Thermodynamically Consistent Coarse Graining of Biocatalysts beyond Michaelis--Menten

Starting from the detailed catalytic mechanism of a biocatalyst we provide a coarse-graining procedure which, by construction, is thermodynamically consistent. This procedure provides stoichiometries, reaction fluxes (rate laws), and reaction forces (Gibbs energies of reaction) for the coarse-grained level. It can treat active transporters and molecular machines, and thus extends the applicability of ideas that originated in enzyme kinetics. Moreover, we identify the conditions under which a relation between fluxes and forces holds at the coarse-grained level as it holds at the detailed level. In doing so, we clarify the speculations and broad claims made in the literature about such a general flux--force relation.


I. INTRODUCTION
Catalytic processes are ubiquitous in cellular physiology. Biocatalysts are involved in metabolism, cell signalling, transcription and translation of genetic information, as well as replication. All these processes and pathways involve not only a few but rather dozens to houndreds, sometimes thousands of di erent enzymes. Finding the actual catalytic mechanism of a single enzyme is di cult and time consuming work. To date, for many enzymes the catalytic mechanisms are not known. Even if such detailed information was at hand, including detailed catalytic machanisms into a large scale model is typically unfeasable for numerical simulations. erefore, larger biochemical reaction networks contain the enzymes as single reactions following enzymatic kinetics. is simpli ed description captures only the essential dynamical features of the catalytic action, condensed into a single reaction. e history of enzyme kinetics [1] stretches back more than a houndred years. A er the pioneering work of Brown [2] and Henri [3], Michaelis and Menten [4] layed the foundation for the systematic coarse graining of a detailed enzymatic mechanism into a single reaciton. Since then, a lot of di erent types of mechanisms have been found and systematically classi ed [5]. Arguably, the most important catalysts in biochemical processes are enzymes -which are catalytically active proteins. However, also other catalytic molecules are known, some of them ocurr naturally like catalytic RNA (ribozymes) or catalytic anti-bodies (abzymes), some of them are synthetic (synzymes) [5]. For our purposes it does not ma er which kind of biocatalyst is being described by a catalytic mechanism -we treat all of the above in the same way.
From a more general perspective, also other scienti c elds are concerned with the question of how to properly coarse grain a process. While in most applications the focus lies on the dynamics, or kinetics, of a process, it turned out that thermodynamics plays an intricate role in this question [6]. A process occurring at thermodynamic equilibrium usually has a natural coarse graining -a er all, the very foundation of equilibrium thermodynamics is concerned with reduced desriptions of physical phenomena [7]. Chemical reaction sys- * artur.wachtel@uni.lu tems in closed containers are examples of systems relaxing to thermodynamic equilibrium. To the contrary, biological systems are open systems exchanging particles with reservoirs and as such they are inherently out of equilibrium. Nonequilibrium processes, in general, do not have a natural coarse graining. Suggested coarse grainings typically underestimate the dissipation in a nonequililibrium process [8,9] -although also overestimations may occur [10].
Understanding the nonequilibrium thermodynamics of catalysts is a crucial step towards incorporating thermodynamics into large-scale reaction networks. ere is ongoing e ort in the la er [11][12][13] which very o en is based on the connection between thermodynamics and kinetics [14,15]. Here, we elucidate this relation by building on recent developments in the nonequilibrium thermodynamics of open chemical networks [16,17].
In this paper we show how to coarse grain the description of any biocatalyst in a thermodynamically consistent way -extending the applicability of such simpli cations even to molecular motors [18,19] and active membrane transport [20]. e starting point is the catalytic mechanism described as a reversible chemical reaction network with mass-action reaction uxes (kinetic rate laws), ϕ ± ρ , and reaction forces (negative Gibbs free energies of reaction), −∆ ρ G = RT ln ϕ + ρ ϕ − ρ , for each reaction step ρ. From there we construct a reduced set of reactions with e ective reaction uxes ψ ± α and net forces −∆ α G. By construction, our coarse graining procedure captures the entropy production rate (EPR) [21,22] of the underlying catalytic mechanism: erefore, it is applicable in nonequilibrium situations, such as biological systems. In fact, the above equation is exact under steady-state conditions. In transient and other timedependent situations the coarse graining is still a reasonable approximation. We elaborate this point further in last section.
Secondly, we work out the condition for this coarse graining to reduce to a single reaction α. In this case, we prove that the following ux-force relation holds true for this coarse-grained reaction: In the past, such a ux-force relation has been used in the literature [23,24] a er its general validity was claimed [14] and later questioned [12].
is paper is structured as follows: First we present our results. en, we illustrate our ndings with two examples: e rst is enzymatic catalysis of two substrates into one product. is can be reduced to a single reaction, for which we verify the ux-force relation at the coarse-grained level. e second example is a model of active membrane transport of protons, which is a prototype of a biocatalyst that cannot be reduced to a single reaction. Finally, we sketch the proofs for our claims and discuss the case of time-dependent situations. Rigorous proofs are provided in the supplementary information.

II. RESULTS
Our main result is a systematic procedure for a thermodynamically consistent coarse graining of catalytic processes. ese processes may involve several substrates, products, modifyers (e.g. activators, inhibitors) that bind to or are released from a single molecule -the catalyst. e coarse graining involves only a few steps: (1) Consider the catalytic mechanism in a closed box and identify the internal cycles of the system. An internal cycle is a sequence of reactions leaving the state of the system invariant. Formally, these cycles constitute the nullspace of the full stoichiometric matrix, S.
(2) Consider the concentrations of all substrates, modifyers, and products (summarized as Y ) constant in time -therefore reduce the stoichiometric matrix by exactly those species. As a consequence, the reduced stoichiometric matirx, S X , has a larger nullspace: new cycles emerge in the system. Choose your favourite set of linearly independent emergent cycles C α .
(3) Identify the net stoichiometry, S Y C α , together with the sum, −∆ α G, of the forces along each cycle α.
(4) Calculate the apparent uxes ψ ± α along the emergent cycles at steady state. Now for every emergent cycle α, you have a new reversible reaction with net stoichiometry S Y C α , net force −∆ α G, and net uxesψ ± α . is procedure preserves the EPR and, therefore, is thermodynamically consistent.
Our second result is a consequence of the main result: We prove that the ux-force relation is satisi ed at the coarsegrained level by any catalytic mechanism for which only one single cycle emerges in step 2 of the presented procedure.

A. Enzymatic catalysis
Let us consider an enzyme, E, that is capable of catalyzing a reaction of two substrates, S 1 and S 2 , into a single product molecule, P. Let us assume furthermore that the binding order of the two substrates does not ma er. Every single one of these reaction steps is assumed to be microscopically reversible and to follow mass-action kinetics. For every reaction we adopt a reference forward direction. Overall, the enzymatic catalysis can be represented by the reaction network in Fig. 1.
We apply our main result to this enzymatic scheme and thus construct a coarse-grained description for the net catalytic action. We furthermore explicitly verify our second result by showing that the derived enzymatic reaction uxes satisfy the ux-force relation.
(1) Closed system -internal cycles When this system is contained in a closed box, no molecule can leave or enter the reaction volume. e dynamics is then described by the following rate equations: where we introduced the concentration vector and the current vector, , e substrates can bind in arbitrary order. We adopt a reference direction for the indivinual reactions: forward is from le to right, as indicated by the arrows. e backward reactions are from right to le , thus every single reaction step is microscopically reversible. as well as the stoichiometric matrix , In the dynamical equations, only the currents J (z) depend on the concentrations, whereas the stoichiometric matrix S does not. e stoichiometric matrix thus imposes constraints on the possible steady-state concentrations that can be analyzed with mere stoichiometry: At steady state the current has to satisfy 0 = S J (z ss ) or, equivalently, J (z ss ) ∈ ker S. In our example, the null-space ker S is one-dimensional and spanned by C int = 1 −1 −1 1 0 0 . Hence, the steady-state current is fully described by a single scalar value, J (z ss ) = int C int . e vector C int represents a series of reactions that leave the system state unchanged: the two substrates are bound along reactions 1 and 4 and released again along reactions −3 and −2. In the end, the system returns to the exact same state as before these reactions. erefore, we call this vector internal stoichiometric cycle. Having identi ed this internal cycle renders the rst step complete. In the following we explain why this step is important. e closed system has to satisfy a constraint that comes from physics: A closed system necessarily has to relax to a thermodynamic equilibrium state -which is characterized by the absence of currents of extensive quantities on any scale. us thermodynamic equilibrium is satis ed if int = 0. One can show that this requirement is equivalent to Wegscheider's condition [25]: e product of the forward rate constants along the internal cycle equals that of the backward rate constants, Furthermore, irrespective of thermodynamic equilibrium, the steady state has to be stoichiometrically compatible with the initial condition: ere are three linearly independent vectors in the cokernel of S: For each such vector, the scalar L ≡ · z evolves according to ∂ ∂t · z = · S J (z) = 0, and thus is a conserved quantity. We deliberately chose linearly independent vectors with a clear physical interpretation. ese vectors represent conserved moieties, i.e. a part of (or an entire) molecule that remains intact in all reactions. e total concentration of the enzyme moiety in the system is given by L E . e other two conservation laws, L 1 and L 2 , are the total concentrations of moieties of the the substrates, S 1 and S 2 , respectively. Given a set of values for the conserved quantities from the initial condition, Wegscheider's condition on the rate constants ensures uniqueness of the equilibrium solution [25].
(2) Open system -emergent cycles So far we only discussed the system in a closed box that will necessarily relax to a thermodynamic equilibrium.
We now open the box and assume that there is a mechanism capable of xing the concentrations of S 1 , S 2 and P to some given levels. ese three species therefore no longer take part in the dynamics. Formally, we divide the set of species into two disjoint sets: e rst are the internal species, X , which are subject to the dynamics. e second are the chemosta ed species, Y , which are exchanged with the environment. We apply this spli ing to the stoichiometric matrix, and the vector of concentrations, z = (x, ). Analogously, the rate equations for this open reaction system split into e Equation (7) is merely a de nition for the ma er current I , keeping the species Y at constant concentrations. e actual dynamical rate equations, the Eqs. (6), are a subset of the original equations for the closed system, treating the chemostats as constant parameters. Absorbing these la er concentrations into the rate constants, we arrive at a linear ODE system with new pseudo-rst-order rate constantsk( ). For these rate equations, one needs to reconsider the graphical representation of this reaction network: Since the chemosta ed species now are merely parameters for the reactions, we have to remove the chemostated species from the former vertices of the network representation and associate them to the edges. e resulting graph representing the open network is drawn in Fig. 2.
e steady-state current J ss = J (x ss , ) of Eq. (6) needs to be in the kernel of the internal stoichiometric matrix S X only.
is opens up new possibilities. It is obvious that ker S is a subset of ker S X , but ker S X is in fact bigger. In our example we now have two cycles, We give a visual representation on the right of Fig. 2. e rst cycle is the internal cycle we identi ed in the closed system already: this cycle only involves reactions that leave the closed system invariant, thus upon completion of this cycle not a single molecule is being exchanged. e second cycle is di erent: upon completion it leaves the internal species unchanged but chemosta ed species are exchanged with the environment. Since this type of cycle appears only upon chemosta ing, we call them emergent stoichiometric cycles. Overall, the steady-state current is a linear combination of these two cycles: e cycles are not the only structural object a ected by the chemosta ing procedure: the conservation laws change as well. In the enzyme example we have merely one conservation law le -that of the enzyme moiety, L E . e substrate moieties are being exchanged with the environment, which renders L 1 and L 2 broken conservation laws. Overall, upon adding three chemostats two conservation laws were broken and one cycle emerged. In fact, the number of chemosta ed species always equals the number of broken conservation laws plus the number of emergent cycles [26].
(3) Net stoichiometries and net forces e net stoichiometry of the emergent cycle is S 1 + S 2 P . is represents a single microscopically reversible reaction describing the net catalytic action of the enzyme. For a complete coarse graining, we still need to identify the uxes and the net force along this reaction. Its net force is given by the sum of the forces along the emergent cycle: One could also ask about the net stoichiometry and net force along the internal cycle. However, since the internal cycle does not interact with the chemostats, we have S C int = 0 and thus no net stoichiometry to talk about. Moreover, the net force along the internal cycle is by virtue of Wegscheider's condition.

(4) Apparent uxes
We now determine the apparent uxes along the two cycles of the system. To that end, we rst calculate the steady-state concentrations and the steady-state currents.
From the method of King and Altman [27] we know that the concentrations of the enzyme containing species x, scaled by the total enzyme-moiety concentration, L E , are normalized like probabilities. erefore, the rate equations can be interpreted as a continuous-time master equation which is linear in these probabilities. Its steady-state solution is known from probability theory: each x i /L E can be expressed as a sum of twelve terms, each term being a product of pseudo-rst-order rate constants. e total sum over all these sums serves as a normalizing denominator, N E . Note that N E depends on all the rate constants, as well as on the chemosta ed concentrations [S 1 ], [S 2 ] and [P]. We give the exact expressions in the SI.
From Schnakenberg [28] and Hill [29] we know how to express the currents in a multicyclic system with a corresponding diagrammatic method: We choose the reactions 1, 3, 4, and 5 as a tree spanning all vertices of the reaction graph in Fig. 2. e two cycles now can be associated with the remaining two reactions. e internal cycle contains the reaction 2 (in reverse direction) and other reactions from the tree. In contrast, the emergent cycle involves reaction 6 (in forward direction). us we can read o the reaction currents of the cycles from the reaction currents of the respective reaction.
For a concise presentation, we will represent products of pseudo-rst-order rate constants as connected graphs (cf. Fig. 2). ese graphs always contain a circuit (drawn in blue) whose orientation is indicated by an arrow. e remaining black edges are directed towards the circuit. Furthermore, a collection of diagrams is a sum of these products.
With this shorthand notation we express the internal current as Accordingly, the current along the emergent cycle is As expected, the current along the emergent cycle ext is not zero, provided that its net force is not zero. However, note that the current along the internal cycle does not vanish either, even though its own net force is zero. Both currents vanish only when the net force, −∆ ext G, vanishes -which is at thermodynamic equilibrium. Finally, we decompose the current ext = ψ + − ψ − into the apparent uxes Flux-force relation With the explicit expressions for the net force, Eq. (9), and the apparent uxes, Eq. (11), of the emergent cycle we explicitly verify the ux-force relation at the coarse-grained level: is ux-force relation implies that the reaction current is always aligned with the net force along this reaction: ext > 0 ⇔ −∆ ext G > 0. In other words, the reaction current directly follows the force acting on this reaction.
In fact, in this case we can connect the ux-force relation to the second law of thermodynamics. e EPR reads With this representation, it is evident that the ux-force relation ensures the second law: σ ≥ 0. Moreover, we see explicitly that the EPR is faithfully reproduced at the coarsegrained level. is shows the thermodynamic consistency of our coarse-graining procedure.

B. Active Membrane Transport
We analyze a proton pump model similar to the one presented in Ref. [20]. Our proton pump models a membrane protein pumping protons from one side of the membrane (side a) to the other (side b). e membrane protein itself is assumed to be charged to facilitate binding of the protons and to have di erent conformations M − and − M where it exposes the binding site to the two di erent sides of the membrane. Furthermore, when a proton is bound, it can either bind another substrate S when exposing the proton to side a -or the respective product P when the proton is exposed to side b. e la er could be some other ion concentrations on either side of the membrane -or an energy rich compound (ATP) and its energy poor counterpart (ADP). e reactions modelling this mechanism are summarized in Fig. 3.
In order to nd a coarse-grained description for this transporter we apply our result.
(1) Closed system -internal cycles is closed system has no cycle, therefore Wegscheider's conditions do not impose any relation between the reaction rate constants. ere are three conservation laws in the closed system, ey represent the conservation of membrane protein, proton, and substrate moieties, respectively, showing that these three are conserved independently. For any initial condition, the corresponding rate equations will relax to a unique steadystate solution satisfying thermodynamic equilibrium, J (z) = 0.
eir visual representation is given on the right of Fig. 4 .
(3) Net Stoichiometry and net forces e rst emergent cycle has the net stoichiometry H + a +S H + b + P. is is the active transport of a proton from side a to side b, under catalysis of one substrate into one product. e net force of this reaction is e second cycle has net stoichiometry H + b H + a . is represents the slip of one proton from side b back to side a with net force For later reference, we note that the sum C cat = C tr + C sl has net stoichiometry S P, which represents pure catalysis with net force . Only two of them are independent and we choose C tr and C sl as a basis in the main text. e third is their sum C cat = C tr + C sl .

(4) Apparent Fluxes
Following King and Altman [27], we have a solution for the steady-state concentrations with normalizing denominator N M . e exact expressions are given the SI. With the steadystate concentrations, we calculate the contributions of both cycles to the steady-state current: J (x ss , ) = tr C tr + sl C sl . According to Hill's work [29] this reads as following: With the abbreviations we can express the apparent uxes as

Breakdown of the ux-force relation
We see that the abbreviated terms ξ appear symmetrically in the forward and backward uxes. erefore, when the net forces are zero, necessarily the currents vanish and the system is at thermodynamic equilibrium. However, in general, the currents do not vanish. Moreover, the concentrations of the chemostats appear in the four di erent uxes in di erent combinations -indicating that both net forces couple to both coarse-grained reactions. Due to this coupling, it is impossible to nd nice ux-force relations for the two reactions independently. Nonetheless, the EPR is correctly reproduced at the coarse-grained level: Since this is the correct entropy-produtcion rate of the full system at steady state, we know that it is always non-negative -and that the coarse-graining procedure is thermodynamically consistent.

IV. CYCLE-BASED COARSE GRAINING
From the perspective of a single biocatalyst, the rest of the cell (or cellular compartment) serves as its environment, providing a reservoir for di erent chemical species. Our coarse graining exploits this perspective to disentangle the interaction of the catalyst with its environment -in the form of emergent cycles -from the behavior of the catalyst in a (hypothetical) closed box at thermodynamic equilibrium -in the form of the internal cycles. From the perspective of the environment, only the interactions with the catalyst ma er, i.e. the exchanging particle currents. is is the fundamental reason why we can replace the actual detailed mechanism of the catalyst with a set of coarse-grained reactions that reproduce the exchanging currents. A formal version of this reasoning, including all necessary rigor and a constructive prescription to nd the apparent uxes, is provided in the SI.
In our examples we illustrated the fundamental di erence between the case where a catalyst can be replaced with a single coarse-grained reaction and the case where this is not possible. In the rst case, such a catalyst interacts with substrate and product molecules that are directly coupled by mass. We support the term simple catalyst for this case. Especially, any transferase, kinase, and isomerase falls under that condition. Whether or not the catalysis is additionally modi ed by activators or inhibitors, does not interfere with this condition. A er all, the modifyers are neither consumed nor produced.
us they appear only in the normalizing denominators of the steady-state concentrations and a ect the kinetics while leaving the thermodynamics untouched. Furthermore, if there is only one single emergent cycle in a catalytic mechanism, any product of pseudo-rst-order rate constants along any cycle in the network will either (i) satisfy Wegscheider's conditions or (ii) reproduce (up to sign) the net force, −∆ α G, of the emergent cycle. Ultimately, this is why the ux-force relation holds in this case. A formal version of this proof, including all necessary rigor, is provided in the SI.
In the case where we have to provide two or more coarsegrained reactions, the catalytic mechanism couples several processes that are not directly coupled by mass. We support the term molecular engine or free-energy transducer for this case. In particular, molecular motors, active transporters and other processes hydrolyzing ATP fall under this category. In this case the ux-force relation does not hold in general, as we proved with our counter-example. A er all, when several pro-cesses are coupled, the force of one process can overcome the force of the second process to drive the second current against its natural direction. is transduction of energy would not be possible at a coarse-grained level, if the ux-force relation was always true. e presented coarse-graining procedure is exact in steadystate situations, arbitrarily far from equilibrium. When dealing with time dependent situations, such as initial transients or driven systems, the coarse graining can still be put to use. Basically, it assumes a time-scale separation [30]: e catalystcontaining species have to reach a quasi-steady state before the concentrations of the substrates, products, or modifyers change considerably. Otherwise, the dissipation associated with the relaxation of the catalyst-containing species may be important but cannot be captured by our coarse graining.

V. SUMMARY
We have presented a coarse-graining procedure for biocatalysts and have shown that it is thermodynamically consistent. During this coarse graining procedure, a detailed catalytic mechanism is replaced by a few net reactions. e stoichiometry, kinetics and net forces for the coarse-grained reactions are calculated from the detailed mechanism -ensuring that at steady state the detailed mechanism and the net reactions have the same entropy-production rate.
Furthermore, we have shown that in the case where a detailed mechanism is replaced by a single reaction, this net reaction satis es a ux-force relation. In the case where a detailed mechanism has to be replaced with several net reactions, the ux-force relation does not hold for the net reactions due to cross-coupling.

VI. SUPPLEMENTARY INFORMATION
A. Steady-state concentrations for the enzymatic catalysis e steady-state concentrations for the enzymatic catalysis example are given by the following directional diagrams: Here, the diagrams are directed towards the vertices highlighted by circles and the normalizing denominator N E equals the sum of all the 60 directional diagrams given above.
B. Steady-state concentrations for the active transporter e steady-state concentrations for the active transporter example are given by the following directional diagrams: Here, the diagrams are directed towards the vertices highlighted by circles and the normalizing denominator N M equals the sum of all the 90 directional diagrams given above.

C. Kinetic rate laws for the coarse-grained reactions
We consider a catalytic mechanism with a catalyst and several substrates, products, inhibitors or activators. e mechanism is resolved down to elementary reactions following mass-action kinetics.
Upon chemosta ing all the substrates, products, inhibitors and activatiors -summarized as -we are le with rate equations that are linear in the catalyst-containing speciessummarized as x. is open system still has a conservation law for the total catalyst-moiety concentration L = i x i . Moreover, it has a simple representation as a graph G where all the catalyst-containing species i form the vertices V and the reactions ρ∪−ρ form bidirectional edges R. e rate equations for the normalized concentrations x i /L are equivalent to a master equation describing a biased random walk on this graph. Using probability theory, we calculate the steady-state concentrations x * explicitly: Here, N ( ) = i τ ∈T i ρ ∈τkρ ( ) is a normalizing denominator, T i is the set of spanning trees rooted in vertex i, and k ρ ( ) is the pseudo-rst-order rate contant of reaction ρ. A rooted spanning tree is a spanning tree with its edges oriented such that all edges point towards the root.
In order to calculate the steady-state currents we choose a special spanning tree of the graph: (1) We start with the closed system and determine its internal cycles ker S. We take the set I ⊂ R of edges that the internal cycles are supported on. (2) Chose a spanning tree τ I on I ⊂ G. en every edge in I \ τ I gives rise to an internal cycle. (3) Complete τ I to a spanning tree τ * of G. All the edges not contained in the spanning tree are the chords. Now this spanning tree has some chords whose fundamental cycle are the emergent cycles. All other fundamental cycles are internal cycles. is construction of a spanning tree keeps the internal cycles and the emergent cycles separated.
We furthermore, choose an orientation of the chords. e currents on the chords then are identical to the steady-state currents along the fundamental cycles of the chords.
Let j → i = η be the chord of an emergent cycle. en the current through that chord is Next, we note that a lot of terms cancel by taking this di erence: all the spanning trees that contain the edge i → j or j → i, respectively, appear with both plus and minus sign: A er cancelling these spanning tree contributions, we de ne the apparent uxes as We obviously have i j = ψ i j − ψ ji . us the apparent uxes serve as kinetic rate laws for the coarse-grained reactions.

D. Proof of the ux-force relation
Before we proove the ux-force relation, we re-write the apparent uxes for the emergent cycles derived in Eq. (17).
is simpli es the nal proof considerably. To that end, we observe that adding a chord to a spanning tree not containing this chord always creates a cycle with coe cients in {−1, 0, 1} on the edges. is is a special kind of cycle: it can be interpreted as a subgraph, thus there is only a nite number of them. In the following we use the term circuit for this kind of cycle. Circuits can be easily represented visually, as done with the basis vectors in the main text. Since in Eq. (17) we sum over all possible spanning trees, the same circuits re-appear in several summands. We now re-sort the sums to rst run over distinct circuits, and then sum over the remainders of the spanning trees. For that we need some notation.
For any circuit c we abbreviate the product of pseudo-rstorder rate constants along it as w(c) = ρ ∈ckρ ( ) . e net force along a circuit thus is concisely wri en as For any circuit, c, we furthermore de ne F (c) to be the set of subforests of G not containing any edge of c but spanning the rest of the graph. Formally, Analogously to the product of rate constants along a circuit, for this set of subforests we denote the sum of products of rate constants as ξ (c) = f ∈ F(c) ρ ∈fkρ ( ). By construction, ξ (c) = ξ (−c) since the set F (c) does not depend on the orientation of c. Recall that the circuits appearing in Eq. (17) contain the chord j → i = η giving rise to the emergent cycle. Let C i j be this set of circuits. ese circuits can in fact be constructed with the spanning tree τ * : Since j → i is the chord of the emergent cycle, any such circuit is given by a self avoiding path from i to j through the rest of the graph. Either this path is entirely contained in τ * , in which case it is the emergent cycle itself -or it traverses one or more chords of the other internal or emergent cycles.
With this notation we rewrite the apparent uxes in the following way: is re-writing is not limited to the case of a single emergent cycle.
We now prove the ux-force relation -under the assumption that there is exactly one emergent cycle c η with chord η = j → i. Let −∆ η G be the net force along this cycle and let η be its current at steady state. Having only one emergent cycle means that for every circuit c ∈ C i j we have one of the following cases: • e circuit is formed by following the spanning tree, in which case it is exactly the emergent cycle: c = c η • e circuit is formed by traversing more chords, in which case it can be wri en as c = c η + γ where γ ∈ ker S is an internal cycle. In this case we have w (c) w (−c η ) due to Wegscheider's conditions.
In any case we can write w(±c) = ζ (c)w(±c η ) where ζ (c) = ζ (−c) is a symmetric factor. Overall, the apparent uxes for the emergent cycle are By construction, ξ and ζ are symmetric and also any sum over these terms is symmetric. Consequently, the apparent forward and backward uxes of the emergent cycle satisfy which, together with Eq. (18), concludes the proof.