Estimation of single-molecule blinking parameters using photon counting histogram

Chemical Physics Letters 470 (2009) 363–366

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Estimation of single-molecule blinking parameters using photon counting histogram Marta Hajdziona, Andrzej Molski * ´ , Poland Adam Mickiewicz University, Laboratory for Dynamics of Physicochemical Processes, Grunwaldzka 6, 60-780 Poznan

a r t i c l e

i n f o

Article history: Received 24 October 2008 In final form 31 January 2009 Available online 5 February 2009

a b s t r a c t We have used computer simulations to explore the photon counting histogram (PCH) as a statistic for estimating kinetic parameters of a blinking immobilized fluorophore. Fluorescence trajectories made up of 500 on–off cycles allow recovery of the escape rates kon ; koff with 20% error, when the bin size h is in an optimal range. The corresponding estimates of the intensities Ion ; Ioff improve with increasing resolution kon h; koff h of the on- and off-time, respectively. PCH analysis requires that the intensities Ion ; Ioff be well separated. The parameter estimates improve with increasing fluorescence trajectory length. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Fluorescence from single immobilized molecules exhibits often jumps between discrete intensity levels. A common example is triplet blinking of organic fluorophores where sojourns into nonemissive triplet states lead to fluorescence jumps between a bright (on) and a dark (off) fluorescence levels [1–3]. A simple model of on–off blinking is a two-state model, kon

on  off koff

ð1Þ

where kon and koff are the escape rates from on- and off-state, respectively. The two states have different fluorescence intensities, Ion and Ioff . Assuming a Markovian jump process from the on- and off-states, the corresponding on- and off-waiting time distributions are exponential. Several methods of statistical analysis can be used to estimate kinetic parameters from fluorescence trajectories of immobilized molecules, including on–off analysis, the intensity correlation function, and the photon counting histogram [4,5]. As pointed out in [4] on–off analysis is sensitive to the choice of the bin size. A recent contribution of Watkins and Yang [6] falls into the category of on–off analysis and has an advantage that the problems with an arbitrary choice of the bin size are avoided. To recover the blinking parameters from the intensity correlation function one needs additional information that can be provided, for instance, by the moments of the fluorescence intensity [7]. In this work we focus on the photon counting histogram (PCH) as a means for recovering the parameters kon ; koff ; Ion and Ioff . Ryden et al. [8,9] used an experimental fluorescence trace and the corresponding simulation experiment to demonstrate that PCH analysis * Corresponding author. E-mail address: [email protected] (A. Molski). 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.01.084

can be useful for recovering the on–off blinking parameters. Here we extend that work by applying a Monte Carlo method to study statistical properties of PCH parameter estimates. The present study is intended as an aid at planning and interpreting experiments where single-molecule blinking is quantified. A key limiting factor in single-molecule blinking is the photobleaching effect which limits the number of available photons. In this work we address this issue by exploring fluorescence trajectories with finite numbers of on–off cycles. 2. PCH model For the two-state model the distribution PðijhÞ of photon counts i in an interval of length h can be derived by combining Mandel’s formula in connection with the distribution of time spent in each state [10,11]. This theory was generalized in [12]. Here we limit ourselves to two-state system where each state has a different fluorescence intensity so that

PðijhÞ ¼

Z

1

expðIx hÞ

0

ðIx hÞi pðxjhÞdx i!

ð2Þ

where x is the random fraction of the integration time h spent by the molecule in the on-state, and

Ix ¼ xIon þ ð1  xÞIoff

ð3Þ

eq pðxjhÞ ¼ peq on pðxjh; onÞ þ poff pðxjh; offÞ

ð4Þ

with

koff kon ; peq ð5Þ off ¼ kon þ koff kon þ koff pðxjh; onÞ ¼ dð1  xÞ expðkon hÞ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ kon h I0 ðyÞ þ I1 ðyÞ koff x=½kon ð1  xÞ expðkx hÞ ð6Þ peq on ¼

M. Hajdziona, A. Molski / Chemical Physics Letters 470 (2009) 363–366

pðxjh; offÞ ¼ dðxÞ expðkoff hÞ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ koff h I0 ðyÞ þ I1 ðyÞ kon ð1  xÞ=ðkoff xÞ expðkx hÞ ð7Þ kx ¼ xkon þ ð1  xÞkoff qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y ¼ 2h kon koff xð1  xÞ

ð8Þ ð9Þ

and I0 ; I1 are modified Bessel functions of order 0 and 1, respectively. 3. Simulations and data analysis

0.06

0.06

κon = 0.57 κoff = 1.00 ιon = 1.10 ιoff = 1.12

0.03

counts

frequency

0.00

κon = 0.63 κoff = 1.00 ιon = 1.00 ιoff = 0.81

0.03 0.00

0

10 20 30 40 50 60 70

counts

counts

0.00

30 0 60

2

2

30 0

κoff

κon = 0.67 κoff = 0.73 ιon = 1.00 ιoff = 1.00

0.10

counts

0.00

60

Fig. 2 shows statistical properties of the scaled estimates of on– off blinking parameters jon ; joff ; ion ; ioff as functions of the lifetime resolution kh for the case of equal on- and off-lifetimes kon ¼ koff ¼ k. The scaled intensities were kept constant at Ion h ¼ 40; Ioff h ¼ 10. One can see that there exists an optimal value of kh corresponding to the best estimates of the escape rates kon and koff . Away from the optimal value kh  0:3 the estimates get worse. The estimates of the intensities Ion and Ioff improve as kh gets smaller. This behavior can be explained qualitatively as follows. The bins in a fluorescence intensity trajectory can be classified into three categories according to the fluorescence state during bin intervals: on-bins, off-bins, and mixed-bins where at least one intensity jump occurred. Fluorescence counts in on- and off-bins carry information only on the intensities Ion and Ioff . As kh gets smaller the number of on- and off-bins increases, whereas the number of mixedbins is limited by the trajectory length L. Thus the fraction of onand off-bins increases and the recovery of the intensities improves. On the other hand information on the kinetic parameters kon and

κon

0.03

4. Results

1

1

60 0 0.1

30

1

0 0.1

10

1

10

1

10

0 60 30 0 0

2

2

ιon

κon = 0.88 κoff = 0.99 ιon = 1.00 ιoff = 1.08

counts

0.06

frequency

frequency

frequency

The primary simulation parameters of a two-state on–off system include the blinking parameters kon ; koff ; Ion ; Ioff , the bin size h, and the trajectory length L. In this work the trajectory length L is determined by the total number of on–off cycles. The dimension of parameter space is reduced by one when the nondimensional parameters hkon ; hkoff ; hIon ; hIoff are introduced. This is equivalent to choosing the time unit such that the bin size h ¼ 1. The scaled rates hkon and hkoff determine the resolution of the dwell times 1 son ¼ k1 on and soff ¼ koff , respectively, i.e. the average number of bins per dwell time. The scaled intensities, hIon and hIoff determine the signal strength (=average photon count per bin) in the on- and off-states, respectively. We simulated photon arrivals assuming Poisson counts with intensities Ion and Ioff , and a Markovian jump process with escape rates kon and koff from the on- and off-states, respectively. The state trajectory was produced by generating dwell times drawn from exponential distributions with rates kon and koff . Photon arrivals were produced by generating photon waiting times drawn from an exponential distribution with rate Ion or Ioff , depending on the current state. When the next photon arrival occurres after a state jump, the arrival is discarded, a photon waiting time is drawn with a new rate, and that waiting time is added to the jump time. The simulation procedure can be generalized to several states i with different intensities Ii and transition rates kij . A state trajectory can be produced based on exponential dwell times determined

P by the escape rates ki ¼ j–i kij . At each state jump the destination state is drawn from a discrete distribution determined by individual transition rates kij, j – i. Given a state i, photon waiting times are exponentially distributed with rate Ii . Photon arrivals were binned into consecutive bins of size h, and then a histogram of photon counts was calculated. Distribution (2) was fitted to the photon counting histogram using a nonlinear ^ ; ^Ion and ^I were ^on ; k least-squares procedure. The estimates k off off ^ =k ; jon scaled by the corresponding true values joff ¼ k off off ^on =kon ; i ¼ ^I =I ; ion ¼ ^Ion =Ion to facilitate comparison of dif¼k off off off ferent simulation experiments. Fig. 1 shows examples of PCH analysis. The positions of maxima on a PCH correspond to the average photon counts per bin in the on- and off-states. The height of a maximum decreases as the resolution of the corresponding average dwell time decreases. The simulation and fitting protocol was repeated 103 times for each simulation parameter set. The parameter estimates were accepted when the reduced v2 6 3, which happened in about 99% of the fits. We characterize statistical properties of the distribution of accepted parameter estimates using the sample average, standard deviation, minimum and maximum values.

250

time

ιoff

364

1

1

500

Fig. 1. Examples of PCH analysis for trajectories of length L ¼ 500. Left panels show simulated PCHs (points) and the fitted PCHs (lines) together with the scaled parameter estimates. Right panels show initial portions of the analyzed trajectories. Simulation parameters: ðAÞ kon h ¼ 0:1; koff h ¼ 0:1; Ion h ¼ 40; Ioff h ¼ 10: ðBÞ kon h ¼ 6; koff h ¼ 0:1; Ion h ¼ 40; Ioff h ¼ 10: ðCÞ kon h ¼ 0:1; koff h ¼ 6; Ion h ¼ 40; Ioff h ¼ 10: ðDÞ kon h ¼ 0:1; koff h ¼ 0:1; Ion h ¼ 40; Ioff h ¼ 20.

0 0.1

1

kh

10

0 0.1

kh

Fig. 2. Average (squares), standard deviation (error bars), maximum and minimum (solid lines) of 103 scaled estimates of on–off blinking parameters kon ; koff ; Ion ; Ioff as functions of the lifetimes resolution (kh) for the case of equal on- and off-lifetimes kon ¼ koff ¼ k. Simulation parameters: Ion h ¼ 40; Ioff h ¼ 10; L ¼ 500.

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M. Hajdziona, A. Molski / Chemical Physics Letters 470 (2009) 363–366

koff is contained in the mixed-bins, whose content is also determined by the intensities. Good estimates of the escape rates kon and koff can be obtained when the trajectory contains not only mixed-bins but also on- and off-bins. This leads to an optimum value of kh for the recovery of the rates. Fig. 3 shows statistical properties of the scaled parameter estimates as functions of the on-lifetime resolution kon h at constant koff h ¼ 0:1, and Ion h ¼ 40; Ioff h ¼ 10. One can notice trends similar to those in Fig. 2. However, the optimal region of kon h is broader than in the kon ¼ koff case presented in Fig. 2. As kon h increases so does the fraction of off-bins in the histogram. The corresponding slight improvement in the recovery of Ioff is hard to notice at the resolution of Fig. 3. Fig. 4 shows similar behavior of the scaled parameter estimates when the off-time resolution is varied at kon h ¼ 0:1. Note that this time the statistical properties of the estimates ^Ion are practically unchanged. The trajectory length in Figs. 2–4 was fixed at L ¼ 500. The effect of varying the trajectory length is illustrated in Table 1. When

1 0

0.1

1

1

10

50 500 5000

joff

Min.

Avg. (sdv)

Max.

Min.

Avg. (sdv)

Max.

0.28 0.80 0.903

0.99 (0.35) 1.06 (0.16) 1.002 (0.037)

2.28 1.80 1.113

0.60 0.881 0.920

1.37 (0.34) 1.06 (0.11) 1.005 (0.031)

2.86 1.53 1.081

2

ιoff

ιon

jon

L

1 0 0.1

10

2 1 0

Table 1 Dependence of scaled parameter estimates jon and joff on the trajectory length L. Simulation parameters kon h ¼ 1; koff h ¼ 0:1; Ion h ¼ 40; Ioff h ¼ 10.

2

κoff

κon

2

the trajectory length L increases by a factor of 102 the standard deviation of the kinetic parameter estimates reduces by a factor 10, as expected. PCH analysis requires that the intensities Ion ; Ioff be different. Table 2 illustrates the effect of the distance between the on- and offintensities. When Ioff gets closer to Ion the estimates of kinetic parameters get worse. As exemplified in Table 2 the ratio Ioff =Ion < 0:5 is needed for PCH analysis to resolve the kinetic parameters. As an illustrative example we show in Fig. 5a PCH analysis of a blinking system that has been studied in the literature using a different approach. In Ref. [7] Burrato and co-workers measured fluorescence of DiIC12 adsorbed on a solid substrate. They employed the intensity correlation function combined with analytical expressions for the mean and skewness coefficient of the intensity distribution to recover the blinking parameters. Here we used exemplary parameters from [7], kon ¼ 0:1 ms1 ; koff ¼ 0:025 ms1 ; Ion ¼ 15 kHz; Ioff ¼ 3 kHz, to generate 103 normalized parameter estimates whose histograms are presented in Fig. 5. To

0.1

1

konh

0

10

Table 2 Dependence of scaled parameter estimates jon and joff on the off-intensity Ioff at a constant Ion h ¼ 40. Simulation parameters kon h ¼ 0:1; koff h ¼ 0:1; L ¼ 500.

1

0.1

1

10

Ioff h

konh

Fig. 3. Average (squares), standard deviation (error bars), maximum and minimum (solid lines) of 103 scaled estimates of on–off blinking parameters kon ; koff ; Ion ; Ioff as functions of the on-lifetime resolution ðkon hÞ. Simulation parameters: koff h ¼ 0:1; Ion h ¼ 40; Ioff h ¼ 10; L ¼ 500.

0 10 20

jon

joff

Min.

Avg. (sdv)

Max.

Min.

Avg. (sdv)

Max.

0.829 0.52 0.24

1.02 (0.063) 1.06 (0.28) 1.15 (0.42)

1.217 2.11 2.74

0.877 0.58 0.26

1.016 (0.052) 1.09 (0.30) 1.174 (0.44)

1.206 2.18 3.11

2

3×10

1 0 0.1

1

1 0

10

events

2

κoff

κon

2

0.1

1

2

2×10

κon

κoff

ιon

ιoff

2

1×10

0

10

3

1

koffh

10

events

1 0 0.1

1×10

2

ιoff

ιon

2

1 0 0.1

1

koffh

10

Fig. 4. Average (squares), standard deviation (error bars), maximum and minimum (solid lines) of 103 scaled estimates of on–off blinking parameters kon ; koff ; Ion ; Ioff as functions of the off-lifetime resolution ðkoff hÞ. Simulation parameters: kon h ¼ 0:1; Ion h ¼ 40; Ioff h ¼ 10; L ¼ 500.

2

5×10

0

0.75

1

1.25

0.75

1

1.25

Fig. 5. Histograms of 103 normalized parameter estimates jon ; joff ; ion ; ioff for a blinking fluorophore with kon ¼ 0:1 ms1 ; koff ¼ 0:025 ms1 ; Ion ¼ 15 kHz; Ioff ¼ 3 kHz; h ¼ 3:36 ms, corresponding to kon h ¼ 0:336; koff h ¼ 0:084; Ion h ¼ 50:4; Ioff h ¼ 0:08. The trajectory length L ¼ 1000 corresponds to an average of 2.7  105 photons.

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M. Hajdziona, A. Molski / Chemical Physics Letters 470 (2009) 363–366

optimize the PCH analysis we used the bin size h ¼ 3:36 ms, which is 30 times larger than the bin size used for the correlation function analysis in [7]. The corresponding dimensionless parameters are kon h ¼ 0:336; koff h ¼ 0:084; Ion h ¼ 50:4; Ioff h ¼ 10:08. The simulated trajectory length was L ¼ 1000 which corresponds to an average of 2.7  105 photons. Fig. 5 shows that PCH analysis can be used to recover the blinking parameters when a proper binning of the intensity trajectory is applied. 5. Summary In this work we have used computer simulations to study statistical properties of parameter estimates from the PCH of an immobilized molecule. PCH analysis allows to recover all four parameters kon ; koff ; Ion ; Ioff of a two-state blinking (1). The accuracy and precision of parameter estimates improves with increasing trajectory length L. A fluorescence trajectory made up of 500 on– off cycles allows recovery of the escape rates kon ; koff within 20% of the true values, when the bin size h is in an optimal range.

The corresponding estimates of the intensities Ion ; Ioff improve with increasing resolution of the off- and on-time koff h; kon h. PCH analysis requires that the intensities Ion ; Ioff be well separated. References [1] T. Ha, T. Enderle, D.S. Chemla, P.R. Selvin, S. Weiss, Chem. Phys. Lett. 271 (1997) 1. [2] W.-T. Yip, D. Hu, J.J. Yu, D.A. Vanden Bout, P.F. Barbara, J. Phys. Chem. A 102 (1998) 7564. [3] J.A. Veerman, M.F. Garcia-Parajo, L. Kuipers, N.F.v. Hulst, Phys. Rev. Lett. 83 (1999) 2155. [4] M. Lippitz, F. Kulzer, M. Orrit, Chem. Phys. Chem. 6 (2005) 770. [5] R. Verberk, M. Orrit, J. Chem. Phys. 119 (2003) 2214. [6] L.P. Watkins, H. Yang, J. Phys. Chem. B 109 (2005) 617. [7] K. Weston, P. Carson, J.A. DeAro, S. Burrato, Chem. Phys. Lett. 308 (1999) 58. [8] T. Burzykowski, J. Szubiakowski, T. Ryden, Proc. SPIE 5258 (2003) 171. [9] T. Burzykowski, J. Szubiakowski, T. Ryden, Chem. Phys. 288 (2003) 291. [10] A. Molski, Chem. Phys. Lett. 324 (2000) 301. [11] A.M. Berezhkovskij, A. Szabo, G.H. Weiss, J. Chem. Phys. 110 (1999) 9145. [12] I. Gopich, A. Szabo, J. Chem. Phys. 122 (2005) 14707.