Far-From-Equilibrium Kinetic Processes

DOI 10.1515/jnet-2015-0059 Ë J. Non-Equilib. Thermodyn. 2015; 40 (4):275–281

Research Article

J. Miguel Rubí and Agustin Pérez-Madrid* Far-from-equilibrium kinetic processes

Abstract: We analyze the kinetics of activated processes that take place under far-from-equilibrium conditions, when the system is subjected to external driving forces or gradients or at high values of anities. We use mesoscopic non-equilibrium thermodynamics to show that when a force is applied, the reaction rate depends on the force. In the case of a chemical reaction at high anity values, the reaction rate is no longer constant but depends on anity, which implies that the law of mass action is no longer valid. This result is in good agreement with the kinetic theory of reacting gases, which uses a Chapman–Enskog expansion of the probability distribution.

Keywords: Far-from-equilibrium, kinetic processes, mesoscopic non-equilibrium thermodynamics

ËË J. Miguel Rubí: Departament de Física Fonamental, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain, e-mail: [email protected] *Corresponding author: Agustin Pérez-Madrid: Departament de Física Fonamental, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain, e-mail: [email protected]

1 Introduction

Processes that need a minimum amount of energy to proceed are usually referred to as activated processes, which in many cases take place under the inuence of external driving forces. The presence of these forces modies the free energy barrier that the system has to surmount, which leads to changes in activation rates. Many examples can be found in physicochemical and biological systems. Mechanical forces may provide the energy that reactants need to transform into products [1, 2] and can also induce chemical changes in polymeric materials [3]. Forces may induce activation of covalent bonds [4]. They can also aect the kinetics of single-molecule reactions [5]. The amount of tension applied at the ends of an RNA molecule progressively breaks the bonds, giving rise to new congurations of the molecule [6]. The kinetics of nucleation processes is also aected by the presence of gradients imposed to the metastable phase [7, 8], which results in changes in nucleation rate. Activation kinetics is governed by the law of mass action, i.e. the rate of a chemical reaction is proportional to the product of the masses of the reactants. This law enables one to formulate a set of coupled dif- ferential equations for all components whose solution leads to the evolution in time of the concentrations. Although the law can be applied systematically to chemical reactions, biochemical cycles, and genetic net- works, we will show in this article that its validity is limited to suciently small values of the anities [9]. In this article, we present a general formalism to treat activated processes under far-from-equilibrium conditions under high thermodynamic or applied forces. In our approach, at short time scales, the system is in mesoscale local equilibrium [10, 11] and undergoes a diusion process through a potential barrier [12, 13]. This irreversible process entails dissipation due to changes in the conformation of the system, which can be measured through the entropy production computed in the space of the reaction coordinate. From the probability current, one obtains the law of mass action [14], for which the current is proportional to the dierence between the fugacities at the initial and nal states of the process. The presence of a constant driving force is incorporated in the chemical potential through the work made by this force. The eect of this force is to tilt the energy landscape, and thus, detailed balance principle is preserved. In activated processes taking place at higher anity values, the diusion coecient in the space of the reaction coordinate is not a constant. This fact leads to corrections to the law of mass action.

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The article is structured as follows. In Section 2, we present the mesoscopic non-equilibrium thermodynamics approach to activated process, showing how the law of mass action can be derived from the entropy production of the system. Section 3 is devoted to the study of the eect of an applied force on the activation kinetics. In Section 4, we show how the law of mass action is not valid at high anity values. Finally, in Section 5, we present our main conclusions.

2 Activation kinetics from mesoscopic non-equilibrium thermodynamics

Processes such as cluster formation [15], protein ligand binding [16], self-assembly [17], adsorption of a single molecule on a substrate [18], and chemical reactions [19] need minimum energy to proceed. They can be modeled by a particle crossing a free energy barrier that separates two well-dierentiated states located at the minima at each side of the barrier [20, 21] and are generically referred to as activated processes [22]. Their intrinsic non-linear nature renders them untreatable by linear non-equilibrium thermodynamics [23]. Non-equilibrium thermodynamics oers a partial description of the process performed in terms of the initial and nal positions, ignoring the transient states. This results in a linear behavior of the current in terms of the anity, which only agrees with the law of mass action for small anity values [24]. If we consider the process at shorter time scales, the state of the system, instead of jumping suddenly from the initial to the nal state, progressively transforms by passing through successive molecular congurations. These congurations can be parametrized by a reaction coordinate . At these time scales, one may assume that the system undergoes a diusion process through a potential barrier separating the initial from the nal states. At observational time scales, the system is mostly found atã the minima of potentials and . In the quasi- 1 2 stationary limit, when the energy barrier is much higher than the thermal energy and intra-well relaxation has already taken place, the probability distribution is strongly peaked at these valuesã andã almost zero everywhere else. Under these conditions, the Fokker–Planck description leads to a kinetic equation in which the net reaction rate satises the mass action law. P(ã, t) Using Kramers’ model, an activated process in which a substance or a state A transforms into another B is assimilated to a diusion process along a reaction coordinate. The corresponding entropy production written in terms of the ux-force pair is then given by

(1)

1 àì where the chemical potential is given by ò = − J , T àã B with as the Boltzmann constant and as the reaction potential. The current obtained from this expres- B sion can be rewritten in terms of the local fugacityì = k denedT ln P + alongΦ, the reaction coordinate B as k Φ(ã) z(ã) ≡ exp ì(ã)/k T B which can be expressed as 1 àz J = −k L , z àã where the diusion coecient is related to the Onsager coecientàz by means of the equalities J = −D , àã −Φ/kBT (2) B B We now assume that is constant and is equal to L andL integrate from to to obtain the non-linear kinetic D = k 0= k e . law for the average current [14]: z P

D 2 D 1 2 ì2/kBT ì1/kBT 0 2 1 0 1 J ≡ X Jdã = −D (z − z ) = −D (e − e ).

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This equation can also be expressed as A/kBT (3) 0 where and is the anity. This equation corresponds to the law of mass action 0 0 1 B 1 2 J = J (1 − e ), usually found in dierent activated processes [25–27]. TheJ scheme= D exp presented(ì /k T) reproducesA = ì − theì results of the rate theory [14, 20]. Equation (3) can be written in terms of the forward and backward reaction rates, and , as + −

k− 2 k+ 1 where the reaction rates are given in terms of theJ = reactionk n − k potential:n ,

Φ2/kBT Φ1/kBT + 0 − 0 e e Here, for are the populationsk = D of the, minima, k = D with . 2 Φ/kBT . á á Δ Δ 1 In equilibrium, the current vanishes and one obtains detailed balance condition or, equiva- 1,eq 2,eq lently,n = P /Δ á = 1, 2 Δ = ∫ e dã

+ 2,eq ΔΦ/kBT z = z − 1,eq n with , showing that the forwardk and backward reaction constants cannot be chosen indepen- 2 1 = = e , dently. k n WeΔΦ can= Φ also− Φ analyze the case in which the transformation is coupled to another process , in which the concentration of the components can be controlled. The current is in this case given by A ྲ¯ B C ྲ¯ D 0 2 1 0 B D A C where the initial and nal fugacitiesJ are= −D (z − z ) = −D (z z − z z ),

(ìA+ìC)/kBT (ìB+ìD)/kBT 1 A C 2 B D

When both reactions are in equilibrium,z = e the detailed= z z , balance z = e condition= imposesz z .

+ A,eq C,eq − B,eq D,eq k n n Changes in the populations of states C and D drive= the system. away from equilibrium to a state in which k n n detailed balance is not fullled. The previous scheme can be generalized to the case of open and closed triangular reactions [28]. Fluctu- ations in the population densities at both wells can be analyzed by means of uctuating hydrodynamics [14]. The coarse-graining of the description leads to violations of the uctuation-dissipation theorem [29, 30].

3 Activation kinetics in the presence of a force

To show how the activation kinetics changes when an external driving force is applied, let us consider the case of a single molecule in a solvent that undergoes transformations from an initial state to a nal state. Examples are the stretching of DNA or RNA molecules or a protein suspended in a liquid. In the rst case, a constant force is applied, whereas in the second, a ligand binds to the protein. In addition to the position of its center of mass , the macromolecule is characterized by an additional uctuating variable , which might represent, for instance, its size or its orientation or, in general, a parameter accounting for the conformation of the molecule. Forx small values of the force, when the macromolecule is close to equilibrium,ã local equilibrium in the -space holds in such a way that we can formulate the Gibbs equation expressed in dierential form x (4)

Tds(x) = −ì(x)dñ(x) − FdΓ(x), Brought to you by | Universitat de Barcelona Authenticated Download Date | 3/31/17 6:50 PM 278 Ë J. M. Rubí and A. Pérez-Madrid, Far-from-equilibrium kinetic processes where is the entropy, is the chemical potential, is the applied force, and (a displacement) is the average value of the variable dened as s(x) ì(x) F Γ ã

Non-equilibrium thermodynamics oers a descriptionΓ(x) = X ãP(x, in terms ã)dã. of mean values that correspond to the mo- ments of the probability distribution. From (4), one would obtain the entropy production, and from it, Fick’s law, in the presence of the driving force inserted in the mass conservation law, gives rise to the diusion equation. To describe the dynamics of the uctuating quantities, we will use the denition of the chemical potential in the -space,

(x, ã) and the expression for the entropy, ì(x)äñ(x) = X ì(x,̃ ã)äP(x, ã)dã,

One then arrives from the Gibbs equation inäs(x) the -space= X äs(x,̃ (4) ã)dã. to the Gibbs equation in -space:

x (x, ã)

The chemical potential is according to theäs(x, previous̃ ã) = − equationì(x,̃ ã)äP(x, given ã). by

B where the potential containsì(x, thẽ ã) eect= k ofT ln theP(x, applied ã) + Φ,p force through the work exerted to modify the conformation up to the value . Following the steps indicated in Section 2, we can arrive at the expression of the current Φp = Φ + Fã ã A/k̃ BT 0 where and the anity is . The current can also be written as 0 1 B J = J2̃ (1 −1 e ),

J̃ = D exp(ì̃ /k T) Ã = ì̃ − ì2̃ + 1 where the reaction rates are given in terms of theJ = reactionk̃ n − k̃ potentialn , ,

p p Φ2/kBT Φ1/kΦpBT + 0 − 0 e e The ratio between both rates is now givenk̃ = D by , k̃ = D . Δ̃ Δ̃

Φp2/kBT + ΔΦ/kp BT Φp1/kBT − k̃ e which shows that detailed balance is preserved.= = e , k̃ e

4 Activation kinetics beyond the law of mass action

In Section 2, we have shown that the law of mass action follows when the diusion coecient in the diusion process along a reaction coordinate is a constant. According to the identication of the diusion coecient with the Onsager coecient of the diusion process given in (2), this assumption means that the coecient is linear in the probability density, a property that holds when the system is not too far from equilibrium. In this section, we will study how activation kinetics proceeds when the system is far from equilibrium, at high values of the anity. Our analysis will be performed for the case of reactive gases.

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Application of an external force drives equilibrium systems to non-equilibrium states. The evolution is in fact the result of the competition between two factors: the applied force and the countless collisions between particles that tend to cancel the increase of momentum of the particles due to the force. The solution of the Boltzmann equation for a reacting gas through the Chapmann–Enskog expansion enables one to describe the transition toward non-equilibrium states [9]. The expansion establishes that the zeroth-order term corresponds to the Maxwellian velocity distribution, whereas the rst-order term is proportional to the anity. For high anity values, the law of mass action is no longer valid. To obtain the reaction rates at higher values of the anity, we rst expand the Onsager coecient in powers of the probability density. The diusion coecient along the reaction coordinate dened in (2) is then given by

B −Φ/kBT B −Φ/kBT 0 1 This expression is equivalent to an expansionk L in the fugacities:k D = e = (L + L P)e . P P −Φ/kBT 0 1 0 1 where −Φ/kBT and −Φ/kBT and the fact that Φ/kT has been used. The diusion 0 B 0 1 DB =1 D + D P = D + D e z, current is then given by

D = k L e /P D = k L e /P −Φ/kBT z = Pe 0 1 àz Following the procedure indicated in SectionJ(ã, t) = 2, − in(D the+ quasi-stationaryD e z) . in which the current does not depend on the reaction coordinate, one can integrate in the coordinate to obtainàã

2 2 0 2 1 1 2 1 1 1 or, equivalently, J(t) = −D (z − z ) − D (z − z ) Δ 2 e 2 1 where the eective diusion coecient depends on the driving force and is given by e J(t) = −D (z − z ), 2 1

D 1 (z − z ) e 0 1 2 1 0 1 D 1 As in the rate theory, the current alsoD can= D be expressed1 + in2z terms+ (z of forward− z ). and backward currents in the form Δ D 2

+ 2 − 1 where the forward and backward reaction ratesJ(t) are= k giveñ n − byk̃ n ,

1 Φ2/kBT Φ2/kBT + 0 1 2 1 e 0 2 D 1 k̃ = D 1 + 1 z + (z − z )eΦ1/kBT = D eΦ1/kBT, − 0 Δ D 1 4 2 1 e 0 2 D 1 We can then see that at higherk̃ = D values1 + of thez chemical+ (z − drivingz )e force,= theD reactione . rates depend on anity Δ D 4 through the driving force ì1/kBT A/kBT , 2 1

ì1/kBT (z − z ) = −e (1 − e1 ) A/kBT e 0 0 2 D e 1 and therefore, the law of massD action= D ceases1 + to be valid. Therefore,1 − 1 − e the detailed, balance principle is not sat- Δ D 4 ised. The previous equations generalize the rate theory to the case of high anities. Our analysis based on the entropy production in the reaction coordinate space (1) is consistent with the kinetic theory of gases, which shows that the entropy production can still be expressed in terms of ux-force pairs up to rst order in the Chapmann–Enskog expansion [9]. This property is not guaranteed for higher values of the driving force when one has to consider higher-order terms in the expansion.

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5 Conclusions

In this article, we have analyzed activated processes under far-from-equilibrium conditions. The rst case treated is activation in the presence of an applied constant force. We have shown that the symmetry breaking inherent to the presence of the force implies that when xing, say, the forward reaction rate, the backward one can be controlled upon variations of the force. However, since the eect of this symmetry breaking reduces to tilting the potential, the detailed balance principle is preserved. The second situation analyzed is that of high anities. We have shown that the law of mass action is obtained when the diusion coecient in the diusion process along the reaction coordinate is a constant. When it depends on the dierence between the fugacities of the initial and nal states, the reaction rates depend on the anities and the law of mass action is no longer valid. This result obtained from mesoscopic non-equilibrium thermodynamics is in good agreement with the same result obtained from the kinetic theory of reacting gases. The formalism presented provides a general scheme to analyze far-from-equilibrium-activated processes. It oers applications to a wide variety of situations including chemical and biochemical reactions, nucleation, and self-assembly processes.

References

[1] C. R. Hickenboth, J. S. Moore, S. R. White, N. R. Sottos, J. Baudry and S. R. Wilson, Biasing reaction pathways with mechanical force, Nature 446 (2007), 423–427. [2] B. M. Rosen and V. Percec, A reaction to stress, Nature 446 (2007), 381–382. [3] M. M. Caruso, D. A. Davis, Q. Shen, S. A. Odom, N. R. Sottos, S. R. White and J. S. Moore, Mechanically-induced chemical changes in polymeric materials, Chem. Rev. 109 (2009), 5755–5798. [4] E. M. Lupton, C. Bräuchle and I. Frank, Understanding mechanically induced chemical reactions, in: NIC Symposium 2006, NIC Ser. 32, John von Neumann Institute for Computing, Jülich (2006), 57–64. [5] I. Tinoco Jr. and C. Bustamante, The eect of force on thermodynamics and kinetics of single molecule reactions, Biophys. Chem. 101/102 (2002), 513–533. [6] J. M. Rubí, D. Bedeaux and S. Kjelstrup, Unifying thermodynamic and kinetic descriptions of single-molecule processes: RNA unfolding under tension, J. Phys. Chem. B 111 (2007), 9598–9602. [7] D. Reguera and J. M. Rubí, Homogeneous nucleation in inhomogeneous media. I. Nucleation in a temperature gradient, J. Chem. Phys. 119 (2003), 9877–9887. [8] D. Reguera and J. M. Rubí, Homogeneous nucleation in inhomogeneous media. II. Nucleation in a shear flow, J. Chem. Phys. 119 (2003), 9888–9893. [9] J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys. 35 (1961), 19–28. [10] A. Pérez-Madrid, J. M. Rubí and P. Mazur, Brownian motion in the presence of a temperature gradient, Physica A 212 (1994), 212–231. [11] J. M. G. Vilar and J. M. Rubí, Thermodynamics “beyond” local equilibrium, Proc. Nat. Acad. Sci. 98 (2001), 11081–11084. [12] D. Reguera, J. M. Rubí and J. M. G. Vilar, The mesoscopic dynamics of thermodynamic systems, J. Phys. Chem. B 109 (2005), 21502–21515. [13] J. M. Rubí, The long arm of the second law, Sci. Am. 299 (2008), 62–67. [14] I. Pagonabarraga, A. Pérez-Madrid and J. M. Rubí, Fluctuating hydrodynamics approach to chemical reactions, Physica A 237 (1997), 205–219. [15] S. Pollack, D. Cameron, M. Rokni, W. Hill and J. H. Parks, Charge-exchange and cluster formation in an rf Paul trap: Interac- + Chem. Phys. Lett. 256 tion of alkali atoms with C60, (1996), 101–108. [16] J. A. Winkles, W. H. Phillips and R. M. Grainger, Drosophila ribosomal RNA stability increases during slow growth conditions, J. Bio. Chem. 260 (1985), 7716–7120. [17] M. F. Hagan, O. M. Elrad and R. L. Jack, Mechanisms of kinetic trapping in self-assembly and phase transformation, J. Chem. Phys. 135 (2011), 1–13. [18] Y. Liu and L. Shen, From Langmuir kinetics to rst- and second-order rate equations for adsorption, Langmuir 24 (2008), 11625–11630. [19] R. S. Berry, S. A. Rice and J. Ross, Physical Chemistry, 2nd ed., Oxford University Press, Oxford, 2000. [20] H. Kramers, Brownian motion in a eld of force and the diusion model of chemical reactions, Physica 7 (1940), 284–304.

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[21] S. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes: The Kinetics of Chemical Reactions, Viscosity, Diu- sion and Electrochemical Phenomena, McGraw-Hill, New York, 1941. [22] P. Hänggi, P. Talkner and M. Borkevec, Reaction-rate theory: Fifty years after Kramers, Rev. Mod. Phys. 62 (1990), 251–341. [23] T. L. Hill, Free Energy Transduction and Biochemical Cycle Kinetics, Dover, New York, 1989. [24] S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover, New York, 1984. [25] S. Kjelstrup, J. M. Rubí and D. Bedeaux, Energy dissipation in slipping biological pumps, Phys. Chem. Chem. Phys. 7 (2005), 4009–4018. [26] G. Gomila and J. M. Rubí, Relation for the nonequilibrium population of the interface states: Eects on the bias depen- dence of the ideality factor, J. Appl. Phys. 81 (1997), 2674–2681. [27] J. M. Rubí and S. Kjelstrup, Mesoscopic nonequilibrium thermodynamics gives the same thermodynamic basis to Butler– Volmer and nernst equations, J. Phys. Chem. B 107 (2003), 13471–13477. [28] J. M. Rubí, D. Bedeaux, S. Kjelstrup and I. Pagonabarraga, Chemical cycle kinetics: Removing the limitation of linearity of a non-equilibrium thermodynamic description, Int. J. Thermophys. 34 (2013), 1214–1228. [29] A. Pérez-Madrid, D. Reguera and J. M. Rubí, Origin of the violation of the fluctuation-dissipation theorem in systems with activated dynamics, Physica A 329 (2003), 357–364. [30] J. W. Dufty and J. M. Rubí, Generalized Onsager symmetry, Phys. Rev. A 36 (1987), 222–225.

Received September 12, 2015; revised September 28, 2015; accepted October 2, 2015.

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