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1

Interplay of energy, dissipation, and error in kinetic proofreading: Control via

concentration and binding energy

Premashis Kumar,1 Kinshuk Banerjee,2 and Gautam Gangopadhyay1, ∗

1S. N. Bose National Centre For Basic Sciences,

Block-JD, Sector-III, Salt Lake, Kolkata 700 106, India2Department of Chemistry, Acharya Jagadish Chandra Bose College, Kolkata 700 020, India

(Dated: December 23, 2021)

Kinetic proofreading mechanisms explain the extraordinary accuracy observed in central biologicalevents in terms of the enhanced specificity of substrate selection networks under a nonequilibriumenvironment. The nonequilibrium steady state theory incorporated with a chemical thermodynamicframework is implemented to execute a systematic investigation of dynamic and thermodynamicfeatures of the proofreading network under continuous fuel consumption. We have identified thatthe dissipation-error trade-off domain of the network has a one-to-one correspondence with thedeeper portion of the basin-like error rate profile depicted here. Further, quantifying the energy costthrough concentration control of the chemical fuel aids in unveiling the association of the energy andchemical work with the optimal operating region of the biological error-correcting mechanism. It isshown that the proper energy content of the system, the semigrand Gibbs free energy, approaches anominal value in the trade-off regime, whereas it gets considerably minimized in the domain with alack of trade-off. We have also introduced a performance measuring entity corresponding to differentenergetic discrimination magnitudes as a product of average dissipation and coefficient of variationfor the whole range of chemical fuel. Besides invoking a different perspective to the experimentaland theoretical assessment of biological processes like DNA replication or protein synthesis, ourfindings can be advantageous in designing more efficient synthetic biological architectures.

I. INTRODUCTION

Biosynthesis, a complex process comprising a seriesof chemical reactions, supplies essential composite prod-ucts required for various necessary processes related tothe survival, regulation, and growth of living organismswith extraordinary accuracy. The observation of remark-ably high accuracy in biosynthesis processes is rational-ized by the strategy known as kinetic proofreading pro-posed independently by J. J. Hopfield and J. Ninio [1, 2].The proofreading mechanism asserts that the high fi-delity in selecting the right substrate over wrong onescan be adopted in the reaction network model of corebiological events like DNA replication [3, 4], RNA tran-scription [5, 6] or protein synthesis [7] by strongly driv-ing the network out of equilibrium. Most importantly,biological systems utilize the chemical energy of energy-bearing biomolecules to drive the process. Therefore, inthe simplest form, the kinetic proofreading mechanismof the biosynthesis process can be cast into the cyclicreceptor-ligand binding model [8, 9] coupled with Adeno-sine Tri-Phosphate(ATP) hydrolysis. Interestingly, a re-action network driven out of equilibrium approaches anonequilibrium steady state(NESS) characterized by thenonzero driving force and reaction flux which generatesdissipation [10] in the system. Dissipation plays a cru-cial role in sustaining or enhancing the performance of areaction network. Particularly, in the context of kineticproofreading network, a trade-off between the dissipation

∗ gautam@bose.res.in

and error rate can exist [11–13] and a proper investigationof this trade-off scenario aids to detect optimal operat-ing regimes of the network and biological systems havingsuch network. Nevertheless, the lowest error rate canbe achieved for vanishingly small dissipation for the en-ergetic discrimination scenario implying the difference ofbinding energy among intermediates [1, 2] of the network.A generalized model [12, 14, 15] of error-correcting mech-anism also exploited the energetic discrimination scenarioto investigate substantial complex and essential featuresof the proofreading scheme, including error-dissipationrelation.

The error-dissipation constraint of the proofreadingscheme has been studied in different variants of proof-reading networks with the context of relevant biologicalprocesses to reveal the guiding role of dissipation [13, 16].In this regard, the overall energy cost of the mechanismplays a pivotal role in shaping and carrying out these bio-logical process [17, 18]. Thus the energetic expenditure ofdetecting the right substrate would impose an importantconstraint on the proper functioning of the network giventhe limited energy budget in a biological environment.Although much attention has been dedicated to under-standing the interplay between error and dissipation, theexplicit energy utilization and modulation in a chemicalfuel-driven error-correcting mechanism is not addressedproperly. However, the recent development in nonequi-librium thermodynamics of open chemical reaction net-work(CRN) [19, 20] provides a rigorous basis to quantifythe proper energy content of an arbitrarily driven sys-tem under the concentration and flux control [21] of theexchanged species. Such energy quantification concern-ing externally-controlled chemical fuel in the context of

2

proofreading schemes can provide crucial insight into theconcept of chemical work and the thermodynamic effi-ciency of biological information processing mechanisms.Additionally, as the existence of multiple regimes of dif-ferent discriminatory ability has been reported [14] re-cently, it will be very interesting to explore how the sys-tem’s energy under a nonequilibrium environment is as-signed to different discriminatory regimes having vari-ous possible trade-off situations. The resemblance of dif-ferent proofreading regimes with stages of microtubulegrowth [12] and nonequilibrium sensing [22] seeks a com-plete thermodynamic picture of such state to unveil theunderlying general thermodynamic principles applicableto biological information processing [23].

Here, we aim to unveil the overall performance ofthe kinetic proofreading network in terms of error, dis-sipation, and proper energy representation of the opensystem and establish a connection among these entitieswithin different discriminatory regimes. In this regard,our work further intends to shed light on accessing thedifferent error rates and associated dissipation by merelycontrolling the concentration of certain species and mag-nitude of the energetic discrimination without changingthe network structurally and then evaluating the per-formance of the proofreading network. We have consid-ered a simple two-cycle network of kinetic proofreadingto systematically employ recently developed conservationlaw [24, 25] based analytically tractable chemical ther-modynamic framework [19, 20] to quantify the networkenergetics properly. Hence, along with the conventionalnonequilibrium Gibbs free energy, we would evoke thesemigrand Gibbs free energy [19, 20] to identify the roleof system energetics in the performance of the network.In this regard, measuring energy and dissipation by steer-ing the concentrations of certain chemical fuels like ATPwould be more natural in a driven system like this. More-over, we will introduce an overall performance measur-ing quantity that can assess the basis of overall error-dissipation trade-off scenarios for different energetic dis-crimination magnitudes. Besides theoretical importance,our approach of manipulating error, dissipation, and en-ergy with varying concentrations of energy-rich speciescan indeed open up various implementation opportunitiesin synthetic reaction networks and biological systems.

The paper is organized as follows. First, we have de-scribed the kinetic network of the proofreading mecha-nism and corresponding NESS dynamics in sec.II. Theerror rate of the network as a function of externally con-trollable parameters and the binding energy differenceis also discussed in the section. In the next section, wehave captured the thermodynamic response of the net-work due to ATP concentration variation. We have alsoformulated the semigrand Gibbs free energy by exploit-ing the network topology. Then, the error-dissipation-semigrand Gibbs free energy profile and quantification ofthe network performance are depicted in sec.IV. Finally,the summary and conclusions are provided.

II. KINETIC NETWORK: NONEQUILIBRIUM

STEADY STATE CHARACTERIZATION AND

ERROR RATE VARIATION

The proofreading network is described here with anominal synthesis rate. Indeed, in the presence of en-ergetic discrimination only, Hopfield’s model [1] showsminimal error at zero velocity. A schematic diagram ofthe kinetic network of the proofreading scheme is illus-trated in fig. 1(a). One cycle of the double-cycle proof-reading model corresponds to the ‘right’(‘R’) pathway,whereas another cycle belongs to the ‘wrong’(‘W’) one.The network depicts that a protein E binds with sub-strate ‘R’ (or ‘W’) to form intermediate complexes ‘ER’,‘ER*’ (or ‘EW’, ‘EW*’) and then disassociates into en-zyme and product. The discrimination between ‘R’ and‘W’ substrates in a living system usually occurs throughunbinding rather than binding [26]. Hence, it is com-prehensible to consider that all rate constants, exceptthose corresponding to unbinding, are equal for ‘R’ and‘W’ pathways because of the structural similarity of thesetwo substrates. Thus, it is sufficient to assume that dif-ference in affinity due to only rate constants kR−1 6= kW−1

and kR3 6= kW3 in fig. 1(a). Furthermore, with correct sub-strate(‘R’) having more residence time, kR−1 and kR3 areconnected with the corresponding rates of ‘W’ reaction

cycle through the relationkR−1

kW−1

=kR3

kW3

= exp−∆ = σ < 1

where ∆ is the binding energy difference between ‘W’and ‘R’ intermediate complexes. Now depending on theconcentration dynamics of the species, we can divide allthe species of the proofreading network in fig. 1(a) intotwo disjoint sets of ‘I’ and ‘C’,

{E,ER,ER∗, EW,EW ∗}I

⋃ {R,ATP,ADP,Rp,W,Wp}C

,

where ‘I’ is a set of intermediate species having dynamicconcentration, and ‘C’ is a set of chemostatted specieswith a constant concentration within the time scale ofinterest. In other words, elements of the set ‘I’ formthe vertices in the graphical representation of a reactionnetwork, whereas elements of set ‘C’ are associated withthe edge. To keep the approach more general, Rp and Wp

instead of commonly used R andW have been consideredin the third step of the proofreading network.

Now the schematic diagram, fig. 1(a) yields eR∗+eR+eW + eW ∗ + e = e0 with e0 being the total enzyme con-centration and eR, eR∗, eW, eW ∗ being concentrations ofspecies ‘ER’, ‘ER*’, ‘EW’, ‘EW*’, respectively. We as-sume that total enzyme concentration can be representedas e0 = eR0 + eW0 to simplify the problem considerablyand generally eR0 6= eW0 . This assumption results ineR∗ + eR + eR = eR0 and eW + eW + eW ∗ = eW0 wheree = eR + eW . With these considerations, we can math-ematically map this two cycle proofreading model intoa single cycle model as shown in right panel of fig. 1.Hence kinetics of the cycle involving right substrate can

3

FIG. 1. kinetic model: In fig. 1(a), two distinct cycles correspond to two very similar substrates ‘right’(‘R’) and ‘wrong’(‘W’).

‘R*’ and ‘W*’ are related to the synthesis rate, which is assumed to be sufficiently low to be neglected here. Species ‘W’,‘R’, ‘ATP’, ‘ADP’, ‘Rp’ and ‘Wp’ are assumed to have constant concentrations over the time scale of interest and can beabsorbed within rate constants. In fig. 1(b), the hypothetical image of the free energy landscape for the kinetic networkhas been considered here. Right: Any cycle of the kinetic proofreading network can be represented by a single cyclic modelshown in the upper panel. By exploiting pseudo-first-order rate constants k1 = k0

1(S), k−3 = k0−3(Sp), k+2 = k0

+2(ATP ) andk−2 = k0

−2(ADP ), the single cyclic model in the upper panel can be effectively mapped into a simple three-state system asshown in the lower panel, with seemingly no explicit material exchange with the environment.

be expressed as

∂eR

∂t= kR1 (e

R0 − eR∗ − eR)− (kR−1 + kR+2)(eR) + kR−2(eR

∗)

(1a)

∂eR∗

∂t= kR−3(e

R0 − eR∗ − eR)− (kR−2 + kR+3)(eR

∗) + kR+2(eR).

(1b)

Here k1 = k01(R), k−3 = k0−3(Rp), k+2 = k0+2(ATP )and k−2 = k0−2(ADP ) are pseudo-first-order rate con-stants [8, 9] and concentration of ATP, ADP, R, andRp are held at constant levels. Exploiting the conditioneR∗ + eR + eR = eR0 , we acquire following steady stateconcentrations from eq. (1a) and (1b),

4

eRss =(kR−2k

R−3 + kR1 k

R3 + kR1 k

R−2)e

R0

(kR1 kR−2 + kR1 k

R3 + kR1 k

R2 ) + (kR−1k

R−3 + kR2 k

R−3 + kR−2k

R−3) + (kR−1k

R−2 + kR−1k

R3 + kR2 k

R3 )

(2a)

eR∗ss =

(kR1 kR2 + kR−1k

R−3 + kR2 k

R−3)e

R0

(kR1 kR−2 + kR1 k

R3 + kR1 k

R2 ) + (kR−1k

R−3 + kR2 k

R−3 + kR−2k

R−3) + (kR−1k

R−2 + kR−1k

R3 + kR2 k

R3 )

(2b)

eRss =(kR−1k

R−2 + kR−1k

R3 + kR2 k

R3 )e

R0

(kR1 kR−2 + kR1 k

R3 + kR1 k

R2 ) + (kR−1k

R−3 + kR2 k

R−3 + kR−2k

R−3) + (kR−1k

R−2 + kR−1k

R3 + kR2 k

R3 )

. (2c)

Now, unlike a equilibrium steady state, there will bea clockwise steady state flux for the cyclic reactionnetwork at NESS. Each cycle of the proofreading net-work comprises of three elementary reactions and net

fluxes of these reactions elementary can be obtained asj1 = k1(e) − k−1(es); j2 = k2(es) − k−1(es

∗)andj3 =k3(es

∗) − k−3(e). Therefore, the steady state cycle fluxin clock-wise direction for a single cycle is represented as

jRss =(kR1 k

R2 k

R3 − kR−1k

R−2k

R−3)e

R0

(kR1 kR−2 + kR1 k

R3 + kR1 k

R2 ) + (kR−1k

R−3 + kR2 k

R−3 + kR−2k

R−3) + (kR−1k

R−2 + kR−1k

R3 + kR2 k

R3 )

. (3)

In the case of wrong substrate pathway, eWss, eW∗ss, e

Wss

and jWss will have the same expression with all the super-script ‘W’ instead of ‘R’ in each term. As required bya thermodynamic viewpoint, all the reaction steps haveto be reversible in nature. The driving force associatedwith the NESS cycle flux is the net chemical potentialaround the single cycle, µL = µe,es + µes,es∗ + µes∗,e =

ln k1k2k3

k−1k−2k−3

= ln γ. As the rate constants k1, k±2, k−3

are equal in both ‘W’ and ‘R’ pathways and k−1 andk3 are connected through a common factor exp−∆, boththe ‘W’ and ‘R’ cycles are subjected to equal drivingforce. We have set Boltzmann’s constant kB = 1 andtemperature T = 1 to slim our notations throughoutthis study. Unless otherwise stated, we have imple-mented following rate constants throughout our study,kR1 = kW1 = 5, kR3 = 1, kR−1 = 50, kR−2 = kW−2 = 10.−3

and kR−3 = kW−2 = 10.−3 [27]. Rate constants with equalmagnitudes for ‘R’ and ‘W’ cycle have been used withouta superscript in future discussion.

A. Error Rate Profile

Initially we would examine the error rate profile byvarying multiple pseudo-first-order rate constants simul-taneously. We have expressed the error rate of the proof-reading scheme by mapping the two-cycle model into asingle cycle. In the limit of slow synthesis rate, the errorrate of the kinetic proofreading network at NESS can bedefined as [14]

η =eW ∗

ss

eWss

eRsseR∗

ss

(4)

where eW ∗ss and eR∗

ss are steady-state concentrations of‘EW*’ and ‘ER*’ respectively and concentration of en-zyme corresponding ‘R’ and ‘W’ cycles is given by eRss andeWss . Under equilibrium condition, the error rate of thekinetic proofreading network would result in η = exp−∆.However, an error rate as low as η = exp−2∆ can beachieved within a nonequilibrium environment.Now using eq. (2b) and (2c), we can express the error

rate from eq. (4) as,

η =(kW1 kW2 + kW−1k

W−3 + kW2 kW−3)(k

R−1k

R−2 + kR−1k

R3 + kR2 k

R3 )

(kR1 kR2 + kR−1k

R−3 + kR2 k

R−3)(k

W−1k

W−2 + kW−1k

W3 + kW2 kW3 )

.

(5)Natural dominance of reaction step 2 rather than step 3of the proofreading network in production of substrate-enzyme complex eR∗ or eW ∗ implies k1 >> k−3 in fig. 1.Further, preference of step 3 in the formation of ‘Rp’ (or‘Wp’) from eR∗ (or eW ∗) suggests k3 >> k−2. Takingvery small values of k−3 and k−2 ensure that correspond-ing steps of reaction network are almost irreversible. Nowunder assumptions k1 >> k−3, and k3 >> k−2, we wouldanalyze the modification of the specificity of proofread-ing network due to the relative strength between rateconstants k−1 and k+2. For this purpose, we have takenthree different situations on the basis of relative strengthbetween rate constants k−1 and k+2 and identified ordersof the specificity in those situations.

• k−1 >> k+2; γ =kR1kR2kR3

kR−1

kR−2

kR−3

≈ 1: The general

expression of the error rate in eq. (5) becomes η1 =kW−1

kR−1

(kR−1

kR3)

(kW−1

kW3

)= exp−∆ = σ.

• k−1 ≃ k+2; γ >> 1: The error rate in eq. (5)

5

FIG. 2. In fig. 2(a), a basin-like characteristic of error rate is obtained by varying only k2 from 10−6 to 106. Other rateconstants(or pseudo-first-order rate constants) of the proofreading network are fixed at k1 = 5, kR

3 = 1, kR−1 = 50, k−2 = 10−3

and k−3 = 10−3. To visualize the characteristics of the error rate more lucidly, we have taken the log scale of the k+2. Infig. 2(b), a 3D counterpart of error basin is illustrated by changing both k−1 and k+2 simultaneously. k+2 is varied from 10−5

to 105, while k−1 is changed within the range, 0.5− 50. Other rate constants are the same as in fig. 2(a). The binding energydifference, ∆ = 3 in both figures.

results in η2 =(kR

−1kR3)

(kW−1

kW3

)

(1+kR2

kR−1

)

(1+kW2

kW−1

)= σ2

(1+kR2

kR−1

)

(1+kW2

kW−1

)= σ2.

• k−1 << k+2; γ >> 1: Finally this condition yields

error rate as, η3 =kR3

kW3

= σ.

This result reveals the subtle role of the relative strengthof rate constants k−1 and k+2 in the modification of er-ror. We can use either k2 or k−1 as a control parameterof this proofreading scheme to access different discrim-inatory regimes. Most importantly, for the conditionk−1 << k+2, although the system is far from equilib-rium, we do not obtain any improvement over the errorrate near equilibrium.We have obtained an error rate profile in fig. 2(a) from

the general expression of the error rate in eq. (5) by vary-ing only rate constant k+2 at fixed values of binding en-ergy difference ∆ and all other rate constants. As we pro-ceed from k−1 >> k+2 to k−1 ≃ k+2 regime by varyingk+2, it is evident from fig. 2(a) that there is an abrupt fallin the error and eventually the error will reach its mini-mum value σ2. The plateau region in fig. 2(a) representsthe k−1 ≃ k+2 regime σ2. Then, while we move in pa-rameter space from the k−1 ≃ k+2 region to k−1 << k+2,a basin-like characteristics of the error emerges as the er-ror rate approaches the equilibrium error rate value σwithin nonequilibrium environment. This graphical rep-resentation of the general error rate in fig. 2(a) depictsthe analytical results under three conditions, as discussedearlier in this section. In fig. 2(b), we have obtained a 3Dcounterpart of the error basin in fig. 2(a) by varying bothk−1 and k+2 simultaneously. This 3D figure more clearly

illustrates the modulation of error rate due to relativestrength between k−1 and k+2. Thus, this representa-tion will be crucial to ensure maximum accuracy of theproofreading scheme by controlling either k−1 or k2 orboth rate constants at the same time for a fixed ∆(= 3here).The central observation from this analysis is that error

rate η is not constant under the nonequilibrium conditionbut strongly depends on the relative strength betweenrate constants k−1 and k+2. We assert that the kineticproofreading network can have two different regimes withdistinct discrimination ability depending on the relativestrength of rate constants k−1 and k+2 under nonequi-librium conditions. As the pseudo-first-order rate con-stants contain the concentration of chemostatted speciesby definition, we can also modify the error rate curve inthe same way by controlling the concentration of thesespecies.

B. Impact of Binding Energy Difference

To capture the impact of binding energy difference,∆ on the error rate of the kinetic proofreading network,we have shown variation of log(η) with respect to ∆,separately for three different regimes corresponding tothe relative strength between k−1 and k+2. Then akin tothe discriminatory index in ref. [14], we have defined an

index, ν = ∂ log(η)∂∆ , i.e., the slope of the log(η) versus ∆

plot. For example, this definition yields ν = −1 for theequilibrium condition as η = exp−∆ in this regime.Now we would analyze the error rate, η in terms of the

6

FIG. 3. The impact of the binding energy difference, ∆ on the error rate, η for three different regimes: log η vs ∆ is exhibitedhere for three regions corresponding to k−1 >> k+2, k−1 ≃ k+2 and k−1 << k+2 in fig. 3(a), fig. 3(b) and fig. 3(c), respectively.

The corresponding variations of the discriminatory index, ν = ∂ log(η)∂∆

over the range of ∆ for k−1 >> k+2, k−1 ≃ k+2 andk−1 << k+2 are depicted in fig. 3(d), fig. 3(e) and fig. 3(f), respectively.

newly defined quantity, ν, for three previously mentionedregimes. For the regime corresponding to k−1 >> k+2,log(η) follows line corresponding to exp−∆ over thewhole range of ∆ as shown in fig. 3(a). As all otherparameters are kept fixed at constant values, we can as-sert that the error rate in this regime is not effected by∆. The same is evident from fig. 3(d) as ν = −1 for thewhole range of ∆. For the regime, k−1 ≃ k+2, log(η)curve in fig. 3(b) fairly follows the line corresponding toexp−2∆ up to a certain extent of ∆ magnitude and thenit deviates from this line upon further increment of ∆. Itis clear from the fig. 3(e) that ν in this regime switchesfrom −2 to −1 for higher ∆ value. Hence, the accu-racy is less than its maximum level for the parameterregion representing the relatively higher binding energydifference. The error rate in such case can be generallyrepresented by η = exp−(1 + β)∆ with 0 < β < 1. Fi-nally, for the regime, k−1 << k+2, log(η) at first followsthe line of exp−∆ in fig. 3(c) as expected from the errorbasin in fig. 2(a). However, as the ∆ is further extendedover a certain value, the log(η) line gradually deviatesfrom the exp−∆ line to the exp−2∆. This deviationfor higher values of ∆ suggests that it is possible to havea relatively more effective proofreading(with respect tothe lower ∆ counterpart) even for regime correspondingto k−1 << k+2. The error rate curve again moves to-wards the exp−∆ line on further increment of ∆ abovea certain value. The whole observation in fig. 3(c) is alsoconfirmed by the variation of the ν between −1 and −2in fig. 3(f). Thus, the overall graphical analysis in fig. 3reveals that it is possible to expand, shrink, or shift theparameter space corresponding to a particular error rate

by using binding energy difference as a control parame-ter.

III. THERMODYNAMIC RESPONSE DUE TO

CONCENTRATION CONTROL

Any change in concentrations of chemostatted speciescan be a source of massive modification in the system’sentropic and energetic response under a nonequilibriumenvironment [28, 29]. The pseudo-first-order rate con-stants are defined in terms of concentrations of thosechemostatted species present in the open chemical re-action network representation of the kinetic proofread-ing. Therefore, rather than changing the rate constants,the variation in the pseudo-first-order rate constants canbe realized by altering the chemostatted concentrations.In the following study, we will analyze the thermody-namic evolution of the proofreading network controlledby the influx of the chemostatted species, especially thefuel molecule ATP, and establish the association of thisthermodynamic response with the dynamics of the net-work.

A. Conservation Laws and Emergent Cycles

The reaction network of the kinetic proofreading infig. 1(a) comprises of six elementary chemical reactionsand eleven species. Hence, the proofreading network canbe presented by the following 11 × 6 stoichiometric ma-

7

trix [30]

Sσρ =

R1 R2 R3 R4 R5 R6

E −1 0 1 −1 0 1

ER 1 −1 0 0 0 0

ER∗ 0 1 −1 0 0 0

EW 0 0 0 1 −1 0

EW ∗ 0 0 0 0 1 −1

R −1 0 0 0 0 0

ATP 0 −1 0 0 −1 0

ADP 0 1 0 0 1 0

W 0 0 0 −1 0 0

Rp 0 0 1 0 0 0

Wp 0 0 0 0 0 1

. (6)

The left null linear independent vectors correspondingto left null space of this stoichiometric matrix Sσ

ρ areknown as the conservation laws [30] and can be expressedas,

∑

σ lλσS

σρ = 0 where {lλσ} ∈ R

(σ−w)×σ, w = rank(Sσρ ).

For the stoichiometric matrix (6) of the proofreading net-work, the following five vectors represent the conserva-tion laws of the closed reaction network,

lλ=1σ =

(

E ER ER∗ EW EW ∗ R ATP ADP W Rp Wp

1 1 1 1 1 0 0 0 0 0 0)

lλ=2σ =

(

E ER ER∗ EW EW ∗ R ATP ADP W Rp Wp

0 0 0 0 0 0 1 1 0 0 0)

lλ=3σ =

(

E ER ER∗ EW EW ∗ R ATP ADP W Rp Wp

0 1 0 1 0 1 −1 0 1 0 0)

lλ=4σ =

(

E ER ER∗ EW EW ∗ R ATP ADP W Rp Wp

0 1 1 0 0 1 0 0 0 1 0)

and

lλ=5σ =

(

E ER ER∗ EW EW ∗ R ATP ADP W Rp Wp

−1 −2 −1 −1 0 −1 1 0 0 0 1)

.

These conservation laws of the reaction network furtherspecify conserved quantities of the network known asComponents of the system and are defined as, Lλ =∑

σ lλσzσ such that d

dt

∫

drLλ = 0. Thus, componentscorresponding to those five conservation laws are L1 =e + eR + eR∗ + eW + eW ∗; L2 = ATP + ADP ; L3 =eR+ eW +R−ATP +W ; L4 = eR+ eR∗+R+Rp andL5 = −e− 2eR− eR∗ − eW − R+ATP +Wp.

While a closed system is opened by chemostatting,the stoichiometric matrix Sσ

ρ breaks into two parts: SIρ

referring to intermediate species and SCρ belonging to

chemostatted species. Therefore, conservation laws inan open system are characterized as

lλI SIρ + lλCS

Cρ = 0

{

lλb

I SIρ 6= 0 broken CL,

lλu

I SIρ = 0 unbroken CL

(7)

with u and b being labels corresponding to unbroken andbroken ones, respectively and {lλ} = {lλb}∪{lλu}. In thisreaction network, conservations laws, lλ=2

σ , lλ=3σ , lλ=4

σ and

8

lλ=5σ of the closed system can be broken due to chemostat-ting. Depending on whether chemostatted species breaka conservation law or not, the set of chemostatted speciescan be divided into two subsets {C} = {Cb} ∪ {Cu}. Itis apparent from eq. (7) that the broken conservationlaws are not left null vectors of SI

ρ for at least one re-action of the network. Hence, components of the opensystem related to the broken conservation laws are nolonger a global conserved quantities and are denoted asLλb

. The unbroken conservation law of the open systemof the proofreading are obtained from the stoichiometricmatrix of intermediate species, SI

ρ as,

lλu=1I =

(

E ER ER∗ EW EW ∗

1 1 1 1 1)

and corresponding component Lλu= e+eR+eR∗+eW+

eW ∗ refers to the total concentration of the enzyme, aglobal conserved quantity of this open system. Right nullvectors of the stoichiometric matrix, Sσ

ρ cnσ represent in-

ternal cycles which restore all the species’ concentrationto its prior state upon completion. However, this kineticproofreading network has no internal cycle. Instead, thisopen reaction network contains following two indepen-dent emergent cycles [24],

c1 =

1 1

2 1

3 1

4 0

5 0

6 0

and c2 =

1 0

2 0

3 0

4 1

5 1

6 1

.

A complete emergent cycle keeps the states of intermedi-ate species unchanged but chmeostatted species are ex-changed between system and chemostats [19]. The netstoichiometry of the emergent cycle c1 and c2 are R +ATP Rp + ADP and W + ATP Wp + ADP,respectively. The steady-state current vectors for theemergent cycle c1 and c2 are

jc1 =

1 jc1

2 jc1

3 jc1

4 0

5 0

6 0

and jc2 =

1 0

2 0

3 0

4 jc2

5 jc2

6 jc2

,

respectively. Each of these flux can be given by anyof the elementary reactions of the proofreading networkby inserting steady state concentrations of intermediatespecies within it,

jc1 = kR1 (eRss)−kR−1(eRss) and jc2 = kW1 (eWss )−kW−1(eWss).

Both jc1 and jc2 have mathematical expression same aseq.(3) with two different superscript ‘R’ and ‘W’ respec-tively for rate constants.

B. Dissipation of The Network

We can represent total dissipation [31] of the reac-tion network using a flux-force relation. The overallsteady state flux of kinetic proofreading network is ac-quired as the sum of fluxes of two emergent cycles, Jss =jc1 + jc2 . Now the chemical force, reaction affinities[31]acting along the cycles are represented by emergent affini-ties [24] which obeys following equation

µǫ = cǫ lnkρ

k−ρ

z−Sc

ρc , (8)

with zc and Scρ being concentrations and stoichiometric

elements of chemostatted species, respectively. There-fore, the chemical affinities along c1 and c2 are µ1 =

lnkR1kR2kR3

kR−1

kR−2

kR−3

and µ2 = lnkW1

kW2

kW3

kW−1

kW−2

kW−3

. Any non-zero posi-

tive values of affinities µ1 and µ2 will drive the system outof equilibrium. Our assumptions for reaction rates heremake sure that µ1 = µ2 = µ. Thus, total dissipation ofthe proofreading network is obtained as, Σ = Jssµ. Thesecond law of thermodynamics demands this dissipationto always be positive.A change in ATP concentration from very low to a

high level brings modulation in dissipation as illustratedin fig. 4. This change in the ATP concentration, keepingthe concentration of ADP fixed at a certain level, impliesa gradual increase in the ATP to ADP concentration ra-tio of the system. We have presented two breakdownfigures fig. 4(a) and fig. 4(b) to understand the dissipa-tion curve with respect to ATP concentration, and cor-responding error variations are shown in fig. 4(c) andfig. 4(d). The range of ATP concentration variation onthe left panel of fig. 4 drives the system from its ini-tial near-equilibrium state to nonequilibrium one. Dueto the concentration change, the error in fig. 4(c) grad-ually decreases from near-equilibrium to the nonequilib-rium regime. This error rate variation is expected fromthe error basin illustration in fig. 2(a) and correspondingdiscussion of section IIA. Now from fig. 4(c) and fig. 4(a),we can observe that dissipation becomes nonzero and in-creases monotonically as the error enters the nonequilib-rium error regime from near-equilibrium. This dissipa-tion curve reveals the trade-off nature of error and dissi-pation in this range of ATP concentration. For a muchhigher range of ATP concentration on the right panel offig. 4, the error increases from its minimum value and ap-proaches the equilibrium magnitude within the nonequi-librium environment in fig. 4(d). By comparing fig. 4(b)and fig. 4(d), it is observed that the corresponding dis-sipation curve qualitatively reflects the same dynamicsas the error rate of the proofreading mechanism in thisregime. A simultaneous increase in both error and dis-sipation in this range of ATP concentration hints at the

9

FIG. 4. In fig. 4(a) and fig. 4(c), dissipation and error rate of the kinetic proofreading for the range of ATP concentration10−6

− 100 is shown. As error rate deceases over the higher end of the concentration, dissipation increases. The dissipationand error rate of the kinetic proofreading for the higher range of ATP concentration 100 − 106 is illustrated in fig. 4(b) andfig. 4(d), respectively. In this range of the concentration, dissipation qualitatively reflects the profile of the error rate. The rateconstants of proofreading network are fixed at k1 = 5,k0

2 = 1 kR3 = 1, kR

−1 = 50, k−2 = 10−3 and k−3 = 10−3.

lack of trade-off between error and dissipation under thenonequilibrium environment.

C. Semigrand Gibbs Free Energy

We now need to incorporate a proper nonequilibriumthermodynamic potential of the open system to analyzethe energy expenditure of the proofreading network atthe nonequilibrium regime. For this purpose, we havecharacterized each chemical species thermodynamicallyby chemical potential µσ. The expression of the chemicalpotential is µσ = µo

σ+ln zσz0

with solvent concentration z0and standard-state chemical potential µo

σ. The standard-state quantities with notation ‘o′ are taken at standardpressure p = po and molecular concentration and chemi-cal potential to set a baseline for substances. Moreover,a constant term like ln z0 can also be absorbed withinµoσ of the chemical potential. Thus, the nonequilibrium

Gibbs free energy of a chemical reaction network can bewritten in terms of the chemical potential as [32]

G = G0 +∑

σ 6=0

(zσµσ − zσ) (9)

with G0 = z0µo0. However, the proper entity to capture

the energetics of the open system like the proofreadingnetwork is semigrand Gibbs free energy [20]. The sem-igrand Gibbs free energy of an open system is obtainedfrom the nonequilibrium Gibbs free energy by a Legendretransformation [19]

G = G−∑

λb

µλbMλb

. (10)

where Mλb=

∑

Cblλb

−1

CbLλb

represents moieties [33] ex-changed between chemostats and system through the ex-ternal flow of the chemostatted species.The variation in the Gibbs free energy, the semigrand

Gibbs free energy, and its slope due to change in ATP

10

FIG. 5. The variation of error rate, Gibbs free energy, semigrand Gibbs free energy and slope of semigrand Gibbs free energyfor lower range of ATP concentration (10−6

− 100) is plotted in (i), (ii), (iii) and (iv) of fig. 5(a). Whereas response curvecorresponding to higher range of ATP concentration is shown in (v), (vi), (vii) and (viii) of fig. 5(b). For both the cases, bindingenergy difference, ∆ = 3 has been considered. The rate constants of proofreading network are fixed at k1 = 5,k0

2 = 1 kR3 = 1,

kR−1 = 50, k−2 = 10−3 and k−3 = 10−3.

concentration over two different error rate regimes is pre-sented in fig. 5(a) and fig. 5(b). The Gibbs free en-ergy of the system exhibits similar qualitative charac-teristics as the error profile of the system for both thelower and higher range of ATP concentration in (ii) and(vi) of fig. 5(a) and fig. 5(b), respectively. For the con-centration range corresponding to minimum error rate,when dissipation starts increasing from zero(fig. 4(a)),the Gibbs free energy lowers accordingly. However, forthe higher range of ATP concentration, as the error growsafter the plateau region of minimum error, the dissipa-tion further enhances(fig. 4(b)) within a more stronglydriven nonequilibrium environment, and the Gibbs freeenergy rapidly moves towards a very high positive value.The sum of nonequilibrium Gibbs free energy and dissipa-tion of proofreading network quantifies the chemical workperformed on the system [19]. Thus we assert that theproofreading network performs with optimal error ratefor small and finite chemical work. The chemical workdiverges rapidly for the strongly driven nonequilibriumregime having lower accuracy.

Figure 5(a) suggests that as more ATP accumulates inthe system, the semigrand Gibbs free energy, G, movestowards zero from a negative value. At the same time,the error rate goes towards a minimum. The corre-sponding semigrand Gibbs free energy slope remains con-stant initially and then goes from one to zero, as the er-ror changes from the equilibrium value to the minimumone. For a comparatively higher ATP concentration infig. 5(b), semigrand Gibbs free energy remains near zeroover an extended range of ATP concentration and sus-

tains the minimum error within a nonequilibrium envi-ronment. However, as the error curve deviates from itsminimum value due to further change in ATP concentra-tion, a sharp and vast change towards negative value canbe seen in this semigrand Gibbs free energy profile. Thismassive change is also evident from the correspondingslope shown in (viii) of fig. 5(b). The enormous mag-nitude of Gibbs free energy and semigrand Gibbs freeenergy in fig. 5(b) suggests that the system is far fromequilibrium. Further, we conclude that the proofread-ing system operates at optimal accuracy and dissipationwhen semigrand Gibbs free energy has a value near zero.

IV. ERROR-DISSIPATION-SEMIGRAND

GIBBS FREE ENERGY PROFILE AND

PERFORMANCE OF PROOFREADING

NETWORK

A trade-off between error and dissipation is present ifany modification in a parameter cannot simultaneouslyimprove both. However, this trade-off feature can becompletely absent in other parts of the same process.We have investigated the trade-off nature of the dissipa-tion and related error for different discrimination regimesthat correspond to externally controlled ATP concentra-tion variation. Association of Semigrand Gibbs free en-ergy with this trade-off scenario is also demonstrated.Additionally, in the presence of distinct binding energydifference ∆, we have unveiled the overall performanceof the reaction network in terms of error and dissipationfor the whole range of ATP concentration.

11

FIG. 6. Dissipation with respect to the error rate is shown in fig. 6(a) for the range of ATP concentration, 10−6− 104. The

binding energy difference ∆ = 3 has been considered. Figure 6(e) corresponds to the regime(k−1 >> k+2) where the error rateis equal to the equilibrium error rate and dissipation is negligible. As the error rate decreases from equilibrium magnitude tothe minimum one(in k−1 ≃ k+2 regime), a sharp change in dissipation is shown in fig. 6(d). In fig. 6(c), both the error rate anddissipation increase (the k−1 << k+2 regime). Figure 6(b) illustrates the simultaneous variation of dissipation and semigrandGibbs free energy concerning the error rate over the same ATP concentration range.

Here, fig. 6(a) illustrates a trade-off curve of error anddissipation with the error being exploited as an inde-pendent parameter. The figure suggests that dissipa-tion is infinitesimally small till the minimum error isreached. From fig. 6(e), we can see the infinitesimal dis-sipation change near the equilibrium error regime. Thena sharp and finite increase in the dissipation near theminimum error rate can be seen from fig. 6(d). The fast-increasing part implies that as we further decrease the er-ror rate, the dissipation inevitably increases as the virtueof dissipation-error trade-off in this regime. Then the er-ror rate again approaches the magnitude of the equilib-rium error(for a fixed lower ∆) due to a further increase inthe external driving force and dissipation monotonicallyincreases owing to that external driving force in fig. 6(c).Thus, it is observed that the error value does not exclu-sively determine the dissipation as the dissipation can bedifferent for the same error depending on the distancefrom the equilibrium. Further, fig. 6(b) exhibits the si-multaneous variation of the dissipation and semigrandGibbs free energy with the error rate and thus reveals theallocation of the proper energy content of the system tothe error-dissipation trade-off scenarios. The red dot inthe figure indicates the optimal working condition of thenetwork having the minimal semigrand Gibbs free energy,dissipation, and error rate. In fig. 7(a), the dissipation-error profile similar to fig. 6(a) is presented for four ar-bitrary magnitudes of the binding energy difference, ∆.Although the error rate significantly decreases for higher∆ values, the maximum dissipation for the same rangeof ATP concentration decreases. The characteristic dif-

ference between the dissipation curves corresponding tohigher and lower values of ∆ is also evident from fig. 7(a).In the case of higher ∆ values, the error rate of the proof-reading scheme within the nonequilibrium environmentgets stuck at a lower value for a much wider parameterspace of ATP concentration. Thus, a more rapidly in-creasing nonzero portion of the dissipation plot for ∆ = 3than the same part of ∆ = 12 is observed owing to thenonequilibrium regime with no improvement in error rateover the equilibrium error for relatively lower ∆ values.It is also important to note from fig. 7(a) that the dissipa-tion corresponding to the minimum error rate for higher∆ is greater than the one corresponding to the lower ∆.As we have considered equal driving forces for different∆ values, these differences of dissipation curves are solelydue to the different fluxes for distinct ∆ values.

In a recent study [34], the performance of the kineticproofreading network representing biological operationhas been studied with the aid of bound set by the ther-modynamic uncertainty relation [35]. However, we haveanalyzed the NESS of the kinetic proofreading network ata fixed time with a variation in ATP concentration andenergetic discrimination. To quantify the performanceof the network in terms of both error and dissipation inthe presence of different binding energy differences, wehave introduced a performance measuring entity at theNESS of the deterministic system for the variation ofATP concentration. For this purpose, we have set theerror rate, Ao = η as the observable of our interest, andit changes with the concentration of ATP. Now at theNESS, we have denoted the mean and standard deviation

12

FIG. 7. In fig. 7(a), the dissipation with respect to the error is shown for four different values of the binding energy difference,∆(= 3, 6, 9 and 12). For all the cases, the range of ATP concentration varies from 10−6 to 104. All other parameters arefixed. Q value is defined as a product of the average dissipation and the coefficient of variation for the whole range of ATPconcentration, as shown for ∆(= 3, 6, 9 and 12) in fig. 7(b). The lowest value of the Q at ∆ = 9 hints at comparatively goodtrade-off between error and dissipation for the whole range of the ATP concentration. Inset: Change in Q factor and coefficientsof variation(COV) due to the continuous variation of ∆. The Q factor profile is scaled by a factor 0.1 to obtain a proper fit.

associated with Ao as 〈Ao〉 and√

(〈A2o〉 − 〈Ao〉2), respec-

tively. Then, the coefficient of variation across the wholerange ATP concentration in this case can be expressed as,

ǫ =

√(〈A2

o〉−〈Ao〉2)

〈Ao〉. Implementing the coefficient of varia-

tion is useful here as we have drastically different meanscorresponding to distinct ∆ values. For a particular ∆,we now define the performance measuring factor Q as aproduct of the average dissipation and the coefficient ofvariation, Q = 〈σ〉 ǫ.

In fig. 7(b), the factor Q decreases to a minimum valuefor ∆ = 9 after having almost equal magnitude for ∆ = 3and ∆ = 6. It is also evident from the fig. 7(b) that thisQ factor for ∆ = 12 is even higher than the value corre-sponding to ∆ = 3. The Q factor and coefficients of vari-ation corresponding to a continuous variation of ∆ areillustrated in the inset of fig. fig. 7(b). Despite the coeffi-cients of variation corresponding to the error rate beinglower for ∆ = 3 and ∆ = 6 than the other higher ∆, ahigh average dissipation values for those two ∆ push thecorresponding Q values above its value for ∆ = 9. How-ever, for ∆ = 12, even a much lower average dissipationis not enough to compensate for the higher coefficientof variation, and its’ Q value is well above all other ∆values. Therefore, although average dissipation for thewhole range of ATP concentration decreases monotoni-cally with ∆, the Q factor shows a non-monotonic be-havior with ∆. We assert that a lower Q value betteradjusts error and dissipation to maintain optimal overallperformance concerning different operating regimes overthe whole ATP concentration range. Therefore, for a

specific velocity and a fixed energy budget of the kineticproofreading network, this Q term aids us in revealinga particular ∆ that has optimal overall performance interms of error and dissipation.

V. CONCLUSIONS

We have described the kinetic proofreading throughthe substrate selection network from the standpoint ofenergetic discrimination and thereby exploited the chem-ical thermodynamic framework to analyze the operationof the network systematically. The main findings andstrengths of our theoretical analysis include identifica-tion and characterization of the impact of the controlparameters on the dynamic and thermodynamic entitiesof the network and the necessary formal development ofproper energy content of a generic biological network.Further, our estimation of the nonequilibrium thermody-namic quantities aid in exploring the optimal regime ofthe proofreading in terms of error, dissipation, and sem-igrand Gibbs free energy. Finally, we have unveiled theoverall performance of the proofreading network by defin-ing a new entity, Q, in terms of the average dissipationand coefficient of variation for the whole range of a spe-cific chemical fuel concentration for a particular bindingenergy difference.

Initially, it is shown that the proofreading kinetics canhave different discrimination regimes depending on therelative strength between the pseudo-first-order rate con-stants. The association of the chemostatted species con-

13

centration with these pseudo-first-order rate constantsmakes it possible to access all different discriminationregimes at will by the concentration control of corre-sponding chemostatted species. Furthermore, the errorbasin found by changing the relative strength betweenrate constants experiences a modification depending onthe magnitude of binding energy difference, ∆. There-fore, without changing the hardcore network structure,one can shift or modify the error regimes by the magni-tude of the binding energy difference. This dependenceon the binding energy magnitudes will also be vital if afamily of competing substrates is present in a biologicalsystem. Further, in a biomolecular system, the impact ofenhanced binding discrimination proves to be decisive inmaintaining the system’s operation close to the thermo-dynamic uncertainty relation [35] bound or further fromthe bound [34].By changing the concentration of chemical fuel, ATP,

we have demonstrated that dissipation has the samequalitative variation as the error rate curve only for ahigher concentration. On the contrary, the nonequilib-rium Gibbs free energy of the system exhibits similarqualitative characteristics as the error profile of the sys-tem for both the lower and higher range of the concen-tration. The variation of these two quantities in differentdiscrimination regimes establishes a relation between thechemical work and the error rate. Besides the conven-tional nonequilibrium Gibbs free energy, the semigrandGibbs free energy provides important intuition about thenetwork. It has been identified that the optimal networkaccuracy and dissipation are associated with near-zerosemigrand Gibbs free energy. This finding in a sim-ple proofreading network can be crucial and advanta-

geous for searching the optimal specificity and dissipa-tion regime in more complex and general proofreadingnetworks and copying schemes [12, 14, 15]. This con-centration control approach will be useful to estimatethe energy cost of general biochemical feedback controlmechanisms [23, 36].Additionally, the binding energy difference resulting in

a lower Q value can be recommended for a certain pur-pose within a specific ATP concentration range. Thus,the Q factor can be an effective entity to be measuredin synthetic biochemical networks to achieve a certaintarget with a concentration control. Indeed, the con-straints and purposes of synthetic and biological engi-neered systems [37–39] are different from their naturalcounterparts.Controlling discriminatory proofreading regimes and

associated energetics through the concentrations of cer-tain species and magnitude of binding energy differencewould have crucial implications in designing more effi-cient synthetic biological architectures. This investiga-tion can even be relevant for systematic performanceanalysis of self-assembled synthetic system [40, 41]. Theregimes of trade-off and non-trade-off within the nonequi-librium environment hold the immense possibility to lookinto new control parameters and the concept of newmechanisms depending on the structure and energetics ofthe network, which can be captured via externally con-trolled chemostatted species. Knowledge of the properenergy consumption and storage of the network undercontinuous fuel consumption would also open the pos-sibility and perspective for a more effective experimen-tal design of tRNA Charging [42], translation by Ribo-somes [43], and similar biological events.

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