THE JOURNAL OF CHEMICAL PHYSICS 127, 145103 共2007兲
Modulation of electron transfer kinetics by protein conformational
fluctuations during early-stage photosynthesis
Srabanti Chaudhury and Binny J. Cherayila兲
Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560012, India
共Received 29 June 2007; accepted 20 August 2007; published online 9 October 2007兲
The kinetics of electron transfer during the early stages of the photosynthetic reaction cycle has
recently been shown in transient absorption experiments carried out by Wang et al. 关Science 316,
747 共2007兲兴 to be strongly influenced by fluctuations in the conformation of the surrounding protein.
A model of electron transfer rates in polar solvents developed by Sumi and Marcus using a
reaction-diffusion formalism 关J. Chem. Phys. 84, 4894 共1986兲兴 was found to be successful in fitting
the experimental absorption curves over a roughly 200 ps time interval. The fits were achieved using
an empirically determined time-dependent function that described protein conformational
relaxation. In the present paper, a microscopic model of this function is suggested, and it is shown
that the function can be identified with the dynamic autocorrelation function of intersegment
distance fluctuations that occur in a harmonic potential of mean force under the action of fractional
Gaussian noise. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2783845兴
I. INTRODUCTION
Fluctuations in the conformations of enzymes and other
biological macromolecules that control chemical reactions
are well known to modulate their kinetics.1 The effect of
these fluctuations can become especially pronounced when
the time scales of the fluctuations are comparable to the time
scales of the reaction. Under such conditions, the rate of the
reaction can fluctuate as well, leading to the phenomenon of
dynamic disorder.2
Within existing theoretical models,3 it was generally
necessary to assume that one of the key steps in the early
stages of the photosynthetic reaction cycle—the transfer of
an electron from a chlorophyll donor to a pheophytin
acceptor—occurred much more slowly than the structural relaxation of the surrounding protein matrix. Experiments by
Wang et al.4 now suggest that this assumption may not be
entirely valid. These experiments recorded changes to the
transient absorption at 930 and 280 nm of the chromophores
in the photosynthetic reaction center of wild-type and mutant
species of Rhodobacter sphaeroides following initiation of
electron transfer by photoexcitation at 860 nm. The signal at
930 nm monitored the kinetics of electron transfer from donor to acceptor, while the signal at 280 nm, which originates
in 39 tryptophan residues located around the reaction center
that are sensitive to changes in their environment, was used
as a measure of the dynamics of the protein during the electron transfer process.
A number of interesting findings emerged. Over an interval of time extending from about 1 to 200 ps, the 930 nm
decay profiles 共arising from electron transfer兲 of the wildtype and 14 different mutants were all different, in addition
to being highly nonexponential, whereas over the same interval of time, the 280 nm decay profiles 共ascribed to protein
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2007/127共14兲/145103/6/$23.00
dynamics兲 of the same set of species were all the same, although still nonexponential. The insensitivity of the 280 nm
signal to mutations that strongly affected electron transfer
was interpreted to mean that on the picosecond time scale,
the dynamics of the protein did not track the dynamics of
electron transfer 共as might have been expected had the
equilibration of the protein been much faster than electron
transfer兲, and were therefore independent of them. This in
turn suggested that the protein dynamics were actually
“slow” on this time scale, and could therefore potentially
modulate electron transfer rates. To test this possibility,
Wang et al. sought to fit the 930 nm decay curves to theoretical curves of the time-dependent survival probability of
the excited state electron donor, which were calculated from
a model developed by Sumi and Marcus to describe electron
transfer in polar solvents.5 The theoretical model was defined, among other things, by a time-dependent “dielectric
function” C p共t兲 characterizing the thermal fluctuations of the
solvent and by the free energy difference ⌬G0 between donor
and acceptor states of the system. By fitting C p共t兲 to a triexponential that was in semiquantitative agreement with the
280 nm decay curves, Wang et al. found that the 930 nm
signals could be successfully reproduced by adjustment of
⌬G0 alone.
The degree of agreement between the experimental and
theoretical decay curves is actually quite remarkable, given
the simplicity of the Sumi-Marcus model and the complexity
of the system to which it is applied. In the context of protein
dynamics, however, the function C p共t兲 lacks a clear physical
interpretation, and its identification with an empirically determined quantity, the transient absorption signal at 280 nm,
seems somewhat arbitrary. In this paper, we would like to
present a microscopic model of these dynamics that in conjunction with the Sumi-Marcus description of electron transfer rates also provides excellent fits between experimental
and theoretical decay curves for all the 15 species of bacte-
127, 145103-1
© 2007 American Institute of Physics
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145103-2
J. Chem. Phys. 127, 145103 共2007兲
S. Chaudhury and B. J. Cherayil
rium considered in Ref. 4. Our model also contains a function analogous to C p共t兲, but its microscopic origins are fairly
well defined, and it can be calculated independently from
within the model itself. To fit the results of the model to the
experimental data, it turns out that one additional parameter,
aside from ⌬G0, must be adjusted; this parameter is a time
scale ␶ associated with fluctuations of the protein’s conformations.
The section below presents details of our model, while a
final concluding section discusses its theoretical predictions.
具P共t兲典 ⬅ S共t兲 = P共0兲具exp共−
dt⬘k共x共t⬘兲兲兲典.
共2兲
0
To calculate the average, we expand the exponential in a
series of cumulants, truncating at second order.8 This yields
冋冕
t
S共t兲 ⬇ P共0兲exp −
+
1
2
冕 冕
t
0
dt1具k共x共t1兲兲典
0
t
dt2兵具k共x共t1兲兲k共x共t2兲兲典 − 具k共x共t1兲兲典
dt1
0
II. THEORETICAL BACKGROUND
In the experiments described in Ref. 4, the intensity of
the transient absorption signal at different instants of time t is
directly proportional to the probability P共t兲 that the excited
state that leads to electron transfer survives up to that time.
This state decays with a rate k that is itself time dependent as
a result of dynamic disorder originating in slow fluctuations
of the protein conformational coordinates. Such fluctuations
are ultimately the result of motions of individual atoms or
groups of atoms and are intrinsically multidimensional, span
a wide range of time scales, and are highly correlated.6 To
couple such motions to reaction dynamics, these multiple
molecular degrees of freedom are typically projected onto an
abstract reaction coordinate x that evolves stochastically in a
potential of mean force U共x兲 connecting reactant to product
minima across a free energy barrier.7
A definite model of chemical reactivity in the presence
of dynamic disorder is specified by specifying the rate equation for P共t兲, the x dependence of k, and the time evolution
of x共t兲. We shall define our model of electron transfer by
discussing each of these elements in turn.
冕
t
册
⫻具k共x共t2兲兲典其 .
共3兲
We further assume that the correlation functions in Eq. 共3兲
are stationary 共as they are likely to be experimentally兲. The
second cumulant in this equation then becomes a function
only of 兩t1 − t2兩, which makes it possible to simplify the
double integral to 2兰t0dt⬘共t − t⬘兲f共t⬘兲, where f共t兲 is defined as
f共t兲 = 具k共x共t兲兲k共x共0兲兲典 − 具k共x共t兲兲典具k共x共0兲兲典. Our general expression for the survival probability S共t兲 is therefore given by
冋冕
t
S共t兲 = P共0兲exp −
dt⬘具k共x共t⬘兲兲典 +
0
冕
t
dt⬘共t − t⬘兲
0
册
⫻兵具k共x共t⬘兲兲k共x共0兲兲典 − 具k共x共t⬘兲兲典具k共x共0兲兲典其 ,
共4兲
and it is expected to be valid under conditions where the rate
fluctuations are small and fast. Nonperturbative techniques
can be applied to treat other dynamical regimes.9
B. The electron transfer rate
A. The survival probability
Assuming that the decay of the excited state in the experiments of Ref. 4 follows first-order kinetics, we see that
the probability density function P共t兲 must satisfy the following equation:
⳵ P共t兲
= − k共x共t兲兲P共t兲,
⳵t
共1兲
where, as a result of dynamic disorder, the first-order rate
constant k is taken to depend on the random variable x.2 In
the model of Sumi and Marcus 共which is discussed further in
the following section兲, x is a scalar component of the solvent
polarization induced by charge redistribution.5 In the present
model, x no longer has this connotation, but it is likewise a
projection onto a single dimension of the coordinates that
define the instantaneous state of the medium, the medium in
this case being a protein. Although a more precise identification of x is probably unwarranted, it is clearly a measure of
the distance between the residues whose fluctuations are
principally responsible for modulating the kinetics of electron transfer.
The solution of Eq. 共1兲, averaged over the distribution of
x 共the average being denoted by angular brackets兲, is the
quantity that will actually be compared with the measured
transient absorption curve; it is given by
The model of electron transfer introduced by Sumi and
Marcus involves a space of two dynamical variables, a “fast”
coordinate q associated with the vibrational modes of the
donor-acceptor pair and a “slow” coordinate x associated
with the polarization of the surrounding solvent. The effective free energy surfaces V共q , x兲 of the reactant 共r兲 and product 共p兲 states of the system are assumed to be harmonic and
diagonal in these variables; that is,
Vr共q,x兲 = 21 aq2 + 21 bx2 ,
共5a兲
V p共q,x兲 = 21 a共q − q0兲2 + 21 b共x − x0兲2 + ⌬G0 ,
共5b兲
where a and b are defined, respectively, as ␮␯2 and m␻2,
with ␮ and m being effective masses, and ␯ and ␻ being
angular frequencies; aq20 / 2 is the fast component, ␭ f , of the
total reorganization energy ␭, and bx20 / 2 is the corresponding
slow component, ␭s, such that ␭ f + ␭s = ␭; and ⌬G0 is the
standard free energy of the reaction. From a comparison of
Eqs. 共5a兲 and 共5b兲 with the second-order expansion of the
electronic energy around the equilibrium state of the reactant
and product, the variables x and x0 are given specifically by10
x2 =
4␲
c
冕
ex
dr兩Pex共r兲 − Pr,0
共r兲兩2
共6a兲
and
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145103-3
x20 =
4␲
c
冕
ex
2
dr兩Pex
p,0共r兲 − Pr,0共r兲兩 ,
共6b兲
where Pex共r兲 is the difference between the total polarization
of the solvent, P共r兲, and its electronic polarization, P⬁共r兲
ex
共r兲 and Pex
关i.e., Pex共r兲 = P共r兲 − P⬁共r兲兴, and Pr,0
p,0共r兲 are the
corresponding differences for the reactant and product, respectively, at equilibrium; the constant c is defined as 1 / ␧⬁
− 1 / ␧0, where ␧⬁ and ␧0 are the infinite and zero frequency
values of the permittivity.
These two free energy surfaces intersect along the locus
of points defined by Vr共q* , x兲 = V p共q* , x兲, which yields q*
= 共␭ + ⌬G0 − x冑2␭sb兲 / 冑2␭ f a. The free energy barrier, ⌬G*共x兲,
between the reactant and product states is Vr共q* , x兲
2
− Vr共0 , x兲, so from Eq. 共5a兲 we have ⌬G*共x兲 = aq* / 2. The
rate of electron transfer k共x兲 共a function of the state of the
environment兲 is therefore given by k共x兲 = ␯q exp
共−⌬G*共x兲 / kBT兲, where ␯q is a frequency factor, which can be
determined from quantum mechanical perturbation theory.11
The rate k共x兲 finally takes the form
k共x兲 =
J2
ប
冑
␲
exp关− 共␮ − ␥x兲2兴,
␭ f k BT
共7兲
where J is a matrix element that couples electronic states of
the reactant and product, ប is Planck’s constant divided by
2␲, kB is Boltzmann’s constant, T is the temperature, and ␮
and ␥ are defined by the relations ␮ = 共⌬G0 + ␭兲 / 冑4␭ f kBT and
␥ = 冑␭sm␻2 / 2␭ f kBT.
The survival probability defined in Eq. 共4兲 will be evaluated using the above expression for k共x兲.
C. The dynamics of the reaction coordinate
d2x共t兲
=−␨
dt2
冕
t
0
dt⬘K共t − t⬘兲ẋ共t⬘兲 −
dU共x兲
+ ␰共t兲.
dx
K共兩t − t⬘兩兲 = 共1/␨kBT兲具␰共t兲␰共t⬘兲典.
共8a兲
Here ␨ is the friction coefficient of the particle, ␰共t兲 is a noise
term representing the effects of protein conformational fluctuations, and K共t兲 is a memory function, which is related to
the noise through a fluctuation-dissipation theorem, i.e.,
共8b兲
The potential U共x兲 in Eq. 共8a兲 is the same function that appears in Eq. 共5a兲, and is therefore given by U共x兲 = m␻2x2 / 2.
The random variable ␰共t兲, following Kou and Xie,12 is chosen to coincide with the stochastic process referred to as
fractional Gaussian noise.13 With this assumption, the
memory function becomes
K共兩t − t⬘兩兲 = 2H共2H − 1兲兩t − t⬘兩2H−2 ,
共9兲
where H, the Hurst index,13 is a real number lying between
1 / 2 and 1 that is a measure of the temporal correlations in
the noise. The value H = 3 / 4 appears to be somewhat special,
as it yields good fits to relaxation data from several distinct
protein systems, including flavin reductase,14 the fluoresceinantifluorescein complex,15 ␤-galactosidase,16 and cholesterol
oxidase.17
Since the environment in which the variable x evolves is
liquidlike, it is reasonable to assume overdamped conditions
and to neglect the inertial contribution 共md2x / dt2兲 to Eq.
共8a兲. As shown elsewhere,18–20 the resulting equation can
then be transformed exactly to the following Smoluchowski
equation for the probability density function P共x , t兲:
冋
册
⳵ P共x,t兲
⳵
k BT ⳵ 2
= ␩共t兲
x+
P共x,t兲.
⳵t
⳵x
m␻2 ⳵x2
共10兲
Here ␩共t兲 is a time-dependent diffusion constant that is defined as
␩共t兲 = −
In order to apply the Sumi-Marcus model of solventmediated electron transfer to the experiments of Ref. 4,
where the environment around the donor and acceptor moieties is a protein, we shall identify x with the distance separating some pair of amino acid residues involved in the
modulation of the electron transfer reaction, rather than with
the state of polarization of the solvent. With this identification, the dynamics of x can be modeled by a particle of mass
m moving stochastically under the action of thermal noise in
a potential of mean force U共x兲. This model was originally
developed by Kou and Xie12 to explore the anomalous dynamics of intersegment distance fluctuations in single proteins, and we shall use it here to explore the effects of dynamic disorder on electron transfer. The model is defined by
the following generalized Langevin equation for the evolution of x:
m
J. Chem. Phys. 127, 145103 共2007兲
Modulation of electron transfer kinetics by protein fluctuations
d
ln ␹共t兲,
dt
共11兲
and ␹共t兲 is proportional to the dynamic autocorrelation function of x, specifically, ␹共t兲 = 具x共t兲x共0兲典 / 共kBT / m␻2兲. Precisely
this form of the diffusion equation has been used by Wang
et al.4 to model their data, but there the diffusion constant
analogous to ␩共t兲 关denoted C p共t兲 in that work兴 is determined
empirically and is expressed as a sum of exponentials. In the
related work of Zhu and Rasaiah,10 which applies the SumiMarcus formalism to electron transfer in non-Debye solvents
and is elaborated around exactly the same kind of diffusion
equation as Eq. 共10兲, the corresponding time-dependent diffusion constant is identified with the time correlation function of the solvation energy.
Unlike the dielectric function C p共t兲 of Ref. 4, the function ␹共t兲 can be calculated within the framework of the
model itself. Such a calculation yields ␹共t兲 = E2−2H
共−共t / ␶兲2−2H兲, where Ea共z兲 is the Mittag-Leffler function of
index a,21 and ␶ is a decay constant, defined as ␶ = 共␨⌫共2H
+ 1兲 / m␻2兲1/共2−2H兲, ⌫共x兲 being the gamma function. The function ␹共t兲 is therefore intrinsically multiexponential.
As may be verified by direct substitution, the solution of
Eq. 共10兲 subject to the initial condition x = x0 at time t = 0 is
the Gaussian distribution
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145103-4
J. Chem. Phys. 127, 145103 共2007兲
S. Chaudhury and B. J. Cherayil
P共x,t兩x0,0兲 =
冑
m␻2
2␲kBT共1 − ␹2共t兲兲
冋
⫻exp −
册
m␻2共x − x0␹共t兲兲2
.
2kBT共1 − ␹2共t兲兲
共12兲
At long times t → ⬁, this expression should evolve to the
equilibrium distribution Ps共x兲. Since ␹共t兲 → 0 as t → ⬁, one
easily sees that
冑
Ps共x兲 =
冋
册
m␻2
m ␻ 2x 2
.
exp −
2 ␲ k BT
2kBT
共13兲
Given these two distributions, one can now determine the
correlation functions 具k共x共t兲兲k共x共0兲兲典 and 具k共x共t兲兲典
= 具k共x共0兲兲典 from the relations
具k共x共t兲兲k共x共0兲兲典
冕 冕
⬁
=
⬁
dx
dx0k共x兲P共x,t兩x0,0兲k共x0兲Ps共x0兲
共14a兲
−⬁
−⬁
and
具k共x共t兲兲典 = 具k共x共0兲兲典 =
冕
⬁
dxk共x兲Ps共x兲.
共14b兲
FIG. 1. Comparison of the survival probability S共t兲 as calculated from Eqs.
共15a兲 and 共15b兲 共full lines兲 with the transient absorbance changes as determined from experiment 共Ref. 4兲 共symbols兲. Details of the values assigned to
the various parameters in Eqs. 共15a兲 and 共15b兲 are provided in Sec. III. The
experimental curves correspond to data from the wild type and five different
mutants of Rhodobacter sphaeroides, and reproduce the curves in Fig. 3共A兲
of Ref. 4. The codes identifying different mutants are explained in Ref. 4.
−⬁
Using the expression for k共x兲 in Eq. 共7兲, the integrals in Eqs.
共14a兲 and 共14b兲 can be easily evaluated in closed form; the
results, after substitution into Eq. 共4兲 关with P共0兲 set to 1兴
finally yield
冋 冑
S共t兲 ⬵ exp − ␣t
+ ␣2
冕
冋
c
c␮2
2 exp −
c+␥
c + ␥2
册
t
dt⬘共t − t⬘兲g共t⬘兲 ,
0
册
共15a兲
where
␣ ⬅ 共J2/ប兲冑␲/␭ f kBT, c = m␻2/2kBT,
and
g共t兲 =
c
冑␥2共c + ␥2共1 − ␹2共t兲兲兲 + c共c + ␥2兲
exp
−
冋
2c␮2
c + ␥2共1 + ␹共t兲兲
冋
册
册
2c␮2
c
exp
−
.
c + ␥2
c + ␥2
共15b兲
The right-hand side of Eq. 共15a兲 共which does not appear to
be reducible to a simpler closed form, and which must therefore be evaluated numerically兲 is the quantity we shall compare with experiment. The results of this comparison are discussed in the next section.
III. RESULTS AND DISCUSSION
The evaluation of Eq. 共15a兲 begins by assigning definite
values to various fixed parameters in the model. These parameters include J, ␭ f , ␭s, T, m␻2 / kBT, and H. The values of
the first three 共J, ␭ f , and ␭s兲 are taken directly from Ref. 4
and are given by 39 cm−1, 280 meV, and 70 meV, respectively. The temperature T is chosen to correspond to room
temperature and is set to 300 K. The parameter m␻2 / kBT is
set to 0.48 Å−2, a value that we have found yields the best
results when adjusting other parameters to fit the experimental data; this assignment fixes the value of m␻2, but this
value is not needed. Finally, the Hurst index H, which specifies the nature of the noise correlations, is chosen to be 3 / 4.
There are two reasons for this choice: one, it allows the
Mittag-Leffler function to be represented in terms of simpler
functions 关specifically, E2−2H共z兲 becomes E1/2共z兲 which can
be expressed as exp共z2兲erfc共−z兲, where erfc共z兲 is the complementary error function兴, and two, based on our earlier work,
it can be expected to provide a satisfactory description of
dynamic distance correlations in proteins. 共The fundamental
constants h and kB are assigned their usual values, viz.,
6.626⫻ 10−34 J s and 1.381⫻ 10−23 J K−1, respectively.兲 With
these assignments, the parameter ␣ 共defined as
J2冑␲ / ប冑␭ f kBT兲 has the value of 0.74 ps−1, and the parameter
␥ 关cf. Eq. 共7兲兴 has the value of 0.25 Å−1.
Two parameters of the model remain unspecified. They
are the free energy ⌬G0 关or equivalently the parameter ␮, cf.
Eq. 共7兲兴 and the decay constant ␶ that appears in the expression for the correlation function ␹共t兲 关which is defined in
terms of the Mittag-Leffler function in the paragraph after
Eq. 共11兲兴. These two parameters, ␮ and ␶, are now adjusted
for best fit of S共t兲 关evaluated numerically from Eq. 共15a兲兴
with each of 13 selected experimental decay curves. 共Two of
the experimental curves have been omitted because they lie
very close to each other, making it difficult to distinguish
them.兲 The nature of these fits is shown in Figs. 1 and 2, and
the best fit values of ␮ and ␶ are shown in Table I. From the
values of ␮ so obtained, one calculates the corresponding
values of ⌬G0. Table I shows these ⌬G0 values relative to the
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145103-5
Modulation of electron transfer kinetics by protein fluctuations
FIG. 2. Comparison of the survival probability S共t兲 as calculated from Eqs.
共15a兲 and 共15b兲 共full lines兲 with transient absorbance changes as determined
from experiment 共Ref. 4兲 共symbols兲 for a second set of R. sphaeroides
mutants. The theoretical curves are constructed with the same parameter
values as used in Fig. 1 共see Sec. III for details兲. The experimental curves
reproduce seven of the curves in Fig. S2共A兲 of the online supporting material of Ref. 4, two of the original curves being omitted for clarity.
⌬G value of the wild type taken as a reference, both for the
0
fits obtained in Ref. 4 共denoted ⌬⌬Gexp
兲 and for those ob0
tained here 共denoted ⌬⌬Gtheor兲.
The theoretical and experimental curves of Figs. 1 and 2
are seen to be in close agreement, the quality of the fits being
about the same as that obtained in Ref. 4. The orders of
magnitude of the respective ⌬⌬G values are also very similar. A notable difference, though, is that the fits in Ref. 4 are
achieved through the use of a single 共empirical兲 relaxation
function, C p共t兲, whereas those in the present calculation are
achieved through the use of multiple relaxation functions,
0
TABLE I. Estimates of the parameters ␶ and ␮ for 15 reaction center mutants of R. sphaeroides as obtained from best fits of S共t兲 关Eqs. 共15a兲 and
0
is
共15b兲兴 to experimental transient absorbance data 共Figs. 1 and 2兲. ⌬⌬Gexp
the free energy change relative to the wild type as estimated in Ref. 4.
0
is the corresponding free energy change as calculated from the
⌬⌬Gtheor
definition of ␮ given after Eq. 共7兲 using the best-fit ␮.
Species
L131LH+ M160LH+ M197FH
L153HD
L131LH+ M197FH
L131LH+ M160LH
M203GL
L131LH
L153HF
M197FH
L153HS
L153HV
M160LH
Wild type
L170ND
L168HE
L168HF
␶ 共ps兲
␮
0
⌬⌬Gexp
共meV兲
210
190
185
180
165
154
89
70
60
55
50
40
31
21
6
1.87
1.73
1.65
1.64
1.51
1.45
1.25
1.17
1.12
1.11
1.10
0.96
0.95
0.7
0.6
180
148
140
136
105
99
57
40
37
28
27
0
−7
−48
−75
0
⌬⌬Gtheor
共meV兲
155
131
118
117
93
85
50
37
28
26
25
0
−0.7
−44
−60
J. Chem. Phys. 127, 145103 共2007兲
FIG. 3. Comparison of the function ␹共t兲 = E1/2共−共t / ␶兲1/2兲 at two different
values of the decay constant ␶ 共full lines兲 with the function C p共t兲 = ae−t/3 ps
+ be−t/10 ps + ce−t/190 ps at two different sets of a, b, and c values 共dotted
lines兲. In the lower curves 共A兲, ␶ is 6 ps, and a, b, and c are, respectively,
0.45, 0.3, and 0.25. In the upper curves 共B兲, ␶ is 35 ps, and a, b, and c are,
0.25, 0.25, and 0.5, respectively.
␹共t兲, having different decay times ␶. The implications of this
difference are not entirely clear, but the following considerations may be germane to its understanding. As indicated in
the online supplementary material of Ref. 4, the relative
weights of the three exponentials used to construct C p共t兲 may
be varied by as much as a factor of 2 without undue change
to the fits and only “minor changes in the driving force,”
even though the change to C p共t兲 itself is quite significant.
Within this factor of 2 window, a set of C p共t兲’s may be constructed that track, qualitatively, the decay profiles of ␹共t兲’s
whose ␶ values lie between about 6 and 35 ps. Figure 3
illustrates this point by comparing two such C p共t兲 functions
with two of the ␹共t兲’s used in Figs. 1 and 2. The magnitude
of the minor changes in the driving force alluded to above
has not been specified, but it appears to be on the order of
±30 meV, since this is the range within which agreement is
claimed between the ⌬⌬G0 values obtained from the fit to
the absorption data and those estimated independently from
electrochemical measurements.4 This range encompasses the
differences between our estimates of ⌬⌬G0 and the estimates
reported in Ref. 4, so for at least a subset of mutants, the two
sets of calculations extract essentially the same information
from the experimental data, and there is in effect no real
difference between using a single C p共t兲 function or multiple
␹共t兲 functions to do so.
However, our fits to the experimental data also use ␹共t兲’s
in which the ␶ values are as high as 210 ps. For C p共t兲 to track
the decay of such functions, the relative ratio of slow
共190 ps兲 to fast 共3 ps兲 components would need to be made
much higher, and it is then no longer clear whether under
such conditions good fits to the data could still be achieved
with only about ±30 meV changes to ⌬⌬G0.
Nevertheless, we believe that these results suggest that
the previously unknown dielectric function C p共t兲 can be
identified with the function ␹共t兲, and should therefore be interpreted in terms of intersegment distance fluctuations that
Downloaded 10 Oct 2007 to 203.200.43.195. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
145103-6
J. Chem. Phys. 127, 145103 共2007兲
S. Chaudhury and B. J. Cherayil
take place in a harmonic potential of mean force in the
neighborhood of the photosynthetic reaction center. These
fluctuations are the result of fractional Gaussian noise originating in the many-body dynamics of the protein, and they
share exactly the same statistical properties as the distance
fluctuations found in a number of other protein systems.14–17
Furthermore, although the decay of the fluctuations is described by a single functional form 共the Mittag-Leffler function兲, there is no reason, a priori, to expect the fluctuations
of different mutants to decay with exactly the same decay
constant ␶. So the fact that it is necessary to use different ␶’s
共and different ⌬G0’s兲 for different mutants in our model is
physically reasonable. But more incisive single-molecule experiments may be needed to reveal whether distance fluctuations in different mutants do in fact conform to the predictions of this model.
Single-molecule experiments that exploit electron transfer between donor and acceptor species in a protein environment to probe such fluctuations have actually been carried
out,14,15 but in these experiments the modulation of the electron transfer rates is caused not by ⌬G0 fluctuations but by
fluctuations in J 关cf. Eq. 共7兲兴, the electronic coupling matrix
element. The latter quantity depends exponentially on the
distance x between donor and acceptor,11 and it has been
shown that the same model of conformational dynamics
above provides a satisfactory description of the timedependent correlations in the fluctuations of this distance.
To conclude, we have shown how a fairly realistic microscopic model of protein dynamics based on the generalized Langevin equation with fractional Gaussian noise accounts in detail for the effects of dynamic disorder on
electron transfer kinetics during the early stages of photosynthesis. That this model also provides a unified description of
a number of other phenomena, including barrier crossing,18,22
structural relaxation in liquid crystals,23 intermittency,17共b兲
and enzyme kinetics24 seems quite remarkable.
ACKNOWLEDGMENTS
The authors are grateful to Professor Neal Woodbury and
his group at Arizona State University for making available
their original data on the transient absorption profiles of
wild-type and mutant species of Rhodobacter sphaeroides.
These data were used in the comparisons of theory and experiment shown in Figs. 1 and 2. One of the authors 共S.C.兲
acknowledges financial support from the Council of Scientific and Industrial Research 共CSIR兲, Government of India.
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1
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