- Email: [email protected]
Theory of time-resolved single-molecule fluorescence spectroscopy
25 February 2000
Chemical Physics Letters 318 Ž2000. 325–332 www.elsevier.nlrlocatercplett
Theory of time-resolved single-molecule fluorescence spectroscopy Andrzej Molski a
a,b,)
, Johan Hofkens a , Thomas Gensch a , Noel ¨ Boens a, Frans De Schryver a
Department of Chemistry, Katholieke UniÕersiteit LeuÕen, Celestijnenlaan 200F, B-3001 HeÕerlee, Belgium Adam Mickiewicz UniÕersity, Department of Physical Chemistry, Grunwaldzka 6, 60-780 Poznan, Poland
b
Received 8 September 1999; in final form 4 January 2000
Abstract The statistics of time-resolved photon detection in single-molecule continuous-excitation fluorescence spectroscopy is examined for three- and two-state models of a dye molecule. A unified description is provided in terms of stationary stochastic point processes. The pair distribution function and the distribution of interdetection times are related to the parameters of the models, and explicit expressions are given for the two-state model. The effect of the detected background is taken into account. Based on an analysis of triplet blinking, we argue that time-resolved photon detection offers an alternative to fluorescence intensity spectroscopy in unravelling the underlying single-molecule processes. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Recently there has been much interest in fluorescence spectroscopy of individual molecules Žfor recent reviews see Refs. w1–4x.. The observed fluctuating fluorescence intensity is a realization of a stochastic process reflecting the underlying singlemolecule dynamics. Hence single-molecule spectroscopy has a potential to unravel molecular processes that are obscured by the ensemble averaging inherent in bulk measurements. Note that the relation )
Corresponding author. Adam Mickiewicz University, Department of Physical Chemistry, Grunwaldzka 6, 60-780 Poznan, Poland. Fax: q48-61-8769653; e-mail: [email protected]
between the observed emission and the single-molecule dynamics is complicated by the fact that only a fraction of the emitted photons is detected. Another complicating factor is the detection of scattered light and the dark noise of the detector. Several approaches has been developed to connect the observed fluorescence intensity fluctuations to the triplet state dynamics w5–10x and other singlemolecule processes w8,11–16x. In fluorescence correlation spectroscopy the intensity correlation function is fitted to a multiexponential function to give the correlation times and the rate constants of the underlying processes w6,8,10,11,16x. Correlation between different trajectories, rather than between different times on the same trajectory, can also be informative
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 0 4 0 - 3
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A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
w13x. If an intensity threshold can be set between different fluorescence intensity levels, e.g. onroff levels, then duration histograms can be generated and quantified w8–12,14,15x. For instance, the correlation function of the on-times was used identify the presence of fluctuations in the rate of a enzymatic reaction w14,15x. Similarly, the correlation function of the off-times was used to identify the static Žs environmental., and dynamic Žs temporal. disorder via an analysis of the triplet state lifetimes w9x. In a typical continuous-excitation measurement individual photon detection events are integrated over a constant time window Žs bin time. to produce the time-dependent fluorescence intensity, defined as the number of counts per bin time. However, currently available technology makes it possible to register individual detection events w17,18x, so that the loss of information due to the integration of counts in bins can be avoided. In this Letter we explore the statistics of time-resolved photon detection of fluorescence from a single immobile molecule under the conditions of continuous excitation. We focus on the three-state model of a dye molecule and on its limiting case where it reduces to the two-state model ŽScheme 1.. The purpose of the Letter is three-fold. First, we demonstrate that the dynamics of the single-molecule processes and the statistics of time-resolved photon detection can be conveniently described in terms of renewal stochastic processes
w19,20x. This approach allows a unified description of the three- and two-state models of the molecule and the corresponding detection statistics. Second, we explore the relation of the distribution of interdetection times to the parameters of the models and to the pair distribution function of detection events. In particular we present explicit expressions for the distribution of interdetection times for the two-level model. Finally, we comment briefly on the effect of the detected background on the statistics of photon counting.
2. Three-state model Let us consider an immobile molecule under continuous excitation. The molecule undergoes transitions between the ground state G, the excited single state S, and the excited triplet state T. The transitions between the states are described by the rate constants for fluorescence, k f , nonradiative deactivation k nr , inter-system crossing k isc , deactivation of the triplet state, k GT, and excitation k exc . We assume that the activation–deactivation processes in the three-state model in Scheme 1 is a continuous time Markov jump process w21x, whose kinetics are described by the rate equations for the probabilities Pi Ž t . that the molecule is in state i s G, S, T w6x: d dt
PG Ž t . s yk exc PG Ž t . q Ž k f q k nr . PS Ž t . q k GT PT Ž t . ,
d dt
Ž 1.
PS Ž t . s k exc PG Ž t . y Ž k f q k nr q k isc . PS Ž t . ,
Ž 2. d dt
Scheme 1. The three-state model Župper panel. and the corresponding two-level model of a dye molecule.
PT Ž t . s k isc PS Ž t . y k GT PT Ž t . .
Ž 3.
Note that photobleaching is neglected in Eqs. Ž1. – Ž3., which is appropriate as a first approximation for photostable organic dye molecules. Let p 1Ž t 1 .d t 1 be the probability that a detection event occurred in the interval Ž t 1 , t 1 q d t 1 ., and
A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
p 1Ž t 2 .d t 2 be the probability that a detection event occurred in the interval Ž t 2 , t 2 q d t 2 .. Let p 2 Ž t 1 , t 2 .d t 1d t 2 be the joint probability that one detection event occurred in the interval Ž t 1 , t 1 q d t 1 ., and some other Žnot necessarily the next one. occurred in the interval Ž t 2 , t 2 q d t 2 .. The pair distribution function g Ž t 1 , t 2 . is defined as w21x:
g Ž t1 , t 2 . s
p 2 Ž t1 , t 2 . p1Ž t1 . p1Ž t 2 .
.
Ž 4.
The probability density of detecting a photon at t s t 2 , conditional on the fact that a photon was detected at t s t 1 , is p 2 Ž t 2 < t 1 . s p 2 Ž t 1 , t 2 .rp1Ž t 1 . s gŽ t 1 , t 2 . p 1Ž t 2 .. For stationary processes of interest here the detection probability density is constant p 1Ž t 1 . s p 1Ž t 2 . s p 1 , and equal to the average detection rate. Moreover, for a stationary process one has p 2 Ž t 1 , t 2 . s p 2 Ž< t 2 y t 1 <., and p 2 Ž t 2 < t 1 . s p 2 Ž< t 2 y t 1 < <0., so that the pair distribution function depends on time difference t s < t 2 y t 1 <: g Ž t . s p 2 Ž t . rp12 s p 2 Ž t <0 . rp1 .
™
Ž 5.
At long times p 2 Ž t <0. p 1 , and g Ž`. s 1. From the experimental point of view the pair distribution function g Ž t . can be defined as a quantity proportional to the number of pairs of detection events where the separation time falls into the interval Ž t, t q d t .. The properly normalized pair distribution function is then recovered by requiring that g Ž t . becomes unity at long times. We first consider the case where the detection of the background photons can be neglected. Let us assume that a photon was detected at t s 0. This implies that the molecule relaxes to the ground state at t s 0, and its further evolution is described by Eqs. Ž1. – Ž3. with the initial conditions PG Ž0. s 1, and PS Ž0. s PT Ž0. s 0. Assuming the same detection probability, wdet , for each emitted photon, wdet k f PS Ž t .d t gives the probability that a photon is detected in the interval Ž t, t q d t ., conditional on the fact that a photon was detected at time t s 0. At long times the probabilities Pi Ž t . tend to their limiting values Pi Ž`., and the detection probability density becomes p 1 s wdet k f PS Ž`.. Thus, in the case of
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background-free detection, one can write for the pair distribution function: g Ž t . s PS Ž t . rPS Ž ` . ,
Ž 6.
where PS Ž t . is determined by Eqs. Ž1. – Ž3., and PS Ž`. is the long time limit of PS Ž t .. In fluorescence intensity spectroscopy the detection events are integrated in bins to obtained the time-dependent intensity trace, I Ž t .. One of the statistical characteristics of the bin-averaged fluorescence intensity is the correlation function GŽ t . s I 2 g I Ž t . s ² I Žt . I Žt q t .:, where I Žt . is the instantaneous intensity at time t , and I is the average fluorescence intensity. When the bin time is small compared to the characteristic time of the processes affecting fluorescence emission, the intensity correlation function g I Ž t . is a good approximation to the pair distribution function g Ž t ., and g Ž t . can be replaced by g I Ž t . in Eq. Ž6.. Let EŽ t . be the probability that the first photon will be emitted no later than at time t, given that the molecule was in the ground state at time t s 0. Note that the molecule may undergo several excitationrnonradiative deactivation cycles before it emits a photon. The probability EŽ t . defines the distribution of the first emission time, conditioned on the molecule being in the ground state at t s 0. Stated differently EŽ t . is the distribution of the first passage time from the ground state G to a state E which is accessed from the excited state S radiatively Žsee Scheme 2.. Let Q i Ž t . denote the probability that the molecule is in state i s G, S, T in Scheme 2. The rate equations describing Scheme 2 are different from
Scheme 2. Modification of the three-state model to calculate the distribution EŽ t . of the first emission time. At t s 0 the molecule is in the ground state G. To register the moment of the first emission, an absorbing state E is introduced, whose population is the probability EŽ t . that the first emission event has occurred by the time t.
A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
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Eqs. Ž1. – Ž3. since now state E acts as an absorbing state: d Q Ž t . s yk exc QG Ž t . q k nr QS Ž t . q k GT Q T Ž t . , dt G Ž 7. d dt
QS Ž t . s k exc QG Ž t . y Ž k f q k nr q k isc . QS Ž t . ,
Ž 8. d dt
where fˆŽ z . denotes the Laplace transform of a function f Ž t .: Dˆ Ž z . s
Given DŽ t ., or its Laplace transform Dˆ Ž z ., the average waiting time t between adjacent detection events can be calculated as: `
Q T Ž t . s k isc QS Ž t . y k GT Q T Ž t . .
Ž 9.
The distribution of the first emission time EŽ t . is related to the probability QS Ž t . as: EŽ t . s kf
t
H0 Q Ž t . d t ,
Ž 10 .
S
where QS Ž t . is determined by Eqs. Ž7. – Ž9. with the initial conditions QG Ž0. s 1, and QS Ž0. s Q T Ž0. s 0. Let DŽ t . denote the probability that the lag between two adjacent detection events does not exceed time t, i.e. DŽ t . is the distribution of the waiting time between successive detection events. Each time a photon is emitted, the system returns to the ground state, and its future evolution does not depend on the previous emission events. In the terminology of point processes, emission moments are regeneration moments of a renewal process w19,20x. Given the distribution of the first emission time EŽ t ., the distribution of interdetection times DŽ t . can be obtained from the following renewal type equations w19,20x: D Ž t . s wdet E Ž t . q Ž 1 y wdet .
t
X
H0 D Ž t y t .
d E Ž tX .
d tX , Ž 11 . d tX where the r.h.s results from conditioning the probability DŽ t . on the detection of the first emitted photon. The first term originates from the events where the first emitted photon is detected, and the second term gives the contribution from the events where the first photon emitted at some tX - t is not detected, but another photon emitted during the time interval t y tX is. Eq. Ž11. can be solved using Laplace transforms, =
Dˆ Ž z . s
wdet Eˆ Ž z . 1 y Ž 1 y wdet . zEˆ Ž z .
,
Ž 12 .
`
H0 exp Ž yzt . f Ž t . d t .
ts
H0 t
d DŽ t . dt
dtsy
d zDˆ Ž z . dz
.
Ž 13 .
xs 0
Note that the detection probability density, or, equivalently, the average detection rate is p 1 s ty1 . In order to relate the pair distribution function, g Ž t ., to the distribution of interdetection times, DŽ t ., let us note that not only the emission events but also the detection events form a renewal process. Moreover, the distribution DŽ t . of interdetection times is also the distribution of the first detection time, conditioned on the molecule being in the ground state at t s 0. This allows one to write the following renewal type equation: y1
t
gŽ t. s
DŽ t . dt
y1
qt
t
X
H0 g Ž t y t .
d D Ž tX . d tX
d tX .
Ž 14 . In order to derive Eq. Ž14. one can consider two contribution to the conditional probability density ty1 g Ž t . of detecting a photon at time t, given that a photon was detected at time t s 0. The first contribution, d DŽ t .rdt, is the conditional probability density that the first photon after t s 0 is detected at time t, and the second contribution comes from the events where the first photon was detected at some tX - t, and some other photon was detected at time t. Eq. Ž14. can be solved using Laplace transforms to give: gˆ Ž z . s
t zDˆ Ž z . 1 y zDˆ Ž z .
.
Ž 15 .
Thus, in the absence of background signal, the distribution of interdetection times DŽ t ., the pair distribution function g Ž t ., and the average detection rate, ty1 , are related through the renewal type equations Ž14. and Ž15..
A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
So far we have dealt with the case where the background emission was absent. Now we consider the superposition of photons coming from the molecule and those coming from the background. This leads to a new detection process, whose pair distribution function and the interdetection time distribution will be denoted as g ) Ž t ., and D ) Ž t ., respectively. When the background photons are superimposed on the true single-molecule emission, the detection events do not form a renewal process, and g ) Ž t ., and D ) Ž t . do not satisfy the relationships Ž14. and Ž15.. Nevertheless, one can relate g ) Ž t ., and D ) Ž t . to the true emission and to that of the background photons. Here we consider the case where the detection events of the background photons form a Poisson process with intensity ty1 b : D b Ž t . s 1 y exp Ž ytrt b . .
Ž 16 .
The overall detection probability density p 1) Žs the average detection rate. in the presence of background is p 1) s ty1 q ty1 b . The effect of the background photon can be assessed from the formula:
`
Ht
1 y D Ž tX . d tX ,
5
and `
tyt exp Ž y1rt b .
Ht
1 y D Ž tX . d tX ,
= 1 y x y xty1 r Ž z q ty1 b . ,
Ž 18 .
where SˆŽ z . is the Laplace transform of SŽ t . s 1 y DŽ t ., and DŽ t . is given by Eq. Ž11.. Now let us consider the probability Žty1 b q y1 . ) Ž . t g t d t that a photon will be detected in the interval Ž t, t q d t ., conditional on the fact that a photon was detected at t s 0. The probability that the photon at t s 0 came from the background is x, and that it came from the molecule is 1 y x. Similarly, the photon detected in the interval Ž t, t q d t . may come either from the background or from the molecule. This leads to four possibilities as to the origin of the pair Žbackground or the molecule.. Collecting the contributions from the four cases one gets:
½
g ) Ž t . s x x q Ž 1 y x . 1 y D Ž t . q ty1 Ž 1 y x . t
1 y D Ž tX . g Ž t y tX . d tX
5
qŽ 1 y x . x q Ž 1 y x . g Ž t . 4
Ž 17 .
y1 . Ž y1 where x s ty1 is the relative backb r t b qt ground emission intensity, i.e. the ratio of the average background count rate and the long time average total detection rate. The l.h.s. of Eq. Ž17. is the probability that the waiting time for the next detection event after one at t s 0 will be greater than t. This probability can be conditioned on the fact that the photon detected at t s 0 originated from the molecule, which happens with probability 1 y x, or on the fact that it was a background photon, which happens with probability x. The conditional probabilities of those two events are
exp Ž ytrt b . 1 y D Ž t .
y1 ˆ Dˆ ) Ž z . s zy1 q xr Ž z q ty1 b . ySŽ zqt b .
H0
½
qxty1
respectively, which leads to Eq. Ž17.. The Laplace transform Dˆ ) Ž z . of D ) Ž t . is:
=
1 y D ) Ž t . s exp Ž ytrt b . Ž 1 y x . 1 y D Ž t .
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Ž 19 .
which simplifies to: 2
g ) Ž t . sx Ž2yx . q Ž1yx . g Ž t . .
Ž 20 .
Eq. Ž20. means that the pair distribution function g Ž t . can be readily recovered from g ) Ž t . when the relative background intensity, x, is known.
3. Two-state model and triplet blinking The equations of the three-state model presented in the previous section can be readily solved using Laplace transforms. However, the resulting expressions simplify substantially when one takes into account the fact that at low excitation powers the depopulation rate of the excited singlet state, k f q k nr q k isc , is much higher than the excitation rate in the three-state model of Scheme 1. This means that the occupancy of S is much lower than that of the
A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
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ground state G and the triplet state T. This can be taken into account by putting in Eqs. Ž1. – Ž3.: Ž k f q k nr q k isc . ` and PS 0 such that Ž k f q k nr q k isc . PS stays finite. Applying this limit to Eq. Ž2. one gets Ž k f q k nr q k isc . PS s k exc PG . Now the rate equations Ž1. – Ž3. reduce to:
™
™
parameters uniquely. When the background is present Eq. Ž20. gives: g ) Ž t . s1q Ž1yx .
2
a isc k exc k GT
=exp y Ž a isc k exc q k GT . t , d dt d dt
PG Ž t . s ya isc k exc PG Ž t . q k GT PT Ž t . ,
Ž 21 .
PT Ž t . s a isc k exc PG Ž t . y k GT PT Ž t . ,
Ž 22 .
where a f s k frŽ k f q k nr q k isc . and a isc s k iscrŽ k f q k nr q k isc .. The rate equations Ž21. and Ž22. describe transition between the ground state G, and the triplet state T. While in the ground state, the molecule can get excited and then emit a photon. The average emission rate is a f k exc . While in the triplet state the molecule does not fluoresce. Thus Eqs. Ž21. and Ž22. describe fluctuations between the bright Žemissive. and the dark Žnonemissive. states of the molecule ŽScheme 1.. Since the dark state is the triplet state, these fluctuations have been termed triplet blinking. Triplet blinking is a common phenomenon, and has become a paradigm of a molecular process that can be studied by observing single-molecule fluorescence w3,5,6x. Here we are interested in the relation between the triplet blinking kinetics and the distribution of the interdetection times, DŽ t ., and in the comparison of the information content of DŽ t . and that of the pair distribution function g Ž t .. In the limiting case of the two-state model Eq. Ž6. becomes g Ž t . s PG Ž t .rPG Ž`., where PG Ž t . satisfies Eqs. Ž21. and Ž22. with the initial conditions PG Ž0. s 1 and P T Ž0. s 0, and PG Ž`. is the long time limit of PG Ž t .. Solving Eqs. Ž21. and Ž22. for PG Ž t . one obtains the well-known result for the pair distribution function: g Ž t. s1q
a isc k exc k GT
Ž 24 .
so that the background emission rescales the preexponential factor, but, nevertheless, the rate constants a isc k exc and k GT are identifiable when the relative background intensity, x, is known. This conclusion is consistent with Ref. w5x, but is at odds with Ref. w10x, where it was argued that in the presence of background emission the rate constants a isc k exc and k GT are not uniquely identifiable. In the limiting case of the two-state model, the first emission time distribution EŽ t . is related to the occupancy of the ground state as: E Ž t . s a f k exc
t
H0 Q
G
Ž t. dt .
Ž 25 .
The rate equations Ž7. – Ž9. reduce to: d dt
QG Ž t . s y Ž a f q a isc . k exc QG Ž t . q k GT Q T Ž z . ,
Ž 26 . d dt
Q T Ž t . s a isc k exc QG Ž t . y k GT Q T Ž t . .
Ž 27 .
Solving Eqs. Ž26. and Ž27. one gets for the Laplace transform of DŽ t .: Dˆ Ž z . s
wdet a f k exc Ž z q k GT . z Ž z y g1 . Ž z y g2 .
,
Ž 28 .
where g 1 and g 2 are the roots of the quadratic equation: z 2 q Ž wdet a f q a isc . k exc q k GT z
exp y Ž a isc k exc q k GT . t .
q wdet a f k exc k GT s 0 ,
Ž 23 . Note that the pair distribution function g Ž t . is mono-exponential, and the rate constants a isc k exc and k GT are identifiable, i.e. g Ž t . determines those
Ž 29 .
so that
g 1 g 2 s wdet a f k exc k GT ,
Ž 30 .
g 1 q g 2 s y Ž wdet a f q a isc . k exc y k GT .
Ž 31 .
A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
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Inverting Eq. Ž28. one gets: DŽ t . s 1 q
wdet a f k exc g1 yg2
Ž 1 q k GT rg 1 . exp Ž g 1 t .
y Ž 1 q k GT rg 2 . exp Ž g 2 t . .
Ž 32 .
Thus the distribution of interdetection times is biexponential. When the exponents g 1 and g 2 , and the ratio of the prexponential factors are known, the quantities k GT , wdet a f k exc , and a isc k exc can be determined. If the detection efficiency wdet is known, the branching ratio a fra isc can also be determined. In Ref. w22x interdetection times were measured to reveal the correlations between fluorescence photons Žphoton antibunching. at short Žnanosecond. times under the conditions of high excitation intensities. Under the experimental conditions of Ref. w22x, the density of the distribution of interdetection times, d DŽ t .rd t, and the scaled pair distribution function, ty1 g Ž t ., were indistinguishable. Here we are interested in the low excitation regime, and the submillisecond and millisecond time scales corresponding to the intervals between photon arrivals in the bright state, and to the duration of the dark periods. In this regime the distribution of interdetection times DŽ t . is different from the pair distribution function g Ž t . Žsee Fig. 1.. Nevertheless, as Eq. Ž14. demonstrates,
Fig. 2. The distribution of interdetection times DŽ t . Žsolid line, Eq. Ž32.. for the same parameters as in Fig. 1. For comparison the broken line shows the distribution of interdetection times D ) Ž t . y1 ŽEq. Ž17.. when the background detection rate is ty1 , b s 200 s which corresponds to the relative background intensity x s 0.12.
the collective information provided by the pair distribution function g Ž t . and the average detection rate ty1 is equivalent to that provided by the distribution of interdetection times DŽ t .. This holds true in the absence of background photons. Note that in the presence of background signal, D ) Ž t . has, in general, a different shape then that of DŽ t . Žcf. Eq. Ž17. and Fig. 2..
4. Summary and conclusion
Fig. 1. The density of the distribution of interdetection times d DŽ t .rd t Žbroken line, Eq. Ž32.., and the scaled correlation function ty1 g Ž t . Žsolid line, Eq. Ž23.. as a function of time, calculated for the following parameters: k exc s1=10 5 sy1 , k f s 0.85=10 9 sy1 , k nr s 0.1=10 9 sy1 , k isc s 0.05=10 9 sy1 , k GT s1=10 3 sy1 . The curves calculated for the three-state model and the two-state model in Scheme 1 overlap within the resolution of the graph.
In this Letter we have studied the standard threestate model of a molecule under constant low-power excitation, where the transitions between the states are described by the rate constants of a Markov jump process ŽScheme 1.. When individual detection events are recorded, the concept of a stochastic point process provides a unified description of the molecular processes and the detection events. The distribution DŽ t . of the interdetection times is determined by the distribution EŽ t . of the emission of the first photon Žcf. Eq. Ž11... The distribution EŽ t . can be calculated by solving appropriate kinetic equations for the three-state model, Eqs. Ž7. – Ž9., and for the two-state model, Eqs. Ž26. and Ž27.. Eq. Ž14. relates the distribution of interdetection times DŽ t . to the pair distribution function g Ž t .. The effect of background signal is quantified by Eqs. Ž18. and Ž20..
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When the rate of the excited singlet state deactivation is much higher than the excitation rate, the three-state system reduces to an effective two-state system, which fluctuates between a bright state and a nonemissive dark state Žs the triplet state. ŽScheme 1.. For the two-state model Eq. Ž32. gives an explicit expression for the distribution of the interdetection times DŽ t . as a function of the rate constants for triplet depopulation k GT , ground state depopulation a isc k exc , and the bright period emission rate wdet a f k exc . Since these constants are identifiable, we conclude that not only the pair distribution function g Ž t . but also the distribution of interdetection times DŽ t . can be useful in quantifying the kinetics of triplet blinking. Triplet blinking is just one example of a molecular process that can be observed via single-molecule fluorescence spectroscopy. Thus the question arises how other dynamic process Že.g., conformational changes. can be quantified by means of the statistical characteristics of time-resolved photon detection. Another question is to what extent the finite lifetime of dye molecules limits the information content of those characteristics. Although those questions have not yet been addressed the present analysis suggests that time-resolved photon detection may be a useful alternative to fluorescence intensity spectroscopy in unravelling single-molecule processes.
Acknowledgements A.M. is grateful to the K.U. Leuven University Research Fund for a Senior Research fellowship. J.H. thanks the FWO for a post-doctoral fellowship. N.B. is an Onderzoeksdirecteur of the Fonds voor Wetenschappelijk Onderzoek ŽFWO.. This work was supported by the FWO, the Flemish Ministry of Education through GOAr1r96, the EC through the
TMR Sistomas and TMR Marie Curie, the VW Stifung, and the DWTC ŽBelgium. through IUAP-411. References w1x S. Nie, R. Zare, Annu. Rev. Biophys. Biomol. Struct. 26 Ž1997. 567. w2x X.S. Xie, J.K. Trautman, Annu. Rev. Phys. Chem. 49 Ž1998. 441. w3x S. Weiss, Science 283 Ž1999. 1676. w4x W.E. Moerner, M. Orrit, Science 283 Ž1999. 1670. w5x J. Bernard, L. Fleury, H. Talon, M. Orrit, J. Chem. Phys. 98 Ž1993. 850. ¨ Mets, R. Rigler, J. Phys. Chem. 99 Ž1995. w6x J. Widengren, U. 13368. w7x T. Ha, Th. Enderle, D.S. Chmela, P.R. Selvin, S. Weiss, Chem. Phys. Lett. 271 Ž1997. 1. w8x W.-T. Yip, D. Hu, J. Yu, V. Bout, P. Barbara, J. Phys. Chem. 102 Ž1998. 7564. w9x J.A. Veerman, M.F. Garcia-Parajo, L. Kuipers, N.F. Van Hulst, Phys. Rev. Lett. 83 Ž1999. 2155. w10x K.D. Weston, P.J. Carson, J.A. DeAro, S.K. Burrato, Chem. Phys. Lett. 308 Ž1999. 58. w11x K.D. Weston, S.K. Burrato, J. Phys. Chem. A 102 Ž1998. 3635. w12x K.D. Weston, P.J. Carson, H. Metiu, S.K. Burrato, J. Chem. Phys. 109 Ž1998. 7474. w13x J.N. Onuchic, J. Wang, P.G. Wolynes, Chem. Phys. 247 Ž1999. 175. w14x H.P. Lu, X.S. Xie, Z. Phys. Chem. 212 Ž1999. 1877. w15x H.P. Lu, X.S. Xie, Science 282 Ž1998. 1877. w16x L. Edman, Z. Foeldes-Papp, S. Wennmalm, R. Rigler, Chem. Phys. 247 Ž1999. 11. w17x S. Nie, D.T. Chiu, R.N. Zare, Science 266 Ž1994. 1018. w18x J.R. Fries, L. Brand, C. Eggeling, M. Kollner, C.A.M. ¨ Seidel, J. Phys. Chem. 102 Ž1998. 6601. w19x D.R. Cox, V. Isham, Point Processes, Chapman, London, 1980. w20x S.M. Ross, Applied Probability Models with Optimization Applications, Dover, New York, 1992. w21x N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland Publ., Amsterdam, 1992. w22x Th. Basche, W.E. Moerner, M. Orrit, H. Talon, Phys. Rev. Lett. 69 Ž1992. 1516.
Chemical Physics Letters 318 Ž2000. 325–332 www.elsevier.nlrlocatercplett
Theory of time-resolved single-molecule fluorescence spectroscopy Andrzej Molski a
a,b,)
, Johan Hofkens a , Thomas Gensch a , Noel ¨ Boens a, Frans De Schryver a
Department of Chemistry, Katholieke UniÕersiteit LeuÕen, Celestijnenlaan 200F, B-3001 HeÕerlee, Belgium Adam Mickiewicz UniÕersity, Department of Physical Chemistry, Grunwaldzka 6, 60-780 Poznan, Poland
b
Received 8 September 1999; in final form 4 January 2000
Abstract The statistics of time-resolved photon detection in single-molecule continuous-excitation fluorescence spectroscopy is examined for three- and two-state models of a dye molecule. A unified description is provided in terms of stationary stochastic point processes. The pair distribution function and the distribution of interdetection times are related to the parameters of the models, and explicit expressions are given for the two-state model. The effect of the detected background is taken into account. Based on an analysis of triplet blinking, we argue that time-resolved photon detection offers an alternative to fluorescence intensity spectroscopy in unravelling the underlying single-molecule processes. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Recently there has been much interest in fluorescence spectroscopy of individual molecules Žfor recent reviews see Refs. w1–4x.. The observed fluctuating fluorescence intensity is a realization of a stochastic process reflecting the underlying singlemolecule dynamics. Hence single-molecule spectroscopy has a potential to unravel molecular processes that are obscured by the ensemble averaging inherent in bulk measurements. Note that the relation )
Corresponding author. Adam Mickiewicz University, Department of Physical Chemistry, Grunwaldzka 6, 60-780 Poznan, Poland. Fax: q48-61-8769653; e-mail: [email protected]
between the observed emission and the single-molecule dynamics is complicated by the fact that only a fraction of the emitted photons is detected. Another complicating factor is the detection of scattered light and the dark noise of the detector. Several approaches has been developed to connect the observed fluorescence intensity fluctuations to the triplet state dynamics w5–10x and other singlemolecule processes w8,11–16x. In fluorescence correlation spectroscopy the intensity correlation function is fitted to a multiexponential function to give the correlation times and the rate constants of the underlying processes w6,8,10,11,16x. Correlation between different trajectories, rather than between different times on the same trajectory, can also be informative
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 0 4 0 - 3
326
A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
w13x. If an intensity threshold can be set between different fluorescence intensity levels, e.g. onroff levels, then duration histograms can be generated and quantified w8–12,14,15x. For instance, the correlation function of the on-times was used identify the presence of fluctuations in the rate of a enzymatic reaction w14,15x. Similarly, the correlation function of the off-times was used to identify the static Žs environmental., and dynamic Žs temporal. disorder via an analysis of the triplet state lifetimes w9x. In a typical continuous-excitation measurement individual photon detection events are integrated over a constant time window Žs bin time. to produce the time-dependent fluorescence intensity, defined as the number of counts per bin time. However, currently available technology makes it possible to register individual detection events w17,18x, so that the loss of information due to the integration of counts in bins can be avoided. In this Letter we explore the statistics of time-resolved photon detection of fluorescence from a single immobile molecule under the conditions of continuous excitation. We focus on the three-state model of a dye molecule and on its limiting case where it reduces to the two-state model ŽScheme 1.. The purpose of the Letter is three-fold. First, we demonstrate that the dynamics of the single-molecule processes and the statistics of time-resolved photon detection can be conveniently described in terms of renewal stochastic processes
w19,20x. This approach allows a unified description of the three- and two-state models of the molecule and the corresponding detection statistics. Second, we explore the relation of the distribution of interdetection times to the parameters of the models and to the pair distribution function of detection events. In particular we present explicit expressions for the distribution of interdetection times for the two-level model. Finally, we comment briefly on the effect of the detected background on the statistics of photon counting.
2. Three-state model Let us consider an immobile molecule under continuous excitation. The molecule undergoes transitions between the ground state G, the excited single state S, and the excited triplet state T. The transitions between the states are described by the rate constants for fluorescence, k f , nonradiative deactivation k nr , inter-system crossing k isc , deactivation of the triplet state, k GT, and excitation k exc . We assume that the activation–deactivation processes in the three-state model in Scheme 1 is a continuous time Markov jump process w21x, whose kinetics are described by the rate equations for the probabilities Pi Ž t . that the molecule is in state i s G, S, T w6x: d dt
PG Ž t . s yk exc PG Ž t . q Ž k f q k nr . PS Ž t . q k GT PT Ž t . ,
d dt
Ž 1.
PS Ž t . s k exc PG Ž t . y Ž k f q k nr q k isc . PS Ž t . ,
Ž 2. d dt
Scheme 1. The three-state model Župper panel. and the corresponding two-level model of a dye molecule.
PT Ž t . s k isc PS Ž t . y k GT PT Ž t . .
Ž 3.
Note that photobleaching is neglected in Eqs. Ž1. – Ž3., which is appropriate as a first approximation for photostable organic dye molecules. Let p 1Ž t 1 .d t 1 be the probability that a detection event occurred in the interval Ž t 1 , t 1 q d t 1 ., and
A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
p 1Ž t 2 .d t 2 be the probability that a detection event occurred in the interval Ž t 2 , t 2 q d t 2 .. Let p 2 Ž t 1 , t 2 .d t 1d t 2 be the joint probability that one detection event occurred in the interval Ž t 1 , t 1 q d t 1 ., and some other Žnot necessarily the next one. occurred in the interval Ž t 2 , t 2 q d t 2 .. The pair distribution function g Ž t 1 , t 2 . is defined as w21x:
g Ž t1 , t 2 . s
p 2 Ž t1 , t 2 . p1Ž t1 . p1Ž t 2 .
.
Ž 4.
The probability density of detecting a photon at t s t 2 , conditional on the fact that a photon was detected at t s t 1 , is p 2 Ž t 2 < t 1 . s p 2 Ž t 1 , t 2 .rp1Ž t 1 . s gŽ t 1 , t 2 . p 1Ž t 2 .. For stationary processes of interest here the detection probability density is constant p 1Ž t 1 . s p 1Ž t 2 . s p 1 , and equal to the average detection rate. Moreover, for a stationary process one has p 2 Ž t 1 , t 2 . s p 2 Ž< t 2 y t 1 <., and p 2 Ž t 2 < t 1 . s p 2 Ž< t 2 y t 1 < <0., so that the pair distribution function depends on time difference t s < t 2 y t 1 <: g Ž t . s p 2 Ž t . rp12 s p 2 Ž t <0 . rp1 .
™
Ž 5.
At long times p 2 Ž t <0. p 1 , and g Ž`. s 1. From the experimental point of view the pair distribution function g Ž t . can be defined as a quantity proportional to the number of pairs of detection events where the separation time falls into the interval Ž t, t q d t .. The properly normalized pair distribution function is then recovered by requiring that g Ž t . becomes unity at long times. We first consider the case where the detection of the background photons can be neglected. Let us assume that a photon was detected at t s 0. This implies that the molecule relaxes to the ground state at t s 0, and its further evolution is described by Eqs. Ž1. – Ž3. with the initial conditions PG Ž0. s 1, and PS Ž0. s PT Ž0. s 0. Assuming the same detection probability, wdet , for each emitted photon, wdet k f PS Ž t .d t gives the probability that a photon is detected in the interval Ž t, t q d t ., conditional on the fact that a photon was detected at time t s 0. At long times the probabilities Pi Ž t . tend to their limiting values Pi Ž`., and the detection probability density becomes p 1 s wdet k f PS Ž`.. Thus, in the case of
327
background-free detection, one can write for the pair distribution function: g Ž t . s PS Ž t . rPS Ž ` . ,
Ž 6.
where PS Ž t . is determined by Eqs. Ž1. – Ž3., and PS Ž`. is the long time limit of PS Ž t .. In fluorescence intensity spectroscopy the detection events are integrated in bins to obtained the time-dependent intensity trace, I Ž t .. One of the statistical characteristics of the bin-averaged fluorescence intensity is the correlation function GŽ t . s I 2 g I Ž t . s ² I Žt . I Žt q t .:, where I Žt . is the instantaneous intensity at time t , and I is the average fluorescence intensity. When the bin time is small compared to the characteristic time of the processes affecting fluorescence emission, the intensity correlation function g I Ž t . is a good approximation to the pair distribution function g Ž t ., and g Ž t . can be replaced by g I Ž t . in Eq. Ž6.. Let EŽ t . be the probability that the first photon will be emitted no later than at time t, given that the molecule was in the ground state at time t s 0. Note that the molecule may undergo several excitationrnonradiative deactivation cycles before it emits a photon. The probability EŽ t . defines the distribution of the first emission time, conditioned on the molecule being in the ground state at t s 0. Stated differently EŽ t . is the distribution of the first passage time from the ground state G to a state E which is accessed from the excited state S radiatively Žsee Scheme 2.. Let Q i Ž t . denote the probability that the molecule is in state i s G, S, T in Scheme 2. The rate equations describing Scheme 2 are different from
Scheme 2. Modification of the three-state model to calculate the distribution EŽ t . of the first emission time. At t s 0 the molecule is in the ground state G. To register the moment of the first emission, an absorbing state E is introduced, whose population is the probability EŽ t . that the first emission event has occurred by the time t.
A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
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Eqs. Ž1. – Ž3. since now state E acts as an absorbing state: d Q Ž t . s yk exc QG Ž t . q k nr QS Ž t . q k GT Q T Ž t . , dt G Ž 7. d dt
QS Ž t . s k exc QG Ž t . y Ž k f q k nr q k isc . QS Ž t . ,
Ž 8. d dt
where fˆŽ z . denotes the Laplace transform of a function f Ž t .: Dˆ Ž z . s
Given DŽ t ., or its Laplace transform Dˆ Ž z ., the average waiting time t between adjacent detection events can be calculated as: `
Q T Ž t . s k isc QS Ž t . y k GT Q T Ž t . .
Ž 9.
The distribution of the first emission time EŽ t . is related to the probability QS Ž t . as: EŽ t . s kf
t
H0 Q Ž t . d t ,
Ž 10 .
S
where QS Ž t . is determined by Eqs. Ž7. – Ž9. with the initial conditions QG Ž0. s 1, and QS Ž0. s Q T Ž0. s 0. Let DŽ t . denote the probability that the lag between two adjacent detection events does not exceed time t, i.e. DŽ t . is the distribution of the waiting time between successive detection events. Each time a photon is emitted, the system returns to the ground state, and its future evolution does not depend on the previous emission events. In the terminology of point processes, emission moments are regeneration moments of a renewal process w19,20x. Given the distribution of the first emission time EŽ t ., the distribution of interdetection times DŽ t . can be obtained from the following renewal type equations w19,20x: D Ž t . s wdet E Ž t . q Ž 1 y wdet .
t
X
H0 D Ž t y t .
d E Ž tX .
d tX , Ž 11 . d tX where the r.h.s results from conditioning the probability DŽ t . on the detection of the first emitted photon. The first term originates from the events where the first emitted photon is detected, and the second term gives the contribution from the events where the first photon emitted at some tX - t is not detected, but another photon emitted during the time interval t y tX is. Eq. Ž11. can be solved using Laplace transforms, =
Dˆ Ž z . s
wdet Eˆ Ž z . 1 y Ž 1 y wdet . zEˆ Ž z .
,
Ž 12 .
`
H0 exp Ž yzt . f Ž t . d t .
ts
H0 t
d DŽ t . dt
dtsy
d zDˆ Ž z . dz
.
Ž 13 .
xs 0
Note that the detection probability density, or, equivalently, the average detection rate is p 1 s ty1 . In order to relate the pair distribution function, g Ž t ., to the distribution of interdetection times, DŽ t ., let us note that not only the emission events but also the detection events form a renewal process. Moreover, the distribution DŽ t . of interdetection times is also the distribution of the first detection time, conditioned on the molecule being in the ground state at t s 0. This allows one to write the following renewal type equation: y1
t
gŽ t. s
DŽ t . dt
y1
qt
t
X
H0 g Ž t y t .
d D Ž tX . d tX
d tX .
Ž 14 . In order to derive Eq. Ž14. one can consider two contribution to the conditional probability density ty1 g Ž t . of detecting a photon at time t, given that a photon was detected at time t s 0. The first contribution, d DŽ t .rdt, is the conditional probability density that the first photon after t s 0 is detected at time t, and the second contribution comes from the events where the first photon was detected at some tX - t, and some other photon was detected at time t. Eq. Ž14. can be solved using Laplace transforms to give: gˆ Ž z . s
t zDˆ Ž z . 1 y zDˆ Ž z .
.
Ž 15 .
Thus, in the absence of background signal, the distribution of interdetection times DŽ t ., the pair distribution function g Ž t ., and the average detection rate, ty1 , are related through the renewal type equations Ž14. and Ž15..
A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
So far we have dealt with the case where the background emission was absent. Now we consider the superposition of photons coming from the molecule and those coming from the background. This leads to a new detection process, whose pair distribution function and the interdetection time distribution will be denoted as g ) Ž t ., and D ) Ž t ., respectively. When the background photons are superimposed on the true single-molecule emission, the detection events do not form a renewal process, and g ) Ž t ., and D ) Ž t . do not satisfy the relationships Ž14. and Ž15.. Nevertheless, one can relate g ) Ž t ., and D ) Ž t . to the true emission and to that of the background photons. Here we consider the case where the detection events of the background photons form a Poisson process with intensity ty1 b : D b Ž t . s 1 y exp Ž ytrt b . .
Ž 16 .
The overall detection probability density p 1) Žs the average detection rate. in the presence of background is p 1) s ty1 q ty1 b . The effect of the background photon can be assessed from the formula:
`
Ht
1 y D Ž tX . d tX ,
5
and `
tyt exp Ž y1rt b .
Ht
1 y D Ž tX . d tX ,
= 1 y x y xty1 r Ž z q ty1 b . ,
Ž 18 .
where SˆŽ z . is the Laplace transform of SŽ t . s 1 y DŽ t ., and DŽ t . is given by Eq. Ž11.. Now let us consider the probability Žty1 b q y1 . ) Ž . t g t d t that a photon will be detected in the interval Ž t, t q d t ., conditional on the fact that a photon was detected at t s 0. The probability that the photon at t s 0 came from the background is x, and that it came from the molecule is 1 y x. Similarly, the photon detected in the interval Ž t, t q d t . may come either from the background or from the molecule. This leads to four possibilities as to the origin of the pair Žbackground or the molecule.. Collecting the contributions from the four cases one gets:
½
g ) Ž t . s x x q Ž 1 y x . 1 y D Ž t . q ty1 Ž 1 y x . t
1 y D Ž tX . g Ž t y tX . d tX
5
qŽ 1 y x . x q Ž 1 y x . g Ž t . 4
Ž 17 .
y1 . Ž y1 where x s ty1 is the relative backb r t b qt ground emission intensity, i.e. the ratio of the average background count rate and the long time average total detection rate. The l.h.s. of Eq. Ž17. is the probability that the waiting time for the next detection event after one at t s 0 will be greater than t. This probability can be conditioned on the fact that the photon detected at t s 0 originated from the molecule, which happens with probability 1 y x, or on the fact that it was a background photon, which happens with probability x. The conditional probabilities of those two events are
exp Ž ytrt b . 1 y D Ž t .
y1 ˆ Dˆ ) Ž z . s zy1 q xr Ž z q ty1 b . ySŽ zqt b .
H0
½
qxty1
respectively, which leads to Eq. Ž17.. The Laplace transform Dˆ ) Ž z . of D ) Ž t . is:
=
1 y D ) Ž t . s exp Ž ytrt b . Ž 1 y x . 1 y D Ž t .
329
Ž 19 .
which simplifies to: 2
g ) Ž t . sx Ž2yx . q Ž1yx . g Ž t . .
Ž 20 .
Eq. Ž20. means that the pair distribution function g Ž t . can be readily recovered from g ) Ž t . when the relative background intensity, x, is known.
3. Two-state model and triplet blinking The equations of the three-state model presented in the previous section can be readily solved using Laplace transforms. However, the resulting expressions simplify substantially when one takes into account the fact that at low excitation powers the depopulation rate of the excited singlet state, k f q k nr q k isc , is much higher than the excitation rate in the three-state model of Scheme 1. This means that the occupancy of S is much lower than that of the
A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
330
ground state G and the triplet state T. This can be taken into account by putting in Eqs. Ž1. – Ž3.: Ž k f q k nr q k isc . ` and PS 0 such that Ž k f q k nr q k isc . PS stays finite. Applying this limit to Eq. Ž2. one gets Ž k f q k nr q k isc . PS s k exc PG . Now the rate equations Ž1. – Ž3. reduce to:
™
™
parameters uniquely. When the background is present Eq. Ž20. gives: g ) Ž t . s1q Ž1yx .
2
a isc k exc k GT
=exp y Ž a isc k exc q k GT . t , d dt d dt
PG Ž t . s ya isc k exc PG Ž t . q k GT PT Ž t . ,
Ž 21 .
PT Ž t . s a isc k exc PG Ž t . y k GT PT Ž t . ,
Ž 22 .
where a f s k frŽ k f q k nr q k isc . and a isc s k iscrŽ k f q k nr q k isc .. The rate equations Ž21. and Ž22. describe transition between the ground state G, and the triplet state T. While in the ground state, the molecule can get excited and then emit a photon. The average emission rate is a f k exc . While in the triplet state the molecule does not fluoresce. Thus Eqs. Ž21. and Ž22. describe fluctuations between the bright Žemissive. and the dark Žnonemissive. states of the molecule ŽScheme 1.. Since the dark state is the triplet state, these fluctuations have been termed triplet blinking. Triplet blinking is a common phenomenon, and has become a paradigm of a molecular process that can be studied by observing single-molecule fluorescence w3,5,6x. Here we are interested in the relation between the triplet blinking kinetics and the distribution of the interdetection times, DŽ t ., and in the comparison of the information content of DŽ t . and that of the pair distribution function g Ž t .. In the limiting case of the two-state model Eq. Ž6. becomes g Ž t . s PG Ž t .rPG Ž`., where PG Ž t . satisfies Eqs. Ž21. and Ž22. with the initial conditions PG Ž0. s 1 and P T Ž0. s 0, and PG Ž`. is the long time limit of PG Ž t .. Solving Eqs. Ž21. and Ž22. for PG Ž t . one obtains the well-known result for the pair distribution function: g Ž t. s1q
a isc k exc k GT
Ž 24 .
so that the background emission rescales the preexponential factor, but, nevertheless, the rate constants a isc k exc and k GT are identifiable when the relative background intensity, x, is known. This conclusion is consistent with Ref. w5x, but is at odds with Ref. w10x, where it was argued that in the presence of background emission the rate constants a isc k exc and k GT are not uniquely identifiable. In the limiting case of the two-state model, the first emission time distribution EŽ t . is related to the occupancy of the ground state as: E Ž t . s a f k exc
t
H0 Q
G
Ž t. dt .
Ž 25 .
The rate equations Ž7. – Ž9. reduce to: d dt
QG Ž t . s y Ž a f q a isc . k exc QG Ž t . q k GT Q T Ž z . ,
Ž 26 . d dt
Q T Ž t . s a isc k exc QG Ž t . y k GT Q T Ž t . .
Ž 27 .
Solving Eqs. Ž26. and Ž27. one gets for the Laplace transform of DŽ t .: Dˆ Ž z . s
wdet a f k exc Ž z q k GT . z Ž z y g1 . Ž z y g2 .
,
Ž 28 .
where g 1 and g 2 are the roots of the quadratic equation: z 2 q Ž wdet a f q a isc . k exc q k GT z
exp y Ž a isc k exc q k GT . t .
q wdet a f k exc k GT s 0 ,
Ž 23 . Note that the pair distribution function g Ž t . is mono-exponential, and the rate constants a isc k exc and k GT are identifiable, i.e. g Ž t . determines those
Ž 29 .
so that
g 1 g 2 s wdet a f k exc k GT ,
Ž 30 .
g 1 q g 2 s y Ž wdet a f q a isc . k exc y k GT .
Ž 31 .
A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
331
Inverting Eq. Ž28. one gets: DŽ t . s 1 q
wdet a f k exc g1 yg2
Ž 1 q k GT rg 1 . exp Ž g 1 t .
y Ž 1 q k GT rg 2 . exp Ž g 2 t . .
Ž 32 .
Thus the distribution of interdetection times is biexponential. When the exponents g 1 and g 2 , and the ratio of the prexponential factors are known, the quantities k GT , wdet a f k exc , and a isc k exc can be determined. If the detection efficiency wdet is known, the branching ratio a fra isc can also be determined. In Ref. w22x interdetection times were measured to reveal the correlations between fluorescence photons Žphoton antibunching. at short Žnanosecond. times under the conditions of high excitation intensities. Under the experimental conditions of Ref. w22x, the density of the distribution of interdetection times, d DŽ t .rd t, and the scaled pair distribution function, ty1 g Ž t ., were indistinguishable. Here we are interested in the low excitation regime, and the submillisecond and millisecond time scales corresponding to the intervals between photon arrivals in the bright state, and to the duration of the dark periods. In this regime the distribution of interdetection times DŽ t . is different from the pair distribution function g Ž t . Žsee Fig. 1.. Nevertheless, as Eq. Ž14. demonstrates,
Fig. 2. The distribution of interdetection times DŽ t . Žsolid line, Eq. Ž32.. for the same parameters as in Fig. 1. For comparison the broken line shows the distribution of interdetection times D ) Ž t . y1 ŽEq. Ž17.. when the background detection rate is ty1 , b s 200 s which corresponds to the relative background intensity x s 0.12.
the collective information provided by the pair distribution function g Ž t . and the average detection rate ty1 is equivalent to that provided by the distribution of interdetection times DŽ t .. This holds true in the absence of background photons. Note that in the presence of background signal, D ) Ž t . has, in general, a different shape then that of DŽ t . Žcf. Eq. Ž17. and Fig. 2..
4. Summary and conclusion
Fig. 1. The density of the distribution of interdetection times d DŽ t .rd t Žbroken line, Eq. Ž32.., and the scaled correlation function ty1 g Ž t . Žsolid line, Eq. Ž23.. as a function of time, calculated for the following parameters: k exc s1=10 5 sy1 , k f s 0.85=10 9 sy1 , k nr s 0.1=10 9 sy1 , k isc s 0.05=10 9 sy1 , k GT s1=10 3 sy1 . The curves calculated for the three-state model and the two-state model in Scheme 1 overlap within the resolution of the graph.
In this Letter we have studied the standard threestate model of a molecule under constant low-power excitation, where the transitions between the states are described by the rate constants of a Markov jump process ŽScheme 1.. When individual detection events are recorded, the concept of a stochastic point process provides a unified description of the molecular processes and the detection events. The distribution DŽ t . of the interdetection times is determined by the distribution EŽ t . of the emission of the first photon Žcf. Eq. Ž11... The distribution EŽ t . can be calculated by solving appropriate kinetic equations for the three-state model, Eqs. Ž7. – Ž9., and for the two-state model, Eqs. Ž26. and Ž27.. Eq. Ž14. relates the distribution of interdetection times DŽ t . to the pair distribution function g Ž t .. The effect of background signal is quantified by Eqs. Ž18. and Ž20..
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A. Molski et al.r Chemical Physics Letters 318 (2000) 325–332
When the rate of the excited singlet state deactivation is much higher than the excitation rate, the three-state system reduces to an effective two-state system, which fluctuates between a bright state and a nonemissive dark state Žs the triplet state. ŽScheme 1.. For the two-state model Eq. Ž32. gives an explicit expression for the distribution of the interdetection times DŽ t . as a function of the rate constants for triplet depopulation k GT , ground state depopulation a isc k exc , and the bright period emission rate wdet a f k exc . Since these constants are identifiable, we conclude that not only the pair distribution function g Ž t . but also the distribution of interdetection times DŽ t . can be useful in quantifying the kinetics of triplet blinking. Triplet blinking is just one example of a molecular process that can be observed via single-molecule fluorescence spectroscopy. Thus the question arises how other dynamic process Že.g., conformational changes. can be quantified by means of the statistical characteristics of time-resolved photon detection. Another question is to what extent the finite lifetime of dye molecules limits the information content of those characteristics. Although those questions have not yet been addressed the present analysis suggests that time-resolved photon detection may be a useful alternative to fluorescence intensity spectroscopy in unravelling single-molecule processes.
Acknowledgements A.M. is grateful to the K.U. Leuven University Research Fund for a Senior Research fellowship. J.H. thanks the FWO for a post-doctoral fellowship. N.B. is an Onderzoeksdirecteur of the Fonds voor Wetenschappelijk Onderzoek ŽFWO.. This work was supported by the FWO, the Flemish Ministry of Education through GOAr1r96, the EC through the
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