error analysis of simple algorithms for determining fluorescence lifetimes in ultradilute dye...

Error Analysis of Simple Algorithms for Determining Fluorescence Lifetimes in Ultradilute Dye Solutions

S T E V E N A. SOPER* and B E N J A M I N L. L E G E N D R E , JR. Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803-1804

We have evaluated the use of two simple a lgor i thms for determining the decay parameters describing a single exponential process for dyes with nanosecond and subnanosecond fluorescence lifetimes in the limit of low concentrat ions and high backgrounds from scattered photons generated by the solvent us ing exper imental and Monte Carlo s imulat ion results. These algorithms, the maximum likelihood estimator (MLE) and the rapid lifetime determination (RLD), are computat ional ly easy to perform, allowing the evaluation of large amounts of data quickly and ef- ficiently. The M L E and RLD methods were used to calculate the fluorescence lifetimes of three near- IR dyes with lifetimes spanning the range of 0.57 ns to 1.12 ns. For low-concentration conditions and high background-to-fluorescence ratios, the M L E method resulted in larger errors when compared to RLD, a l though both methods yielded comparable s tandard deviations. However, when the interval over which the lifetime was calculated within the decay profile was shifted to latter times in order to reduce the amoun t of scattered photons included in the calculation, s ignificant improvements in the accuracy were observed with the use of M L E . Shif t ing the s tar t channel of the calculation to latter time channe l s within the decay profile did not affect the lifetime with the use of RLD. Inclusion of large amoun t s of scat ter ing photons was found to bias the calculated lifetime to lower values, reducing the accuracy of the determination. The relative standard deviations for M L E and RLD were found to be approximate ly 2 -3% at a background-to-fluorescence ratio of 0.5. The absolute relative error in the methods at the 0.50 background- to-fluorescence ratio ranged from 14 to 27% for M L E and 8 to 18% for the RLD method when the calculation was initiated at t = 0. Th i s error was found to decrease to < 1% with the use of M L E when the calculation was initiated at t ~ 100 ps.

Index Headings: Time-resolved fluorescence; Fluorescence lifetimes; Near- IR fluorescence.

INTRODUCTION

Time-resolved fluorescence spectroscopy has become an important tool for studying various photophysical phe- nomena in chemistry and biochemistry--for example, structure and dynamics in proteins, l rotational diffusion in restricted environmentsfl and excited-state proton- transfer reactions? In addition, fluorescence lifetime determinations have been used in many analytical appli- cations such as liquid chromatography, 4 fluorescence mi- croscopy, 5-9 deferminat ion of adsorption modes on chromatographic stationary phases, l° and measurement of fluorescence lifetimes of single molecular events.'1,12 The utility of the spectroscopy results from the fact that the fluorescence lifetime is very sensitive to the imme- diate environment of the lumiphore. In addition, the wide- spread use of time-resolved fluorescence has evolved from rapid advancements made in instrumentation--for example, the development of solid-state lasers and detec-

Received 21 September 1993; revision received 7 December 1993. * A u t h o r to w h o m correspondence should be sent.

tors, which have reduced the complexity and cost of the instrumentation needed to perform the measurement.

Determination of the parameters which describe an exponential decay process of the form

I(t) = ~ Aie -'/'~o (1)

where n represents the number of components in the decay, A is the preexponential factor, and rf is the fluorescence lifetime, has been accomplished by a variety of methods, one common procedure being a nonlinear least-squares algorithm, accompanied by convolution or deconvolution of the instrument response function with the goodness of the fit determined by the value of x2.13 The difficulty associated with this approach is the exten- sive amount of computational time associated with the determination, which can be a severe limitation when large amounts of data must be processed.

For the case where n = 1 (single exponential decay), there are several simple algorithms for determining both the preexponential and the exponential factors which describe a decay process. One such method is the maximum likelihood estimator, MLE. '4-17 In this algorithm, the lifetime can be calculated via the relationship 16

1 + (e T / I f - 1) -l - m(e mT/'f - 1) - 1 = N; -l ~ iNi (2)

where m is the total number of time channels in the decay spectrum, T is the time width of each channel, Nt is the total number of photocounts used in the calculation, and Ni represents the number of photocounts in the ith time channel. The left-hand side of Eq. 2 is not dependent upon the data and is a function only ofT-f, while the right- hand side of Eq. 2 is determined from the experimental data. The lifetime can be abstracted from the data with the use of graphical, tabulation, or reiterative techniques. This algorithm has been successfully used to calculate the fluorescence lifetimes of single molecules with high accuracy and precision.",'2 The relative standard deviation in the lifetime determination with the use of MLE can be evaluated from the expression 18

__crTf _- NcV2 1 - e -T/T~ (3) rf [(1 - e-T/~gz -- (T/zf)2e-T/~] l/2"

Another simple algorithm that can be used to extract the decay parameters describing a single exponential process is the rapid lifetime determination (RLD) procedure? 9 This method is a variation of the maximum likelihood method and involves binning the data into two contiguous areas, with the lifetime calculated with the use of the expression

400 Volume 48, Number 3, 1994 0003-702s/94/4803-040052.00/0 APPLIED SPECTROSCOPY © 1994 Society for Applied Spectroscopy

FIG. 1.

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IR.1IS

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IR-132

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Molecular structures of the NIR fluorescent dyes.

-At (4)

rr ln(Dl/Do )

where Do and Dl represent two areas under the decay profile of time width At and are evaluated by summing the number of photocounts in each time bin within the boundaries defining Do and D~. For the RLD method, the relative standard deviation may be calculated with the use of the expression

- A t [a~o 2 \1/2 __cr'f __ | ffb,~

Tf [ln(Ol/Oo)] 2 ~Do 2 + O~/ ( 5 )

where aDo and ao, are the standard deviations in Do and D~, respectively. In photon counting experiments, aOo and trD, can be determined by taking the square root of the total number of counts in each interval.

We have examined the use of these algorithms for calculating the fluorescence lifetimes of several dyes with nanosecond and subnanosecond lifetimes in the limit of ultradilute concentrations and short integration times (i.e., low number of photocounts in the decay profile). In par- ticular, the effect of the background in the form of scattering photons was investigated. A recent report 18 evaluated the performance of the MLE procedure for decay profiles constructed with the use of Monte Carlo simulations consisting of approximately 10-25 photocounts which were free of background photocounts. The major results of this study indicated that maximum likelihood estimators of r c showed extreme bias when the reduced time, F (P = T t / T f , where T, is the total time interval of the calculation), was less than 4 and the number of fluorescence counts comprising the decay profile ranged from 10 to 20. However, this bias was small when P > 6. For small reduced times, the reciprocal o f r f (k) was concluded to be a better estimator due to the small biases observed in the simulations for finite T,. In addition, small reduced times were shown to give large relative errors in both zf and k values. When the data were binned and 7"f was calculated via the RLD method, the Monte Carlo results indicated a slight increase in the standard deviation when compared to the minimal binning case, and also little was gained by using unequal bin widths.

The lifetimes in the present study were calculated for

three NIR fluorescent dyes, IR- 125, IR- 132, and dithia- tricarbocyanine iodide (DTTCI)--whose structures are shown in Fig. 1--under ultradilute conditions with the use of MLE and RLD methods. The advantage of NIR fluorescence excitation and detection as compared to visible fluorescence is that contributions from fluorescent impurities in the solvent are significantly reducedfl °-3° Since these algorithms make no distinction between mul- tiexponential and single exponential decays, minimal impurity contribution is a necessity in the case of ultradilute dye solutions. Inclusion of significant amounts of impurity counts into the determination would result in large biases. In addition, the scattering contribution, in the form of Raman photons, is reduced in the NIR as compared to the visible due to a/./4 dependence of the Raman cross section. The result is larger fluorescence observation windows free from solvent Raman bands in the NIR. When included in the determination, scattering photons, which have early arrival times, can introduce bias, especially when the relative numbers of these photons are high with respect to the number of fluorescence photons. The lifetimes of these NIR dyes were calculated with the use of MLE and RLD over a concentration range of 10-1~ M to 10 -~2 M, resulting in various background-to-fluorescence ratios and different integration times (1 and 10 s). The efficiency of these algorithms was evaluated in terms of their relative accuracies and precisions under ultradilute conditions and short data accumulation times.

EXPERIMENTAL

Instrumentation. The excitation source of the NIR time- resolved fluorescence spectrometer consisted of a self- mode-locked Ti: sapphire laser pumped by the all-lines output of an argon-ion laser (Mira 900-F and Innova 310, respectively; Coherent Lasers, Palo Alto, CA). This laser generated nearly transform-limited pulses, with a tem- poral width of 120 fs (FWHM) and a bandwidth of approximately 11 nm at a repetition rate of 76 MHz. The Ti : sapphire laser was set at an operating wavelength of 780 nm and was vertically polarized at the flow cell. The NIR light was focused into a square-bore capillary tube, serving as the flow cell, with the use of a laser diode singlet lens (Melles Griot, Irvine, CA), with a 1/e 2 beam waist of approximately 10 tim. The fluorescence was collected with the use of a 40 ×, 0.85 NA epi-fluorescence microscope objective (Nikon, Natick, MA) and imaged onto a slit serving as a spatial filter to reduce the amount of scattered photons generated at the air/glass interface of the flow cell from reaching the photodetector. The slit width was set at 0.4 mm. The effective sampling volume, assuming a cylilndrical probe volume, was calculated to be 3.14 pL. The fluorescence was further isolated from the scattered photons with the use of a bandpass filter (Omega Optical, Brattleboro, VT) with a center wavelength of 850 nm and a half-bandwidth of 30 nm. The collected light was then focused onto the photodetector with 6.3 x microscope objective. The photodetector was a single photon avalanche diode (SPAD, EG&G Opto- electronics Canada, Vaudreuil, Canada) mounted on a thermoelectric cooler. The pulses generated by the SPAD were amplified by a 2-GHz amplifier (Phillips Scientific, Mahwah, N J) and conditioned with the use of a constant-

APPLIED SPECTROSCOPY 401

fraction discriminator (CFD, Tennelec TC 754, Oak Ridge, TN). The CFD pulses from the SPAD were sent into the gate and stop inputs of a time-to-amplitude con- verter (TAC, Tennelec TC 863). The stop pulse for the TAC was generated by an intracavity photodiode monitoring the pulse train from the Ti:sapphire laser and conditioned by the CFD. The fluorescence decay profiles were collected into 4096 time bins with the use ofa PCA- II A/D board and software (Tennelec Nucleus) on a PC486 computer. Calibration of the time bins in the pulse height analyzer yielded a value of 2.88 ps per channel. The instrument response function of the NIR time-resolved fluorescence spectrometer, measured in the absence of a fluorescent dye, was determined to be 164 ps (FWHM).

All data analysis software was written in Turbo Pascal. Unless otherwise stated, the fluorescence lifetimes ((rf)) were calculated with the use of r = 10 x rr for MLE and r = 5 x rr for RLD. Lifetimes calculated with the use of MLE and a reduced time of 10 x rr result in small biases when one is calculating (rf) directly• 18 Calculation of (Tf) with r = 5 x rf with the use of RLD has previously been demonstrated to yield small relative standard deviations in the determination. 19 The lifetimes for MLE were determined with the use of the experimental data to evaluate the right-hand-side of Eq. 2, with the root of the left- hand-side of Eq. 2 evaluated with the use of a Newton- Ralphson reiterative technique. For RLD, the lifetimes were calculated directly from Eq. 4. Monte Carlo simulations were constructed by convolving the experimental instrument response function with the calculated decay using the "true" decay parameter (r0. The correct number of background counts and of fluorescence counts was Poisson distributed into the appropriate time bins within the convolved function.

In order to effectively sample the same molecules in the ultradilute experiments, the volumetric flow rate was interrupted during the measurement. The laser beam im- pinging on the flow cell was blocked and the volumetric flow rate started in order to sweep a fresh sample into the detection zone. The flow was then stopped and the solution allowed to reach quiescence, after which the excitation light was unblocked and data acquisition allowed to commence. Under these conditions, only random diffusion and thermal convection caused by local heating due to molecular absorption of light resulted in fresh molecules entering the probe volume during the measurement. The number of molecules sampled during a typical measurement could be estimated from the size of the probe volume and the concentration of the dye.

The "true" fluorescence lifetimes (rr) for the NIR dyes in ethanol were evaluated with the use of a reiterative, nonlinear least-squares algorithm, with the goodness of the fit determined by the value of X 2. A typical decay profile is shown in Fig. 2 for IR-125, along with the weighted residuals• The concentration of the dye was ad- justed so that the background was less than 1% of the total counting rate. In order to eliminate anisotropies due to rotational diffusion, a Glan-Thompson polarizing prism was placed in the optical train during these measurements and set at the magic angle (54.7°). The rf values calculated in this fashion were found to be: IR-125, 0.57 _+ 0.01 ns (x 2 = 1.25); IR-132, 0.76 + 0.01 ns (X 2 = 1.30); DTTCI, 1.12 + 0.01 ns (x 2 = 1.33). Previous studies using phase-

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O

• . o . . ° ~ ° o ~

i , i , i , i , I

11 00 1 700 2500 2900 3500

C h a n n e l N u m b e r ( / 2 . 8 8 ps)

-1

- 2

- 4 1 250 1 600 1 950 2500 2650 5000

C h a n n e l N u m b e r ( / 2 . 8 8 ps)

FIG. 2. (a) Fluorescence decay profile for the NIR dye IR-125 in ethanol, and the weighted residuals (b). The profile was collected until approximately 10,000 counts had been accumulated in the channel with the maximum number of counts. The solid line represents the best fit function to the experimental data, determined with the use of a reiterative-convolution, nonlinear least-squares procedure.

resolved fluorescence indicated a lifetime for DTTCI in ethanol of 1.33 + 0.02 ns (x 2 = 21). 31 The difference in our lifetime and that observed previously may arise from varying amounts of water present in the ethanol solvent, since the lifetimes of a number of NIR dyes have been shown to be sensitive to the presence of trace amounts of water. 32

Reagents and Chemicals. The NIR dyes were obtained from Kodak Chemicals (Rochester, NY) and used as received. Spectroscopic-grade ethanol was used for all solutions and obtained from Mallinckrodt (Paris, NY). Stock solutions of the dyes (1 uM) were made monthly and stored in the dark in a refrigerator at 10°C. Over this period of time, no degradation in the stock solutions was apparent, as determined by monitoring the fluorescence emission intensity of the dye solutions. The dilute dye solutions were made from serial dilutions of the stock solutions and prepared daily.

RESULTS AND DISCUSSION

In Fig. 3 are shown (rf) values for the three NIR dyes determined by MLE and RLD as a function of the background-to-fluorescence ratio (B/F). In addition, the lifetimes calculated with the use of MLE for Monte Carlo

402 Volume 48, Number 3, 1994

1.1 5 DTTCI

0.95

0.75

o 0.55

1R-132

1R-125

0.35 r , , , , , 0.00 0.11 0.22 0.33 0,44 0.55

Background/Fluorescenee Ratio

FIG. 3. Calculated lifetimes as a function of the B/F ratio for the NIR fluorescent dyes IR-125 (circles), IR-132 (squares), and DTTCI (dia- monds) with the use of MLE (closed symbols) and RLD (open symbols). Monte Carlo simulation results (x) as calculated by MLE are also shown as a function of the B/F ratio. The B/F ratio was set by adjusting the dye concentration. The laser power used in these measurements was 10 mW (measured at the flow cell). The integration time used in all cases was 1 s. The dashed line represents the " t rue" fluorescence lifetime of the NIR dyes.

simulations are presented which were constructed with the appropriate background and fluorescence counts to match experimental conditions. Fair agreement between the experimental data and the simulation results was observed. In these determinations, the calculation was initiated at t = 0, which was assumed to occur at the channel within the decay profile with the maximum number of counts. As can be seen from this figure, decreases in (rf) were observed with respect to rf as the B/F ratio was increased. The lifetimes at the high B/F ratios (50%) were consistently lower for MLE when compared to RLD values. At the 50% B/F ratio, the relative errors were found to be - 18% and -27% for IR-125; - 12% and - 16% for IR-132; and -8.7% and -14% for DTTCI with the use of RLD and MLE, respectively. These relative errors were found to consistently increase as the B/F ratio was increased. The larger relative errors at high B/F ratios result from large contributions of scattered photons in the calculation. Since the arrival time within the decay profile of these photons is at early times, inclusion of these photons biases (rf) to lower values at high B/F ratios where their relative contribution in the calculation is high. The larger relative errors for the MLE determination indicate a greater sensitivity to the presence of these scattered photons. The errors were also found to be greater for IR- 125, which has the shortest lifetime, with the longer- lifetime dye, DTTCI, demonstrating smaller relative errors.

A plot of the number of occurrences versus (rr) for these dyes was found to yield a normal distribution for the MLE and RLD methods. The standard deviations associated with these distributions were found to be in good agreement with those predicted by Eqs. 3 and 5. As can be seen through inspection of Eqs. 3 and 5, lower dye concentrations result in fewer fluorescence counts and larger relative standard deviations. The relative standard deviations were found to be approximately 0.8-1.0% for MLE and RLD at a B/F ratio of 1% and increased to approximately 2.0-2.7% at a B/F ratio of 50%.

The source of background photons from the solvent can originate from two sources, Raman and Rayleigh scattered photons and/or fluorescence photons from impurities present in the solvent system. Ultrasensitive fluorescence experiments using visible excitation and detection have shown that the detection efficency is limited primarily by the presence of impurity components in the solvent? ~,33 When similar experiments were performed in the NIR, the detection efficiency was significantly im- proved due to the smaller background level arising from these impurity components. 34 In the present application, these impurities can cause significant errors in the calculated lifetime, especially at low dye concentrations (high B/F ratios) since these algorithms make no distinction between a multi- and monoexponential decay process. The calculated lifetime would thus represent a weighted average of the lifetimes associated with the target dye and the impurity components being sampled. Calculation of (rf) with the use of MLE for a profile consisting of 10,000 photocounts from the solvent blank (ethanol) resulted in a value of0.11 ns. In addition, an autocorrelational analysis, which is sensitive to correlated bursts of photons from fluorescence components traveling through the detection zone, 3s for the solvent blank yielded no observable nonrandom correlation. The results indicate the absence of long-lived fluorescence components in this solvent, and thus the dominate background source consists primarily of Raman and Rayleigh scattered photons.

In order to increase the accuracy at high B/F ratios, the calculation was initiated at a channel later in the decay profile to reduce the relative contribution of scattered photons into the determination. Figure 4 shows (rf) calculated by MLE and RLD as a function of the start channel for DTTCI and IR-125. The lifetimes calculated by MLE demonstrated a gradual increase until a channel shift of approximately 30 channels (86 ps) for DTTCI and 35 channels (100 ps) for IR-125 was reached, after which fairly constant lifetime values in good agreement with rf were obtained. Since the experimental data make no distinction between background and fluorescence photons, Monte Carlo simulations were used to determine the relative contribution of background photons in the calculation at time shifts yielding lifetime values in agreement with rr. The simulation results indicated that, at a B/F ratio of 50%, the relative contribution of background photons in the lifetime determination at a time shift of 100 ps was approximately 15% for IR-125 and 10% for DTTCI. When the channel shift was 86 ps, the relative contribution from background photons was 21% for IR- 125 and 14% for DTTCI. The larger relative background contributions for IR-125 at these time shifts compared to DTTCI arise from the shorter lifetime associated with this dye. The results indicate that, for MLE determinations, the contribution of the background photons must be significantly reduced in order to achieve lifetime values with high accuracy. In addition, dyes with longer lifetimes require smaller time shifts due to the smaller relative contribution of scattered photons in the calculation. How- ever, the total number of counts included in the calculation is reduced with a corresponding increase in the relative standard deviation (loss of precision) when one is shifting the start channel to later times within the decay profile. The relative standard deviation at a B/F ratio of


1 15

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[] [.3 D

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"~ 0 .51 o o 0

oO ~ , ~ o ° • u • • o o o

0 . 3 5 ~ ~ , t ~ 1 0 2O 3 0 4O 5 0

Channel Shift ( /2 .88 ps)

FIG. 4. Calculated fluorescence lifetimes for the dyes IR-125 (circles) and DTTCI (squares) as a function of the start channel in the calculation at a B/F ratio of 50% with the use of MLE (open symbols) and RLD (closed symbols). The integration time was 1 s, and 10 m W of laser power was used for excitation. The dashed line represents the " t rue" fluorescence lifetime of the NIR dyes.

50% was found to increase from 2.1%, at a zero time shift, to approximately 5.6% for a time shift of 100 ps (35 channels) for IR-125. In the case of DTTCI, the relative standard deviation was 1.9% for a time shift of 0 ps and increased to 4.0% at a time shift of 86 ps.

Shifting the initial channel in the calculation to later times within the decay spectrum can also result in biases due to the exclusion of fluorescence photons with early arrival times. In order to evaluate these biases, Monte Carlo simulations were performed to construct decay profiles and use MLE for evaluating (~'r) for dyes with rr values of 1.12 ns, 0.76 ns, and 0.57 ns. The decay profiles were constructed with 0 background photons and 10,000 fluorescence photons. The results of these simulations indicated that the relative error in (rf) for r f = 1.12 ns was 4.6% at time shifts of 250 ps, with smaller time shifts yielding smaller biases. When rf = 0.76 ns, the relative error was found to be 10.1% for a time shift of 130 ps (45 channels), and for rf = 0.57 ns, the relative error was 10.2% for a time shift of 100 ps (35 channels). For shorter lifetimes, the early-arriving photons make a significant contribution into the calculation, while for longer lifetimes, the relative contribution is less. Elimination of these photons introduces biases into the MLE estimators for short T f values, while for longer lifetimes, larger shifts can be tolerated without introducing significant biases. The lifetimes calculated as a function of channel shift (see Fig. 4) were corrected for this bias.

Lifetimes calculated by RLD at a B/F ratio of 50% were found to be insensitive to the time shift and yielded consistently lower (r r) estimators as compared to T f (see Fig. 4). Monte Carlo simulation results with the use of background-free decay profiles constructed from 10,000 fluorescence counts also demonstrated an insensitivity of ( r f ) to the channel shift for r r values ranging from 0.57 to 1.12 ns. A shift in the calculation to later time channels within the decay profile does not increase the accuracy of the determination with the use of RLD, but does result in a loss of precision due to decreases in the number of photocounts included in the calculation (see Eq. 5). Therefore, RLD methods should nominally be used with

TABLE I. Fluorescence lifetimes for IR-125, IR-132, and DTTCI in ethanol at a concentration of 5 x 10 -~ and integration times of 1 and 10 s as calculated by MLE. All data have been corrected for biases introduced by shifting the initial channel in the calculation to later times within the decay spectrum to reduce relative contributions of scattering photons into the determination.

Int. time (S) 7"f (ns) a Counts b a~ (ns) ~ aa (ns) d

IR-125" 1 0.58 2250 +0.04 +0.02 10 0.57 21775 +0.02 _+0.007

IR-132 ¢ 1 0.75 2625 _+0.04 _+0.02 10 0.76 25150 _+0.01 _+0.006

DTTCP 1 1.12 3440 _+0.03 _+0.02 10 1.12 31235 _+0.01 +0.005

Represents the average of six replicate measurements. b Photocounts included in the lifetime determination. ° Standard deviations calculated from six replicate measurements. d Standard deviations calculated with Eq. 3. ° Lifetime values calculated with a time shift of 125 ps. r Lifetime values calculated with a time shift of 100 ps.

zero time shifts and high dye concentrations (low B/F ratios) in order to obtain fairly accurate results with good precision.

In Table I, (rr) values calculated with the use of MLE for these NIR dyes at a concentration of 5 x 10 - n M and integration times of 1 and 10 s are presented, and in Fig. 5 a typical decay profile for DTTCI and IR-125 at this dye concentration and a 1-s integration period is shown. At this concentration, the approximate number of molecules sampled (Nm) was estimated with the use of

Nm = CNA Vp (6)

where C is the dye concentration, NA is Avogadro's number, and V~ is the size of the probe volume. At this dye concentration and a 3.14-pL probe volume, Nm = 10. The effective number of molecules sampled is independent of the integration time since the volumetric flow rate was interrupted during the measurement. In addition, the B/F ratio at this concentration (7.0) is independent of the integration time. It was noticed that, during data acquisition, the steady-state count rate decrease when the dyes

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C h a n n e l N u m b e r ( / 2 8 . 8 p s )

FIG. 5. Decay profiles for IR-125 (open circles) and DTTCI (closed circles) at 5 x 10-n M and a 1-s integration period. The data were binned by 10, yielding approximately 28.8 ps per channel. At this integration time and dye concentration, the total number of counts comprising each decay profile was 17,465 for IR- 125 and 16,814 for DTTCI. Each profile was collected with the use of a laser power of 10 mW.

404 Volume 48, Number 3, 1994

were present in the detection zone resulted from photo- bleaching of the sampled molecules. In the case o f DTTCI, significant reductions in the counting rate were observed during data accumulat ion when compared to IR-125 and IR-132, indicating a poorer photochemical stability. I f no new molecules are swept into the detection zone during data acquisition, the counting rate should approach the rate observed for the solvent blank. Since the counting rate remained above the background level in all cases, new molecules were assumed to have been swept into the detection zone through random diffusion and/or thermal gradients. The number o f sampled molecules calculated above should therefore be considered provisional. For IR-125 and IR-132, time shifts o f 125 ps were sufficient to give (~r) values in good agreement with Tf values for MLE, while for DTTCI , a time shift o f 100 ps was required. Similar time shifts were required at both integration times in order to obtain (~-r) values with small errors. At this dye concentration and appopfiate time shift, lifetime values were in excellent agreement with rf for the dyes investigated. For IR-125, 0-r) was 0.58 ns for a 1-s integration time, and when the integration time was increased to 10 s, (rf) was determined to be 0.57 ns. For DTTCI , similar lifetime values were obtained at the 1- and 10-s integration times with small errors. These results indicate that longer integration times do not necessarily result in better accuracy. The accuracy in MLE is determined, in part, by the background in the form of scattering photons which are included in the determination and is set by the B/F ratio (i.e., dye concentration) and the time shift necessary to reduce the number o f scattered photons in the determination to an acceptable level. However, the precision in the determination was found to improve with increasing integration time due to the fact that more photocounts were being accumulated into the decay profile. For IR-125, the relative standard deviation was 6.9% for a 1-s integration time and decreased to 3.5% at a 10-s integration time, while for DTTCI , the relative standard deviations were 2.7% and 1.0% at 1- and 10-s integration times, respectively.

C O N C L U S I O N S

Fluorescence lifetimes associated with single exponential decays can be calculated with little computat ional effort and a high level o f accuracy and precision with the use o f MLE and R L D algorithms. When the background- to-fluorescence ratio is small, both algorithms yield similar levels of accuracy and precision for dyes with nanosecond and subnanosecond lifetimes. In the limit o f low dye concentrations (high B/F ratios) where the background in the form of scattering makes a large relative contribution into the decay profile, MLE was shown to be the method of choice under appropriate calculation conditions. The accuracy was found to be better for MLE when compared to RLD with the use of a time shift within the decay profile in order to reduce the amount of scattered photons included in the calculation. For ultradilute solutions and short integration times, the lifetimes calculated via MLE were found to agree favorably with those obtained in the high concentration limit and evaluated with the use o f a nonlinear least-squares method. The negligible contribution o f fluorescence impurities with long

lifetimes arising from the solvent blank with the use of N I R excitation and detection should allow the determination o f lifetimes in complex matrices for dilute concentrations o f N I R fluorescent dyes and the use o f short integration times.

ACKNOWLEDGMENT The authors would like to thank the Louisiana Educational Quality

Support Fund (LEQSF) for partial financial support of this research.

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3. See, for example, (a) G. W. Robinson, J. Phys. Chem. 95, 10,386 (1991); (b) H. T. Yu, W. J. Colucci, M. L. McLaughlin, and M. D. Barkley, JACS 114, 8449 (1992).

4. D. J. Desilets, P. T. Kissinger, and F. E. Lytle, Anal. Chem. 59, 1830 (1987).

5. T. A. Louis, G. Ripamonti, and A. Lacaita, Rev. Sci. Instrum. 61, 11 (1990).

6. X. F. Wang, T. Uchida, D. M. Coleman, and S. Minami, Appl. Spectrosc. 45, 360 (1991).

7. A. S. Verkman, M. Aemijo, and K. Fushimi, Biophys. Chem. 40, 117 (1991).

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error analysis of simple algorithms for determining fluorescence lifetimes in ultradilute dye...

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