948
Langmuir 1989, 5 , 948-953
Simultaneous Analysis of Time-Resolved Fluorescence-Quenching Data in Micellar Systems: Application to Mobile Quenchers Steven Reekmans, Noel Boens,* Mark Van der Auweraer, Hongwen Luo, and Frans C. De Schryver* Department of Chemistry, Katholieke Universiteit Leuven, Celestijnenlaan 200F, B-3030 Heverlee, Belgium Received November 1, 1988. In Final Form: March 9, 1989 Global nonlinear least-squaresanalysis, using synthetic fluorescence decays of an excited probe quenched by a mobile quencher in micellar solutions, may be used to optimize experimental conditions in single-photon timing experiments. The global analysis shows an improvement in accuracy and precision, if the number of data channels included in the analysis is increased at a given time window. The improvement is asymptotic, indicating that at least 1/4Kdata points are to be included in the analysis. Moreover, the larger the time window, the more accurate the recovered parameters become. Logically, as the number of experiments included in the analysis increases, the recovered decay parameters approach the true data. The quencher concentration range has no influence on the accuracy of the estimated parameters. The practical applicability of global analysis is demonstrated by using real micellar decay data. Fluorescence decays of 1-methylpyrene quenched by the mobile quencher m-dicyanobenzene in sodium dodecyl sulfate micelles are measured at lo3 and lo4 peak counts by using the single-photon timing technique with reference convolution. From analysis of the decay curves both individually and simultaneously, the superior accuracy of the recovered parameters by the latter method is evident. 1. Introduction Fluorescence quenching in micellar systems is a powerful tool in the study of static and dynamic aspects of solubilization' and the binding of counterions to micelles.2 Especially when the single-photon timing technique is applied, information on micellar growth and on the size and shape of the aggregates3 can be obtained. A kinetic expression for the fluorescence quenching in micellar systems adequate to describe fluorescence quenching by neutral and strongly hydrophobic quenchers was deduced by Tachiya4 and Infeltaa5 The resulting equations were later extended to account for static quenchings and polydispersity effect^.^ The fluorescence &response function f , ( t )for a Poisson distribution of quenchers over the micelles is given by5p6y8
f s ( t )= al exp[-A2t - - 4 3 0 - exp(-Ad))l
(1)
with al = the intensity of fluorescence at time t = 0 (1) (a) Turro, N. J.; Graetzel, M.; Braun, A. M. Angew. Chem. 1980, 92,712. (b) Attik, S. S.; Wam, M.; Singer, L. Chem. Phys. Lett. 1979,67, 75. (c) Zana, R.; Lianos, P. J. Phys. Chem. 1980,84, 3339. (d) Infelta, P. Chem. Phys. Lett. 1979, 61, 88. (2) (a) Takayanagi, T.; Nagemura, T.; Matsuo, T. Ber. Bunsen-Ges. Phys. Chem. 1980, 84, 1125. (b) Rodgers, M. A. J.; Da Silva, M. F.; Wheeler, F. Chem. Phys. Lett. 1978,53,165. (c) Henglein, A.; Scheerer, D. Ber. Bunsen-Ges. Phys. Chem. 1978, 82, 1107. (d) Beunen, J. A.; Ruckenstein, E. J. J. Colloid. Znsterfoce Sci. 1983, 96, 469. (3) (a) Porte, G. J. Phys. Chem. 1983, 87, 3541. (b)Israelachvilli, J. N. h o c . Int. Sch. Phys. 'Enrico fermi" 1985, 90,25. (c) Quirion, F.; Magid, L. J. J.Phys. Chem. 1986, 91, 5435. (4) (a) Tachiya, M. Chem. Phys. Lett. 1975,33, 289. (b) Tachiya, M. J. Chem. Phys. 1983, 78(8), 5282. (5) Kalyanasundaram, K. Chem. SOC.Rev. 1978, 7 , 453. (6) Van der Auweraer, M.; Dederen, J. C.; Gelade, E.; De Schryver, F. C. J. Chem. Phys. 1981, 74(2), 1140. (7) Almgren, M.; Loefroth, J. E. J. Chem. Phys. 1982, 76, 2734.
(8)(a) Van der Auweraer, M.; Dederen, J. C.; Palmans-Windels, C.; De Schryver, F. C. J. Am. Chem. SOC. 1982,104, 1800. (b) Dederen, J. C.; Van der Auweraer, M.; De Schryver, F. C. Chem. Phys. Lett. 1979, 68, 451. (c) Dederen, J. C.; Van der Auweraer, M.; De Schryver, F. C. J. Phys. Chem. 1981,85,1198. (d) Henglein, A.; Proske, T.; Schnecke, W. Ber. Bunsen-Ges. Phys. Chem. 1982,82, 956. (e) Almgren, M.; Linse, P.; Van der Auweraer, M.; De Schryver, F. C. J. Phys. Chem. 1984,88,289. (0 Goesele, U.; Klein, A.; Hauser, M. Chem. Phys. lett. 1979,68, 291. (9) Van der Auweraer, M.; De Schryver, F. C. Chem. Phys. 1987,111, 105.
0743-7463/89/2405-0948$01.50/0
with ko = the deactivation rate constant of the fluorescent probe in the absence of quenching
A4
= k,,
+ k-
(4)
with k+ = the entrance rate constant for a quencher into a micelle (s-l M-'), k- = the exit rate constant for a quencher from a micelle (s-'), k,, = the first-order intramicellar quenching rate constant for a micelle with one quencher and one probe (s-l), K = k + / k - (the binding constant of the quencher to the micellar surface), [ Q ] = the quencher concentration, and [MI = the micellar concentration. Details of the derivation of eq 1have been published in ref 4 and 5. An important assumption is that k- and k+ are both independent of the number of quenchers in the m i ~ e l l e . ~This assumption implies Poisson statistics for the distribution of quenchers over the micelles and accounts for the nonexponential fluorescence decay observed when a fluorescent probe and a quencher are added to a solution of a micelle-forming surfactant.'c.dv8bWhen the quencher is able to enter or leave the micelle during the lifetime of the excited probe, as is the case for the compounds discussed in this paper, the desired parameters k,, k+, and k- can be obtained through successive iterations.1° Depending on the shape and size of the micelle, several extensions of the model have been proposed to describe (9) Hunter, T. Chem. Phys. Lett. 1980, 75, 152. (10) Roelants, E.; Gelade, E.; Smid, J.; De Schryver, F. C. J. Colloid. Interface Sci. 1985, 207(2), 337.
(11) (a) Croonen, Y.; Gelade, E.; Van den Zegel, M.; Van der Auweraer, M.; Vandendriessche, H.; De Schryver, F. C.; Almgren, M. J.Phys. Chem. 1983,87,1426. (b) De Schryver, F. C.; Croonen, Y.; Gelade, E.; Van der Auweraer, M.; Dederen, J. C.; Roelants, E.; Boens, N. h o c . Int. S m p . on Surfactants in Solution; Mittal, K. L., Lindmann, B., Eds.; Plenum: New York, 1984.
0 1989 American Chemical Society
Langmuir, Vol. 5, No. 4 , 1989 949
Time-Resolved Fluorescence Quenching in Micelles
the underlying physical process i n the aggregate. T h e choice of t h e correct model for a micelle of certain dimensions m a y be facilitated b y collecting decay curves under different experimental conditions (e.g., quencher concentration, ionic strength, surfactant concentration). B y linking the common decay parameters k,, S2,S3,a n d A4 of multiple decay curves i n the analysis, we can t a k e full advantage of the relationship between the individual decay curves. T h i s global analysis n o t only results in a more accurate recovery of model parameters but also i n a better distinction between competing models. T h e present paper describes the global iterative nonlinear least-squares analysis of multiple fluorescence decay curves where t h e model is a nonexponential decay function corresponding t o a Poisson distribution of decay times as found for the fluorescence quenching i n micellar systems.15 The performance of the global data analysis will be examined, using b o t h synthetic generated data and real micellar decay data. Simulated decay data are applied t o define optimal experimental conditions. Whether or n o t the quencher concentration range influences the parameter values will be investigated as well as t h e importance of maintaining the reference lifetime at its known value in the calculations. The practical application of global analysis will be shown b y real experimental decay data. 2. Experimental Section 1. Reference Convolution Method i n Analysis of Single-Photon Timing Data. In ideal single-photon timing experiments, the time-resolved fluorescence profile of the sample, d,(&,&,,t), obtained by excitation a t wavelength A, and observed a t emission wavelength A, is the convolution product of the instrument response function l(Aex,Aem,t) and the fluorescence 6 response of the sample, fs(Aex,Aem,t):12
(5) where @ denotes the convolution operator. In the case of a mobile quencher, f,(t) is explicitly given by eq 1,with S2and S3given by eq 2 and 3. Since it is experimentally impossible to measure 1( h,,,A,,,t) directly,12 the instrument response function l(Aex,Aex,t) or l(Aem,Aem,t) is usually measured and used instead of l(Aex,Aem,t) in the model-fitting calculations. Usually the instrument response function l ( t ) is wavelength dependent.16 Consequently, the use of l(Aex,Aex,t) or l(Aem,Aem,t) instead of l(Aex,Aem,t) in eq 5 can lead to poor fits. The best method to correct for this wavelength dependence of the instrument response function is the reference convolution method,12J3in which the decay of a single-exponential is measured a t the same inreference compound, dr(Aex,Aem,t), strumental settings as used for the sa_mple. The parameters of a modified sample response function f,(t) are obtained from the measured decay profiles d,(t) and d,(t) (eq 6) by using a standard deconvolution m e t h ~ d . ' ~ , ' ~
If the reference compound decays with single-exponential decay kinetics (12) (a) Van den Zegel, M.; Boens, N.; Daems, D.; De Schryver, F. C. Chem. Phys. 1986, 101, 311. (b) Boens, N.; Van den Zegel, M.; De Schryver, F. C. Chem. Phys. Lett. 1984,Ill, 340. (13) (a) Gauduchon, P.; Wahl, P. Biophys. Chem. 1978, 8, 78. (b) Wijnaendts van Resandt, R. W.; Vogel, R. H.; Provencher, S. W. Reu. Sci. Instrum. 1982, 53, 1392. (c) Zuker, M.; Szabo, A. G.; Bramall, L.; Krajcarski, D. T.; Selinger, B. Reu. Sci. Instrum. 1985,56, 14. (14) Boens, N.; Luo, H.; Van der Auweraer, M.; Reekmans, S.; De Schryver, F. C. Chem. Phys. Lett., in press. (15) O'Connor, D. V.; Ware, W. R.; Andre, J. C.J . Phys. Chem. 1979, 83, 1333. (16) Bebelaar, D. Reu. Sci. Instrum. 1986,57, 1116. (17) (a) McKinnon, A. E.; Szabo, A. G.; Miller, D. R. J . Phys. Chem. 1977,81,1564. (b) Jennrich, R. I.; Ralston, M. L. Ann. Reu. Biophys. Bioeng. 1979,8 , 195.
/r(t) = ar exp(-t/T,)
(7)
where T~ denotes the decay time of the reference material and a, the corresponding scaling f a ~ t o r , ' ~the ~ Jmodified ~ impulseresponse function /&) corresponding to eq 1 is given by f&t) = A1[6(t) + ( 1 / ~ (ko ~ + SJQ1) - SJQIA4 x exp(-A,t)) exp(-(ko + SAQl)t - SdQl(1- exp(-A4t)))l (8) with Al = a l / a , and 6 ( t ) = the Dirac 6 function. In global analysis, where the decay data are fitted to eq 8, values for ko, S2,S3,and A4 are estimated. f&t) = A1[6(t) + O / T -~1/A2 &A4 exp(-Ad)) exp(-t/Az - Ad1 - exp(-Ad)))l (9) In single-curve analysis, where the decay data are fitted to eq 9, values for A2, AB,and A4 are estimated. 2. Synthetic Data Generation. Synthetic decay data were generated by convolution of the appropriate impulse-response function f ( t ) with a measured instrument response function l(t). Decay data were simulated at different time increments. For each time increment, nine different instrument response functions were measured experimentally in order to obtained instrument response functions l(t)of different size and shape. In this way, real decay data were mimicked as closely as possible. Decays containing 104 peak channel counts were simulated, the preexponential factors al and a, being adjusted to obtain the desired number of counts in the peak channel. Each generated d,(t), d,(t) pair had independent Poisson noise. Several channel widths (0.125 ns, 0.2 ns, 0.25 ns, 0.5 ns, 1 ns, 2 ns) were selected in order to compare at a constant time window different combinations of the time width per channel with the number of data channels. At each specific time increment, synthetic data were generated by using nine different quencher concentrations, (1-10) X lo4 M. Values of 5 X lo4 s-l (ko),5 X lo9 s-l M-' (S2),1 X lo7 s-l (kqm),lo00 M-' (S3), M (micellar concentration), and 10 ns (the reference lifetime) were assumed. Several fluorescence decays with different quencher concentrations were analyzed simultaneously, and use was made of the relations between the different decay curves by linking the decay parameters, ko, Sz, S3, and A4 (which are independent of the quencher concentration, eq 8). Estimates of the fitting variables were computed by a global iteratively reweighted reconvolution program based on the Marquardt algorithmlg for nonlinear least squares. The entire decay profiles, including the rising edge, were analyzed. The convoluted curve was compared with the measured sample fluorescence decay, d,(t), until a satisfactory fit was found.l& A rigorous statistical assessment of acceptability of fits is possible since in the absence of systematic errors the number of counts in a particular channel follows a Poisson distribution.12a*20The global reduced x 2 statistic x2 and its normal deviate ZXzgwere the statistical tests used to jucfge the goodness-of-fit in the global analysis of decay data. Fits with IZ,z I < 3 were considered acceptable. Moreover, the goodness-otfit was examined for the individual decay curves by the calculation of the ordinary runs test?la the Durbin-Watson the local reduced x 2 value x21,and its normal deviate Zxz,. The graphical methods included plots of surfaces23of the autocorrelation function24values versus the experiment number and of the weighted residuals versus the (18) (a) Ameloot, M.; Beechem, J. M.; Brand, L. Biophys. Chem. 1986, 23, 155. (b) Boens, N.; Ameloot, M.; Yamazaki, I.; De Schryver, F. C. Chem. Phys. 1988,121,73. (19) Marquardt, D. W. J . SOC.Indust. Appl. Math. 1963, 11, 431. (20) Boens, N.; Van den Zegel, M.; De Schryver, F. C.; Desie, G. In From photophysics to photobiology;Favre, A.; Tyrrell, R.; Cadet, J.,Eds.; Elsevier: Amsterdam, 1987; p 93. (21) (a) Gunst, R. F.; Mason, R. L. Regression Analysis and its Application, A Data-Oriented Approach; Marcel Dekker: New York, 1980. (b) Draper, N. R.; Smith, H. Applied Regression Analysis, 2nd ed.; Wiley: New York, 1981. (22) (a) Durbin, J.; Watson, G. S. Biometrika 1950, 37, 409. (b) Durbin, J.; Watson, G. S. Biometrika 1951, 38, 159. (c) Durbin, J.; Watson, G. S. Biometrika 1971,58, 1. (23) (a) Beechem, J. M.; Ameloot, M.; Brand, L. Anal. Instrum. 1985, 14, 379. (b) Mauer, R.; Vogel, J.; Schneider, S. Photochem. Photobiol. 1987,46, 247. (c) Mauer, R.; Vogel, J.; Schneider, S. Photochem. Photobiol. 1987,46, 255. (24) Grinvald, A.; Steinberg, I. Z. Anal. Biochem. 1974,59, 583.
950 Langmuir, Vol. 5, No. 4, 1989 channel number versus the experiment number. 3. Chemicals. 1-Methylpyrene (1-MePy) was purified by recrystallization from ethanol in the presence of carbon black, followed by column chromatography on silica gel with benzene as eluent. m-Dicyanobenzene (mDCB, Aldrich) was purified by recrystallization from methanol, followed twice by sublimation. Sodium dodecyl sulfate (SDS, Merck fur biochemische Zwecke) was purified by recrystallization from methanol in the presence of carbon black. No fluorescentimpurities could be detected from the blank SDS solutions under the experimental conditions used in this paper. N-Isopropylcarbazole( T ~= 14.6 ns) in butyronitrile served as a reference. The fluorescence decay curves obtained by excitation at 300 nm and observed at 390 nm were measured with the single-photon timing technique using a synchronouslypumped, cavity-dumped,frequency-doubled Rhodamine 6G dye laser as an excitation source. The details of the picosecond time-resolved fluorimeter used for the fluorescence decay measurements reported here are given elsewhere.20The surfactant concentrationwas the same in all samples (1X IO-' M). The critical micellar concentration of SDS is 7.8 X lo4 M. The probe concentration was kept M) to avoid excimer formation. All samples were delow ( gassed by repeated freeze-pump-thaw cycles before the measurement.
3. Results In section 3.1., global analysis is used for defining the optimal experimental conditions by means of synthetically generated data. Section 3.2. describes the fluorescence decay analysis of 1-MePy in SDS micelles. These real decay data sets are analyzed both individually (single-curve approach) and simultaneously (global approach), and the results are compared. 1. Use of Global Analysis of Synthetically Generated Decay Data To Define the Experimental Conditions. a. Influence of the Time Window on the Accuracy and Precision of the Recovered Decay Parameters. A comparison between several series of multiple decay curves allows investigation of the influence of the time increment and the number of data channels on the accuracy and the precision of the recovered parameters. The time window is defined as the number of data channels multiplied by the time width per channel. The larger the number of data channels, the smaller will be the time increment required to ensure a given time window. In Figure 1, a time window equal to 256 ns is obtained by combining the following time increments and number of data points: 2 ns and 1/8K, 1 ns and 1/4K, 0.5 ns and 1/2K, and 0.25 ns and 1K. Consequently, the results of four data analyses can be compared at a given time window. Figure 1 shows the average estimated fitting parameters hot S2, s,,A4 in function of the number of data channels. All decays contained lo4 peak channel counts and different Poisson counting noise. Each point plotted in Figure 1 is a average value of three independent global analysis estimates. For each time increment and quencher concentration, three decay curves with different Poisson noise were generated. For a given time window, the results obtained from three independent simultaneous analyses of eight decays with different quencher concentrations ( [ Q ] = (1-10) X 10"' M) indicate an improvement in accuracy and precision as the number of data points increases, as shown by (e.g.) the S , parameter values in Table I, representing the results for different combinations of the time increment and the number of data points at a time window equal to 256 ns. The globally determined parameter values are inaccurate and imprecise when the decays contain only 1/8K data points. If the time window becomes smaller (e.g., 128 ns), the recovered model parameters become less accurate and precise, and the standard deviation increases markedly,
Reekmans et al.
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Figure 1. Decay parameters ko, S1, S,, and A4, recovered by simultaneous analysis of eight decay curves ([Q] = (1-10) X lo4 M), are plotted vs the number of data channels. Equation 8 was used to fit the data. The filled symbols represent results for a time window of 256 ns whereas the open symbols represent results for a time window of 128 ns. The true values are shown as horizontal lines. The standard deviations calculated from the parameter covariance matrix are drawn as vertical bars.
as shown by the error bars in Figure 1. If a time window equal to 128 ns is taken (which can be obtained by combining the following time increments and number of data points: 1ns and 1/8K, 0.5 ns and 1/4K, 0.25 ns and 1/2K, 0.125 ns and lK), the precision on parameter S3 is respectively 2990, 2690, 12%, and 7%, and the accuracy approximates 91% , 26%, 1 2%, and 9%, respectively. It should be emphasized that global analysis of all the decays presented in this paper gives statistically acceptable fits (0.887 5 x2g 5 1,109; -2.284 5 ZX2, 5 1.965). The standard deviations u mentioned above are calculated from the diagonal elements of the parameter covariance matrix obtained from the analysis. b. Influence of Quencher Concentration Range on Accuracy and Precision of Recovered Decay Parameters. Since in global analysis multiple decay curves are collected at different quencher concentrations, the question arises whether or not the quencher concentration range may influence the recovered decay parameter values. In Figure 2, six combinations of three decay curves and one combination of eight decay curves containing lo4 peak counts and 1/2K data points are analyzed: (i-j-k) means that three decay curves with [Q] = i X 10" M, j X lo4 M,
Langmuir, Vol. 5, No. 4, 1989 951
Time-Resolved Fluorescence Quenching in Micelles 5.10
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2 ns, 1/8K 1713 f 610 1118 f 246 1724 f 538 1518 f723 18% 52%
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*
0.25 ns, 1K 975 f 31 1000 f 30 970 f 30 982 f27 1% 2%
For different combinations of the time increment and the number of data points at a time window equal to 256 ns.
-
--
Table I. Global Analysis Estimates for S 3 (=lOOO)'
Table 11. Estimated Fitting Parameter Values Obtained by Global Analysis of Eight Sample Decays4 instrument rr linked response (free T. fixedb convolution runnindb k,, (io6S-1 j- 5.000f 0.006 5.006 f 0.005 5.004 0.006 S2(loss-l M-l) 5.055 f 0.045 5.070 f 0.045 5.086 f 0.046 S3 (M-') 985 f 16 976 f 15 975 f 16 A~ (107s-1) 1.004 f 0.019 1.026 & 0.019 1.034f 0.024 T, (ns) 10 9.935 f 0.031
7
*
11 00
[Q]= (1-10)X lo4 M according to eq 5 and eight sample decays according to eq 6. All decays contained lo4 peak counts and 1/2K data points. The third column represents parameter estimates when the reference lifetime is treated as a variable parameter in the fitting curve calculation. Reference convolution.
'r1000 m
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Figure 2. Decay parameters ko, Sz, S3, and A4 recovered from multiple-curve analysis are shown graphically as a function of number combinations which indicate the experiments included in the global analysis. The true values are shown as horizontal 1 ns (0),and 0.5 ns (A)were lines. Time increments of 2 ns (O), investigated. The standard deviations calculated from the parameter covariance matrix are drawn as vertical bars. and k X lo-' M are analyzed simultaneously. As can be seen in Figure 2, the globally determined decay parameters are recovered with equal accuracy independent of the quencher concentration range. The standard deviations are smaller when the time increment is equal to 1 ns, as shown by the errors bars in Figure 1, which emphasizes once more the importance of a suitable dynamic range. For a time increment equal to 1ns, the average standard deviations for ko,S2,Ss, and A4 are small when three decay curves are simultaneously analyzed, being approximately 0.5%,5%, 4 % , and 4%, respectively, independent of the quencher concentration range. Increasing the number of experiments incorporated in a global analysis causes the accuracy of the recovered parameters to increase. However, since the results at 1/2K data points are already quite accurate, the accuracy and the standard deviations improve only slightly when eight decay curves are simultaneously analyzed, as can be seen in Figure 2. For a time window equal to 1ns, the average standard deviations for ko,S2, SB, and A , are approximately 0.1%, 3%, 3%, and 2%, when eight decay curves are simultaneously analyzed. Moreover, the parameter values calculated by global analysis are randomly scattered around the true parameter values, which are indicated by a horizontal line. Conse-
quently, one may conclude that systematic errors in the calculations are absent. Up to this point, the recovery of the decay parameters is performed by making use of the reference convolution method (eq 6). Since in this section all the sample decays d,(t) were synthetically generated by using a particular instrument response function l ( t ) ,a comparison can be drawn between the results given by global analysis of a sample decay d , ( t ) according to eq 5 and a sample decay d , ( t ) according to eq 6. The results obtained by both methods are equivalent as far as the parameter values, the accuracy, and the standard deviations are concerned. In Table 11, the results obtained by global analysis of eight decay curves containg 1/2K data poifits and lo4 peak counts are shown for a time increment equal to 1ns. When T , (the fluorescence lifetimes of the sample) and 7, are closely spaced or when the reference lifetime is rather small as compared to the time increment,18b more precise parameter estimates may be obtained by keeping 7,constant at its true value in the course of the calculations, as has been done in all the experiments discussed above. Consequently, the problem of the accurate determination of the reference lifetime arises. Estimates of the fluorescence parameters of synthetically generated decay curves are compared with and without fixing the lifetime of the reference compound in global analysis. It should be noticed that the reference lifetime used in the data analyses presented in this paper always exceeded double the time increment.IEb All the calculations presented in Figure 2 were repeated, treating T , as a variable parameter in the curve-fitting calculation, which resulted in an accurate parameter estimate of T , and 7,as shown by the results in Table 11. 2. Simultaneous Analysis of Real Decay Data. The fluorescence decays of 1-MePy, solubilized in aqueous SDS micellar systems and quenched by the mobile quencher mDCB, are described by eq 1 and were measured by the single-photon timing technique at various quencher concentrations. All decays contained 1/2K data points and either lo3 or lo4 peak channel counts. The experimental data were analyzed by individual and global analysis in which the four decay parameters ko,S3, Ad, and S2were
952 Langmuir, Vol. 5, No. 4 , 1989
Reekmans et al.
2.5
0
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Figure 4. Decay parameter A4 recovered from statistically acceptable single (open symbols) and multiple (filled symbols on a horizontal line) decay curve analyses of the fluorescence decays of 1-MePyquenched by mDCB in SDS micelles plotted vs [Q]. All sample and reference decays contained lo4peak counts and 1/2K data points. The standard deviations are calculated from the parameter covariance matrix and are drawn as vertical bars. The globally estimated values for k0, Sz, and S3 are respectively (4.94 i 0.02) X lo6 s-l, (3.32 f 0.03) X IO9 s-l M-l, and (441 f 3) M-I.
L
12
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16
20
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Figure 3. (a) Decay parameter A2 recovered from statistical acceptable single-curve analyses according to eq 9 are plotted vs
quencher concentration [Q]. All sample and reference decays contained lo4 peak counts and 1/2K data points. The linear regression analysis (correlation coefficient = 0.9946) gave AZ = (4.60 f 0.30) X lo6 + (3.76 0.23) X 10e[Q]. (b) Single-curve estimates of parameter A3 from the experiments presented in part a are plotted vs [Q]. The linear regression analysis (correlation coefficient = 0.9981) gave A, = -(0.85 f 0.021) + (444 i 16)[Q].
*
linked. The reference lifetime T , was kept fixed a t its known value (14.6 ns) in both single and global analysis. As can be seen in Figure 3a, the dependence of the A2 values, obtained by single-curve analysis, on the quencher concentration clearly demonstrates the importance of the exit and entrance probability of the quencher during the lifetime of the excited probe.8aJ1a The recovered decay parameters A2 and A3 as a function of [Q]are graphically shown respectively in parts a and b of Figure 3. For the studied probe/quencher system, it has been confirmedllb that k- is independent of the quencher concentration and that therefore the parameter A4 remains constant. As can be seen in Figure 4 (lo4peak counts), the parameter values calculated by single-curve analysis are randomly scattered around the estimated global parameters. If all five decays (lo4peak channel counts) are combined in a simultaneous analysis, statistically acceptable fits are obtained as judged by xZg(1.094), Z,z (3.0201, the local statistical tests x21 and ZXzl,the Durbin-%Vatson test statistic, the ordinary runs statistic, the residual plots, and the autocorrelation function. The globally determined decay parameters were ko = (4.94 f 0.02) X lo6 s-l, S2 = (3.32 f 0.03) X lo9 s-l M-l, S3 = (441 f 3) M-l, and A4 = (3.81 f 0.07) X lo7 s-l. By analyzing the individual decay curves ( lo4 peak counts) separately, satisfactory fits were obtained (1.032 5 x2g I 1.125; 0.468 IZ,zg I 1.753). However, the standard deviation of the parameters is larger in single-curve analysis than in global analysis, as can be seen by the following results: ko = (4.60 f 0.30) X lo6 s-l, s2= (3.76 f 0.22) x 109 s-l M-' , s - (444 f 16) M-l, and
A , = (3.67 f 0.20) X lo7 s-l. A reduction of the peak channel counts from lo4 to lo3 increases the standard deviation and decreases the accuracy in the recovery (results not shown). The following parameter values were obtained from global analysis (lo3 peak counts): ko = (4.60 f 0.06) X lo6 s-l, Sz = (4.02 f 0.07) X lo9 s-l M-I, S3 = (423 f 9) M-l, and A , = (4.71 f 0.24) X lo7s?. The parameter estimates from single-curve analysis were (lo3 peak counts) KO = (4.04 f 0.28) X lo6 s-l, S , = (401 f 41) M-l, A4 = (5.91 f 0.63) X lo7 s-l, and S2 = (4.63 f 0.18) X lo9 s-l M-l. Decays containing lo3 and lo4 peak channel counts were analyzed by individual and global analysis with the parameter A2 kept constant ( A , = 0) in the course of the calculations, assuming that the quencher is immobile. The following parameter estimates were obtained from single-curve analysis ( lo4 peak counts and lo3 peak counts, respectively): ko = (9.64 f 0.03) X lo6 s-l, S3 = (415 f 6) M-l, A4 = (3.97 f 0.13) X lo7 s-l, and K O = (9.58 f 0.09) X lo6 s-l, S3 = (454 f 19) M-l, and A , = (4.93 f 0.46) X lo7 s-l. By analyzing the individual decay curves (IO4 peak counts and lo3 peak counts) separately, we obtained satisfactory fits (0.995 I x2gI 1.125; -0.079 5 ZXz,5 1.701). When two decay curves were simultaneously analyzed (lo4 or lo3 peak counts), unacceptable fits were obtained, as judged by the statistical tests.
4. Discussion The results presented in section 3.1. clearly demonstrate that the simultaneous analysis approach of multiple synthetic fluorescence decays is an excellent method to gain insight in the importance of some experimental conditions such as the number of data channels included in the analysis, the time increment, the time window used to view the decay, or the effect of the applied quencher concentration range on the accuracy and precision of the estimated decay parameters. As can be seen in Figure 1, there is no doubt that a larger time window produces better results. In order to recover highly accurate decay parameters, it is more important for a given time window to increase the number of data channels than to take a larger time increment. Increasing the number of channels will improve significantly the accuracy and precision of the recovered parameters, although this tendency assumes an asymptotic form. Taking into account these results, it is
Time-Resolved Fluorescence Quenching in Micelles advisable to include a t least 1/4K data points in the fluorescence decay analysis. However, the more data channels that are included in the analysis, the greater the collection time will become, and consequently the probability of systematic (nonrandom) errors due to the instability of the excitation source might increase. Figure 2 indicates, as one could expect, that the accuracy improves by incorporating more experiments in the global analysis. The accuracy of the recovered decay parameters is not influenced by the choice of the quencher concentration range applied in the global analysis, provided that the number of quenchers per micelle does not exceed certain values leading to a perturbation of the micellar structure. In most of the single-photon timing experiments in which the reference convolution method is applied, the reference lifetime 7, is kept constantlsb at its known value in the calculations. However, if T, is more than twice the time increment and sufficiently different from the l / ( A 2 + A3A4) values, it is not required to maintain 7, a t its known value, since the lifetime of the reference compound can be obtained with great accuracy (
