all the species.Naturally, such knowledge is very seldom achieved.Some approaches seek to develop reduced models of CRNs o en based on eliminating the fast-evolving species [11][12][13][14].Other approaches such as ux balance analysis impose a complicated mix of constraints (physical and experimental) and objective functions (enforcing biologically desired results) to determine the currents through the CRN and avoid using kinetic information about the system [15][16][17] (see also Sec.V).In both cases ensuring the thermodynamic consistency of the schemes has been a major topic of concern in recent years [18][19][20][21][22][23].
In this paper we present a novel approach: a thermodynamically consistent circuit theory of CRNs, inspired by electronic circuit theory.In CRNs elementary reactions transform chemical species into each other, while in electrical circuits devices transfer charges between conductors.But electronic devices are complex objects and the charge transfers are not characterized at an elementary level but instead in terms of current-voltage (I-V) curves which are o en determined experimentally or may also be computed using a more detailed description of the inner workings of the device.We do the same for CRNs.We group elementary reactions into chemical modules that are then solely characterized by their current-concentration curves between terminal species.e current-concentration curve of a chemical module thus corresponds to the I-V curve of an electronic device, but di ers from it in an important point.While the electric currents only depend on the di erence between the electrostatic potentials applied to the terminals of the devices, the chemical currents are functions of the concentrations and, consequently, depend on the absolute value of the chemical potentials of the terminal species.Another di erence between the two circuit theories is that conservation laws in CRNs are signi cantly more complicated than in electronic circuits where only charge conservation is involved.Chemical circuit theory may become an important tool to study and design complex CRNs, in the same way that electronic circuit theory for electrical circuits has become the cornerstone of electrical engineering.To get there, experimental methodologies to determine current-concentration curves should be developed.is should be within reach thanks to recent developments in micro uidics and systems chemistry [9].
In order to present our theory, we adopt a two-fold strategy.In the main text, we adopt an informal style and present the theory by examples, as we simplify the description of the CRN depicted in Fig. 1a into the one depicted in Fig. 1b, rst identifying the chemical modules (Sec.II) and then characterizing them in terms of their current-concentration characteristic (Sec.III).e formal theory is instead presented in App. A. e more mathematically inclined readers may want to start from there before turning to the main text for illustrations.
In Fig. 1a, the outer black box de nes the boundary of the entire open CRN and the species with arrows crossing that boundary have their concentration controlled by the environment.e colored boxes inside the CRN denote the chemical modules and the corresponding colored arrows denote the (elementary and reversible) reactions inside those modules.Note that the colored (internal) species change solely due to reactions in the module of the same color, while the black (terminal) species are involved in reactions of di erent modules.In Fig. 1b, the reactions within the modules are lumped into a minimal number of e ective reactions called emergent cycles.As we will see, an emergent cycle de nes a combination of elementary reactions that upon completion do not interconvert the internal species of a module, but exchange terminal species with other modules.ey have been originally introduced because they capture the entire dissipation of open CRNs at steady state [24][25][26].
e current along the emergent cycles of a module as a function of the concentrations of its terminal species de nes the currentconcentration curve of the module.ree strategies (Sec.III) can be used to determine the current-concentration characteristic.
e rst two (illustrated in App.B for some of the modules in Fig. 1a) are theoretical and require the detailed knowledge of the kinetic properties of the reactions inside the module.e third one (detailed in Sec.III) is experimental and requires the control of the concentrations of the terminal species as well as measuring their consumption/production rates.It is analogous to the way the I-V curve of an electronic device is determined.Finally, based on the currentconcentration characteristics of each module, a closed dynamics for the terminal species is obtained in Eq. ( 19) providing a simpli ed description of the original open CRN.Crucially, this coarse-grained dynamics is thermodynamically consistent (Subs.IV A) and satis es the chemical equivalent of Kirchho 's current and potential laws (Subs.IV B): the sums of currents involving each terminal species vanish at steady state and the sum of the Gibbs free energy of reaction along any closed cycle is zero, respectively.e limitations and extensions of our circuit theory are discussed in Sec.V and illustrated in more detail in App.C for the CRN in Fig. 1a.To be valid beyond steady-state conditions, our theory requires a time-scale separation between the dynamics of terminal species and the internal dynamics of the modules in such a way that the la er is uniquely determined by the former, but multistability (Subs.C 1) can be treated anyway.Modules may be merged into a super-module (Subs.C 2) or split into submodules under certain conditions (Subs.C 3).Finally, the e ective reactions can be experimentally determined without knowing the internal stoichiometry of the modules (Subs.C 4).

II. CHEMICAL MODULES
To explain how to reduce the description of a complex open CRN in terms of chemical modules, we will use the CRN depicted in Fig. 1a and reduce it to Fig. 1b.e formal description of this procedure is given in App. A. In particular, App.A 1 gives a formal de nition of modules, while in Apps.A 2 and A 3 their reduced description is derived.
In Fig. 1a, the arrows denote both the chemical reactions of the network as well as the exchange processes with the environment.e la er are represented by (gray) arrows entering the CRN from the outside and involve the exchanged species (S, F, W, P  , P  , P  , and P  ).e direction of the arrows is arbitrary (set by convention) as all reactions are assumed to be reversible.e boxes inside the CRN in Fig. 1a are the modules.Each module is a subnetwork composed of a unique set of internal species (drawn inside the module) reacting among themselves and potentially also with other species, named terminal species (drawn outside the module).For instance, the (blue) module  in Fig. 1a interconverts the internal species E  , E  S , E  S 2 and the terminal species S and N  via the chemical reactions (1) represented by the (blue) arrows labeled  1 ,  2 , and  3 (also speci ed in Fig. 2).
Chemical modules are the chemical analog of the electronic components (for example diodes, transistors, or microchips) of an electric circuit, and the terminal species are the analog of the electrical contacts or pins of each component.Arrows in Fig. 1a should however not be compared to cables or connections between components in an electronic circuit diagram.Instead, the analog of electrical connections between contacts of di erent electronic components is the chemical species shared between chemical modules, i.e., the terminal species.But while electronic components are spatially separated, chemical modules do not have to be.Assuming homogeneous solutions for simplicity, the de nition of the modules as well as their representation is based on the network of reactions and does not require any spatial organization.Situations involving spatial organization will be discussed in Sec.V.
In the circuit description depicted in Fig. 1b, each module ends up being coarse grained into e ective reactions (denoted by the arrows through the boxes) between its terminal species.e coarse graining reduces the (blue) module  to the single e ective reaction S   N  . ( e coarse-graining procedure is based on the stoichiometry of the module and starts from its stoichiometric matrix [16] also speci ed in Fig. 2, whose entries have a clear physical meaning: they encode the net variation of the number of molecules of each species (identi ed by the matrix row) undergoing each reaction (identi ed by the matrix column) [16]. is matrix is split into the substoichiometric matrices for the internal species  only nds the single emergent cycle: also reported in Fig. 2. e sequence of reactions encoded in the emergent cycle interconverts (upon completion) only the terminal species while leaving the internal species unaltered.By multiplying the substoichiometry matrix for the terminal species    in (3) and the emergent cycle    in (4), one obtains the variation of the number of molecules of terminal species along the emergent cycle, i.e., the stoichiometry of the corresponding e ective reaction (2): e general and formal discussion of the coarse-grained procedure based on the use of the emergent cycles is given in App.A 2 and App.A 3.
We now turn to the (green) module , whose internal species E  , E  F, E  W, and E *  react via the chemical reactions  1 ,  2 ,  3 ,  4 , and  5 (see Fig. 2) with the terminal species N  , N  , F, and W. From the corresponding stoichiometric matrix   (in Fig. 2), we identify two emergent cycles    and    (in Fig. 2) which correspond to the following e ective reactions between the terminal species respectively.Note that Z M E k J J J 3 F I / g K e 7 8 9 Q h 9 g 9 6 M U t 7 U k F y U g 4 P A 7 H 7 / z w y O S g j N t w v D f r c 6 V j 6 5 2 P 7 7 2 S e / 6 j U 8 / + / z m 9 q 0 X O l 8 o C m c 0 5 n 7 M R d J 3 6 K s f f v x p j P p 3 y + 3 U R q 6 P J n 7 7 3 d p 2 j T 1 C L o 4 3 q P x p v 2 0 q 7 d e V 9 t + r t I 7 V J e 7 V J e 4 1 J H w G T 9 9 t P i w 3 r M s N G 3 S + E 3 x y H a f g q 6 x o u j w + j e 4 e i 7 0 e 6 D 7 5 t n 4 k b w f v B B c C e I g s + D B 8 H T 4 H l w F v C e 7 v 3 S + 7 X 3 W / / n / u / 9 P / p / r q n X e o 3 m v a C 1 + n / 9 C 2 i B E r M = < / l a t e x i t > z n a w t 9 1 / 1 l 9 / 8 N v i m b j p f e X d 9 u 5 4 g f e 9 9 8 B 7 4 h 1 5 p x 7 v / N H 5 u / O m 8 3 b l z 5 V / u j e 6 3 T m 1 c 2 0 x 5 0 u v 8 X V X / w U q l I G t < / l a t e x i t > x u P X 4 p + q a W P M + 8 z 7 3 7 n m B 9 9 B 7 7 D 3 3 j r w T j / Z + 6 / 3 Z + 7 v 3 z 9 r v a / + u 3 1 6 / M 0 + 9 e a O a 8 6 l X a + s b / w G q A 8 2 k < / l a t e x i t > x W e c D J T O r n Q r a U o x 9 8 5 P u f z F + e Y 5 J w p 7 f v / n T j Z O X X 6 j z P d s 7 1 z 5 y / 8 e f H S 5 S t v V T a W F L Z p x j P 5 n m A F n K W w r Z n m 8 D 6 X g A X h 8 I 7 s P i 3 8 7 / Z A K p a l W 3 q S w 0 D g J s Z k r j A 4 6 I 5 B P V W 0 D T e r i w H q 8 v 9 f / s L j z / M 2 k T X u + H d 9 G 5 7 g f f A e + y 9 8 D a 9 b Y 9 2 P n e + d b 5 3 f p z 5 0 r 3 Q v d a 9 P g 0 9 e W K 2 5 q p X G d 2 / f g I r e 5 w f < / l a t e x i t > a t e x i t s h a 1 _ b a s e 6 4 = " P 5 e g 2 j F S m E n S t z 13 14 15     / w H t m Q q 3 g = = < / l a t e x i t > a t e x i t s h a 1 _ b a s e 6 4 = " J 7 T R L 5 X o P t s 9 e j E J r N F u g T  2. Chemical reactions, stoichiometric matrix, and cycles of the modules in Fig. 1a.e black horizontal line splits the stoichiometric matrix of each module   into the substoichiometric matrix for the internal species    and for the terminal species    .
is also a right-null vector of    , which corresponds to the e ective reaction But it is linearly dependent on the other two and is thus excluded from the circuit description.Any other pair of these three emergent cycles could also have been chosen.We consider now the (aqua green) module  whose internal species E  , E *  , and E  S are involved in the chemical reactions  1 ,  2 ,  3 ,  4 , and  5 (see Fig. 2) with the terminal species S and N  .Its stoichiometric matrix is speci ed in Fig. 2 and, unlike the previous modules, the substoichiometry matrix for the internal species    admits the right-null vectors    and    , called internal cycles, that are also rightnull vectors of the substoichiometry matrix for the terminal species    .ese internal cycles are sequences of reactions that upon completion leave all the species (both internal and terminal) unaltered.us, they do not correspond to any effective reaction between terminal species.However, the substoichiometry matrix for the internal species admits also the emergent cycle    which corresponds to the following e ective reaction By following the same procedure for the remaining modules, one obtains the following e ective reactions  (10) for the (orange) module ; for the (purple) module ; and for the (red) module  .Note that a chemical module  with |  | terminal species can have a maximum of |  | −1 (linearly independent) emergent cycles and, therefore, e ective reactions.is is analogous to the fact that an electronic component with |  | contacts can have at most |  |−1 independent electrical currents at steady state [27].is follows directly from the existence of at least one conservation law, namely, mass conservation law in CRNs or electric charge conservation law in electronic circuits.
e existence of additional conservation laws (involving the terminal species in CRNs or the contacts in electronic circuits) reduces the number of emergent cycles.In the case of electronic circuits, the only kind of conservation law is the charge, and the only way to have additional conservation laws beyond that of the total charge is for a component to consist of smaller subcomponents that do not interchange any charge (although they might still in uence each other).
is is not the case for CRNs, where conservation laws (involving the terminal species) can be more complicated [24,25].
ey identify parts of (or entire) molecules, named moieties, that are not modi ed by the chemical reactions.Mathematically, they correspond to le -null vectors of the full stoichiometric matrix   of a module whose restrictions to the internal species are not le -null vectors of the stoichiometric matrix    for the internal species.It can be seen that if |  | and |  | are the number of independent conservation laws and emergent cycles, respectively, then is, together with the existence of at least one conservation law (mass conservation law in CRNs or charge conservation in electronic circuits), explains why the number of emergent cycles is at most |  | − 1.Note that these conservation laws are said to be broken because they de ne quantities that are only conserved in the closed system (CRNs [25] or electronic circuits [27]).

III. CURRENT-CONCENTRATION CHARACTERISTIC
In electronic circuits, the steady-state behavior of electronic components is given by their current-voltage characteristics, or "I-V curves", which specify how the value of all independent currents of an electronic component depends on the voltages applied to its contacts.We now apply the same strategy to chemical modules.
e current-concentration characteristic of a chemical module speci es how the (e ective) reaction currents depend on the concentrations of the terminal species only, by assuming that the internal species have already relaxed to steady state (see App.A 2).
When the kinetic constants of the internal reactions of a module are known, the current-concentration characteristic can be derived analytically if the internal reactions of a module are pseudo-rst-order reactions, or otherwise numerically.e procedures to do so are respectively described in App.B 1 and App.B 2 and applied to some of the modules in Fig. 1a.However, in practice, a complete characterization of the kinetics of the internal reactions is seldom achieved.
e real power of the circuit theory is that the current-concentration characteristic can be determined experimentally, as discussed next.
One possible way may be to resort to membrane reactors [28].We describe the procedure using the (blue) module  and the (green) module  in Fig. 1a.e formal theory is detailed in App.A 5.
e setup to characterize the (blue) module  is illustrated in Fig. 3. e concentrations of S and N  are held constant thanks to the exchange processes whose currents are  S and  N  satisfying: us, the e ective reaction current    can be determined by measuring the exchange current  S (or equivalently  N  ) for every value of the concentrations [S] and [N  ]:

Na EaS2 Ea EaS
FIG. 4. Elementary (a) and circuit description (b) of the (green) module  in Fig. 1 in a reactor, similar to the membrane reactor used in Ref. [28], where the concentrations of N  , N  , F and W are controlled by exchange processes whose currents are speci ed by  N  ,  N  ,  F , and  W .
e setup for the (green) module  in Fig. 1a is illustrated in Fig. 4.
e module has now two e ective reactions, (6a) and (6b), but the general strategy remains the same.e concentrations of the terminal species (N  , N  , F, W) are held constant thanks to the exchange processes whose currents are  N  ,  N  ,  F , and  W satisfying: and thus the e ective reaction currents    and   are given by is operation can be repeated for every module.In App.A 5, we formally derive the general expression (Eq.(A22)) of the e ective reaction currents in terms of the exchange currents.

IV. CIRCUIT DESCRIPTION
Having determined the e ective reactions and the currentconcentration characteristic of each module, we can nally formulate the circuit description of the CRN in Fig. 1b.e general formulation is presented in App.A 6.
To do so, modules are connected by sharing their terminal species.For instance, the terminal species S is involved in the two e ective reactions   and   (given in ( 2) and ( 9), respectively) as a reagent and also exchanged with the environment.Its concentration thus evolves according to where  S is the exchange current of S with the environment.Analogously, the terminal species N  is the product of the effective reaction   (given in ( 9)) and the reagent of the e ective reactions   and   (given in ( 11) and ( 12), respectively).Its concentration thus follows which accounts for the fact that 2 molecules of N  are consumed every time reaction   occurs, namely, the stoichiometry of the e ective reaction   .

A. ermodynamics
We emphasize that our circuit theory is thermodynamically consistent, contrary to many other coarse-graining schemes. is means that the entropy production rate of the coarse-grained description is the same as the entropy production of the original full description of the CRN.In other words, the reduction scheme preserves the entropy production rate of the CRN.
e chemical potential of a chemical species  in a homogeneous solution [29] is given by where  •  is the standard chemical potential,  is the gas constant, and  is the temperature of the solution.e Gibbs free energy change in a homogeneous CRN caused by a reaction  is given by where    is the net stoichiometric coe cient of the  species in the  reaction.For example, the Gibbs free energy changes of the internal reaction of the (blue) module  in Fig. 1a are given by At the elementary level of description of CRNs, the local detailed balance property must hold.It relates thermodynamics to the log ratio of the forward and backward reaction uxes  ± (given by Eq. ( A4)) contributing to the reaction currents   =  + −  − .Furthermore, the entropy production rate of elementary CRNs (also called the total dissipation) reads and quanti es the entropy change per unit of time in CRNs as well as in the thermal and chemical reservoirs [25].Together with the local detailed balance property (26), the entropy production rate can be rewri en in a manifestly non-negative form thus mathematically ensuring the validity of the second law.e summation over  in Eq. ( 27) runs over all the reactions excluding the exchange processes with the environment (i.e., all arrows but the grey ones in Fig. 1a).Vanishing entropy production de nes thermodynamic equilibrium where all reaction currents vanish.Together Eqs. ( 26)-( 28) ensure a thermodynamically consistent descriptions of elementary CRNs.
Our circuit theory allows for a thermodynamically consistent description of CRNs made of e ective reactions because the entropy production rate ( 27) can be expressed as where the summation now only runs over the e ective reactions  of the modules (all arrows but the grey ones in Fig. 1b) and the free energy along the e ective reaction  is given by where  is the set of all the terminal species.For instance, the Gibbs free energy change along the single e ective reaction of the (blue) module  reads e remarkable reduction from ( 27) to (29) arises because the emergent cycles de ne a minimal set of e ective reactions which preserve the exact evolution of the terminal species while carrying the full dissipation of modules, as long as there is a time-scale separation between the dynamics of the internal and terminal species.
is has been proven in Ref. [19].We emphasize, however, that the entropy production rate at the circuit level (29) cannot be, in general, expressed in a form reminiscent of (28) as shown in Ref. [18].We also note that, unlike our theory, most coarse-graining schemes underestimate the exact dissipation even in presence of a time-scale separation (see for instance Ref. [30]) because the fast degrees of freedom which have been eliminated are still out-of-equilibrium and contribute to the dissipation.

B. Kir ho 's Laws
We now show that the dynamics of our circuit theory (19) satis es the chemical equivalent of Kirchho 's laws in electrical circuits.Here, we present these laws for the CRN in Fig. 1b, while their general formulation and derivation are given in App.A 6. We emphasize that Kirchho 's laws for CRNs at the level of the elementary dynamics are not new (see for instance Ref. [31][32][33][34]).
e novelty of our approach is that Kirchho 's laws are recovered at the coarsegrained/circuit level.Kirchho 's current law states that, at steady state, the sum of the currents entering into a node of an electronic circuit is equal to the sum of the currents exiting it.
e terminal species correspond to the nodes of an electrical circuit in our circuit theory.Hence, Kirchho 's current law can be expressed for the CRN in Fig. 1b in terms of the steady-state conditions (denoted by the overline) imposing that the sum of the currents (both e ective and exchange) a ecting the concentration of each terminal species vanishes.is is formally derived by imposing that the lehand-sides of Eq. ( 19) vanish and by using the stoichiometric matrix given in Eq. ( 20) and the currents given in Eq. ( 21).
On the other hand, Kirchho 's potential law states that the sum of potential di erences along any closed loop is zero.In our circuit description of CRNs, loops correspond to the internal cycles of Ŝ (introduced in Sec.II and detailed in App.A 2) and potential di erences to the variations of the Gibbs free energy along the (e ective or not) reactions (e.g., {Δ   , Δ   , Δ   , . . .} for the e ective reactions in Fig. 1b).Since, the stoichiometric matrix (20) admits only one internal cycle, Kirchho 's potential law can be expressed as for the CRN in Fig. 1b, which using Eq. ( 24) is indeed true since is is formally derived by imposing that the sum of the variations of the Gibbs free energy along the (e ective or not) reactions (e.g., {Δ   , Δ   , Δ   , . . .} multiplied by the corresponding entry of the internal cycles in Eq. ( 33) vanishes.

V. DISCUSSION AND PERSPECTIVES
We start by discussing how apparent limitations of our circuit theory may be overcome.
e fact that the current-concentration characteristic of a chemical module is evaluated assuming that the module is in a steady state (based on the time scale separation assumption mentioned before and formally discussed in App.A 2) may give the impression that oscillations in the concentrations of the species internal to a module compromise the theory.However, we prove in App.A 3 that this is not the case and that such oscillations can be treated as long as their period is much shorter than the time scale of the terminal species dynamics.
In certain situations, the current-concentration characteristic may be such that the e ective reaction currents are not uniquely de ned in terms of the concentrations of the terminal species.is will happen for modules with nonlinear chemical reactions displaying multistability.In such cases, hysteresis e ects may arise creating a dependence on the past history of the network, but the network theory is still applicable.An explicit example of such a situation is worked out in App.C 1.
We presented the circuit theory starting from elementary reactions that we grouped into modules.But naturally, modules can be further grouped into higher-level modules.We examine this in App.C 2 by showing that the entire CRN depicted in Fig. 1a/b can be treated as a module and its exchanged species become terminal species.
is also raises the question of what are the conditions under which a module can be decomposed into smaller modules.e answer is quite simple: as long as the e ective reactions belonging to a smaller module are independent of those of another module, i.e., when their emergent cycles do not share internal species.Such a decomposition is discussed in App.C 3 for the (orange) module  in Fig. 1a.
When discussing the experimental characterization of the current-concentration characteristic, we implicitly assumed that the stoichiometry of the internal reactions of the module is known.However, even when this is not the case, recovering that stoichiometry is not too complicated experimentally.We illustrate how such a procedure might be implemented in App.C 4 for some of the modules in Fig. 1a.
Our circuit theory was presented here for ideal homogeneous solutions, but these conditions can easily be relaxed.Non-ideal solutions can be treated within mean-eld theories [26] and introducing spatially organized compartments is straightforward.It su ces to treat the chemical species in the di erent compartments as di erent dynamical variables and add reactions amongst them to describe (passive or active) exchanges across compartments.Adding di usion by promoting the description of some or all species from homogeneous concentrations to space-dependent concentration elds is also in principle not an issue.In such cases diffusion is treated within Fick's law and contributes to the dissipation in the CRN [35][36][37].
Our approach is fundamentally di erent from ux balance analysis ones.
ese la er are designed to determine the steady-state currents in a CRN.In the space of all possible steady-state currents, they select those that satisfy a set of constraints the system is supposedly subjected to.ese can range from thermodynamic constraints [38][39][40] or limits imposed by the environment [23] to presumed aims like growth maximization for some cells [40].e resulting steady-state currents will naturally depend on the enforced constraints.
ese approaches are thus top-down.Some of the more teleological constraints, such as maximizing growth, may only be justi ed in complex systems such as living systems shaped by evolution.Identifying the constraints predicting the steadystate currents of the CRN in Fig. 1 for a given set of thermodynamically consistent kinetic constants would be more complicated than solving the full dynamics.Instead, our circuit theory may be de ned as bo om-up.Indeed, it is built to be compatible with a microscopic description of the dynamics.As explained in the main text, the current-concentration characteristics of the modules result from the full dynamics and can be used as an input to our theory to predict the correct dynamics and thermodynamics of the terminal species (Sec.IV).
Our work shares some conceptual similarities with the work by Oster and coworkers which, in the seventies, de-veloped a very general network thermodynamics describing networks made of any type of thermodynamics systems [31,41].eir intent was to describe coupled thermodynamics processes arising in biophysics involving di erent forces such as mechanical, electrical, and chemical forces.
eir theory makes use of bond graphs, a graphical representation inspired by electric diagrams used in electrical circuit theories.e generality of the theory turns however into a disadvantage in the context of CRNs, since the theory is not tailored for them.e bond graph representation of simple CRNs quickly becomes very cumbersome [42,43]. is also explains why the use of the theory has remained limited to simple CRNs.In contrast, our formalism is algebraic and based on the representation of modules in terms of emergent cycles.e la er identify the minimal set of currents needed to de ne the current-concentration characteristics of a module and to determine its dissipation.ey also provide an intuitive description of modules in terms of e ective chemical reactions which can be easily represented in terms of hypergraphs.
e theory by Oster and coworkers do not exploit that reduction.Furthermore, one of the main purposes of our theory is to provide a simpli ed (i.e., coarse-grained) description of the dynamics of CRNs.e theory by Oster and coworkers instead has been mostly used as a formalization and representation tool, not as a reduction tool.
We now discuss interesting perspectives raised by our work.
Electronic engineering makes extensive use of circuit theory to design circuits with intended functionalities, such as computing operations.Similarly, one should explore how to make use of the chemical circuit theory to design useful chemical functions.is may be particularly relevant in the context of chemical computing, a eld increasingly raising a ention [44][45][46][47].
Our work focused on the deterministic description of CRNs, but in many instances such as cellular biology, extending the theory to stochastic descriptions of CRNs would be important.is may be challenging because the statistics of the e ective chemical reactions is not trivially related to the Poisson statistics of the elementary reactions, see for instance Ref. [48].
We presented our circuit theory for open CRNs exchanging ma er and heat with the surrounding.But other forms of energy may be incorporated in the description, such as energy provided by thermal light [49], electrical energy, and osmotic pressure.Indeed, the concept of emergent cycles is a general feature of thermodynamics when taking into account conservation laws [50]. is is why circuit theories have the potential to provide a powerful and realistic characterization of the dynamics and thermodynamics of complex systems.
e key point is that, as for electric circuits, the currentpotential characteristics provide an empirical characterization of complex modules that would otherwise be very hard to determine.
As shown implicitly in Ref. [51], but clearly retrospectively, the circuit theory underlies the fact that central metabolism can be decomposed into modules (glycolysis, Krebs cycles,. . .).But what is true at the level of cellular metabolism still holds true at higher levels, namely, whenever one is dealing with open CRNs coupled to each other by the exchange of terminal species.A food web for instance can be seen as a collection of modules representing the metabolisms of the di erent living systems feeding on each other and ultimately powered by solar energy.In ecology, like previously in biochemistry, tracking the movement of di erent types of atoms across a network under di erent molecular forms is nowadays used to reconstruct CRNs up to global scales, as for instance in biogeochemistry [6,7].Measuring or estimating current-potential characteristics may not be easy in such a context, but is conceivable and worth trying given the importance of these networks.
Circuit theories may even provide a proper framework to formulate models in ecological economics (also called steadystate economics) where minimizing the dissipation arising in the use and recycling of natural resources is a major concern [52].and {   }  ∈   ∈  , respectively.Here, the terminal current vector of the module   accounts for all the processes a ecting the concentrations   besides the reactions  ∈   .Note that Eq. ( A7a) and (A7b) coincide with Eq. (A6a) and (A6b) when the module is treated as an open CRN, and the species   and   are identi ed as  and  , respectively.

E ective Modules at asi Steady State
Modules can be coarse grained into e ective reactions interconverting the terminal species   . is can be done when two conditions are satis ed.
e rst is the existence, for every concentration vector   , of a unique steady-state concentration vector   (  ) for the internal species of the module (see App. C 1 for an explicit example where this does not hold, but the theory can still be applied).
e second is the equivalence between the actual concentration vector   and the steady-state one   . is obviously happens at the steady state to which the module relaxes when the concentrations   are kept constant by the other reactions  ∉   and the exchange processes, i.e., when It also happens to a very good approximation when the chemical species evolve over two di erent time scales such that the concentrations of the   species quickly relax to the steady state corresponding to the values of   .Indeed, a zero-order expansion of the concentrations   in the ratio between the fast time scale of the internal species and the slow time scale of the terminal species leads to is physically occurs when i) the elementary reactions and the exchange processes involving only the terminal species are slower than the elementary reactions involving only the internal species and ii) the abundance of the terminal species is very large compared to the abundance of the internal species which therefore changes much more quickly [48] (indeed, when the terminal and internal species are involved in the same reaction, on the same time scale the concentrations of the internal species dramatically change, the concentrations of the terminal species remain almost constant).Note that describing electronic components in terms of their I-V curves also requires a time-scale separation between their internal dynamics and the dynamics of the voltages on their contacts or pins.
When those two conditions are satis ed, the reaction current vector of the module e so-called internal cycles {   } are also right-null vectors of    , i.e.,       = 0. ey thus represent sequences of reactions that upon completion leave also the concentrations   unchanged (for instance, the (aqua green) module  in Fig. 1a has two internal cycles as speci ed in Fig. 2).
e others {   } are called emergent cycles.
e internal and emergent cycles of the modules in Fig. 1a are reported in Fig. 2.
By employing the steady-state current (A11) in Eq. (A7b) and the spli ing of the cycles into internal and emergent ones, we obtain an e ective and closed dynamical equation for the   species Each vector       speci es the net variation of the number of molecules for each   species along the   emergent cycle.Namely, it speci es the stoichiometry of an e ective reaction.
e e ective reactions of the modules in Fig. 1a are discussed in Sec.II.Correspondingly, the emergent cycle current    speci es the current of this e ective reaction.
Equation (A12b) can be rewri en in a more compact way, introducing the e ective stoichiometric matrix Ŝ  and the e ective current vector ψ .Here, each   column of Ŝ  is given by       and ψ = (. . .,   , . . . ) does not, in general, satisfy mass-action kinetics.
Note that each module is de ned as a subnetwork with a unique set of internal species because this ensures that   and, consequently, ψ are functions of the concentrations   of its terminal species only. is is a necessary condition to obtain the closed dynamical equation (A13) for the terminal species.

E ective Modules with Internal Oscillations
Modules can be coarse grained into the same e ective reactions de ned by the emergent cycles {   } even if the concentrations of the internal species oscillate.is can be done when the concentration vector   relaxes instantaneously to an oscillating dynamics q (  , ) (formally, we use again a zero-order expansion of   in the ratio between the relaxation time scale of the internal species and the time scale of the terminal species leading to   = q (  , )), whose period  is much shorter than the time scale Δ of the terminal species, i.e., /Δ 1.
When these conditions are satis ed, the reaction current vector of the module  A12b), also Eq. (A17) can be rewri en as Eq.(A13) by using the e ective stoichiometric matrix Ŝ  and collecting { ψ  } into an e ective current vector.
is physically means that on the time scale Δ of the terminal species, the internal dynamics is averaged over many (∼ Δ/ 1) oscillations and acts, in practice, as an e ective steady state.

E ective Currents via the Elementary Me anism
We show here how to determine the e ective currents, i.e., the function ψ (  ) in Eq. (A13), from the elementary dynamics, given in Eq. (A7a) and (A7b), by assuming that the steady-state concentration vector   (  ) can be computed for every   (either analytically or numerically).
is approach is then illustrated in App.B for the (blue) module , the (green) module  and the (purple) module  of Fig. 1a.
We start by recognizing that   (  ) can be obtained using its de nition (A10) and   (  ).We then rewrite Eq. (A11) as where we introduced the cycle current vector   (  ) = (. . .,   (  ), . . . ) , which includes also the internal cycle currents unlike ψ (  ) in Eq. (A13) which includes only the emergent cycles, and the matrix ℂ  whose columns are the cycles {   }.Since the cycles {   } are linear independent, the matrix ℂ  ℂ  can be inverted, and we thus obtain 5. E ective Currents via the Terminal Currents We now discuss how to determine the e ective currents, i.e., the function of ψ (  ) in Eq. (A13), by assuming that i) the e ective stoichiometric matrix Ŝ  is known and ii) the concentrations   can be kept equal to arbitrary and constant values by controlling the terminal currents   according to Eq. (A8). is approach is illustrated in Sec.III of the main text.
When the concentrations   are constant, the module relaxes instantaneously towards a nonequilibrium steady state.By using Eq.(A13), the steady-state terminal currents of the module read We now use Eq.(A20) to express ψ (  ) in terms of   (  ).
To do so, we recognize that the e ective stoichiometric matrix Ŝ  has no right-null vectors, as already discussed in Ref. [19].Indeed, suppose that there is a vector  = (. . .,  Once modules are fully characterized, namely, their e ective reactions {      } and currents { ψ } are known, they can be connected by sharing the terminal species.e result is a circuit theory where the dynamics of all terminal species emerges from combining the dynamical equation (A13) of the modules: Here,  = (. . ., [], . . . )  ∈ is the concentration vector of all terminal species (with  the set of all terminal species where and D  equals the sum of all terms between parentheses in Eqs.(B6), (B7), (B8), and (B9).e steady-state reaction currents of reactions  1 ,  2 ,  3 ,  4 , and  5 .can thus be computed using again massaction kinetics: (B10) e corresponding current vector   = (   1 ,   2 ,   3 ,   4 ,   5 ) can be wri en as a linear combination of the two emergent cycles    and    in Fig. 2 using the two e ective reaction currents    and    as coe cients, i.e., which leads to and In general, the diagrammatic method [58,59] provides the steady-state concentrations of the internal species of a module and then, by applying mass-action kinetics, its steadystate current vector.

Numerical Strategy
When the internal reactions are nonlinear (i.e., not pseudorst-order reactions), but the kinetic constants of the internal reactions are known, the current-concentration characteristic can be determined numerically.We illustrate this procedure for the (purple) module  in Fig. 1a, where the internal species M, M * , A 2 and A * 2 react via the chemical reactions  1 ,  2 ,  3 , and  4 with the terminal species N  and N  .Reaction  2 and  4 are bimolecular reactions in M * and M, respectively.When the kinetic constants of the internal reactions are known, one can numerically compute the steady-state concentrations of the internal species for di erent concentrations of the terminal ones, namely, ) for every value of ( [N  ], [N  ]).To do so, one can either use algorithms that directly determine the xed point of the rate equation, or simulate the evolution of the internal concentrations until steady state is reached for xed concentrations of the terminal species.
en, one can repeat the steps of App.B 1. First, the steady-state current vector   = (   1 ,   2 ,   3 ,   4 ) is determined for every value of the concentrations ( [N  ], [N  ]) using mass-action kinetics and the numerical determined values of ( ).Second,   is wri en as a linear combination of cycles.In this case the stoichiometric matrix admits one emergent cycle    (given in Fig. 2) whose corresponding e ective reaction is speci ed in (11).Hence,   =       , which leads to the e ective reaction current shown in Fig. 5 for a speci c set of kinetic constants { ±  } =1,2,3,4 .
Appendix C: Underlying Assumptions and Limitations of the Circuit eory 1. Multistability e circuit description given in Sec.IV and Eq. ( 19) implicitly assumes that the e ective reaction currents are fully 1 2 3 4 5 6 7 8 9 [N ]  (11) for di erent values of the concentrations of the terminal species N  and N  .We use 1/ − 1 and  − 1 / + 1 as units of measure for time and concentration, respectively.We assume < l a t e x i t s h a 1 _ b a s e 6 4 = " q 0 g u i 9 J t u 7 l y i g f with the internal species X and a single e ective reaction A is is the well known Schlögl model [60], displaying bistability far from equilibrium when  + 2 1.7 (using speci c units of measure such that  + 1 Indeed, when the chemical potential di erence Δ between the terminal species A and B is small enough, e.g., Δ < 2.6 if  + 2 = 2, the steady state concentration [X] of internal species has the unique value [X] 1 represented by the blue line in Fig. 7a.Correspondingly, the e ective reaction current    has the unique value  1 represented by the blue line in Fig. 7b.On the other hand, when Δ > 2.6 if  + 2 = 2, there are two possible stable steady state concentrations [X] 1 and [X] 2 represented by the blue and orange line in Fig. 7a, respectively, and one unstable steady state concentration [X] 3 represented by the green line in Fig. 7a.Correspondingly, the e ective reaction current    can have three di erent values  1 ,  2 and  3 represented by the blue, orange and green line in Fig. 7b, respectively.is implies that the steady state to which the internal species relaxes and, consequently, the e ective reaction current are not uniquely determined by the terminal species.To see this, immagine that the initial concentration of internal species is [X] (0) = 12.5 and the concentration of the terminal species are such that Δ = 3.75 (red point in Fig. 7a).en, assuming the time scale separation holds, a er an rapid transient (do ed line in Fig. 7a), the concentration of internal species reaches the steady state [X] = [X] 2 = 19 and the effective current becomes    =  2 37.If the concentrations of the terminal species change (because of the dynamics of the module and the coupling with other possible modules in a large CRN) in such a way that Δ decreases until Δ = 2.6, the steady-state concentration [X] and e ective current    will follow the black dashed lines overlapping the orange lines in Fig. 7a and Fig. 7b, respectively.Once Δ < 2.6, the steady state concentration [X] jumps from [X] 2 to [X] 1 .If nally the concentrations of the terminal species change in such a way that Δ is increased back until Δ = 3.75, the steady-state concentration [X] and e ective current    will follow the black dashed lines overlapping the blue lines in Fig. 7a and Fig. 7b, respectively.
In general, this kind of evolution of a module cannot be obtained by a circuit description accounting only for the terminal species since di erent values of the current correspond to the same values of the concentrations of the terminal species.Nevertheless, if the current-concentration characteristic resolves the multiple steady states (like we have done above for the Schlögl model), the circuit description still holds.

Open CRNs as Modules
In the circuit description of the open CRN in Fig. 1b, the species S, F, W, P  , P  , P  , and P  are exchanged with the environment.Let us assume now that the environment is constituted by other chemical processes.In this case, S, F, W, P  , P  , P  , and P  are involved in the chemical reactions of both the CRN in Fig. 1b and the environment.Namely, they play the role of terminal species coupling the CRN to the environment, and hence the CRN in Fig. 1b can be treated as a module like in Fig. 8a.
As done for the modules in Fig. 1a, also the module in Fig. 8a can be further coarse grained into what we could call a second-order circuit description given in Fig. 8b (assuming that the time scale separation between internal and terminal species holds).To do so, we follow the same strategy as before.First, we determine the e ective reactions by looking for the emergent cycles of the stoichiometric matrix in Eq. (20), where black horizontal line now splits Ŝ into the substoichiometric matrix Ŝ for the internal species (i.e., N  , N  , N  , N  , N  , and G) and the substoichiometric matrix Ŝ for the terminal species (i.e., S, F, W, P  , P  , P  , and P  ).e right-null vectors of Ŝ include the internal cycle (33)   Note that the current-concentration characteristic of these reactions cannot be determined using the diagrammatic method [58,59] as the dynamics of the module in Fig. 8a (given in Eq. ( 19)) does not follow mass-action kinetics, and one should therefore rely on the numerical B 2 or the experimental strategy (see Sec. III).

Further Decomposition of the Modules
In the circuit description, each module is coarse grained into at least one e ective reaction between terminal species.When a module (like the (orange) module  in Fig. 1a and 9a) has more than one e ective reaction (given in Fig. 9b), one can ask if it can be split into independent (sub)modules corresponding to a single e ective reaction each (like in Fig. 9d).
is can be done only if each (sub)module has a unique set of internal species.Indeed, the procedure to split a module into submodules is exactly the same as the one to split a generic CRN into modules and must satisfy the same assumptions.
One can also determine if a module can be split into submodules by examining the currents of the e ective reactions without analyzing the internal species.When the currents of some e ective reactions depend only on a subset of terminal species, they involve a unique set of internal species and constitute, by de nition, an independent (sub)module.To see this, we consider the e ective reaction currents (derived using the diagrammatic method [58,59]) of the (orange) module  in Fig. 1b: respectively.On the one hand, the reaction current    depends only on the concentration of the terminal species N  and P  .On the other hand, the reaction current    (resp.   ) does not only depend on the concentration of N  and P  (resp.N  and P  ), but also on the concentration of N  and P  (resp.N  and P  ) via D  .us, only reaction   can be treated as an independent module (as represented in Fig. 9c).e coupling between reaction   and   is a direct consequence of sharing the internal species E  : whether or not this species is available for one e ective reaction depends on how much is involved in the other.
e two reactions   and   must therefore be considered as part of the same module despite closely resembling a Michaelis and Menten mechanism.

Experimental Derivation of the E ective Reactions
e e ective reactions of the modules have been identied by deriving the emergent cycles from the stoichiometry of the elementary reactions in Sec.II and App.A 2. If the elementary stoichiometry is not known, the e ective reactions can still be determined by using a similar approach as the one implemented in Sec.III.
Consider for instance to transfer the (blue) module  in Fig. 1 in a reactor where the concentration of the terminal species S and N  can be maintained constant via the exchange currents  S and  N  as done in Fig. 3.By measuring these exchange currents, one would observe that they always satisfy  S = − N  . (C7) Since these exchange currents, as already pointed out, balance the variations of the concentrations due to the e ective reaction, Eq. (C7) means that every time the current  S provides (resp.extracts) 1 molecule of S because consumed (resp.produced) by the e ective reaction, the current  N  extracts (resp.provides) 1 molecule of N  . is implies that the net stoichiometry of the e ective reaction must be the one represented in (2).
When the same approach is applied to the (purple) module  in Fig. 1a, one would observe that which physically means that every time the current  N  provides (resp.extracts) 2 molecules of N  because consumed (resp.produced) by the e ective reaction, the current  N  extracts (resp.provides) 1 molecule of N  .us, the net stoichiometry of the e ective reaction must be the one represented in (11).

FIG. 1 .
FIG.1.Elementary (a) and circuit description (b) of a complex CRN enclosed in gray boundaries using hypergraph notation.Gray arrows crossing the network boundaries denote the exchange processes with the environment.Colored boxes are chemical modules.eir internal species and (elementary or e ective) reactions are represented by chemical symbols and arrows of the same color, respectively.eir terminal species are represented by black chemical symbols.All the chemical reactions are assumed to be reversible even though only the forward reactions are represented.E ective reactions are coupled when connected by black lines.
s D a o e k a o x d L 1 i F J 0 8 J y D Z I 6 k g W L l E D S a T y C L 7 A 3 3 C P s c m + w I 9 l r L S k j I O D w O x + / 8 6 N D s i J L t f H 9 v 9 Y 6 1 z 7 5 t H v 9 x m e 9 m 5 9 / c e v L 2 + t 3 X u t 8 p j i c 8 g r b x e m 8 3 e L T b / 7 2 / 8 e z t 8 p m 4 4 X 3 t 3 f P u e 4 H 3 o / f M e + m 9 8 s 4 8 v v Z P 5 6 s O 6 X z T v d X t d 3 / q P l 1 Q O 2 v L M 3 e 9 2 u o O / g V e K z p H < / l a t e x i t > e 3 a O T X P f e m w q R H E 6 L r 6 y T Z x F m 3 P n P U 3 t 3 3 3 r 7 l E w a z O n M A s y / G 3 Q O T s 0 6 P 9 7 5 1 9 8 m S n 3 9 9 z S 2 m 5 o l k C 7 5 k P H u 3 6 / n I m r g T Z O T F 4 / C g o i b 1 1 s n d K Z S F A k 2 e 5 q R 7 L X r i 4 J 2 z H o b N n v 5 g x c l j e n 8 V 8 e S Y P 9 9 0 g G J 5 x r R v Z 1 a D J E p I 6 S 1 s g c 5 a 1 Q D w 8 v A V i M 6 I W i P u G F h g 7 G 7 d A 7 a x u g W X f k l F b I Y l L P C 4 f h 6 D 5 F L S D 5 1 u b w c 5 m / 8 f + 6 s N X 8 2 f i s n f L + 8 y 7 7 Q X e f e + h 9 9 Q 7 8 I 4 9 3 v m l 8 7 b z r v N r 9 0 3 3 s s 6 c w 1 L m / l o g c f b J 6 c l v z j 5 + f H 8 4 P H Y b a b n C M o O 3 z M O H R 2 G 4 m e k j g X x D j B 4 9 j E p i b 4 C O L 7 A o O G j 0 N D f V K 9 W L 1 2 v y 7 R g 5 e 3 n F h K C R c 3 V / O Z O j E z e O J p d c 6 6 Z 2 J 2 q y u M D O 4 h Z I n C U t 0 A 8 P b Y G + G U k L 9 H V D C 0 y d T V u g d l a 3 w L J v 2 b S t k K U l n p a P Q 9 R 8 C t r G i / 2 9 6 P 7 e 8 P f h z o M / V s / E t e B 2 8 G V w J 4 i C 7 4 M H w Z P g e X A W 0 K 3 / O r c 7 / c 6 g e 6 t 7 2 P 2 5 e 3 R J 7 W y t z n w R 1 F b 3 1 / 8 B o 3 A 7 2 g = = < / l a t e x i t > 2 y P e 9 z 2 H 4 Z R s l x b H 2 k y A o Y u D y / P P T y k S Z J z p r T v / 3 H j 5 q 3 e 7 f f u r K 3 3 3 / / g w 4 8 + 3 t j 8 5 J X K p p L C C c 1 4 J t 8 Q r I C z F E 4 0 0 x z e 5 B K w I B x e k 7 P v i v H X 5 y A V y 9 J j P c t h J H C S s p h R r B 0 0 3 u w d m 1 A J r F 8 e m p A L a u 2 3 f R S e k U x G I B 0 s 2 Y X p o 0 b 7 I i z r m s Q W U 8 Z B E 9 h t 9 L 9 p 9 I e N / o N G f y + k s i y 6 h O n E H I z N 3 W L 4 r r U N O f e D 8 o O C j h 7 y q 8 8 V n C 9 c B B c 1 7 o

8 5 2
b j w / J O V P a 9 7 8 v d c 6 d v 9 C 9 e O l y 7 8 r V P / 6 8 d n 3 5 x g e V T S W F P Z r x T H 4 i W A F n K e x p p j l 8 y 8 e 5 5 g f f Y e + a 9 8 t 5 7 e x 5 d + t n p d w a d + 9 3 b 3 c 3 u 6 + 6 b u 2 2 B p / + Y 5 v D 5 J L J I y r l j s w T j N 3 L B B Z 8 3 R z 4 5 k 1 T 5 6 s d r v r 9 l C z T O I 0 g Z n l v Y d r v n + 4 p Y s E 6 b 5 h 8 P h h U B h 2 F t H 6 H h Y 5 B 4 W e Z 7 p 8 D T v h 3 z U V r 6 I 1 k y M m B P W s r e q L m e x t 2 O 2 g P 7 E 1 d m x i t s h a 1 _ b a s e 6 4 = " r / g m P r 0 d q z y 4 l M 7 M I S 6 u 9 O u f 4 + v j 6 z i U V Z 9 r E 8 c f e l a s r q / 0 f 1 q 4 N r t / 4 8 e a t 9 Y 3 b J 7 q c K g r H t O S l e k u w B s 4 kH B t m O L y t F G B B O L w h Z 7 8 H / s 0 5 K M 1 K e W R m F Y w F L i T L G c X G Q 5 P 1 T z a l M 8 p h Y l M Q o P w f F 5 l z 7 v E A p W e k V B k o g Y 1 i F 3 a A W u t u S lU N p r U N W 7 i w d 5 I s W J T M u W W a e 7 U g n i u X S 3 6 7 l O R b m i H 6 f p 7 7 S z R u s r 4 Z 7 8 b 1 Q t 0 g W Q S b 0 W K 9 m m y s / J F m J Z 0 K k I Z y r P U o i S s z t l g Z 5 q v n B u l U Q 4 X p G S 5 g 5 E O J B e i x r Z 0 4 t O W R D O W l 8 j 9 p U I 1 e 3 m G x 0 H o 0 y N b 4 l B 4 x h 6 C s I 7 F F O O G 7 e x h A g P P A N / b Q U H 3 s I e r 0 4 x A W P T Y K j 6 m z r 7 1 1 H i b B 0 J Z 5 l p l s O 3 J p P F j g b B S M m z s e W M g K + 7 6 4 D e M n T A 3 N m 8 A 2 p n d Q c M D 1 N M u h m K P O B 5 m D 5 J e 9 Z 0 g 5 N 7 u 8 m D 3 e H r 4 e a T d 4 s 5 t B b 9 H N 2 J t q M k e h g 9 i V 5 E r 6 L j i P Z G v X 9 6 / / Y + r H 5 c / d z v 9 9 f m 0 i u 9 x Z 6 f o s b q b t e x i t s h a 1 _ b a s e 6 4 = " u e m t n L P 3 4 w R N M z 7 s L n i o a 3 w r I e y S 4 c J Y 3 E P J M 6 S H u i 7 k / Z A n + 2 8 B 3 r L 0 A M L Z 4 s e q J 3 V P T A U p h z 3 T yi L g B f O T 5 + 0 O 2 v 6 w d t b q + n 6 6 t q b t a V H 7 + d z a C G 6 G l 2 L b k R p d D d 6 F L 2 I X k c 7 E R 1 c H N w Z P B g 8 H H 4 a f h l + H X 6 b S U 8 N 5 n u u R K 0 1 / O 8 7 z 0 c W m g = = < /l a t e x i t > x i t s h a 1 _ b a s e 6 4 = " + 6 d g 5 h Z e y W 5 3 s L M C 2 l 9 l r u s B p N s S S g g s c + v / D L y 5 1 d m r r p 9 T i 9 y f z p u a Q + F V f + l C 6 2 X P n E 0 2 b y T u q 7 h S w a z N P I S Z r H x Z I H f 2 4 e 7 2 P 8 4 + e H B z N L r v l t I q h W UJ / z N v 3 d l K k u V M f x P I L 8 T 0 3 p 0 0 E O M h u n + M R c 1 B o 8 e V a U Z B n H 3 7 J m / H j r P z E h O C d p x r 5 0 N P 7 m y 7 / X Q 8 5 1 o 3 s e t p l 9 V M l n B M K H z o 8 I o j 2 i W F 2 q p A Y l 5 J N x l m U s h B r R m v Z D f N B X Y W 9 0 D i L O m B v j t p D / R u 5 z 3 Q S 4 Y e W Dh b 9 E D t r O 6 B o T D l p H 9 C W Q S 8 c H 7 6 p N 1 Z 0 w + e X d t M b 2 6 O n o 7 W b / + 3 m E O n o 4 v R p e h q l E Z / R b e j h 9 G T a C + i g 5 e D N 4 O 3 g 3 c r 7 1 z p g b 4 7 a Q / 0 1 c 5 6 o L c M P T B 3 N u + B 2 l n d A 8 P D F J N + h i I P e B 6 m T 9 K d N f 3 g / d Z m s r 0 5 + m W 0 9 u R T M 4 d W o p v R r e h u l E Q P o y f R q + h d d B D R w e f B r 4 P f B r 8 v / b H 0 9 7 L / S s y l 5 w b N n h t R a y 2 v / g P E S C o F < / l a t e x i t > x i t s h a 1 _ b a s e 6 4 = " Y p j y T y Z I j 5 A y b k d 9 s B T p 6 p x j + / j e m 0 s q z r S J 4 8 + D U 6 f P D P / 4 8 + y 5 0 V / n L / z 9 z 8 r F S 6 9 0 O V M U 9 m n J S / W G Y A 2 c S d g 3 z H B 4 U y n A g n B 4 T Y 4 e B P 7 1 M S j N S r l n 5 h V M B C 4 k y x n F x k P T l Y 8 p n n f 5 7 y + 1 d n r r s + p h r v h f F I z y L 3 r H 0 1 o v e 2 5 s / H G z d j 9 N F c o m L e V R z C X p S 8 L Z M 4 + 2 d t 5 5 u z j x 7 f G 4 0 d u q a x U W B b w v / L O / e 0 4 X q 7 0 N 4 H 8 I U w e 3 k + C c L S G H p 1 g U X H Q 6 L / S 1 P / 4 K P 3 1 T T 4 d u 8 4 u S k w I 2 n W u z Y e e 3 N 1 x B 8 l k o b V u a l e T r q q

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FIG. 3 .
FIG.3.Elementary (a) and circuit description (b) of the (blue) module  in Fig.1in a reactor, similar to the membrane reactor used in Ref.[28], where the concentrations of S and N  are controlled by exchange processes whose currents are speci ed by  S and  N  .
r L 7 2 2 f 6 A / p v + m s y F 9 L N C R o n y a + c b z + f O w U g r r 4 v j n 1 H T r 1 u 0 7 d 2 f u t e 8 / e P j o 8 e z c / L E t K s P h i B e y M K e M W p B C w 5

1 [X] 2 [X] 3 [FIG. 7 .
FIG. 7. Steady-state concentration (a) and e ective current (b) of the Schlogl module 6 for di erent values of the chemical potential difference between the terminal species A and B, i.e., Δ =  A −  B .e red dot represents an initial concentration [X] (0) of the module which relaxes along the do ed line towards the corresponding steady state.e dashed lines specify the value of the steady state concentration [X] and e ective current    when the value of Δ is decreased from 3.75 to 2.6 and then increased back to 3.75 assuming that [X] (0) = 12.5.We use units of measure such that + 1 [A] = − 2 [B] =1which, together with the local detailed balance condition, imposes  − 1 =  + 2  −Δ/ .We assume  + 2 = 2. and

FIG. 8 .FIG. 9 .
FIG.8.First (a) and second (b) order circuit description of the CRN in Fig.1a.Note that P  is not interconverted by the e ective reactions (C4) and thus no arrows connect it to the module in the second-order circuit description.
= 0 and   dependent coe cients {   }.e vectors {   } are called cycles because they represent sequences of reactions that upon completion leave the concentrations   unchanged.Each coe cient    represents the current along the cycle   .ecycles can be split into two disjoint sets, i.e., {   } = {   } ∪ {   }.
j ≡   ( q ,   )       + ∑︁         + ∑︁         , (A15) where we used the internal {   } and emergent {   } cycles as well as the (linearly independent) vectors {   }, named cocycles, generating the orthogonal complement of ker(       + ∑︁         +   (A16) by using       = 0 and assuming that the concentrations of all terminal species are almost constant in the time interval Δ, namely, d    (  ( + Δ) −   ())/Δ and ∫  +Δ  d   /Δ   .We now show that Eq. (A16) simpli es to a closed dynamical equation for the   species similar to Eq. (A13).To do so, we consider that i) the internal dynamics completes  oscillations (with Δ/ − 1 <  ≤ Δ/) in the time interval Δ; ii)  Δ/ when the time scale separation is satis ed, , . . . ) such that Ŝ   = 0. is means that         is a right-null vector of both    and    , i.e., an internal cycle.Since         is a linear combination of emergent cycles, we can conclude that Ŝ  has no right-null vectors. is implies that the columns of Ŝ  are linearly independent and the matrix Ŝ (  ) .