PHYSICAL REVIEW E 93, 062409 (2016)

Towards stable kinetics of large metabolic networks: Nonequilibrium potential function approach

Yong-Cong Chen,1,2 Ruo-Shi Yuan,1 Ping Ao,1 Min-Juan Xu,1 and Xiao-Mei Zhu1,3,*

1Key Laboratory of Systems Biomedicine, Ministry of Education, Shanghai Center for Systems Biomedicine,Shanghai Jiao Tong University, Shanghai, 200240, China

2SmartWin Technology, 67 Tranmere Avenue, Carnegie, VIC 3163, Australia3GeneMath, 5525 27th Avenue N.E., Seattle, Washington 98105, USA

(Received 8 April 2015; revised manuscript received 1 March 2016; published 9 June 2016)

While the biochemistry of metabolism in many organisms is well studied, details of the metabolic dynamicsare not fully explored yet. Acquiring adequate in vivo kinetic parameters experimentally has always been anobstacle. Unless the parameters of a vast number of enzyme-catalyzed reactions happened to fall into veryspecial ranges, a kinetic model for a large metabolic network would fail to reach a steady state. In this work weshow that a stable metabolic network can be systematically established via a biologically motivated regulatoryprocess. The regulation is constructed in terms of a potential landscape description of stochastic and nongradientsystems. The constructed process draws enzymatic parameters towards stable metabolism by reducing the changein the Lyapunov function tied to the stochastic fluctuations. Biologically it can be viewed as interplay betweenthe flux balance and the spread of workloads on the network. Our approach allows further constraints suchas thermodynamics and optimal efficiency. We choose the central metabolism of Methylobacterium extorquensAM1 as a case study to demonstrate the effectiveness of the approach. Growth efficiency on carbon conversionrate versus cell viability and futile cycles is investigated in depth.

DOI: 10.1103/PhysRevE.93.062409

I. INTRODUCTION

Large-scale metabolic networks attract interests from de-veloping diagnostic markers for human diseases to metabolicengineering of bioproducts with economic values [1–4]. Inthe literature, steady states are often assumed when thesesystems are analyzed. The existence of reachable steadystates on a given metabolic network is apparently consistentwith experimental observations [5–9]. On the other hand,steady states for a set of chemical reaction equations areusually difficult to achieve, as many dynamical system theoriesand previous modeling works have experienced [10–12]. Tocomplicate the issue further, it is impractical to constructa kinetic model purely based on current experimental datadue to the unavailability of a large number of parameters[13–17]. Retrofitting to obtain parameters from in vivoexperiments [11,12] is normally limited to laboratory bacteria.

To avert the difficulty, metabolic analyses mostly relyon static methods that employ stoichiometric matrices andincorporate mass conservation, thermodynamic constraints,and optimization hypotheses [7,18–20]. These approacheswould be sufficient if they could adequately reproducetopologies for the underlying networks. Unfortunately, formany systems this is not the case. For example, the fluxdistribution of central glucose metabolism varies widely acrossspecies of bacteria, while the chemical reactions on themetabolism remain conserved [21]. An elementary flux modemethod would have yielded the simplest flux mode under theglucose growth condition. Nature, nevertheless, prefers somecomplex routes for the simple carbon source. Toxic metabolitesbuildup was suggested as a contributing factor [22]. To verifythe hypothesis, a kinetic model is necessary as detailed

*To whom correspondence should be addressed: xiaomeizhu@yahoo.com

knowledge of concentration variations becomes essential.Another example that calls for better modeling is the bistabilityobserved in the Escherichia coli population after a glucose-gluconeogenic substrate shift [24]. In principle, dynamicalaspects of metabolism, such as flux oscillations [25], requireproper kinetic models. To date environmental influence on fluxdistribution remains largely unexplained [21,23].

We investigate the dynamics of a metabolic network byplacing metabolism in an adaptive process. The structure ofthe metabolic network, represented by its entire set of reactionparameters, is assumed to change in time: In our approach,there is a generic “regulatory” dynamics in the parametricspace that acts to improve the network stability. The key ideais to employ a dynamical landscape and its associated potentialfunction that can depict the stability of a stochastic network.The existence and construction of the potential function innongradient systems [26–29] are established via a stochasticdynamical decomposition. The “regulation” draws under thelandscape the enzymatic parameters towards the space inwhich the metabolic network stabilizes. When thermodynamicconstraints on chemical reactions are taken into account, therange of the parametric space can be further specified to offera realistic metabolic model.

The construction of potential functions from generalstochastic differential equations in Ref. [26] has been ex-plicitly demonstrated in typical complex nonlinear dynamics,including fixed points [36,37], limit cycles [29,37], and chaoticsystems [28]. The stochastic dynamical decomposition leadsto a stochastic integration (A-type) which is different fromtraditional Ito’s and Stratonovich’s types. The approach addsa unique advantage connecting determinacy and stochasticitythrough dual roles of the potential functions [27,38]. Experi-ments confirm that some processes in nature do correspondto the A-type integration [27]. The present work intendsto extend the analyses into practically useful computationaltools.

2470-0045/2016/93(6)/062409(12) 062409-1 ©2016 American Physical Society

CHEN, YUAN, AO, XU, AND ZHU PHYSICAL REVIEW E 93, 062409 (2016)

Our work is biologically motivated. A living cell differsfrom an isolated chemical reactor in many ways. Enzymes areconstantly synthesized and degraded, activated and inactivatedby metabolites and kinases. Metabolites move in and out ofthe cell in a regulated manner. The concept of regulationsis truly generic and everywhere. In this work, we attemptto place the various sources and mechanisms of biologicalregulations under a unified mathematical description, with aclear goal. The unified regulation aims to identify the regionsin the parametric space that can harbor stable steady-statesolutions for the metabolic network in question. The stabilityhere refers to the existence of steady-state solutions, and moreimportantly, the solutions are stable against random noisesthat are present in any biological system. To achieve thegoal, we explicitly introduce stochastic fluctuations at someintermediate steps. At the computational stage, the expressionsare simplified with respect to the physiological conditions ofa living cell. Finally, in addition to being a tool for metabolicmodeling, the mathematics developed in our study may help toclarify other biological regulations that are similar in nature.

II. GENERIC RATE EQUATION

Let us start with the simplest formula for an enzyme-catalyzed reaction, the so-called Michaelis-Menten equa-tion [30],

dy

dt= Vmaxx

KM + x, (1)

where x, y, Vmax, and KM are, respectively, the concentrationsof the substrate and the product, the maximum reaction rate,and the Michaelis-Menten constant. Vmax is determined byboth the concentration of the enzyme and its catalytic rate.In an organism, a reaction is often modified via allostericregulations by additional chemical which binds to the sameenzyme and acts to inhibit or activate the enzyme. This canresult in effective Vmax and KM that differs from Eq. (1).For example, the presence of a competitive inhibitor withconcentration xI changes the Michaelis-Menten constant to

KM ⇒ KM

(1 + xI

KI

). (2)

Quasisteady states on the intermediate complexes are assumedin Eqs. (1) and (2).

For kinetic modeling of large-scale metabolic networks,there are several obstacles to overcome concerning the enzymereaction rates. First of all, the modifier(s) to a reaction isnot always known. Second, even when the information doesexist, the exact form of the reaction rate can be obscured bymany unknown parameters. We address the second obstacleby observing that the rate equation of some well-known mod-ification mechanisms, Uni-Uni mechanism, Bi-Uni randommechanism, Bi-Bi ordered mechanism [31], ordered Uni-Bimechanism, random Bi-Bi mechanism, Ping Pong Bi-Bimechanism [32] can all be cast into a generic form of enzymaticrate equation. For a typical chemical reaction inside a cell,

S1 + S2 + · · · Sm

VF�VB

P1 + P2 + · · · Pn, (3)

where each substrate Si or product Pi can be the same ordifferent chemicals, the generic rate format representing thereaction can be written as

ν(xi,yj )

=VF

∏mi=1

xi

Ki− VB

∏nj=1

yj

K ′j

f1(VF ,VB)∏m

i=1

(1+ xi

Ki

) + f2(VF ,VB)∏n

j=1

(1+ yj

K ′j

) ,

(4)

where xi and yj are, respectively, the concentrations of thesubstrates and the products. In Eq. (4), VF and VB are theforward and backward maximum reaction rates. Ki and K ′

j arethe apparent Michaelis-Menten constants for each substrateand product. f1 and f2 are normalization factors given byf1(VF ,VB ) = V 2

F /(V 2F + V 2

B ), f2(VF ,VB ) = V 2B/(V 2

F + V 2B ).

The generic rate equation is symmetrical in both directionsfor a reversible reaction and is formally exact under thequasi-steady-state condition. While the decision to employour simplified rate equation or different forms is a choiceof preference [11,12,33], the simplification is convenient forboth analytical and computational studies. In the Appendix,an explicit instance of the generic rate equation is derived forthe ordered Uni-Bi mechanism.

III. METABOLIC NETWORK

We now consider a metabolic network of N metaboliteswith their concentrations represented by a N × 1 vector, xT =(x1,x2, . . . ,xN ). The superscript T stands for transverse of amatrix. A set of kinetic questions can be readily written in theform

x = Sν(x) + b(x,t) = F (x,V,t), (5)

where V denotes a vector in the parameter space coveringall the reactions on the network, i.e., VF and VB [Eq. (4)]for all reactions, and S is a (N × M) stoichiometric matrixconnecting the reactions to the metabolites. ν(x) is a (M × 1)matrix with M being the total number of reactions. Eachcomponent of ν(x) takes the form of Eq. (4). b(x,t) is a(N × 1) vector representing fluxes in or out of the network. Ifb(x,t) is not explicit in t , it can be absorbed into the first termby adding virtual reactions to or from a fictitious “external”metabolite. This will be discussed in Sec. V B. Hereafter, weassume b(x,t) = b(x) and F (x,V,t) = F (x,V ).

A dynamical system under Eq. (5) is usually unstable: Somemetabolites will quickly accumulate and some others depleteas demonstrated by a simple example in Fig. 1. Rather thanregarding the selection of reaction parameters as a modelingproblem, we need to incorporate both the biological and themathematical aspects of the model for our study. Here theassociated biological question is how a living cell adjustsits parameters and maintains the metabolic stability underfluctuations of concentrations in both metabolites and enzymes(reaction rates). In fact a living cell is assisted by complexbiological processes [35]: At a short timescale, metabolites canbe transferred in and out of the cell and enzymes are modulatedby metabolites. At a longer timescale, enzymes are constantlycreated and degraded. Note that mass conservation is stillsatisfied at the coarse-grained level for tissues or colonies.

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TOWARDS STABLE KINETICS OF LARGE METABOLIC . . . PHYSICAL REVIEW E 93, 062409 (2016)

0 100 200 300 4000

5

10

15

20

0 100 200 300 4000

2

4

6

8

10

ABC

0 100 200 300 4000

2

4

6

8

10ABCV

1F

V1B

V2F

V2B

0 100 200 300 4000

2

4

6

8

10

V1

V2

A1

C3

B 2

Regulation on V1 V2&Regulation on V1 V2&

Constant V1 V2&

Constant V1 V2&

(a) (b)

(c) (d)

FIG. 1. A simple network of two reactions with input and output fluxes. The network is shown in the inset of (a). When V1F , V1B , V2F , andV2B are constant, the existence of steady-state solution depends on their values. (a) is calculated for V1F = 3, V1B = 0.1, V2F = 1, V2B = 0.1and (b) for V1F = 0.3, V1B = 0.1, V2F = 0.5, V2B = 0.1. (c) and (d) are calculated with regulation, starting from the same initial valuescorresponding to (a) and (b), respectively. The regulation dynamics of Eq. (18) is found to help the system reaching steady states. The solutionsare not unique on the network.

Mathematically, the stoichiometric matrix and therefore theform of Eq. (5) remains unaffected. We may regard the param-eters in the generic reaction rate Eq. (4), VF , VB , Ki , and K ′

j asin vivo parameters, to be decided by experimental observationsand other constraints. We will handle VF and VB first.

IV. CONSTRUCTION OF STABLEMETABOLIC DYNAMICS

To find out how the enzymatic parameters in Eq. (4)should be regulated in order to achieve stability across thenetwork, we return to Eq. (5) with an addition of randomnoise ξ , representing stochastic fluctuations over the metabolic

network:

x = F (x,V ) + ξ (x,V,t), (6)

〈ξ (x,V,t)ξ (x,V,t ′)T 〉 = 2D(x,V )δ(t − t ′), (7)

where D stands as the diffusion matrix for the noises, andT denotes transpose of a matrix. Note that by stability, wemean that the steady-state solutions can remain stable againstrandom perturbations. The latter are present in any biologicalsystem. Thus, having their presence in the construction of theregulation is quite logical.

Although fluctuations exist in any biological system,Eqs. (6) and (7) are not derived explicitly from a microscopic

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CHEN, YUAN, AO, XU, AND ZHU PHYSICAL REVIEW E 93, 062409 (2016)

model, hence questions might arise if the equations areappropriate for the present study. In arriving at the equations,Markov process is explicitly imposed in the time-relatedpart δ(t − t ′) in Eq. (7). Such an assumption is widelyadopted in stochastic studies related to colloidal particles. Inour context, we primarily focus on the metabolic reactions,which are chemical reactions in nature and should be welldescribed by the model. Our choice of the diffusion matrixD(x,V ) is not limited to specific forms. In the lack ofany specific information, it can be taken as proportional tothe concentrations of reactants and reaction velocities. Notethat some biological fluctuations might not fall into Markovprocess and should be further studied. Finally, there is a fineline between stochastic fluctuations and purposeful biologicalregulations. The latter is precisely the main area of study inthis work.

To proceed further, we can use a recently developedmethod to identify a Lyapunov function for the nongradientsystem [27]. Following the mathematical analysis developedearlier [26,27], there exists a generalized potential φ(x,V ), aLyapunov function for Eq. (6) such that

[�(x,V ) + �(x,V )]F (x,V ) = −∇xφ(x,V ), (8)

[�(x,V ) + �(x,V )]D(x,V )[�(x,V ) − �(x,V )] = �(x,V ),

(9)

where �(x,V ) is a nonnegative and symmetric matrix, and�(x,V ) is an antisymmetric matrix. Note that both �(x,V )and �(x,V ) are determined by the diffusion matrix D. Theprecise meaning of the potential φ becomes transparent near astable fixed point, where φ can be used to define a Boltzmann-like stationary distribution [36] [cf. below Eq. (15)]. Such adistribution can carry divergence-free probability currents. Inthe linear case where F (x,V ) is a linear function of x, andD(x,V ) is constant, the force can be split into two parts, oneof which gives rise to a detailed balance with diffusive motions,the other induces cyclic motions on the surface of constant φ.A stable metabolic network solution corresponds to a localminimum in the potential φ(x,V ) in our definition. This doesnot exclude φ(x,V ) from more complex topologies such as alimit cycle when φ remains constant in the cycle [29].

We next define a nonnegative cost function ψ(x,V ),

ψ(x,V ) = F (x,V )T �(x,V )F (x,V ), (10)

which is related to the change in the potential for x (underfixed V ) in the following way:

For the deterministic part of the dynamics we have

dφ(x,V )

dt

∣∣∣∣V =const

= ∇xφ(x,V ) · F (x,V ) = F (x,V )T ∇xφ(x,V )

= −F (x,V )T [�(x,V ) + �(x,V )]F (x,V )

= −F (x,V )T �(x,V )F (x,V )

= −ψ(x,V ). (11)

In the above we have used the antisymmetric property of �.We further introduce a regulation on V that aims to minimize

ψ(x,V ) by

W (x,V )dV

dt= −∇V ψ(x,V ). (12)

Here W (x,V ) is a matrix whose symmetric part has positiveeigenvalues. The form of W (x,V ) chosen for our metabolicmodeling will be discussed later. We will show below thatregulation on V will work toward stabilizing the originalnetwork of x.

Let us look at this regulation when the dynamics of themetabolites x and the parameters V are uncoupled. Note thatφ is in fact a Lyapunov function of the original network whenV remains constant [37,38]. On the trajectory of x(t), as aproperty of Lyapunov function, φ monotonically decreases intime. This is indeed confirmed by Eq. (11) as expected. If x

is near a steady state, then dφ(x,V )/dt → 0−. Therefore, thegoal of regulation is to vary V so that dφ(x,V )/dt can bebrought to 0−. When x is held constant and a dynamics in V

is imposed in the form of Eq. (12), the V dynamics suppressesψ , which in turn reduces the change in φ via Eq. (12) when x

is allowed to change. In other words, it leads the dynamics inx into a more stable parametric space as a result.

We can illustrate explicitly that a coupled dynamics on x

and V is assisted by Eq. (12) to become more stable. Since� is symmetric and at least semipositive, minimizing ψ viaEq. (12) moves toward F (x,V ) = 0, and the dynamical systemis stable when only V is allowed to vary. When both x and V

are allowed to change, the system is not necessarily stable. Forthe stability, we need to have dψ/dt � 0 along the trajectoryof x(t) and V (t), i.e., to decrease the absolute value of F (x,V ).For simplicity we set W in Eq. (12) to be an identity matrix(the result is essentially the same for different W ). Using therelationships shown in Eqs. (6) and (8), we have

dψ

dt= ∇V ψ · dV

dt+ ∇xψ · dx

dt

= ∇V ψ · dV

dt+ ∇xψ · (F + ξ )

= ∇V ψ · dV

dt+ ∇x(FT �F ) · F + ∇xψ · ξ

= ∇V ψ · dV

dt+ ∇x(FT (� + �)F ) · F + ∇xψ · ξ

= −∇V ψ · ∇V ψ − ∇x(FT ∇xφ) · F + ∇xψ · ξ

= −∇V ψ · ∇V ψ − FiFj∂xixjφ

− (∂xiφ)(∂xj

Fi)Fj + ∇xψ · ξ. (13)

In the final step of Eq. (13), the first three terms control thedeterministic part of dψ/dt . If V is unregulated and constant,the first term is zero. To consolidate the argument, we shallshow explicitly that the summation of the second and the thirdterms is negative near a stable fixed point in the unregulatedcase. Therefore, the first term, arising from the regulation,contributes to stabilize the network with another negative term.Assuming that x0 is a fixed point, F (x0,V ) = 0, then linearizeF (x0,V ) = F (x − x0), φ = (x − x0)T U (x − x0). Here F andU are constant matrices. Equations (8) and (9) can be written as(� + �)F = −U and F = −(D + Q)U , where (D + Q) =(� + �)−1 [26]. Q and � are antisymmetric matrices. The

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TOWARDS STABLE KINETICS OF LARGE METABOLIC . . . PHYSICAL REVIEW E 93, 062409 (2016)

total contribution of the second and third terms to Eq. (13) is

−FiFj∂xixjφ − (∂xi

φ)(∂xjFi)Fj

= −(x − x0)T U (D − Q)U (D + Q)U (x − x0)

− (x − x0)T U (D + Q)U (D + Q)U (x − x0)

= −2(x − x0)T UDUDU (x − x0) + 2(x − x0)T

×UQUQU (x − x0) − (x − x0)T

× (UDUQU − UQUDU )(x − x0)

= T1 + T2 + T3. (14)

Let DU (x − x0) = u1, QU (x − x0) = u2, then T1 and T2 inEq. (14) become T1 = −2uT

1 Uu1 and T2 = −2uT2 Uu2. Note

that � and Q are a symmetric and an antisymmetric matrix,respectively. For a stable fixed point, eigenvalues of U arepositive. Therefore, both T1 and T2 are negative. The thirdterm has no definite sign. But we can calculate the averagevalue of T3. The steady-state probability distribution is givenby [26]

ρ0 = 1

Zexp

[−1

2(x − x0)T U (x − x0)

]. (15)

We have

〈T3〉 =∫

dx1dx2 . . . dxN (x − x0)T (UDUQU − UQUDU )

× (x − x0)1

Zexp

[−1

2(x − x0)T U (x − x0)

]= tr[(U−1)(UDUQU − UQUDU )]

= tr(DUQU − QUDU ) = 0. (16)

Thus, for a stable fixed point without regulation, dψ/dt < 0as expected. The first term in Eq. (13) adds a negative termto dψ/dt < 0, to further stabilize the network. It is possiblethat the contribution from the regulation is sufficient to bringan unstable fixed point back to stable. This is shown in thesimple example given below. However, the above analysisrelies heavily on the Boltzmann-like distribution, which isonly valid near a stable point. Finally, it is worth mentioningthat the regulation moves towards ψ = 0 by lowering ψ , butit does not guarantee to ever reach it.

A further consideration can be made for practical modelingpurposes. If x values are known, V and the stable metabolicnetwork may be obtained from Eq. (11). In the genericenzyme reaction rate equation, x/K is invariant. Since thein vivo values of the apparent Michaelis-Menten constants ofK are mostly unavailable, we can set x/K = 1 as a startingpoint. Although from a network point of view the restrictionmight appear too strong, in realistic biological systems that isusually the case for a large range of metabolite concentrations.Mathematically, if x/K values are set differently, different VF

and VB are obtained with no qualitative difference.

A. Network stability, a simple example

Before turning to realistic biologic problems of interest,let us use a simple reaction network model to demonstrateexplicitly the idea of this work with just two reactions A +C

V1−→ B and BV2−→ C. The two reactions form a simple cycle

in which one unit of metabolite A is lost per cycle, similar to afutile cycle in biology. We also add an input and output to thenetwork. The dynamics of the network is given by

d[A]

dt= v1 − V1 = fA,

d[B]

dt= V1 − V2 − v2 = fB,

d[C]

dt= −V1 + V2 − v3 = fC,

with

V1 =V1F

[A]KA

[C]KC

− V1B[B]KB

V 21F

V 21F +V 2

1B

(1 + [A]

KA

)(1 + [C]

KC

) + V 21B

V 21F +V 2

1B

(1 + [B]

KB

) ,

V2 =V2F

[B]KB

− V2B[C]KC

V 22F

V 22F +V 2

2B

(1 + [B]

KB

) + V 22B

V 22F +V 2

2B

(1 + [C]

KC

) . (17)

Here [A], [B], and [C] are the concentrations of metabolitesA, B, and C. V1F , V1B , V2F , V2B , KA, KB , and KC are theparameters for the forward and backward maximum reactionrates and the apparent Michaelis-Menten constants as definedin Eq. (4). For demonstration purposes we set v1 = 0.2 tometabolite A, v2 = 0.2[B]/(1 + [B]) from metabolite B, andv3 = 0.1 to metabolite C, KA = KB = KC = 1. The choicesare arbitrary and they do not change the qualitative behaviorof the system. In Figs. 1(a) and 1(b), we show that the stabilityof the two-reaction network indeed depends on the choice ofV1F , V1B , V2F , and V2B . Next, we include a simplified versionof the regulatory dynamics Eq. (12). Note that solving Eqs. (8)and (9) for a given diffusion matrix D is not essential forpractical purposes because detailed in vivo information of D

is largely unavailable. Instead, we can choose an approximate� to start with. In general, though, we may not arbitrarilychoose � while treating D as to be determined reversely byEqs. (8) and (9). In the case of limit cycle � approaches zeroon the cycle regardless of D [29].

On the given example, we choose � and W so that Eq. (12)becomes

dV1F

dt= −V1F ∂V1F

(ωAf 2

A + ωBf 2B + ωCf 2

C

),

dV1B

dt= −V1B∂V1B

(ωAf 2

A + ωBf 2B + ωCf 2

C

),

(18)dV2F

dt= −V2F ∂V2F

(ωAf 2

A + ωBf 2B + ωCf 2

C

),

dV2B

dt= −V2B∂V2B

(ωAf 2

A + ωBf 2B + ωCf 2

C

),

with fA, fB , fC given by Eq. (17),and ωA = 1/

√([A][C]V1F /KAKC)2 + ([B]V1B/KB)2,

ωB = ωC = 1/(√

([A][C]V1F /KAKC)2 + ([B]V1B/KB)2 +√([B]V2F /KB)2 + ([C]V2B/KC)2). The matrix W is assumed

to be diagonal. At the limit of small �, Eq. (9) implies that� scales with the inverse of D. The diffusion matrix D

in turn scales with the reaction rates. This is the reasonto choose the above ωA, ωB , and ωC . We can further set[A]/KA = [B]/KB = [C]/KC = 1 in their expressions. Withthe inclusion of the added regulation, Eqs. (5) and (12) are

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CHEN, YUAN, AO, XU, AND ZHU PHYSICAL REVIEW E 93, 062409 (2016)

then solved simultaneously. We see that in Fig. 1(d) when theinitial parameters are chosen to be the same as in Fig. 1(b),the network eventually stabilizes, in contrast to what it doesin Fig. 1(b).

Given the simple example, we now briefly discuss itspossible connections to realistic biological regulations. Theregulation of Eqs. (10) and (12) minimizes a cost functionthat is essentially the multiplication of the rate of changeF = synthesis − utilization for each metabolite and �. Theminimization of F is nothing but product inhibition andsubstrate activation, as is often encountered in organisms. Forexample, elevated blood sugar level in a person triggers theenzyme transcription of glucokinase and fructose 6-phosphatein the liver [30]. The dependence of � is subtler. The increaseon the diffusion matrix D is better for reactions to be evenlyspread and to avoid near-zero reaction rates. This feature mighthave an implication in the understanding of evolution: Arethe reaction topologies of different species such as those inRef. [21] evolved along this direction?

Note that in the above construction the fluctuations-relatedD only plays an auxiliary role and can be set to zero at the end ifonly the stability at the deterministic level is concerned. Whilethe construction of the regulation is formal and analyticallyshown to be effective towards stability, the approximationused later in this paper may require further validations intheir own context. We need to handle the fluctuations morerealistically if we wish to explore the relationship between themathematically constructed regulation and the biological onesfound in cells. The latter can be complicated by both chemicalreactions [39] and biological processes involved [40].

V. LARGE-SCALE METABOLIC MODELING

Let us consider how the regulation process can be appliedto the forward and backward maximum reaction rates VF , VB

for metabolic modeling. We would like to find a simplifiedversion of Eqs. (11) and (12) that is both computationallystraightforward and biochemically transparent. In general, weexpect the diffusion matrix D to scale with the magnitudeof the fluxes, i.e., D ∼ F (x,V ) ∼ (x/K)VF [or (x/K)VB],representing the level of an intrinsic stochastic process [39].As a result of Eqs. (8) and (9), � (and �) should scaleinversely with D. We further simplify the cost function ψ

by taking � matrix diagonal for each VF (or VB), i.e., aweighted square summation of the related fluxes. The weightshould scale with the inverse of metabolite concentrations.Since the dominant part of metabolism is centered aroundthe usage of organic carbons, we simplify the requirement byscaling the weight with the carbon numbers in the chemicalformulas; for example, CH3COOH has a carbon number =2. The use of carbon number accounts for the stoichiometricrelation that a larger compound (with a high carbon number)is synthesized from smaller ones, hence is less abundant ina cell. It is also due to the majority of metabolic reactionsbeing centered around the carbon routes. Moreover, we letW (x,V ) to scale as 1/V 2

F (or 1/V 2B ) to counter balance the

factor 1/VF (or 1/VB ) in ψ . This sets a preference to smallerbut positive (forward and backward) maximum reaction ratesover large ones. In addition, the nonlinear factors before andafter the partial differentiation ∂νk

ensures that the solutions

fall into small but positive νk . Finally, we assume that thecost function is evaluated at the physiological values of theratios {xi/Ki,yj /K

′j } as appear in Eq. (4). The ratios are

either taken from experiments or set to 1. This is based onobservations that under normal physiological conditions thereactions in a cell are catalyzed by enzymes that are mosteffective when the concentrations match the affinities of theunderlying enzymes [30].

To summarize the above simplifications in terms of mathe-matics, we have

τdVk

dt= −V 2

k ∂Vk

[(1

VkNk

)∑n

(1 + cn)F 2n (x,V )

]

Vk ∈ {VF ,VB}. (19)

Here τ is a characteristic timescale for the regulation process,the summation is over the Nk number of metabolites affectedby Vk , and cn is the carbon number of the nth metabolite.Note that any flux balance solution F (x,V ) = 0 is also asolution to Eq. (19), therefore Eq. (19) can be consideredas an enhanced flux balance analysis that attempts to excludeunstable solutions. The overall flow diagram of the modelingprocedure is presented in Fig. 2.

A. Thermodynamic constraints

We now revisit Eq. (4) and examine closely the impactof the laws of thermodynamics on a metabolic network. Thebalance of a chemical reaction is governed by the differencein the apparent Gibbs free energy between the products andreactants, which in turn determines an equilibrium constantKeq for each reaction. Therefore,

VF

∏nj=1 K ′

j

VB

∏mi=1 Ki

= Keq ∝ exp

(− G

RT

). (20)

Since we do not require Eq. (4) to be an elementary reaction ourKeq is not always dimensionless. G may be considered as theone taken from the rate-deciding step in the apparent chemicalreaction when it involves several stepwise reactions. A reactionis irreversible if the drop in free energy is sufficiently large.Equation (20) holds under quasistationary states. If we assumemost of the reactions are operated under that condition,then for each set of {VF ,VB}, Eq. (20) provides a set ofoptimization conditions for the apparent Michaelis-Mentenconstants {K,K ′}. In real metabolism, the changes in theirvalues can be the results of inhibition or activation by any third-party metabolites, e.g., via Eq. (2). Note that the invarianceof Eq. (4) under {xi/Ki}, {yj/K

′j } implies that if the ratios

are kept constant, the solutions obtained for {VF ,VB} remainunaffected.

Our operational process for the optimization is as follows.We start with an initial set of {x0i/K0i}, {y0j /K

′0j }, then

optimize K and K ′ for the best fit to Eq. (20) while keepingthe ratios of {x/K} and {y/K ′} fixed. In practice, we use acost function,

∑r

{[VF

∏nj=1 K ′

0j /y0j

VB

∏mi=1 K0i/x0i

]·[∏n

j=1 yj∏mi=1 xi

]− Keq

}2

, (21)

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TOWARDS STABLE KINETICS OF LARGE METABOLIC . . . PHYSICAL REVIEW E 93, 062409 (2016)

Metabolic Model

Modeling Procedure

Enzyme concentration dependent

Modifiable bymultisubstrate

kinetics

ParametersParameters

Thermodynamics

Steady state concentration

Metabolic

input/outputPhysiological

concentration

Stability

Efficiency & Stability Initial

Acetyl-CoA

Isocitrate

α-KG

Succ-CoA

Fumarate

Citrate Malate

OAA

Succinate

FIG. 2. A summary of the kinetic modeling method employed in the present work. The in vivo forward and backward maximum reactionrate VF , VB and apparent Michaelis-Menten constants are determined by stability, thermodynamics constraints, and experimental data. Somebiological information on the metabolic input or output is required for the metabolite exchanges between the network in the model and the restof the cell. The exchanges are treated as reactions with an external pool and the maximum reaction rates to be determined in the same manneras that for the normal VF , VB via Eq. (19). When initial values of VF , VB are smaller, the stable states of the network obtained are more efficientin growth (via carbon conversion rate) but with larger relaxation times. The experimental measured efficiency may be used to pick the mostappropriate solution. Without adequate knowledge of in vivo enzyme kinetics, steady-state metabolite concentrations are assumed to be on theorder of Michaelis-Menten constants, and this assumption is used in Eq. (19). Such an approach can be adjusted when further experimental dataare available. In our model, thermodynamic constraints are imposed, independent of the stability requirement, to determine the steady-statemetabolite concentrations and the in vivo apparent Michaelis-Menten constants.

to determine the optimized concentrations. K and K ′ is thenobtained from {x/K} and {y/K ′} ratio by {x/K = x0i/K0i}and {y/K ′ = y0i/K

′0i}. The summation r runs over all the

reactions.

B. Boundary conditions

When only a subset of the complete metabolic reactions ina cell is selected for study, we need to deal with the fluxesbetween the selected part and the rest of the cell as well asthe cellular environment. We employ three types of couplingsto model the external interactions in Eq. (5). The first type isto allow a metabolite such as CO2 or phosphate to maintaina constant concentration. This applies when the metaboliteis involved in a large number of reactions, or can transfer inand out of a cell freely or is buffered by other mechanisms.The second type is for certain pair of metabolites, usually

coenzymes, e.g., adenosine triphosphate (ATP) and adenosinediphosphate (ADP), where metabolites in the pair alwaysappear on different sides of reactions in which they participate.As a result, the pair’s total concentration remains conserved onthe network. We handle this type by adding a virtual recyclingreaction for each pair to emulate the boundary flow. A thirdand the most common type of boundary conditions concernsnutrient intakes and biomass productions by the network.They are again handled by fictitious reactions to and from an“external” metabolite, which represents adjustable exchangesbetween these metabolites and the part of metabolism notincluded in the model.

C. Initial values

The systematic approach proposed in this paper is intendedfor a large metabolic network without sufficient parametric

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CHEN, YUAN, AO, XU, AND ZHU PHYSICAL REVIEW E 93, 062409 (2016)

details. In the absence of reliable experimental data, we canset some zeroth-order initial values as the starting point. Themodel can be improved over time when more accurate valuesare obtained. In the zeroth order and when no further infor-mation is available, the initial concentration of a metabolite isset to 1 mM. Similarly, the initial apparent Michaelis-Mentenconstants Ki and K ′

j of Eq. (4) are set to the respectiveinitial concentrations of the substrates and products involved;i.e., xi/Ki = xj/K

′j = 1. The thermodynamic equilibrium

constants Keq are set to 1 for all reactions that are known to bereversible under the physiological condition in question. Wehave experimented different choices of initial values VF andVB and found that the final flux distribution is only sensitiveto the overall scale of the initial values. The regulation processstarts with small initial VF and VB for all reactions, Eq. (19)works VF and VB up and settles into a steady state. Thisprocess is similar to switch on a cell from constitutivelyinactivate enzymes for better metabolism efficiency. Whenfurther biological information is available, the choice of VF

and VB can be changed to represent either a constitutivelyinactive enzyme (small initial value) or an active enzyme (largeinitial value) on an individual basis. The above zeroth-orderapproximation is used in our model study below.

D. Relaxation time

As an application, we show that the dynamical modelcan be used to estimate dynamical properties of a network.One of them is the relaxation time when deviated metabolicconcentrations return to a steady state. Given a set of metaboliteoutputs from the network (based on the growth rate obtainedexperimentally), we usually can obtain a range of steady-statesolutions. This is done by starting from different sets ofinitial reaction rates and letting them evolve under Eq. (19).To evaluate the stability of a solution, we can estimate the“relaxation time” reacting to small fluctuations from a givensteady state. Biologically shorter relaxation time correspondsto faster response and adaptation to environmental changes. Toquantify the analysis, we can linearize Eq. (5) at a steady stateto obtain the eigenvalues λi of the dynamics near the solution.We define the average or typical response time as

1

N

(∑i

1

λi

), (22)

where N is the number of nonzero eigenvalues. Zero ones areexcluded from the calculation as they correspond to redundantdegrees of freedom in the linearized equations. The number ofnonzero values is determined by the rank of the stoichiometricmatrix S in Eq. (5). The stability of a solution is justified whenthere is no eigenvalue with negative real part.

VI. APPLICATION TO METHYLOBACTERIUMEXTORQUENS AM1

We have successfully applied our methodology to thecentral metabolism of Methylobacterium extorquens AM1, amethylotrophic and environmental important bacterium thathas been extensively studied over the past two decades. Inparticular, the flux distributions were carefully studied [41].Methylobacterium extorquens AM1 grows on both single and

multiple carbon unit compounds. The central metabolism hasmany well-known metabolic cycles, they adjust in differentconditions by changing expressions of relevant enzymes.The metabolic pathways necessary for the growth undermethylotrophic conditions are included in our study. It coversformaldehyde metabolism, serine cycle, citric acid cycle,pentose phosphate pathway, poly-hydroxybutyrate (PHB)synthesis, glyoxylate regeneration cycle, gluconeogenesis,serine biosynthesis, and finally electron respiratory chain.The metabolic network used in this work is taken from aprevious work [42]. See Supplemental Material for additionalinformation [43]. A summary of the metabolic pathways andthe calculated fluxes are depicted in Fig. 3. The results inFig. 3(a) quantitatively agree across all cycles to that presentedin Ref. [42], the latter were constructed from experimentallyavailable data. Figure 4 presents selected real-time plots ofconcentrations returning to the steady state after a perturbation.Especially, it illustrates a rapid detoxification of formaldehydein the cell.

Efficiency versus viability

We can explore, as a biological application, the efficiency ofgrowth versus the viability of a cell for M. extorquens AM1. Inour numerical calculation, the target biomass production andthe type of nutrient intake are set first. Starting with small initialVF and VB for all reactions the regulation process, Eq. (19),works the way upwards until VF and VB are settled into asteady state. Varying the initial values allows us to obtain arange of solutions and compare them to experiments. Largerinitial VF and VB lead to larger fluxes across the network withlower carbon efficiency (measured by the nutrient intake). Theflux topologies remain qualitatively similar. Hence, certainamount of energy appears wasted in some apparent futilecycles. But, as shown in Fig. 3, the lower efficiency solutionhas a faster relaxation time hence is more stable against randomfluctuations. Furthermore, the increase in the futile fluxes atthe expense of carbon efficiency smoothes the relative spreadof steady-state metabolic concentrations. The spread is anunavoidable consequence of the thermodynamics constraints.Biologically, extreme metabolite concentrations cause fatalityas they put severe burdens on osmotic pressure, maintenance,and cellular transport.

The experimental data in Ref. [42] support the lessefficient solution shown in Fig. 3. The result indicates thatthe bacteria do not go with the most efficient metabolism.Instead, they choose to settle into a growth rate that, underthe physiological condition, has the optimal balance betweenefficiency and viability. Finally, the better viability arises fromthe futile cycles in metabolism for the reasons explainedabove.

VII. CONCLUSION

To summarize the work, a regulatory dynamics for en-zymatic parameters on a large-scale metabolic network hasbeen proposed. It is constructed to draw a vast number ofparameters into the phase spaces that stabilize the network.Under a stabilized kinetic metabolic model, thermodynamicalconstraints, the efficiency of growth and the viability of

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TOWARDS STABLE KINETICS OF LARGE METABOLIC . . . PHYSICAL REVIEW E 93, 062409 (2016)

0.25

OAA(5.9)

Malate(1.1) Citrate

Succ-CoA

Succinate -KG -KGKG

Isocitrate Fumarate

(1.1C

Isocitrateit t

itrateitrat

)

MMalate(

TCA Cycle

0.01

0.02

0.10

0.09

0.02

0.02

0.20

0.09

Mena-CoA Prop-CoA

But_CoA

Crot_CoA

3HB-CoA L3HB-CoA

oA

BBut_CoA

t CoA

But

HB-CoA

oA

c-CoASuc A0.10

0.41

0.04

0.09

0.090.09

PHB

Acetoacetate-CoA

3HB

Acetyl-CoA Acetoacetate

Succinate

Succ-CoA

3HHB CoA

0.0

HB

te

c-CoA

3HB3HB

0.01

0.01

Malyl-CoA(2.0)

cetoaceta

0.11

Crot

3HC A

Crot

HB C

0.50

L3HB-C

t Ct

HB C

0.50

CoA

PH

C

0.37

cetoacetaa

0.10

Su

Succ

10.0.

((

2

y

0.29

Ac

yl CoA Ac

0)

cetoace

00.0

cetoace

11

20

yl-CoA(2.

29.20.

Glyoxylate(4.1) ty

Maly

Acetate(4.1)

Glyoxylate Regeneration

Glutamate ate

2PG(4.0)

Glycerate(0.13)

Serine(0.64)

PEP(3.3)

Acetyl-CoA

Glycine(2.1)

OA.02

)

CoACoA

0.25

00.00AA

H-pyruvate(0.62)

0 20.255PE

Glyyoxylacine(2.1)

pyruvat0.32

0.32

2PG(4.0)

0.32

ycerate(0.13) G

H p

Glyc

pyr

(0

0.32

Methylene-H4F(0.06)Methanol

Methylene-H4MPT(280)

Formaldehyde(0.01)

Methenyl-H4MPT(6.5) Formyl-H4MPT(0.04) Formylmethanofuran(3.5 10 )

Formate(300) CO2 Formyl-H4F(2.3) Methenyl-H4F(0.85)

NAD NADH

Glycine(2.1)

e(Seriinete(0.62)

)

) NAD NADH

mate((3(3 CO2

0 35

F2.3)0.29

5)0.29

0)) Met0.64

hylene-H4F(0.06

Methenyl-H4

0.29

0.29

For0.64

) For0.64

ylene-H4MP

( )

0.64

aldehydey

0.64

Formaldehyde Metabolism

rmylmethano

300)

For

0 64

3PG(3.8) PHP(11) 3Pser(0.97)

Serine(0.64)

Glutamate

0.0723.8)

0.07

-KG

3(11)

Fru1,6BP

Glc6P

Fru6P

1,3BPG

TP

6PG

DHAP

3PG(3((3(3 8)

1,3BPG

0.02

TP

1 3BPG

0.02

Fru1,6BP

0.01

Fru6P

Fru1 6BP

0.0

0.01

Glc6P

Fru6P

0.0

6PG

Glc6P

7)

GlutamKG

0.050.050 97

Seri

7)

DH

2

P

00 020.01TPTP DPP

0.01

0.29

Serine Biosynthesis

PPP

Nutrient intake flux: 0.64 mM/second Metabolic network relaxation: 468 seconds

0.12

0.33

OAA(4.6)

Malate(1.1) Citrate

Succ-CoA

Succinate -KG -KGKG

Isocitrate Fumarate

(1.C

Isocitrateit t

itrateitrat

)

MMalate(

TCA Cycle

0.04

0.05

0.22

0.22

0.05

0.05

0.26

0.22

Mena-CoA Prop-CoA

But-CoA

Crot-CoA

3HB-CoA L3HB-CoA

oA

BBut-CoA

t-CoACrot

But-C

HB-CoAHB C

oA

c-CoASuc A0.22

0.51

0.06

0.19

0.190.19

PHB

Acetoacetate-CoA

3HB

Acetyl-CoA Acetoacetate

Succinate

Succ-CoA

3HHB CoA

0.0

HB

ate

c-CoASuc

3HB3HB

0.02

0.03

Malyl-CoA(1.2)

0.70

0.70

CoAC

PH0.45

cetoacetaa

((

2

y

0.47

Ac

yl CoA Ac

2)

cetoace

26

yl-CoA(1.

47.40.

Glyoxylate(2.0) ty

Maly

Aceate(2.0)

Glyoxylate Regeneration

Glutamate ate

2PG(3.0)

Glycerate(0.28)

Serine(1.1)

PEP(2.5)

Acetyl-CoA

Glycine(1.2)

OA05

)

CoACoA

0.33

000.0AA

H-pyruvate(0.97)

0 30.333PE

Glyyoxylacine(1.2)

pyruvat0.53

0.53

2PG(3.0)

0.5

ycerate(0.28) G

H p

Glyc

pyru

(0

0.53

Methylene-H4F(0.16) Methanol

Methylene-H4MPT(4.3)

Formaldehyde(0.01)

Methenyl-H4MPT(2.5) Formyl-H4MPT(0.33) Formylmethanofuran(0.09)

Formate(35) CO2 Formyl-H4F(0.89) Methenyl-H4F(0.60)

NAD NADH

Glycine(1.2)

eSeriinte(0.97)

)

NAD NADH0.43

.89)0.41

F60)0.41

) Me0.84

hylene-H4F(0.16

Methenyl-H4

0.41

0.44

For0.84

FoFo0.84

ylene-H4MP

( )

0.84

aldehyde(y (

0.84

Formaldehyde Metabolism

0.84

3PG(2.7) PHP(7.7) 3Pser

Serine(1.1)

Glutamate

Serine Biosynthesis

0.202.7)

0.20

-KG

37.7)

Fru1,6BP

Glc6P

Fru6P

1,3BPG

TP

6PG

DHAP

3PG(2((2(222 7)

1,3BPG

0.05

TP

1 3BPG

0.0

Fru1,6BP

0.03

Fru6P

Fru1 6BP

0.0

0.06

Glc6P

Fru6P

0.0

6PG

Glc6P

Per

GlutamKG

P0.150.15

3Ps

Ser

er

DH

5

P

0 00.03TPTP DPP

0.03

PPP

0.22 0.25

0.47

Nutrient intake flux: 0.84 mM/second Metabolic network relaxation: 37 seconds

0.05

(a)

(b)

tatt t

S

etoace

20..0.27

FIG. 3. Calculated steady-state fluxes in the dynamical model for Methylobacterium extorquens AM1 using two different sets of initialvalues. See Supplemental Material [43] for detailed results, in the excel files. The set of parameters supporting a stable metabolic network isnot unique. However, the dynamical performance for the network with different parameters can be evaluated and compared with experiments.The more efficient network (b), i.e., with less carbon intake on the set biomass output, is less viable due to a larger relaxation time and a widerspread of metabolite concentrations. It also has less fluxes wasted in futile cycles. The fluxes given in (a) agree well with experiments. Thefluxes are in red; the unit is mM/second. The concentrations of selected metabolites are in green; the unit is mM. Note that some fluxes arecombination of multiple reactions and some fluxes and reactions are not shown in the figure.

062409-9

CHEN, YUAN, AO, XU, AND ZHU PHYSICAL REVIEW E 93, 062409 (2016)

Time (second)0 10 15 20 25 30 35 40 45 50

Met

abol

ite C

once

ntra

tion

(mM

)

0

0.5

1

e

5,10-Methylene-tetrahydrofolateNicotinamide adenine dinucleotide phosphateGlycineSerineHydroxypyruvat

Perturbation

from steady state

5

1.5

90 100

Met

abol

ite C

once

ntra

tion

(mM

)

0

0.2

0.4

0.6

0.8

15,10-Methylene-tetrahydrofolate5,10-Methenyl-tetrahydrofolateNicotinamide adenine dinucleotide phosphateSerineGlycerate

Perturbation

from steady state

Time (second)80706050403020100

(a) (b)

FIG. 4. Time-dependent relaxation of metabolites when perturbed from their steady-state values are shown in (a) and (b), calculated fromparameters corresponding to Figs. 3(a) and 3(b). It confirms that the parameters in Fig. 3(b) indeed results in a longer relaxation time.

the organism can all be studied. Our construction is builtupon a previously developed stochastic decomposition, thelatter offers a potential landscape via a Lyapunov functionon the network [26,38,44]. The regulatory dynamics can besimplified in the biochemical context for practical purposes.Moreover, as network stability is a generic topic in many fields,our work might have implications beyond a simple math-ematical method for establishing kinetic metabolic models[45–47].

Biologically, we aim to provide a useful tool to analyzeand extract properties of large metabolic networks whoseparameters are difficult to obtain from in vivo measurements.Our approach differs from the usual type of methods thatare based on static stoichiometric matrices plus conservationsand/or optimizations. Our results offer a range of kineticmetabolic models (solutions) all having stable steady statesunder which dynamical metabolic properties can be analyzed.One such property is the relaxation time of a steady state, theinverse of which can be interpreted as the viability of a cell. Itallows us to quantify the tradeoff between the growth efficiency(on carbon conversion rate) and the viability (as inverse ofthe relaxation time, futile cycles) of an organism. Hence wecan sift through the solutions to identify the most appropriateone under a given physiological condition. Finally, large scalemetabolic modeling may also offer global perspective onmetabolic engineering [48].

ACKNOWLEDGMENTS

The work was partially supported by the Natural ScienceFoundation of China No. 91329301 and No. 91529306 (PA),No. 81273404 and No. 81473105 (MJX). We thank Prof. A.J. Leggett for many insightful discussions during his visitsto Shanghai Jiao Tong University. One of us (Y.-C. Chen) isgrateful for the hospitality of Shanghai Center for SystemsBiomedicine during his visits.

APPENDIX: GENERIC RATE EQUATIONFROM ORDERED UNI-BI MECHANISM

We can explicitly derive the generic rate equation inthe ordered Uni-Bi mechanism as a demonstration [32].The ordered Uni-Bi mechanism is schematically shown inFig. 5. There are two products P and Q [34]. The finalsteady-state rate equation for the ordered Uni-Bi mechanism,when the intermediate enzyme states are eliminated, is givenby [34]

u = N1A − N2PQ

D1 + D2A + D3P + D4Q + D5AP + D6PQ

(A1)

under quasi-steady-states assumption. Here A and P , Q arethe concentrations of the substrate and products in the reaction.The eight quantities N1, N2, and D1–D6 are functions ofthe six rate constants ν±1, ν±2, ν±3 as well as the enzymeconcentration e0:

N1 = ν1ν2ν3e0,

N2 = ν−1ν−2ν−3e0,

D1 = ν2ν3 + ν−1ν3,

D2 = ν1ν2 + ν1ν3,

FIG. 5. Ordered Uni-Bi mechanism.

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TOWARDS STABLE KINETICS OF LARGE METABOLIC . . . PHYSICAL REVIEW E 93, 062409 (2016)

D3 = ν−1ν−2,

D4 = ν−3ν2 + ν−3ν−1,

D5 = ν1ν−2,

D6 = ν−2ν−3.

The numerator in Eq. (A1) can be rewritten as

A (N1/D1)︸ ︷︷ ︸VF /KA

−PQ (N2/D1)︸ ︷︷ ︸VB/K ′

P K ′Q

= VF

A

KA

− VB

PQ

K ′P K ′

Q

,

where the underbraces indicate that N1/D1 has been rewritteninto VF /KA and likewise for the other term. After dividing byD1, the denominator of Eq. (A1) becomes

1 + D′2A + D′

3P + D′4Q + D′

5AP + D′6PQ, (A2)

where D′2 = D2/D1... D′

6 = D6/D1. Defining f1 + f2 = 1and regrouping Eq. (A2), we have

1 + D′2A + D′

3P + D′4Q + D′

5AP + D′6PQ = f1 + D′

2A + f2 + P (D′3 + D′

5A)︸ ︷︷ ︸D′′

3

+D′4Q + D′

6PQ

= f1

[1 + A (D′

2/f1)︸ ︷︷ ︸1/KA

]+ f2

[1 + P D′′

3/f2︸ ︷︷ ︸g

+Q (D′4/f2)︸ ︷︷ ︸h

+PQ (D′6/f2)︸ ︷︷ ︸l

]

= f1(1 + A/KA) + f2(1 + P (g + Qlα) + Q(h + P lβ) + PQl(1 − α − β))

= f1(1 + A/KA) + f2

[1 + P (g + Qlα)︸ ︷︷ ︸

1/K ′P

][1 + Q (h + P lβ)︸ ︷︷ ︸

1/K ′Q

]

= f1

(1 + A

KA

)+ f2

(1 + P

K ′P

)(1 + Q

K ′Q

).

In the above we have defined D′′3 = D′

3 + D′5A, 1/KA = D′

2/f 1, g = D′′3/f 2, h = D′

4/f 2, l = D′6/f2, 1/K ′

P = g + Qlα, and1/K ′

Q = h + P lβ. K ′P and K ′

Q are defined if (g + Qlα)(h + P lβ) = l(1 − α − β), the latter should be regarded as the equationfor α and β. Since A, P , and Q are positive and so are all the Di , g, h, and l must be positive. As a result, the sign of the term P lβ

follows the sign of β. Similarly, the sign of the term Qlα follows the sign of α. If gh = l, α = β = 0. If gh < l, then a solutionexists with α > 0 and β > 0. If gh > l, there is a solution with α < 0 and β < 0 [but (g + Qlα) > 0 hence (h + P lβ) > 0]. Tothis end we have successfully transformed the reaction rate Eq. (A1) into the form of Eq. (4) in the sense that all the parametersdefined in the latter are positive. Since Eq. (A2) does not have the exact polynomial expression as the denominator in Eq. (4),our derivation reveals the implicit approximation when regarding the apparent Michaelis-Menten constants as constant. Strictlyspeaking, they can partially depend on the concentrations.

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