The dynamics of single enzyme reactions: A reconsideration of

THE JOURNAL OF CHEMICAL PHYSICS 129, 075104 共2008兲

The dynamics of single enzyme reactions: A reconsideration of Kramers’ model for colored noise processes Srabanti Chaudhury, Debarati Chatterjee, and Binny J. Cherayila兲 Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore-560012, India

共Received 29 April 2008; accepted 21 July 2008; published online 21 August 2008兲 The utility of an approximate heuristic version of Kramers’ theory of reaction rates that was earlier used 关Chaudhury and Cherayil, J. Chem. Phys. 125, 024904 共2006兲兴 to successfully describe the nonexponential waiting time distributions of the enzyme ␤-galactosidase is reassessed. The original model, based on the Smoluchowski equation, is reformulated in terms of the phase space variables of the reaction coordinate, without neglecting inertial contributions. A new derivation of the Fokker– Planck equation 共FPE兲 that describes the dynamics of this coordinate is presented. This derivation, based on functional methods, provides a more direct alternative to the existing distribution function approach used by Hanggi and Mojtabai 关Phys. Rev. A 26, 1168 共1982兲兴. The time-dependent coefficients in the FPE, when incorporated into the exact expression for the transmission coefficient obtained from a reactive-flux formalism 关Kohen and Tannor, J. Chem. Phys. 103, 6013 共1995兲兴, are found to yield virtually the same results as the earlier heuristic model. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2969767兴 I. INTRODUCTION

In an early and incomplete effort at understanding observations of stochasticity in the reactions of single enzymes,1 we had proposed a simple model of catalytic activity based on the generalized Langevin equation 共GLE兲 and Kramers’ theory of thermally activated barrier crossing.2 The model proved to be surprisingly successful in reproducing nonexponentiality in the measured waiting time distributions of the enzyme ␤-galactosidase.3 Unlike Kramers’ approach, which was formulated in terms of the phase space dynamics of a reaction coordinate evolving under white noise near a harmonic barrier, our own approach employed a reduced description, in which the reaction coordinate evolved in position space alone 共under overdamped conditions兲, its motion governed by power-law correlated random forces that were intended to reflect the multiple time scale dynamics of the surrounding protein.4 In this approach, the calculation of the waiting time distribution was reduced to the calculation of a time-dependent barrier crossing rate from the generalized diffusion equation to which the GLE could be exactly transformed. The rate was calculated as the ratio of the flux of particles over the barrier to the population of particles in the metastable well, exactly as in Kramers’ method, except that these quantities were no longer evaluated in the stationary limit. This meant that the flux and population in our method were time dependent, and their ratio—the barrier crossing rate—was time dependent as well 共and hence not strictly a rate at all.兲 However the use of this rate in the calculation of waiting time distributions led to a good agreement with data from the experiments on ␤-galactosidase by English et al.3 The various approximations in this approach were largely ad hoc, but we did show that in the long-time limit its a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected].

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results agreed qualitatively with the corresponding results of two exact formulations of a non-Markovian generalization of Kramers’ barrier crossing model: one by Hanggi and Mojtabai5 and the other by Kohen and Tannor.6 We also showed,7 in an extension of our approach, that a model based on the subdiffusive dynamics of a particle in a double well potential led to an expression for the waiting time distribution that reproduced its earlier scaling structure, but this second calculation also used approximations 共including a shorttime limit兲 whose validity was not entirely clear. The aim of the present note is to show, first of all, how the functional methods of our earlier calculations can be used to exactly re-express the GLE model as an equivalent phase space Fokker–Planck equation 共FPE兲 without neglecting inertial contributions. Such methods are amongst the most direct and economical routes to the derivation of diffusion equations from Langevin equations, and their importance as a calculational tool can be gauged by the number of reviews that have been written about them.8 Their application to the GLE model leads to a FPE that is identical 共as it should be兲 to the equation obtained by Hanggi and Mojtabai5 using a different approach based on the distribution function of the phase space variables.9 Our derivation of the FPE is new and sets down what we believe are important guidelines for the proper conversion of the coupled equations for positions and momenta to the equivalent equation for their joint probability density when these variables are governed by Gaussian colored noise. A second aim is to reconcile the results of our earlier approximate calculations with the results of Hanggi and Mojtabai5 and Kohen and Tannor.6 In particular it is to show that when the results of the FPE are combined with the exact results of Kohen and Tannor6 to calculate the transmission coefficient, the expression that is then obtained is virtually identical to the expression obtained from the original flux-overpopulation method, suggesting that the approxima-

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© 2008 American Institute of Physics

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tions used in its derivation were not necessarily as severe as previously believed. Waiting time distributions determined from the approximate transmission coefficient compare as favorably with data from Ref. 3 as those determined from the present expression. There are three sections in this paper: Section II A below briefly recapitulates the main results of Ref. 1, Sec. II B describes the results of the present calculations, and Sec. III discusses their implications. The Appendix provides details of the derivation of the phase space FPE using functional methods. The Appendix may be omitted by readers not interested in the technical minutiae of the calculations.

H had to be assigned the value of 3 / 4. For this special value of H, the function ␹共t兲 was found 关from Eq. 共2b兲兴 to equal14 E1/2共冑t / ␶兲, where E␣共z兲 is the Mittag–Leffler function and ␶ ⬅ 共3␨冑␲ / 4m␻2b兲2 is some relaxation time. From the solution of Eq. 共2a兲, under the initial condition P共x , 0兲 = ␦共x − x0兲, a time-dependent flux and a timedependent population were determined, from which, using Kramers’ method, the transmission coefficient ␬共t兲 was found to be

␬共t兲 = −

A. Recapitulation of some results from Reference 1

Assuming that the fluctuations of a single stochastic distance variable x共t兲 are the underlying cause of the fluctuations in the rate at which single enzymes convert substrates to products during one catalytic turnover cycle, the time evolution of enzyme activity can be described by the time evolution of x共t兲, which we assume is governed by the following GLE10 for a particle of mass m moving in a potential U共x兲:

冕

dt⬘K共t − t⬘兲v共t⬘兲 + ␪共t兲.

册

⳵ P共x,t兲 ⳵ k BT ⳵ 2 = ␩共t兲 共x − xb兲 − P共x,t兲, ⳵t ⳵x m␻2b ⳵x2

ˆ 共s兲 − m␻2 s␨K b

.

冕

t

册

dt⬘␬共t⬘兲 ,

0

共4兲

where kTST is the barrier crossing rate calculated from transition state theory. The above distribution, after setting H to 3 / 4 and adjusting the other phenomenological parameters in kTST and ␬共t兲 for best fit, was found to agree very well with the waiting time distribution determined from the singlemolecule experiments on ␤-galactosidase.3

B. Exact treatment of Equation „1…

Without neglecting the inertial term in Eq. 共1兲, and assuming only that ␪共t兲 is Gaussian, with the same moments as defined above, the phase space FPE analogous to Eq. 共2a兲 共after setting xb to 0 for convenience, but without loss of generality兲, is shown in the Appendix 共and by Hanggi and Mojtabai in Ref. 5兲 to be given by

⳵ ⳵P ⳵P ⳵P k BT ⳵ 2 P + ⌶ v P − ⍀ 2x =−v + 共⍀2 − ␻2b兲 ⳵x ⳵v ⳵v ⳵t m␻2b ⳵v⳵x +⌶

k BT ⳵ 2 P , m ⳵v2

共5兲

where P ⬅ P共x , v , t兲, and ⌶ and ⍀2 are time-dependent coefficients, which, from the definitions given in the Appendix, can be shown to be ⌶共t兲 =

ត共t兲 − ␹˙ 共t兲␹¨ 共t兲 ␹共t兲␹ ␹˙ 共t兲2 − ␹共t兲␹¨ 共t兲

共6a兲

ត共t兲 − ␹¨ 共t兲2 ␹˙ 共t兲␹ , 2 ␹˙ 共t兲 − ␹共t兲␹¨ 共t兲

共6b兲

and 共2a兲

where ␩共t兲 ⬅ −␹˙ 共t兲 / ␹共t兲, with ␹共t兲 the inverse Laplace transform of the function

␨Kˆ共s兲

冋

d exp − kTST dt

共1兲

Here v共t兲 = x˙共t兲 is the velocity of the particle, ␨ is its friction coefficient, and ␪共t兲 is a Gaussian colored noise with the moments 具␪共t兲典 = 0 and 具␪共t兲␪共t⬘兲典 = ␨kBTK共兩t − t⬘兩兲, where K共t兲 is a memory function, kB is Boltzmann’s constant, and T is the temperature. In Ref. 1, the potential was chosen to correspond to the inverted parabola U共x兲 = U共xb兲 − m␻2b共x − xb兲2 / 2, where xb is the location of the barrier top, ␻b is a frequency, and ␪共t兲 was chosen to correspond to fractional Gaussian noise11 共to model the effects of protein conformational fluctuations兲, so the memory function was given by12 K共兩t − t⬘兩兲 = 2H共2H − 1兲兩t − t⬘兩2H−2 共for t ⫽ t⬘兲, with H a phenomenological parameter lying between 1 / 2 and 1 that is a measure of the extent of temporal correlations in the noise. Furthermore, it was assumed that the friction was sufficiently large that the inertial term in Eq. 共1兲, mx¨共t兲, could be neglected. The resulting equation could then be exactly transformed to the following Smoluchowski equation for the probability density P共x , t兲 that the particle is at x at time t,

␹ˆ 共s兲 =

f共t兲 = −

t

0

冋

共3兲

The distribution of waiting times, f共t兲, between barrier crossing events was then calculated from

II. DYNAMICS OF THE ENZYMATIC REACTION COORDINATE

mv˙ 共t兲 = − U⬘共x兲 − ␨

␩共t兲 . ␻b

共2b兲

Here the Laplace transform gˆ共s兲 of a function g共t兲 is defined by the relation gˆ共s兲 = 兰⬁0 dte−stg共t兲. It was found that to fit the results of this model to the dynamic distance data of the protein complex fluorescein-antifluorescein,13 the parameter

⍀2共t兲 =

where ␹共t兲 is the inverse Laplace transform of the function

␹ˆ 共s兲 =

s + ␨Kˆ共s兲/m , s2 + s␨Kˆ共s兲/m − ␻2

共7a兲

b

and the dots on ␹共t兲 denote derivatives with respect to time. Equation 共7a兲 is the generalization to the inertial regime of the function ␹ˆ 共s兲 introduced in the preceding section 关see Eq. 共2b兲兴 and reduces to it in the overdamped limit.

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The dynamics of enzyme reactions

As shown by Hanggi and Mojtabai,5 Kramers’ fluxoverpopulation method can be applied directly to Eq. 共5兲 to determine the barrier crossing rate k in the stationary limit. From their analysis, it immediately follows that

␻0 k= 2␲␻b

冋冑

⌶⬁2 4

+

⍀⬁2

册

⌶⬁ − exp共− ⌬E/kBT兲, 2

共7b兲

where ␻0 is the oscillator frequency at the bottom of the well, ⌬E is the barrier height, and ⌶⬁ and ⍀⬁2 are the t → ⬁ limits of ⌶共t兲 and ⍀2共t兲, respectively. Furthermore, since x共t兲 and v共t兲 are Gaussian random variables, the solution of Eq. 共5兲 is given by P共x, v,t兲 =

冑C 2␲

再

1 exp − 关D11共x − ¯x共t兲兲2 + 2D12共x − ¯x共t兲兲 2

冎

⫻共v − ¯v共t兲兲 + D22共v − ¯v共t兲兲2兴 ,

共8a兲

␴2x 共t兲 = − ␴2v共t兲 =

冉

C␴2v共t兲

− C␴xv共t兲

− C␴xv共t兲

C␴2x 共t兲

冊

,

with C = det D = 关␴2x 共t兲␴2v共t兲 − ␴2xv共t兲兴−1. The time-dependent coefficients in Eqs. 共8a兲 and 共8b兲 are the moments of P; i.e., ¯x共t兲 = 兰dx兰dvxP共x , v , t兲 and ¯v共t兲 = 兰dx兰dvv P共x , v , t兲, while ␴2x 共t兲 = 兰dx兰dvx2 P共x , v , t兲 −¯x共t兲2, ␴2v共t兲 = 兰dx兰dvv2 P共x , v , t兲 − ¯v共t兲2, and ␴xv共t兲 = 兰dx兰dvxv P共x , v , t兲 −¯x共t兲¯v共t兲. From Eq. 共5兲, these moments are seen to satisfy the following equations:

⳵¯x共t兲 = ¯v共t兲, ⳵t

共9a兲

⳵¯v共t兲 ¯ 共t兲, = − ¯v共t兲⌶共t兲 + ⍀2共t兲x ⳵t

共9b兲

⳵␴2x 共t兲 = 2␴xv共t兲, ⳵t

共9c兲

⳵␴2v共t兲 2kBT = − 2⌶共t兲␴2v共t兲 + 2⍀2共t兲␴xv共t兲 + ⌶共t兲, ⳵t m

共9d兲

␴xv共t兲 =

⳵␴xv共t兲 k BT 2 = ␴2v共t兲 + ⍀2共t兲␴2x 共t兲 − ⌶共t兲␴xv共t兲 + 共⍀ 共t兲 ⳵t m␻2b 共9e兲

By directly solving Eq. 共1兲 for x共t兲 and v共t兲, as done, for example, in recent papers by Viñales and Despósito,15 these equations can be shown to be satisfied by ¯x共t兲 = x0␹共t兲 +

¯v共t兲 = x0␹˙ 共t兲 +

v0

␻2b v0

␻2b

␹˙ 共t兲,

共10a兲

␹¨ 共t兲,

共10b兲

册

冋

共10c兲

共10d兲

册

1 k BT 1 ␹共t兲␹˙ 共t兲 − 4 ␹˙ 共t兲␹¨ 共t兲 , m ␻2b ␻b

共10e兲

where x0 and v0 are, respectively, the initial position and velocity of the particle. Kohen and Tannor6 showed how an exact expression for the transmission coefficient ␬共t兲 can be derived from a bivariate Gaussian distribution function having the structure of Eq. 共8a兲 using the reactive flux formalism;16 their results immediately establish that

␬共t兲 =

冑

␹˙ 共t兲 ␹˙ 共t兲2 m 2 + ␴ 共t兲 k BT x ␻4b

,

共11兲

which from Eq. 共10c兲 simplifies to

␬共t兲 =

␹˙ 共t兲

␻b␹共t兲冑1 − ␹共t兲−2

.

共12兲

Equations 共7b兲 and 共12兲 are among the main results of the paper. III. DISCUSSION

From the expressions for ⌶共t兲 and ⍀2共t兲 given in Eqs. 共6a兲 and 共6b兲, it can be shown that in the overdamped limit— for the special case of fractional Gaussian noise—with H = 3 / 4 关and ␹共t兲 therefore given by E1/2共冑t / ␶兲兴, Eq. 共7b兲 reduces to k=

and

− ␻2b兲.

冋

␻2b 共8b兲

册

1 1 k BT 1 + 2 ␹˙ 共t兲2 − 4 ␹¨ 共t兲2 , m ␻b ␻b

and

where the constants Dij are the elements of the matrix D=

冋

1 2 k BT 1 ˙ 2 4 ␹共t兲 − 2 共␹ 共t兲 − 1兲 , m ␻b ␻b

␻0 exp共− ⌬E/kBT兲, 2␲␻b␶

共13兲

where ␶ is defined after Eq. 共2b兲. This is exactly the expression that the barrier crossing rate in Ref. 1 also reduces to, but in that reference it had not been possible to show that the limit was identical to the result obtained from the Hangi– Mojtabai rate expression. Further, comparing Eqs. 共12兲 and 共3兲 关and recalling that ␩共t兲 ⬅ −␹˙ 共t兲 / ␹共t兲兴, one sees that apart from the factor of 冑1 − ␹共t兲−2, the two expressions for the transmission coefficient are also of the same form 关the ␹共t兲’s in these expressions are, of course, different兴, and they become exactly the same when the overdamped limit is taken in Eq. 共7a兲 and t is made large. In this limit, then, both the flux overpopulation method and the reactive flux formalism yield a ␬共t兲 that scales as ␹˙ 共t兲 / ␹共t兲. If the distribution of barrier crossing times, f共t兲, is now determined from Eq. 共4兲 共which is a consequence of defining the rate as a flux over a population1兲, with ␬共t兲 having the form above, one sees that f共t兲 ⬃ ␹˙ 共t兲 / ␹共t兲␮, where ␮ is some nonuniversal constant that contains details of the potential, the temperature, etc. This is also exactly the scaling structure of f共t兲 that we had obtained from another model of barrier crossing;7 that model used an

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Chaudhury, Chatterjee, and Cherayil

approximate Smoluchowski equation to determine f共t兲 as the solution to a first passage time problem for the dynamics of an overdamped particle evolving under fGn in a continuous symmetric double well potential in which the barrier height was fixed by a single real number A 共where −1 / 2 ⬍ A ⬍ 0兲. The present calculations therefore establish quantitatively what it had earlier only been possible to suggest qualitatively, that Eq. 共3兲 is entirely consistent with the results of the Hanggi–Mojtabai barrier crossing rate,5 the Kohen– Tannor transmission coefficient,6 and the first passage time distribution for the double well potential.7 They also confirm, therefore, that the approximations used earlier to simplify Kramers’ treatment of barrier crossing were reasonable and well motivated. Furthermore, by showing that the methods of functional calculus can be extended without approximation to phase space variables in the inertial regime, the calculations also complete and unify the program begun in Refs. 1 and 7 to describe single enzyme dynamics in the framework of Kramers’ theory of chemical reaction rates. However, these remarks must be qualified by an important caveat. With f共t兲 in our calculations given by N␹˙ 共t兲 / ␹共t兲␮, N being a time-independent prefactor, the mean barrier crossing time 具t典 = 兰⬁0 dttf共t兲 is 具t典 =

N N lim t␹共t兲1−␮ − 1 − ␮ t→⬁ 1−␮

冕

⬁

dt␹共t兲1−␮ .

共14兲

0

Now for both the inverted parabolic potential of Ref. 1 共which is also the potential used in the present calculations兲 and the symmetric double well potential of Ref. 7, the factor 1 − ␮ is negative and the function ␹共t兲, given by E2−2H共共t / ␶兲2−2H兲, grows as an exponential at long times; so 具t典 in Eq. 共14兲 is finite. This means that, asymptotically, a well-defined barrier crossing rate exists. However in a recent exact calculation,17 Goychuk and Hanggi showed that in overdamped conditions the survival probability of a particle driven by 1 / f noise in a cusp-shaped parabolic potential is asympotically a power law in the time and that the escape time out of the potential, 具t典GH, is therefore infinite, implying the absence of a well-defined barrier crossing rate. The divergence of 具t典GH in these calculations has an interesting source: it comes from the integral shown below, 具t典GH ⬃

冕

⬁

dt␹GH共t兲␯ ,

共15兲

0

where ␹GH共t兲 = E␣共−共t / ␶D兲␣兲, and ␣, ␯, and ␶D are constants. The parallels between Eqs. 共14兲 and 共15兲 are obvious, but the difference in the sign of the arguments of the two Mittag– Leffler functions 共which reflects the differences in the shapes of the respective potentials兲 leads to drastically different prezⰇ1

dictions because, on the one hand, Ea共−z兲 → z−1, whereas on zⰇ1

FIG. 1. Comparison of the theoretical waiting time distribution f共t兲, normalized to a time t0, calculated from Eq. 共12兲 共full lines兲 with the corresponding experimental data of English et al. 共Ref. 3兲 on the enzyme ␤-galactosidase 共symbols兲 at four different substrate concentrations 共10 ␮M, open squares; 20 ␮M, open triangles; 50 ␮M, plus signs; and 100 ␮M, open circles兲. The theoretical curves were constructed by adjusting the well frequency ␻0 and the barrier height ⌬E / kBT for best fit to these data after fixing m␻2b to 1 共in appropriate units兲 and ␨ / m␻2b to 0.714 s1/2. The best fit values of ␻0 and ⌬E / kBT at the four substrate concentrations are given in the text. Also shown are normalized curves 共dashed lines兲 derived from a fitting function discussed in Ref. 3.

the other, Ea共z兲 → exp共z1/a兲. A meaningful rate description is recovered from the Goychuk–Hanggi model if an exponential cutoff at low frequencies is incorporated into the expression for the memory kernel, in which case the survival probability decays exponentially at long times 关as in Eq. 共14兲兴 and the integral in

Eq. 共15兲 no longer diverges. One way of ensuring finiteness of the barrier crossing rate is to increase the height of the barrier, which would entail a clear separation of time scales between barrier crossing and equilibration within the metastable well. It is presumably exactly this condition that is implicit in the equation 关Eq. 共4兲兴 that we have used to determine f共t兲. These facts must be borne in mind whenever one considers escape processes governed by subdiffusive dynamics. Other theoretical developments in the field of singlemolecule enzyme dynamics have occurred over the past few years, including extensions of our model1 that consider reaction and diffusion within the Michaelis–Menten mechanism18 and that consider particle dynamics on free energy surfaces of two dimensions,19 to cite two recent examples. By way of conclusion, and for the sake of completeness, we now use Eq. 共12兲 in the overdamped limit to recompute f共t兲 from Eq. 共4兲 for the case of fractional Gaussian noise, with H chosen to be 3 / 4 as before and ␹共t兲 = E1/2共冑t / ␶兲. The calculated curves of the distribution are shown as full lines in Fig. 1, where they are compared with data on ␤-galactosidase from Ref. 3 at four different concentrations of added substrate: 10 ␮M, open squares; 20 ␮M, open triangles; 50 ␮M, plus signs; and 100 ␮M, open circles. In constructing these curves 关which have been normalized by the value of f共t兲 at some t0 chosen to ensure that they coincide with the experimental data at the same initial time兴, the well frequency ␻0 and the barrier height ⌬E were adjusted for a best fit to the data, keeping m␻B2 and ␨ / m␻B2 fixed, respectively at 1 共in suitable units兲 and 0.714 s1/2. 共The decay constant ␶ thus assumes the fixed value of 0.9 s.兲 The

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best fit values of ␻0 and ⌬E / kBT were found to be, respectively, 3.8␲ ⫻ 1011 s−1 and 25.0 共for the 10 ␮M data兲, 8␲ ⫻ 1011 s−1 and 24.8 共for the 20 ␮M data兲, 1.45␲ ⫻ 1012 s−1 and 24.67 共for the 50 ␮M data兲, and 2␲ ⫻ 1012 s−1 and 24.67 共for the 100 ␮M data兲. These values are physically reasonable and very close to the corresponding values determined earlier.1 Also shown in the figure are curves obtained from the fitting function discussed in Ref. 3 共dashed lines兲. The theoretical curves obtained from Eq. 共12兲 are seen to be in very good agreement with the experimental data at all concentrations, and their fits are indistinguishable, for all practical purposes, from the fits found in the corresponding calculations of Ref. 1. Owing to the limited time span over which the data on ␤-galactosidase have been collected, the agreement between experiment and theory in Fig. 1 is not necessarily a validation of our model, and other simpler models might do equally well in fitting the data. However, the present results should be seen in the context of the success that the model 共or its variants兲 has had in characterizing the dynamics of other complex soft matter system.20 ACKNOWLEDGMENTS

S.C. acknowledges financial support from the Centre for Scientific and Industrial Research 共CSIR兲, Government of India. The authors are grateful to Brian English for the data on ␤-galactosidase and to Sunney Xie, Sam Kou, and Wei Min of Harvard University for fruitful discussions on single enzyme kinetics. APPENDIX: THE PHASE SPACE FOKKER–PLANCK EQUATION

To derive the phase space FPE shown in Eq. 共5兲 using functional methods,8 the definition v共t兲 = x˙共t兲 is first substituted into Eq. 共1兲, and the equation then solved for x共t兲, using Laplace transforms. With U共x兲 an inverted parabola, the result is

冕 ⬘ 冕 ⬘ t

x共t兲 = x0␹共t兲 + v0

dt ␹共t − t⬘兲␰共t⬘兲 −

xb␻2b

0

1 ⫻␤共t⬘兲 + m

冕

t

dt⬘␹共t − t⬘兲

0

t

dt ␹共t − t⬘兲␥共t⬘兲,

共A1兲

0

ˆ 共s兲 / m兴, ␤ˆ 共s兲 = ␰ˆ 共s兲 / s, ␥ˆ 共s兲 = ␰ˆ 共s兲␪ˆ 共s兲, where ␰ˆ 共s兲 = 1 / 关s + ␨K and ␹ˆ 共s兲 = 1 / 关s − ␻2b␰ˆ 共s兲兴. An equation for x˙共t兲 in which x0 has been eliminated is now derived by multiplying all but the first term on the right hand side of Eq. 共A1兲 by ␹共t兲 / ␹共t兲 and then differentiating the equation with respect to t. This yields x˙共t兲 = − ␩共t兲x共t兲 + v0¯␰共t兲 − xb␻2b¯␤共t兲 + ¯␥共t兲.

共A2兲

Here ␩共t兲 = −␹˙ 共t兲 / ␹共t兲, ¯␰共t兲 / ␹共t兲 = 共d / dt兲⌬␰共t , t⬘兲, ¯␤共t兲 / ␹共t兲 = 共d / dt兲⌬␤共t , t⬘兲, and ¯␥共t兲 / ␹共t兲 = 共1 / m兲共d / dt兲⌬␥共t , t⬘兲, with ⌬z共t , t⬘兲 ⬅ 1 / ␹共t兲兰t0dt⬘␹共t − t⬘兲Z共t⬘兲 and Z共t兲 standing for ␰共t兲, ␤共t兲, or ␥共t兲. The function ¯␰共t兲 can be simplified to ¯␰共t兲 = −␹共t兲␩˙ 共t兲 / ␻2b by evaluating 兰t0dt⬘␹共t − t⬘兲␰共t⬘兲 from

L−1␹ˆ 共s兲␰ˆ 共s兲 共L−1 being the inverse Laplace operator兲 after replacing ␰ˆ 共s兲 by its expression in terms of ␹ˆ 共s兲. In the same way, ¯␤共t兲 can be reduced to ¯␤共t兲 = −␩共t兲 / ␻2b. Equation 共A2兲 is the defining equation for the evolution of x共t兲. A related equation for the evolution of v共t兲 in which v0 has been eliminated is obtained by multiplying all but the second term on the right hand side of Eq. 共A2兲 by ¯␰共t兲 /¯␰共t兲 and then differentiating the equation with respect to t; after substituting ¯␤共t兲 = −␩共t兲 / ␻2b into the equation and collecting terms, it is found that v˙ 共t兲 = − ⌶共t兲v共t兲 + ⍀2共t兲共x共t兲 − xb兲 + ␭共t兲,

共A3兲

˙ where ⌶共t兲 = ␩共t兲 −¯␰共t兲 /¯␰共t兲, ⍀2共t兲 = −¯␰共t兲共d / dt兲关␩共t兲 /¯␰共t兲兴, and ␭共t兲 is a new random variable: ␭共t兲 ⬅¯␰共t兲共d / dt兲 关¯␥共t兲 /¯␰共t兲兴. Equation 共A3兲 is the defining equation for v共t兲. The equations for x˙共t兲 and v˙ 共t兲 are substituted into the expression obtained after differentiating 共with respect to t兲 the definition of the phase space probability density, P共x , v , t兲 ⬅ 具␦共x − x共t兲兲␦共v − v共t兲兲典; the use of Novikov’s theorem21 in the result yields

⳵P ⳵P ⳵ ⳵ P ⳵2 P + ⌶共t兲 v P − ⍀2共t兲共x − xb兲 =−v Y1 + ⳵x ⳵t ⳵v ⳵v ⳵v⳵x +

⳵2 P Y2, ⳵v2

共A4兲

where Y 1 ⬅ 兰t0dt⬘具␭共t兲␭共t⬘兲典关␦x共t兲 / ␦␭共t⬘兲兴 and Y2 t ⬅ 兰0dt⬘具␭共t兲␭共t⬘兲典关␦v共t兲 / ␦␭共t⬘兲兴. The functional derivatives in Y 1 and Y 2 are obtained as follows. First, Eq. 共A2兲 is functionally differentiated with respect to ␭共t⬘兲, producing 共d / dt兲关␦x共t兲 / ␦␭共t⬘兲兴 = − ␩共t兲关␦x共t兲 / ␦␭共t⬘兲兴 + ␦¯␥共t兲 / ␦␭共t⬘兲. The equation for ␭共t兲 is next rewritten as d¯␥共t兲 / dt ˙ = 关¯␰共t兲 /¯␰共t兲兴¯␥共t兲 + ␭共t兲 and similarly differentiated with re˙ spect to ␭共t⬘兲 to give 共d / dt兲关␦¯␥共t兲 / ␦␭共t⬘兲兴 = 共¯␰共t兲 /¯␰共t兲兲 ⫻关␦¯␥共t兲 / ␦␭共t⬘兲兴 + ␦共t − t⬘兲. The solution of this equation is ␦¯␥共t兲 / ␦␭共t⬘兲 = H共t − t⬘兲exp关兰t ds关¯␰˙ 共s兲 /¯␰共s兲兴兴, where H共x兲 is t⬘

the step function. From these equations, it is easily shown that

冋冕

冕

t 1 ␦x共t兲 = ds¯␰共s兲exp − ␦␭共t⬘兲 ¯␰共t⬘兲 t⬘

册

t

ds⬘␩共s⬘兲 .

s

共A5兲

The relation ¯␰共s兲 = −␹共s兲␩˙ 共s兲 / ␻2b is now substituted into the term I1 ⬅ 兰tt⬘ds¯␰共s兲exp关−兰stds⬘␩共s⬘兲兴, and the result is integrated by parts to produce I1 = 共1 / ␻2b兲共␹˙ 共t兲 − ␹共t兲 ⫻关␹˙ 共t⬘兲 / ␹共t⬘兲兴兲, which is then substituted into the integral Y 1, along with Eq. 共A5兲 and the definition of the random variable ␭共t兲 关defined after Eq. 共A3兲兴. This leads to Y1 =

1

␻2b ⫻

冓

␭共t兲

1 ¯␰共t⬘兲

冉

冕

t

dt⬘¯␰共t⬘兲

0

␹˙ 共t兲 − ␹共t兲

冉

d ¯␥共t⬘兲 dt⬘ ¯␰共t⬘兲

␹˙ 共t⬘兲 ␹共t⬘兲

冊冔

.

冊

共A6兲

After the integration of Eq. 共A6兲 by parts, substitution of

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075104-6

Chaudhury, Chatterjee, and Cherayil

¯␥共0兲 = 0 共as indicated later, ¯␰共0兲 = 1兲, and use of the relation d␩共t兲 / dt = −␻2b关¯␰共t兲 / ␹共t兲兴, it is found, after some algebra, that

冓

冉 冕

d 1 Y 1 = ¯␰共t兲␹共t兲 dt ¯␰共t兲

t

dt⬘

0

¯␥共t兲¯␥共t⬘兲 ␹共t⬘兲

冊冔

− 具¯␥共t兲2典. 共A7兲

冋

1 mkBT ␹˙ 共t⬘兲 2 ␹共t兲␹共t⬘兲 m ␻2b ␹共t⬘兲2

再

⫻ −

1 d ␹˙ 共t兲 ␹共兩t − t⬘兩兲 2 ␹共兩t − t⬘兩兲 + ␹共t兲 ␹共t兲 dt

再

冎

+

1 d d mkBT ␹共兩t − t⬘兩兲 − ␩˙ 共t兲␩˙ 共t⬘兲 . ␹共t兲 dt dt⬘ ␻4b

册

From the definition of ¯␰共t兲, it can also be shown that ¯␰共0兲 = ␰共0兲 = 1 and that ␹¨ 共0兲 = ␻2b. So, from Eq. 共A8兲, we eventually find 共A9兲

− t1兲␹共t − t2兲␥共t1兲␥共t2兲典 by substituting the definitions of ¯␥共t兲 and ¯␥共t⬘兲 and rearranging. The application of double Laplace transforms to the result yields I2 = 共kBT / m␻2b兲关␹˙ 共t兲 / ␹共t兲2兴 − 共kBT / m␻4b兲␹共t兲␩共t兲␩˙ 共t兲. Substituting I2 into Eq. 共A7兲, using ¯␰共t兲 = −␹共t兲␩˙ 共t兲 / ␻2, and simplifying further, we have b

冕

t

0

=

dt⬘具␭共t兲␭共t⬘兲典

冋

␦x共t兲 ␦␭共t⬘兲

册

ត共t兲 − ␹¨ 共t兲2 kBT ␹˙ 共t兲␹ − ␻2b . m␻2b ␹˙ 共t兲2 − ␹共t兲␹¨ 共t兲

共A10兲

The above relation leads to the identification, ⍀2共t兲 ត共t兲 − ␹¨ 共t兲2兴 / 关␹˙ 共t兲2 − ␹共t兲␹¨ 共t兲兴. For the purely Markov= 关␹˙ 共t兲␹ ian case, the memory function in Eq. 共1兲 is delta correlated, and ␹共t兲 is the sum of two exponentials; in this case, ⍀2共t兲 reduces to ␻2b, so the mixed derivative contribution to the phase space FPE vanishes, as it should. The evaluation of Y 2 关see Eq. 共A4兲兴 is carried out along much the same lines. Since x˙共t兲 = v共t兲, the expressions for d¯␥共t兲 / dt, ␦x共t兲 / ␦␭共t⬘兲, and I1 together yield ¯␰共t兲 1 ␩共t兲 ␦v共t兲 =− 2 ␹共t兲关− ␩共t兲 + ␩共t⬘兲兴 + ␪共t − t⬘兲 . ␦␭共t⬘兲 ¯␰共t⬘兲 ␻b ¯␰共t⬘兲 共A11兲 Therefore,

dt

0

冊

d ¯␥共t⬘兲 关␩共t兲2␹共t兲 dt⬘ ¯␰共t⬘兲

冔

⬅ IA − IB + IC ,

共A12兲

Y2 =

冕

t

0

dt⬘具␭共t兲␭共t兲⬘典

冉

冊

¯␰˙ 共t兲 ␦v共t兲 k BT 2 ␻b , =− − ␩共t兲 ␦␭共t⬘兲 ¯␰共t兲 m␻2b 共A13兲

which leads, finally, to the identification ⌶共t兲 = ␩共t兲 ˙ ត共t兲 − ␹˙ 共t兲␹¨ 共t兲兴 / 关␹˙ 共t兲2 − ␹共t兲␹¨ 共t兲兴. For the −¯␰共t兲 /¯␰共t兲 = 关␹共t兲␹ purely Markovian case, ⌶共t兲 reduces to the constant ␨ / m, as it should. S. Chaudhury and B. J. Cherayil, J. Chem. Phys. 125, 024904 共2006兲. H. A. Kramers, Physica 共Amsterdam兲 7, 284 共1940兲; P. Hanggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 共1990兲; R. F. Grote and J. T. Hynes, J. Chem. Phys. 73, 2715 共1980兲. 3 B. P. English, W. Min, A. M. van Oijen, K. T. Lee, G. Luo, H. Sun, B. J. Cherayil, S. C. Kou, and X. S. Xie, Nat. Chem. Biol. 2, 87 共2006兲. 4 P. Debnath, W. Min, X. S. Xie, and B. J. Cherayil, J. Chem. Phys. 123, 204903 共2005兲; J. Tang and R. Marcus, Phys. Rev. E 73, 022102 共2006兲; J. Xing and K. S. Kim, ibid. 74, 061911 共2006兲; P. Vallurupalli and L. E. Kay, Proc. Natl. Acad. Sci. U.S.A. 103, 11910 共2006兲; H. Chen, E. Rhoades, J. S. Butler, S. N. Loh, and W. W. Webb, ibid. 104, 10459 共2007兲. 5 P. Hanggi and F. Mojtabai, Phys. Rev. A 26, 1168 共1982兲. 6 D. Kohen and D. J. Tannor, J. Chem. Phys. 103, 6013 共1995兲; see also J. D. Bao, ibid. 124, 114103 共2006兲; R. Chakrabarti, ibid. 126, 134106 共2007兲 for other derivations of the Kohen–Tannor expression for the transmission coefficient. 7 S. Chaudhury and B. J. Cherayil, J. Chem. Phys. 125, 114106 共2006兲. 8 P. Hanggi, Z. Phys. B 31, 407 共1978兲; M. San Miguel and M. Sancho, J. Stat. Phys. 22, 605 共1980兲; A. Hernández-Machado, J. M. Sancho, M. San Miguel, and L. Pesquera, Z. Phys. B: Condens. Matter 52, 335 共1983兲; P. Hänggi, in Stochastic Processes Applied to Physics, edited by L. Pasquera and M. Rodriguez 共World Scientific, Philadelphia, 1985兲, pp. 69–95; R. F. Fox, Phys. Rev. A 33, 467 共1986兲; P. Hänggi, in Noise in Nonlinear Dynamical Systems, edited by F. Moss and P. V. E. McClintock 共Cambridge University Press, New York, 1989兲, Vol. 1, Chap. 9, pp. 307–328; J. Luczka, Chaos 15, 026107 共2005兲. 9 S. A. Adelman, J. Chem. Phys. 64, 124 共1976兲. 10 D. A. McQuarrie, Statistical Mechanics 共University Science Books, Sausalito, 2000兲; R. Zwanzig, Nonequilibrium Statistical Mechanics 共Oxford University Press, New York, 2001兲. 11 B. Mandelbrot and J. van Ness, SIAM Rev. 10, 422 共1968兲; S. C. Lim and S. V. Muniandy, Phys. Rev. E 66, 021114 共2002兲; K. S. Fa and E. K. Lenzi, ibid. 71, 012101 共2005兲. 12 S. C. Kou and X. S. Xie, Phys. Rev. Lett. 93, 180603 共2004兲. 13 W. Min, G. Luo, B. J. Cherayil, S. C. Kou, and X. S. Xie, Phys. Rev. Lett. 94, 198302 共2005兲. 14 A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher 1

The integral I2 ⬅ 兰t0dt⬘关1 / ␹共t⬘兲兴具¯␥共t兲¯␥共t⬘兲典 in Eq. 共A7兲 is now written as I2 = 共1 / 2m2兲具␹共t兲共d / dt兲关1 / ␹共t兲2兴兰t0dt1兰t0dt2␹共t

Y1 =

冕 ⬘冉

C

␹˙ 共t兲 d mkBT 1 − ␹共兩t − t⬘兩兲 ␹共t兲2 dt⬘ ␻2b ␹共t⬘兲

k BT k BT 2 2 ␹共t兲2␩˙ 共t兲2 . 2 关␩共t兲 − ␻b兴 − m␻b m␻4b

冓

where IA can be found at once, IA = 共1 / ␻2b兲具␭共t兲␩共t兲2␹共t兲 ⫻关¯␥共t兲 /¯␰共t兲兴典. IB is evaluated by first substituting ¯␰共t兲 = −␹共t兲␩˙ 共t兲 / ␻2b into the expression, after which it is written as IB = IA + IB共1兲, where IB共1兲 = 具␭共t兲␩共t兲␹共t兲兰t0dt⬘关¯␥共t⬘兲 / ␹共t⬘兲兴典. This integral, after introducing ␭共t兲 and simplifying, produces IB共1兲 =¯␰共t兲␩共t兲␹共t兲共d / dt兲关1 / ␰共t兲兴I2 − ␩共t兲具¯␥共t兲2典. Here, I2 is the same integral defined after Eq. 共A9兲. Finally, IC can be ˙ immediately reduced to I = −关¯␰共t兲 /¯␰共t兲兴具¯␥共t兲2典 + 共1 / 2兲共d / dt兲

共A8兲

具¯␥共t兲2典 = −

␻2b

t

␭共t兲

具¯␥共t兲2典. After putting these results for IA, IB, and IC back into the expression for Y 2, substituting for 具¯␥共t兲2典, and simplifying, the result is

−

冎

1

− ␩共t兲␹共t兲␩共t⬘兲 + ␻2b¯␰共t兲兴

Double Laplace transforms are introduced into the definition of ¯␥共t兲 关defined after Eq. 共A2兲兴 to evaluate the correlation function 具¯␥共t兲¯␥共t⬘兲典. In this way it can be shown that 具¯␥共t兲¯␥共t⬘兲典 =

Y2 =

J. Chem. Phys. 129, 075104 共2008兲

2

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075104-7

J. Chem. Phys. 129, 075104 共2008兲

The dynamics of enzyme reactions

Transcendental Functions 共McGraw-Hill, New York, 1953兲, Vol. 3; I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products 共Academic, San Diego, 1980兲. 15 A. D. Viñales and M. A. Despósito, Phys. Rev. E 73, 016111 共2006兲; M. A. Despósito and A. D. Viñales, ibid. 77, 031123 共2008兲. 16 D. Chandler, J. Chem. Phys. 68, 2959 共1978兲; J. A. Montgomery, D. Chandler, and B. J. Berne, ibid. 70, 4056 共1979兲; D. Chandler, Introduction to Modern Statistical Mechanics 共Oxford, New York, 1987兲. 17 I. Goychuk and P. Hanggi, Phys. Rev. Lett. 99, 200601 共2007兲.

S. Chaudhury and B. J. Cherayil, J. Chem. Phys. 127, 105103 共2007兲; X. Xue, F. Liu, and Z.-c. Ou-Yang, Phys. Rev. E 74, 030902共R兲 共2006兲. 19 W. Min, X. S. Xie, and B. Bagchi, J. Phys. Chem. B 112, 454 共2008兲. 20 S. Chaudhury and B. J. Cherayil, “A model of anomalous chain translocation dynamics,” J. Phys. Chem. B 共to be published兲; J. Chem. Phys. 127, 145103 共2007兲; 125, 184505 共2006兲; S. Chaudhury, S. C. Kou, and B. J. Cherayil, J. Phys. Chem. B 111, 2377 共2007兲. 21 E. A. Novikov, Sov. Phys. JETP 20, 1290 共1965兲. 18

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