Article
pubs.acs.org/JPCC
Single Turnover Measurements of Nanoparticle Catalysis Analyzed
with Dwell Time Correlation Functions and Constrained Mean Dwell
Times
Maicol A. Ochoa,† Peng Chen, and Roger F. Loring*
Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, United States
ABSTRACT: Single turnover measurements of a fluorogenic
reaction at the surface of a nanoparticle provide a detailed view
of reaction dynamics at a catalyst with multiple heterogeneous
active sites. This picture must be extracted from a fluorescence
trajectory of one particle, which records individual reaction and
desorption events. We have previously proposed analyzing
fluorescence trajectories with constrained mean dwell times in
either light or dark states, which are averaged over a subensemble of
events in which the dwell time in the previous state satisfies a
criterion of being less than or greater than a specified time. We have
shown that these quantities can be used to distinguish between
correlated and independent fluctuations at multiple active sites.
Here we show that this analysis is complementary to calculating
dwell time correlation functions, whose decay with turnover index quantifies dynamical disorder in the underlying kinetics. We
analyze a measured fluorescence trajectory from a gold nanoparticle in terms of both constrained mean dwell times and dwell
time correlation functions. The analysis demonstrates that the minimal kinetic model with discrete states that is qualitatively
consistent with the data allows active sites to fluctuate among at least three substates with distinct adsorption and reaction rates.
I. INTRODUCTION
Single-molecule measurements of enzyme catalysis1−9 have the
capacity to discern the differing enzymatic activity of distinct
conformational states of the protein. Such measurements probe
a reaction which either creates or destroys a fluorescent species,
generating a binary fluorescence trajectory showing transitions
between a fluorescent state that we denote L (light) and a
nonfluorescent state that we denote D (dark). Each transition
between these states represents a molecular event, either
reaction or regeneration of the initial state of the enzyme. A
variety of statistical measures have been devised10−21 to deduce
the kinetic mechanism that underlies such data. These include
the autocorrelation functions of dwell times in states D or L as
a function of the number of turnovers separating the dwell
times.1,10,19 The decay of this correlation function reflects
conformational dynamics of the enzyme.
Chen and co-workers22−29 have extended the single-turnover
study of a fluorogenic reaction to catalysis by gold and platinum
nanoparticles. The nanoparticles catalyze the reductive Ndeoxygenation of the nonfluorescent reactant resazurin or the
oxidative N-deacetylation of the nonfluorescent reactant amplex
red, with both reactions producing the fluorescent product
resorufin. In a study of the former reaction at gold
nanoparticles,22 dwell times in both D and L states were
resolved, with the transition from D to L representing reaction
and the transition from L to D representing desorption of
product. Autocorrelation functions of dwell times in states D
and L are nonzero, indicating the presence of disorder in rate
© 2013 American Chemical Society
constants for reaction and for product desorption. These
autocorrelation functions decay with increasing number of
turnovers, showing the existence of dynamical disorder,30,31
that is, transitions among substates of an active site that affect
reaction and desorption. This dynamical disorder, which plays a
role analogous to that of conformational changes in an enzyme,
is ascribed22,29 to dynamic restructuring of the metal surface,32−40 either spontaneously or induced by adsorbed
molecules.
A single turnover study of catalysis by a nanoparticle with
multiple heterogeneous active sites raises questions that have
not arisen in the analysis of single enzyme kinetics. In
particular, do the dynamical processes that affect catalytic
activity of active sites occur independently at different sites or
in a correlated manner affecting multiple sites simultaneously?
Since a spatially resolved measurement is not feasible for a
small nanoparticle, can fluorescence trajectories from the entire
particle be analyzed to reveal the extent of correlation between
fluctuations at different active sites? To address the existence of
correlation among these fluctuations, we have proposed41
calculating constrained mean dwell times in states D and L. The
unconstrained mean dwell time in state D, for example, is the
reciprocal of the averaged rate constant for reaction, as would
be determined in a bulk measurement. The constrained mean
Received: July 2, 2013
Revised: August 14, 2013
Published: September 4, 2013
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dwell time tD̅ <(τL) is the mean time between a desorption event
and the subsequent reaction event for a constrained ensemble
of events in which the preceding dwell time in L is less than τL.
The complementary quantity tD̅ >(τL) is computed from dwell
times in D with the constraint that the preceding dwell time in
L exceeds τL. Constrained mean dwell times in L are similarly
defined, and the information carried by the dependence on τD
or τL of the four constrained mean dwell times is analyzed in ref
41. We have computed constrained mean dwell times for
kinetic models of multiple sites that change discrete substates in
either a correlated way or independently from each other, and
have demonstrated that these quantities are sensitive to
correlations among active sites. In particular, for a large system,
tL̅ <(τD) is predicted to decay with τD more slowly than tL̅ >(τD)
for correlated fluctuations, with the reverse relation holding for
independent fluctuations. This qualitative criterion allows the
assessment of experimental data without the need for
quantitative fitting of data to a model. From our calculations
of constrained mean dwell times for gold nanoparticles of 6 nm
diameter under conditions of saturating substrate concentration, we have concluded that the data are better described by
a model of spatially correlated fluctuations than by a picture of
independent fluctuations. The interpretation of the data as
reflecting fluctuations at active sites that are correlated
throughout the nanoparticle is consistent with a surface
reconstruction mechanism, as these dynamics can affect the
entire particle.32−40
In ref 41, we showed that the τ dependence of the four
constrained mean times could be described qualitatively with a
model in which each active site fluctuates between two
substates with different catalytic activity and product desorption
rates. This represents the minimal kinetic model with discrete
states consistent with these four functions. Here we construct a
minimal kinetic model that is qualitatively consistent both with
the set of four constrained mean dwell times and with the set of
four auto and cross correlation functions of dwell times. We
demonstrate that the two substate model of ref 41 is not
qualitatively consistent with measured correlation functions and
that at minimum three active site substates are required to
describe the data. In doing so, we show that the constrained
mean dwell times and the dwell time correlation functions
emphasize different aspects of the underlying dynamics and that
consistency with both sets of quantities is a rigorous criterion
for the suitability of a proposed kinetic model. In section II, we
review the definitions of dwell-time correlation functions1,10,19
and constrained mean dwell times41 and describe the
calculation of these quantities for a kinetic model41 of a
catalyst with multiple active sites undergoing correlated
transitions among discrete10 substates. Correlation functions
and constrained mean dwell times are computed for an
experimental fluorescence trajectory in section III, and the
results are analyzed with the kinetic model. Our conclusions are
summarized in section IV.
describe their calculation within the particular kinetic model of
ref 41. We define the joint dwell time distributions f (m)
AA (t, t′) for
two dwell times in state A separated by m intervening
occupancies of state B and f (m)
AB (t, t′) for a dwell time in A
following one in B with m intervening occupancies of state B.
Here A and B denote D or L and the turnover index m satisfies
(m)
m ≥ 1 for f (m)
AA (t, t′) and m ≥ 0 for f AB (t, t′). For uncorrelated
dwell times, each distribution would factor into a product of
distributions of single dwell times, e.g., f (m)
AB (t, t′) → fA(t)f B(t′).
We therefore define the contributions to these distributions
that embody correlations between dwell times
(m)
(m)
Δf AA
(t , t ′) ≡ f AA
(t , t ′) − fA (t )fA (t ′)
(1)
(m)
(m)
Δf AB
(t , t ′) ≡ f AB
(t , t ′) − fA (t )fB (t ′)
(2)
1,10,42
The correlation function
CA(m) quantifies the statistical
relation between an initial dwell time in A and a subsequent
dwell time in that state, with m ≥ 1 intervening occupancies of
B
CA(m) ≡ σA −2
∫0
∞
∫0
dt
∞
(m)
dt ′t t ′Δf AA
(t , t ′)
(3)
with σA the standard deviation of dwell times in state A. The
analogous cross correlation function CAB (m) describes
correlations between an initial dwell time in B and a later
dwell time in A after m ≥ 0 intervening residences in B
CAB(m) ≡ (σAσB)−1
∫0
∞
dt
∫0
∞
(m)
dt ′t t ′Δf AB
(t , t ′)
(4)
1,10
We follow previous authors
in defining dwell time
correlation functions that depend on a discrete turnover
index, rather than a continuous time variable. This definition
has proven its utility in analyzing fluorescence trajectories that
record discrete events, as shown in the discussion of eq 22
below.
The constrained mean dwell time41 tA̅ <(τB) is defined to be
the dwell time in state A averaged over a restricted ensemble of
events in which the preceding dwell time in B is less than τB.
The difference between this quantity and the unconstrained
mean dwell time is related to one- and two-time distributions
by
∞
Δ tA̅ <(τB) ≡
τ
(0)
(t , t ′)
∫0 dt t ∫0 B dt ′Δf AB
τ
∫0 B dt ′fB (t ′)
(5)
The complementary quantity ΔtA̅ >(τB) reflects an average over
a constrained ensemble of events in which the occupancy of A
is preceded by a dwell time in B that exceeds τB
∞
Δ tA̅ >(τB) ≡
∞
(0)
(t , t ′)
∫0 dt t ∫τ dt ′Δf AB
B
∞
∫τ dt ′fB (t ′)
B
II. OBSERVABLES AND KINETIC MODEL
The quantities we determine from binary fluorescence
trajectories, dwell time correlation functions and constrained
mean dwell times, reflect correlations between pairs of dwell
times. The experiments are treated as probing equilibrium
fluctuations in an ergodic system, in which case these quantities
may be calculated from equilibrium joint distributions of two
dwell times in the D or L states.10,12,13,19 We begin by defining
these distributions independently of any kinetic model and then
(6)
We will evaluate the four auto and cross correlation functions
defined by eqs 3 and 4 and the four constrained mean dwell
times defined by eqs 5 and 6 for a kinetic model described in
ref 41. The model catalyst contains N active sites that promote
the reaction of an adsorbed nonfluorescent reactant to generate
a fluorescent product. The reactant concentration in solution is
sufficiently high that the adsorption rate of reactant is large
compared to desorption rates of both reactant and product, so
that each active site can be assumed to be always occupied by
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The joint distribution of two dwell times in A separated by m
≥ 1 occupancies of state B is similarly10 given by
an adsorbate. Each active site may thus occupy one of two
states associated with adsorption of reactant or product. In
addition, each site is taken to have S chemically distinct internal
substates, giving each site 2S possible states. The rate constants
for reaction and desorption are substate dependent, and the
substates interconvert according to another set of rate
constants. In keeping with experimental conditions,22 we
consider only states of the catalyst in which at most one
fluorescent product molecule is adsorbed. We considered41 two
versions of this model: one in which each site fluctuates
independently among substates and a second in which the
entire catalyst changes substate so that these transitions occur
simultaneously at all active sites. We demonstrated that the
constrained mean dwell times in the fluorescent state ΔtL̅ <(τD)
and ΔtL̅ >(τD) have qualitatively different dependences on τD for
large N in these two models. In the model of independent
fluctuations, ΔtL̅ <(τD) decays more rapidly than ΔtL̅ >(τD),
while, for the model of completely correlated fluctuations, the
reverse holds. As shown in Figure 7 of ref 41 and below in
Figure 4, the nanoparticle data show the scenario more
consistent with the model of correlated fluctuations. We adopt
the model of correlated site fluctuations here.
The catalyst therefore has S nonfluorescent substates
collectively corresponding to the D state and NS fluorescent
substates collectively representing the L state, as the fluorescent
product molecule may be adsorbed at any one of N active sites.
If the catalyst is in the fluorescent state, it makes transitions
from substate α to substate γ with rate constant lγα, while this
substate transition occurs with rate constant dγα if the catalyst is
in the nonfluorescent state. The rate constant for the process L
→ D when the catalyst occupies substate α is kLα, which has the
physical significance of the rate constant for product desorption
from a single site. The rate constant for D → L in substate α is
kDα, which is the product of the number of active sites N and
the reaction rate constant at a single active site, as reaction can
occur at any site.
For this model, the equilibrium distribution of dwell
times10,13 is given by
fA (t ) = 5 −1⟨1|wA gA(t )wA |pA ⟩
5 = ⟨1|kA|pA ⟩
(m)
f AA
(t , t ′) = 5 −1⟨1|wA gA(t )TA m − 1mBkAgA(t ′)wA |pA ⟩
(9)
mB ≡ k BwB−1
(10)
TA ≡ mBmA
(11)
The dependence of the distribution on m is associated with the
turnover matrix TA defined in eq 11 which represents
integrated time evolution in state A, followed by transition to
state B, integrated time evolution in state B, and transition back
to state A.42−44 The turnover matrices TD and TL are related by
a similarity transformation and so share eigenvalues {λn}, with n
= 1, ... , S. Probability conservation dictates that one of these
eigenvalues, to which we assign n = 1, is unity, λ1 = 1, giving TA
the spectral representation
S
TA = |R1(A)⟩⟨L1(A)| +
∑ |R n(A)⟩λn⟨Ln(A)|
n=2
(12)
Since TA is not in general symmetric, we allow for the existence
of distinct right and left eigenvectors associated with each
(A)
(A)
(A)
eigenvalue, obeying TA|R(A)
n ⟩ = λn|Rn ⟩ and ⟨Ln |TA = ⟨Ln |λn.
The eigenvectors associated with the unit eigenvalue are
|R1(A)⟩ = wA |pA ⟩/5
(13)
⟨L1(A)| = ⟨1|
(14)
Substituting this representation of TA into eq 9 yields the
decomposition into a product of single-time distributions fA(t)
fA(t′), produced by the first term in eq 12, and Δf (m)
AA (t, t′),
which results from the second term in eq 12,
S
(m)
Δf AA
(t , t ′) =
∑ λnm⟨1|kAgA(t )|R n(A)⟩⟨Ln(A)|wAgA(t′)|R1(A)⟩
n=2
(15)
The joint distribution of dwell times in different states is
obtained similarly to be
(7)
S
(m)
Δf AB
(t , t ′) =
(8)
∑ λnm+ 1⟨1|kAgA(t )wAkA −1|R n(B)⟩
n=2
× ⟨Ln(B)|wBg B(t ′)|R1(B)⟩
For notational efficiency, we adopt bra-ket notation and
represent a vector by |v⟩ and its transpose (and in principle
complex conjugate) by ⟨v|. The S-dimensional vector of
equilibrium populations in state A is |pA⟩, and the Sdimensional matrix of rate constants for leaving state A, kA,
has elements δαγkAα. The S-dimensional matrix wA governs
dynamics within the A substates, and has elements (wA)αγ 
δαγ(kAα + ∑β≠α aβα) + (δαγ − 1)aαγ, with a representing l or d.
The propagator gA(t)  exp(−wAt) describes time evolution
within the manifold of A substates. The S-dimensional vector
with unit elements in the basis of substates {α} is denoted ⟨1|,
S
so that ⟨1|v⟩  ∑α=1
vα. The form of the dwell-time
distribution in eq 7 is derived by Cao in ref 10. The significance
of the form of eq 7 may be summarized by reading the
expression from right to left. 5 −1wA|pA⟩ represents the
equilibrium flux into state A from state B, gA(t) represents
time evolution among substates within state A for dwell time t,
and the leftmost application of wA represents the transition out
of state A into state B.
(16)
The dwell-time correlation functions are obtained by
performing the time integrations in eqs 3 and 4 using the
distributions in eqs 15 and 16
S
CA(m) = σA −2 ∑ λnm⟨1|wA −1|R n(A)⟩⟨Ln(A)|wA −1|R1(A)⟩
n=2
(17)
S
CAB(m) = σA −1σB−1 ∑ λnm + 1⟨1|kA −1|R n(B)⟩⟨Ln(B)|wB−1|R1(B)⟩
n=2
(18)
The constrained mean dwell times are similarly obtained from
eqs 5 and 6 using Δf (0)
AB (t, t′) in eq 16
S
Δ tA̅ <(τB) =
∑ λn
n=2
⟨1|kA −1|R n(B)⟩⟨Ln(B)|(I − g B(τB))|R1(B)⟩
⟨1|(I − g B(τB))|R1(B)⟩
(19)
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S
Δ tA̅ >(τB) =
∑ λn
n=2
Article
⟨1|kA −1|R n(B)⟩⟨Ln(B)|g B(τB)|R1(B)⟩
⟨1|g B(τB)|R1(B)⟩
0.5, kL1 = 1, kL2 = 3, d21 = d12 = l12 = 0.02, and l21 = 0.0333 in an
arbitrary time unit. This parameter set satisfies the condition11
of detailed balance, kD1l21kL2d12 = kD2l12kL1d21. These parameters
exemplify the case in which the substate with faster reaction,
substate 2, also has faster desorption of product, leading to
positive cross correlations. These calculations also depict the
scenario in which substate changes are slow relative to reaction
and desorption processes. For clarity, we plot dwell-time
correlation functions as though the discrete turnover index m
were a continuous variable, although m is only defined for
integer values. The dashed curves are determined from a
parameter set in which the values of kD1 and kD2 used for the
solid curves are interchanged, and the value of l21 is adjusted to
satisfy detailed balance, l21 = 0.00133. The cross correlations are
now negative, as the substate with faster reaction, substate 1,
has slower desorption of product. For S = 2 and slow substate
transition rates, the amplitude of each autocorrelation function
CA(m) is proportional to the square of the difference in rate
constants for A → B transitions, e.g., CD(m) ∝ (kD1 − kD2)2,
while the amplitudes of both cross correlation functions scale as
(kD1 − kD2)(kL1 − kL2). In keeping with eq 22, all correlation
functions for a given parameter set display the same decay, and
consistent with eq 23, all have comparable amplitude. The
calculations in Figure 1 illustrate that the decay rates of
correlation functions reflect dynamics of transitions among
substates,1,10 while their amplitudes reflect the relative
magnitudes of rate constants for reaction and desorption.
Constrained mean dwell times are shown in Figure 2,
calculated from eqs 19 and 20 for the same parameters used in
(20)
with I the identity in S dimensions.
The information content of the correlation functions and
constrained mean times may be explored and compared by
considering the simplest nontrivial case of two substates. For S
= 2, the turnover matrices TD and TL have a single nonunit
eigenvalue13,19
λ2 =
[(1 + d 21k D1−1 + l 21k L1−1)(1 + d12k D2−1 + l12k L 2−1)]−1
(21)
Each of the terms in this expression represents the ratio of the
rate constant for loss from a particular substate from a substate
change to the rate constant for loss from that same substate
because of reaction or desorption. For example, d21kD1−1 is the
ratio of the rate constant for the transition D1 → D2 through
change of substate to the rate constant for the transition D1 →
L1 through chemical reaction. It follows from eqs 17 and 18
that for S = 2 all correlation functions share the same singleexponential decay governed by the eigenvalue λ2,
C D(m)
C (m )
C (m )
C (m )
= L
= DL
= LD
= λ 2m − 1
C D(1)
C L(1)
C DL(1)
C LD(1)
(22)
with amplitudes that are related by
C D(1)C L(1) = λ 2−1C LD(1)C DL(1)
(23)
According to eqs 21 and 22, if substate transition rate constants
are small compared to rate constants for reaction or desorption,
then all correlation functions will decay exponentially in m with
a decay rate d21kD1−1 + l21kL1−1 + d12kD2−1 + l12kL2−1 that is
controlled by rate constants for substate interconversion. The
relation among amplitudes in eq 23 shows that if both
autocorrelation functions are nonzero, both cross correlation
functions cannot vanish.
The solid curves in Figure 1 show dimensionless dwell time
correlation functions for S = 2 for parameters kD1 = 0.1, kD2 =
Figure 2. Constrained mean dwell times in D (dark) and L (light)
states for a catalyst with two substates are shown for the parameters
used in Figure 1. Solid lines are calculated for a case in which the
substate with faster reaction has faster product desorption, while
dashed lines show a case in which the substate with faster reaction has
slower desorption. The time unit is arbitrary.
Figure 1. All times are given in the same arbitrary units that
specify the rate constants. The information content of this
quantity was extensively analyzed in ref 41 and will be briefly
summarized here. For example, ΔtD̅ <(τL) describes the mean
time for reaction, given that the preceding time to product
desorption was smaller than the specified value τL. The dashed
curve for ΔtD̅ <(τL) shows the case in which the substate with
rapid product desorption has slow reaction. For small values of
τL, ΔtD̅ <(τL) is dominated by contributions from events in
which the preceding dwell time in state L is unusually short.
This implies that the catalyst likely occupies substate 2, since
Figure 1. Dwell time correlation functions in D (dark) and L (light)
states are plotted versus turnover index for a catalyst with two
substates. Solid lines are calculated for a case in which the substate
with faster reaction has faster product desorption, while dashed lines
show a case in which the substate with faster reaction has slower
desorption.
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kL2 > kL1. Since this substate has the smaller rate constant for
reaction, the mean dwell time in D is longer than that for an
unconstrained average, giving ΔtD̅ <(τL) a positive amplitude.
This quantity decays to zero as the constraint that the dwell
time in L is shorter than τL becomes less significant. This decay
occurs approximately with rate kL1, the smaller of the two
product desorption rate constants, which controls the rate at
which the initially selected subensemble of events approaches
the unconstrained ensemble. Interpretation of this quantity is
analogous to measurements such as hole-burning spectroscopy
that probe the evolution of an initially selected subensemble
into a full ensemble.45 The complementary quantity ΔtD̅ >(τL)
vanishes at τL = 0, for which the constrained and unconstrained
ensembles are identical. As τL is increased, the dashed curve for
ΔtD̅ >(τL) becomes negative, because the system is more likely
to be found in substate 1, which has the faster rate of reaction,
so that the constrained mean time is less than the unconstrained mean, giving rise to a negative difference. This quantity
reaches its asymptote with an approximate rate of kL2, the larger
of the two desorption rate constants, since, for kL2τL ≫ 1,
further increase of τL does not change the subensemble. The
constrained mean times in state L may be interpreted with the
same reasoning, which applies in the limit in which rate
constants for desorption and for reaction are well separated in
magnitude and in which rate constants for substate change are
small compared to those for reaction and desorption. In this
case, ΔtA̅ <(τB) evolves according to the smallest of {kBj}, while
ΔtA̅ >(τB) evolves according to the largest of {kBj}. The two
complementary constrained mean times thus provide distinct
information. Like the cross correlation functions in Figure 1,
the algebraic signs of the amplitudes of the constrained mean
times carry information about correlations between desorption
and reaction rates. However, the m dependence of the
correlation functions reflects the dynamics of transitions
among substates, while the τ dependence of constrained
mean dwell times originates primarily in rate constants for
reaction and desorption in different substates. These two sets of
quantities thus provide complementary information, as will be
shown in analysis of experimental data in section III.
Figure 3. Discrete symbols show correlation functions of dwell times
in light L and dark D states versus turnover index, determined from an
experimental fluorescence trajectory of a gold nanoparticle with
diameter 6 nm. Lines are calculated for a model of active sites with
three substates undergoing correlated fluctuations. Model parameters
are determined with simultaneous fits to these data and to the
constrained mean dwell times in Figure 4.
Figure 4. Discrete symbols show constrained mean dwell times in light
L and dark D states determined from the experimental fluorescence
trajectory of Figure 3. Lines show simultaneous fits of these data and
those in Figure 3 to a kinetic model with three substates. All times are
given in s.
The experimental correlation functions in Figure 3 show
several distinctive features. The autocorrelation functions
CL(m) and CD(m) show significant amplitude at m = 1 and
decay exponentially with m at different rates. Xu et al.22 have fit
these autocorrelation functions to single exponential decays, CA
∝ exp(−m/m̅ A), with m̅ L ≈ 3 and m̅ D ≈ 13. The cross
correlation functions are zero for all m within experimental
accuracy. These properties are entirely inconsistent with the
two-substate model discussed in section II. According to eqs 22
and 23, for S = 2, all four correlation functions decay at the
same rate, and if CL(m) and CD(m) are appreciable in
magnitude, the cross correlation functions cannot both be
negligibly small.
The constrained mean dwell times computed from the same
trajectory used for Figure 3 are shown by discrete points in
Figure 4. All times are given in s. These results are similar to
those reported in ref 41 for an ensemble of shorter trajectories
for different nanoparticles of the same size and at the same
reactant concentration. For A = D and A = L, ΔtA̅ <(τB) > 0 and
ΔtA̅ >(τB) < 0. The decay rate of ΔtA̅ >(τB) is larger than that of
ΔtA̅ <(τB).
Since the correlation functions in Figure 3 are inconsistent
with S = 2, we consider the next simplest case of a kinetic
III. NANOPARTICLE CATALYSIS
Discrete symbols in Figures 3 and 4 show dwell-time
correlation functions and constrained mean dwell times
determined from a single trajectory including approximately
800 turnovers from one nanoparticle of diameter 6 nm at a
saturating resazurin concentration of 1.2 μM.22 This trajectory
was used to produce the autocorrelation functions in Figure 4
of ref 22 and the cross correlation functions in Figure S11 of
the Supporting Information.22 In ref 41, we reported
constrained mean dwell times from an ensemble of trajectories
from different nanoparticles of this size at the same reactant
concentration. We observe that constrained mean dwell times
show less variation from particle to particle than do the
correlation functions. For this reason, the constrained mean
dwell times shown here for a single trajectory are similar to
those reported in Figure 7 of ref 41 for an ensemble of
trajectories. The correlation functions determined from the
ensemble are qualitatively similar to those shown in Figure 3,
but the amplitudes are substantially reduced, obscuring
differences between auto and cross correlation functions.
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10−5(0.608)m + 0.23(0.954)m. For this case, λL ≈ 0.608 and λD
≈ 0.954. In general, the physical significance of each of the two
eigenvalues and hence of the decay rate of each autocorrelation
function is not as straightforward as in the S = 2 case in eq 21.
An approximate analysis based on the relatively small
magnitudes of rate constants for substate interconversion
provides insight into the processes underlying these eigenvalues. The turnover matrix in eq 11 is usefully written as
TD  I/(I + VD), with VD = dkD−1 + lkL−1 + dkD−1lkL−1.
Because of the small magnitudes of the rate constants for
substate transitions, we neglect the contribution to VD that is
quadratic in these rate constants. The elements of the resulting
approximation quantify transitions between substates. In
particular, the matrix element that reflects the importance of
transitions from substate 2 to substate 3 is small, d32/kD2 +
l32/kL2 ≈ 0.007. We therefore set this quantity to zero and
diagonalize the resulting approximation to VD to obtain
approximations to the nonunit eigenvalues of the turnover
matrix
model with three substates. According to eq 17, the
autocorrelation functions for S = 3 decay biexponentially with
m as
CA(m) = cA(2)λ 2m + cA(3)λ3m
(24)
cA(j)
with
determined from eq 17. The different singleexponential decays of CD(m) and CL(m) in Figure 3 imply
for this model that for A = D one of the coefficients in eq 24 is
negligibly small and that for A = L the other coefficient is
negligible, so that each autocorrelation function is dominated
by a single distinct eigenvalue, CL ∝ λmL and CD ∝ λmD.
The solid curves in Figures 3 and 4 show simultaneous fits
for the S = 3 model to the four correlation functions and the
four constrained mean dwell times. The kinetic scheme with
the resulting rate constant values is shown in Figure 5. The
λ 2 ≈ (1 + l 21k L1−1 + l12k L 2−1 + d 21k D1−1 + d12k D2−1)−1
(25)
λ3 ≈ (1 + l 23k L3
−1
−1 −1
+ d 23k D3 )
(26)
For the rate constants of Figure 5, λ2 ≈ 0.954 and λ3 ≈ 0.627,
which agree well with the correct eigenvalues controlling the
decays of the two autocorrelation functions in Figure 3, λD ≈
0.954 and λL ≈ 0.608. The eigenvalue λ2 corresponds to the
single eigenvalue in eq 21 for an S = 2 system composed of
substates 1 and 2 with associated rate constants. When terms
quadratic in rate constants for substate transitions in the
denominator of eq 21 are neglected, the result is eq 25. This
approximation suggests that the decay of CD(m) is controlled
by the subsystem containing substates 1 and 2, while the decay
of CL(m) reflects transitions from substate 3 to substate 2. This
qualitative interpretation is confirmed by calculations of CL(m),
not shown here, for the S = 2 model consisting of substates 2
and 3 and of CD(m) for the S = 2 subsystem comprising
substates 1 and 2. These correlation functions agree
quantitatively in both magnitude and decay rates with the S
= 3 results for CL(m) and CD(m) in Figure 3. This analysis
shows that the decays of the two autocorrelation functions can
reflect distinct chemical and physical processes.
The fits to the constrained mean dwell times in Figure 4 are
of comparable quality to the fits for an S = 2 model in ref 41.
The S = 3 model is necessary to provide even qualitative
agreement with the correlation functions, but addition of a third
substate is not required to represent the constrained mean
dwell times. As with the correlation functions, the interpretation of the constrained mean dwell times can be simplified by
considering an S = 2 subsystem of the S = 3 model in Figure 5.
In the scheme of Figure 5, the reaction rate constants in two of
the substates are nearly equal, kD2 ≈ kD3. The amplitudes of the
constrained mean dwell times in D are sensitive to
heterogeneity in reaction rate constants.41 For this reason, as
shown in Figure 4.3 of ref 47, ΔtD̅ <(τL) and ΔtD̅ >(τL) are
dominated by contributions from the S = 2 subsystem
composed of substates 1 and 2. The constrained mean dwell
times in L contain significant contributions from substate 3 but
are qualitatively similar to the result from the subsystem of
substates 1 and 2. Therefore, the constrained mean times can
be fit as well with two substates as with three.
Figure 5. A kinetic scheme with three substates is used to
simultaneously fit the four dwell time correlation functions in Figure
3 and the four constrained mean dwell times in Figure 4. All rate
constants are in s−1.
model has 14 rate constants constrained by two detailedbalance11 conditions, kD1l21kL2d12 = d21kD2l12kL1 and kD3l23kL2d32 =
l32kD2d23kL3. Rate constants were determined by a partitioning
optimization strategy46 that divides parameters into subsets and
optimizes the parameters in one subset, with parameters in
other subsets fixed. We employed partitioning choices in which
one subset included rate constants for a kinetically connected
subsystem, as for example the set of rate constants involving
substates 1 and 2, as well as partitioning choices not associated
with kinetically connected subsystems. This procedure was
repeated for a variety of partitioning choices until convergence
was achieved.
The resulting collection of rate constants is shown in units of
s−1 to two significant figures in Figure 5. The rate constants for
reaction are kD1 = 2.469 s−1, kD2 = 0.1754 s−1, and kD3 = 0.1561
s−1, and the rate constants for desorption are kL1 = 1.428 s−1, kL2
= 2.796 s−1, and kL3 = 0.07662 s−1. The rate constants for
substate transitions with adsorbed reactant are d12 = 0.0001688
s−1, d21 = 0.004346 s−1, d23 = 0.01230 s−1, and d32 = 0.000177
s−1. The rate constants for substate transitions with adsorbed
product are l12 = 0.04513 s−1, l21 = 0.04217 s−1, l23 = 0.03957
s−1, and l32 = 0.01849 s−1. Rate constants for substate changes
with adsorbed reactant are generally smaller than rate constants
for substate changes with adsorbed product, and rate constants
for substate changes are generally significantly smaller than rate
constants for either reaction or adsorption.
The autocorrelation functions with this parameter set indeed
have the form of eq 24 with one coefficient negligibly small:
CL(m) ≈ 0.42(0.608)m + 0.0031(0.954)m and CD(m) ≈ 8.0 ×
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The Journal of Physical Chemistry C
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Notes
IV. CONCLUSIONS
A nanoparticle catalyst includes multiple active sites, which
unlike the active site or sites of a biological catalyst are
heterogeneous. Single turnover measurements on gold nanoparticles indicate the existence of dynamical processes, which
like conformational changes in an enzyme affect catalytic
activity. We have analyzed the binary fluorescence trajectories
determined in such measurements with two complementary
statistical measures. These are the four auto and cross
correlation functions of dwell times in fluorescent and
nonfluorescent states and the four constrained mean dwell
times in fluorescent and nonfluorescent states. The decays of
the correlation functions with turnover index reflect the time
scales of dynamical processes that induce changes in the
substate of the active site. The decays of the constrained mean
dwell times reveal aspects of the distribution of rate constants
for reaction and for desorption, and can be used to distinguish
between correlated and independent fluctuations41 at different
active sites.
We have shown that for a model of discrete substates at least
three substates are necessary for qualitative agreement between
calculated and measured dwell time correlation functions and
constrained mean dwell times. The fit of a model with three
substates to the data in Figures 3 and 4 involves varying 12
parameters: the 14 parameters shown in Figure 5 with two
constraints for detailed balance. The fits shown in these figures
need not represent a global minimum in this search. However,
the fit is robust in the sense demonstrated by the analysis
described in the concluding paragraph of section III. This
analysis identifies a pair of substates within the three substate
model with a dominant contribution to the constrained mean
dwell times, thereby explaining why a model with two substates
was consistent with these four quantities, as shown in ref 41.
This analysis relies on a separation of time scales between rate
constants for substate interconversion and rate constants for
reaction and desorption and also on a pair of reaction rates, kD2
and kD3 in Figure 5, having similar magnitudes. Other parameter
sets obeying these conditions can also provide a qualitative fit
to the data in Figures 3 and 4. The fits in Figures 3 and 4 also
demonstrate that the different decay rates of the two dwell time
autocorrelation functions can reflect distinct physical processes,
as shown by the perturbation analysis in eqs 25 and 26. The
principal conclusion to be drawn from the fits reported here is
that a model with three discrete substates is consistent with
these data from nanoparticle catalysts.
Our model of correlated fluctuations is consistent with the
scenario in which substate changes arise from dynamic surface
reconstruction.32−40 However, our phenomenological kinetic
model is not based on any particular microscopic mechanism.
Our analysis of fluorescence trajectories from multisite catalysts
identifies a set of complementary dynamical properties that
must be reproduced by any molecular description of nanoparticle catalysis.
■
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
M.A.O. and R.F.L. acknowledge support from the National
Science Foundation through Grant No. CHE0743299. P.C.
acknowledges support from the Department of Energy (DEFG02-10ER16199), Army Research Office
(W911NF0910232), and National Science Foundation
(CBET-0851257).
■
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected].
Present Address
†
Department of Chemistry and Biochemistry, University of
California, San Diego, CA.
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