�ermodynamically Consistent Coarse Graining of Biocatalysts beyond Mi�aelis–Menten

Artur Wachtel,∗ Riccardo Rao, and Massimiliano EspositoComplex Systems and Statistical Mechanics, Physics and Materials Science Research Unit,

University of Luxembourg, 162a, Avenue de la Faıencerie, 1511 Luxembourg, G. D. Luxembourg(Dated: April 27, 2018)

Starting from the detailed catalytic mechanism of a biocatalyst we provide a coarse-graining procedurewhich, by construction, is thermodynamically consistent. �is procedure provides stoichiometries, reaction�uxes (rate laws), and reaction forces (Gibbs energies of reaction) for the coarse-grained level. It can treatactive transporters and molecular machines, and thus extends the applicability of ideas that originated inenzyme kinetics. Our results lay the foundations for systematic studies of the thermodynamics of large-scalebiochemical reaction networks. Moreover, we identify the conditions under which a relation between one-way�uxes and forces holds at the coarse-grained level as it holds at the detailed level. In doing so, we clarify thespeculations and broad claims made in the literature about such a general �ux–force relation. As a furtherconsequence we show that, in contrast to common belief, the second law of thermodynamics does not requirethe currents and the forces of biochemical reaction networks to be always aligned.

I. INTRODUCTION

Catalytic processes are ubiquitous in cellular physiology.Biocatalysts are involved in metabolism, cell signalling, tran-scription and translation of genetic information, as well asreplication. All these processes and pathways involve not onlya few but rather dozens to hundreds, sometimes thousandsof di�erent enzymes. Finding the actual catalytic mechanismof a single enzyme is di�cult and time consuming work. Todate, for many enzymes the catalytic mechanisms are notknown. Even if such detailed information was at hand, includ-ing detailed catalytic machanisms into a large scale modelis typically unfeasable for numerical simulations. �erefore,larger biochemical reaction networks contain the enzymes assingle reactions following enzymatic kinetics. �is simpli�eddescription captures only the essential dynamical features ofthe catalytic action, condensed into a single reaction.

�e history of enzyme kinetics [1] stretches back more thana hundred years. A�er the pioneering work of Brown [2] andHenri [3], Michaelis and Menten [4] layed the foundationfor the systematic coarse graining of a detailed enzymaticmechanism into a single reaction. Since then, a lot of di�er-ent types of mechanisms have been found and systematicallyclassi�ed [5]. Arguably, the most important catalysts in bio-chemical processes are enzymes – which are catalyticallyactive proteins. However, other types of catalytic moleculesare also known, some of them occur naturally like catalyticRNA (ribozymes) or catalytic anti-bodies (abzymes), some ofthem are synthetic (synzymes) [5]. For our purposes it doesnot ma�er which kind of biocatalyst is being described by acatalytic mechanism – we treat all of the above in the sameway.

From a more general perspective, other scienti�c �elds areconcerned with the question of how to properly coarse graina process as well. While in most applications the focus lieson the dynamics, or kinetics, of a process, it turned out thatthermodynamics plays an intricate role in this question [6].

∗ artur.wachtel@uni.lu

For processes occurring at thermodynamic equilibrium, everychoice of coarse graining can be made thermodynamicallyconsistent – a�er all, the very foundation of equilibriumthermodynamics is concerned with reduced descriptions ofphysical phenomena [7]. Instead, biological systems are opensystems exchanging particles with reservoirs and as such theyare inherently out of equilibrium. Nonequilibrium processes,in general, do not have a natural coarse graining.

When the particle numbers in a reaction network are small,it needs to be described stochastically with the chemical mas-ter equation. Indeed, there is increased interest in the cor-rect thermodynamic treatment of stochastic processes [8, 9].With stochastic processes it is possible to investigate energy-conversion in molecular motors [10–13], error correction viakinetic proofreading [14–16], as well as information process-ing in small sensing networks [17–19]. In this �eld, di�erentsuggestions arose for coarse grainings motivated by thermo-dynamic consistency [20–22]. In these cases, the dissipationin a nonequililibrium process is typically underestimated –although also overestimations may occur [23]. For a generaloverview of coarse-graining in Markov processes, see Ref. [24]and references therein.

For large-scale networks however, a stochastic treatment isunfeasable. On the one hand, stochastic simulations quicklybecome computationally so demanding that they are e�ec-tively untractable. On the other hand, when species appearin large abundances (e.g. metabolic networks) the stochas-tic noise is negligible. �is paper is exclusively concernedwith this la�er case. �e dynamics is governed by determin-istic di�erential equations: the non-linear rate equations ofchemical kinetics. Assuming a separation of time scales inthese equations, model reduction approaches have been devel-oped [25–27]. However, they do not address the question ofthermodynamic consistency. Remarkably, recent developmentin the thermodynamics of chemical reaction networks [28, 29]highlighted the strong connection between the thermodynam-ics of deterministic rate equations and of stochastic processes,including the relation between energy, work, and informa-tion. Unfortunately, these studies were limited to elementaryreactions with mass–action kinetics. �e present paper ad-dresses this constraint, thus extending the theory to kinetics

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of coarse-grained catalysts.Understanding the nonequilibrium thermodynamics of cat-

alysts is a crucial step towards incorporating thermodynamicsinto large-scale reaction networks. �ere is ongoing e�ort inthe la�er [30–32] which very o�en is based on the connectionbetween thermodynamics and kinetics [33–35].

In this paper we show how to coarse grain the determin-istic description of any biocatalyst in a thermodynamicallyconsistent way – extending the applicability of such simpli�ca-tions even to molecular motors [10, 36] and active membranetransport [37]. �e starting point is the catalytic mechanismdescribed as a reversible chemical reaction network whereeach of the M reaction steps ρ is an elementary transitionrepresenting a conformational change of a molecule or anelementary chemical reaction with mass–action kinetics. �ecorresponding rates are given by the �uxes (kinetic rate laws),ϕ±ρ , that incorporate the reaction rate constants and the de-pendence on the concentration of the reactant molecules. �emass–action reaction forces (negative Gibbs free energies ofreaction) are −∆ρG = RT lnϕ+ρ/ϕ−ρ [38]. At this level, thereaction currents, Jρ = ϕ+ρ −ϕ−ρ , of these elementary steps arealigned with their respective reaction forces [39]: when oneis positive, so is the other. From here we construct a reducedset of C reactions with e�ective reaction �uxes ψ±α and netforces −∆αG. As we will see later, there is a limited freedomto choose the exact set of reduced reactions. Nonetheless, thenumber of reduced reactions is independent of this choice.

By construction, our coarse graining procedure capturesthe entropy-production rate (EPR) [39, 40] of the underlyingcatalytic mechanism,

Tσ := −M∑ρ

(ϕ+ρ − ϕ−ρ

)∆ρG = −

C∑α

(ψ+α −ψ−α

)∆αG ≥ 0 ,

even though the number C of e�ective reactions α is muchsmaller than the number M of original reaction steps ρ. �ere-fore, our procedure is applicable in nonequilibrium situations,such as biological systems. In fact, the above equation isexact under steady-state conditions. In transient and othertime-dependent situations the coarse graining can be a rea-sonable approximation. We elaborate this point further in thediscussion .

Secondly, we work out the condition for this coarse grainingto reduce to a single reaction α . In this case, we prove that thefollowing �ux–force relation holds true for this coarse-grainedreaction:

−∆αG = RT lnψ+αψ−α.

A trivial consequence is that the coarse-grained reaction cur-rent, Jα = ψ+α −ψ−α , is aligned with the net force, −∆αG. Inthe past, such a �ux–force relation has been used in the liter-ature [41, 42] a�er its general validity was claimed [33] andlater questioned [31, 34]. From here the belief arises that inevery biochemical reaction network with any type of kineticsthe currents and the forces of each reaction individually needto be aligned , a constraint used especially in �ux balanceanalysis [43–45] . However, as we show in this paper, this

relation does not hold when the coarse-graining reduces thebiocatalyst to two or more coupled reactions.

�is paper is structured as follows: First we present ourresults. �en, we illustrate our �ndings with two examples:�e �rst is enzymatic catalysis of two substrates into oneproduct. �is can be reduced to a single reaction, for whichwe verify the �ux–force relation at the coarse-grained level.�e second example is a model of active membrane transportof protons, which is a prototype of a biocatalyst that cannotbe reduced to a single reaction. A�erwards, we sketch theproofs for our general claims. Finally, we discuss our resultsand their implications. Rigorous proofs are provided in theappendix .

II. RESULTS

Our main result is a systematic procedure for a thermody-namically consistent coarse graining of catalytic processes.�ese processes may involve several substrates, products,modi�ers (e.g. activators, inhibitors) that bind to or are re-leased from a single molecule – the catalyst. �e coarse grain-ing involves only a few steps and is exempli�ed graphicallyin Fig. 1 :

(1) Consider the catalytic mechanism in a closed box andidentify the internal stoichiometric cycles of the system. Aninternal stoichiometric cycle is a sequence of reactions leavingthe state of the system invariant. Formally, internal stoichio-metric cycles constitute the nullspace of the full stoichiometricmatrix, S.

(2) Consider the concentrations of all substrates, modi�ers,and products (summarized as Y ) constant in time – thereforereduce the stoichiometric matrix by exactly those species.�e remaining species, X , represent N di�erent states of thecatalyst. As a consequence, the reduced stoichiometric matrix,SX , has a larger nullspace: New stoichiometric cycles emergein the system. �ese emergent cycles cause a turnover inthe substrates/products while leaving the internal speciesinvariant. Choose a basis, Cα , of emergent stoichiometriccycles that are linearly independent from the internal cycles.

(3) Identify the net stoichiometry, SYCα , together with thesum, −∆αG, of the forces along each emergent cycle α .

(4) Calculate the apparent �uxes ψ±α along the emergentcycles at steady state.

For each emergent stoichiometric cycle α this procedureprovides a new reversible reaction with net stoichiometrySYCα , net force −∆αG, and net �uxes ψ±α . Furthermore, itpreserves the EPR and, therefore, is thermodynamically con-sistent.

Our second result is a consequence of the main result: Weprove that the �ux–force relation is satisi�ed at the coarse-grained level by any catalytic mechanism for which only onesingle cycle emerges in step 2 of the presented procedure , asin example III A. When more cycles emerge, the �ux–forcerelation does not hold as we show in the explicit counter-example III B .

3

Y₁ + X₁ … S₁

X₃ + Y₃

X₂ + Y₂P

Q₄ Q₃ + P

Q₁ H⁺ + Q₂

X₆X₅ + S₂

S + X₄H⁺ + X₇

S₁ + ES₂ES₁S₂ E + PEP

S₂ + ES₁

ES₁E + S₁ ES₂E + S₂

HMS MH + PPMHHM + S–M M–

HM MH–M + H+ M– + H+

Y₁ + X₁ … S₁

X₃ + Y₃

X₂ + Y₂P

Q₄ Q₃ + P

Q₁ H⁺ + Q₂

X₆X₅ + S₂

S + X₄H⁺ + X₇

S₁ + S₂ P

S P

H+ H+

ψ⁺

ψ¯

ψ⁺

ψ¯ψ⁺

ψ¯

ψ₁

ψ₂

ψ₂

ψ₁

FIG. 1. Overview of the coarse-graining procedure: [le�] �e starting point is a reaction network with elementary reactions followingmass–action kinetics in a steady state. �is example contains two catalytic mechanisms (blue boxes) and for illustrative purposes someadditional arbitrary reactions. Each of the two catalyst species, E and M, is conserved throughout the network. �e reaction partners ofthe catalysts re-appear in the rest of the network. From the perspective of the remaining network, only the turnover (blue arrows) of thesemolecules are relevant. �e involved concentrations may be global, as for S, or refer to di�erent well stirred sub-compartments (green box), asfor P. [right] �e procedure provides few coarse-grained reactions (blue arrows) that replace the originally more complicated mechanisms. �ekinetic rate laws,ψ , of the coarse-grained reactions are di�erent from mass–action. We construct them explicitly during the coarse-grainingprocedure, so that the turnover is correctly reproduced. Combined with the coarse-grained reaction forces (Gibbs free energies) also theentropy-production rate is reproduced exactly. We work out the coarse graining of these two catalysts, E and M, in detail in section III.

III. EXAMPLES

A. Enzymatic catalysis

Let us consider the enzyme E that we introduced in Fig. 1.It is capable of catalyzing a reaction of two substrates, S1and S2, into a single product molecule, P. �e binding orderof the two substrates does not ma�er. Every single one ofthese reaction steps is assumed to be reversible and to followmass–action kinetics. For every reaction we adopt a referenceforward direction. Overall, the enzymatic catalysis can berepresented by the reaction network in Fig. 2.

We apply our main result to this enzymatic scheme and thusconstruct a coarse-grained description for the net catalyticaction. We furthermore explicitly verify our second result byshowing 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 moleculecan leave or enter the reaction volume. �e dynamics is thendescribed by the following rate equations:

ddt z = S J (z) , (1)

where we introduced the concentration vector and the currentvector,

z =

©­­­­­­­­­­«

[E][ES1][ES2][ES1S2][EP][S1][S2][P]

ª®®®®®®®®®®¬, J (z) =

©­­­­­­«

k1[E][S1] − k−1[ES1]k2[E][S2] − k−2[ES2]

k3[ES2][S1] − k−3[ES1S2]k4[ES1][S2] − k−4[ES1S2]k5[ES1S2] − k−5[EP]k6[EP] − k−6[E][P]

ª®®®®®®¬,

4

E + S1 E + S2 ES2ES1

ES2 + S1ES1 + S2

ES1S2 EP E + P

1 2

3

465

FIG. 2. An enzymatic scheme turning two substrates into oneproduct. �e substrates can bind in arbitrary order. We adopt areference direction for the indivinual reactions: forward is fromle� to right, as indicated by the arrows. �e backward reactionsare from right to le�, thus every single reaction step is reversible.�is scheme has a clear interpretation as a graph: �e reactions areedges, reactants/products are vertices, where di�erent combinationsof reactants/products are considered di�erent vertices. �is graphhas three disconnected components and contains no circuit.

as well as the stoichiometric matrix ,

S =

©­­­­­­­­­­«

−1 −1 0 0 0 11 0 0 −1 0 00 1 −1 0 0 00 0 1 1 −1 00 0 0 0 1 −1−1 0 −1 0 0 00 −1 0 −1 0 00 0 0 0 0 1

ª®®®®®®®®®®¬. (2)

In the dynamical equations, only the currents J (z) dependon the concentrations, whereas the stoichiometric matrix S

does not. �e stoichiometric matrix thus imposes constraintson the possible steady-state concentrations that can be ana-lyzed with mere stoichiometry: At steady state the currenthas to satisfy 0 = S J (zss) or, equivalently, J (zss) ∈ kerS.In our example, the null-space kerS is one-dimensional andspanned by Cint =

(1 −1 −1 1 0 0

)ᵀ. Hence, the steady-

state current is fully described by a single scalar value, J (zss) =JintCint .�e vector Cint represents a series of reactions thatleave the system state unchanged: �e two substrates arebound along reactions 1 and 4 and released again along reac-tions −3 and −2. In the end, the system returns to the exactsame state as before these reactions. �erefore, we call thisvector internal stoichiometric cycle. Having identi�ed thisinternal cycle renders the �rst step complete.

Note that this stoichiometric cycle does not correspond toa self-avoiding closed path, or circuit, in the reaction graph inFig. 2. �is is due to the fact that combinations of species serveas vertices. If instead each species individually is a vertex,then also each cycle corresponds to a circuit.

In the following we explain why the �rst step of the proce-dure is important. �e closed system has to satisfy a constraintthat comes from physics: A closed system necessarily has torelax to a thermodynamic equilibrium state – which is char-acterized by the absence of currents of extensive quantitieson any scale. �us thermodynamic equilibrium is satis�ed ifJint = 0. One can show that this requirement is equivalent toWegscheider’s condition [46]: �e product of the forward rateconstants along the internal cycle equals that of the backwardrate constants,

k1k4k−3k−2 = k−1k−4k3k2 . (3)

E

ES2

ES1

ES1S2

EP

1

24

3

5

6 S1

S1

S2 S2P CintCext

FIG. 3. [le�] Enzymatic catalysis as an open chemical network. �especies S1, S2 and P are now associated to the edges of the graph,instead of being part of its vertices as in Fig. 2. �is graph hasonly one connected component and contains three distinct circuits.[center, right] Graphical representation of the two circuits spanningthe kernel of SX . �e lower le� triangle constitutes the third circuit.It can be recovered by a linear combination of the other two circuits.

Furthermore, irrespective of thermodynamic equilibrium,the steady state has to be stoichiometrically compatible withthe initial condition: �ere are three linearly independentvectors in the cokernel of S:

`E =

©­­­­­­­­­­«

E 1ES1 1ES2 1

ES1S2 1EP 1S1 0S2 0P 0

ª®®®®®®®®®®¬, `1 =

©­­­­­­­­­­«

E 0ES1 1ES2 0

ES1S2 1EP 1S1 1S2 0P 1

ª®®®®®®®®®®¬, `2 =

©­­­­­­­­­­«

E 0ES1 0ES2 1

ES1S2 1EP 1S1 0S2 1P 1

ª®®®®®®®®®®¬.

For each such vector, the scalar L ≡ ` · z evolves according to∂∂t ` · z = ` · S J (z) = 0, and thus is a conserved quantity. Wedeliberately chose linearly independent vectors with a clearphysical interpretation. �ese vectors represent conservedmoieties, i.e. a part of (or an entire) molecule that remainsintact in all reactions. �e total concentration of the enzymemoiety in the system is given by LE. �e other two conserva-tion laws, L1 and L2, are the total concentrations of moietiesof the substrates, S1 and S2, respectively.

Given a set of values for the conserved quantities from theinitial condition, Wegscheider’s condition on the rate con-stants ensures uniqueness of the equilibrium solution [46].

(2) Open system – emergent cycles

So far we only discussed the system in a closed box thatwill necessarily relax to a thermodynamic equilibrium.

We now open the box and assume that there is a mechanismcapable of �xing the concentrations of S1, S2 and P to somegiven levels. �ese three species therefore no longer take partin the dynamics. Formally, we divide the set of species intotwo disjoint sets:

{E,ES1,ES2,ES1S2,EP}︸ ︷︷ ︸X

∪ {S1, S2, P}︸ ︷︷ ︸Y

.

�e �rst are the internal species, X , which are subject to thedynamics. �e second are the chemosta�ed species, Y , which

5

are exchanged with the environment. We apply this spli�ingto the stoichiometric matrix,

S =

(SX

SY

),

and the vector of concentrations, z = (x ,y). Analogously, therate equations for this open reaction system split into

∂

∂tx = SX J (x ,y) , (4)

0 ≡ ∂∂ty = SY J (x ,y) + I (x ,y) . (5)

�e Equation (5) is merely a de�nition for the exchange cur-rent I , keeping the speciesY at constant concentrations. Notethat the exchange currents I quantify the substrate/productturnover. �e actual dynamical rate equations, the Eqs. (4),are a subset of the original equations for the closed system,treating the chemostats as constant parameters. Absorbingthese la�er concentrations into the rate constants, we arriveat a linear ODE system with new pseudo-�rst-order rate con-stants k(y). For these rate equations, one needs to reconsiderthe graphical representation of this reaction network: Sincethe chemosta�ed species now are merely parameters for thereactions, we have to remove the chemosta�ed species fromthe former vertices of the network representation and asso-ciate them to the edges. �e resulting graph representing theopen network is drawn in Fig. 3.

�e steady-state current Jss = J (xss,y) of Eq. (4) needs tobe in the kernel of the internal stoichiometric matrix SX only.�is opens up new possibilities. It is obvious that kerS is asubset of kerSX , but kerSX is in fact bigger. In our examplewe now have two stoichiometric cycles,

Cint =

©­­­­­­«

1 12 −13 −14 15 06 0

ª®®®®®®¬and Cext =

©­­­­­­«

1 12 03 04 15 16 1

ª®®®®®®¬. (6)

�e �rst cycle is the internal cycle we identi�ed in the closedsystem already: It only involves reactions that leave theclosed system invariant, thus upon completion of this cyclenot a single molecule is being exchanged. �e second cycleis di�erent: Upon completion it leaves the internal speciesunchanged but chemosta�ed species are exchanged with theenvironment. Since this type of cycle appears only uponchemosta�ing, we call them emergent stoichiometric cycles.Overall, the steady-state current is a linear combination ofthese two cycles: Jss = JintCint + JextCext. �is completes step2.

�ese two stoichiometric cycles correspond to circuits inthe open reaction graph. We give a visual representation onthe right of Fig. 3. As a consequence of working with catalysts,the vertices of the reaction graph for the open system coin-cide with the internal species X . �erefore, for all catalyststhe cycles of the open system correspond to circuits in thecorresponding graph.

�e cycles are not the only structural object a�ected by thechemosta�ing procedure: �e conservation laws change aswell. In the enzyme example we have merely one conservationlaw le� – that of the enzyme moiety, LE. �e substrate moietiesare being exchanged with the environment, which rendersL1 and L2 broken conservation laws. Overall, upon addingthree chemostats two conservation laws were broken and onecycle emerged. In fact, the number of chemosta�ed speciesalways equals the number of broken conservation laws plusthe number of emergent cycles [47].

(3) Net stoichiometries and net forces

�e net stoichiometry of the emergent cycle is S1 + S2 P .�is represents a single reversible reaction describing the netcatalytic action of the enzyme. For a complete coarse graining,we still need to identify the �uxes and the net force along thisreaction. Its net force is given by the sum of the forces alongthe emergent cycle. Collecting the Gibbs energies of reactionin a vector, ∆rG B (∆1G, . . . ,∆6G)ᵀ , this sum is conciselywri�en as

−∆extG := −Cext · ∆rG = RT ln k1k4k5k6[S1][S2]k−1k−4k−5k−6[P]

. (7)

One could also ask about the net stoichiometry and netforce along the internal cycle. However, we have S Cint = 0since the internal cycle does not interact with the chemostats.Moreover, the net force along the internal cycle is

−Cint · ∆rG = RT ln k1k4k−3k−2k−1k−4k3k2

= 0 (8)

by virtue of Wegscheider’s condition.

(4) Apparent �uxes

We now determine the apparent �uxes along the two cy-cles of the system. To that end, we �rst solve the linear rateequations to calculate the steady-state concentrations andthe steady-state currents. For the steady-state concentra-tions we use a diagrammatic method popularized by King andAltman [48] that we summarize in appendix A.

As derived in step 2 of the procedure, the steady-state cur-rent vector is

Jss =

©­­­­­­«

1 Jint + Jext2 −Jint3 −Jint4 Jint + Jext5 Jext6 Jext

ª®®®®®®¬.

Hence the two cycle currents are

Jint = −J2 = k−2[ES2] − k2[E][S2] ,Jext = J6 = k6[EP] − k−6[E][P] .

6

With the explicit steady-state concentrations given in ap-pendix A 1, we �nd (see appendix B 1 for details):

Jint = k−2[ES2] − k2[E][S2]

=LEk2k3NE(y)k1

(k−1k4+ [S2]

)(k−1k−4k−5k−6[P] − k1k4k5k6[S1][S2]) .

and

Jext = k6[EP] − k−6[E][P]

=LEξ (y)NE(y)

(k1k4k5k6[S1][S2] − k−1k−4k−5k−6[P]) . (9)

Here, LE is the total amount of available enzyme, NE(y) is apositive quantity that depends on the chemostat concentra-tions as well as all rate constants, and

ξ (y) = k3[S1] +k2k3[S2]

k1+ k−2 +

k−2k−3k−4

.

As expected, the current along the emergent cycle Jext isnot zero, provided that its net force is not zero. However,note that the current along the internal cycle does not vanisheither, even though its own net force is zero. Both currentsvanish only when the net force, −∆extG, vanishes – which isat thermodynamic equilibrium.

Finally, we decompose the current Jext = ψ+ −ψ− into the

apparent �uxes

ψ+ =LEξ (y)NE(y)

k1k4k5k6[S1][S2] > 0 ,

ψ− =LEξ (y)NE(y)

k−1k−4k−5k−6[P] > 0 .(10)

Here, it is important to note that while

ψ+ −ψ− = k6[EP] − k−6[E][P] ,

there are several cancellations happening in the derivation ofEq. (9) implying that

ψ+ , k6[EP] , ψ− , k−6[E][P] .

We elaborate on these cancellations in this special case inappendix B 1 as well as for the general case in appendix B 3.

Flux–force relation

With the explicit expressions for the net force, Eq. (7), andthe apparent �uxes, Eq. (10), of the emergent cycle we ex-plicitly verify the �ux–force relation at the coarse-grainedlevel:

RT ln ψ+

ψ−= RT ln k1k4k5k6[S1][S2]

k−1k−4k−5k−6[P]= −∆extG .

�is �ux–force relation implies that the reaction current isalways aligned with the net force along this reaction: Jext >

−M + H+a

HM + S

M−

HM

HMS

−M

MH

PMH

M− + H+b

MH + P

1

2 3 4

57

6

FIG. 4. Reaction graph for the mechanism modeling the activetransport of protons from one side of a membrane, H+a , to the otherside, H+b . �e transport is coupled to the catalysis of a substrate, S,to a product, P. �e free transporter itself exists in two di�erentconformations denoted −M and M−, respectively. Again, all reactionsare considered reversible and to follow mass–action kinetics. Areference forward direction is indicated as arrows from le� to right.

0⇔ −∆extG > 0. In other words, the reaction current directlyfollows the force acting on this reaction.

In fact, in this case we can connect the �ux–force relationto the second law of thermodynamics. �e EPR reads

Tσ (xss,y) = −Jss · ∆rG = −JintCint · ∆rG − JextCext · ∆rG

= −Jext ∆extG = RT (ψ+ −ψ−) ln ψ+

ψ−≥ 0 .

With this representation, it is evident that the �ux–force re-lation ensures the second law: σ ≥ 0. Moreover, we seeexplicitly that the EPR is faithfully reproduced at the coarse-grained level. �is shows the thermodynamic consistency ofour coarse-graining procedure.

B. Active Membrane Transport

We now turn to the second example introduced in Fig. 1:A membrane protein, M, that models a proton pump similarto the one presented in Ref. [37]. It transports protons fromone side of the membrane (side a) to the other (side b). �emembrane protein itself is assumed to be charged to facilitatebinding of the protons and to have di�erent conformationsM− and −M where it exposes the binding site to the two dif-ferent sides of the membrane. Furthermore, when a proton isbound, it can either bind another substrate S when exposingthe proton to side a – or the respective product P when theproton is exposed to side b. �e la�er could be some other ionconcentrations on either side of the membrane – or an energyrich compound (ATP) and its energy poor counterpart (ADP).�e reactions modelling this mechanism are summarized inthe reaction graph in Fig. 4.

In order to �nd a coarse-grained description for this trans-porter we apply our result. Since the procedure is alreadydetailed in example III A, we omit some repetitive explana-tions in this example.

(1) Closed system – internal cycles

�is closed system has no cycle, therefore Wegscheider’sconditions do not impose any relation between the reaction

7

1

2

5

4

6

7

3

−M

HM

HMS PMH

MH

M−

H+a

S P

H+bCtr

Csl

Ccat

FIG. 5. [le�] Active membrane transport as a graph representingthe open chemical network. �e proton concentrations H+a and H+b ,as well as the substrate and the product are chemosta�ed, thus areassociated to the edges of the graph. [right] Graphical representa-tions for the three distinct cycles in this graph . Only two of themare independent and we choose Ccat and Csl as a basis in the maintext. �e third is their di�erence Ctr = Ccat −Csl.

rate constants. �ere are three conservation laws in the closedsystem,

`M =

©­­­­­­­­­­­­­­«

−M 1H M 1

H M S 1P M H 1M H 1M− 1H+a 0H+b 0S 0P 0

ª®®®®®®®®®®®®®®¬, `H =

©­­­­­­­­­­­­­­«

−M 0H M 1

H M S 1P M H 1M H 1M− 0H+a 1H+b 1S 0P 0

ª®®®®®®®®®®®®®®¬, `S =

©­­­­­­­­­­­­­­«

−M 0H M 0

H M S 1P M H 1M H 0M− 0H+a 0H+b 0S 1P 1

ª®®®®®®®®®®®®®®¬.

�ey represent the conservation of membrane protein (LM) ,proton (LH) , and substrate moieties (LS) , respectively, show-ing that these three are conserved independently. For anyinitial condition, the corresponding rate equations will relaxto a unique steady-state solution satisfying thermodynamicequilibrium, J (z) = 0.

(2) Open system – emergent cycles

We now �x the concentrations of the protons H+a and H+bin the two reservoirs, as well as the substrate and the productconcentrations. �e reaction network for this open system isdepicted in Fig. 5. �e open system still has a conserved mem-brane protein moiety while the conservation laws of protonsand substrate are broken upon chemosta�ing. Furthermore,there are two emergent cycles now,

Ccat=

©­­­­­­­­«

1 02 13 14 15 06 07 −1

ª®®®®®®®®¬and Csl =

©­­­­­­­­«

1 −12 03 04 05 −16 −17 −1

ª®®®®®®®®¬. (11)

�eir visual representation as circuits is given on the rightof Fig. 5.

(3) Net Stoichiometry and net forces

�e �rst emergent cycle has the net stoichiometry S P,which represents pure catalysis with net force

−∆catG = RT ln k2k3k4k−7[S]k−2k−3k−4k7[P]

. (12)

�e second cycle has net stoichiometry H+b H+a . �isrepresents the slip of one proton from side b back to side awith net force

−∆slG = RT lnk−1k−5k−6k−7[H+b ]k1k5k6k7[H+a ]

. (13)

For later reference, we note that the di�erenceCtr = Ccat−Cslhas net stoichiometry H+a + S H+b + P. �is is the activetransport of a proton from side a to side b, under catalysis ofone substrate into one product. �e net force of this reactionis

−∆trG = −∆catG + ∆slG

= RT lnk1k2k3k4k5k6[H+a ][S]

k−1k−2k−3k−4k−5k−6[H+b ][P]. (14)

(4) Apparent Fluxes

Solving the linear rate equations (see appendix A), we havea solution for the steady-state concentrations. �e exact ex-pressions are given in appendix A 2. With the steady-stateconcentrations, we calculate the contributions of both cyclesto the steady-state current: J (xss,y) = JcatCcat + JslCsl. Eachcurrent contribution is given by a single reaction:

Jcat = J2 =: ψ+cat −ψ−cat , Jsl = −J1 =: ψ+sl −ψ−sl .

With the abbreviations

ξcat := k−6k−5[H+b ] + k1k−5[H+a ][H+b ] + k6k1[H+a ] ,ξsl := k3k4 + k−2k4 + k−3k−2 ,

we can express the apparent �uxes as

NMLM

ψ+cat = k1k2k3k4k5k6[H+a ][S] + ξcatk−7k2k3k4[S] ,

NMLM

ψ−cat = k−1k−2k−3k−4k−5k−6[H+b ][P] + ξcatk7k−2k−3k−4[P] ,

NMLM

ψ+sl = k−1k−2k−3k−4k−5k−6[H+b ][P] + ξslk−1k−5k−6k−7[H+b ] ,

NMLM

ψ−sl = k1k2k3k4k5k6[H+a ][S] + ξslk1k5k6k7[H+a ] ,

�e derivation for these equations is detailed in appendix B 2.Note that NM depends on all rate constants and all chemostatconcentrations.

8

Breakdown of the �ux–force relation

We see that the abbreviated terms ξ appear symmetricallyin the forward and backward �uxes. �erefore, when the netforces are zero, necessarily the currents vanish and the sys-tem is at thermodynamic equilibrium. However, in general,the currents do not vanish. Moreover, the concentrations ofthe chemostats appear in the four di�erent �uxes in di�erentcombinations – indicating that both net forces couple to bothcoarse-grained reactions. Due to this coupling, it is impos-sible to �nd nice �ux–force relations for the two reactionsindependently:

−∆catG, RT lnψ+catψ−cat, −∆slG , RT ln

ψ+slψ−sl. (15)

To the contrary, it is easy to �nd concentrations for thefour chemostats where the catalytic force is so strong that itdrives the slip current againts its natural direction – givingrise to a negative contribution in the entropy-production rate.Nonetheless, the overall EPR is correctly reproduced at thecoarse-grained level:

Tσ = −Jss · ∆rG = −Jcat ∆catG − Jsl ∆slG ≥ 0 .

Since this is , by construction, the correct entropy-productionrate of the full system at steady state, we know that it is alwaysnon-negative – and that the coarse-graining procedure isthermodynamically consistent. �is example shows explicitlythat biochemical reaction networks need not satisfy the �ux–force relation, nor need their currents and forces be alignedto comply with the second law. A�er all, the function of thismembrane protein is to transport protons from side a to sideb against the natural concentration gradient.

IV. CYCLE-BASED COARSE GRAINING

From the perspective of a single biocatalyst, the rest of thecell (or cellular compartment) serves as its environment, pro-viding a reservoir for di�erent chemical species. Our coarsegraining exploits this perspective to disentangle the inter-action of the catalyst with its environment – in the formof emergent cycles – from the behavior of the catalyst ina (hypothetical) closed box at thermodynamic equilibrium– in the form of the internal cycles. From the perspectiveof the environment, only the interactions with the catalystma�er, i.e. the particle exchange currents: �ey prescribethe substrate/product turnover and when combined with thereservoir’s concentrations (chemical potentials) also the dissi-pation. Our coarse graining respects the reservoir’s concentra-tions and incorporates all the emergent cycles that exchangeparticles with the reservoir. It thus correctly reproduces theexchange currents: �is is the fundamental reason why wecan replace the actual detailed mechanism of the catalyst witha set of coarse-grained reactions in a thermodynamically ex-act way . A formal version of this reasoning, including allnecessary rigor and a constructive prescription to �nd theapparent �uxes, is provided in appendix B .

In our examples we illustrated the fundamental di�erencebetween the case where a catalyst can be replaced with a singlecoarse-grained reaction and the case where this is not possible.In the �rst case, such a catalyst interacts with substrate andproduct molecules that are coupled via exchange of mass in aspeci�c stoichiometric ratio. �is is known as tight coupling.Whether or not the catalysis is additionally modi�ed by ac-tivators or inhibitors, does not interfere with this condition.A�er all, the modi�ers are neither consumed nor produced.�us they appear only in the normalizing denominators ofthe steady-state concentrations and a�ect the kinetics whileleaving the thermodynamics untouched. Furthermore, if thereis only one single emergent cycle in a catalytic mechanism,any product of pseudo-�rst-order rate constants along anycircuit in the network will either (i) satisfy Wegscheider’sconditions or (ii) reproduce (up to sign) the net force, −∆αG,of the emergent cycle. Ultimately, this is why the �ux–forcerelation holds in this tightly-coupled case. A formal ver-sion of this proof, including all necessary rigor, is provided inappendix C.

In the case where we have to provide two or more coarse-grained reactions, the catalytic mechanism couples severalprocesses that are not tightly coupled via exchange of mass.To the contrary: �e turnover of di�erent substrates/productsneed not have �xed stoichiometric ratios. In fact, their ratioswill depend on the environment’s concentrations. In this casethe �ux–force relation does not hold in general, as we provedwith our counter-example. A�er all, when several processesare coupled, the force of one process can overcome the forceof the second process to drive the second current against itsnatural direction. �is transduction of energy [12, 49] wouldnot be possible at a coarse-grained level, if the �ux–forcerelation was always true.

We now asses the reduction provided by our procedure: �enumber C of coarse-grained reactions α is always lower thanthe number M of reaction steps ρ in the original mechanism.�is can be understood from the graph representation of theopen system: �e number B of circuits in a connected graphis related to its number N of vertices (catalyst states) and thenumber M of edges (reaction steps) by B = M − N + 1 [50].Some of the circuits represent internal cycles, rendering B anupper bound to the number of emergent cycles C . Since thenumber N of catalyst states is at least two, these numbers areordered: M > B ≥ C . �is proves that our coarse grainingalways reduces the number of reactions.

V. DISCUSSION

�e original work of Michaelis and Menten [4] was basedon a speci�c enzyme that converts a single substrate into asingle product assuming a totally irreversible step. �eir goalwas to determine the rate of production of product molecule.Later progress in enzyme kinetics extended their method todeal with fully reversible mechanisms, as well as many sub-strates, many products and modi�ers [1]. �e focus on theturnover led many people to identify the net e�ect of the en-zyme with a single e�ective reaction, describing its kinetics

9

with the Michaelis–Menten equation (or one of its general-izations). Our coarse-graining indeed incorporates all thesespecial cases: �e Michaelis–Menten equation arises fromcoarse graining a mechanism of the form

S + E ES EP E + P (16)

and assuming that the last reaction step, the release of theproduct, is much faster than the other steps. �en the coarse-grained reaction current is identical to the substrate/productturnover. Importantly, our procedure highlights that there isno direct correspondence between the number of required netreactions and the number of circuits in the reaction graph –even of the open system. Some circuits correspond to inter-nal cycles that play a kinetic role, not leaving a trace in thethermodynamic forces. Only the emergent cycles need to betaken into account for the coarse graining. �us the net e�ectof a multi-cyclic catalyst might be consistently expressed as asingle e�ective reaction, as seen in the example III A.

Likewise, in theoretical studies of biochemical systems,e�ective unimolecular reactions of the form

AX Y

B

are frequently used, where the reaction rate constants satisfy

k+

k−= exp

[µ◦A − µ◦B + µX − µY

RT

].

Here, the chemical potentials, µ, account for the thermody-namic force exerted by X and Y. Even when the actual e�ectivereaction does not follow mass–action kinetics, this equationis assumed, implying that the e�ective reaction �uxes arek+[A] = ψ+ and k−[B] = ψ−, and the “constants” k indeeddepend on some concentrations. �is is only consistent if theimplicit conversion mechanism is tightly coupled by exchangeof mass: When tightly coupled, the di�erences of the chemicalpotentials represent the Gibbs free energy change along thereaction A+X B+Y. In this case, the above equation is the�ux–force relation. Otherwise, our coarse-graining procedurereveals that this is thermodynamically inconsistent: If theimplicitly modelled catalysis is not tightly coupled via theexchange of mass, there is a hidden thermodynamic drivingforce that is independent of the concentrations of A and B,while the turnover of X/Y is not in a stoichiometric ratio tothe turnover of A/B. We have seen in example III B that the�ux–force relation indeed does not hold in this case.

�e failure of the �ux–force relation in the non-tightly cou-pled case does not imply inconsistent thermodynamics. Ourcoarse-graining procedure indeed deals with this case veryeasily. �e resulting �uxes and forces reproduce the entropy-production rate while sacri�cing the �ux–force relation. �ekey di�erence to the original ideas in enzyme kinetics is thatthe substrate/product turnover is split into several e�ectivereactions with their own reaction �uxes and forces, reproduc-ing the entropy-production rate. �is is especially importantfor complex catalysts: Many models for molecular motors andactive transporters are not tightly coupled. �ese free-energytransducers o�en display slippage via futile cycles. While

some enzymes also show signs of slippage, many simple en-zymes are modelled as tightly coupled – which implies theysatisfy the �ux–force relation. Our coarse graining deals withall these cases and in that sense goes far beyond Michaelis–Menten.

Our procedure greatly reduces the number of species and re-actions involved in a network while reproducing the entropy-production rate. �is comes at the cost of complicated e�ective�uxes (rate laws). �ey are rational functions of the involvedconcentrations and thus more complicated than simple mass–action kinetics. Nonetheless, our procedure is constructiveby giving these complicated expressions explicitly. With theexplicit solutions at hand, further assumptions can be madeto simplify the e�ective �uxes – as in the case of the originalMichaelis–Menten equation. Note that these additional sim-pli�cations may have an impact on the entropy-productionrate, in the worst case breaking the thermodynamic consis-tency. �is trade-o� between simplicity and thermodynamiccorrectness needs to be evaluated case by case.

We now discuss the limitations of our approach. �e pre-sented coarse-graining procedure is exact in steady-state situa-tions, arbitrarily far from equilibrium. When the surroundingreaction network is not in a steady state, the coarse grainingcan still be used: �en the coarse-grained reaction �uxes andforces have to be considered instantaneous – they change intime due to the changing substrate/product (or modi�er) con-centrations. Underlying this point of view is a separation oftime scales: When the abundance of substrates and productsis very large, as compared to the abundance of catalyst, thenthe concentrations of the la�er change much more quickly.�is results in a quasi-steady state for the catalyst-containingspecies. Consequently, our coarse graining cannot capture thecontribution to dissipation that arises in this fast relaxationdynamics. It only captures the dissipation due to the conver-sion of substrate into product. �is reasoning can be mademore rigorous: �ere are time-scale separation techniquesfor deterministic rate equations [25, 51] frequently used inbiochemical contexts [26], furthermore stochastic correctionsdue to small copy-numbers [52] and even e�ective memory ef-fects [27, 53] can be incorporated. However, these techniquesdo not explicitly address the question of thermodynamic con-sistency and we think that combining our coarse-grainingwith these techniques is a promising endeavour for the future.

We restricted the entire reasoning in this paper to catalysts.�ey follow linear rate equations when their reaction part-ners have constant concentrations. �is linearity allowed usto give explicit solutions for general catalysts. Focusing onthe emergent cycles to reproduce the correct thermodynamicspaves the way to apply a similar procedure beyond catalysts:Reaction networks that remain non-linear a�er chemostat-ting still have emergent cycles [28]. �ey can be calculatedalgebraically from bases for the nullspaces of the full and thereduced stoichiometric matrices, S and SX . �e cycles innon-linear networks may not have a representation as circuitsin the reaction graph, as we have seen with the internal cy-cle of the enzyme in a closed box. Nonetheless, each of theemergent cycles Cα can serve as an e�ective reaction: It hasa well de�ned stoichiometry, SYCα , and a well de�ned net

10

force, −∆rG ·Cα . �e steady state concentrations as well asthe �uxes, however, need to be determined case by case. Non-linear di�erential equations can be multi-stable, where ourcoarse graining applies to each stable steady state. Some non-linear ODEs exhibit limit cycles, thus never reaching a steadystate. In this case our procedure is no longer applicable.

VI. SUMMARY

We have presented a coarse-graining procedure for biocata-lysts and have shown that it is thermodynamically consistent.During this coarse graining procedure, a detailed catalyticmechanism is replaced by a few net reactions. �e stoichiom-etry, deterministic kinetic rate laws and net forces for thecoarse-grained reactions are calculated explicitly from thedetailed mechanism – ensuring that at steady state the de-tailed mechanism and the net reactions have both the samesubstrate/product turnover and the same entropy-productionrate.

Furthermore, we have shown that in the tightly-coupledcase where a detailed mechanism is replaced by a single re-action, this net reaction satis�es a �ux–force relation. In thecase where a detailed mechanism has to be replaced withseveral net reactions, the �ux–force relation does not holdfor the net reactions due to cross-coupling of independentthermodynamic forces . Ultimately, this cross-coupling allowsthe currents and forces not to be aligned – while complyingwith the second law of thermodynamics.

Overall, we have shown that coarse-graining schemeswhich preserve the correct thermodynamics far from equilib-rium are not out of reach.

ACKNOWLEDGMENTS

�is work is �nancially supported by the National ResearchFund of Luxembourg in the frame of AFR PhD Grants No.7865466 and No. 9114110. Furthermore, this research is fundedby the European Research Council project Nano�ermo (ERC-2015-CoG Agreement No. 681456).

Con�ict of Interest

�e authors declare that they have no con�ict of interest.

Appendix A: Diagrammatic method for explicit steady states oflinear reaction networks

We consider a catalytic mechanism with a catalyst and sev-eral substrates, products, inhibitors or activators. �e mech-anism is resolved down to elementary reactions followingmass–action kinetics.

Upon chemosta�ing all the substrates, products, inhibitorsand activators – summarized as y – we are le� with rateequations that are linear in the catalyst-containing species –summarized as x . While the steady-state equations alone,0 = SX J (x ,y), are under-determined and linearly dependent,the open system still has a conservation law for the totalcatalyst-moiety concentration L =

∑i xi , which again is a

linear equation. We can replace the �rst line of the steady-stateequations with this constraint to arrive at linear equationsLe1 = M(y)x , where e1 = (1, 0, . . . ) is the �rst Cartesian unitvector and M(y) is an invertible square matrix that dependson the chemostat concentrations. According to Cramer’s rulethe unique solution to this problem is given by

x∗iL=

detMi (y)detM(y) , (A1)

where Mi (y) is identical to M(y) just with the ith column re-placed by e1. We now provide a diagrammatic method to rep-resent this solution. �is diagrammatic method is frequentlya�ributed to King and Altman [48] or Hill [54], while an equiv-alent approach was already employed by Kirchho� [55] tosolve problems in electric networks. We give the diagram-matic method in the language of graph theory [50, 56], forwhich we need some de�nitions.

�e open pseudo-�rst-order reaction network has a simplerepresentation as a connected graph G where all the catalyst-containing species i form the vertices V and the reactionsρ∪−ρ form bidirectional edges R. �e reduced stoichiometricmatrix SX is the incidence matrix for this graph.

A closed self-avoiding path in a graph is a circuit and can beidenti�ed with a vector c ∈ RR over the edges, whose entriesare in fact restricted to {−1, 0, 1}. Since a circuit is a closedpath, it satis�es SXc = 0 and reaches as many vertices as itcontains edges. A graph not containing any circuit is calledforest, a connected forest is called tree.

A connected subgraph τ ⊂ G is called spanning tree if itspans all the vertices but contains no circuit. �e set T ofspanning trees of a �nite graph is always �nite. A rootedspanning tree is a tree where all the edges are oriented alongthe tree towards one and the same vertex, called the root.

With these notions set, the determinants in Eq. (A1) can bewri�en as

detMi (y) =∑τ ∈Ti

∏ρ ∈τ

kρ (y) ,

detM(y) =∑i

∑τ ∈Ti

∏ρ ∈τ

kρ (y) C N (y) .

Here, Ti is the set of spanning trees rooted in vertex i , andkρ (y) is the pseudo-�rst-order rate constant of reaction ρ.

11

Overall, Kirchho�’s formula for the solution to the linearproblem is

x∗iL=

1N (y)

∑τ ∈Ti

∏ρ ∈τ

kρ (y) . (A2)

From this result it is easy to con�rm that the solution existsand is always unique as long as the chemostat concentra-tions are �nite and positive. Furthermore, the steady-stateconcentrations are expressed as sums of products of positivequantities, thus themselves are always positive.

While this formula is very compact and abstract, it is notobviously convenient for practical calculations. However, therooted spanning trees appearing in this formula can be visu-ally represented as diagrams, as we will see in the followingexamples. �ese diagrams are intuitive enough to make prac-tical calculations with this formula feasible.

1. Steady-state concentrations for the enzymatic catalysis

�e enzymatic catalysis example in the main text, whenopen, is represented by the graph in �gure 3. �is graph has�ve vertices and six edges. It contains three distinct circuitsand twelve di�erent spanning trees.

A visual representation of Kirchho�’s formula (A2) forits steady-state concentrations is given by the following dia-grams:

[E]LE=

1NE

[ES1]LE=

1NE

[ES2]LE=

1NE

[ES1S2]LE

=1NE

[EP]LE=

1NE

Here, each diagram represents a product of pseudo-�rst-order rate constants over a spanning tree that is rooted inthe (circled) vertex associated with the species we want tosolve for (le� hand side). �us, the concentrations are sumsof twelve diagrams each, normalized by a denominator NEthat equals the sum of all the 60 diagrams given above.

2. Steady-state concentrations for the active transporter

�e active membrane transporter example in the main text,when open, is represented by the graph in �gure 5. �is graph

has six vertices and seven edges. It contains three distinctcircuits and 15 di�erent spanning trees.

A visual representation of Kirchho�’s formula (A2) forits steady-state concentrations is given by the following dia-grams:

[−M]LM

=1NM

[HM]LM

=1NM

[HMS]LM

=1NM

[M−]LM

=1NM

[MH]LM

=1NM

[PMH]LM

=1NM

Here, each diagram represents a product of pseudo-�rst-order rate constants over a spanning tree that is rooted in the(circled) vertex associated with the species we want to solvefor (le� hand side). �us, the concentrations are sums of 15diagrams each, normalized by a denominator NM that equalsthe sum of all the 90 diagrams given above.

Appendix B: Kinetic rate laws for the coarse-grained reactions

We now explicitly construct the kinetic rate laws as ap-parent cycle �uxes. First, we make use of the diagrammaticmethod to derive the coarse-grained kinetic rate laws for thetwo example systems of the main text. �en we generalizethese examples to generic catalysts.

1. Kinetic rate laws for the enzymatic catalysis

As shown in the main text, the cycle currents are

Jint = −J2 = k−2[ES2] − k2[E][S2] ,Jext = J6 = k6[EP] − k−6[E][P] .

12

Plugging in the diagrams (appendix A 1) for the steady-stateconcentrations of the enzyme-containing species we arrive at

NELE

Jint = k−2

−k2[S2]

,NELE

Jext = k6

−k−6[P]

.Next, we multiply the remaining pseudo-�rst-order rate con-stants into the diagrams and highlight them in blue. �is leadsus to

NELE

Jint =

−

,NELE

Jext =

−

.Note how some of the diagrams did not contain that edgebefore, leading to a circuit in the new diagrams. �e newpseudo-�rst-order rate constant carries an arrowhead to high-light the orientation of that edge. �e black edges remainoriented along the other black edges towards the circled ver-tex. �e remaining diagrams already contained the reversepseudo-�rst-order rate constant for the newly incorporatededge. �e product of these forward and backward pseudo-�rst-order rate constants is highlighted as a dashed blue edgewithout arrowhead. �e la�er tree diagrams appear on bothsides of the minus signs and can be cancelled. �us the cur-rents areNELE

Jint =[ ]

−[ ]

,

NELE

Jext =[ ]

−[ ]

.

Here, we highlight the entire circuits in blue to emphasize thecommon factors in the remaining terms. Note that the squarerepresenting the internal cycle remained in the internal cyclecurrent on both sides of the minus sign. However, Wegschei-der’s conditions, Eq. (3), ensure that these terms cancel as well.

Furthermore, Wegscheider’s conditions allow us to expressthe diagrams containing the lower triangle with the uppertriangle:

= × =k−2k−3k−1k−4

,

= × =k2k3k1k4

.

Overall, the currents expressed with rate constants andconcentrations are

Jint =LEk2k3NEk1

(k−1k4+ [S2]

)(k−1k−4k−5k−6[P] − k1k4k5k6[S1][S2]) ,

Jext =LENE

(k3[S1] +

k2k3[S2]k1

+ k−2 +k−2k−3k−4

)×

(k1k4k5k6[S1][S2] − k−1k−4k−5k−6[P]) .

2. Kinetic rate laws for the active transporter

We proceed analogously to the previous calculation for theenzymatic catalysis: Plug the tree diagrams from appendix A 2into

Jcat = J2 = k2[S][HM] − k−2[HMS] ,Jsl = −J1 = k−1[HM] − k1[H+a ][−M] ,

and cancel all diagrams that do not contain a circuit. �isleads us to

NMLM

Jcat =[ ]

−[ ]

,

NMLM

Jsl =[ ]

−[ ]

.

Since this membrane transporter mechanism does not havean internal cycle, we cannot exploit Wegscheider’s conditionsto cancel more terms. Nonetheless, we see that we can factorthe circuits out of some of the terms. Overall, we arrive at thecycle currents

Jcat =: ψ+cat −ψ−cat , Jsl =: ψ+sl −ψ−sl .

with the �uxesNMLM

ψ+cat = k1k2k3k4k5k6[H+a ][S] + ξcatk−7k2k3k4[S] ,

NMLM

ψ−cat = k−1k−2k−3k−4k−5k−6[H+b ][P] + ξcatk7k−2k−3k−4[P] ,

NMLM

ψ+sl = k−1k−2k−3k−4k−5k−6[H+b ][P] + ξslk−1k−5k−6k−7[H+b ] ,

NMLM

ψ−sl = k1k2k3k4k5k6[H+a ][S] + ξslk1k5k6k7[H+a ] ,

13

where we used the abbreviations

ξcat := k−6k−5[H+b ] + k1k−5[H+a ][H+b ] + k6k1[H+a ] ,ξsl := k3k4 + k−2k4 + k−3k−2 .

3. Kinetic rate laws for generic catalysts

By making use of the graph theory notation introducedin appendix A, we can generalize the above calculations togeneric catalysts.

Before proceeding with calculations, we need a generalmethod to determine the cycle currents from individual reac-tion currents. To that end, we construct a special spanningtree τ ∗ for the graph G of the open system: (1) We start withthe closed system and determine its internal cycles kerS. Wetake the set I ⊂ R of edges that the internal cycles are sup-ported on. (2) Consider this set of edges I ⊂ G as a subgraphof the open network. Choose a spanning tree τI for this sub-graph. (3) Complete τI to a spanning tree τ ∗ of G. All theedges not contained in the spanning tree are the chords.

�ere is a special connection between chords and circuits�rst highlighted by Schnakenberg [57]: �e spanning treealone, by de�nition, does not contain any circuit. Adding achord to the spanning tree gives rise to a circuit composed ofthe chord together with edges from the spanning tree. Fur-thermore, by construction every chord gives rise to a di�erentcircuit and the set of these circuits form a basis of the cy-cle space kerSX . In this context the circuits associated tochords are also called fundamental cycles. �e currents on thechords then are identical to the steady-state currents alongthe fundamental cycles of the chords [57].

�e special spanning tree τ ∗ that we constructed is sepa-rating the chords into two sets: Each chord in I gives riseto an internal cycle, while the chords not in I give rise tothe emergent cycles. �is construction provides a basis forthe entire cycle space, yet keeps the internal cycles and theemergent cycles separated. �erefore we call it a separatingspanning tree.

It is worth noting that not every basis of circuits can beexpressed as fundamental cycles of a spanning tree. �is tech-nical detail, however, has no impact on our results. Di�erentbases are just di�erent representations of the same space. Inthe following we assume a spanning tree mainly for conve-nience.

Let j → i be the chord of an emergent cycle. �en thecurrent through that chord is

Ji j = ki j (y)x j − kji (y)xi

=L

N (y)

ki j (y)∑τ ∈Tj

∏ρ ∈τ

kρ (y) − kji (y)∑τ ∈Ti

∏ρ ∈τ

kρ (y) .

Next, we note that a lot of terms cancel by taking this di�er-ence. All the spanning trees that contain the edge i → j or

j → i , respectively, appear with both plus and minus sign:

Ji j =L

N (y)

ki j (y)∑τ ∈Tji→j ∈τ

∏ρ ∈τ

kρ (y) − kji (y)∑τ ∈Tij→i ∈τ

∏ρ ∈τ

kρ (y)

︸ ︷︷ ︸=0

+L

N (y)

ki j (y)∑τ ∈Tji→j<τ

∏ρ ∈τ

kρ (y) − kji (y)∑τ ∈Tij→i<τ

∏ρ ∈τ

kρ (y)

.A�er cancelling these spanning tree contributions, we de�nethe apparent cycle �uxes as

ψi j := L

N (y) ki j (y)∑τ ∈Tji→j<τ

∏ρ ∈τ

kρ (y) . (B1)

We obviously have Ji j = ψi j − ψji . �us the apparent cy-cle �uxes serve as kinetic rate laws for the coarse-grainedreactions.

�ere is, technically speaking, no strict necessity to can-cel the spanning tree contributions in order to arrive at ex-pressions that can serve as coarse-grained kinetic rate laws.Keeping the spanning tree contributions results in the ap-parent �uxes of the substrates/products that are being pro-duced/consumed along the chord. �is is a natural choice fordealing with data from isotope labelling experiments. Withthis de�nition for kinetic rate laws , however, the �ux–forcerelation is not satis�ed – even in the case of a single emergentcycle [34]. In contrast, our de�nition of apparent �uxes resem-bles the apparent cycle �uxes, rather than apparent exchange�uxes. Comparing the apparent cycle �uxes with the net forcealong the emergent cycle, we do have a �ux–force relation, asshown in the next section.

Appendix C: Proof of the �ux–force relation

Before we prove the �ux–force relation, we rewrite theapparent �uxes for the emergent cycles derived in Eq. (B1).�is simpli�es the �nal proof considerably. To that end, weobserve that adding a chord to a spanning tree not containingthis chord always creates a circuit . Since in Eq. (B1) we sumover all possible spanning trees, the same circuits re-appearin several summands. We now re-sort the sums to �rst runover distinct circuits, and then sum over the remainders ofthe spanning trees. For that we need some notation.

For any circuit c we abbreviate the product of pseudo-�rst-order rate constants along it as w(c) =∏

ρ ∈c kρ (y) . �e netforce along a circuit thus is concisely wri�en as

−∆cG = RT∑ρ ∈c

lnkρ (y)k−ρ (y)

= RT ln w(c)w(−c) . (C1)

Here, −c refers to traversing the circuit c with reversed ori-entation. For any circuit, c , we furthermore de�ne F (c) to be

14

the set of subforests of G that (i) do not contain any edge ofc , (ii) span the rest of the graph, and (iii) are directed towardsthe circuit c . Analogously to the product of rate constantsalong a circuit, for this set of subforests we denote the sum ofproducts of rate constants as

ξ (c) B∑

f ∈F(c)

∏ρ ∈f

kρ (y) .

By construction, ξ (c) = ξ (−c) since the set F (c) does notdepend on the orientation of c . Let Ci j be the set of circuitstraversing the edge j → i . Note that these circuits are exactlythe ones appearing in Eq. (B1) .

With this notation we rewrite the apparent cycle �uxes inthe following way:

ψi j =L

N (y)∑c ∈Ci j

w(c)ξ (c) .

�is rewriting is not limited to the case of a single emergentcycle. In fact, we used this form to express the apparent cycle�uxes of the active membrane transporter in appendix B 2.

We now prove the �ux–force relation – under the assump-tion that there is exactly one emergent cycle cη with chordη = j → i . Let −∆ηG be the net force along this cycle andlet Jη be its current at steady state. Let furthermore τ ∗ be aseparating spanning tree, as we de�ned in appendix B 3.

Having only one emergent cycle means that for every cir-cuit c ∈ Ci j we have one of the following cases:

• �e circuit is formed by following the separating span-ning tree from vertex i back to j , in which case it isexactly the emergent cycle: c = cη .

• �e circuit is formed by traversing more chords, inwhich case it can be wri�en as c = cη + γ where γ ∈kerS is an internal cycle. In this case we have w (c)

w (−c) =w (γ )w (−γ )

w (cη )w (−cη ) =

w (cη )w (−cη ) due to Wegscheider’s conditions.

In any case we can write w(±c) = ζ (c)w(±cη) where ζ (c) =ζ (−c) is a symmetric factor. Overall, the apparent �uxes for

the emergent cycle are

ψi j =L

N (y)∑c ∈Ci j

w(c)ξ (c) = L

N (y)

∑c ∈Ci j

ξ (c)ζ (c) w(cη) .

By construction, ξ and ζ are symmetric and also any sumover these terms is symmetric. Consequently, the apparentforward and backward �uxes of the emergent cycle satisfy

ψi j

ψji=

LN (y)

[∑c ∈Ci j ξ (c)ζ (c)

]w(cη)

LN (y)

[∑c ∈Ci j ξ (−c)ζ (−c)

]w(−cη)

=w(cη)w(−cη)

which, together with Eq. (C1), concludes the proof.From this proof it is evident, why the �ux–force relation

breaks down once there are several emergent cycles with non-zero forces: In the case where a circuit c ∈ Ci j is not identicalto the emergent cycle cη , we can still write it as c = cη + γ .However, now γ need not be an internal but might be anotheremergent cycle. �erefore, Wegscheider’s condition does notapply to it, thus w(γ ) and hence ζ (c) need not be symmetric.As a consequence, the ratio of apparent forward and backwardcycle �uxes cannot be expressed by the force of the emergentcycle −∆ηG alone.

�e proof also shows why the choice of a separating span-ning tree is mainly for convenience. In the case of a singleemergent cycle, the exact basis for the internal cycles doesnot ma�er and you can always �nd an appropriate separatingspanning tree. In the case of several emergent cycles, thereis no simple and direct relation between the force and the�uxes of a cycle. �e only consistency requirement is theentropy-production rate. However, the entropy-productionrate is a scalar and thus invariant under change of basis. Fur-thermore, it involves only the forces and the currents of thecycles. �is imposes no restrictions on the individual forwardand backward �uxes.

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