Propensity approach to nonequilibrium thermodynamics of a chemical reactionnetwork: Controlling single E-coli -galactosidase enzyme catalysis through theelementary reaction stepsa)Biswajit Das, Kinshuk Banerjee, and Gautam Gangopadhyay

Citation: The Journal of Chemical Physics 139, 244104 (2013); doi: 10.1063/1.4844195 View online: http://dx.doi.org/10.1063/1.4844195 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Optimization of collective enzyme activity via spatial localization J. Chem. Phys. 139, 135101 (2013); 10.1063/1.4823504 A probability generating function method for stochastic reaction networks J. Chem. Phys. 136, 234108 (2012); 10.1063/1.4729374 Entropic estimate of cooperative binding of substrate on a single oligomeric enzyme: An index of cooperativitya) J. Chem. Phys. 136, 154502 (2012); 10.1063/1.3703505 Stochastic theory of large-scale enzyme-reaction networks: Finite copy number corrections to rate equationmodels J. Chem. Phys. 133, 195101 (2010); 10.1063/1.3505552 Kinetics of autocatalysis in small systems J. Chem. Phys. 128, 015101 (2008); 10.1063/1.2815800

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THE JOURNAL OF CHEMICAL PHYSICS 139, 244104 (2013)

Propensity approach to nonequilibrium thermodynamics of a chemicalreaction network: Controlling single E-coli β-galactosidase enzymecatalysis through the elementary reaction stepsa)

Biswajit Das,1,b) Kinshuk Banerjee,2 and Gautam Gangopadhyay1,c)

1S. N. Bose National Centre For Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700 098, India2Department of Chemistry, University of Calcutta, 92 A.P.C. Road, Kolkata 700 009, India

(Received 18 June 2013; accepted 22 November 2013; published online 26 December 2013)

In this work, we develop an approach to nonequilibrium thermodynamics of an open chemical reac-tion network in terms of the elementary reaction propensities. The method is akin to the microscopicformulation of the dissipation function in terms of the Kullback-Leibler distance of phase space tra-jectories in Hamiltonian system. The formalism is applied to a single oligomeric enzyme kinetics atchemiostatic condition that leads the reaction system to a nonequilibrium steady state, characterizedby a positive total entropy production rate. Analytical expressions are derived, relating the individ-ual reaction contributions towards the total entropy production rate with experimentally measurablereaction velocity. Taking a real case of Escherichia coli β-galactosidase enzyme obeying Michaelis-Menten kinetics, we thoroughly analyze the temporal as well as the steady state behavior of variousthermodynamic quantities for each elementary reaction. This gives a useful insight in the relativemagnitudes of various energy terms and the dissipated heat to sustain a steady state of the reactionsystem operating far-from-equilibrium. It is also observed that, the reaction is entropy-driven at lowsubstrate concentration and becomes energy-driven as the substrate concentration rises. © 2013 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4844195]

I. INTRODUCTION

Chemical kinetics of small reaction systems requires astochastic approach to accommodate the significant fluctu-ation in species concentration.1–12 The theoretical tools arewell-established in the literature in the context of chemicalmaster equation.4–6 Recent experiments on single molecules,e.g., temporal fluctuation of catalytic rate of a singleenzyme,13, 14 keep the interest growing in this field. In recentyears, this theoretical approach has been enormously used todescribe mesoscopic systems, e.g., single bio-molecular pro-cesses, in which the time evolution is considered as arandom-walk in the conformational state space.11, 12, 15–17 Themaster equation approach also provides the stochastic ther-modynamic description of such systems, starting froma definition of the system entropy in terms of stateprobabilities.10, 18–29 The total entropy production rate (EPR)due to the chemical reactions can be calculated which is al-ways non-negative and becomes zero at equilibrium.10 It hasbeen established that, closed systems tend to such equilibriumwhereas, open systems can reach a nonequilibrium steadystate (NESS),30–36 characterized by a positive total EPR.12

In chemical reaction systems, such NESS can be attained inpresence of chemiostatic condition, where concentrations ofsome species are kept fixed throughout the reaction.10 This

a)The authors have dedicated this article to Professor Deb Shankar Ray onhis 60th birthday.

b)Current address: Beijing International Center for Mathematical Researchand Biodynamic Optical Imaging Center, Peking University, Beijing,100871, People’s Republic of China.

c)Electronic mail: gautam@bose.res.in

is the simplest assumption that can model an open chemicalsystem, e.g., a living cell in homeostatis.22

Recently, Kawai et al.23 gave a phase space picture of dis-sipation in terms of the relative entropy or Kullback-Leibler(KL) distance33, 34 of probabilities of realizing forward and(time-) reversed phase space trajectories. This formulationgoes to the core of the realization of the development ofmacroscopic irreversibility from the microscopic reversibilityand the arrow of time.24, 25 In the same spirit, here our mainaim is to formulate an expression of total EPR in a chem-ical reaction network, using suitable trajectory probabilitiesin population state space in terms of the propensities of eachelementary reaction as introduced by Gillespie.4, 5 The evolu-tion of the reaction system involves jump from one populationstate to another along trajectories in forward and backwarddirections. Now, the characteristic feature of such a systemis that, one or more, but not all, of the elementary reactionscan be responsible for a particular state change. This auto-matically leads one to construct the theory on the basis of in-dividual elementary reactions. Determination of the role ofeach reaction event in governing the nonequilibrium thermo-dynamics of an open chemical reaction network should givenew insights in the temporal energetics, complementing thekinetic knowledge. In this regard, we thoroughly analyze boththe first and second laws for nonequilibrium system, devel-oped from a master equation description of the system.26–29

The theory has been applied on a single oligomeric enzymecatalysis reaction reaching NESS.11, 12, 15, 22

The paper is organized as follows. In Sec. II, we discusson the formulation of total EPR using the elementary reac-tion probability densities, denoted here as propensities, of an

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244104-2 Das, Banerjee, and Gangopadhyay J. Chem. Phys. 139, 244104 (2013)

open nonequilibrium reaction network. The theory is appliedon a single oligomeric enzyme kinetics in Sec. III, givinganalytical relations between the EPR and experimental ob-servables, like net reaction velocity. Section IV is devoted toa thorough analysis of the contributions of elementary reac-tions in various state functions, like internal energy, free en-ergy, and also path functions, like heat dissipation and irre-versible work. In this context, we take a real example in theform of Escherichia coli β-galactosidase enzyme. The paperis concluded in Sec. V.

II. EPR IN AN OPEN REACTION SYSTEM:ELEMENTARY REACTION PROPENSITIES

Following Gaspard10 we consider a reaction network

a∑i=1

νi<μAi +

b∑j=1

νj<μZj

kμ

⇀↽k−μ

a∑i=1

νi>μAi +

b∑j=1

νj>μZj. (1)

Here Zj are the intermediate species and Ai are species whoseconcentrations are kept constant by connecting the reactionsystem to reservoirs. This is known as the chemiostatic con-dition, leading the system to a NESS. The system is assumedto be at thermal equilibrium with the reservoir, kept at tem-perature T. An elementary reaction is denoted by the indexμ with μ = ±1, ±2, . . . , ±k. The condition of microscopicreversibility is the fundamental requirement for the construc-tion of the nonequilibrium thermodynamic theory, as it needsto consider the (time-)reversed process. Thus, every transitionin the population state space due to the reaction μ has a coun-terpart in the reverse transition through the reaction −μ. Also,none of the rate constants of the elementary reactions can beexactly zero. The stoichiometric coefficient, ν

jμ of the species

Zj involved in the μth reaction is defined as

νjμ = νj

>μ − νj<μ, (j = 1, . . . , b). (2)

We denote the population vector of the intermediate speciesby Z ( = {Zj}, j = 1, . . . , b) and the stoichiometric coefficientvector of the μth reaction is given by νμ. The chemical mas-ter equation governing the stochastic reaction events is givenby10, 19–21

dP (Z, t)

dt=

±k∑μ=±1

[wμ(Z − νμ|Z)P (Z − νμ, t)

−w−μ(Z|Z − νμ)P (Z, t)]. (3)

Here P(Z, t) is the probability of having population vectorZ of the intermediate species at time t and wμ(Z|Z + νμ) isthe transition probability for the jump Z → Z + νμ along astochastic trajectory via reaction μ.

Let us now consider an evolution of the system’s distribu-tion given as P (Z, t) → P (Z + νμ, t + �t). This can be de-scribed as a transition from the state-Z to the state-(Z + νμ)at time t and then considering a time interval �t such that nofurther transition takes place in this interval. According to thedefinition of stochastic system entropy given by Seifert,30–32

the system entropy production for this transition due to the

elementary reaction μ can be written as

�s(μ)sys = −lnP (Z + νμ, t + �t) − (−lnP (Z, t))

= lnP (Z, t)

P (Z + νμ, t + �t). (4)

The medium entropy production or entropy flow associatedwith this transition is defined as28, 30–32

�s(μ)m = ln

wμ(Z|Z + νμ)

w−μ(Z + νμ|Z). (5)

Then, the corresponding total entropy production becomes

�s(μ)tot = �s(μ)

sys + �s(μ)m = ln

Rμ(Z, t |Z + νμ)

R−μ(Z + νμ, t + �t |Z). (6)

Here we introduce the propensity, Rμ(Z, t |Z + νμ)= wμ(Z|Z + νμ)P (Z, t), in the same spirit of the re-action probability density given by Gillespie.4–6 Thus,Rμ(Z, t |Z + νμ)dt is the probability that the system at time twill go from state-Z to state-(Z + νμ) due to the occurrenceof an elementary reaction μ in the next infinitesimal timeinterval dt.

Now, all the above expressions of entropy production arefor a specific transition as indicated by the lower case “s.”To get the overall entropy production due to an elementaryreaction, we need to average over all the transitions due tothe μth reaction considering all possible values of the pop-ulation vector Z. This averaging is done with the probabil-ity, Rμ(Z, t |Z + νμ)�t . Here, �t is considered to be smallenough so that Rμ remains constant over this interval. Thenwe have

�S(μ)tot = ⟨

�s(μ)tot

⟩ =∑

Z

(Rμ(Z, t |Z + νμ)�t)ln

× Rμ(Z, t |Z + νμ)

R−μ(Z + νμ, t + �t |Z). (7)

The expression of the total EPR follows from Eq. (7)

lim�t→0

�S(μ)tot

�t= S

(μ)tot

=∑

Z

Rμ(Z, t |Z + νμ)lnRμ(Z, t |Z + νμ)

R−μ(Z + νμ, t |Z).

(8)

The structure of Eq. (8) is similar to the relative entropy orthe Kullback-Leibler (KL) distance of two probability distri-butions but not identical to it. In our case, the KL probabilitydistance function, D(ρ||ρ) = 〈ln ρ

ρ〉, also called relative en-

tropy of ρ versus ρ, gets replaced by the propensity distancefunction for the μth reaction, 〈ln Rμ(Z,t |Z+νμ)

R−μ(Z+νμ,t |Z) 〉 where the av-erage is over the forward process, Rμ(Z, t |Z + νμ). Thus, aKL distance-type relation in terms of the forward and reversepropensities, Rμ and R−μ, respectively, gives rise to the totalEPR due to the μth reaction in the forward direction and −μthreaction in the backward direction. However, as Rμ and R−μ

are not normalized, so S(μ)tot can be negative also. The splitting

of S(μ)tot in Eq. (8) into system and medium contributions is

straightforward.

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244104-3 Das, Banerjee, and Gangopadhyay J. Chem. Phys. 139, 244104 (2013)

Recently, Esposito and Van den Broeck27, 28 introducedanother separation of the total EPR, Stot (t) into adiabaticand nonadiabatic parts, Sa(t) and Sna(t), respectively. In ourscheme of reaction network, they become

S(μ)a (t) =

∑Z

Rμ(Z, t |Z + νμ)lnRss

μ (Z, t |Z + νμ)

Rss−μ(Z + νμ, t |Z), (9)

and

S(μ)na (t) =

∑Z

Rμ(Z, t |Z + νμ)lnP (Z, t)P ss(Z + νμ)

P ss(Z)P (Z + νμ, t). (10)

Here Rssμ (Z, t |Z + νμ) = wμ(Z|Z + νμ)P ss(Z) with Pss(Z)

being the steady state probability of having the populationvector Z. Here and throughout the paper, the “ss” super-script denotes the quantities at NESS. Unlike the system andmedium contributions, the adiabatic and nonadiabatic partsare always individually positive. Recently, Ge and Qian26 dis-cussed the nonequilibrium thermodynamics of a system gov-erned by a master equation, characterizing the rates of internalenergy change, U and free energy change, F . They also con-structed the first law in terms of the rate of heat dissipation, re-lated to the medium EPR, Sm and the rate of irreversible workdone, given by the adiabatic EPR, Sa(t). The nonadiabaticEPR is found to be negative of the temperature-scaled freeenergy dissipation rate, Sna(t) = −F (t)/T . The rate of in-ternal energy change, given as U (t)/T = Sa(t) − Sm(t), thenbehaves like first law.

III. SINGLE OLIGOMERIC ENZYME CATALYSIS

To get some physical understanding, here we take thesimplest case of a single oligomeric enzyme kinetics witha single intermediate. A generalized version is given in Ap-pendix A. The reaction of an active site of the enzyme is writ-ten following a reversible Michaelis-Menten (MM) scheme

E + Sk1

′

⇀↽k−1

ESk−2

⇀↽k′

2

E + P. (11)

Under chemiostatic condition,10, 12, 22 with concentrations ofsubstrate, S and product, P maintained constant, the abovereaction scheme can be reduced to

EK1

⇀↽K2

ES, (12)

where K1 = k1 + k2 and K2 = k−1 + k−2. The pseudo first-order rate constants, k1 and k2, are given by k1 = k′

1[S] andk2 = k′

2[P ]. We consider the enzyme to consist of nT numberof identical non-interacting subunits where each subunit hasone active site. During the reaction, the active sites whichhave already formed the ES-complex are referred to as occu-pied sites and those lying vacant at that moment are called thevacant sites. With the total number of sites being conserved,the reaction system can be described in terms of a single ran-dom variable, the number of occupied sites, n, at a particulartime t. The corresponding master equation is written as15, 16

dP (n, t)

dt=

±2∑μ=±1

[wμ(n − νμ|n)P (n − νμ, t)

−w−μ(n|n − νμ)P (n, t)], (13)

where P(n, t) being the probability of having n number ofoccupied states at time t. The stoichiometric coefficients, νμ

= 1 for μ > 0 and νμ = −1 for μ < 0. The rate constant ofthe elementary reaction, μ is denoted by kμ. The transitionprobabilities are defined as follows:

wμ(n − νμ|n) = kμ(nT − (n − νμ)), μ > 0

and

wμ(n − νμ|n) = kμ(n − νμ), μ < 0. (14)

Then Eq. (13) can be rewritten as

dP (n, t)

dt=K1(nT −n+1)P (n−1, t)+K2(n+1)P (n+1, t)

−K1(nT − n)P (n, t) − K2nP (n, t). (15)

Solution of the master equation, obtained using the generatingfunction method,35, 36 is given as15, 16

P (n, t) = nT !

n!(nT − n)!XnYnT −n, (16)

where X = K1(1−exp(−(K1+K2)t))K1+K2

, Y = K2+K1exp(−(K1+K2)t)K1+K2

,assuming that initially all the active sites are vacant.

Now we give a description of the total EPR, Stot for theenzyme kinetics scheme (11), based on the KL distance typeformalism, described in Sec. II. The elementary reactions μ

= ±1 and μ = ±2 form the reaction-“counter” reaction pairson the basis of ES-complex formation and dissociation. Thetransition between two states of the system, say state m andstate l, is governed by Eq. (13). As νμ = ±1, so l = m ± 1. Interms of state variable, n, we can write m, l = (n − 1), n and(n + 1). In a forward transition, μ > 0 and l > m whereas, fora backward transition, μ < 0 and l < m. Then one can definethe total EPR at time t due to the reaction μ in terms of theforward and backward reaction propensities as

S(μ)tot (t) =

∑m,l

Rμ(m, t |l)ln(

Rμ(m, t |l)R−μ(l, t |m)

)δl,m+1, μ > 0

=∑m,l

Rμ(m, t |l)ln(

Rμ(m, t |l)R−μ(l, t |m)

)δl,m−1, μ < 0.

(17)

Using Eqs. (14) and (16) in Eq. (17), we get the desired EPRin a concise form as

S(μ)tot (t) = (kμnT Y )ln

(kμnT Y

k−μnT X

), μ > 0

= (kμnT X)ln

(kμnT X

k−μnT Y

), μ < 0. (18)

The total EPR of the whole reaction system is obviously

Stot (t) =±2∑

μ=±1

S(μ)tot (t). (19)

We clarify here that, using the usual master equation de-scription starting from the definition of Shannon entropy, onewill get the same expression as given in Eq. (19). However, asalso mentioned in Sec. I, the construction of various EPRs interms of the forward and backward reaction propensities has

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244104-4 Das, Banerjee, and Gangopadhyay J. Chem. Phys. 139, 244104 (2013)

deeper physical meaning, giving a microscopic description ofthe second law of thermodynamics. Also, use of the standardformulation can give rise to a technical problem in the deter-mination of the individual reaction contributions towards totalEPR. No such confusion arises in the case of Eq. (8). This isdiscussed in Appendix B. It has been proved that, both theadiabatic and the nonadiabatic entropy productions (but nei-ther system nor medium entropy production) can be writtenas a relative entropy or KL distance of suitably defined trajec-tory probabilities.28 We show in Appendix C that, followingour formulation, the nonadiabatic EPR can be written in sucha form.

A. EPR of the elementary reactions at NESS

In this subsection, we focus on the total EPR due tothe elementary reactions of the enzyme kinetics scheme (11)and their relations, if any, with experimental observables.We mention that the system EPR and the nonadiabatic EPRare zero at NESS both being state functions. So, one getsSss

tot = Sssm = Sss

a .

1. Michaelis-Menten analogues for total EPR

The kinetics scheme in Eq. (11) tends to the traditionalMM one for k2 → 0. The net velocity of the MM enzymecatalysis at NESS can be defined as

vssnet = k−2〈n〉ss = nT k−2X

ss = nT k−2[S]

KM + [S]. (20)

This is the famous MM equation of enzyme catalysis,shown here for the single oligomeric enzyme case, whereKM = k−1+k−2

k′1

= K2k′

1is the usual MM constant.

Now at NESS, from Eq. (18) with μ = 1, we get

S1(ss)tot = (k1nT Y ss)ln

(k1nT Y ss

k−1nT Xss

)

= nT K2lnf [S]

KM + [S], f = K2

k−1, k′

2 ≈ 0. (21)

Similar treatment for other reactions gives

S−1(ss)tot = −nT k−1lnf [S]

KM + [S]; S

2(ss)tot = nT k′

2KM lnf ′[P ]

KM + [S];

S−2(ss)tot = −nT k−2lnf ′[S]

KM + [S], (22)

where f ′ = k′2K2[P ]

k′1k−2[S] . It is clear from the above equations that

the total EPR for the reactions μ = ±1 show hyperbolic de-pendence on [S] at NESS. For the reactions μ = ±2, thisdependence is not clear, although the structure of the equa-tions are similar. Thus, Eqs. (21) and (22) can be designated asthe “MM analogues” for the total EPR due to the elementaryreactions.

2. Relation between the net velocity and total EPR

It will be very important to have a relationship betweenthe experimentally measurable vss

net and the total EPR. Using

Eqs. (21) and (22) and neglecting S2(ss)tot , we have

S(ss)tot = S

1(ss)tot + S

−1(ss)tot + S

−2(ss)tot = vss

net lnγ, (23)

the desired relationship, with γ = k1k−2

k−1k2. At the condition of

detailed balance, γ = 1 one obtains, S(ss)tot = 0, corresponding

to equilibrium.

IV. RESULT AND DISCUSSION

In this section, we have numerically studied the individ-ual reaction contributions to various thermodynamic quan-tities. To this end, a single Escherichia coliβ-galactosidaseenzyme is considered, which is tetrameric39–41 and followsthe MM kinetics.13, 14 We have taken the rate constant datafrom the single molecule experiment of English et al.13 Theyare as follows: k′

1 = 5 × 107 M−1 s−1, k−1 = 18 300 s−1, andk−2 = 730 s−1, In this regard, we want to clarify that herewe have used the value of k−2 without the dynamic dis-order, taken in their study.13 We set the rate constantk2 = 1 × 10−5 s−1, as the calculation of the nonequilibriumthermodynamic quantities does not permit any rate constant tobe exactly zero. However, the rate constant is small enough toensure that the effect of this reaction step on the overall kineticand thermodynamic properties is negligible, as will be shownbelow. The substrate concentration is set at [S] = 100 μM.

We have plotted the time evolution of the elementaryreaction contributions to the total EPR for μ = ±1, ±2 inFigs. 1(a)–1(d). The plots reveal four main features: (i) S

(1)tot (t),

S(−2)tot (t) remain positive over the time course whereas, S(−1)

tot (t)remains negative throughout. (ii) Initially, the major contribu-tion to Stot (t) comes from S

(1)tot (t). This is because contribu-

tions of elementary reactions μ = −1, −2 to total EPR startsfrom zero. (iii) At NESS, S(1)

tot and S(−1)tot almost cancel out each

other and hence the total EPR is governed by S(−2)tot . (iv) At in-

termediate times, S(1)tot (t) and S

(−2)tot (t) continuously decreases

and increases, respectively, whereas S(−1)tot (t) passes through a

minimum. This gives rise to the minimum in Stot (t). S(2)tot (t)

remains insignificant relative to others. In Appendix E, wegive the condition for S

(μ)tot (t) to change its sign during time

evolution.

0 0.0005 0.001time (sec)

0

25000

50000

0 0.0005 0.001time (sec)

-6000

-4000

-2000

0

0 0.0005 0.001time (sec)

0

10000

20000

0 0.0005 0.001time (sec)

-0.00055

-0.0005

-0.00045

Stot(1).

totS(-1).

S tot(-2).

Stot(2).

(a) (b)

(c) (d)

FIG. 1. (a)–(d) Contributions of the individual reactions to the total EPR,S

(μ)tot as a function of time, t. The rate constants are k′

1 = 5 × 107 M−1s−1,k−1 = 18 300 s−1, k−2 = 730 s−1, k2 = 1 × 10−5 s−1. The substrate con-centration is set at [S] = 100 μM.

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244104-5 Das, Banerjee, and Gangopadhyay J. Chem. Phys. 139, 244104 (2013)

0 1000 20000

1000

2000

3000theorynumerical

0 1000 2000-3000

-1500

0

0 1000 20000

25000

50000

0 1000 2000

-0.0004

-0.0002

0

0 1000 2000

[S]

0

25000

50000

0 1000 2000

[S]

0

200

400

600

Sto

t. 1(

ss)

Sto

t

. -1(s

s)

tot

. S-2

(ss)

Sto

t2(

ss)

.

. tot

S V net

(a) (b)

(c) (d)

(e) (f)

(ss)

(ss)

μM μM

FIG. 2. (a)–(f) The individual reaction contributions in total EPR, S(μ)tot and

net velocity of reaction are plotted as a function of substrate concentration, [S]at NESS. The rate constants are: k′

1 = 5 × 107 M−1 s−1, k−1 = 18 300 s−1,k−2 = 730 s−1, k2 = 1 × 10−5 s−1.

We have calculated Sμ(ss)tot as a function of substrate con-

centration, [S] from Eqs. (21) and (22). We have also de-termined the quantities directly from the master equation,Eq. (15). Results obtained from the two approaches tally verywell as depicted in Figs. 2(a)–2(d). In Figs. 2(a)–2(c), it isobserved that the variation of S

1(ss)tot , S

−1(ss)tot , and S

−2(ss)tot as

a function of [S] are hyperbolic in nature. Also, S−1(ss)tot re-

mains negative throughout, its magnitude almost same as thatof S

1(ss)tot . The value of S

2(ss)tot remains negligible throughout,

as shown in Fig. 2(d). Therefore, S(ss)tot is mainly governed by

reaction μ = −2. In Figs. 2(e) and 2(f), we have plotted thetotal EPR, S

(ss)tot and the net velocity, vss

net , both exhibiting hy-perbolic dependence on [S] at NESS.

Next, we focus on various entropy productions, deter-mined by integrating the corresponding rates over a time in-terval starting from t = 0 to a fixed final time, when the systemresides in NESS. We have plotted the individual reaction con-tributions as a function of [S] at NESS shown in Figs. 3(a)–3(f). In the figure, we have not shown the contribution of re-action μ = 2 due to its smallness. First of all, we investigatehow the irreversible work done in sustaining the NESS is uti-lized by the reactions. We remind the reader that, the adiabaticentropy production, �Sa is the measure of the (temperature-scaled) irreversible work done. �S

(μ)a is plotted as a function

of [S] at NESS, depicted in Fig. 3(a). In figure, it is evidentthat, �S(1)

a remains positive whereas, �S(−1)a remains nega-

tive and they are close in magnitude. So, the irreversible workdone on the system is mostly consumed by the reactionμ = −2, more so at higher [S]. Now according to first law,some amount of the irreversible work done on the systemchanges its internal energy and rest of it dissipates as heatto the surroundings. We have plotted the measure of heatdissipation, �S

(μ)m and the temperature-scaled internal en-

ergy change of the system, �U(μ)/T as a function of [S] inFigs. 3(b) and 3(c), respectively. In Fig. 3(b), it is observedthat �S(1)

m is negative at small values of [S], indicating heat ab-sorption by the system from surroundings. As [S] is increasedhowever, �S(1)

m becomes positive and increases linearly, indi-cating growing heat dissipation to surroundings. For reaction

0 1000 20000

100

200

0 1000 2000-200

0

200

0 1000 2000-200

0

200

0 1000 2000-200

0

200

0 1000 2000

[S]

-9-6-303

0 1000 2000

[S]

0

100

200

ΔS

tot

Δ

μ

mS

S sys

M

U

μ

F SΔ

a

(a) (b)

(c) (d)

(e) (f)

μ= 1μ=-1μ=-2

(μ)

(μ)

(μ)

(μ)

(μ)

(μ)

TT

ΔΔΔ

M

FIG. 3. (a)–(f) The individual reaction contributions in the various entropyproductions, free energy change and the internal energy change are plottedas a function of [S]. The rate constants are: k′

1 = 5 × 107 M−1 s−1, k−1 =18 300 s−1, k−2 = 730 s−1, k2 = 1 × 10−5 s−1.

μ = −1, the trend is almost exactly opposite and cancels outthe contribution of the reaction μ = 1. Hence, again the re-action μ = −2 governs the total heat dissipation, as �S(−2)

m

remains positive and rises with [S] to saturation.Interestingly, the trend of variation of �S(−2)

m is quan-titatively similar to �S(−2)

a , as one can see from Figs. 3(a)and 3(b). This indicates that the irreversible work consumedby the reaction μ = −2 almost entirely dissipates as heatand therefore, the contribution of this reaction to internal en-ergy change, �U(−2)/T is little. This is indeed the case, asshown in Fig. 3(c). The variation of the corresponding con-tributions of reactions μ = ±1 towards the internal energychange, �U(±1)/T show opposite trends as a function of [S]compared to the variation of �S(±1)

m (see Fig. 3(b)). This isexpected as heat absorption by the system due to an elemen-tary reaction leads to an increase in internal energy whereas,heat released by the system results in fall of its internal en-ergy. Same is true for the reaction contributions in the systementropy change, �S

(μ)sys plotted in Fig. 3(d). The difference be-

tween �U(μ)/T and �S(μ)sys gives the temperature-scaled free

energy change, �F(μ)/T(which is also the nonadiabatic en-tropy production, �S

(μ)na ), shown in Fig. 3(e). The magnitude

of �F(μ)/T is small compared to the other quantities due tothe quantitative similarity of �U(μ)/T and �S

(μ)sys . We see from

Fig. 3(e), the contribution of reactions μ = −1, −2 are pos-itive as well as small and the major negative contribution inthe total free energy change comes from the reaction μ = 1.Finally, we have plotted �S

(μ)tot as a function of [S] in Fig. 3(f).

As the magnitude and trend of variations of �S(μ)m and �S

(μ)sys

are opposite for the reactions μ = 1 and μ = −1, so the valueof �S

(±1)tot remains small compared to the reaction μ = −2

over the range of [S].After the detailed analyses regarding the role of the el-

ementary reactions in governing the nonequilibrium thermo-dynamics of the enzyme catalysis, next we explore the overallchanges. For this purpose, we have determined the changesin the state functions, �Ssys, �F/T, �U/T shown in Fig. 4(a)as well as the changes in the path functions �Stot, �Sm, and�Sa plotted in Fig. 4(b) as a function of [S]. In Fig. 4(a), we

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244104-6 Das, Banerjee, and Gangopadhyay J. Chem. Phys. 139, 244104 (2013)

0 1000 2000[S]

-6

-4

-2

0

2

ΔU/T

, ΔF

/T a

nd Δ

S sys

ΔU/TΔ F/TΔS sys

0 1000 2000[S]

0

50

100

150

200

Δ S

tot

, Δ S

m a

nd Δ

S a

ΔS totΔ S mΔS a

(a) (b)

μM μM

FIG. 4. Variations of (a) �Ssys, �U/T, �F/T and (b) �Stot, �Sm, �Sa areplotted as a function of [S]. The rate constants are k′

1 = 5 × 107 M−1 s−1,k−1 = 18 300 s−1, k−2 = 730 s−1, k2 = 1 × 10−5 s−1.

find out that �F/T becomes more and more negative with [S],as expected. On the other hand, �Ssys remains positive andpasses through a maximum. Now, from the same plot, we ob-serve that at low [S], �U/T is positive. So in this region, thereaction is entropy-driven. As [S] increases, �U/T decreasessteadily, changes sign and runs parallel to �F/T. Therefore, athigher [S], the reaction gradually becomes energy-driven. InFig. 4(b), one can see that, all the path functions are almost thesame at low [S] and begin to differ as [S] increases. Their mag-nitudes are much greater than the corresponding state functionquantities, again indicating the fact that the irreversible workdone on the system dissipates almost entirely as heat.

V. CONCLUSION

We have theoretically formulated the total EPR in anopen chemical reaction system in terms of a KL-type distancebetween forward and backward elementary reaction propensi-ties, equivalent to the reaction probability density, introducedby Gillespie.4, 5 The formalism is inspired by a recent study onthe dissipation function in phase space.23 A general nonequi-librium thermodynamic scenario is discussed in presence ofchemiostatic condition that allows the system to sustain aNESS. The theory has been applied to the stochastic kinet-ics of a single oligomeric enzyme obeying reversible MMkinetics. Analytical expressions of the contributions of eachelementary reaction to the total EPR are derived, providingimportant insights in the connection of these theoretical quan-tities to experimental observables, like the net velocity of thereaction at NESS.

As a real case study, we have taken the single moleculekinetics of the tetrameric Escherichia coliβ-galactosidase en-zyme under chemiostatic condition. The variation of totalEPR for individual reactions is analyzed with time as well aswith substrate concentration at NESS. In both situations, wefind that the contributions of the enzyme-substrate binding re-action to form the ES-complex and its reverse reaction, i.e., μ

= ±1 balance out each other and the overall dissipation ismostly governed by the product-formation reaction, μ = −2.We have proceeded further in determining the various entropyproductions due to these reactions over the time course of theenzyme reaction from start to reaching NESS. The adiabaticentropy production, representing the irreversible work done indriving the system to NESS and then to sustain it, is shown to

be almost entirely consumed by the product formation step.This, in turn, mostly dissipates as heat. So, the internal en-ergy change and entropy change of the system come out tobe much lower in magnitude. Again the contributions of reac-tions μ = ±1 cancel out each other over the range of substrateconcentration studied. Interestingly, the free energy change ofthe overall process is governed by the enzyme-substrate bind-ing reaction, μ = 1. Exploring the energy and entropy con-tributions to the overall free energy change, we have foundthat the reaction is entropy-driven at low substrate concentra-tion and becomes energy-driven as the substrate concentrationrises. Therefore, a detailed understanding of the mechanismof energy expenditure of the open reaction system is achievedunder conditions arbitrarily away from thermodynamic equi-librium. In the light of the intimate relation between detailedbalance and total EPR, we have shown a similar connectionbetween circular balance and nonadiabatic EPR, that can alsobe written in the propensity distance formalism. Furthermore,we point out that with k′

2 ≈ 0, the MM kinetics is applicableto several other real cases, e.g., acid-base equilibrium occur-ring ahead of a first-order or pseudo first-order step and uni-molecular reactions in the gas phase described by Lindemannmechanism.37, 38 However, to simulate the processes inside asmall volume, fluctuations in all the concentrations should betaken into account more accurately. Recent experimental andtheoretical studies also indicate that the mechanism of sub-strate input has a significant effect in governing the overallstochastic kinetics and thermodynamics.42–45 Specifically, in-vestigations on stochastic substrate generation in bursts revealnovel phenomena, like concentration inversion,46 and providephysical insights into the mechanism of single cell rhythms.47

Usually inside cells substrate cannot be constant and is pro-duced via translation; what is really constant is the gene con-centration and translation gives rise to production of substrateand other proteins in bursts. These works unequivocally con-firm the need to go beyond the chemiostatic condition and thepresent formalism can be extended to more generic ways ofintroducing substrates through a bursty way to understand therole of stochasticity in biological systems.

ACKNOWLEDGMENTS

K.B. acknowledges the University Grants Commission(UGC), India for Dr. D. S. Kothari Fellowship. B.D. acknowl-edges S. N. Bose National Centre for Basic Sciences, SaltLake, Kolkata, India, for an extended SRF where most of thework was completed.

APPENDIX A: TOTAL EPR IN A GENERAL CYCLICREACTION SCHEME

For the sake of completeness, we consider a cyclic reac-tion scheme as given below

E0

k1

⇀↽k−1

E1

k2

⇀↽k−2

E2

k3

⇀↽k−3

. . . . . .kn

⇀↽k−n

En

k(n+1)

⇀↽k−(n+1)

. . . . . . ⇀↽ Er

k(r+1)

⇀↽k−(r+1)

E0.

(A1)

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244104-7 Das, Banerjee, and Gangopadhyay J. Chem. Phys. 139, 244104 (2013)

Here number of intermediate species are taken to be r. Thereaction index is μ = ±1, ±2, . . . , ±(r + 1) with eachelementary reaction having a reverse counterpart. The rateconstant, kμ of reaction μ can either be simple first orderor can be pseudo first order. In the latter case, the reac-tion system contains species other than En(n = 0, 1, . . . ,r) (not shown in the above scheme) which react with thespecies En and/or are generated due to the reactions but

their concentrations are assumed fixed under chemiostaticcondition. The number of molecules of different intermedi-ate species, En (n = 1, 2, . . . , r) are collectively denotedby the vector Z = {Zn} and the corresponding vector ofthe stoichiometric coefficients associated with the μth reac-tion is denoted by νμ with νn

μ = ±1. In this case the to-tal EPR due to the μth reaction can be explicitly writtenas

S(μ)tot (t) =

∑Z,Z′

Rμ(Z1, . . . , Zr, t |Z′1, . . . , Z

′r )ln

(Rμ(Z1, . . . , Zr, t |Z′

1, . . . , Z′r )

R−μ(Z′1, . . . , Z

′r , t |Z1, . . . , Zr )

)δZ′

1,Z1±1, μ = ±1

=∑Z,Z′

Rμ(Z1, . . . , Zr, t |Z′1, . . . , Z

′r )ln

(Rμ(Z1, . . . , Zr, t |Z′

1, . . . , Z′r )

R−μ(Z′1, . . . , Z

′r , t |Z1, . . . , Zr )

)δZ′

r ,Zr∓1, μ = ±(r + 1)

=∑Z,Z′

Rμ(Z1, . . . , Zr, t |Z′1, . . . , Z

′r )ln

(Rμ(Z1, . . . , Zr, t |Z′

1, . . . , Z′r )

R−μ(Z′1, . . . , Z

′r , t |Z1, . . . , Zr )

)δZ′

n−1,Zn−1∓1δZ′n,Zn±1 ,

μ = ±n(n = 2, 3, . . . , r). (A2)

APPENDIX B: TOTAL EPR FROM SHANNON ENTROPY

Starting from the definition of system’s Shannonentropy10, 16, 18–21

Ssys(t) = −∑

Z

P (Z, t)lnP (Z, t), (B1)

and using Eq. (3), we get the total EPR as

Stot (t) = 1

2

∑Z,μ

[wμ(Z − νμ|Z)P (Z − νμ, t)

−w−μ(Z|Z − νμ)P (Z, t)]

× lnwμ(Z − νμ|Z)P (Z − νμ, t)

w−μ(Z|Z − νμ)P (Z, t). (B2)

Now, Eq. (B2) implies that contribution of reaction μ to-wards Stot (t) is the same as that of reaction −μ. This is notphysically meaningful. Actually, this problem is encounterednot only in the case of total EPR but also for all the otherEPRs. However, the individual reaction contributions to thetotal EPR are distinct from each other following the propen-sity distance formalism, as given in Eq. (17).

APPENDIX C: CIRCULAR BALANCE AND FREEENERGY DISSIPATION RATE IN SINGLE ENZYMECATALYSIS

Splitting of total EPR into adiabatic and nonadiabaticcontributions is particularly relevant when the system doesnot satisfy detailed balance. This is so in our case, with theenzyme kinetics in Eq. (11) under chemiostatic condition.

Now the question arises: can the formalism of constructingthe total EPR from forward and backward reaction propen-sities also be used to express the adiabatic and nonadiabaticparts as KL distances? Here we show that, at least the nona-diabatic EPR, which is the negative of the temperature-scaledfree energy dissipation rate, can be cast in such a form. Forthe steady state solution of the master equation, Eq. (13), onecan use two kinds of balance conditions: the detailed balanceand the circular balance. The former gives the equilibriumstate whereas the latter gives the NESS. The detailed balancecondition is given as wμ(m|l)P (m, t) = w−μ(l|m)P (l, t), i.e.,Rμ(m, t|l) = R−μ(l, t|m). As the total EPR becomes zero atequilibrium, so the expression of the total EPR must be builtfrom these two reaction propensities in a relative entropy kindof structure as is already shown in Sec. III. This gives theclue to construct the appropriate reaction propensities for thedescription of the nonadiabatic EPR. As, by definition (seeEq. (10)), the nonadiabatic EPR becomes zero at the station-ary state (whether equilibrium or NESS), the balance condi-tion that is valid for NESS (and obviously for equilibrium)must be linked to this EPR. This leads us to the circular bal-ance condition which is given in terms of the overall forma-tion and dissociation reaction propensities of the ES-complexas

Rf (n, t |n + 1) = Rd (n + 1, t |n), (C1)

where Rf(n, t|n + 1) = K1(nT − n)P(n, t) and Rd(n + 1, t|n)= K2(n + 1)P(n + 1, t) with K1, K2 already defined inSec. III. Then the contribution of the formation reaction to the

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244104-8 Das, Banerjee, and Gangopadhyay J. Chem. Phys. 139, 244104 (2013)

nonadiabatic EPR, Sna is written as

S(f )na (t) = 1

2

[ ∑n−1

Rf (n − 1, t |n)ln

(Rf (n − 1, t |n)

Rd (n, t |n − 1)

)

+∑

n

Rf (n, t |n + 1)ln

(Rf (n, t |n + 1)

Rd (n + 1, t |n)

) ]

= (K1nT Y )ln

(K1nT Y

K2nT X

). (C2)

Here X and Y are the same as those defined in Sec. III. Sim-ilarly, we get the contribution of the dissociation reaction as

S(d)na (t) = (K2nT X)ln

(K2nT X

K1nT Y

), (C3)

with

Sna(t) = S(f )na (t) + S(d)

na (t). (C4)

So, the nonadiabatic EPR or the free energy dissipation ratecan be written in the same relative entropy kind of formalismin terms of the overall formation (forward) and dissociation(backward) reactions. Thus, for total EPR, the detailed bal-ance gives the appropriate reaction propensities whereas, forthe nonadiabatic EPR, the circular balance provides the suit-able choices.

APPENDIX D: INEQUALITIES IN Sμ(ss)tot IN THE MM

KINETICS

For the MM kinetics with k′2 ≈ 0, we have the following

inequalities:

S1(ss)tot

S−1(ss)tot

= − K2

k−1= −(1 + k−2

k−1) < −1, (D1)

and

S−2(ss)tot

S2(ss)tot

= −k−2K1

k2K2= −k−2[S]

k2KM

� −1. (D2)

APPENDIX E: SIGN INVERSION OF S(μ)tot (t)

Let us consider the equality kμY = k−μX in Eq. (18) todetermine the time, t∗μ when the curve of S

(μ)tot (t) crosses the

time axis, if at all. It is given by

t∗μ = 1

K1 + K2ln

(K1(kμ + k−μ)

k−μK1 − kμK2

). (E1)

Hence S(μ)tot (t) can change sign with time if k−μK1 > kμK2.

With our parameters, this condition gets satisfied only for thereaction, μ = 2. Hence, the total EPR due to the reactionsμ = ±1, −2 remain positive or negative throughout the timeevolution up to the NESS, as shown in Fig. 1. For reaction

μ = 2, t∗2 comes out to be ∼10−12 s which is beyond ournumerical precision (time-steps of 10−7 s). Hence, the signchange is not detected in Fig. 1(d).

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