single turnover measurements of nanoparticle catalysis...

Single Turnover Measurements of Nanoparticle Catalysis Analyzedwith Dwell Time Correlation Functions and Constrained Mean DwellTimesMaicol 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 fluorogenicreaction at the surface of a nanoparticle provide a detailed viewof reaction dynamics at a catalyst with multiple heterogeneousactive sites. This picture must be extracted from a fluorescencetrajectory of one particle, which records individual reaction anddesorption events. We have previously proposed analyzingfluorescence trajectories with constrained mean dwell times ineither light or dark states, which are averaged over a subensemble ofevents in which the dwell time in the previous state satisfies acriterion of being less than or greater than a specified time. We haveshown that these quantities can be used to distinguish betweencorrelated and independent fluctuations at multiple active sites.Here we show that this analysis is complementary to calculatingdwell time correlation functions, whose decay with turnover index quantifies dynamical disorder in the underlying kinetics. Weanalyze a measured fluorescence trajectory from a gold nanoparticle in terms of both constrained mean dwell times and dwelltime correlation functions. The analysis demonstrates that the minimal kinetic model with discrete states that is qualitativelyconsistent 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 thecapacity to discern the differing enzymatic activity of distinctconformational states of the protein. Such measurements probea reaction which either creates or destroys a fluorescent species,generating a binary fluorescence trajectory showing transitionsbetween a fluorescent state that we denote L (light) and anonfluorescent state that we denote D (dark). Each transitionbetween these states represents a molecular event, eitherreaction or regeneration of the initial state of the enzyme. Avariety of statistical measures have been devised10−21 to deducethe kinetic mechanism that underlies such data. These includethe autocorrelation functions of dwell times in states D or L asa function of the number of turnovers separating the dwelltimes.1,10,19 The decay of this correlation function reflectsconformational dynamics of the enzyme.Chen and co-workers22−29 have extended the single-turnover

study of a fluorogenic reaction to catalysis by gold and platinumnanoparticles. The nanoparticles catalyze the reductive N-deoxygenation of the nonfluorescent reactant resazurin or theoxidative N-deacetylation of the nonfluorescent reactant amplexred, with both reactions producing the fluorescent productresorufin. In a study of the former reaction at goldnanoparticles,22 dwell times in both D and L states wereresolved, with the transition from D to L representing reactionand the transition from L to D representing desorption ofproduct. Autocorrelation functions of dwell times in states Dand L are nonzero, indicating the presence of disorder in rate

constants for reaction and for product desorption. Theseautocorrelation functions decay with increasing number ofturnovers, showing the existence of dynamical disorder,30,31

that is, transitions among substates of an active site that affectreaction and desorption. This dynamical disorder, which plays arole analogous to that of conformational changes in an enzyme,is ascribed22,29 to dynamic restructuring of the metal sur-face,32−40 either spontaneously or induced by adsorbedmolecules.A single turnover study of catalysis by a nanoparticle with

multiple heterogeneous active sites raises questions that havenot arisen in the analysis of single enzyme kinetics. Inparticular, do the dynamical processes that affect catalyticactivity of active sites occur independently at different sites orin a correlated manner affecting multiple sites simultaneously?Since a spatially resolved measurement is not feasible for asmall nanoparticle, can fluorescence trajectories from the entireparticle be analyzed to reveal the extent of correlation betweenfluctuations at different active sites? To address the existence ofcorrelation among these fluctuations, we have proposed41

calculating constrained mean dwell times in states D and L. Theunconstrained mean dwell time in state D, for example, is thereciprocal of the averaged rate constant for reaction, as wouldbe determined in a bulk measurement. The constrained mean

Received: July 2, 2013Revised: August 14, 2013Published: September 4, 2013

Article

pubs.acs.org/JPCC

© 2013 American Chemical Society 19074 dx.doi.org/10.1021/jp4065246 | J. Phys. Chem. C 2013, 117, 19074−19081

pubs.acs.org/JPCC

dwell time tD<(τL) is the mean time between a desorption eventand the subsequent reaction event for a constrained ensembleof events in which the preceding dwell time in L is less than τL.The complementary quantity tD>(τL) is computed from dwelltimes in D with the constraint that the preceding dwell time inL exceeds τL. Constrained mean dwell times in L are similarlydefined, and the information carried by the dependence on τDor τL of the four constrained mean dwell times is analyzed in ref41. We have computed constrained mean dwell times forkinetic models of multiple sites that change discrete substates ineither a correlated way or independently from each other, andhave demonstrated that these quantities are sensitive tocorrelations among active sites. In particular, for a large system,tL<(τD) is predicted to decay with τD more slowly than tL>(τD)for correlated fluctuations, with the reverse relation holding forindependent fluctuations. This qualitative criterion allows theassessment of experimental data without the need forquantitative fitting of data to a model. From our calculationsof constrained mean dwell times for gold nanoparticles of 6 nmdiameter under conditions of saturating substrate concen-tration, we have concluded that the data are better described bya model of spatially correlated fluctuations than by a picture ofindependent fluctuations. The interpretation of the data asreflecting fluctuations at active sites that are correlatedthroughout the nanoparticle is consistent with a surfacereconstruction mechanism, as these dynamics can affect theentire particle.32−40

In ref 41, we showed that the τ dependence of the fourconstrained mean times could be described qualitatively with amodel in which each active site fluctuates between twosubstates with different catalytic activity and product desorptionrates. This represents the minimal kinetic model with discretestates consistent with these four functions. Here we construct aminimal kinetic model that is qualitatively consistent both withthe set of four constrained mean dwell times and with the set offour auto and cross correlation functions of dwell times. Wedemonstrate that the two substate model of ref 41 is notqualitatively consistent with measured correlation functions andthat at minimum three active site substates are required todescribe the data. In doing so, we show that the constrainedmean dwell times and the dwell time correlation functionsemphasize different aspects of the underlying dynamics and thatconsistency with both sets of quantities is a rigorous criterionfor the suitability of a proposed kinetic model. In section II, wereview the definitions of dwell-time correlation functions1,10,19

and constrained mean dwell times41 and describe thecalculation of these quantities for a kinetic model41 of acatalyst with multiple active sites undergoing correlatedtransitions among discrete10 substates. Correlation functionsand constrained mean dwell times are computed for anexperimental fluorescence trajectory in section III, and theresults are analyzed with the kinetic model. Our conclusions aresummarized in section IV.

II. OBSERVABLES AND KINETIC MODELThe quantities we determine from binary fluorescencetrajectories, dwell time correlation functions and constrainedmean dwell times, reflect correlations between pairs of dwelltimes. The experiments are treated as probing equilibriumfluctuations in an ergodic system, in which case these quantitiesmay be calculated from equilibrium joint distributions of twodwell times in the D or L states.10,12,13,19 We begin by definingthese distributions independently of any kinetic model and then

describe their calculation within the particular kinetic model ofref 41. We define the joint dwell time distributions f AA

(m)(t, t′) fortwo dwell times in state A separated by m interveningoccupancies of state B and f AB

(m)(t, t′) for a dwell time in Afollowing one in B with m intervening occupancies of state B.Here A and B denote D or L and the turnover index m satisfiesm ≥ 1 for f AA

(m)(t, t′) and m ≥ 0 for f AB(m)(t, t′). For uncorrelated

dwell times, each distribution would factor into a product ofdistributions of single dwell times, e.g., f AB

(m)(t, t′) → fA(t)f B(t′).We therefore define the contributions to these distributionsthat embody correlations between dwell times

Δ ′ ≡ ′ − ′f t t f t t f t f t( , ) ( , ) ( ) ( )m mAA( )

AA( )

A A (1)

Δ ′ ≡ ′ − ′f t t f t t f t f t( , ) ( , ) ( ) ( )m mAB( )

AB( )

A B (2)

The correlation function1,10,42 CA(m) quantifies the statisticalrelation between an initial dwell time in A and a subsequentdwell time in that state, with m ≥ 1 intervening occupancies ofB

∫ ∫σ≡ ′ ′Δ ′−∞ ∞

C m t t t t f t t( ) d d ( , )mA A

2

0 0 AA( )

(3)

with σA the standard deviation of dwell times in state A. Theanalogous cross correlation function CAB(m) describescorrelations between an initial dwell time in B and a laterdwell time in A after m ≥ 0 intervening residences in B

∫ ∫σ σ≡ ′ ′Δ ′−∞ ∞

C m t t t t f t t( ) ( ) d d ( , )mAB A B

1

0 0 AB( )

(4)

We follow previous authors1,10 in defining dwell timecorrelation functions that depend on a discrete turnoverindex, rather than a continuous time variable. This definitionhas proven its utility in analyzing fluorescence trajectories thatrecord discrete events, as shown in the discussion of eq 22below.The constrained mean dwell time41 tA<(τB) is defined to be

the dwell time in state A averaged over a restricted ensemble ofevents in which the preceding dwell time in B is less than τB.The difference between this quantity and the unconstrainedmean dwell time is related to one- and two-time distributionsby

∫ ∫∫

τΔ ≡′Δ ′

′ ′

τ

τ<

∞

tt t t f t t

t f t( )

d d ( , )

d ( )A B

0 0 AB(0)

0 B

B

B

(5)

The complementary quantity ΔtA>(τB) reflects an average overa constrained ensemble of events in which the occupancy of Ais preceded by a dwell time in B that exceeds τB

∫ ∫

∫τΔ ≡

′Δ ′

′ ′τ

τ

>

∞ ∞

∞tt t t f t t

t f t( )

d d ( , )

d ( )A B

0 AB(0)

B

B

B (6)

We will evaluate the four auto and cross correlation functionsdefined by eqs 3 and 4 and the four constrained mean dwelltimes defined by eqs 5 and 6 for a kinetic model described inref 41. The model catalyst contains N active sites that promotethe reaction of an adsorbed nonfluorescent reactant to generatea fluorescent product. The reactant concentration in solution issufficiently high that the adsorption rate of reactant is largecompared to desorption rates of both reactant and product, sothat each active site can be assumed to be always occupied by

The Journal of Physical Chemistry C Article

dx.doi.org/10.1021/jp4065246 | J. Phys. Chem. C 2013, 117, 19074−1908119075

an adsorbate. Each active site may thus occupy one of twostates associated with adsorption of reactant or product. Inaddition, each site is taken to have S chemically distinct internalsubstates, giving each site 2S possible states. The rate constantsfor reaction and desorption are substate dependent, and thesubstates interconvert according to another set of rateconstants. In keeping with experimental conditions,22 weconsider only states of the catalyst in which at most onefluorescent product molecule is adsorbed. We considered41 twoversions of this model: one in which each site fluctuatesindependently among substates and a second in which theentire catalyst changes substate so that these transitions occursimultaneously at all active sites. We demonstrated that theconstrained mean dwell times in the fluorescent state ΔtL<(τD)and ΔtL>(τD) have qualitatively different dependences on τD forlarge N in these two models. In the model of independentfluctuations, ΔtL<(τD) decays more rapidly than ΔtL>(τD),while, for the model of completely correlated fluctuations, thereverse holds. As shown in Figure 7 of ref 41 and below inFigure 4, the nanoparticle data show the scenario moreconsistent with the model of correlated fluctuations. We adoptthe model of correlated site fluctuations here.The catalyst therefore has S nonfluorescent substates

collectively corresponding to the D state and NS fluorescentsubstates collectively representing the L state, as the fluorescentproduct molecule may be adsorbed at any one of N active sites.If the catalyst is in the fluorescent state, it makes transitionsfrom substate α to substate γ with rate constant lγα, while thissubstate transition occurs with rate constant dγα if the catalyst isin the nonfluorescent state. The rate constant for the process L→ D when the catalyst occupies substate α is kLα, which has thephysical significance of the rate constant for product desorptionfrom a single site. The rate constant for D → L in substate α iskDα

, which is the product of the number of active sites N andthe reaction rate constant at a single active site, as reaction canoccur at any site.For this model, the equilibrium distribution of dwell

times10,13 is given by

= ⟨ | | ⟩−f t t pw g w( ) 1 ( )A A AA1

A (7)

= ⟨ | | ⟩pk1 A A (8)

For notational efficiency, we adopt bra-ket notation andrepresent a vector by |v⟩ and its transpose (and in principlecomplex conjugate) by ⟨v|. The S-dimensional vector ofequilibrium populations in state A is |pA⟩, and the S-dimensional matrix of rate constants for leaving state A, kA,has elements δαγkAα

. The S-dimensional matrix wA governsdynamics 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 evolutionwithin the manifold of A substates. The S-dimensional vectorwith unit elements in the basis of substates {α} is denoted ⟨1|,so that ⟨1|v⟩ ∑α=1

S vα. The form of the dwell-timedistribution in eq 7 is derived by Cao in ref 10. The significanceof the form of eq 7 may be summarized by reading theexpression from right to left. −1wA|pA⟩ represents theequilibrium flux into state A from state B, gA(t) representstime evolution among substates within state A for dwell time t,and the leftmost application of wA represents the transition outof state A into state B.

The joint distribution of two dwell times in A separated by m≥ 1 occupancies of state B is similarly10 given by

′ = ⟨ | ′ | ⟩− −f t t t t pw g T m k g w( , ) 1 ( ) ( )m mA A A B A A AAA

( ) 1 1A (9)

≡ −m k wB B B1

(10)

≡T m mA B A (11)

The dependence of the distribution on m is associated with theturnover matrix TA defined in eq 11 which representsintegrated time evolution in state A, followed by transition tostate B, integrated time evolution in state B, and transition backto state A.42−44 The turnover matrices TD and TL are related bya similarity transformation and so share eigenvalues {λn}, with n= 1, ... , S. Probability conservation dictates that one of theseeigenvalues, to which we assign n = 1, is unity, λ1 = 1, giving TAthe spectral representation

∑ λ= | ⟩⟨ | + | ⟩ ⟨ |=

R L R LTn

S

n n nA 1(A)

1(A)

2

(A) (A)

(12)

Since TA is not in general symmetric, we allow for the existenceof distinct right and left eigenvectors associated with eacheigenvalue, obeying TA|Rn

(A)⟩ = λn|Rn(A)⟩ and ⟨Ln

(A)|TA = ⟨Ln(A)|λn.

The eigenvectors associated with the unit eigenvalue are

| ⟩ = | ⟩R pw /A1(A)

A (13)

⟨ | = ⟨ |L 11(A)

(14)

Substituting this representation of TA into eq 9 yields thedecomposition into a product of single-time distributions fA(t)fA(t′), produced by the first term in eq 12, and Δf AA

(m)(t, t′),which results from the second term in eq 12,

∑ λΔ ′ = ⟨ | | ⟩⟨ | ′ | ⟩=

f t t t R L t Rk g w g( , ) 1 ( ) ( )m

n

S

nm

n nA A A AAA( )

2

(A) (A)1(A)

(15)

The joint distribution of dwell times in different states isobtained similarly to be

∑ λΔ ′ = ⟨ | | ⟩

× ⟨ | ′ | ⟩=

+ −f t t t R

L t R

k g w k

w g

( , ) 1 ( )

( )

m

n

S

nm

n

n

A A A A

B B

AB( )

2

1 1 (B)

(B)1(B)

(16)

The dwell-time correlation functions are obtained byperforming the time integrations in eqs 3 and 4 using thedistributions in eqs 15 and 16

∑σ λ= ⟨ | | ⟩⟨ | | ⟩−

=

− −C m R L Rw w( ) 1n

S

nm

n nA AA A2

2

1 (A) (A) 11(A)

(17)

∑σ σ λ= ⟨ | | ⟩⟨ | | ⟩− −

=

+ − −C m R L Rk w( ) 1n

S

nm

n nA BAB A1

B1

2

1 1 (B) (B) 11(B)

(18)

The constrained mean dwell times are similarly obtained fromeqs 5 and 6 using Δf AB

(0)(t, t′) in eq 16

∑τ λτ

τΔ =

⟨ | | ⟩⟨ | − | ⟩

⟨ | − | ⟩<

=

−

tR L R

R

k I g

I g( )

1 ( ( ))

1 ( ( ))n

S

nn nA B

BA B

2

1 (B) (B)B 1

(B)

B 1(B)

(19)



∑τ λτ

τΔ =

⟨ | | ⟩⟨ | | ⟩

⟨ | | ⟩>

=

−

tR L R

R

k g

g( )

1 ( )

1 ( )n

S

nn nA B

BA B

2

1 (B) (B)B 1

(B)

B 1(B)

(20)

with I the identity in S dimensions.The information content of the correlation functions and

constrained mean times may be explored and compared byconsidering the simplest nontrivial case of two substates. For S= 2, the turnover matrices TD and TL have a single nonuniteigenvalue13,19

λ =

+ + + +− − − − −d k l k d k l k[(1 )(1 )]2

21 D1

21 L1

12 D1

12 L1 1

1 1 2 2

(21)

Each of the terms in this expression represents the ratio of therate constant for loss from a particular substate from a substatechange to the rate constant for loss from that same substatebecause of reaction or desorption. For example, d21kD1

−1 is theratio of the rate constant for the transition D1 → D2 throughchange of substate to the rate constant for the transition D1 →L1 through chemical reaction. It follows from eqs 17 and 18that for S = 2 all correlation functions share the same single-exponential decay governed by the eigenvalue λ2,

λ= = = = −C mC

C mC

C mC

C mC

( )(1)

( )(1)

( )(1)

( )(1)

mD

D

L

L

DL

DL

LD

LD2

1

(22)

with amplitudes that are related by

λ= −C C C C(1) (1) (1) (1)D L 21

LD DL (23)

According to eqs 21 and 22, if substate transition rate constantsare small compared to rate constants for reaction or desorption,then all correlation functions will decay exponentially in m witha decay rate d21kD1

−1 + l21kL1−1 + d12kD2

−1 + l12kL2−1 that is

controlled by rate constants for substate interconversion. Therelation among amplitudes in eq 23 shows that if bothautocorrelation functions are nonzero, both cross correlationfunctions cannot vanish.The solid curves in Figure 1 show dimensionless dwell time

correlation functions for S = 2 for parameters kD1= 0.1, kD2

=

0.5, kL1 = 1, kL2 = 3, d21 = d12 = l12 = 0.02, and l21 = 0.0333 in anarbitrary time unit. This parameter set satisfies the condition11

of detailed balance, kD1l21kL2d12 = kD2

l12kL1d21. These parametersexemplify the case in which the substate with faster reaction,substate 2, also has faster desorption of product, leading topositive cross correlations. These calculations also depict thescenario in which substate changes are slow relative to reactionand desorption processes. For clarity, we plot dwell-timecorrelation functions as though the discrete turnover index mwere a continuous variable, although m is only defined forinteger values. The dashed curves are determined from aparameter set in which the values of kD1

and kD2used for the

solid curves are interchanged, and the value of l21 is adjusted tosatisfy detailed balance, l21 = 0.00133. The cross correlations arenow negative, as the substate with faster reaction, substate 1,has slower desorption of product. For S = 2 and slow substatetransition rates, the amplitude of each autocorrelation functionCA(m) is proportional to the square of the difference in rateconstants 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, andconsistent with eq 23, all have comparable amplitude. Thecalculations in Figure 1 illustrate that the decay rates ofcorrelation functions reflect dynamics of transitions amongsubstates,1,10 while their amplitudes reflect the relativemagnitudes 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

Figure 1. All times are given in the same arbitrary units thatspecify the rate constants. The information content of thisquantity was extensively analyzed in ref 41 and will be brieflysummarized here. For example, ΔtD<(τL) describes the meantime for reaction, given that the preceding time to productdesorption was smaller than the specified value τL. The dashedcurve for ΔtD<(τL) shows the case in which the substate withrapid product desorption has slow reaction. For small values ofτL, ΔtD<(τL) is dominated by contributions from events inwhich 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 twosubstates. Solid lines are calculated for a case in which the substatewith faster reaction has faster product desorption, while dashed linesshow a case in which the substate with faster reaction has slowerdesorption.

Figure 2. Constrained mean dwell times in D (dark) and L (light)states for a catalyst with two substates are shown for the parametersused in Figure 1. Solid lines are calculated for a case in which thesubstate with faster reaction has faster product desorption, whiledashed lines show a case in which the substate with faster reaction hasslower desorption. The time unit is arbitrary.



kL2 > kL1. Since this substate has the smaller rate constant forreaction, the mean dwell time in D is longer than that for anunconstrained average, giving ΔtD<(τL) a positive amplitude.This quantity decays to zero as the constraint that the dwelltime in L is shorter than τL becomes less significant. This decayoccurs approximately with rate kL1, the smaller of the twoproduct desorption rate constants, which controls the rate atwhich the initially selected subensemble of events approachesthe unconstrained ensemble. Interpretation of this quantity isanalogous to measurements such as hole-burning spectroscopythat probe the evolution of an initially selected subensembleinto a full ensemble.45 The complementary quantity ΔtD>(τL)vanishes at τL = 0, for which the constrained and unconstrainedensembles are identical. As τL is increased, the dashed curve forΔtD>(τL) becomes negative, because the system is more likelyto be found in substate 1, which has the faster rate of reaction,so that the constrained mean time is less than the uncon-strained mean, giving rise to a negative difference. This quantityreaches 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. Theconstrained mean times in state L may be interpreted with thesame reasoning, which applies in the limit in which rateconstants for desorption and for reaction are well separated inmagnitude and in which rate constants for substate change aresmall compared to those for reaction and desorption. In thiscase, ΔtA<(τB) evolves according to the smallest of {kBj

}, while

ΔtA>(τB) evolves according to the largest of {kBj}. The two

complementary constrained mean times thus provide distinctinformation. Like the cross correlation functions in Figure 1,the algebraic signs of the amplitudes of the constrained meantimes carry information about correlations between desorptionand reaction rates. However, the m dependence of thecorrelation functions reflects the dynamics of transitionsamong substates, while the τ dependence of constrainedmean dwell times originates primarily in rate constants forreaction and desorption in different substates. These two sets ofquantities thus provide complementary information, as will beshown in analysis of experimental data in section III.

III. NANOPARTICLE CATALYSIS

Discrete symbols in Figures 3 and 4 show dwell-timecorrelation functions and constrained mean dwell timesdetermined from a single trajectory including approximately800 turnovers from one nanoparticle of diameter 6 nm at asaturating resazurin concentration of 1.2 μM.22 This trajectorywas used to produce the autocorrelation functions in Figure 4of ref 22 and the cross correlation functions in Figure S11 ofthe Supporting Information.22 In ref 41, we reportedconstrained mean dwell times from an ensemble of trajectoriesfrom different nanoparticles of this size at the same reactantconcentration. We observe that constrained mean dwell timesshow less variation from particle to particle than do thecorrelation functions. For this reason, the constrained meandwell times shown here for a single trajectory are similar tothose reported in Figure 7 of ref 41 for an ensemble oftrajectories. The correlation functions determined from theensemble are qualitatively similar to those shown in Figure 3,but the amplitudes are substantially reduced, obscuringdifferences between auto and cross correlation functions.

The experimental correlation functions in Figure 3 showseveral distinctive features. The autocorrelation functionsCL(m) and CD(m) show significant amplitude at m = 1 anddecay exponentially with m at different rates. Xu et al.22 have fitthese autocorrelation functions to single exponential decays, CA∝ exp(−m/mA), with mL ≈ 3 and mD ≈ 13. The crosscorrelation functions are zero for all m within experimentalaccuracy. These properties are entirely inconsistent with thetwo-substate model discussed in section II. According to eqs 22and 23, for S = 2, all four correlation functions decay at thesame rate, and if CL(m) and CD(m) are appreciable inmagnitude, the cross correlation functions cannot both benegligibly small.The constrained mean dwell times computed from the same

trajectory used for Figure 3 are shown by discrete points inFigure 4. All times are given in s. These results are similar tothose reported in ref 41 for an ensemble of shorter trajectoriesfor different nanoparticles of the same size and at the samereactant concentration. For A = D and A = L, ΔtA<(τB) > 0 andΔtA>(τB) < 0. The decay rate of ΔtA>(τB) is larger than that ofΔtA<(τB).Since the correlation functions in Figure 3 are inconsistent

with S = 2, we consider the next simplest case of a kinetic

Figure 3. Discrete symbols show correlation functions of dwell timesin light L and dark D states versus turnover index, determined from anexperimental fluorescence trajectory of a gold nanoparticle withdiameter 6 nm. Lines are calculated for a model of active sites withthree substates undergoing correlated fluctuations. Model parametersare determined with simultaneous fits to these data and to theconstrained mean dwell times in Figure 4.

Figure 4. Discrete symbols show constrained mean dwell times in lightL and dark D states determined from the experimental fluorescencetrajectory of Figure 3. Lines show simultaneous fits of these data andthose in Figure 3 to a kinetic model with three substates. All times aregiven in s.



model with three substates. According to eq 17, theautocorrelation functions for S = 3 decay biexponentially withm as

λ λ= +C m c c( ) m mA A

(2)2 A

(3)3 (24)

with cA(j) determined from eq 17. The different single-

exponential decays of CD(m) and CL(m) in Figure 3 implyfor this model that for A = D one of the coefficients in eq 24 isnegligibly small and that for A = L the other coefficient isnegligible, so that each autocorrelation function is dominatedby a single distinct eigenvalue, CL ∝ λL

m and CD ∝ λDm.

The solid curves in Figures 3 and 4 show simultaneous fitsfor the S = 3 model to the four correlation functions and thefour constrained mean dwell times. The kinetic scheme withthe resulting rate constant values is shown in Figure 5. The

model has 14 rate constants constrained by two detailed-balance11 conditions, kD1

l21kL2d12 = d21kD2l12kL1 and kD3

l23kL2d32 =

l32kD2d23kL3. Rate constants were determined by a partitioning

optimization strategy46 that divides parameters into subsets andoptimizes the parameters in one subset, with parameters inother subsets fixed. We employed partitioning choices in whichone subset included rate constants for a kinetically connectedsubsystem, as for example the set of rate constants involvingsubstates 1 and 2, as well as partitioning choices not associatedwith kinetically connected subsystems. This procedure wasrepeated for a variety of partitioning choices until convergencewas achieved.The resulting collection of rate constants is shown in units of

s−1 to two significant figures in Figure 5. The rate constants forreaction 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 forsubstate transitions with adsorbed reactant are d12 = 0.0001688s−1, d21 = 0.004346 s−1, d23 = 0.01230 s−1, and d32 = 0.000177s−1. The rate constants for substate transitions with adsorbedproduct are l12 = 0.04513 s−1, l21 = 0.04217 s−1, l23 = 0.03957s−1, and l32 = 0.01849 s−1. Rate constants for substate changeswith adsorbed reactant are generally smaller than rate constantsfor substate changes with adsorbed product, and rate constantsfor substate changes are generally significantly smaller than rateconstants 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 ×

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 twoeigenvalues and hence of the decay rate of each autocorrelationfunction is not as straightforward as in the S = 2 case in eq 21.An approximate analysis based on the relatively smallmagnitudes of rate constants for substate interconversionprovides insight into the processes underlying these eigenval-ues. The turnover matrix in eq 11 is usefully written asTD I/(I + VD), with VD = dkD

−1 + lkL−1 + dkD

−1lkL−1.

Because of the small magnitudes of the rate constants forsubstate transitions, we neglect the contribution to VD that isquadratic in these rate constants. The elements of the resultingapproximation quantify transitions between substates. Inparticular, the matrix element that reflects the importance oftransitions from substate 2 to substate 3 is small, d32/kD2

+

l32/kL2 ≈ 0.007. We therefore set this quantity to zero anddiagonalize the resulting approximation to VD to obtainapproximations to the nonunit eigenvalues of the turnovermatrix

λ ≈ + + + +− − − − −l k l k d k d k(1 )2 21 L1

12 L1

21 D1

12 D1 1

1 2 1 2

(25)

λ ≈ + +− − −l k d k(1 )3 23 L1

23 D1 1

3 3 (26)

For the rate constants of Figure 5, λ2 ≈ 0.954 and λ3 ≈ 0.627,which agree well with the correct eigenvalues controlling thedecays of the two autocorrelation functions in Figure 3, λD ≈0.954 and λL ≈ 0.608. The eigenvalue λ2 corresponds to thesingle eigenvalue in eq 21 for an S = 2 system composed ofsubstates 1 and 2 with associated rate constants. When termsquadratic in rate constants for substate transitions in thedenominator of eq 21 are neglected, the result is eq 25. Thisapproximation suggests that the decay of CD(m) is controlledby the subsystem containing substates 1 and 2, while the decayof CL(m) reflects transitions from substate 3 to substate 2. Thisqualitative interpretation is confirmed by calculations of CL(m),not shown here, for the S = 2 model consisting of substates 2and 3 and of CD(m) for the S = 2 subsystem comprisingsubstates 1 and 2. These correlation functions agreequantitatively in both magnitude and decay rates with the S= 3 results for CL(m) and CD(m) in Figure 3. This analysisshows that the decays of the two autocorrelation functions canreflect 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 qualitativeagreement with the correlation functions, but addition of a thirdsubstate is not required to represent the constrained meandwell times. As with the correlation functions, the interpreta-tion of the constrained mean dwell times can be simplified byconsidering an S = 2 subsystem of the S = 3 model in Figure 5.In the scheme of Figure 5, the reaction rate constants in two ofthe substates are nearly equal, kD2

≈ kD3. The amplitudes of the

constrained mean dwell times in D are sensitive toheterogeneity in reaction rate constants.41 For this reason, asshown in Figure 4.3 of ref 47, ΔtD<(τL) and ΔtD>(τL) aredominated by contributions from the S = 2 subsystemcomposed of substates 1 and 2. The constrained mean dwelltimes in L contain significant contributions from substate 3 butare qualitatively similar to the result from the subsystem ofsubstates 1 and 2. Therefore, the constrained mean times canbe fit as well with two substates as with three.

Figure 5. A kinetic scheme with three substates is used tosimultaneously fit the four dwell time correlation functions in Figure3 and the four constrained mean dwell times in Figure 4. All rateconstants are in s−1.



IV. CONCLUSIONS

A nanoparticle catalyst includes multiple active sites, whichunlike the active site or sites of a biological catalyst areheterogeneous. Single turnover measurements on gold nano-particles indicate the existence of dynamical processes, whichlike conformational changes in an enzyme affect catalyticactivity. We have analyzed the binary fluorescence trajectoriesdetermined in such measurements with two complementarystatistical measures. These are the four auto and crosscorrelation functions of dwell times in fluorescent andnonfluorescent states and the four constrained mean dwelltimes in fluorescent and nonfluorescent states. The decays ofthe correlation functions with turnover index reflect the timescales of dynamical processes that induce changes in thesubstate of the active site. The decays of the constrained meandwell times reveal aspects of the distribution of rate constantsfor reaction and for desorption, and can be used to distinguishbetween correlated and independent fluctuations41 at differentactive sites.We have shown that for a model of discrete substates at least

three substates are necessary for qualitative agreement betweencalculated and measured dwell time correlation functions andconstrained mean dwell times. The fit of a model with threesubstates to the data in Figures 3 and 4 involves varying 12parameters: the 14 parameters shown in Figure 5 with twoconstraints for detailed balance. The fits shown in these figuresneed not represent a global minimum in this search. However,the fit is robust in the sense demonstrated by the analysisdescribed in the concluding paragraph of section III. Thisanalysis identifies a pair of substates within the three substatemodel with a dominant contribution to the constrained meandwell times, thereby explaining why a model with two substateswas consistent with these four quantities, as shown in ref 41.This analysis relies on a separation of time scales between rateconstants for substate interconversion and rate constants forreaction and desorption and also on a pair of reaction rates, kD2

and kD3in Figure 5, having similar magnitudes. Other parameter

sets obeying these conditions can also provide a qualitative fitto the data in Figures 3 and 4. The fits in Figures 3 and 4 alsodemonstrate that the different decay rates of the two dwell timeautocorrelation functions can reflect distinct physical processes,as shown by the perturbation analysis in eqs 25 and 26. Theprincipal conclusion to be drawn from the fits reported here isthat a model with three discrete substates is consistent withthese data from nanoparticle catalysts.Our model of correlated fluctuations is consistent with the

scenario in which substate changes arise from dynamic surfacereconstruction.32−40 However, our phenomenological kineticmodel is not based on any particular microscopic mechanism.Our analysis of fluorescence trajectories from multisite catalystsidentifies a set of complementary dynamical properties thatmust be reproduced by any molecular description of nano-particle catalysis.

■ AUTHOR INFORMATION

Corresponding Author*E-mail: [email protected].

Present Address†Department of Chemistry and Biochemistry, University ofCalifornia, San Diego, CA.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSM.A.O. and R.F.L. acknowledge support from the NationalScience Foundation through Grant No. CHE0743299. P.C.acknowledges support from the Department of Energy (DE-F G 0 2 - 1 0 E R 1 6 1 9 9 ) , A r m y R e s e a r c h Offi c e(W911NF0910232), and National Science Foundation(CBET-0851257).

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