Far-from-equilibrium kinetic processes

Transcription

1 DOI /jnet J. Non-Equilib. Thermodyn. 2015; 40 (4): 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 affinities. 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 affinity values, the reaction rate is no longer constant but depends on affinity, which implies that the law of mass action is no longer valid.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 J. Miguel Rubí: Departament de Física Fonamental, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, Barcelona, Spain, mrubi@ub.edu *Corresponding author: Agustin Pérez-Madrid: Departament de Física Fonamental, Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, Barcelona, Spain, agustiperezmadrid@ub.edu 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 influence of external driving forces. The presence of these forces modifies 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 affect 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 configurations of the molecule [6]. The kinetics of nucleation processes is also affected 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 differential 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 networks, we will show in this article that its validity is limited to sufficiently small values of the affinities [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 diffusion 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 difference between the fugacities at the initial and final 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 effect of this force is to tilt the energy landscape, and thus, detailed balance principle is preserved. In activated processes taking place at higher affinity values, the diffusion coefficient in the space of the reaction coordinate is not a constant. This fact leads to corrections to the law of mass action.

2 276 J. M. Rubí and A. Pérez-Madrid, Far-from-equilibrium kinetic processes 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 effect of an applied force on the activation kinetics. In Section 4, we show how the law of mass action is not valid at high affinity 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-differentiated 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 offers a partial description of the process performed in terms of the initial and final positions, ignoring the transient states. This results in a linear behavior of the current in terms of the affinity, which only agrees with the law of mass action for small affinity 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 final state, progressively transforms by passing through successive molecular configurations. These configurations can be parametrized by a reaction coordinate γ. At these time scales, one may assume that the system undergoes a diffusion process through a potential barrier separating the initial from the final states. At observational time scales, the system is mostly found at the minima of potentials γ 1 and γ 2. In the quasistationary limit, when the energy barrier is much higher than the thermal energy and intra-well relaxation has already taken place, the probability distribution P(γ, t) is strongly peaked at these values and almost zero everywhere else. Under these conditions, the Fokker Planck description leads to a kinetic equation in which the net reaction rate satisfies the mass action law. Using Kramers model, an activated process in which a substance or a state A transforms into another B is assimilated to a diffusion process along a reaction coordinate. The corresponding entropy production written in terms of the flux-force pair is then given by where the chemical potential is given by σ = 1 μ J T γ, (1) μ = k B T ln P + Φ, with k B as the Boltzmann constant and Φ(γ) as the reaction potential. The current obtained from this expression can be rewritten in terms of the local fugacity defined along the reaction coordinate z(γ) exp μ(γ)/k B T as J = k B L 1 z z γ, which can be expressed as J = D z γ, where the diffusion coefficient is related to the Onsager coefficient by means of the equalities L D = k B z = k L B P e Φ/k BT. (2) We now assume that D is constant and is equal to D 0 and integrate from 1 to 2 to obtain the non-linear kinetic law for the average current [14]: 2 J Jdγ = D 0 (z 2 z 1 ) = D 0 (e μ 2/k T B e μ 1/k T B ). 1

3 J. M. Rubí and A. Pérez-Madrid, Far-from-equilibrium kinetic processes 277 This equation can also be expressed as J = J 0 (1 e A/k BT ), (3) where J 0 = D 0 exp(μ 1 /k B T) and A = μ 1 μ 2 is the affinity. This equation corresponds to the law of mass action usually found in different activated processes [25 27]. The scheme presented reproduces the results of the rate theory [14, 20]. Equation (3) can be written in terms of the forward and backward reaction rates, k + and k, as J = k n 2 k + n 1, where the reaction rates are given in terms of the reaction potential: k + = D 0 e Φ 2/k B T Δ, k = D 0 e Φ1/kBT Δ. Here, n α = P α /Δ for α = 1, 2 are the populations of the minima, with Δ = 2 1 eφ/kbt dγ. In equilibrium, the current vanishes and one obtains detailed balance condition z 1,eq = z 2,eq or, equivalently, k + = n 2,eq = e ΔΦ/kBT, k n 1,eq with ΔΦ = Φ 2 Φ 1, showing that the forward and backward reaction constants cannot be chosen independently. We can also analyze the case in which the transformation A B is coupled to another process C D, in which the concentration of the components can be controlled. The current is in this case given by where the initial and final fugacities are J = D 0 (z 2 z 1 ) = D 0 (z B z D z A z C ), z 1 = e (μ A+μ C )/k B T = z A z C, z 2 = e (μ B+μ D )/k B T = z B z D. When both reactions are in equilibrium, the detailed balance condition imposes k + k = n A,eqn C,eq n B,eq n D,eq. Changes in the populations of states C and D drive the system away from equilibrium to a state in which detailed balance is not fulfilled. The previous scheme can be generalized to the case of open and closed triangular reactions [28]. Fluctuations in the population densities at both wells can be analyzed by means of fluctuating hydrodynamics [14]. The coarse-graining of the description leads to violations of the fluctuation-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 final state. Examples are the stretching of DNA or RNA molecules or a protein suspended in a liquid. In the first 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 x, the macromolecule is characterized by an additional fluctuating variable γ, which might represent, for instance, its size or its orientation or, in general, a parameter accounting for the conformation of the molecule. For small values of the force, when the macromolecule is close to equilibrium, local equilibrium in the x-space holds in such a way that we can formulate the Gibbs equation expressed in differential form Tds(x) = μ(x)dρ(x) FdΓ(x), (4)

4 278 J. M. Rubí and A. Pérez-Madrid, Far-from-equilibrium kinetic processes where s(x) is the entropy, μ(x) is the chemical potential, F is the applied force, and Γ (a displacement) is the average value of the γ variable defined as Γ(x) = γp(x, γ)dγ. Non-equilibrium thermodynamics offers a description in terms of mean values that correspond to the moments 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 diffusion equation. To describe the dynamics of the fluctuating quantities, we will use the definition of the chemical potential in the (x, γ)-space, μ(x)δρ(x) = μ(x, γ)δp(x, γ)dγ, and the expression for the entropy, δs(x) = δs(x, γ)dγ. One then arrives from the Gibbs equation in the x-space (4) to the Gibbs equation in (x, γ)-space: δs(x, γ) = μ(x, γ)δp(x, γ). The chemical potential is according to the previous equation given by μ(x, γ) = k B T ln P(x, γ) + Φ, where the potential Φ = Φ + Fγ contains the effect of the applied force through the work exerted to modify the conformation up to the value γ. Following the steps indicated in Section 2, we can arrive at the expression of the current J = J 0 A/k (1 e T B ), where J 0 = D exp( μ 1 /k B T) and the affinity is A = μ 2 μ 1. The current can also be written as J = k n 2 k + n 1, where the reaction rates are given in terms of the reaction potential Φ, The ratio between both rates is now given by which shows that detailed balance is preserved. eφ 2 /k B T eφ k + = D 0 Δ, 1 /k B T k = D 0 Δ. k + = e Φ 2 /k B T k = eδφ/k T B, eφ 1 /k B T 4 Activation kinetics beyond the law of mass action In Section 2, we have shown that the law of mass action follows when the diffusion coefficient in the diffusion process along a reaction coordinate is a constant. According to the identification of the diffusion coefficient with the Onsager coefficient of the diffusion process given in (2), this assumption means that the coefficient 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 affinity. Our analysis will be performed for the case of reactive gases.

5 J. M. Rubí and A. Pérez-Madrid, Far-from-equilibrium kinetic processes 279 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 first-order term is proportional to the affinity. For high affinity values, the law of mass action is no longer valid. To obtain the reaction rates at higher values of the affinity, we first expand the Onsager coefficient in powers of the probability density. The diffusion coefficient along the reaction coordinate defined in (2) is then given by D = k BL P e Φ/k BT = k B P (L 0 + L 1 P)e Φ/kBT. This expression is equivalent to an expansion in the fugacities: D = D 0 + D 1 P = D 0 + D 1 e Φ/k BT z, where D 0 = k B L 0 e Φ/k BT /P and D 1 = k B L 1 e Φ/k BT /P and the fact that z = Pe Φ/kT has been used. The diffusion current is then given by J(γ, t) = (D 0 + D 1 e Φ/k BT z) z γ. Following the procedure indicated in Section 2, in the quasi-stationary in which the current does not depend on the reaction coordinate, one can integrate in the coordinate to obtain or, equivalently, J(t) = D 0 (z 2 z 1 ) 1 Δ D (z2 2 z2 1 ) J(t) = D eff (z 2 z 1 ), where the effective diffusion coefficient D eff depends on the driving force (z 2 z 1 ) and is given by D eff = D 0 {1 + 1 D 1 (2z Δ D (z 2 z 1 ))}. As in the rate theory, the current also can be expressed in terms of forward and backward currents in the form J(t) = k + n 2 k n 1, where the forward and backward reaction rates are given by D 1 k + = D 0 {1 + 2 (z Δ D (z 2 z 1 ))}e Φ 2/k T B = D eff e Φ 2/k T B, D 1 k = D 0 {1 + 2 (z Δ D (z 2 z 1 ))}e Φ 1/k T B = D eff e Φ 1/k T B. We can then see that at higher values of the chemical driving force, the reaction rates depend on affinity through the driving force (z 2 z 1 ) = e μ 1/k B T (1 e A/k BT ), D eff = D 0 {1 + 2 D 1 e μ 1/k B T (1 1 Δ D 0 4 (1 ea/k BT ))}, and therefore, the law of mass action ceases to be valid. Therefore, the detailed balance principle is not satisfied. The previous equations generalize the rate theory to the case of high affinities. 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 flux-force pairs up to first 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.

6 280 J. M. Rubí and A. Pérez-Madrid, Far-from-equilibrium kinetic processes 5 Conclusions In this article, we have analyzed activated processes under far-from-equilibrium conditions. The first 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 fixing, say, the forward reaction rate, the backward one can be controlled upon variations of the force. However, since the effect of this symmetry breaking reduces to tilting the potential, the detailed balance principle is preserved. The second situation analyzed is that of high affinities. We have shown that the law of mass action is obtained when the diffusion coefficient in the diffusion process along the reaction coordinate is a constant. When it depends on the difference between the fugacities of the initial and final states, the reaction rates depend on the affinities 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 offers 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), [2] B. M. Rosen and V. Percec, A reaction to stress, Nature 446 (2007), [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), [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), [5] I. Tinoco Jr. and C. Bustamante, The effect of force on thermodynamics and kinetics of single molecule reactions, Biophys. Chem. 101/102 (2002), [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), [7] D. Reguera and J. M. Rubí, Homogeneous nucleation in inhomogeneous media. I. Nucleation in a temperature gradient, J. Chem. Phys. 119 (2003), [8] D. Reguera and J. M. Rubí, Homogeneous nucleation in inhomogeneous media. II. Nucleation in a shear flow, J. Chem. Phys. 119 (2003), [9] J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys. 35 (1961), [10] A. Pérez-Madrid, J. M. Rubí and P. Mazur, Brownian motion in the presence of a temperature gradient, Physica A 212 (1994), [11] J. M. G. Vilar and J. M. Rubí, Thermodynamics beyond local equilibrium, Proc. Nat. Acad. Sci. 98 (2001), [12] D. Reguera, J. M. Rubí and J. M. G. Vilar, The mesoscopic dynamics of thermodynamic systems, J. Phys. Chem. B 109 (2005), [13] J. M. Rubí, The long arm of the second law, Sci. Am. 299 (2008), [14] I. Pagonabarraga, A. Pérez-Madrid and J. M. Rubí, Fluctuating hydrodynamics approach to chemical reactions, Physica A 237 (1997), [15] S. Pollack, D. Cameron, M. Rokni, W. Hill and J. H. Parks, Charge-exchange and cluster formation in an rf Paul trap: Interaction of alkali atoms with C + 60, Chem. Phys. Lett. 256 (1996), [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), [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), [18] Y. Liu and L. Shen, From Langmuir kinetics to first- and second-order rate equations for adsorption, Langmuir 24 (2008), [19] R. S. Berry, S. A. Rice and J. Ross, Physical Chemistry, 2nd ed., Oxford University Press, Oxford, [20] H. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940),

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