Global analysis of unmatched polarized fluorescence decay curves

J . Phys. Chem. 1993,97, 8133-8145

8133

Global Analysis of Unmatched Polarized Fluorescence Decay Curves Marc Crutzen,t Marcel Ameloot,'v* N d l Boens,t R. Martin Negri,' and Frans C. De Schryver'*t Department of Chemistry, Katholieke Universiteir Leuven, B-3001 Heverlee, Belgium. and Limburgs Universitair Centrum, B-3590 Diepenbeek. Belgium Received: October 23, 1992; Zn Final Form: May 20, I993

The performance of the simultaneous analysis of unmatched polarized fluorescence decay curves collected under various experimental conditions, including reference convolution, is investigated systematically for a number of nonassociative (homogeneous) and associative (heterogeneous) model cases with zero limiting anisotropy at long times. The study has been carried out using an improved and rapidly converging algorithm based on the constraint for equal area under the observed and fitted curves. The efficiency of changing experimental conditions such as the orientation of the emission polarizer, the emission/excitation wavelength, and the timing calibration is discussed with respect to the parameter recovery and the power to discriminate between alternative models for the fluorescence anisotropy. When the decay of the total fluorescence and the anisotropy are monoexponential, the combination of decay traces collected at the two extreme orientations of the emission polarizer is sufficient for the parameter recovery when the rotational relaxation time is not too much different from the fluorescence relaxation time and is larger than the timing calibration of the channels. For a single compound system with a multiexponential r(t)the use of different excitation wavelengths is recommended. Different timing calibrations have to be utilized when the relaxation times in the anisotropy decay are widely different. For a mixture of compounds the best results are obtained by using different excitation/emission wavelengths. As an example of the latter a mixture of resorufine and cresyl violet in 1-propanol at 20 OC has been analyzed using reference convolution.

Introduction

It will be assumed in this paper thatflt) can be written as

Time-resolved fluorescence depolarization measurements are an important tool in the study of the dynamic behavior of molecules. When a sample is excited with a 6 pulse of polarized light, the molecules with the transition dipole of absorption near the directionof the polarization of the excitation beam are excited preferentially. Rotational motion of the excited molecules redistributes the emission dipoles of the excited molecules and hence depolarizes the emitted light. The information on the reorientational dynamics of the transition dipoles is contained in the fluorescence anisotropy decay r(t)

whereil(t) and il(t) denoterespectively the polarized fluorescence decays with the emission polarizer set parallel and perpendicular to the polarization direction of the excitation light, which is perpendicular to the excitation-detection plane. The polarized intensities can be written as

+ 2rWl i l ( t ) = (1/31f(t)[l -4t)l

i&t) = (1/31f(t)[l

(2)

(3)

with At) being the decay of the total fluorescence. For an excited species in a single type of isotropic environment, r(t)is, in general, given by a linear combination of exponentially decaying functions (4)

4

Katholieke Universiteit Leuven. Limburgs Universitair Centrum.

0022- 3654/93/ 2097-81 33$04.OO/ 0

n

i- 1

In the case of a mixture of species and/or environments, in(r) and il(t) can generally be written2 as

n

m

where the factors L,, are equal to unity if the relaxation time 7, is associated with the correlation time 3 and zero otherwise. Two cases have to be considered. In the nonassociative (homogeneous) case, the relaxation times Ti are associated with each of the rotational correlation times a,. In the associative (heterogeneous) case however, each 7,corresponds to a particular set of Discriminationbetween the two models and parameter recovery can be very difficult even when the correct model is known.132 Polarized intensity decays can be collected by the single-photon timisg techniquein the time domain and by phase and modulation fluorometry in the frequency domain.3--5 The use of mode-locked, synchronously pumped, cavity dumped dye lasers and fast detectors have enhanced the performance of both techniques,With these techniques high quality polarized fluorescence decay data are now available, allowing a rigorous data analysis. This paper focuses on the single-photon timing technique. In this method the collected polarized fluorescence decays, Zl(i) and ZL(r),areconvolutionproductsofill(t)andiL(r)withthemeasured excitation pulse. In all cases considered here, the functions ill(r) and i L ( t ) are sums of exponentials. Traditionally, the analysis Q 1 993 American Chemical Society

8134

The Journal of Physical Chemistry, Vol. 97, No. 31. 1993

of the polarized fluorescence data has involved the construction of two related decay curves, the sum curve S(t)and the difference curve D(t)9 S(t) = Z&f)

+ 2KZ,(t)

where K is a matching factor which accounts for the polarization bias of the apparatus, as well as for fluctuations in excitation pulse intensity and differences in collection time between Za(t) and Z,(f). The sum curveS(t) is analyzed to give the parameters of the total fluorescencedecayflt). The resulting values are then kept fixed during the analysis of the difference curve D(t) to obtain the parameters of the anisotropy decay r(t). This method has some drawbacks. For a proper deconvolution of such S(r) and D ( f ) the shapes of the instrumental response functions corresponding to Z,(t) and Z,(t) have to be essentially (within inherent noise) identical. Therefore,Zy (t), ZL(r), and the excitation pulse profile need to be measured in alternation. The correct weighting factors should be calculated for each data point,+'' the determination of the parameters of r ( f )depends on the previously estimated parameters of f l t ) , and an accurate knowledge of the matching factor K between the two curves is an absolute prerequisite. The global analysis approach, in which different sets of experimental data are analyzed simultaneously by linking parameters which are common to different fitting functions, have proved to be very efficient in the analysis of total fluorescence d e c a y ~ . ' ~ -The l ~ global analysis is now also the method of choice for the analysis of fluorescence depolarization data.lIJ3J5-22 Simultaneousanalysisof the measured decays Zl(t) and Z,(t) has a number of advantages compared to the sequential analysis mentioned above. The excitation profiles of the two curves may be different, allowing for several simultaneous detectionchannels. The usual weighting factors for single-photon timing experiments based upon the Poisson (counting) statistics can be used directly. All parameters of bothflt) and r(t)are simultaneously optimized in a single-step analysis. If desired, the preexponential factors at in the expression of polarized fluorescence decay curves can be linked through the use of matching factors K . The matching factor K does not need to be known beforehandI7*20J2but the value can be determined experimentally9to reduce the number of fitting parameters. However, no rigorous analysis of the method of unmatched polarized fluorescence decayshas been described yet. Polarized intensity decays can be collected under various conditions.13J7.20.21 The resulting decay data surface can then be analyzed globally to obtain optimum parameter estimates.13.21 The experimental axes that can be used will be described in the theoretical section. The purpose of this paper is to evaluate the global analysis method of unmatched polarized fluorescence decays. We shall investigate the relative importance of different experimental conditions with respect to the accuracy and the precision of the estimated parameters and the model testing capability when a global analysis approach is used for related but unmatched polarized fluorescencedecays. Special attention will be given to the discrimination between associative and nonassociative models. Both the convolution with a measured instrumental response function and with the decay of a reference compound will be discussed. The reference convolution method has proven its merits in the analysis of the total fluorescence decays,1**15.23.24 but has rarely been used in the analysis of fluorescence depolarization data.15.19 Theory: General Equations and Possible Linkages

The anisotropy decay of a general ellipsoid in an isotropic environment is given by a sum of five e~ponentials.2s-2~ However, in practice no more than three exponentials will be observed.28 If the molecules can be approximated as a rigid symmetric rotor,

Crutzen et al. the emission anisotropy becomes a sum of three exponentials (eq 4) with

e, - 1) (3 cos2 e, - 1) f12 = 0.3 sin' e, sin2 8, cos 2$,, p3 = 1.2 sin e, cos 0, sin 8, cos 0, cos $, p1 = 0.1 (3

(8a)

(8c) 6, and 0, are the angles made by the absorption and emission dipoles with respect to the symmetry axis and $,,.is the angle between their projections in the plane perpendicular to the symmetry axis. The value of depends on the excitation wavelength but is independentof temperature and viscosity. The sum of the factors gives the initial (zero time) anisotropy,r(o) = ro. The value of ro depends only on the angle between the absorption and emission dipoles. It can be shown that (e.g. see ref 9) -0.2 Iro I0.4. The corresponding rotational correlation times are given by

@,

@,

Qi,

= (all,)-'

+ 40,)-1 a3 = ( 5 0 , + 0,)-1 = (20,

D , and Di are respectively the rotational diffusion constants for rotation around an axis perpendicular to the symmetry axis and around the symmetry axis. The rotational correlation times are functionsof theshapeand theelongation of therotor, theviscosity of the solvent, and the temperature, but are independent of the excitation and emission wavelengths.29 This means that in a global analysis of polarized fluorescence decays the following linking scheme can be utilized. The preexponential factors fl, of r(t) can be linked along any experimental axis except the wavelength axes when different transitions are involved. However, along the excitation and emission wavelength axes the rotational correlation times of a given rotor can be linked. This approach has proven its usefulness in resolving closely spaced rotational correlation times.13 The ratios of all or some of the preexponential factors arofflt) of a heterogeneous system generally depend on either or both the excitation and the emission wavelengths. When different timing calibrations are used to improve the recovery of widely different relaxation times, all the parameters T ~Bt,, , &j (i.e. 3 /, and 4, for a given emitter), and the ratios of af can be linked. It has been suggested that a similar improvement can be obtained on reducing the fluorescence lifetime by quenching.30 In this case though, only the anisotropyparameters for a given emitter can be linked, unless enough quenching concentrations are included to allow fitting for the parameters that describe the effect of the quencher on T ~ . Consider the situation where the polarizing angles of the excitation and emission polarizer with respect to the normal to the excitation-emission plane are given by $ and B respectively. The corresponding polarized fluorescence decay i($,O,f)can then be expressed (see Appendix A) as i($,e,t) = (1/3)f(t)[l + (3 cos2 $ cos2 B - l)r(t)] (loa) Equation 10a shows that the effect of changing the excitation polarizer or the emission polarizer is equivalent. It is assumed in this report that = Oo so that the indication of the angle can be omitted. In that case the polarized intensity is given by

+

+

i(e,t) = (1/3)f(t)[l + (3 cos2 B - l)r(t)] (lob) This means that iH(t) = i(Oo,t)and that i l ( t ) = i(90°,r). At B = 54.7O, theso-calledmagicangle,i(54.7°,t) = f l t ) / 3 . Alldecay curves observed with a different polarizing angle of the emission polarizer are determined by the same set of fitting parameters, apart from a matching factor, so that a full linkagebetween them

Fluorescence Decay Curves is possible. The use of more values of B in addition to Oo and 90° was suggested by Cross and Fleming" and has been implemented by Flom and Fendlerezo

Materials and Methods Materials. Cresyl violet perchlorate (Laser grade) and resorufine were obtained from Aldrich. Both dyes yielded a single spot upon thin-layer chromatography and were used without further purification. 1-propanol was of spectroscopic grade (Aldrich) and was used as received. Water was filtered using a Milli-Q system. The samples were not degassed. The concentration of the dyes was less than lo" M. The experiments were done at room temperature using 1-cm cells. Instrumentation. The anisotropy decays were obtained by the single-photon timing technique. The samples were excited by the output of a cavity-dumped Rhodamine 6G dye laser synchronously pumped by a mode-lockedargon ion laser. Full details of the experimental setup are given elsewhere.31 The polarized fluorescence decays of the sample can be measured at any angle of the emission polarizer, which is controlled by the computer. The decay traces of the reference compound were measured with the emission polarizer set at 54.7O. A circular variable neutraldensityfilter(RosOpticsZ06,011, mat FS,OD0-3) in theemission path was adjusted by the computer so that all decay traces were collectedat thesamedetectionrate. Thedecay dataofthesample and the reference were collected alternately for preset dwell times. All calculations and synthetic data generations were performed on an IBM 6150-125 computer in single precision. Data Analysis. The analysis program allows for a maximum of three exponentials inflt) and r(t). To maximize the flexibility of the analysis program, the polarized intensity i(B,t) is written as n

Within the program a matrix L* = [15*~,]is associated with the matrix L = [&,I. L*, can take the values 0, 1, 2, and 3. If L, = 0, then L*i, = 0. If L,, = 1, then L*, > 0. If L*~J= 1 the corresponding&value is kept constant at a set value. The program has the possibility to constrain the sum of the &values to a given value for ro. ro can be different for each T ( . For each given value of ro, the matrix element L*,,of one 3/, is then equal to 2. The 3/, parameter with L*,,= 2 is not optimized during the analysis, but takes a value so that the sum of the appropriate /3 values equals the given ro value. If L*,, = 3 the associated #?I is freely adjustable during the analysis. K ( 6 ) in Eq 11 is a matching factor for thepolarizedfluorescence decay collected at the orientation 0 of the emission polarizer. Therefore, the parameters aiof the decay traces taken at different analyzer angles can be linked. If all the decays are collected for the same number of absorbed excitation photons, K ( e ) has only to compensatefor the polarization bias of the deprotection system. This bias can be measured so that K(B) can be kept fixed during the analysis. Alternatively, K ( B ) can be a freely adjustable parameter. Several implementations are possible, (a) all ai and all K ( B ) but one are freely adjustable; the remaining K ( B ) is fixed at an arbitrary value; (b) all ai but one and all K ( e ) are freely adjustable; the remaining ai is fixed at some arbitrary value. It has been pointed out that for each decay trace the total number of counts under the observed and adequately matched calculated decay curve must be the same.20 We have implemented this constraint in the analysis program to enhance the rate of convergence. In this new implementation the fitting parameters are no longer independent of each other. The implications of the

The Journal of Physical Chemistry, Vol. 97, No. 31, 1993 8135 constraint on the derivatives of the fitting function are described in Appendix B. When r(t) andflt) are respectively given by eqs 4 and 5, then in an ideal single-photon timing experiment, the polarized fluorescence decay of the sample Z(B,Xex,Xe",f), obtained by excitation at wavelength Xex and observed at wavelength X a , can be written as z(e,r,vm,t)

= K u ( ~ e x , ~ e m , si(e,Xex,Am,t ) - s)

(12)

where the shape of the instrument response function (IRF) u(X=,Xm,t) is taken to be independent of 8. To correct for the wavelength dependence of the instrument response function, the reference convolution method14JsJ9.23."was used. In this method eq 12 is replaced by z(e,vX,vm,t)

= ~ ~ r ( X ~ x , ~ ~ , s ) ~ ( e -, X 8) edr ~ , ~(13) em,t

where dr(Xex,Xem,t) is the total fluorescence decay of a reference compound measured under identical instrumental conditions as for the sample but with the emission polarizer set at 54.7'. If the reference decay is monoexponential

f,(t)=

exP(-t/rr)

(14)

;(t) is given by23

i ( t ) = ar-'[i(O)S(t)

+ i'(t) + i ( t ) / r r ]

(15) where 6 ( t ) is the Dirac delta function and i'(t) denotes the time derivative. The anisotropy analysis program is part of a global analysis program which has been described earlier in detail.32 A global, reweighted iterative reconvolutionprogram based on the nonlinear least-squares algorithm of M a r q ~ a r dwas t ~ ~used to estimate the unknown parameters. The fitting parameters were determined by minimizing the global reduced :x l4

where the index k sums over q experiments, and the index i sums over the appropriate channel limits for each individualexperiment, and wWare the weighting factors.34 PkiandZ'&,denoterespe!ctively the observed (experimentally measured or computer generated) and calculated values corresponding to the ith channel of the kth experiment. v represents the number of degrees of freedom for the entire multidimensional fluorescence decay surface. The statistical criteria to judge the quality of the fit included both graphical and numerical tests. The graphical methods comprised plots of surfaces ("carpets") of the autocorrelation function values vs experiment number and of the weighted residuals vs channel number vs experiment number. A good fit should produce "carpets" free of pronounced "creases". The numerical statistical tests included the calculation of the global reduced chi-square statistic :x and its corresponding Z-2 ,-

Since Z , 2 is standard normally distributed, theoretical probabilities ofZ,: values occurring within a given range can be easily obtained from the cumulativestandard normal distribution. Using Z*z the goodness of fit of analyses with different v can be readily compared.

Synthetic Data Generation The synthetic sample decays were generated by convolution of the expressionsfor i(B,t) with a nonsmoothed measured IRF. The full width at half-maximum was about 0.7 ns. Poisson noise was

8136 The Journal of Physical Chemistry, Vol. 97, No. 31, 1993

Crutzen et al.

TABLE I: Recovery of the Parameters of a Monoexponential r(t) and flt) by a Global Analysis of I(Oo,t) and I(90°,t) at Time Increments per Channel of 47 psa 7 (ns) 0 (ns) a U 0' (ns) ip. (4 u (4 r,' r, 5.0 5.0 5.03 4.99 5.006 5.000 4.995 4.998 4.998 4.999 4.994 4.994 4.996

0.4 0.2 0.01 0.02 0.006 0.004

0.004 0.004 0.004 0.004 0.004

0.004 0.004

OSb

OSb 0.22 0.37 0.341 0.352 0.342 0.355 0.34 0.38 0.42 0.38 0.35

4.5 1.3 0.04 0.04 0.005 0.002 0.004 0.008 0.02 0.05 0.09 0.09 0.09

,

200 100 50 25 10 2 0.5 0.25 0.1 0.05 0.02 0.01 0.005

303 161 28 26 9.6 2.0 0.511 0.245 0.100 0.045 0.01 0.003 0.002

4160 630 6 4 0.3 0.2 0.009 0.007 0.008 0.008 0.02 111 7 x 107

0.341 0.333 0.318 0.292 0.233 0.1 0.0318 0.0167 0.00686 0.00347 0.00139 0.000699 0.000350

0.491 0.484 0.184 0.306 0.224 0.101 0.0317 0.0166 0.00673 0.00338 0.00100 0.000229 0.000141

T I = 5 ns. rot = 0.35. r,' and r, denote the true and recovered steady-state anisotropy, respectively. b Upper boundary for ro in the program was set to 0.5.

added to the simulated curves. For each curve a different seed for the random number generator was used. Reference decays were synthesized by convolution of the same IRF with a single exponential. Unless noted otherwise, all simulated decays had 1/2K data channels and lo4counts in the peak channel, implying different K ( 0 ) values. The time increment per channel was adjusted to obtain a decay over at least two decades. More details of the decay data simulations are given el~ewhere.3~In the tables the simulation parameters are indicated by the superscript t.

Results and Discussion A. Computer Generated Data. Severalsyntheticdata sets were globally analyzed in different ways: a) with freely adjustable K ( 0 ) according to the method described by Flom and Fendler20 b) with freely adjustable K ( 0 ) according to our method (see Appendix B). In general, these methods yielded parameter estimates with practically identical accuracy and precision. However, the rate of convergencefor method b was less dependent on the initial guesses. Analyses with known and fixed matching factors were carried out with method a. The standard deviations u on the recovered parameter values were estimated from the diagonal elements of the parameter covariance matrix available from the nonlinear least-squares analysis at the point of convergence. The standard deviations were also estimated from the distribution of the parameter values obtained from 20 global analyses of data sets generated with the same simulation parameters but with different seeds for the random number generator. Quite similar values for the standard deviations were obtained (results not shown), indicating that the estimates for the standard deviations obtained by the analysis program are reliable. A.1. Monoexponential fit) and Monoexponential r(t) with IRF Convolution. To investigate the recovery of 9 and ro as a function of the ratio @ / T , a number of unmatched polarized fluorescence decay sets were simulated. Each set consisted of I(Oo,t) and I(90°,t). For all the simulated decays T was set to 5 ns. Q, ranged from 200 ns to 5 ps so that the corresponding ratio 9 /was ~ varied between 40 and 0.001, The values for the initial anisotropy ro were 0.35 and 0.1. The corresponding steady state anisotropy, r, PO r, = -

(18)

E+ 1

4

ranged from 0.341 to 0.00035 for ro = 0.35 and from 0.0974 to 0.0001 for ro = 0.1. All decays were simulated with a time increment per channel of 47 ps. All sets were analyzed by global analysis with all parameters freely adjustable except for the linked a parameter which was fixed to 1. The results obtained for ro = 0.35 and for ro = 0.1

showed the same tendency. The recovered parameter values are only tabulated for ro = 0.35 (Table I). The results can be summarized as follows. The recovery of 9and ro was dependent on @ / T and ro. The estimatesfor 9and ro were accurate and with relative errors smaller than 10% for 0.01 I @ / T I5 at ro = 0.35 and for 0.05 I@ / T I2 at ro = 0.1. For these @ / T ratios the fluorescence lifetime is recovered to good accuracy (better than 0.2%) and precision (better than 0.3%). The estimates obtained for 4 and ro at the longer 9 values were highly inaccurate and imprecise. (Note that rovalues of 0.5in Table I correspond with the upper boundary level for ro set within the analysis program. This upper level was chosen to allow for maximal flexibility in the minimalization process.) Indeed, for very large @/T values there is no observable anisotropy decay. This means that essentially ro has to be determined so that the knowledge of the matching factors is required. For the largest 9 values (50, 100, and 200 ns) and the smallest 9 values (0.02,0.01, and 0.005 ns) at ro = 0.35 the calculated values for r, deviatesubstantially from the true value. In the cases where the calculated steady-state anisotropy essentially corresponds with the true value, also the area under the calculated r(t) = (=ro9) is correct because the fluorescence lifetime is well recovered. This implies that when the recovered value for the rotational correlation time was lower than the true value, the recovered value for the initial anisotropy was higher than the true value and vice versa. It is not evident that with freely adjustable matching factors the steady-state anisotropy is recovered. When ro was kept fixed at its true value, the recovered value for the steady-state anisotropy was, in most cases, accurate, indicatingthat in those cases the values obtained for the matching factors were accurate also. Note that similar results were also obtained for the longer 9 values but not for the three shortest 9 values being smaller than half the time increment per channel. The estimates for 9 were accurate and had percentage errors smaller than 10% for 0.002 I O / T I40 at ro = 0.35 (see Table 11) and for 0.01 5 @/T I 20 at ro = 0.1 (results not shown). These results can be rationalized if i(0,t) (eq 11 with n = 1 and m = 1) is rewritten as a double exponential. The ratio of the preexponential factors in i(6,t) is given by (3 cos2 0 - 1)rO. The two decay times are given by T and ( 1 / ~+ 1/9)-l. At low @/T values one has T >> @ and (1/ T + 1/9)-1 * 9.The performance of the analysis decreases when at the very low @ / T ratios 9is less than the time increment per channel. Values of 9 less than the time increment per channel can be determined by fixing ro at its truevalue. In thatcasenotonlyis thenumberoffitting parameters reduced but the ratio of the two preexponential factors in i(0,t) is fiied. It has been shown previously that the simultaneous analysis of fit) traces obtained at different timing calibrations can resolve widely differing decay rates.z13J4 A global analysis using different timing calibrations should then lead to improve-

The Journal of Physical Chemistry, Vol. 97, No. 31, 1993 8137

Fluorescence Decay Curves

TABLE IL Analysis with Fixed ro of the Data Discussed in Table I (ns) 4.994 5.006 4.993 4.995 4.998 5.001 4.995 4.998 4.998 5.000 4.995 4.994 4.996

dt (ns) 0.004 200 0.004 100 0.004 50 0.004 25 10 0.004 2 0.004 0.004 0.5 0.004 0.25 0.004 0.1 0.004 0.05 0.02 0.004 0.004 0.01 0.004 0.005

Q

(4

(ns)

191 101 50 24.5 10.0 2.02 0.498 0.249 0.098 0.050 0.022 0.015 0.003

u (ns)

r,'

rl

12 4 1 0.3 0.1 0.02 0.006 0.004 0.002 0.002 0.002 0.003 34

0.341 0.333 0.318 0.292 0.233 0.1 0.0318 0.0167 0.00686 0.00347 0.00139 0.000699 0.000350

0.341 0.333 0.318 0.291 0.234 0.101 0.0317 0.0166 0.00673 0.00347 0.00153 0.00105 0.000210

TABLE IIk Recovery of the Parameters of a Monoexponential r(t) and 4t) at Low @ / T Values by a Global Analysis of Z(Oo,t) and I(90°,t), Each Beiig Simulated at Time Increments of 47 and 5 ps' T (ns) u (ns) ro u W(ns) d (ns) u(ns) 4.999 4.994 4.995 4.995 (I

it =

0.004 0.004 0.004 0.004

0.38 0.35 0.38 0.35

0.02 0.03 0.02 0.02

0.050 0.020 0.010 0.005

0.044 0.020 0.008 0.005

0.003 0.002 0.001 0.001

5 ns. rot = 0.35.

ment of the accuracy and precision of the recovered parameters at low O / T values. To check this, additional I(Oo,t) and Z(90°,t) curves with ro = 0.35 were simulated with a time increment per channel of 5 ps. The IRF had 500 ps full width at half-maximum. These two curves were analyzed together with the two curves generated with the same simulation parameters at a time increment per channel of 47 ps. The results are shown in Table 111. 9 values as low as 5 ps ( 9 / r = 0,001, r, = 0.000 35) and ro were accurately recovered with good precision. When at the 5-ps time increment a 10times narrower IRF (50 ps full width at half-maximum) was used instead, slightly more accurate and precise results were obtained than with the broader (500 ps) IRF. Quenching of the fluorescence lifetime has been proposed to improve the recovery of a short correlation time.30 Indeed, when the fluorescence lifetime is decreased, @ / T increases and the parameters can be determined more easily. A global analysis using different timing calibrations provides an elegant alternative that has the advantage that no sample manipulation is required. The lower quality of the results obtained at the high @ / T values (Table I) can also be rationalized by considering the polarized fluorescencedecay as a double exponential. At high @ / T ratios, one has @ >> T and ( 1 / ~ l/@)-l 7 . Hence, i(0,t) can be considered as a biexponential decay with closely spaced decay times. This problem has already been extensively studied for total intensity decays.12-14 It was found that the parameter recovery is improved by the simultaneous analysis of decay curves with widely different preexponentialfactors and linked relaxation times, The ratio of the preexponential factors in i(0,t) can be changed by using different excitation wavelengths, Le. different ro, and/or different angles for the emission polarizer. To study the effect of the ro value, Z(O",t) and Z(90°,r) curves were generated with ro = -0.2 and T = 5 ns. The values for @ ranged from 25 to 200 ns so that the corresponding @ / rratios varied from 5 to 40. The data were analyzed with freely adjustable and fixed ro (see Table IV). The results arevery similar to those shown in Tables I and 11. The data simulated with ro = -0.2 were then combined with the correspondingdata with ro = 0.35. The results obtained with the global analysis of the four polarized decay curves for each of the considered O / r ratios are shown in Table V. For all @ / 7 ratios the simultaneousanalysis of the four polarized fluorescence

+

TABLE n7: Recovery of the Parameters of a Monoexponential r(t) and qt) by a Global Analysis of Z(Oo,t) and Z(90°,t) at a Time Increment per Channel of 47 ps* 5.00 4.992 5.00 4.992 4.99 4.992 4.992 4.992

0.02 0.004 0.02 0.004 0.02 0.004 0.008 0.004

-0.1 -0.26 -0.2 -0.26 -0.196 -0.2b -0.20 -0.26

0.3 0.2 0.05 0.02

200 200 100 100 50 50 25 25

100 192 80 97 47 49 25 24.6

300 16 105 4 21 1 3 0.5

= 5 ns. rot = -0.2. b Value fixed in the analysis.

(I

TABLE V: Recovery of the Parameters of a Monoexponential r(t) and fit) at Hi@ @ / rValues by Global Analysis of Z(Oo,t) and Z(9Oo,t) with ro = 0.35 and ro = -0.2' (ns)

u(ns)

4.996

0.007

4.993

0.008

4.998

0.008

4.995

0.006

T

ro 0.3 -0.18 0.44 -0.24 0.33 -0.20 0.35 -0.199

u

@t(ns)

d (ns)

u(ns)

0.1 0.06 0.06 0.03 0.03 0.02 0.02 0.007

200

179

72

100

134

28

50

47

6

25

25

2

= 5 ns. Time increment per channel = 47 ns.

it

TABLE VI: Recovery of the Parameters of a Monoexponential r(t) and 4t) at Hifh @ / T values by global analvsis of two Z(0O.t) and two Z(90 .t) curves with rn = -0.2' T (is) (ns) a u W(ns) @(ns) u(ns) ~~

~

~~~~~

Q

5.00 4.992 5.00 4.992 4.993 4.992 4.99 1 4.992

0.01 0.003 0.01 0.003 0.009 0.003 0.005 0.003

-0.1 -0.2* -0.2 -O).Zb -0.196 -0).2b -0.20 -0.26

0.2 0.1 0.04 0.01

200 200 100 100 50 50 25 25

102 192 80 97 47 49 25 24.6

219 11 74 3 15 1 3 0.4

it = 5 ns. Time increment per channel = 47 ns. Value fixed in the analysis. (I

decays with the same value of @ yielded a more accurate and precise parameter recovery as compared to the analysis of the curves at ro = 0.35 (Table I) or at ro = -0.2 (Table IV). To verify whether these improvements were due to the combination of two different ro values or to the increased number of curves in the total decay data surface, additional simulations were done with ro = -0.2 so that a global analysis could be performed of a surface consisting of two Z(Oo,t) and two Z(90°,t) curves. The obtained results are tabulated in Table VI (four curves) and should be compared with those in Table IV (two curves). The results shown in these tables are quite comparable. The same behavior of the parameter recovery was obtained with ro = 0.35 (results not shown). To investigate whether some other combinations of ro values would give a better parameter recoveryZ(OO,t)andZ(90°,t) curves with T = 5 ns and @ = 50 ns ( @ / T = 10) were synthesized with different ro values at a time increment per channel of 47 ps. As indicated above, this @ / rratio was rather unfavorable for accurate parameter recovery. A global analysis was performed by combining the polarized fluorescence decay curves with two different ro values (Table VII). All data surfaces had approximately the same number of counts. Combinations that included a set with a high ro value (ro = 0.4) gave better results then those that combined two lower ro values. Nevertheless, the results of the latter combinations were quite good when compared to the

8138 The Journal of Physical Chemistry, Vol. 97, No. 31, 1993

TABLE M: Recovery of the Panmeters of a 9/7 value (W7 = tial r(t) d4t) at a

M-xr 7

10) by Mal Analysis of f(O",t) md (W",t)at different ro valuesa

Crutzen et al.

TABLE M: Effect of the Counts in the Peak chrnwl on the Recovery of the Parameters of a Mowexponential r(t) and at) by Global Anrlvsis of Z(Oo.t) md 1(90°,t) counts at peak T (ns) 0 (ns) u @ (ns) u(ns) ro 4.2 x 103

5.00

0.03

5.01

0.02

4.99

0.01

0.4 0.4 0.4 0.35

4.999

0.005

4.996

0.004

4.998

0.004

4.995

0.007

4.994

0.005

5.000

0.005

4.995

0.010

4.991

0.006

4

TI

0.4 0.3 0.4 0.2 0.4 0.1 0.4 -0.1 0.4 -0.2 0.1 0.1 0.1 0.2 0.1 -0.1 0.1 -0.2

0.38 0.38 0.36 0.31 0.41 0.30 0.39 0.20 0.40 0.098 0.39 -0.096 0.40 -0.21 0.10 0.09 0.14 0.27 0.13 -0.12 0.13 -0.24

0.08 0.08

47

12

0.05 0.05

43

8

0.04 0.04 0.02 0.02 0.02 0.008 0.02 0.007 0.03 0.01 0.06

50

7

1.1

4.9 x lo4 9x10' 47

4

49

3

48

3

50

4

43

28

66

13

64

62

68

17

0.05

0.03 0.05

0.01 0.09 0.03 0.03

= 5 ns. 0' = 50 ns.

TABLE VIIk Recovery of the Panmeters of a Mowexponential r(t) and fit) at a high WT value (@/T = 10) by a Global Analysia of Polarized Intensity Decay Curves at Different Polarizer Aodesa polarizer angles (ded T (ns) u(ns) ro u @(ns) u (ns) 0,90 0,54.7 15,75

30,60 0,15 0.30 60,75 60,90 75,90 54.7,90 0,90,0,90 0,15,75,90 0,30,60,90 15,30,60,75 0,90,0,90,0,90 0,15, 30,60,75,90

5.03 4.990 4.97 4.99 4.820 4.98 4.991 4.997 5.00

4.990 5.02 5.00 4.999 4.995 5.01 4.998

0.01 0.005 0.07 0.02 0.004 0.03 0.007 0.008 0.02 0.005

0.02 0.01 0.005

0.005 0.01 0.004

0.22

0.04 0.6 OSb 0.4 0.7 OSb 0.005 0.007 0.16 0.09 0.25 0.05 0.36 0.09 0.33 0.06 0.27 0.04 0.28 0.05 0.32 0.04 0.33 0.02 0.37 0.04 0.29 0.05 0.34 0.02 OS6

x lo4

28 65 75 74 0.2 29 28 54 46 31 38 44 46 53 39 48

6 43 74 87 0.4 10 9 2 14 8 8 7 4 6 8 4

T( = 5 ns. 0' = 50 ns. rot = 0.350. The time increment was 47 p. Upper boundary for ro in the program was set to 0.5.

situation where only one set of Z(O",t) and Z(90°,r) with ro = 0.1 was analyzed. The other parameter which effects the ratio of the preexponentials is the angle 8 of the emission polarizer. For the cases where @ / T < 1, Flom and Fendlerzo found a small decrease in the spread of the parameter estimates when polarized fluorescence decay traces at more than two values of 8 were included. To investigate this approach for higher @ / T ratios, Z(t9,t) curves were simulated for various values of 8 with T = 5 ns and @ = 50 ns ( @ / T = 10) and ro = 0.350 at a time increment per channel of 47 ps (Table VIII). For the combinations with two different angles 8, the simultaneous analysis of Z(OO,t) and Z(90°,t) did not give the best results. Sets with both angles 8 larger than 5 4 . 7 O gave better results. For these angles the ratio of the preexponentials is negative. When more than two different angles were included in the analysis the parameters were recovered with a higher accuracy and precision. When ro or the matching factors are fixed to their proper value, a better parameter recovery is achieved (results not shown).

4.9 4.984 4.95 4.993 5.00 4.997 5 .00 4.998

0.1 0.006 0.09 0.003 0.01 0.002 0.01 0.001

0.56

0.5

0.3Y 0.3 OS6 0.3Y 0.34 0.05 0.3S 0.36 0.05 0.39

79 48 82 51 48 49.5 51 50.3

116 2 84 1 9 0.5

9 0.4

TI = 5 ns. Qit = 50 11s. rot = 0.35. Time increment per channel is 48 p. Upper boundary for ro in the program was set to 0.5. e Kept fmed in the analysis.

The effect of increasing the number of countswithout increasing the number of curves was investigated by simultaneouslyanalyzing Z(O",t) and1(90°,t) curves withdifferent countsat thepeakchanne1 (+ = 5 ns, @t = 50 ns, ro = 0.35, time increment per channel of 48 ps). The results are displayed in Table IX. There is a pronounced effect of the number of counts in the peak channel on the accuracy and the precision of the recovered parameters. A.2. Monoexponentialflt) and Monoexponential r(t) with Reference Convolution. It has been shown that the reference convolution method for the analysis of total intensity decays performs well under the following condition~:1~3~~ the reference lifetime should be as follows: (a) severaltimes the time increment per channel; (b) smaller than the largest relaxation time inflt) ofthesample; (c) not tooclose tooneofthefluorescencerelaxation times. If one of the decay times of the sample lies close to the reference lifetime this decay time can be masked by the reference.l4~z4This problem can be eliminated by fixing the reference lifetime at its known value. In the analysis of time-resolved fluorescence anisotropy experiments the reference convolution method appears only to have been used for matched polarized fluorescence decays.lsJ9 To test the performance of the reference convolution method for the analysis of unmatched fluorescence depolarization data, experiments were simulated with different reference lifetimes. Z(Oo,t) and Z(90°,t) curves were simulated at a time increment per channel of 48 ps with T = 5 ns, ro = 0.35, and @ = 500 ps or @ = 100 ps. The reference lifetimes ranged from 100 ps to 3 ns. Using the method of Flom and Fendlerzo the simulated data surfaces were analyzed in three different ways: (A) fixed T , and fiied matching factors; (B) freely adjustable T, and fiied matching factors; (C) freely adjustable T~ and matching factors. The results of the analyses are shown in Tables X and XI. For the simulations with CP = 500 ps the recovery of the parameters was equally accurate and precise for the three different methods of analysis and independent of the value of the reference lifetime. The value recovered for the reference lifetime was always slightly lower than the simulation value. For the second set of experiments with @ = 100ps (Table XI), the recovered ro and @ values exhibited some dependence on the reference lifetime for all three different modes. The recovered ro values tended to decrease while the associated @ values tended to increase with increasing value of the reference lifetime. These results can be accounted for by the low @ / T ratio and the fact that ( 1 / ~+ 1/@)-l is only twice the time increment per channel. For fixed or correctly recovered matching factors, the steadystate anisotropy is recovered within a few percent. This can be accounted for by the fact that the least-squares procedure used adjusts the parameters in such a way that the areas under the collected and the fitted curvesis the same.3$ Therefore, a decrease in ro will be compensated by an increase in @, and vice versa. A.3. Monoexponentialflt) and Biexponential r(t) with ZRF Convolution. When f l t ) is monoexponential and r(t) is biex-

Fluorescence Decay Curves

The Journal of Physical Chemistry, Vol. 97, No. 31, 1993 8139

TABLE X Recovery of the Parameters of a Monoexponential r(t) and 4t) by Global Analysis of Z(Oo,t) and Z(90°,t) Using the Reference Convolution Method' Case A Fixed 7 , and Fixed Matching Factors 4.997 4.995 4.998 5.OOO 5.002

0.004 0.004 0.004 0.004 0.005

0.345 0.349 0.351 0.350 0.348

0.004 0.005 0.005 0.006 0.006

0.497 0.4% 0.49 0.50 0.50

0.009 0.009 0.01 0.01 0.01

0.10 0.25 1.00 2.00 3.00

Case B: Freely Adjustable 7, and Fixed Matching Factors

ro

~ ( n s ) u(ns)

u

@(ns) u(ns) r?(ns)

0.004 0.353 0.005 0.482 0.009 0.004 0.353 0.005 0.488 0.009

4.989 4.990 4.991 4.981 4.97

0.005 0.353 0.006 0.49 0.008 0.351 0.006 0.49 0.01 0.349 0.006 0.50

0.01 0.01 0.01

0.10 0.25 1.00 2.00 3.00

r,(ns) u(ns) 0.093 0.246 0.993 1.983 2.97

0.001 0.001 0.003 0.006 0.01

Case C: Freely Adjustable T , and Matching Factors u(ns) ro 0.004 0.352 0.004 0.353 0.005 0.353 0.008 0.351 0.01 0.349

r(ns) 4.989 4.990 4.991 4.981 4.97

u

0.005 0.005 0.006 0.006 0.006

@ (ns) u(ns) r?(ns) s,(ns)

0.49 0.49 0.49 0.49 0.50

0.01 0.01 0.01 0.01 0.01

0.10 0.25 1.00 2.00 3.00

u(ns) 0.001 0.001 0.003 0.006 0.01

0.093 0.245 0.993 1.983 2.97

071 = 5 ns. = 500 ps. rot = 0.350. The time increment was 48 p. The different reference lifetimes are indicated by T,.

TABLE XI: Recovery of the Parameters of a Monoexponential r(t) and 4t) by Global Analysis of Z(Oo,t) and Z(90°,t) Using the Reference Convolution Method' Case A Fixed T , and Fixed Matching Factors 4.995 4.996 4.999 5.001 5.002

0.004 0.004 0.004 0.004

0.004

0.40 0.41 0.36 0.31 0.28

0.03 0.04 0.02 0.02 0.02

Case B: Freely Adjustable

T,

0.089 0.088 0.108 0.125 0.138

0.008 0.007 0.007 0.008 0.009

0.10 0.25 1.00 2.00 3.00

and Fixed Matching Factors

ro u @ (ns) u (ns) r?(ns) 0,004 0.42 0.03 0.086 0.007 0.1 0.004 0.42 0.04 0.087 0.007 0.25

~ ( n s ) u(ns)

r,(ns)

u (ns)

4.988 4.991 4.992 4.982 4.97

0.094 0.246 0.994 1.983 2.97

0.001 0,001 0.003 0.006 0.01

0.005 0.36 0.02 0.107 0.008 0.31 0.02 0.124 0.01 0.28 0.02 0.137

0.007 0.008 0.009

1.00 2.00 3.00

TABLE W. Recovered Parameter Values of a Biexponential r(t) and a Monoexponential 4t) Obtained by Global Analysis of Z(Oo,t) and Z(90°,t) at a Single ro Value When r(t) is Erroneously Taken To Be Monoexponential' Panel A r0~(xslL1)= 0.4 (81' = 0.1, Bzt = 0.3) 01 (ne) A(%) ~ ( n s ) u(ns) ro u o(ns) u Zy-a 0.670 30 4.996 0.004 0.395 0.004 0.557 0.008 -0.551 40 5.002 0.004 0.399 0.004 0.552 0.008 -0.062 0.730 50 4.997 0.004 0.399 0.004 0.559 0.008 -1.843 0.790 60 4.996 0.004 0.398 0.003 0.571 0.008 -2.301 0.850 70 4.994 0.004 0.391 0.003 0.597 0.008 -2.305 0.914 80 5.000 0.004 0.391 0.003 0.620 0.009 -1.923 0.978 90 4.99s 0.004 0.383 0.003 0.638 0.009 -2.039 1.044 1.112 100 4.997 0.004 0.379 0.003 0.660 0.009 -0.982 1.875 200 4.993 0.004 0.364 0.003 0.774 0.011 0.322 4.768 300 4.995 0.340 2.857 0.892 Panel B: r o t ( k ) = -0.2

(81' 0.1, ,921 -0.3) 5

5

(ns) A(%) r(ns) u(ns) ro u @(ns) u Zx.a 0.730 40 4.993 0.004 -0.205 0.004 0.402 0.011 -1.946 so 4.994 0.004 -0.209 0.004 0.380 0.010 -2.045 0.790 60 4.998 0.004 -0.211 0.004 0.359 0.010 -2.502 0.850 70 4.991 0.004 -0.207 0.004 0.360 0.011 -0.889 0.914 80 4.994 0.004 -0.212 0.004 0.346 0.010 -0.993 0.978 90 4.991 0.004 -0.231 0.005 0.300 0,010 1.132 1.044 1.112 100 4.990 0.004 -0.224 0.005 0.310 0.010 1.001 1.875 200 4.967 -0.244 0.262 10.269 300 4.965 2.857 -0.255 0.258 23.223

@1

it= 5ns. @zt= 5 0 0 p . @1~isdeterminedbyA(seeeq20). Standard deviations for the unacceptable fits are not shown. (I

that contain the rotational correlation times, a global analysis over different polarizer angles can still be useful (see section A.l and Table VIII). On the other hand, the relative contributions of the preexponentialscan be varied by changing 81 and 8 2 . This can be accomplished by measuring at different excitation wavelengths. This approach has already been used for matched I( e,t) traces. 13.29936 To test the usefulness of this methodology in the global analysis of unmatched Z(0,t) curves, a number of simulations were performed. Sets of Z(Oo,t) and Z(90°,t) curves were synthesized with two rotational correlation times. A fluorescence lifetime of 5 ns and a time increment per channel of 47 ps were used for all the simulated decays. The rotational correlation time was always 500 ps while @I was varied. The relative difference A in the directly observable decay times

Case C: Freely Adjustable 7, and Matching Factors ~ ( n s ) u(ns) 4.988 4.991 4.992 4.982 4.97

0.004 0.004 0.005 0.008 0.01

ro

u

0.42 0.42 0.37 0.33 0.31

0.03 0.04 0.03 0.02 0.02

@ (ns) u (ns) r:(ns)

r,(ns)

u (ns)

0.087 0.088 0.106 0.118 0.12

0.094 0.246 0.994 1.984 2.97

0.001 0.001 0.003 0.006 0.01

0.008 0.008 0.008 0.009 0.01

0.10 0.25 1.00 2.00 3.00

0 rt = 5 ns. 9' = 100 p. rot = 0.350. The time increment was 48 ps. The different reference lifetimes are indicated by rr.

ponential,the polarized fluorescence decay (eq 11 with n = 1 and m = 2) can be written as i(e,r) = K(8)/3 [ae+

+ 4 3 cosz e - 1)fi1e4(1'+1/41) + a ( 3 cosz e - 1)/32e-f(1/T+1'4z)~ (19)

The ability to resolve the two correlation times depends not only on the difference between the correlation times but also on T . To resolve closely spaced decay times, the global analysis should include experiments which differ in the relative values of the composite preexponential factors of the decay times of eq 19.12-l4 Although changing the angle of the emission polarizer does not change the relative amplitudes of the two decay times

ranged from 10%to 300%. The preexponential factor b1was set at 0.1. For each combination of @I and @z, decays were simulated with/32valuesof0.3(ro=0.4) and-0.3 (ro=-0.2). Thesevalues correspond to the case of an oblate rotor with 8, = Be = 90° and with the dipole moment of absorption parallel (82 = 0.3) and perpendicular (82 = -0.3) to the direction of the emission dipole moment. Simultaneous analysis of Z(Oo,t) and Z(90°,r) corresponding to a single excitation wavelength yielded the following results. We found that the fluorescence lifetime was always recovered accurately, regardless of whether the analysis was carried out inappropriately for a monoexponential r(r) model or appropriately for a biexponential one. When ro was freely adjustable,the model of a monoexponentiallydecaying anisotropy gave an acceptable fit if the difference A was smaller than 200% (for ro = -0.2) or even 300% (for ro = 0.4) (Table XII). When a global analysis of the decay traces at the two ro values was performed with a freely adjustable ro, a monoexponential fit was acceptable only if the difference A was less than 80%. When the ro were fixed at their true values, A had to be less than about

8140 The Journal of Physical Chemistry, Vol. 97, No. 31. 1993

Crutzen et al.

TABLE Mm: R e c o d Parmeter Values of a Bie;nrpoaentirl r(t) and a Monoex nenti.l fit) obtriaed by Global Analysis of = -0.2. Z(Oo,t) and Z(90°,t) at Two Excitation Wavelengths Corresponding to rot = 8.4 Panel A All Parameters Freely Adjustable 7

(ns)

4.997

0.003

4.997

0.003

5.00

0.03

4.995

0.003

5.002

0.003

4.997

0.003

4.997

0.003

4.992

0.003

4.997

0.003

@l' (ns)

81

U

0.4 -0.2 0.08 0.07 0.07 0.07 0.09 0.19 0.10 0.09 0.13 0.09 0.13 0.11 0.098 0.096 0.104 0.102

0.2 0.1 0.03 0.03 0.03 0.03 0.06 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.006 0.006 0.004 0.004

~(ns)

(A(%))

@ l W

0.730

0.54

.(4 0.05

(40) 0.790 (50) 0.850 (60) 0.914 (70) 0.978 (80) 1.044 (90) 1.112 (100) 1.875 (200) 2.857 (300)

0.9

0.1

0.9

0.1

0.8

0.1

1.04

0.09

1.03

0.08

1.02

0.09

1.89

0.09

2.6

0.1

82 -0.3 -0.2 0.33 -0.27 0.33 -0.27 0.3 1 -0.4 0.30 -0.29 0.28 -0.30 0.27 -0.31 0.303 -0.296 0.290 -0,303

U

0.2

@~(ns)

u(ns)

0.388

0.009

-1.37

0.49

0.03

-0.927

0.49

0.02

-3.68

0.55

0.05

-2.61

0.49

0.02

-2.23

0.47

0.02

-2.19

0.50

0.02

-1.57

0.50

0.01

-3.23

0.51

0.01

-1.37

0.1 0.04 0.03 0.03 0.03 0.06 0.2 0.03 0.02 0.02 0.02 0.02 0.02 0.007 0.007 0.006 0.005

Panel B: Fixed ro Values, Le. 81Freely Adjustable and 82 Constrained

4.997 4.997

0.003 0.003

4.998

0.003

4.995

0.003

5.002

0.003

4.997

0.003

4.998

0.249 0.135 0.100 0.098 0.107 0.100 0.088 0.129 0.108 0.097 0.101 0.085 0.071 0.099 0.099 0.097 0.099 0.098

'

0.003

4.992

0.003

4.997

0.003

@r&(&l) = 0.4

(81'

= 0.1,

82'

0.004 0.004 0.004 0.004 0.003 0.003 0.003 0.003 0.003 0.003 0.004 0.003 0.004 0.003 0.003 0.003 0.003 0.003

0.730 (40) 0.790 (50) 0.850 (60) 0.914 (70) 0.978 (80) 1.044 (90) 1.112 (100) 1.875 (200) 2.857 (300)

= 0.3) and rot(&)

0.59 0.77

0.02

0.84

0.02

1.01

0.03

1.13

0.03

1.22

0.04

1.88

0.05

2.75

0.08

= -0.2 (81' = 0.1, 82' = -0.3).

81 = -0.2 @1 = 1 ns 82 = 0.3 @2 = 5 ns System B a1 = 1 71 = 0.463 ns 81 = 0.393 @I = 0.344 ns a2 = 1 9 7 2 = 2.850 ns 192 = 0.389 @2 = 0.208 ns The parameters provided for system B corrcapond to the results obtained by Flom and Fendlermfor a mixture of two fluorophores. TI = 2 ns 72

0.02

0.8 1

TABLE xn7: Simulation Parameters of the Associative Systems A and B4 System A a1 = 1 a2 = 0.5

0.01

= 8 ns

@

60% for the monoexponential anisotropy model to give an acceptable fit. The analyses with a biexponential r ( t ) were performed in two different ways (Table XIII). In the first type of analysis, both 81 and 82 were freely adjustable parameters (Table XIIIA). Accurate parameter recovery was obtained for A 2 80%. In the alternative analysis, the sum of & and & was constrained to be equal to the truevalues of the initial anisotropies (Table XIIIB). In that case only 81 was allowed to be freely adjustable and the condition for accurate parameter recovery relaxed to A 2 40%. The tabulated values for the standard deviation on 02 are calculated with the error propagation formula. To investigate how these results depended on the values of the two initial anisotropies, additional Z(Oo,t) and Z(90°,t) decays were synthesized with different ro values. The value for B1 was

0.151 -0.335 0.300 -0.298 0.293 -0.300 0.312 -0.329 0.292 -0.297 0.299 -0,285 0.329 -0.299 0.301 -0.297 0.301 -0.298

0.004

0.478

0.008

-1.37

0.004 0.004

0.496

0.008

-2.93

0.493

0.009

-3.68

0.521

0.009

-2.54

0.49

0.01

-2.23

0.49

0.01

-2.14

0.52

0.01

-1.3 1

0.50

0.01

-3.25

0.51

0.01

-1.35

0.004 0.003 0.003 0.003 0.003 0.003 0.003

0.004 0.003

0.004 0.003 0.003 0.003 0.003 0.003

= 5 ns. 02' = 0.500 p. A is defined by eq 20.

always 0.1 while 8 2 was varied from 4 . 3 to 0.3 in steps of 0.1. The relaxation times were 7 = 5 ns, $1 = 1112 ps (A = loo%), and (Pz = 500ps. We found that thediscriminating power between a mono- and a biexponential model for r(t) improved with increasing difference between the true values for the initial anisotropy. To resolve widely spaced correlation times it has been proposed to use a global analysis of progressively quenched samples.30 The reasoning is as follows. For each rotational correlation time a, there is a value of 7 for which the ratio @ j / ~reaches the value for optimal recovery of a,. By an appropriate change (decrease) in the fluorescence lifetime, another (shorter) correlation time will be well recovered. A global analysisof sampleswith different quencher concentrations can then accurately determine all the rotational correlation times. It should be emphasized that changing the fluorescencelifetime by quenching has a negative effect on the analysis when the rotational correlation times are closely spaced. In general, when the fluorescence lifetime decreases, the difference A decreases and the analysis becomes more difficult. An alternative way to improve the recovery of widely differing rotational correlation times is a global analysis using different timing calibrations (seealsosectionA.1). However, bothmethods have the same limitation: the fluorescence lifetime should be large enough to resolve the largest rotational correlation time. A.4. Models Containing TwoFluorescenceRelaxation Times

The Journal of Physical Chemistry, Vol. 97, No. 31, 1993 8141

Fluorescence Decay Curves

TABLE XV: Recovery of the Parameters of System A (see Table VI) Using a Global Analysis of Z(Oo,t) and Z(90°,t)' peak counts 82 u @2(ns) u(ns) Associative Model

ZQa

~~

10 000 20 000 30 000 40 000 50 000 250 000

1.98 1.99 1.982 2.007 1.996 1.998

0.01 0.01 0.008 0.007 0.007 0.003

0.515 0.505 0.507 0.499 0.503 0.501

0.005 0.003 0.003 0.002 0.002 0.001

7.92 7.97 7.97 8.00 7.98 7.999

0.02 0.02 0.01 0.01 0.01 0.005

10000 20000 30000 40000 50000 250000

1.99 1.99 1.984 2.012 2.000 2.002

0.01 0.01 0.009 0.007 0.007 0.003

0.506 0.504 0.507 0.498 0.503 0.539

0.005 0.003 0.003 0.002 0.002 0.001

7.96 7.98 7.97 8.00 7.98 7.989

0.02 0.02 0.01 0.01 0.01 0.005

-0.205 -0.210 4.207 -0,197 -0.203 -0,200

0.007 0.007 0.005 0.003 0.004 0.001

1.072 1.20 1.04 0.91 1.07 0.99

0.009 0.09 0.06 0.04 0.05 0.02

0.29 0.31 0.307 0.290 0.311 0.297

0.01 0.01 0.007 0.005 0.006 0.002

5.1 4.5 4.9 5.2 4.9 5.01

0.3 0.2 0.2 0.1 0.1 0.05

0.066 -0.098 -0.279 -2.33 -0.761 4.815

1.3 1.5 1.38 1.22 1.40 1.29

0.1 0.1 0.07 0.04 0.05 0.02

0.20 0.21 0.21 0.181 0.209 0.227

0.02 0.02 0.01 0.005 0.009 0.003

6.2 5.6 6.2 7.4 6.3 5.9

0.7 0.5 0.4 0.4 0.3 0.1

0.051 4.160 -0.003 -1.90 -0.593 1.93

Nonassociative Model

a

-0.23 -0.24 -0.24 -0.208 4.234 -0.235

0.02 0.02 0.01 0.006 0.009 0.003

The results are shown for the correct associative and for the incorrect non-associative model.

a1 was held

constant at unity during the analysis.

TABLE XVI: Recovery of the Parameters of System A (See Table VI) Using a Global Analysis of I(0,t) Traces' (deg) TI (ns) u(ns) a2 u 72(ns) u(ns) 81 u @p1 (ns) u(ns) 8 2 u @2(ns) u(ns) AssociativeModel 0,90 15,75 30,60 0,30 60,90

1.98 1.97 1.99 1.97 1.98

0.01 0.01 0.02 0.03 0.02

0.506 0.510 0.509 0.50 0.501'

0.005 0.005 0.005 0.01 0.005

7.95 7.93 7.95 7.95 7.99

0.02 0.02 0.03 0.04 0.03

-0.203 -0.21 -0.19 -0.19 -0.23

0.007 0.01 0.01 0.01 0.02

1.01 1.2 1.0 1.0 0.85

0.09 0.1 0.2 0.1 0.1

0.29 0.32 0.31 0.29 0.29

0, 15, 75,90 0,30,60,90 15,30,60,75

1.97 1.98 1.98

0.01 0.01 0.01

0.508 0.003 0.507 0.003 0.510 0.003

7.94 7.95 7.94

0.02 0.02 0.02

-0.204 -0.201 -0.203

0.006 0.006 0.008

1.08 1.02 1.1

0.08 0.08 0.1

0, 15, 3460,75,90

1.978

0.008

0.508

7.94

0.01

-0.203

0.005

1.07

0.003

0.01 0.02 0.02 0.02 0.03

Zaz

5.0 4.8 5.3 5.1 4.8

0.3 0.3 0.5 0.4 0.5

2.81 -0.151 -0.177 -0.992 -1.42

0.303 0.009 0.30 0.01 0.32 0.01

4.9 5.0 4.9

0.2 0.2 0.3

1.81 1.82 -0.308

0.07

0.304 0.008

5.0

0.2

1.33

Nonassociative Model 0,90 15,75 30,60 0,30 60,90

1.98 1.97 2.00 2.00 1.98

0.02 0.01 0.02 0.04 0.02

0.505 0.510 0.506 0.49 0.500

0.005 0.005 0.005 0.01 0.006

7.95 7.94 7.97 8.04 8.00

0.02 0.02 0.03 0.08 0.03

-0.22 -0.26 -0.21 -0.20 -0.24

0.02 0.03 0.02 0.02 0.04

1.3 1.5 1.3 1.3 1.2

0.1 0.2 0.2 0.1 0.2

0.19 0.23 0.20 0.19 0.19

0.02 0.03 0.01 0.02 0.04

6.5 5.7 8. 9. 6.

0.8 0.8 2. 3. 2.

2.89 4,131 -0.226 -1.04 -1.31

0, 15, 75,90 0,30,60,90 15,30,60,75 0, 15, 30,60,75,90

1.98 1.99 1.98

0.01 0.01 0.01

0.507 0.507 0.509

0.003 0.003 0.003

7.94 7.95 7.95

0.02 0.02 0.02

-0.24 -0.22 -0.24

0.02 0.01 0.02

1.41 1.32 1.5

0.09 0.09 0.1

0.21 0.19 0.22

0.02 0.01 0.02

6.2 6.8 6.4

1.88 1.85 -0,311

1.981

0.008

0.508

0.003

7.95

0.01

-0.23

0.01

1.39

0.08

0.21

0.01

6.4

0.5 0.7 0.7 0.5

0

1.37

The results are shown for the correct associative and for the incorrect nonassociative model. a1 was held constant at unity during the analysis.

and Two Rotational Correlation Times Analyzed with IRF Convolution. In this section the analysis of associative (heterogeneous) and nonassociative (homogeneous) systems is investigated. The considered nonassociative model consists of a biexponentialflt) of a single emitter and a biexponential r(t), Le. all L,,= 1, In the associative model two emitting species each having a monoexponential f l t ) and monoexponential r ( t ) are considered, i.e. L l l = ,522 = 1 and L I Z= 1521 = 0. Associative and nonassociativesystems may be difficult to distinguish from each other. It should be noted here that the parameters obtained for r(t) of different species in a mixture from the analysis of a data set obtained at a single spectroscopic condition, i.e. at a single a ratio for their excited-state decay, are not mathematically unique. However, the incorrect values may be physically eliminable. This is because exactly equivalent degenerate fits can be obtained by considering different possible associations of the ai,fl,, T,, and @,parameters. Twodifferent associativesystems were simulated to investigate these problems. The simulation parameters of the two systems are given in Table XIV. System A can expected to be an easily resolvable one: the @ / T ratios are within the range described in section A. 1 and the decay times (1/q + 1/ a 1 ) - l and (1 /TZ + 1 are well separated. Polarized intensity decays with a different number of counts inthepeakchannel(rangingfr0m l(ylto2.5 X 105) weresimulated at a time increment of 80 ps per channel. Even at lo4counts in the peak channel the simulation parameters were recovered with good accuracy and precision when the correct model was used (Table XV). The results obtained for the incorrect model were

as follows. The Zx2values were comparable to those obtained with the correct model. The parameters of f l t ) were well recovered, except for the simulation with a peak count of 2.5 X 105 where the ratio a l / a 2 was inaccurate. The estimates for the anisotropy parameters differed from the true values and were recovered with precisions that were only slightly lower than those obtained for the associativemodel. However, the nonassociative model cannot be accepted because a biexponential r(t) with preexponential factors of about 0.2 and -0.2 is nonphysical for a nonassociative system. To investigate the possible benefit of a global analysis using different polarizer angles, Z(8,t) curves were generated for system A using various values for 8. The results of the global analyses of I(8,t) curves using various combinations of 8 are shown in Table XVI. Both the correct associative and incorrect nonassociativemodels describethe data adequately. For the associative model the best combination of two polarizer angles was the commonly used Oo and 90° combination. The precision of the parameters decreased as the differencebetween the angles became smaller. In this case, including more angles lead to a marginally higher precision for both models so that not much is gained in performing a global analysis using different polarizer angles. The preexponentials of the additional cross-terms in the nonassociative model, i.e. the terms with L 1 2 = L21 = 1 in eq 11, depend in the same way on the polarizer angle as the preexponentials of the terms that appear also in the expression of the associative model, i.e. the terms with ,511= f.22 = 1 in eq 11; Le., their relative values do not vary on changing the polarizer angle.

Crutzen et al.

8142 The Journal of Physical Chemistry, Vol. 97, No. 31, 1993

TABLE XW: Recovery of the Parameters of System A (See Table VI) Using a Global Analysis of l(Oo,t) and I(!Mo,t) at Different Ratios a~/az* alt/azt 71 (ns) u(ns) az u n ( n d u(ns) 81 u 01(ns) u(ns) 62 u @2((118) U(M) Associative Model 211 1/2 2/1, 112

1.98 1.98 1.99

0.01 0.03 0.01

0.506 2.04 0.501 2.050

0.005 0.03 0.003 0.023

7.95 7.98 7.98

0.02 0.02 0.01

2/1 112 211, 112

1.98 1.98 2.054

0.02 0.03

0.505 2.04 0.521 1.967

0.005 0.03

7.95 7.98 8.006

0.02 0.02

-0.203 -0.20 -0.201

0.007 0.02 0.005

i?,l

1.01 1.06 1.01

0.09 0.23 0.05

0.29 0.299 0.298

0.01 0.008 0.003

5.0 5.0 5.0

0.3 0.2 0.1

2.81 -1.25 1.01

1.3 1.2 1.180

0.1 0.1

0.19 0.27 0.237

0.02 0.01

6.5 5.4 5.919

0.8 0.3

2.89 -1.24 53.6

Nonassociative Model

a

-0.22 -0.13 -1.782

0.02 0.02

The results are shown for the correct associative and for the incorrect nonassociative model. a1 was held constant at unity during the analysis.

TABLE Xwr: Recovered Parameter Value8 for System B (See Table VI) by Global Analysis of I(Oo,t) and l(Wo,t)Using the Correct Assdative Model with Starting Guesses Close to the True Values' case 1 1 (ns) u (ns) a2 u ~ ( n s ) u(ns) BI u 01(ns) 82 u %(ns) u(ns) Z,] a b

0.464 0.464 a

0.001 0.001

0.1113 0.1102

0.0008 0.0008

2.85 2.84

0.01 0.01

0.42 0.4

0.05 0.1

0.32 0.31

0.04 0.06

0.2 0.3

0.4

0.4 0.4

1.

0.4 0.3

-1.41 4643

The upper bound value for A in the program was set at 0.5 in case a and 0.4 in case b.

This explainswhy a global analysis over different polarizer angles is not helpful in discriminating between the associative and the nonassociative models. The differences between the associative and the homogeneous model functions are the most pronounced in the early part of the polarized decay. We found that a global analysis of the polarized fluorescence decays at two time increments (80 and 16 ps) did not allow us to discriminate between the associative and the nonassociative models. When several emitting species are present, a global analysis of polarized fluorescence decays collected at various emission wavelengths may help in discriminating between associative and nonassociative models. Curves taken at different emission wavelengths differ only in the amplitudes ai. Such simulations were carried out using the parameter values of system A. The effect of different emission wavelengths was mimicked by taking ul/u2 = 2 and ul/a2 = 0.5. The time increment was 80 ps per channel. Table XVII shows the results of the analyses with the associativeand the nonassociativeresults of the analyses with the associativeand the nonassociativemodel. When, at each emission wavelength, I(Oo,t) and Z(9O0,t) were simultaneously analyzed, no distinction between the correct and the incorrect model could be made because the ratios of the utdid not change. However, a simultaneousanalysis of the four decay curves at the two different q/a2 ratios resulted in an unacceptably high Z%zvalue ( 2 - 2 = 53) when the incorrect nonassociative model was used. In that case the standard deviations on the parameters recovered are meaningless and are not indicated in Table XVII. The success of this approach can be readily understood on inspection of the expression of the anisotropy in terms of the sums of exponentialsmaking up the polarized fluorescencedecays. For the nonassociative model, each component decay contains six exponentialsmade up from the parameters offlt) and r(t).When such an incorrect nonassociativemodel is used during the analysis at a single emission wavelength, the parameters of At) are recovered well. Then the rotational correlation times and the factors /3 take values such that the combination of the four remaining exponential terms describes the polarized fluorescence decays adequately. As the emission wavelength is varied, the factors a&(3 c o s 2 8 - 1 ) have to adjust to the change in the u-ratio accordingly. To adequately describe this change on the basis of the inappropriate nonassociative model, the values of the rotational correlation times would also have to adjust, which is prevented in the global analysis across the different emission wavelengths over which all the anisotropy parameters are linked. The parameter values of system B (see Table XIV) correspond to the values obtained by Flom and Fendler'O from a mixture of

rhodamine Band merocyanine 544 in ethanol. Polarized intensity decays were simulated with 104 counts in the peak channel using a time increment of 20 ps per channel. The decay traces I(Oo,t) and Z(90°,t) were globally analyzed. The parameters offit) were easily recovered. However,it wasvery difficult toobtainaccurate and precise estimates for the anisotropy parameters, even when the correct model was used. The quality of the fits with a monoexponential and a associative "biexponential" r(t) was practically equal, as would be expected considering the closeness of the two rotational correlation times. The estimates for the anisotropy parameters using the 'biexponential" model exhibited a large dependence on the initial guesses. The values shown in TableXVIIIa were obtained when the initial guesses were slightly different from the true parameter values. The parameters were easily recovered with acceptable precision except for 82 and @2. However, because the estimate obtained for 81 was not physically acceptable, the data were reanalyzed with the upper bound for 8, set to 0.4 (Table XVIIIb). The accuracy and precision for @2 was maintained, and the accuracy for 8 2 improved but only at the expense of a substantial decrease in precision. Including additional polarizer angles in the global analysis did not improve the parameter recovery. Similar results were obtained with simulated polarized fluorescence decays with 5 X 104 counts at the peak. The resultsobtained for system B can be rationalized as follows. The fluorescencerelaxation times are well recovered because the shorter fluorescence relaxation time r1is well separated from 72 and is associated with the larger preexponential factor ut. The poor recovery of & and dp2 can be accounted for in the following way. The polarized fluorescence decay curves can be written as a sum of four exponentials

+

+

i(6.t) = (K")/3)[qe-'/'' a!2e-'/" (3 6 - l)(u e-W71+~/W+ 1 4

-Wn+V*d)] (21)

2B2e

Since (l/rl + l/@])-l = 194 ps and ( 1 / 7 2 + l/@z)-l = 197 ps, the polarized fluorescence decay can be approximated by i(e,t) = ( ~ ( 6 ) / 3 ) [ ( ~ ~ e - ' / a ~21e+2 +

+

e - i)(crlpl + with 73 = (1/71 + 1/@1)-l= ( 1 / 7 2 + 1/@2)-l. (3 cos'

(22) As could be expected, the parameters of f l t ) were recovered easily when analyzed with this constraint. This means that when the data are analyzed for the correct associative model, the third exponential term in eq 22 is to be fitted with two exponential terms. The values of the two relaxation times 41 and $2 and the splitting of

The Journal of Physical Chemistry, Vol. 97, No. 31, 1993 8143

Fluorescence Decay Curves

TABLE XM: Recovery of the Parameters of System B (See Table VI) by Global Analysis of I(Oo,t) and 1(90°,t) at Two adaj Ratios Using the Correct Associative ModeP al'laz'

71

911 119 812 218

(ns)

u(m)

~ ( m ) 82

ZQZ

u

*2(ns)

2.846

0.003

'0.392

0.003

0.341

0.006

0.393

0.004

0.207

a@) 0.003

4,605

2.846

0.003

0.394

0.004

0.354

0.008

0.382

0.004

0.209

0.004

0.003

0.002

0.427 2.33

0.002 0.02

2.850

0.003

0.401

0.005

0.33

0.01

0.384

0.006

0.214

0.005

4,190

0.002

0.668 1.49

0.003 0.01

2.851

0.004 0.40

0.01

0.35

0.02

0.39

0.01

0.200

0.009

0.167

0.462

0.001

713 317

0.464

614 416

0.464

a1

(ns)

0

0.001 0.3 0.001 0.07

0.001

0

81

~z(ns)

az

0.111 9.0 0.250 4.15

0.462

0

*I

was held constant at unity during the analysis.

TABLE XX: Global Analysis of I(Oo,t) and 1(W0,t)Traces of Resorufim, Cresyl Violet, and the Mixture of Both in 1-Propanol at 20 OC* 71 (ns) 81 *1(4 7 2 (ns) 82 0 2 (ns) z*z resorufine 4.07 (0.01) 0.36 (0.02) 0.44 (0.02) 2.55 cresyl violet 3.15 (0.01) 0.337 (0.007) 0.60 (0.02) 1.25 mixture (associative model) 4.14 (0.05) 0.37 (0.03) 0.40 (0.03) 3.08 (0.05) 0.36 (0.02) 0.58 (0.03) 2.03 I, Standard deviations are indicated within parentheses. O.'

0.08 A

t 1

b

0.06 .

400

440

480

510

560

600

Wavelength (nm) Fnopn 1. Absorption spectra of resorufine (-) at 1.5 cresyl violet at 5 X 1od M in 1-propanol at 25 'C.

640

X 106

M and

(e-)

(3 cosz 8 - l)(orl81 + adz) in the two amplitudes for the biexponential are unpredictable but are constrained by the fact that the areas under the monoexponential (with 73) and the biexponential (with 41 and 42) should be essentially equal. For system B the contribution of the third exponential term in eq 21 is almost an order of magnitude larger than that of the fourth exponentialterm. In addition the @,/qratios are more favorable for cP1 than for @z. Both conditions account for the poor recovery of 82 and cPz. When the ratio a~/azwas reversed, Le. ctl/aZ = 1/9, the precision and accuracy of all parameters decreased. Global analysis of polarized fluorescence decay traces including additionalvaluesof 8 cannot,of course, helpin thecorrect splitting of the factor (3 cos2 8 - l)(arlfll + a&) in eq 22 or even of the exact corresponding factor in eq 21. However, the ratio orl/az can be changed by collecting the data at different emission wavelengths. Thus, a global analysis of Z(Oo,t) and Z(90°,t) obtained at variousemission wavelengths should result in a better parameter recovery. Additional I(Oo,t) and Z(90°,t) traces were simulated using the simulation parameters of system B but with different q/az ratios. As expected, simultaneousanalysis of Z(Oo,t) andZ(90°,t) correspondingto two emission wavelengthsgave rise to a dramatic improvement in the accuracy and the precision of the recovered parameter values (see Table XIX). The precision and the accuracy depended on the particular combination used in the analysis. Themore theratios cul/or~intheglobal analysis differed, the higher the obtained accuracy and precision. B. ExperimentallyCollected Data. To test the global analysis of unmatched, experimentally collected polarized fluorescence

540

580

610

660

700

740

Wavelength (nm) Figure 2. Corrected emission spectra of resorufine (-) at 1.5 X 1od M and cresyl violet at 5 X 1od M in 1-propanolat 25 'C. Excitation was at 570 nm. (e-)

decays, we studied the time-resolved emission anisotropy of a mixture of cresyl violet and resorufine in 1-propanol at 20 OC. In this temperature region the fluorescence lifetime, the fluorescence quantum yield and the spectra of cresyl violet and resorufine are temperature independent. Cresyl violet in water was used as reference ( T = ~ 2.05 ns). The decay traces were collected in l/zK channels and had 1V counts at the peak. The time increment per channel was 47 p. The excitation wavelength was 570 nm in all experiments. This means that both fluorophores were excited in their long wavelength absorption bands (see Figure 1). The fluorescence emission spectra of both fluorophores are shown in Figure 2. The fluorophoreswere first measured separately. The emission was monitored at 599 nm for resorufine and at 630nm for cresyl violet. The results of the simultaneous analysis of Z(Oo,r) and 1(90°,t) for each chromophoreseparately are given in Table W. The anisotropy decay of each chromophore was described adequately by a monoexponential. The fundamental anisotropy ro obtained for resorufinewas 0.36 (+0.02). This value is within the range of 0.34 (f0.03) to 0.37 (fO.O1) reported for resorufine in alcohols.36 The recovered ro value for cresyl violet was 0.337 (f0.007) whichagreeswellwith thevalueof0.35 (fO.02) obtained by other groups.37,3* The parameters T, 8, and 4 are comparable for the two probes. This means that a mixture of these probes is difficult to analyze, even though both probes have appropriate 4 / r ratios. In the mixture of cresyl violet and resorufinethe concentration ofthedyeswas keptverylow ( < l W M ) tominimizebothradiative

8144 The Journal of Physical Chemistry, Vol. 97, No. 31, 1993

and nonradiativeenergy transfer from resorufineto cresyl violet. Theemission was monitored at 584and 589 nm where theemission of resorufine is dominant and at 624 and 629 nm where the emission of cresyl violet prevails. Not surprisinglyin view of the discussion above, global analysis of the polarized fluorescence decays measured at a single emission wavelength failed. Also the analysis of data simulated with the same factor LY for both probes yielded poor results when only one emission wavelength was considered. By contrast, the global analysis of the decay traces collected at the four emission wavelengths was successful (Table XX). The recovered parameter values are in good agreement with the values found for each of the chromophores separately. Analysis of the multiwavelength data surface of the mixture using the inappropriate nonassociative model yielded Z,a = 4.27.

Conclusion The results in this study provideevidence that thesimultaneous analysis of unmatched polarized fluorescencedecaycurves Z(Oo,t) and Z(90°,t) is able to recover successfully the parameters of the time-dependent anisotropyif the rotational relaxation time is not too different from the fluorescence relaxation time and larger than the time increment per channel. The matching factors can then be obtained from the analysis. In the more difficult cases, additionaldecay curves collected at other settings of the emission polarizer and excitation and/or emission wavelength have to be included in the data surface. The method works with IRF and reference convolution. The fact that matching factors are no longer required allows one to simultaneously analyze all related decay curves measured under any possible combinations of experimental conditions without the procedures to determine the matching factors. Usually, it is relatively easy to match two polarized fluorescence decays. When the sample is macroscopically isotropic, excitation with horizontally polarized light should yield, in principl, identical polarized intensity data irrespective of the setting of the emission polarizer. In this way any polarization bias in the detection system can be determined. Different collection times between various Z(8,t) can be accounted for by matching factors obtained from steady-state anisotropy experiments. However, this procedure cannot be followed when the Z(0,t) curves are collected in a narrow time window with respect to the relaxation times. In that case, even under ideal experimental conditions, the areas under the collected curves do not have the same relative scaling factor with respect to their steady-state equivalent. To match a whole series of polarized fluorescence decays collected under different experimental conditions can take the same effort as collecting additional decay curves on a laser instrument. The global analysis of related polarized fluorescence decays with or without known matching factors combines flexibility in use with a maximum of parameter linkage over the data surface. For example, the preexponential factors af can be linked irrespectiveof the orientation of emission polarizer or the collection time of the decay curve. Each of the polarized fluorescence decay curves can be collected for the desired signal-to-noise ratio. Decay data collected on different instruments and/or with different timing calibrations can be combined easily. When several detectors are used simultaneously to speed up the collection of the experimentaldata surface, the different quantum efficiencies and timing characteristics of the various detectors are automatically taken into account when using adjustable matching factors in the analysis. It is further shown that the accuracy and precision of recovery of the parameters offlt) from twoor more appropriatelydifferent polarized emission components are practically independent of whether or not a correct model is used for the emission anisotropy. As in the case of fixed predetermined matching factors the optimum recovery of Ip in the case of a monoexponentialflt) and r ( t ) is obtained when @ and T are of similar magnitude.

Crutzen et al. In most casesZ(OO,t) andZ(90O.t) aresufficient for an adequate parameter recovery but other combination of two polarizer settings may yield better results. The inclusion of polarized fluorescence decays collected at additional orientationsof the emissionpolarizer can lead to improved accuracy and precision. In general, a more accurate parameter recovery and a better model discrimination is obtained when the weights of the exponential terms in the expression of the polarized intensity can be changed. For high @ / T ratios it is beneficial to vary the polarizer setting (even though the relative weight of the exponential terms in the emission anisotropy cannot be changed by varying the orientation of the polarizers) or the excitation wavelength. Fixing the initial anisotropy toa predetermined value results in a better parameter recovery. This is equivalent to fixing the matching factors to their propervalues. Increasing the number of counts in Z(Oo,t) and Z(90°,t) is more efficient than measuring at different settings of the emission polarizer. For a monoexponentialAt) in combination with a multiexponential r(t), the use of different excitation wavelengths (i.e. excitationinto bands of different polarization) is recommended. Different timing calibrations should be utilized when the relaxation times in r(t) are widely different. In the case of an associative system, the relative weights in the anisotropy decay can usually be varied by measuring the polarized fluorescence decays at different excitation and/or emission wavelengths. Acknowledgment. M.C. acknowledges the Belgian National Fonds voor Wetenschappelijk Onderzoek for a predoctoral fellowship. N.B. is a Bevoegdverklaard Navorser of the Belgian Fonds voor Geneeskundig WetenschappelijkOnderzoek. R.M. thanks the Katholieke Universiteit Leuven for financial support. The continuing supportof the Ministryof ScientificProgramming through Grant UIAP-11-16 is gratefully acknowledged. The authors wish to express their gratitude to one of the reviewers for useful suggestions for improvement of the manuscript.

Appendices A. Derivationof Eq 1Oa. Consider a macroscopically isotropic system containing a single fluorescent species characterized by the total fluorescenceflt) and the emission anisotropyr(t). Assume completely unpolarized excitation light and no bias in the detection of the polarized components of emission. The fluorescence intensity under this condition, i(*,*.t), can then be expressed in terms of the polarized fluorescenceintensities i w ( t ) ,iVH(t), i ~ r ( t ) and im(t)where the first and second subscript refer to the orientation of the excitation and emission polarizer respectively

+

i(*,*,?) = iw(t) + iVH(t)+ iH&) iHH(t) (Al) For observation at right angles to the direction of propagation of the excitation beam in theexcitation-detectionplane, symmetry considerations yield39

The polarized intensity with the excitation polarizer set at the angle $ to thevertical and without an emission polarizer, i($,*,r), is given by i(+,*,t)

[iw(t)

+ iVH(t)lCOS’ + + [iHV(t)+ iHH(t)] sin’ +

(A41 Upon insertion of an emission polarizer set at the angle 8 to the vertical the polarized intensity i($,O,t) can be writtet as

e + ivH(t) sin’ e] cos’ $ + [iHV(t)cos’ e + iHH(t) sin’ 81 sin’ r ~ ,(AS)

i($,e,t) = [iw(t)cos2

Using eqs 2,3, A2, and A3 in combination with eq A5 yields eq loa.

The Journal of Physical Chemistry, Vol. 97, No. 31, 1993 8145

Fluorescence Decay Curves

B. Calculation of the Derivativesunder the Constraint of Equal Areas under the Observed and Fitted Curves. The polarized fluorescence decay Z(8,t) can be written as Z(e,t) = K(e)G(o,f)

(B1)

where G(8,r) is independent of the matching factor K ( 8 ) . The discrete realizations of Z(8,t) and G(8,t) corresponding to a channel in the multichannelanalyzer will bedenoted respectively by II and GI. The condition that the areas under the calculated series Itc and the observed series Ziohave to be equal leads to

(7) Gratton, E.; Alcala, J. R.; Barbieri, B. In Luminescence Techniques in Chemical and Biochemical Analysis; Baeyens, W. R. G., De Keukeleire, D., Korkidis, K., Eds.; Marcel Dekker: New York, 1991; pp 47-72. (8) Lakowicz, J. R., Ed. Topics in Fluorescence Spectroscopy; Plenum

Press: New York, 1991; Vol. 1. (9) Dale, R. E. In Time-resolved fluorescence spectroscopy in Biochemistry and Biology; Cundall, R. B., Dale, R. E., Eds.; NATO AS1 Series A: Life Sciences; Plenum Press: New York, 1983; Vol. 69, pp 555612. (10) Wahl, Ph. Biophys. Chem. 1979, 10, 91. (1 1) Gilbert, C. W. In Time-resolved Fluorescence Spectroscopy in Biochemistry and Biology; Cundall, R. B.; Dale, R. E., Eds.; NATO AS1 Series A Life Sciences; Plenum Press: New York, 1983; Vol. 69, pp 605606. (12) Knutson, J. R.; Beechem, J. M.; Brand, L. Chem. Phys. Lett. 1983,

102, 501. (13) Beechem, J. M.; Ameloot, M.; Brand, L. Anal. Instrum. 1985, 14,

Equation B2 implies that K depends on (cqr7,,[email protected] x be any freely adjustable parameter different from K . The derivative of the calculated function Z t with respect to x is given by

a4 - a K

- -G: ax ax

+

Because of eq B2 the derivative of

ac,C K-

K

ax is given by

In the program the constraint for equal area is directly embedded in the calculation of the derivatives and is not just a multiplication afterward as has been described by others.40 In each iteration, the increments in the values of the parameters (CY,,T,,~,,@,)are determined by the normal equations used in the least-squares search. The coefficients in these equations are given by eqs B3 and B4. The adjusted values of the matching factors K ( 8 ) are determined by eq B2.

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