Extracting chemical energy by growing disorder: Efficiency at maximum power
Massimiliano Esposito, Katja Lindenberg, Christian Van den Broeck
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Extracting chemical energy by growing disorder: Efficiency at maximum power
Massimiliano Esposito ∗ and Katja Lindenberg Department of Chemistry and Biochemistry and BioCircuits Institute,University of California, San Diego, La Jolla, CA 92093-0340, USA
Christian Van den Broeck
Hasselt University, B-3590 Diepenbeek, Belgium
We consider the efficiency of chemical energy extraction from the environment by the growthof a copolymer made of two constituent units in the entropy-driven regime. We show that thethermodynamic nonlinearity associated with the information processing aspect is responsible fora branching of the system properties such as power, speed of growth, entropy production, andefficiency, with varying affinity. The standard linear thermodynamics argument which predicts anefficiency of 1 / PACS numbers: 05.70.Ln,05.70.-a,05.40.-a
I. INTRODUCTION
Carnot efficiency is one of the cornerstones of thermo-dynamics since it leads to the definition of entropy andthe Second Law of thermodynamics. It expresses a fun-damental limitation on how much work can be extractedfrom a heat flow. A less studied but arguably more rele-vant question for many isothermal chemical and biologi-cal processes is how much chemical energy a system canextract from its environment by increasing the system’sconfigurational entropy. Thermodynamics does, in fact,also prescribe a limit, even though at first sight it ap-pears to be almost trivial: the energy extracted by suchan isothermal transfer can be carried out with 100% effi-ciency. However, there is a crucial additional condition,namely, that this efficiency can only be reached – justas in the case of Carnot efficiency – by a reversible, in-finitely slow process. Hence 100% efficiency is achievedfor a process with zero power output. The question ofefficiencies at finite power should thus be addressed.In the context of thermal machines, a straightfor-ward analysis based on linear irreversible thermodynam-ics teaches us that, as one moves away from the reversibleregime, the power goes through a unique maximum, andthat the efficiency at this maximum is, at most, 50%[1, 2]. The same argument can easily be extended tothe transformation between different forms of chemicalenergy. However, the above prediction may not applyfor several reasons. First, the point of maximum powerdoes not necessarily lie in the linear regime. Second,thermodynamic nonlinear effects can give rise to bifur-cated branches. Finally, the above-mentioned efficiency ∗ Also at Center for Nonlinear Phenomena and Complex Systems,Universit´e Libre de Bruxelles, Code Postal 231, Campus Plaine,B-1050 Brussels, Belgium. is attained upon maximization with respect to the ther-modynamic force associated with the power generatingflux. While this is a natural set-up in many problems, itmay not always correspond to the relevant scenario.In this paper, we investigate the efficiency of a chem-ical entropy-driven process of capital importance in bio-physics, namely, copolymer synthesis [3, 4, 5, 6], see also[7, 8]. As exemplified by the copolymer DNA, guardianof genetic information, such processes are essential forbiological information processing. We will show thatthe above-mentioned complications are present in thisgeneric model. In particular, the thermodynamic nonlin-earity associated with the information processing aspectis responsible for a branching of the systems propertiessuch as power, speed of growth, efficiency and entropyproduction, as one varies the affinity. Furthermore, theregime of maximum power is located either outside of thelinear regime or on the separate bifurcated branch. Fi-nally, it turns out that the thermodynamic force is nota natural control variable in the present model. Whileour (exact) analysis is carried out for the simplest pos-sible model, namely, copolymer synthesis with two con-stituent building blocks, our findings suggest that chem-ical information processing usually operates in the far-from-equilibrium regime, with unique features due to theentropic contribution.In Sec. II we present the basic thermodynamic formu-las that define our system. In Sec. III we present thedetailed kinetic description of our model, whose resultsare discussed in detail in Sec. IV. In particular, it is herethat we exhibit the correct and unexpected results forthe efficiency at maximum power, results that arise en-tirely from the nonlinear nature of the problem. A briefrecapitulation is presented in Sec. V.
II. THERMODYNAMICS
We begin with some well-known relations for isother-mal systems. Consider a spontaneous chemical processinvolving particles of different types labeled by j , withcorresponding particle number N j and chemical potential µ j . The system is in thermal and mechanical equilibriumat temperature T and pressure P . The total Gibbs freeenergy G = U + P V − T S = H − T S = X i µ i N i (1)evolves toward a minimum value, so that dG ≤
0. Alter-natively, to characterize the evolution of the isothermalsystem we write dS = d i S + d e ST d i S = − X j µ j d i N j T d e S = dH − X j µ j d e N j . (2)We have separated the total entropy change into two con-tributions. The first one, d i S , is the always-positive partof the entropy change, called the internal entropy produc-tion. The other is the contribution to the entropy changedue to exchange processes between the system and its en-vironment, and can be positive or negative. Associatedwith these contributions, we have written the change inhe number of particles of type j as dN j = d e N j + d i N j , (3)where the first contribution is due to exchange of parti-cles with the environment, and the second is the internalchange caused by the chemical reaction. We take the sys-tem to be closed, i.e., it exchanges only energy but notparticles with the environment, so that d e N = 0. Thesedefinitions lead to consistency between the statementsthat the system evolves toward a minimum in the Gibbsfree energy and that the internal entropy production ofthis chemical system has to be positive [9], that is, d i S = dS − dHT = − P j µ j dN j T = − dGT ≥ . (4)Obviously, for a reversible transformation with zero in-ternal entropy production d i S = 0.We now turn to the simplest scenario of copolymersynthesis. The system consists of a bulk-phase containingtwo types of monomer units, 1 and 2, which can attachor detach at the endpoint of a single linear copolymer.We identify the four states 1 f , 1 c , 2 f and 2 c . Here jf represent free bulk monomers and jc represent monomersattached to the copolymer. Since the number of eachtype of monomer is conserved, one has dN f = − dN c and dN f = − dN c . The entropy production (4) can thus be written in the familiar bilinear form˙ S i ≡ d i Sdt = ( µ f − µ c ) dN c dt + ( µ f − µ c ) dN c dt = A J + A J , (5)with the affinities A j = ( µ jf − µ jc ) and the conjugatefluxes J j = dN jc /dt .In view of the relation d i S = dS − dH/T , we rewritethe entropy production as˙ S i = (cid:0) s − h T (cid:1) J + (cid:0) s − h T (cid:1) J + D ( J + J ) . (6)Here h j is the change of enthalpy per monomer upontransfer from the bulk to the copolymer. The crucialpoint, which has been discussed in detail in the litera-ture [3, 4, 5, 6], is to realize that the average change ofentropy upon transfer of a monomer from the bulk tothe copolymer contains two contributions. One is themonomer entropy, s j , due to the change in the monomerdegrees of freedom and in the monomer internal structurebetween the free monomer in solution and the monomerinside the copolymer. The other is the configurationalentropy denoted by D , due to the change in the informa-tion stored in the polymer sequence that occurrs when amonomer is added to the copolymer. It is given by theShannon entropy D = − lim l →∞ l X ω P ω ln P ω , (7)where l is the copolymer length in monomer units and P ω is the probability of a copolymer with monomer sequence ω . In the absence of correlations, the Shannon entropyis expressed solely in terms of the monomer abundanceprobabilities p = p and p = 1 − p , D = − p ln p − (1 − p ) ln(1 − p ) . (8)For simplicity, we further assume that monomer entropyand enthalpy changes upon transfer of a monomer fromthe bulk to the copolymer and vice versa have the samevalue for both monomers, that is, ǫ ≡ h T − s = h T − s . (9)We henceforth call T ǫ the monomer “free enthalpy.” In-troducing the net speed of growth of the copolymer, v = J + J , the entropy production can finally be writtenas ˙ S i = Av ≥ , (10)where the total affinity is given by A = D − ǫ. (11)The expression (10) for the entropy production in thesteady state regime of the growing copolymer has beenderived in Refs. [3, 4, 5, 6]. It is interesting to realize thatthe affinity is not an obvious control parameter due to itsdependence on D which is in turn a nontrivial function of ǫ . Only ǫ can be easily controlled externally by changingthe concentration of the monomers in solution.The power at which ǫ , the free enthalpy dividedby temperature, is extracted from the surroundings bycopolymer growth is given by P = ǫv = ( D − A ) v. (12)The efficiency η of the process is defined as the ratio ofthis power over the cost Dv of the entropy growth perunit time, η = ǫvDv = D − AD . (13)In the reversible limit with
A, v →
0, the efficiency ofthe process becomes optimal, η = 1, but the extractedpower goes to zero. The standard prediction from linearthermodynamics that arrives at an efficiency of 50% atmaximum power is obtained upon expanding the velocityin terms of the affinity, v = LA , with L the linear re-sponse coefficient. Within this approximation the powerbecomes P ≈ L ( D − A ) A . Note that this power attainsits maximum for A = D/ η = 1 /
2, if we assume that D is kept constant.However, below we will investigate the more natural op-timization with respect of ǫ , since this is the natural andeasily controllable variable related to the free enthalpyflux. Whatever control variable is used, we will see inSec. IV that the true maximum is beyond the reach ofthis linear expansion (and even of a nonlinear continua-tion of this expansion). III. KINETICS
We now turn to the detailed kinetic description of thecopolymerization process, which will allow us to identifythe expressions for v and p in the context of a full nonlin-ear analysis. Let us call k + j and k − j the rates of insertionand removal, respectively, of monomer j = 1 ,
2. Becausethe free enthalpy of the monomers has been assumed tobe the same, the ratios of the reaction rates are given by k +1 k − = k +2 k − = e − ǫ . (14)The fraction p of monomers of type 1 present in thecopolymer in the regime of steady growth can be deter-mined by the following self-consistency argument. Theratio p/ (1 − p ) of the number of 1 versus 2 monomers inan ensemble of copolymers has to be equal to the ratioof their net rates of attachment. For monomer 1, thisnet rate is the pure rate of attachment, k +1 , minus therate of detachment, which is − k − p . The factor p arisesfrom the fact that detachment is only possible when themonomer at the tip of the copolymer is of type 1, and this occurs with probability p . The net rate of attach-ment for 2 is similarly given by k +2 − k − (1 − p ). Wethus conclude that k +1 − k − pk +2 − k − (1 − p ) = p − p . (15)The solution of the resulting quadratic equation for p reads p = a − p a − k − − k − ) k +1 k − − k − ) , (16)where a = k +1 + k +2 + k − − k − . By a similar argumentwe find that the speed of growth of the copolymer, givenby the rate of attachment k +1 + k +2 minus the rate ofdetachment k − p + k − (1 − p ), is given by v = k +1 − k − p + k +2 − k − (1 − p ) . (17)We note from Eqs. (8), (10), (11), (16) and (17) thatequilibrium, v = 0 and A = 0, occurs at ǫ = ln 2 with p = 1 / D = ln 2. For smaller (larger) values of ǫ , A >
A <
0) and the copolymer is synthesized (de-graded), i.e., v > v <
A > v >
0, but with ǫ > ≤ ǫ ≤ ln 2. Monomers are pumped uphill against thefree enthalpy barrier ǫ ≥ D to the affinity. The power P (en-thalpy per unit time) extracted from the copolymeriza-tion dynamics is positive in this entropy driven regime, cf.Eq. (12), with corresponding efficiency given in Eq. (13). IV. EFFICIENCY AT MAXIMUM POWER
To study the main question of interest, namely, theregime of maximum power and its corresponding effi-ciency, we choose convenient variables. We note thatthe model is described by four kinetic constants, but thelatter are not independent since they obey the relationEq. (14). Furthermore, one of them can be set equal to 1by an appropriate choice of the time unit, e.g., k +1 = 1.As the remaining two degrees of freedom, we choose ǫ and k +2 . We then have explicit functional expressionsfor all the other quantities k − = e ǫ , k − = e ǫ k +2 , p and v , cf. Eqs. (16) and (17), and hence also D , A , P ,˙ S i , and η , see Eqs. (11), (12), (13), (8) and (10). Otherrelations between, for example, P and A , can then be ob-tained by parametric elimination. The quantities P , v , η , A , and ˙ S i can easily be calculated numerically. Theresults are summarized in Figs. 1, 2, 3, and 4. We nextturn to a discussion of these figures, supplemented withcorresponding analytic calculations.The information contained in Fig.1 is detailed in thecaption. The most striking feature in this figure is the P k + = k + = k + = k + = k + = k + ®¥ k + ® v k + = k + = k + = k + = k + = k + ®¥ k + ® A Η k + = k + = k + = k + = k + = k + ®¥ k + ® FIG. 1: (Color online) The full thick curves represent theparametric dependence on 0 ≤ ǫ ≤ ln 2 of the power P , thecopolymerization speed v , and the efficiency η on the ordinateand the affinity A as abscissa. The point ǫ = ln 2 is locatedat the origin of the axes for P and v and at η = 1 and A =0 for η . The small dots along the curves are separated by∆ ǫ = (ln 2) /
14 to indicate how fast ǫ changes along the curves.Different thick curves correspond to different choices of k +2 ,with k +1 ≤ k +2 ≤ ∞ . Without loss of generality we set k +1 =1 (time rescaling). The thin dashed curves intersect the thickcurves where the value of ǫ corresponds to maximum powerwith respect to ǫ . The curves in the inset in the P plot andin the upper inset of the v plot correspond to k +2 = 65, 85,105, 121, 160, and 200. The dashed curves in the lower insetof this plot represent the linear response predictions v = LA for k +2 = 1 , , P Η k + = Ε= k + = Ε= k + = Ε= k + = Ε= k + = Ε= k + ®¥Ε® k + ® Ε® FIG. 2: (Color online) Efficiency η and power P correspondingto the maximum power denoted simply by ǫ in the figure fordifferent values of k +2 . We have set k +1 = 1. - - - - Ε dAd Ε k + ®¥ k + = k + = k + ® Ε A k + ®¥ k + ® FIG. 3: (Color online) Derivative of the affinity with respectto ǫ and affinity (in the inset) as a function of ǫ . The differentcurves correspond to k +2 = 0 ,
11, 65, 85, 105, 121, 160, 200,1001, and ∞ and k +1 = 1. existence of two different branches for the power and ve-locity in terms of the affinity. The transition between thetwo branches occurs when d ( P , v ) dA = d ( P , v ) dǫ (cid:18) dAdǫ (cid:19) − (18)diverges. ( P , v ) indicates P or v . Since ∂ ( P , v ) /∂ǫ isan analytic function of ǫ , the new branch appears when ∂A/∂ǫ touches zero. As long as the latter quantity re-mains positive, which is the case for k +2 smaller thana certain critical value, cf. Fig. 3, the power and ve-locity can be seen as a true function of A . Branch-ing takes place at the critical point, characterized by ∂A/∂ǫ = ∂ A/∂ǫ = 0, resulting in k +2 ≈ .
33 and ǫ = 0 . k +2 larger thanthis critical value, power is no longer a proper functionof A , as two branches appear, with two values of ( P , v )for a given value of A . While along the linear branch andits continuation the affinity decreases with ǫ , the affinity increases with ǫ on the new lower branch, cf. the insetin Fig. 3. This remarkable result implies that we canapproach low values of affinities via a nonlinear branchwhich is distinct from the branch predicted by linear re-sponse theory and its continuation. We note that the en-tropy production itself becomes a bi-valued function interms of the affinity, as can be seen in Fig. 4. Naively, onewould expect that entropy production and affinity bothprovide consistent measures for the distance from equi-librium. This is clearly not the case in the present model,where the entropy production is a decreasing function ofthe affinity on the upper nonlinear branch. In particular,for very large values of k +2 one finds that the entropyproduction becomes very large while the affinity goes tozero. We conclude that the affinity is not a reliable mea-sure for the distance from equilibrium. A S i k + = k + = k + = k + = k + = k + ®¥ k + ® FIG. 4: (Color online) Same type of plot as Fig. 1 but forentropy production.
To explore the region close to equilibrium and, in par-ticular, the linear response regime, we write ǫ = ln 2 − δǫ and expand in powers of δǫ . From Eq. (8) with Eqs. (16)and (14) we find D = ln 2 − α δǫ + O ( δǫ ) , (19)where α = ( k +1 − k +2 ) k +1 + k +2 ) . (20)For the affinity, we find from Eq. (11) that A = δǫ − αδǫ + O ( δǫ ) . (21) The efficiency thus becomes η = 1 − δǫ ln 2 + α ln 2 δǫ + O ( δǫ )= 1 − A ln 2 + O ( δǫ ) . (22)This linear dependence of the efficiency on the affinityclose to equilibrium is clearly identified in the upperleft region of the third affinity plot in Fig. 1, while thecorresponding behavior of the affinity in terms of ǫ , cf.Eq. (21), is observed in the lower left region of the insetof Fig. 3. In this regime close to equilibrium we find thestandard linear response relations v = LA + O ( δǫ ) (23) P = ln 2 LA + O ( δǫ ) , (24)with the Onsager coefficient given by L =4 k +1 k +2 / ( k +1 + k +2 ), cf. the lower left regions ofthe power and speed plots in Fig. 1. Note also that theOnsager coefficient becomes independent of k +2 in thelimit k +1 ≪ k +2 , where L = 4 k +1 .We have seen that linear response predicts an efficiencyat maximum power of 50%. However, as announced ear-lier, this result is not correct. This is seen in Fig. 1 or inFig. 2, where the affinity is clearly above the value 1 / v plot in Fig. 1, where the linear response curves(dashed lines) become inaccurate at maximum power.Furthermore, we note that the point of maximum powermoves onto the nonlinear branch as k +2 grows, now oc-curring at decreasing values of A . So, even though we areapproaching a regime of low power output with decreas-ing affinity, we do so via the nonlinear branch, where theprediction of linear response theory utterly fails. Themain conclusion is that, while there is indeed a regimeof linear response, it is unable to describe the region ofmaximum power, which always occurs outside the regimeof validity of the linear law.To complete our analysis, we explore in detail the lim-iting cases k +2 → k +1 and k +2 /k +1 → ∞ . For trans-parency, we explicitly retain k +1 instead of setting itequal to unity. In the limit where k +2 → k +1 , we findthat p = 12 , v = k +1 (2 − e ǫ ) , D = ln 2 . (25)This leads to an efficiency η = ǫ/ ln 2 = 1 − A/ ln 2, asobserved in Fig. 1. In this limit, the value of ǫ leadingto maximum power is obtained as the solution of thetranscendental equation 2e − ǫ − ǫ = 1, namely, ǫ ≈ . P ≈ . k +1 and η ≈ . , (26)as seen in Figs. 1 and 2. As an immediate consequence,we also find v ≈ . k +1 and A ≈ . k +2 → ∞ , where p = 1 − e − ǫ , v = k +1 e ǫ (2e − ǫ − − ǫ − ,D = e − ǫ ǫ − (1 − e − ǫ ) ln(1 − e − ǫ ) , (27)the efficiency reads η = − ǫ/ [e − ǫ ǫ + (1 − e − ǫ ) ln(1 − e − ǫ )]. The numerical results of Fig. 1 suggest that max-imum power in this limit occurs for ǫ very close tozero. We therefore expand the velocity around ǫ = 0and find v = − p k +1 k +2 + ( k +1 + k +2 ) ǫ/ O ( ǫ ).Using Eq. (12), we find that maximum power occursat ǫ = − p k +1 k +2 / ( k +1 + k +2 ), resulting in P = p k +1 k +2 / ( k +1 + k +2 ). For k +2 → ∞ the latter becomes P = 1 , (28)as observed in Figs. 1 and 2. Similarly, by expanding η to first order around ǫ = 0 and using the value we foundfor ǫ at maximum power, we find that η → , (29)as observed in Figs. 1 and 2. V. CONCLUSIONS
Using a simple model of copolymerization, we haveshown that free enthalpy can be extracted from the en-vironment in response to the entropic force correspond-ing to the information stored in a growing copolymer sequence. The thermodynamic nonlinearity associatedwith the information processing aspect is responsible fora branching of the dependence on the affinity of systemproperties such as power, speed of growth and efficiency.The nonlinear regime occuring after the branching is par-ticularly surprising since the entropy production keeps in-creasing even as the affinity begins to decrease. We iden-tified a regime of linear response where the efficiency ofthe energy extraction is optimal (equal to 1), but where,as usual, the power output goes to zero. Considering in-stead the efficiency at maximum power, we found thatthe universal prediction of linear response theory (effi-ciency equal to 1 /
2) is inappropriate for this model. Thereason is that the copolymerization generating maximumpower occurs far from equilibrium in a region not accessi-ble to linear response theory. Our results suggest a possi-ble self-powering mechanism for nonequilibrium systemsthat can extract chemical energy from their surroundingsby growing their internal structural information.
Acknowledgments
M. E. is supported by the FNRS Belgium (charg´ede recherches) and by the government of Luxembourg(Bourse de formation recherches). This work was par-tially supported by the National Science Foundation un-der grant No. PHY-0354937. [1] C. Van den Broeck, Phys. Rev. Lett. , 190602, (2005);Adv. Chem. Phys. , 189 (2007).[2] M. Esposito, K. Lindenberg and C. Van den Broeck, PRL , 130602 (2009).[3] C. H. Bennett, BioSystems , 85 (1979).[4] D. Andrieux and P. Gaspard, PNAS , 9516 (2008).[5] C. Jarzynski, PNAS , 9451 (2008).[6] D. Andrieux and P. Gaspard, J. Chem. Phys. , 014901 (2009).[7] M.V. Volkenstein, Molecular Biophysics (Academic Press,New York, 1977).[8] E. Smith, J. Theo. Biol. , 198 (2008).[9] D. Kondepudi and I. Prigogine,