THE JOURNAL OF CHEMICAL PHYSICS 125, 024904 共2006兲
Complex chemical kinetics in single enzyme molecules: Kramers’s model
with fractional Gaussian noise
Srabanti Chaudhury and Binny J. Cherayila兲
Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
共Received 21 February 2006; accepted 9 May 2006; published online 11 July 2006兲
A model of barrier crossing dynamics governed by fractional Gaussian noise and the generalized
Langevin equation is used to study the reaction kinetics of single enzymes subject to conformational
fluctuations. The direct application of Kramers’s flux-over-population method to this model yields
analytic expressions for the time-dependent transmission coefficient and the distribution of waiting
times for barrier crossing. These expressions are found to reproduce the observed trends in recent
simulations and experiments. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2209231兴
I. INTRODUCTION
A number of recent single-molecule experiments have
shown that the catalytic activity of single enzymes can
change with time, producing measurable kinetic effects, including time-dependent rate constants, nonexponential waiting time distributions, and correlations between successive
turnover events.1 These effects are believed to be largely the
result of conformational fluctuations between different states
of the enzyme with distinct reactivities.2 The conformational
fluctuations themselves reflect the action of the random,
time-varying forces generated by the interaction of the molecule with a large thermal reservoir. For many systems, these
forces are of such short duration that they can be modeled as
white noise, but for proteins and other biopolymers, their
effects appear to persist over long periods of time,3,4 and a
white noise description appears no longer to apply. It has
been recently shown,4 in the context of a one-dimensional
generalized Langevin equation 共GLE兲 model of intersegment
distance fluctuations in proteins, that the forces are, in fact,
well characterized as fractional Gaussian noise5 共fGn兲, which
has a power law, rather than delta, correlation. Debnath et
al.,6 and independently Tang and Marcus,7 later demonstrated that such power law correlations can emerge naturally
from a discrete or continuum Rouse model8 for protein conformational dynamics.
Under the fGn-GLE framework, Min and Xie9 and Min
et al.10 have shown through numerical simulations that if one
views an enzymatic reaction as the passage of a particle 共representing a reaction coordinate兲 over an energy barrier, as in
Kramers’s model of reaction rates,11 then the distribution of
waiting times f共t兲 between successive turnovers of a single
enzyme molecule has a nonexponential decay, which could
potentially lead to an explanation of the experimentally observed dispersed kinetics and dynamic disorder12 in single
enzyme activity.
The numerical simulations by Min and Xie9 and Min
et al.,10 while providing useful insights into single enzyme
behavior, were limited to a consideration of low reaction
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2006/125共2兲/024904/9/$23.00
barriers. In this article, therefore, we seek to derive an analytic expression for f共t兲—based on the same fGn-GLE model
of barrier crossing dynamics—that could be compared with
experiment for a wider range of conditions than is generally
accessible to simulations.
The problem of deriving an expression for f共t兲 can be
reformulated as the problem of deriving an expression for
k共t兲, the time-dependent rate at which a particle moving according to the GLE under the action of fGn crosses over a
barrier. As is well known, Kramers’s venerable theory of
chemical reaction rates, proposed in 1940,11 addresses the
related but distinct problem of the escape of a Brownian
particle from a metastable minimum in the t → ⬁ limit. This
escape rate was calculated from the ratio of the stationary
particle flux over the barrier top to the equilibrium particle
population in the region of the potential well. Because this
treatment of barrier crossing is based on a Markovian description of the underlying dynamics, and assumes a clearcut separation of time scales between particle motion and
bath relaxation, it does not adequately describe processes
governed by long time memory, such as those responsible for
protein conformational fluctuations.
There have been numerous extensions and refinements
of Kramers’s theory since then 共a comprehensive review of
the status of the field 50 years after the publication of his
landmark 1940 paper may be found in Ref. 13兲, but one of
the earliest theories of activated barrier crossing governed by
non-Markovian dynamics was put forward by Grote and
Hynes in 1980.14 But this theory, like Kramers’s model, invoked the Markovian assumption of time scale separation.
Subsequently, this restriction was lifted in the theory of
Hanggi and Mojtabai.15 An expanded discussion of this
theory, and of the more general problem of escape from
metastable states, may be found in Ref. 16. The simulations
of Ref. 9 are based on the Hanggi-Mojtabai model,15 but
neglect inertia, so the dynamics of the particle takes place in
coordinate space rather than phase space.
Given the non-Markovian nature of these dynamics, the
proper starting point for developing a theory of the corresponding waiting time distribution is the full phase space
125, 024904-1
© 2006 American Institute of Physics
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024904-2
J. Chem. Phys. 125, 024904 共2006兲
S. Chaudhury and B. J. Cherayil
II. BARRIER CROSSING RATE AND WAITING TIME
DISTRIBUTION
Consider the one-dimensional potential energy surface
sketched in Fig. 1. Imagine that some number of particles n
are initially found in the well centered around xA, and that
they are driven across the barrier by some random process.
Let the probability that a given particle has not crossed the
barrier up to time t be p共t兲. Then the number of particles that
leave the reactant well in the interval of time between t and
t + ␦t is n关p共t兲 − p共t + ␦t兲兴, and the number left behind is np共t兲.
Their ratio, per unit interval of time, is by definition, the
barrier crossing rate k共t兲. Hence,
k共t兲 =
FIG. 1. Schematic representation of the double well potential in which
particle motion takes place. xA and xC are the locations of the well minima,
while xB is the location of the barrier top.
n关p共t兲 − p共t + ␦t兲兴
np共t兲␦t
=−
formulation of Ref. 15, with due attention paid to dynamics
in the region of the metastable minimum. However, there
appears to be no simple way to use this formulation to obtain
a closed form expression for f共t兲. As an alternative, therefore, we now consider an approach based on what may be
referred to as a naive application of the flux-over-population
method to the coordinate space formulation of Ref. 9. By
naive application of the method, we mean its application
away from the t → ⬁ limit, ignoring the complications and
subtleties associated with the occurrence of non-Markovian
dynamics. This approach, though ad hoc, will be seen to
yield correct long time limits, and, in particular, it will be
seen to recover the scaling structure of certain parameters
that appear in the non-Markovian theory of Hanggi and
Mojtabai.15
The following section shows how the waiting time distribution f共t兲 may be related to the barrier crossing rate k共t兲.
Section III then reviews the generalized Langevin equation
formalism that has been used in Ref. 9 and elsewhere4 to
describe the motion of a particle in a potential U共x兲 that is
acted on by fractional Gaussian noise. Since Kramers’s
method assumes that the portion of the potential most relevant to the calculation of the flux is the barrier top, U共x兲 is
chosen to correspond to an inverted parabola. Appendix A
transforms this GLE, in the inertialess limit, into an equation
for the time-dependent probability density of the particle position. This equation has the structure of an ordinary diffusion equation, but the diffusion coefficient is now a function
of time. The flux-over-population method is applied directly
to this equation in Sec. IV, in the sense explained above,
following the approach of Kramers11 and Hanggi and
Mojtabai.15,16 The resulting time-dependent rate k共t兲 is used
to calculate f共t兲 according to the prescription given in Sec. II.
The calculated distribution is compared with experimental
data on single-molecule enzyme activity, and the results are
discussed in Sec. V. Appendix B presents arguments to show
that k共t兲 is consistent with the results obtained from earlier
phase space calculations.
dp共t兲/dt
,
p共t兲
␦t → 0.
共1a兲
共1b兲
This equation is easily solved for p共t兲, the survival probability, yielding the expression
冋
p共t兲 = exp −
冕
t
0
册
dsk共s兲 .
共2兲
The distribution of barrier crossing times, i.e., the waiting
time distribution f共t兲, can then be found from
f共t兲 = −
dp共t兲
.
dt
共3兲
Equation 共3兲, through Eq. 共2兲, relates f共t兲 to k共t兲; k共t兲 itself
corresponds to the ratio of a flux to a population, and can
therefore be determined, in principle, from the general formalism associated with Kramers’s reaction rate theory. However, the flux-over-population method, in its original
formulation,11 is based on an assumption of stationarity, i.e.,
the assumption that barrier crossing occurs at equilibrium.
The rate in Eq. 共1兲, on the other hand, is defined for all times.
To avoid the complications of developing a rigorous timedependent formulation of the non-Markovian barrier crossing problem, we shall instead adopt the simpler heuristic
approach discussed earlier, which is based on the naive application of the flux-over-population method away from the
t → ⬁ limit. Before discussing specific details of the method,
we shall first review the GLE formalism that forms the basis
for the models of conformational dynamics and enzyme kinetics used in Refs. 4, 9, and 10.
III. REVIEW OF GLE FORMALISM
In Sec. II, the source of the randomness that drives particles from one potential minimum to the other is the thermal
fluctuations of the reservoir to which the particles are
coupled. When these fluctuations are temporally correlated,
the evolution equation for the position x共t兲 of a given particle
is described by a generalized Langevin equation 共GLE兲.17,18
For a particle of mass m in equilibrium with a thermal reservoir at temperature T and moving on the one-dimensional
surface of a potential U共x兲, this GLE can be written as4
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024904-3
ẋ共t兲 = v共t兲,
m
J. Chem. Phys. 125, 024904 共2006兲
Complex chemical kinetics in single enzyme molecules
dv共t兲
=−␨
dt
共4a兲
冕
t
dt⬘K共t − t⬘兲v共t⬘兲 −
0
dU共x兲
+ ␪共t兲,
dx
共4b兲
where v共t兲 is the velocity of the particle, ␨ is the friction
coefficient, ␪共t兲 is the noise term that corresponds to the
effects of thermal fluctuations in the medium, and K共t兲 is the
memory function, which is related to the noise through a
fluctuation-dissipation theorem, i.e., K共兩t − t⬘兩兲 = 共1 / ␨kBT兲
⫻具␪共t兲␪共t⬘兲典. In the barrier region of the potential, U共x兲 can
be approximated by an inverted parabola, i.e., U共x兲 ⬇ E
− m␻B2 共x − xB兲2 / 2, where E is the height of the barrier, xB is
the location of the barrier top, and ␻B is the barrier frequency. Equations 共4a兲 and 共4b兲 are then equivalent to the
corresponding equations used in Ref. 15.
Equations 共4a兲 and 共4b兲 are also the starting point for the
treatment of conformational fluctuations in Ref. 4 and barrier
crossing in Ref. 9. Both these works simplify the equations
further by neglecting the inertial contribution mẍ共t兲. This is a
generally valid approximation if the friction is high 共corresponding to the overdamped limit兲, but friction here must be
understood to refer to ␨K̃共␻B兲, where K̃共␻B兲 is the Fourier
transform of the memory function evaluated at the barrier
frequency 共cf. Ref. 13 and references therein兲. Strictly speaking, this approximation of neglecting the inertia is not necessary, but the full phase space formulation of the dynamics
is quite algebraically involved, and the use of the overdamped limit makes it much simpler to derive analytical results. The neglect of inertia in Eqs. 共4a兲 and 共4b兲 leads to
− m␻B2 共x共t兲 − xB兲 = − ␨
冕
t
dt⬘K共t − t⬘兲ẋ共t⬘兲 + ␪共t兲
共5兲
0
⳵ P共x,t兲
⳵
⳵2
k BT
= ␩共t兲 共x − xB兲P共x,t兲 −
2 ␩共t兲 2 P共x,t兲.
⳵t
⳵x
⳵x
m␻B
共9兲
,
IV. CALCULATION OF k„t… AND f„t…
The barrier crossing rate k共t兲 is now determined, following Kramers11 and Hanggi and Mojtabai15,16 共and as nicely
illustrated by Mazo24兲, as the ratio of a flux to a population.
To calculate the flux, Eq. 共7兲 is rewritten as a continuity
equation, i.e., as
冋
⳵ P共x,t兲 ⳵
⳵
k BT
+
P共x,t兲
− ␩共t兲共x − xB兲P共x,t兲 +
2 ␩共t兲
⳵t
⳵x
⳵
x
m␻B
册
共10兲
= 0,
so that the probability current J共x , t兲 共i.e., flux兲 can be identified as
J共x,t兲 = − ␩共t兲共x − xB兲P共x,t兲 +
J共x,t兲 = ␩共t兲
⳵
k BT
P共x,t兲,
2 ␩共t兲
⳵x
m␻B
共11兲
Here ␩共t兲 is defined as
共8兲
where ␹共t兲 is the inverse Laplace transform of the function
␹ˆ 共s兲, which is given by
冋
⳵
k BT
exp共m␻B2 共x − xB兲2/2kBT兲 P共x,t兲
⳵x
m␻B2
册
⫻exp共− m␻B2 共x − xB兲2/2kBT兲 .
共12兲
Proceeding now along the lines discussed by Mazo,24 both
sides of Eq. 共12兲 are multiplied by the Boltzmann factor
exp关−m␻B2 共x − xB兲2 / 2kBT兴 and integrated over x from xA 共the
location of the reactant minimum兲 to xC 共the location of the
product minimum兲 to produce
冕
xC
dx exp共− m␻B2 共x − xB兲2/2kBT兲J共x,t兲
xA
= ␩共t兲
共7兲
␹˙ 共t兲
␹共t兲
共s兲 − m␻2
s␨K
B
with K̂共s兲 the Laplace transform of the memory kernel K共t兲.
When K共t兲 is a power law in time, ␩共t兲 can be determined in
closed form, as shown later.
Equation 共7兲 has the structure of a diffusion equation in
which the quantity −kBT␩共t兲 / m␻B2 can be identified with a
time-dependent diffusion coefficient.21 A similar equation
has been discussed by Hanggi et al. in Ref. 22.
Related time-convolutionless master equations for nonMarkovian dynamics have also been derived,15,23 and, in particular, the rate calculated in Ref. 15 will be used to test the
results derived from Eq. 共7兲.
共6兲
Equations 共5兲 and 共6兲 are the defining equation of the barrier
crossing model considered here.
To proceed further, Eq. 共5兲 is now transformed to an
equivalent equation for the probability density P共x , t兲 that the
particle is at the point x at time t. The transformation, which
is exact, is carried out using standard methods of functional
calculus.19,20 For completeness, details of the calculation are
provided in Appendix A. The final result for P共x , t兲, for dynamics near the barrier top, is given by
␩共t兲 ⬅ −
共s兲
␨K
which in turn can be written identically as
near the barrier top. Further, if the thermal fluctuations are
described by fractional Gaussian noise, then the memory
function is given by4,5
K共兩t − t⬘兩兲 = 2H共2H − 1兲兩t − t⬘兩2H−2 .
共s兲 =
␹
k BT
P共x,t兲exp 兩共− m␻B2 共x − xB兲2/2kBT兲兩xxC .
A
m␻B2
共13兲
At this stage, Kramers’s treatment assumes that the current is
stationary, i.e., that it is a constant, and can therefore be
taken outside the integral sign. In our implementation of
Kramers’s approach we shall continue to assume that J共x , t兲
is independent of x, and can still be taken outside the integral
sign. There are likely to be conditions under which this assumption holds at least approximately, such as high barrier
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024904-4
J. Chem. Phys. 125, 024904 共2006兲
S. Chaudhury and B. J. Cherayil
heights, but the assumption may be difficult to justify rigorously. In its defense, we shall only note that correct limiting
behavior will be recovered when t → ⬁.
A further simplification is introduced at this stage. On
the right hand side of Eq. 共13兲, we set P共xC , t兲 = 0, implying
that the population of particles in the product minimum is
negligible, as would be the case if they were removed from
the well 共by a sink located there, for example兲 immediately
upon arrival. This step is a direct transcription from Kramers’s Markovian analysis, but it is unlikely to be strictly true
for the present non-Markovian treatment, which may require
a nonlocal boundary condition. The introduction of such a
boundary condition would complicate the later calculations,
however, so in the interests of simplicity, the local boundary
condition is retained.
Equation 共13兲 now becomes
J共x,t兲 = −
␩共t兲kBT
P共xA,t兲exp关− m␻B2
m␻B2
⫻共xA − xB兲 /2kBT兴
2
冒冕
xC
dx exp关−
− xB兲2 / 2kBT兲, and the time-dependent correction, ␬共t兲, can be
identified with what is referred to as the transmission
coefficient.9,26 The transmission coefficient is therefore
␬共t兲 ⬅ −
f共t兲 =
␹˙ 共t兲
kTST
kTST/␻B+1 .
␻B ␹共t兲
The next step is the calculation of the population of particles in well A. The original treatment assumes an equilibrium distribution of particles, but we are interested in the
population at some time t, which we denote nA共t兲. We shall
assume that this population can be calculated as25
nA共t兲 ⬇ P共xA,t兲
冕
xA+␦x/2
xA−␦x/2
dx exp共− U共x兲/kBT兲,
共16兲
the integral in 共15兲 is now evaluated approximately by extending the range of integration to plus infinity and minus
infinity 共since the contributions to the integral away from the
minimum are very small兲, and then carrying out the resulting
Gaussian integral exactly. This leads to
nA共t兲 = P共xA,t兲
冑
2 ␲ k BT
m␻A2
.
共17兲
The integral in Eq. 共14兲 is evaluated similarly 共by extending
the limits to plus and minus infinity兲. Dividing the result by
nA共t兲 in Eq. 共17兲, we obtain the following time-dependent
barrier crossing rate k共t兲:
k共t兲 = −
␩共t兲
␻A exp关− m␻B2 共xA − xB兲2/2kBT兴,
2␲␻B
共18兲
which can be expressed in the form
k共t兲 ⬅ kTST␬共t兲,
冋冉 冊 册
2−2H
t
␶
共22兲
,
where E␣ is the Mittag-Leffler function27 关defined, in gen⬁
eral, by the series expansion E␣共x兲 = 兺k=0
xk / ⌫共␣k + 1兲, with
⌫共z兲 the gamma function兴, and the decay constant ␶ is defined as
␶=
冉
␨⌫共2H + 1兲
m␻B2
冊
1/共2−2H兲
.
共23兲
共15兲
where the limits on the integrals indicate that the domain of
integration encompasses only the immediate vicinity of the
reactant minimum. Approximating U共x兲 in this region by a
harmonic well,
U共x兲 = m␻A2 共x − xA兲2/2,
共21兲
This is one of the key results of this paper.
The function ␹共t兲, which is the inverse Laplace transform of ␹ˆ 共s兲, is calculated from Eq. 共9兲. For the power law
memory kernel of Eq. 共6兲 describing fractional Gaussian
noise, the function K̂共s兲 is a power law in s, so the Laplace
inverse of Eq. 共9兲 can be carried out exactly, to produce
xA
共14兲
共20兲
with ␩共t兲 given by −␹˙ 共t兲 / ␹共t兲 关cf. Eq. 共8兲兴. Hence, using Eqs.
共1兲–共3兲, the waiting time distribution f共t兲 is easily found to
be
␹共t兲 = E2−2H
m␻B2
⫻共x − xB兲2/2kBT兴.
␩共t兲
,
␻B
A. The case H = 1 / 2
When H = 1 / 2, fGn reduces exactly to white noise; the
particle dynamics is thus described by the ordinary Langevin
equation with white noise, and the Mittag-Leffler function
reduces to a simple exponential,27 so ␹共t兲 = exp共t / ␶兲 and
␹˙ 共t兲 = ␶−1 exp共t / ␶兲. Hence,
␬共t兲 =
where kTST is the barrier crossing rate obtained from transition state theory, viz., kTST = 共␻A / 2␲兲exp共−m␻B2 共xA
共24兲
independent of t. The substitution of Eq. 共24兲 into Eq. 共19兲
recovers the familiar Kramers’s expression kK for the rate
constant in the high friction limit 共defined essentially as ␨
Ⰷ ␻B兲, viz.,
kK = 共m␻A␻B/2␲␨兲exp关− m␻B2 共xA − xB兲2/2kBT兴.
共25兲
In terms of kK, the waiting time distribution is therefore
given by
f共t兲 = kK exp共− kKt兲.
共26兲
For this case, furthermore, the mean waiting time t̄, defined
as
t̄ =
共19兲
1
m␻B
=
␻ B␶
␨
冕
⬁
dttf共t兲,
共27兲
0
is just the reciprocal of Kramers’s rate constant, i.e.,
t̃ = 1/kK .
共28兲
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024904-5
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Complex chemical kinetics in single enzyme molecules
B. The case H Å 1 / 2
When H ⫽ 1 / 2, the derivative of the Mittag-Leffler function can no longer be expressed in closed form, in general.
However, there are other values of H 共besides H = 1 / 2兲, for
which such closed forms are known. These satisfy the relation 2 − 2H = 1 / n, n = 2 , 3 , 4 , . . ., and they yield the
expressions27
冋
n−1
␹共t兲 = E1/n共共t/␶兲 兲 = exp共t/␶兲 1 + 兺
1/n
k=1
册
␥共1 − k/n,t/␶兲
,
⌫共1 − k/n兲
共29兲
1
␶
␹˙ 共t兲 =
冋兺
n−1
k=1
册
共t/␶兲−k/n
+ E1/n共共t/␶兲1/n兲 ,
⌫共1 − k/n兲
共30兲
where ␥共a , b兲 in Eq. 共29兲 is the incomplete gamma function.
The case H = 3 / 4 共corresponding to n = 2兲 is of special significance. For this value of H, E2−2H共x兲 = E1/2共x兲
= exp共x2兲erfc共−x兲, where erfc共x兲 is the complementary error
function. The Mittag-Leffler function E1/2共−x1/2兲 共note the
negative sign in the argument兲 has been shown to provide a
highly satisfactory fit to experimental data on the time correlation function of distance fluctuations in proteins.4 So it is
of interest to derive an expression for f共t兲 when H = 3 / 4. We
first write down the expression for the transmission coefficient ␬共t兲. This is found from Eqs. 共20兲, 共29兲, and 共30兲 to be
冋
册
1
1
1+
冑␲冑t/␶E1/2共冑t/␶兲 .
␻ B␶
␬共t兲 =
共31兲
Then, using Eqs. 共19兲 and 共1兲–共3兲, f共t兲 is given by
f共t兲 =
␨
kK␬共t兲E1/2共共t/␶兲1/2兲−kTST/␻B .
m␻B
共32兲
For this function, one can readily establish the limits28
f共t/␶ Ⰷ 1兲 ⬃ 2−kTST/␻BkK* exp共− kK* t兲,
f共t/␶ Ⰶ 1兲 ⬃
1
冑␲
kK*
冉冊
t
␶
共33a兲
−1/2
,
共33b兲
where the parameter kK* , defined as
kK* = 共16m␻B2 /9␲␨兲kK ,
共33c兲
can be thought of as an effective or modified Kramers’s rate
constant.
The mean waiting time t̄ for the case H ⫽ 1 / 2 can no
longer be obtained in closed form, but it can be
evaluated numerically from t̄ = −兰⬁0 dtt共d / dt兲␹共t兲−kTST/␻B
= 兰⬁0 dt␹共t兲−kTST/␻B, and when H = 3 / 4, this expression reduces
to
t̄ =
冕
⬁
dt exp共− kK* t兲关1 + erf共冑t/␶兲兴−kTST/␻B ,
共34兲
0
which, if the barrier height is large, so that kTST / ␻B is small,
leads to
FIG. 2. Transmission coefficient ␬共t兲, normalized by ␬0, the value of ␬共t兲 at
the arbitrary initial time t = 0.03 s, as a function of t, as calculated from Eq.
共31兲, for the case H = 3 / 4 at three different values of ␨ / m␻2B 共0.05 s1/2, full
line; 0.9 s1/2, dashed line; and 5.0 s1/2, dotted line兲 at a fixed value of unity
共in appropriate units兲 for m␻2B.
t̄ ⬇ 1/kK*
共35兲
in close analogy to Eq. 共28兲.
V. DISCUSSION
A. The transmission coefficient
Figure 2 shows the time dependence of ␬共t兲, normalized
by the value of ␬共t兲 at the 共arbitrary兲 initial time t = 0.03 s, as
calculated from Eq. 共31兲, for three different values of ␨ / m␻B2
共0.05, 0.9, and 5.0 s1/2兲, when H = 3 / 4 and m␻B2 = 1 in appropriate units. The graphs are qualitatively similar to the results
of the simulation reported in Ref. 9 in that the transmission
coefficient is a constant at small ␨ 共over the indicated time
interval兲, and is time dependent at larger ␨. In the present
calculations, this behavior originates from the ratio of the
time t to the relaxation time scale ␶ 关which is directly proportional to ␨ at fixed values of the remaining constants, cf.
Eq. 共23兲兴: at small ␨, the ratio t / ␶ is large for any fixed
interval of time t, so the system is effectively in the long time
regime, where E1/2共共t / ␶兲1/2兲 behaves as an exponential28
共through its connection to the error function兲, and ␬共t兲 itself
is therefore practically a constant 关cf. Eq. 共31兲兴. At large ␨,
the ratio t / ␶ is small for the same interval of time t, so the
system is in the short time regime, where the behavior is
dominated by the derivative of E1/2共共t / ␶兲1/2兲, which varies as
a power law;28 ␬共t兲 therefore varies as a power law as well,
until t itself is sufficiently large that the system crosses over
into the long time regime, and reduces, effectively, to the
H = 1 / 2 case.
In the region where ␬共t兲 is time dependent 关producing a
time-dependent k共t兲兴, the system is not characterized by a
well-defined rate constant.9,26 But in this region, ␬共t兲 关as
given by Eq. 共31兲兴 has exactly the same analytic structure as
the time-dependent transmission coefficient calculated by
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024904-6
J. Chem. Phys. 125, 024904 共2006兲
S. Chaudhury and B. J. Cherayil
FIG. 3. Comparison of the normalized waiting time distribution
f共t兲 / f共t0兲 共t0 being an initial time兲 vs t 共in seconds兲, as determined from the
experiment 共Ref. 1兲 共open circles兲 and as calculated from Eq. 共32兲 共full line兲
at H = 3 / 4, m␻2B = 1 共in appropriate units兲 and ␨ / m␻2B = 0.714 s1/2. The other
parameters, viz., the time t0, the well frequency ␻A, and the barrier height,
are adjusted for best fit to the data and have been assigned the values 1.6
⫻ 10−4 s, 2␲ ⫻ 1012 s−1, and 24.6kBT, respectively.
Kohen and Tannor in the limit of large 共but finite兲 t using a
phase space reactive flux formalism.26,29 Appendix B provides a demonstration of this equivalence.
B. The waiting time distribution
The expression for the theoretical f共t兲 given in Eq. 共32兲,
and its asymptotic limits 共33a兲 and 共33b兲, demonstrate that
the behavior of f共t兲 is governed by the same factors that
determine the behavior of ␬共t兲; that is to say, in the interval
of time over which the curve is calculated, a ratio of t / ␶
much less than 1 leads to power law decay of f共t兲; a ratio of
t / ␶ much greater than 1 leads to exponential decay. Deviations from exponentiality can therefore occur when conformational fluctuations are slow on the time scale of experimental observations. Thus the combined effects of reactivity
and conformational fluctuations, as embodied in the GLE
dynamics on a bistable potential under fractional Gaussian
noise, lead to one possible microscopic model of nonexponentiality in the waiting time distributions of enzymatic reactions.
Beyond suggesting a physical picture of complex reaction dynamics, the model is also successful in reproducing
the experimental trends in the data of English et al.1 on the
enzyme ␤-galactosidase. The data were collected at four different concentrations 关S兴 of the substrate resorufin␤-D-galactopryanoside 共RGP兲. At the lowest concentration
of RGP 共10 ␮M兲, the measured f共t兲 can be fitted to a single
exponential, while at the highest concentration 共100 ␮M兲, it
shows marked deviations from exponential behavior. All four
data sets have been compared with the predictions of Eq.
共32兲, but for clarity 共and because the fit between theory and
experiment at the three lower concentrations shows exactly
the same trends兲 only the results corresponding to 关S兴
= 100 ␮M are shown here. These results are depicted in Fig.
3 on a log-linear scale of f共t兲 关normalized by f共t兲 at an initial
time t0兴 versus t 共in seconds兲 The open circles correspond to
experimental data points, and the full line to the theoretical
curve. The theoretical curve was drawn by adjusting the values of the initial time t0, the well frequency ␻A, and the
barrier height for best fit after fixing H at 3 / 4, m␻B2 at unity,
and ␨ / m␻B2 at the value of 0.714 s1/2 assigned to it in Ref. 4
to probe protein conformational dynamics. The best fit values of t0, ␻A, and the barrier height were found to be 1.6
⫻ 10−4 s, 2␲ ⫻ 1012 s−1, and 24.6kBT, respectively, which are
physically reasonable.
When ␨ / m␻B2 has the value 0.714 s1/2, the relaxation
time ␶ has the value of 0.9 s 关cf. Eq. 共23兲兴. Since t spans the
range 0.01艋 t 艋 0.16, the ratio t / ␶ is indeed small in this
interval, and nonexponentiality—specifically power law behavior of f共t兲 results.
In the simulations of Ref. 9, the nonexponentiality of the
simulated waiting time distribution is attributed to high values of ␨, which are said to correspond to strong coupling
between the system and its environment, and thereby to significant overlap in the time scales t̄ and ␶ associated with
barrier crossing and conformational relaxation, respectively.
In this interpretation, when t̄ / ␶ ⬃ O共1兲, the waiting time distribution f共t兲 departs from exponential behavior. In the
present work, t̄ can be estimated from Eq. 共34兲 by numerical
evaluation of the integral, and it is found to be approximately
0.003 s. The corresponding ratio t̄ / ␶ is therefore approximately 3 ⫻ 10−3. Hence, we find that f共t兲 becomes nonexponential when conformational fluctuations occur much more
slowly than barrier crossing. Similar analyses of the theoretical curves that best fit the experimental data at the remaining three concentrations indicate that f共t兲 likewise becomes
exponential when conformational fluctuations occur much
more rapidly than barrier crossing.
The GLE model of barrier crossing in the presence of
fractional Gaussian noise therefore provides a highly plausible microscopic description of dynamic disorder in singlemolecule enzymatic dynamics.
ACKNOWLEDGMENTS
The authors would like to acknowledge illuminating discussions with Wei Min, Sam Kou, and Sunney Xie. They
would also like to thank Brian English for single-molecule
data. One of the authors 共S.C.兲 is grateful to the Council of
Scientific and Industrial Research, Government of India, for
financial assistance.
APPENDIX A: DERIVATION OF THE EQUATION
FOR P„x , t… FROM THE GLE
The steps in this derivation are based on methods discussed in Refs. 19 and 20. See also Ref. 21 and references
therein.
Recall that near the barrier top, the GLE with fractional
Gaussian noise 共fGn兲 in the overdamped limit is given by
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024904-7
J. Chem. Phys. 125, 024904 共2006兲
Complex chemical kinetics in single enzyme molecules
− m␻B2 共x共t兲 − xB兲 = − ␨
冕
t
dt⬘K共t − t⬘兲ẋ共t⬘兲 + ␪共t兲,
共A1兲
0
where ␪共t兲 is fGn, which is related to the memory kernel K共t兲
by
具␪共t兲␪共t⬘兲典 = ␨kBTK共兩t − t⬘兩兲
共A2兲
and
¯␪共t兲 ⬅ ␹共t兲 d
dt
冕
t
0
␾共t − t⬘兲
␪共t⬘兲.
␹共t兲
具␦共x共t兲 − x兲¯␪共t兲典 =
冕
共A3兲
=−
Equation 共A1兲 is solved for x共t兲 using Laplace transforms,
1
x共t兲 = x共0兲␹共t兲 −
+ xB
冕
m␻B2
冕
t
dt⬘␾共t − t⬘兲␪共t⬘兲
t
dt⬘具¯␪共t兲¯␪共t⬘兲典
0
dt⬘␾共t⬘兲,
共A4兲
0
where ␹共t兲 and ␾共t兲 are obtained from the inverse Laplace
transforms of, respectively,
共s兲 =
␹
共s兲
␨K
冉冕
1
␹˙ 共t兲
d
x共t兲 −
␹共t兲
ẋ共t兲 =
␹共t兲
dt
m␻B2
冕
冕
␾共t − t⬘兲
dt⬘
␪共t⬘兲
␹共t兲
0
t
dt⬘␩共t⬘兲
0
冋
⫻ x共0兲 −
␾共t⬘兲
.
dt⬘
␹共t兲
0
dt⬘具¯␪共t兲¯␪共t⬘兲典
␦x共t兲
␦¯␪共t⬘兲
冔
共A13兲
.
1
m␻B2
冕
冊
冉冕
t
dt⬘ exp
0
t⬘
dt⬙␩共t⬙兲
0
册
冊
共A7兲
共A9兲
After substituting Eq. 共A7兲 in Eq. 共A9兲, it can be shown that
=−
1
m␻B2
冉冕
exp −
冊
t
t⬘
dt1␩共t1兲 .
共A15兲
1 ⳵2
⳵ P共x,t兲
⳵
= ␩共t兲 共x − xB兲P共x,t兲 + 2 4 2 P共x,t兲D共t兲,
⳵x
⳵x
m ␻B ⳵x
共A16兲
where
D共t兲 =
冕
冉冕
t
dt⬘具¯␪共t兲¯␪共t⬘兲典exp −
0
t
t⬘
冊
dt1␩共t1兲 .
共A17兲
Substituting the definition of the random variable ¯␪共t兲 关Eq.
共A12兲兴 into Eq. 共A17兲, and integrating by parts, it can be
shown that
1
d 1
D共t兲 = ␹2共t兲
2
dt ␹2共t兲
冕 冕
t
t
dt2␾共t − t1兲␾共t − t2兲
dt1
0
0
⫻具␪共t1兲␪共t2兲典.
1 ⳵
⳵ P共x,t兲
⳵
= ␩共t兲 xP共x,t兲 +
具␦共x共t兲 − x兲¯␪共t兲典
⳵t
⳵x
m␻B2 ⳵x
⳵
xB P共x,t兲,
⳵x
␦x共t兲
共A8兲
where the angular brackets denote an average over the statistics of the noise, with x共t兲 regarded as a functional of ␪共t兲.
Differentiating Eq. 共A8兲 with respect to t, and using the
chain rule, we obtain
⳵ P共x,t兲
⳵
= − 具␦共x共t兲 − x兲ẋ共t兲典.
⳵t
⳵x
共A14兲
Substituting Eqs. 共A13兲–共A15兲 into Eq. 共A10兲, we find that
t
P共x,t兲 = 具␦共x共t兲 − x兲典,
冔
where A共t兲 ⬅ xB␹共t兲共d / dt兲兰t0dt⬘关␾共t⬘兲 / ␹共t兲兴. Hence, it can be
shown that
␦¯␪共t⬘兲
The probability density distribution P共x , t兲 of finding the particle at x at time t is defined by
共A18兲
31
共A10兲
Using double Laplace transforms, and the definitions of the
functions ␾共t兲 and K共t兲, the function D共t兲 in Eq. 共A18兲 can
be simplified 共after fairly lengthy algebra兲 to
1
d 1
共1 − ␹2共t兲兲.
D共t兲 = − m␻B2 kBT␹2共t兲
2
dt ␹2共t兲
where
␹˙ 共t兲
␹共t兲
␦共x共t兲 − x兲
t
共A6兲
By definition ␹共0兲 = 1. Equation 共A4兲 can be used to obtain
an equation for ẋ共t兲 in which the initial value x共0兲 has been
eliminated. The result is
␩共t兲 ⬅
␦¯␪共t⬘兲
⫻兵¯␪共t⬘兲 − m␻B2 A共t⬘兲其 ,
共s兲 = 1 − s␹
共s兲.
␾
− ␩共t兲
␦
0
t
x共t兲 = exp −
and
d
+ xB␹共t兲
dt
冓
冓
The functional derivative in the second line of Eq. 共A13兲 is
obtained from the solution of Eq. 共A7兲, which is given by
共A5兲
共s兲 − m␻2
s␨K
B
冕
⳵
⳵x
⫻ ␦共x共t兲 − x兲
t
共A12兲
The new random variable ¯␪共t兲, being linearly related to ␪共t兲,
is also a Gaussian variable, so by Novikov’s theorem30
0
K共兩t − t⬘兩兲 = 2H共2H − 1兲兩t − t⬘兩2H−2 .
and
dt⬘
共A11兲
共A19兲
Carrying out the differentiation in Eq. 共A19兲, using the definition of ␩共t兲, and substituting into Eq. 共A16兲, we get the
desired equation for P共x , t兲,
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024904-8
J. Chem. Phys. 125, 024904 共2006兲
S. Chaudhury and B. J. Cherayil
⳵ P共x,t兲
⳵
⳵2
k BT
= ␩共t兲 共x − xB兲P共x,t兲 −
␩
共t兲
P共x,t兲.
⳵x
⳵x
⳵x2
m␻B2
共A20兲
APPENDIX B: LONG TIME LIMIT OF BARRIER
CROSSING RATE
In Ref. 15, Hanggi and Mojtabai have shown that for
non-Markovian barrier crossing governed by Eqs. 共4a兲 and
共4b兲, the rate r is given by
r = kTST␬* ,
共B1兲
where ␬* is defined as
1
␬ =
␻B
*
冋冉
¯␥2
¯ B2
+␻
4
冊 册
1/2
¯␥
−
2
共B2兲
¯ B are the t → ⬁ limits of the following
and where ¯␥ and ␻
time-dependent functions:
¯␥共t兲 = − ȧ共t兲/a共t兲,
共B3a兲
¯ B2 = − b共t兲/a共t兲,
␻
共B3b兲
␳ˆ u共s兲 =
1
s − ␻B2 + ␮s␭
2
共B7兲
,
where ␮ is a combination of numerical and phenomenological constants. The function ␳ˆ u共s兲 is now expressed as an
infinite series,
⬁
␳ˆ u共s兲 = 兺 ␻B2k
k=0
s−␭k−␭
.
共s + ␮兲k+1
2−␭
共B8兲
Introducing the parameters ␣ ⬅ 2 − ␭ and ␤ ⬅ ␭k + 2, Eq. 共B8兲
is now written as
⬁
with
a共t兲 = ␳y共t兲␳˙ u共t兲 − ␳˙ y共t兲␳u共t兲,
共B4a兲
b共t兲 = ␳˙ y共t兲␳¨ u共t兲 − ␳¨ y共t兲␳˙ u共t兲.
共B4b兲
Here, the functions ␳y共t兲 and ␳u共t兲 are defined, respectively,
by
␳y共t兲 = 1 + ␻B2
冕
t
d ␶ ␳ u共 ␶ 兲
共B5a兲
0
and
␳u共t兲 = L−1
asymptotic scaling dependence, we follow the approach used
recently by Viñales and Despósito32 to obtain correlation
function from the full phase GLE corresponding to Eqs. 共4a兲
and 共4b兲.
This approach is deployed in the following way: suppose
that K共t兲 has the power law form given in Eq. 共6兲, then K̂共s兲,
the Laplace transform of K共t兲, is a power law in s of the
general form s␭−1, with ␭ a known function of the Hurst
index H. In Laplace space, Eq. 共B5b兲 can therefore be written as
1
s2 − ␻B2 + s␨K̂共s兲/m
,
共B5b兲
where L−1 is the operation of taking the inverse Laplace
transform, s being the Laplace variable conjugate to t.
The expression for k共t兲 derived in the present work 关Eq.
共19兲兴 by the direct flux-over-population method in the overdamped limit is a special case of Eq. 共B1兲, and should coincide with it when t → ⬁. In this long time limit, and for the
special choice H = 3 / 4 for which analytic results can be derived, the rate k共t兲 is easily seen from Eqs. 共31兲 and 共23兲 to
have the following behavior 共ignoring all numerical coefficients兲:
␻B2k k!s␣−␤
␣
k+1 .
k=0 k! 共s + ␮兲
␳ˆ u共s兲 = 兺
In this form, using the results given in Ref. 32, the inverse
transform of 共B9兲 can be determined exactly in terms of derivatives of the generalized Mittag-Leffler function. This
yields
⬁
␻B2k ␣k+␤−1 共k兲
t
E␣,␤共− ␮t␣兲,
k!
k=0
␳u共t兲 = 兺
m
␻A␻B3 exp共− ⌬E/kBT兲.
␨2
共B6兲
The corresponding behavior of r 关Eq. 共B1兲兴 has so far only
been determined for K共t兲 = 0 and K共t兲 a delta function.15
When K共t兲 is a power law, as in the present calculations, it
does not appear to be possible to evaluate r exactly, but for
the purposes of comparison with Eq. 共B6兲 it should suffice to
¯ B 共i.e., the nature of
determine the scaling structure of ¯␥ or ␻
their dependence on m, ␨, ␻A, and ␻B, ignoring numerical
prefactors兲 as t becomes very large. To extract this
共B10兲
⬁
where E␣共k兲,␤共y兲 ⬅ dkE␣,␤共y兲 / dy k, and E␣,␤共y兲 ⬅ 兺k=0
y k / ⌫共␣k
+ ␤兲 is the generalized Mittag-Leffler function.
In the same way the function ␳y共t兲 can be obtained as
⬁
␻B2k ␣k+␦−1 共k兲
t
E␣,␦共− ␮t␣兲,
k!
k=0
␳y共t兲 = 1 + ␻B2 兺
共B11兲
where ␦ = ␭k + 3.
In the limit t → ⬁, we have the asymptotic result
E␣,␤共y兲 ⬃ −
1
,
y⌫共␤ − ␣兲
y ⬍ 0,
共B12兲
k!
1
.
k+1
y ⌫共␤ − ␣兲
共B13兲
and therefore
2
k共t → ⬁兲 ⬃
共B9兲
E␣共k兲,␦共y兲 ⬃ 共− 1兲k+1
Using these results and the recursion relation ⌫共x + 1兲
= x⌫共x兲, it is now easy to show that
␳u共t → ⬁兲 ⬃
1 d
E␭共␻B2 t␭/␮兲,
␻B2 dt
共B14兲
where E␭共y兲 is the ordinary Mittag-Leffler function defined
earlier. In the same way
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024904-9
␳y共t → ⬁兲 ⬃ E␭共␻B2 t␭/␮兲.
共B15兲
For the special case H = 3 / 4, the exponent ␭ has the value of
1 / 2 and the various time derivatives involving the functions
a共t兲 and b共t兲 关Eqs. 共B3兲 and 共B4兲兴 that are needed to evaluate
¯␥ can be obtained from 共B14兲 and 共B15兲. Using these relations, along with the identity27
1
d
冑
E1/2共冑z兲 =
冑␲z + E1/2共 z兲
dz
共B16兲
and the asymptotic expansion27
E␯共z兲 ⬃
冉冊
1
1
exp共z1/␯兲 + O
,
␯
兩z兩
z → ⬁,
共B17兲
we can show that
¯␥ = ¯␥共t → ⬁兲 ⬃ m2␻B4 /␨2
共B18兲
共ignoring all constants as before兲. On dimensional grounds
¯ B must scale the same way, and hence, the rate r, from Eq.
␻
共B1兲, can be seen to have the same scaling structure as k共t
→ ⬁兲 关Eq. 共B6兲兴.
Kohen and Tannor, using a reactive flux formalism,26,29
have independently derived an expression for ␬共t兲 starting
from the phase space description of Eqs. 共4a兲 and 共4b兲. This
expression 共in the notation of Ref. 15兲 is given by
␬共t兲 =
关␳2u共t兲
where
A11共t兲 = −
␳u共t兲
,
+ 共m/kBT兲A11共t兲兴1/2
冋
共B19a兲
册
1
k BT 2
␳u共t兲 − 2 兵␳2y 共t兲 − 1其 ,
m
␻B
共B19b兲
which can be reduced to
␬共t兲 =
␻B␳u共t兲
冑␳2y 共t兲 − 1 .
共B20兲
The asymptotic results derived earlier for ␳u共t兲 关Eq. 共B14兲,
along with the identity 共B16兲兴 and for ␳y共t兲 关Eq. 共B15兲兴,
when substituted into Eq. 共B20兲, after specializing to the case
␭ = 1 / 2, are easily shown to lead to exactly the same form for
␬共t兲 as Eq. 共31兲.
1
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