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Chemical Physics 313 (2005) 85–92

A systematic study of global analysis ofdiffusion-mediated fluorescence quenching data

Jacek Kłos, Andrzej Molski *

Laboratory for Dynamics of Physicochemical Processes, Adam Mickiewicz University, Grunwaldzka 6, 60780 Poznan, Poland

Received 30 September 2004; accepted 27 December 2004Available online 21 January 2005

Abstract

The recovery of the Smoluchowski–Collins–Kimball (SCK) model parameters from simulated fluorescence quenching decays andstationary Stern–Volmer (SV) quenching data is studied for the direct method where the instrument response function (IRF) is mea-sured together with a sample, and the reference method where the monoexponential fluorescence decay of a reference compoundprovides information on the shape of the IRF. The zero-time shift was assumed to be an adjustable parameter in the direct method,and a fixed parameter in the reference method. We found that the quality of parameter estimates from the direct method is similar tothat from the reference method. Single-curve analysis, where individual decay traces are fitted to the SCK model, leads to ratherpoor parameter estimates. Better results can be obtained when quenching decays are analyzed globally, i.e., simultaneously. In par-ticular, global analysis of quenching decays with different time resolutions (=channel widths) improves the recovery compared tosingle curve analysis. For decays with high counts at a peak channel the inclusion of stationary SV data into the global analysisdoes not improve significantly parameter recovery.� 2005 Elsevier B.V. All rights reserved.

Keywords: Diffusion-mediated quenching; Smoluchowski–Collins–Kimball model; Global analysis

1. Introduction

In the Smoluchowski–Collins–Kimball (SCK)model of diffusion-mediated collisional quenchingspherical quencher and fluorophore molecules diffusein a continuum solvent and interact when a quenchermolecule encounters an excited fluorophore molecule[1]. When intrinsic quenching is fast compared to diffu-sional relaxation, the deviations of the fluorescence de-cays from first order kinetics carry information aboutthe microscopic quantities characterizing the quenchingprocess. Assuming the SCK model, one can determinethe sum of the diffusion coefficients of the fluorophoreand quencher, D = DF* + DQ, the sum of the radii of

0301-0104/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2004.12.017

* Corresponding author. Tel./fax: +48 61 8759249.E-mail address: amolski@amu.edu.pl (A. Molski).

the fluorophore and quencher, R = RF* + RQ, andthe intrinsic quenching rate coefficient k.

So far studies exploring the estimation of diffusion-mediated quenching parameters from time-correlatedsingle-photon counting data (TCSPC, [2]) have beenconcerned mainly with single-curve analysis, whereindividual decay traces are fitted to the SCK model[3–6]. It has been known from studies on multiexpo-nential decays that global, i.e., simultaneous, analysisof fluorescence decays enhances parameter recovery[7–10]. We have demonstrated recently that globalanalysis of two SCK decays can substantially improvethe accuracy of the recovered parameters compared tosingle-curve analysis [11]. In [11], we focused solely onparameter ranges corresponding to experimental re-ports on quenching of S2-xanthions by hydrocarbonsin perfluorohydrocarbons [12], and used the reference

86 J. Kłos, A. Molski / Chemical Physics 313 (2005) 85–92

method where a monoexponential fluorescence decayof a reference compound is measured instead of theinstrument response function (IRF) [13–15]. In thepresent paper, we describe a systematic study of globalanalysis of fluorescence quenching decays with diffu-sion-mediated transients. We extend our previouswork in two directions. First, we explore the perfor-mance of global analysis for a broad range of param-eters, and, second, we use two methods of fluorescencedecay analysis: the reference method and the direct

method where the instrument response function(IRF) is measured together with a sample.

The paper is organized as follows. In the followingsection, we briefly recall the theory of diffusion-mediatedquenching as applied to the SCK model. In Section 3,we review the method of data synthesis and parameterrecovery used in this work. In Section 4, we presentthe results of Monte-Carlo simulations of parameterestimation. We have found that the direct method andthe reference method give similar results. Thus, to avoidrepetition, we focus the presentation on global analysisas applied to the reference method. The present system-atic study of global analysis was done for a broad rangeof parameters, which allows us to formulate several con-clusions as to the recovery of diffusion-mediatedquenching parameters. Those conclusions should be use-ful for interpreting and planning kinetic experiments ofnano- and pico-second time scales, and are summarizedin the last section.

2. Theory

In TCSPC experiments one measures the counts Dn inchannels n = 1,. . .,Nch [2]. The counts in a channel is arandom quantity whose expectation value hDni is relatedto the decay profile d(t) as

hDni ¼Z tn

tn�1

dðtÞ dt; ð1Þ

where the r.h.s. is integration of the decay profile d(t)over channel n of length h = tn � tn� 1 An equivalentexpression used in this work is [6],

hDni ¼Z tn

0

iðtn � uÞRðuÞ du; ð2Þ

where i(t) is the fluorescence intensity decay function(fluorescence response), and RðtÞ ¼

R tt�h rðuÞ du is the

integral of the instrument response function r(t) overan interval (t, t � h) of length h.

When the instrument response function is measureddirectly, it is often shifted by a zero-time shift s withrespect to r(t), rm(t) = r(t + s), so that the expectedcounts hRni in the nth channel (tn� 1, tn) of the mea-sured instrument response function are given byhRni = R(nh + s).

2.1. Direct method

The time-resolved fluorescence decay profile d(t) of asample is given by the convolution of the fluorescenceintensity decay function i(t) with the instrument re-sponse function r(t) (IRF),

dðtÞ ¼Z t

0

iðuÞrðt � uÞ du; ð3Þ

In the direct method the IRF is measured togetherwith the decay profile, and Eq. (3) is used for the recov-ery of parameters determining the fluorescence intensitydecay function.

For the SCK model of diffusion-mediated quenching,the decay profile i(t) of a fluorophore with lifetime s inthe presence of quencher Q is known explicitly [1],

iðtÞ=ið0Þ ¼ exp �tðs�1 þ pÞ � ðq=c2Þ½expðx2ÞerfcðxÞ�

�1þ 2x=p1=2��; ð4Þ

where

p ¼ ½Q�kkD=ðk þ kDÞ; q ¼ ½Q�k2=ðk þ kDÞ;c ¼ ðk þ kDÞ=ðkDs1=2D Þ ð5Þ

and

x ¼ ð1þ k=kDÞðt=sDÞ1=2; sD ¼ R2=D;

kD ¼ 4pRD: ð6Þ

2.2. Reference method

In the reference method [13–15], instead of measuringthe IRF directly, one measures the fluorescence decayprofile dr(t) of a reference compound,

drðtÞ ¼Z t

0

rðuÞirðt � uÞ du; ð7Þ

where ir(t) is the reference fluorescence response. Theparameters of the fluorescence response of the samplei(t) are obtained from the measured decay profiles ofthe sample d(t) and reference dr(t) through the relation

dðtÞ ¼Z t

0

drðuÞ~iðt � uÞ du; ð8Þ

where ~iðtÞ is the modified fluorescence response of thesample. When the fluorescence response of the referenceir(t) is monoexponential, ir(t) = ir(0) exp(�t/sr), where sris the reference lifetime, then the modified fluorescenceresponse of the sample is

~iðtÞ ¼ ið0Þ½dðtÞ þ iðtÞ=sr þ diðtÞ=dt�=irð0Þ: ð9ÞFor the SCK model, the modified intensity ~iðtÞ in Eq.

(9) can be calculated explicitly since i(t) is given by Eq.(4) and

diðtÞ=dt ¼ �iðtÞ½s�1 þ p þ q expðx2ÞerfcðxÞ�: ð10Þ

J. Kłos, A. Molski / Chemical Physics 313 (2005) 85–92 87

2.3. Stationary Stern–Volmer plots

Stationary fluorescence intensity measurements arerepresented by the SV plots [1],

I0I

� �¼

R10

i0ðtÞ dtR10

iðtÞ dt¼ sR1

0½iðtÞ=ið0Þ� dt

; ð11Þ

where hI0/Ii is the expected ratio of the stationary fluo-rescence intensity I0 in the absence of the quencher,[Q] = 0, to the stationary intensity I at a nonzeroquencher concentration; i0(t), and i(t) are the corre-sponding fluorescence responses, whose initial value isi(0). In actual experiments one measures the quantityI0/I that differs from hI0/Ii by a random error.

0 100 200 300 400 500

0

1x104

2x104

3x104

4x104

5x104

/D

= 0.1, h = 1 ps

referencesample

coun

ts

channel #

0 100 200 300 400 500-4

-2

0

2

4

wei

ghte

dre

sidu

als

channel #

Fig. 1. Reference method (IRF2): top panels show sample and reference decbottom panels show the weighted residuals of the fit. Simulation parameters

0 100 200 300 400 500

0

1x104

2x104

3x104

4x104

5x104

reference

sample

/D

= 1.47, h = 1 ps

coun

ts

channel #

0 100 200 300 400 500-4

-2

0

2

4

wei

ghte

dre

sidu

als

channel #

Fig. 2. Reference method (IRF2): top panels show sample and reference decbottom panels show the weighted residuals of the fit. Simulation parameters

3. Simulations and data analysis

In this work, we employed two model IRFs [6,11],

IRF1 : rðtÞ ¼ a1ðe�t=a � et=bÞ ð12Þ

with a = 1.818 · 10�2 ns, and b = 1.667 · 10�2 ns, and

IRF2 : rðtÞ ¼ a2ðt2e�t=a þ cte�t=bÞ ð13Þwith a = 8.0 · 10�3 ns, b = 4.0 · 10�2 ns, andc = 5.0 · 10�4 ns. The amplitudes a1 and a2 determinethe peak hight. Two scaled and time-shifted ‘‘after-pulses’’ of shape (13) were added to the main peak(13) (see Figs. 1 and 2). The model IRFs were shiftedto provide 92 zero-intensity leading channels.

/D

= 0.1, h = 5 ps

0 100 200 300 400 500

0

1x104

2x104

3x104

4x104

5x104

reference

sampleco

unts

channel #

0 100 200 300 400 500-4

-2

0

2

4

wei

ghte

dre

sidu

als

channel #

ay traces with different time resolutions h = 1 and 5 ps, at s/sD = 0.1,: k/kD = 1, u = 0.5, Dmax = Rmax = 4 · 104, s = 0, Nch = 512.

0 100 200 300 400 500

0

1x104

2x104

3x104

4x104

5x104

reference

sample

/D

= 1.47, h = 5 ps

coun

ts

channel #

0 100 200 300 400 500-4

-2

0

2

4

wei

ghte

dre

sidu

als

channel #

ay traces with different time resolutions h = 1 and 5 ps, at s/sD = 1.47,as in Fig. 1.

88 J. Kłos, A. Molski / Chemical Physics 313 (2005) 85–92

The sample fluorescence counts, Dn, reference fluores-cence counts, Dr,n, IRF counts, Rn, and Stern–Volmerdata, I0/In, were synthesized by algorithms reported pre-viously [6,11]. In short, smooth decays were calculatedusing the trapezoidal rule in connection with Eq. (3)for the direct method, and Eq. (8) for the referencemethod. In the direct method the time-shift was allowedfor by an interpolation method [15]. In the referencemethod the time-shift was kept fixed at s = 0, becausethe wavelength matching for sample and reference re-moves the color effect responsible for the time-shift.The random number generator ran2 from NumericalRecipes [16] was used to produce the noise added tothe smooth decays. Poisson noise was added for countsbelow 50, and Gaussian noise for higher counts. The ref-erence lifetime was sr = 10 ps. The fluorescence was nor-malized to a peak value of Dmax = Dr,max = 4 · 104.

The parameters were recovered by minimizing [11],

v2 ¼Xi

Xn

wni Dni � Dnih ic� �2

þXm

wm I0=Im � I0=Imh ic� �2

; ð14Þ

using a Levenberg–Marquardt algorithm with bounds.For parameters D, R, and k, the lower bound was 0.1of the true value of a parameter and the upper boundwas 10 times the true value of a parameter. An index c

is used to indicate the calculated values for the expectedhDnii and hI0/Imi. In global analysis parameters D, R, kare linked over the data surface (14). The simulated dataare denoted as Dni (counts in channel n in decay i) andI0/Im (the mth SV data point), respectively, and the cor-responding weights are denoted as wni and wm. In the di-rect method, the weights were

w�1n ¼ Dn; ð15Þ

whereas in the reference method the weights wn of thedecay data points were adjusted during the fitting proce-dure [15],

w�1n ¼ Dn þ ½ið0Þ=irð0Þ�2Dr;n: ð16ÞThe standard deviation (SD) of the SV data was

SDSV = 0.0333 · [(I0/Im) � 1], which corresponds to amaximum error of about 10%. The weights of the SVdata were

w�1m ¼ ðSDSVÞ2: ð17ÞIn the reconvolution analysis the trapezoid integra-

tion rule was used for the decays, and Romberg�s inte-gration [16] for the SV data. For each simulationexperiment, the minimization procedure was repeated30 times with different initial parameter guesses in searchfor a global minimum.

Parameter estimates ~D; ~R; ~k were normalized to thecorresponding true (=simulated) values D, R, k:d ¼ ~D=D; q ¼ ~R=R; j ¼ ~k=k. Since bounds on parame-

ter estimates were used the minimum value of d, q,and j is 0.1, and the maximum is 10. The maximum er-ror of a parameter fit is defined here as the absolute va-lue of the maximum deviation from the true value,recorded in 103 simulations. We also used the averageerror of a fit e = (|j � 1| + |d � 1| + |q � 1|)/3, as a com-posite measure of accuracy.

The quenching kinetics can be characterized by thedimensionless quantities k/kD,s/sD, and u = 4pR3[Q]/3.In this work, we simulated decay traces for 13 differentvalues of dimensionless parameters k/kD and s/sD thatwere evenly distributed on the logarithmic scale between0.1 and 10, and for 10 values of u evenly distributed on alinear scale between u = 0.1 and u = 1. The number ofchannels was Nch = 512. We simulated single decaytraces with channel width h = 5 ps, pairs of decay traceswith h = 1 ps and h = 5 ps, and pairs of decay traceswith h = 1 ps and h = 5 ps together with 10 Stern–Vol-mer data points evenly distributed between u = 0.1and u = 1. For each parameter set simulation and fittingwere done using the direct method and the referencemethod.

In the simulations the time-shift was set to zero,s = 0. In the reference method the shift was not opti-mized. In the direct method, the shift was a fitted param-eter, and the initial value of s in the fitting procedure wasset to zero. When s was fitted (direct method) the initialvalue of s in the fitting procedure was set to zero.

Each simulation experiment was repeated 103 times,and the fit results were recorded. For each experiment,we calculated the statistical characteristics of the good-ness of fit: the reduced chi-square, v2 its standard nor-mal variate Zv2 , the Durbin–Watson parameter d, andthe ordinary run test statistics Z. Only those fits were ac-cepted that stayed at least 10% away from the lower andupper parameters bounds (0.11 < d, q, j < 9), and satis-fied the statistical conditions jZv2 j < 3, 1.6 < d < 2.4, and|Z| < 3.

4. Results

The parameter recovery for the SCK model dependson three parameters characterizing quenching kineticsk/kD, s/sD and u. This makes an exploration of param-eter space for two analysis schemes a difficult problem.For the sake of presentation we focus here on the per-formance of the estimation procedure along differentaxis of parameter space around the point k/kD = 1,s/sD = 2, u = 0.5. This particular point was selected inorder to bring out the trends in the performance of theestimation procedure.

Figs. 1 and 2 show examples of sample and referencedecays with different time resolutions h = 1 and 5 ps asused in the global analysis. Simulation parameters werek/kD = 1, u = 0.5, Dmax = Rmax = 4 · 104, s = 0,

J. Kłos, A. Molski / Chemical Physics 313 (2005) 85–92 89

Nch = 512. In Fig. 1 s/sD = 0.1, which does not guaran-tee good parameters fits, whereas in Fig. 2 s/sD = 1.47which is in the optimal parameter range for fitting (cf.Figs. 3 and 6). Figs. 1 and 2 also show the weightedresiduals of the decay curve fits. Note that in bothcases the weighted residuals indicate good fits, althoughthe parameters cannot be reliably recovered whens/sD = 0.1.

Fig. 3 shows correlations between parameter esti-mates d ¼ ~D=D; q ¼ ~R=R; j ¼ ~k=k for global analysisof two curves from Figs. 1 and 2. In the top panelsthe parameter s/sD = 0.1 lies away from the optimal fit-

0 5 10

0

5

10

0

0

5

10

0.5 1.0 1.50.5

1.0

1.5

0.5 10.5

1.0

1.5

r

k

k

r

d

d

Fig. 3. Correlations between parameter estimates d, q and j. Reference meresolutions h = 1 and 5 ps, at s/sD = 0.1 (top panels) and s/sD = 1.47 (bottomscales used in the top and bottom panels.

0.

1

0 .1 1 1 0

0 .1

1

10

k /kD

1E-

1E-

0.0

0.

10 .1 1 1 0

0.1

1

1 0

min

avg

max

k/kD

Fig. 4. Accuracy and precision of parameter estimation as a function of tdifferent time resolutions h = 1 and 5 ps using the reference method. Minimurecovered parameters d ¼ ~D=D; q ¼ ~R=R; and j ¼ ~k=k, and of the average as/sD = 2, u = 0.5, Rmax = Dmax = 4 · 104, s = 0, Nch = 512. Arrows serve as afor comment).

ting range, whereas in the bottom panels s/sD = 1.47,which guarantees good fits. Each point in Fig. 3 showsthe location of a minimum of v2 as determined by thesimulation–optimization routine. Away from the opti-mum region (top panels) parameter estimations show awide spread that indicates failure to locate the trueparameter values. Recall that the true values correspondto d = q = j = 1. Close to the optimal range (bottompanels) the spread of parameter estimates is substan-tially reduced. Moreover, the shape of the distributionof estimates becomes closer to elliptical, which wouldcorrespond to the Gaussian distribution of errors.

5 10 0 5 10

0

5

10/ D=0.1

.0 1.5 0.5 1.0 1.50.5

1.0

1.5/

D=1.47

k r

k r

d

d

thod (IRF 2): global analysis of two decay traces with different timepanels). Simulation parameters as in Figs. 1 and 2. Note the different

0 .1 1 1 0

1

1

0IRF1IRF2

k/kD

0 .1 1 1 04

3

1

1

1

0

k /kD

he dimensionless parameter k/kD: global analysis of two decays withm (min), average (avg), and maximum (max) values of the normalizedbsolute error e = (|j � 1| + |d � 1| + |q � 1|)/3. Simulation parameters:guide for the eye for data points at k/kD = 0.68 (cf. Table 1 and see text

0 .1 1 10

0 .1

1

10

m in

avg

m ax

0 .1 1 10

0 .1

1

10

0 .1 1 10

0 .1

1

10

/D

0 .1 1 101E -4

1E -3

0 .01

0 .1

1

10

IR F 1IR F 2

/D

/D/

D

Fig. 6. Accuracy and precision of parameter estimation as a function of the dimensionless parameter s/sD: global analysis of two decays withdifferent time resolutions h = 1 and 5 ps using the reference method. Simulation parameters: k/kD = 1, u = 0.5, Rmax = Dmax = 4 · 104, s = 0,Nch = 512. Arrows serve as a guide for the eye for data points at s/sD = 0.1 and 1.47 (cf. Figs. 1–3 and see text for comment).

0 .1 1 1 0

0 .1

1

10

m in

<r

m ax

k /kD

0 .1 1 1 0

0 .1

1

10 IR F 1IR F 2

k /kD

0 .1 1 1 0

0 .1

1

10

k /kD

0 .1 1 1 01E -4

1E -3

0 .0 1

0 .1

1

10

k /kD

Fig. 5. Accuracy and precision of parameter estimation as a function of the dimensionless parameter k/kD: single-curve analysis with time resolutionh = 5 ps using the reference method. Simulation parameters same as in Fig. 4. Arrows serve as a guide for the eye for data points at k/kD = 0.68 (cf.Table 1 and see text for comment).

90 J. Kłos, A. Molski / Chemical Physics 313 (2005) 85–92

The standard deviation for the fitting parameters isestimated in this work from Monte-Carlo simulations.This allows us to follow not only the spread of the esti-mates, but also the bias (systematic error) for all param-eter ranges. An alternative approach is to use thecovariance matrix. The square roots of the diagonal ele-ments of the inverse of the covariance matrix provideestimates of parameter errors for good parameter esti-mates. For instance, for simulations in Figs. 1–3 ats/sD = 1.47, the average estimates are d = 1.00,q = 1.01, and j = 1.00, and the standard deviationsare sdv(d) = 0.083, sdv(q) = 0.083, and sdv(j) = 0.033(cf. Fig. 6 at s/sD = 1.47). The corresponding error esti-mates from the covariance matrix are sdv(d) = 0.078,sdv(q) = 0.076, and sdv(j) = 0.030, which is consistentwith the simulations. On the other hand, for

s/sD = 0.1 the average estimates are d = 3.39, q = 0.35,and j = 4.06, and standard deviations are sdv(d) = 1.25,sdv(q) = 0.40, and sdv(j) = 1.80, (cf. Fig. 6 ats/sD = 0.1), whereas the corresponding error estimatesfrom the covariance matrix are sdv(d) = 2.8,sdv(q) = 1.5, and sdv(j) = 0.23. Thus, when the biasand assymetric errors are present, the covariance matrixmay not provide a quantitative measure of the spread ofparameter estimates.

Simulations using the direct method show that theaccuracy of parameter estimates is marginally worsethan that from the reference method. Recall that inthe direct method the time-shift s is fitted. Thus fittingthe time-shift has no major effect on parameter recovery.This conclusion is at odds with [5] and [12] where fixingthe time-shift was strongly advocated. This difference

J. Kłos, A. Molski / Chemical Physics 313 (2005) 85–92 91

may result from the fact that here we consider peakcounts of Dmax = 4 · 104, whereas in [5] and [12] decaycurves were measured with Dmax = 104.

Fig. 4 shows the minimum (min), average (avg), andmaximum (max) values of the normalized estimates d, q,j, and their average error e as a function of the dimen-sionless ratio k/kD for two IRFs (solid and brokenlines). We note that the results of the estimation proce-dure for the two IRFs are quite similar, despite the factthat the effective width of IRF2 is larger, since two after-pulses are present in addition to the mean peak. Fork/kD < 1, the minimum and maximum values of the esti-mates indicate an asymmetric scatter of the estimates.Recall that the values 0.1 and 10 are the lower and upperbound on the relative parameters set by the estimationprocedure. The average values of the normalized esti-mates can deviate from 1, which shows that the fittingprocedure is biased. The recovery of R and D improveswhen k/kD increases. For instance, when k/kD = 1, themaximum error of R and D reduces to about 30%,and reaches about 5% at k/kD = 10. There is an optimalrange of k/kD where the rate constant k is recoveredmost accurately. In Fig. 4 the estimates of k lay withinabout 10% of the true value when 0.5 < k/kD < 5. Theright bottom panel in Fig. 4 shows the minimum, aver-age, and maximum values of the average errore = (|j � 1| + |d � 1| + |q � 1|)/3. Overall, the parame-ters can be recovered most accurately aroundk/kD � 5, where maximum e � 10% (upper curves).

Fig. 4 should be compared with Fig. 5 that gives theaccuracy and precision for the corresponding single-curve analysis with h = 5 ps. Clearly global analysis im-proves the accuracy and precision of parameter recoverycompared to single-curve analysis. Interestingly, whentwo curves are analyzed globally for an optimal rangeof k/kD the intrinsic rate constant k can be recoveredmore accurately than D and R. This is in contrast to sin-gle-curve analysis where k is most difficult to recover [6].We cannot offer an explanation of this behavior.

Table 1Global analysis of two decays with k/kD = 0.68, s/sD = 2, h = 5 ps, N = 512average (avg), and maximum (max) values of the normalized recovered para

IRF d q

min avg (stv) max min avg (stv)

u = 0.3 and 0.7

IRF1 0.59 0.96 (0.11) 1.23 0.74 1.06 (0.13IRF2 0.29 0.94 (0.12) 1.24 0.79 1.08 (0.14

u = 0.5 and 0.25

IRF1 0.40 0.94 (0.13) 1.27 0.69 1.08 (0.16IRF2 0.38 0.92 (0.13) 1.30 0.75 1.11 (0.17

u = 0.5 and 0.5

IRF1 0.45 0.95 (0.11) 1.27 0.70 1.07 (0.13IRF2 0.53 0.94 (0.12) 1.26 0.74 1.08 (0.14

Fig. 6 shows the accuracy and precision of the param-eter estimation as a function of the dimensionlessparameters s/sD. The recovery of the quenching param-eters improves with increasing s/sD and the maximumparameter errors can become as low as 10%. Similarbehavior is seen when the accuracy and precision ofthe parameter estimation is analyzed as a function ofu (data not shown). Those trends correspond to thosefor single-curve analysis [6] but global analysis giveshigher accuracy of parameter estimates.

One can combine in global analysis decays measuredat different quencher concentrations. Table 1 givesan example for simulations parameters: k/kD = 0.68,s/sD = 2, h = 5 ps, N = 512, Dmax = Rmax = 4 · 104

and three different combinations of quencher concentra-tions. The values in Table 1 can be compared with thedata points indicated by arrows in Figs. 4 and 5. Clearlyglobal analysis of decays with different quencher concen-trations offers an improvement over single-curve analy-sis in Fig. 5, but the estimates are less accurate thanthose in Fig. 4. Moreover, the last column in Table 1shows that 5–15% of the fits are rejected. This shouldbe compared to about 2% rejection rate when decayswith different resolutions are analyzed globally. Thus,combining decays measured at different quencher con-centrations and the same time resolution, is not as effec-tive as combining decays with the same [Q] and differenth. The origin of this behavior is unclear. When the anal-ysis in Table 1 is repeated with h = 1 ps one gets poorparameter estimates, which indicates that indeed theaccuracy depends critically on the channel width h asit does in single-curve analysis.

We have repeated calculations in Figs. 4–6 with 10stationary Stern–Volmer data points added to globalanalysis of two decays with h = 1 ps and h = 5 ps. Wehave found no substantial improvement in parameterrecovery, except for k at the lowest u values u = 0.1and u = 0.2 (data not shown). Thus adding a few SVpoints to decay curves with high peak counts

, Dmax = Rmax = 4 · 104 for values of u as indicated. Minimum (min),meters d, q, j. The last column gives the percentage of accepted fits

j %

max min avg (stv) max

) 1.55 0.908 0.987 (0.033) 1.133 92.9) 2.28 0.865 0.982 (0.034) 1.100 88.2

) 1.95 0.878 0.985 (0.041) 1.189 90.9) 1.93 0.881 0.978 (0.041) 1.142 84.5

) 1.84 0.882 0.987 0.035 1.170 94.6) 1.67 0.890 0.984 (0.035) 1.125 90.5

92 J. Kłos, A. Molski / Chemical Physics 313 (2005) 85–92

(Dmax = 4 · 104) may have no practical effect on param-eter recovery.

5. Summary and outlook

In this paper, we have explored the parameter estima-tion of diffusion-mediated fluorescence quenchingaccording to the SCK model. We simulated experimen-tal fluorescence decay traces, and estimated the fluores-cence quenching parameters: the sum of the diffusioncoefficients of the fluorophore and quencher D, thesum of their radii R, and the intrinsic quenching ratecoefficient k. The presentation focused on global analy-sis of two decays with different time resolutions usingthe reference method. However, calculations were car-ried out also for the direct method showing similar re-sults. The result of this and previous studies [6,11] onparameter recovery for the SCK model can be summa-rized as follows:

1. The accuracy of parameter estimates from the di-rect method is comparable to that from the referencemethod. Slightly better results for the reference methodare related to the fact that in the reference method thetime-shift s is held constant at its true value s = 0. 2.The primary parameter determining the quality ofparameter estimates is the signal-to-noise ratio deter-mined by the counts at the peak channel. Low peakcounts (say 104) do not guarantee good parameter esti-mates from single curve analysis unless optimal condi-tions are met. Higher counts (say 4 · 104 or 105) leadto substantially better parameter estimates. In each caseglobal analysis improves parameter estimates. 3. Given anumber of counts at the peak channel, parameter recov-ery is determined by the location of system parametersin parameter space. The best parameter estimates areobtained for long-lived fluorophores at high quencherconcentrations. The estimated R and D are more accu-rate as the intrinsic quenching rate k becomes faster,but the estimation of k shows an optimal range of k val-ues. 4. Global analysis of pairs of decay traces with dif-ferent time resolutions can lead to be2tter estimates forthe intrinsic rate constant k than for the diffusion coef-ficient D, and the reaction radius R. This is in contrastwith the single-curve analysis where k is the parametermost difficult to recover. 5. A strategy alternative to ana-lyzing globally decay traces measured at different chan-nel widths is to combine decays measured at differentquencher concentrations at the same time resolution.This leads to an improvement over single-curve analysisbut is not as effective as combining decays with differenttime resolutions. 6. Adding Stern–Volmer data points toglobal analysis of two curves with different channelswidths may have no significant effect on the analysis athigh peak counts. 7. In the case of rapid diffusion-med-

iated quenching good fits to single fluorescence decaysdo not necessarily entail that the recovered parametersare close to the true ones. More reliable results can beobtained when two or several decays are analyzedglobally.

A direction for potentially interesting future workon parameter recovery for diffusion-mediated reactionsis to explore two-dimensional systems. The parameterrecovery of the SCK model in two dimensions (2D)is important for interpretation of fluorescence quench-ing on cell surfaces. Deterministic identifiability analy-sis shows that in 2D the fluorescence lifetime and theSCK parameters can be recovered from a singlequenching curve [17]. This is in contrast with the 3Dcase where two decays are needed to determine thefluorescence lifetime and the SCK parameters, and sug-gests that parameter recovery in 2D may be differentthan that in 3D.

Acknowledgements

We thank the referees for their comments that im-proved substantially this paper. This work was sup-ported in part by a KBN Grant No. 4 T09A 132 24.

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