Anal. Chem. 1996, 68, 611-620
Maximum Entropy Method for Frequency-Domain Fluorescence Lifetime Analysis. 2. Timing, Mismatched Intensity, and Reference Lifetime Errors Jeremy M. Shaver and Linda B. McGown*
Department of Chemistry, P. M. Gross Chemical Laboratory, Duke University, Box 90346, Durham, North Carolina 27708-0346
The maximum entropy method (MEM) provides a robust and unbiased solution to fluorescence lifetime data through the use of a broad window of decay terms fit by simultaneous minimization of the χ2 goodness-of-fit parameter and maximization of a statistical entropy function. This work investigated the effects of three systematic errors, common in frequency-domain measurements, on fluorescence lifetime recovery by MEM. Through real and simulated data, the expression of the systematic errors in lifetime distributions recovered by MEM was compared to that in standard nonlinear least-squares (NLLS) analysis. Reference lifetime errors in the presence of random noise had similar effects on both MEM and NLLS results. Characteristic changes in the recovered lifetimes, fractional intensities, and peak shapes were related to the identification of the true reference lifetime. Compared to NLLS, MEM afforded significant improvements for the recovery of lifetimes and fractional intensities from data containing timing or mismatched intensity errors. These improvements are linked to the dynamic, self-modeling approach of MEM and the direction provided by the entropy criterion. These results speak to the utility of the maximum entropy approach in frequency-domain fluorescence lifetime recovery as well as in other applications. Successful interpretation of the results of powerful data analysis algorithms such as the maximum entropy method (MEM),1-5 requires a fundamental understanding of their ability to handle both signal and noise in the data. With MEM, this understanding is particularly important, because the algorithm is designed to automatically respond, in the absence of a priori information or assumptions, to the noise and signal present in the data. Some practical considerations for MEM analysis of frequency-domain fluorescence lifetime data were shown in Part 1 of this series,6 including effects of frequency range, number of frequencies, and nonconstant variance on the quality of recovered distributions. In a continuation of the investigation of frequency-domain lifetime (1) Brochon, J. C.; Livesey, A. K.; Pouget, J.; Valeur, B. Chem. Phys. Lett. 1990, 174, 517-522. (2) Skilling, J.; Bryan, R. K. Mon. Not. R. Astron. Soc. 1984, 211, 111-124. (3) Livesey, A. K.; Skilling, J. Acta Crystallogr. 1985, A41, 113-122. (4) Gull, S. F.; Skilling, J. IEE Proc. 1984, 131F, 646-659. (5) McGown, L. B.; Hemmingsen, S. L.; Shaver, J. M.; Geng, L. Appl. Spectrosc. 1995, 49, 60-66. (6) Shaver, J. M.; McGown, L. B. Anal. Chem. 1996, 68, 9-17. 0003-2700/96/0368-0611$12.00/0
© 1996 American Chemical Society
data analysis by MEM, a discussion of common systematic errors is presented here. Three systematic errors that have significant impact on frequency-domain data analysis are timing errors, mismatched sample and reference intensities, and reference lifetime errors. The first two are associated with anomalous changes in detector response,7-11 and the third is usually the result of measurement imprecision, possibly compounded by the first two errors.12,13 Each of these errors introduces artifacts not necessarily expected by the analyst nor well fit by the equations used in the analysis of the data. As a result, analysis by standard methods requiring a priori assumptions can be complicated by the presence of these errors. Because MEM does not utilize preconceived models, it provides unique manifestation and visualization of systematic errors which can be used to avoid inaccurate lifetime results. For this reason, an investigation to determine how the systematic errors affect distributions recovered by MEM, as well as the use of these effects for diagnosis and correction of systematic errors, was undertaken. THEORY Frequency-Domain Measurements and Maximum Entropy Method Analysis. The reader is directed both to Part 1 of this series6 and elsewhere for detailed discussions of frequencydomain lifetime measurements14-16 and the MEM algorithm.1-3 We present here a summary of the theories specifically relevant to the discussion of systematic errors. In frequency-domain lifetime measurements, a fluorescent sample is excited by light with sinusoidally modulated intensity and, as a result, emits fluorescence that is also sinusoidally modulated at the same angular modulation frequency. The fluorescence emission is, however, phase-shifted and demodulated relative to the excitation beam. The phase shift, φ, and modulation (7) Bauer, R. K.; Balter, A. Opt. Commun. 1979, 28, 91-96. (8) Baumann, J.; Calzaferri, G. J. Photochem. 1983, 22, 297-312. (9) Jameson, D. M.; Weber, G. J. Phys. Chem. 1981, 85, 953-958. (10) Berndt, K.; Durr, H.; Palme, D. Opt. Commun. 1983, 47, 321-323. (11) Pouget, J.; Mugnier, J.; Valeur, B. J. Phys. E: Sci. Instrum. 1989, 22, 855862. (12) Barrow, D. A.; Lentz, B. R. J. Biochem. Biophys. Methods 1983, 7, 217. (13) Litwiler, K. S.; Huang, J.; Bright, F. V. Anal. Chem. 1990, 62, 471. (14) Gaviola, E. Z. Phys. 1927, 42, 853. (15) Lakowicz, J. R. Principles of Fluorescence Spectroscopy; Plenum: New York, 1983. (16) Mitchell, G.; Swift, K. in Time-Resolved Laser Spectroscopy in Biochemistry II; Lakowicz, J. R., Ed.; SPIE 1204; SPIEsThe International Society for Optical Engineering: Bellingham, WA, 1990; Part 1, pp 270-274.
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ratio, m, expected at an angular modulation frequency, ω, for a fluorophore with a single exponential decay can be calculated from
φ ) tan-1 (ωτ)
(1)
m ) (1 + ω2τ2)-1/2
(2)
and
where τ is the lifetime of the fluorophore. Although the phase and modulation responses are rigorously measured with respect to the excitation beam, this is difficult in practice. Instead, the sample phase and modulation are measured relative to a reference, such as a scattering solution with an effective lifetime of zero or a fluorophore with known lifetime. Because of this, the measured phase shift, φmeas, and measured modulation ratio, mmeas, include the reference phase and modulation, as described by the equations
φmeas ) φs - φr
(3)
mmeas ) ms/mr
(4)
where the s and r subscripts denote responses for sample and reference, respectively. After the phase and modulation responses of the sample alone have been determined through eqs 3 and 4, they can be used in eqs 1 and 2 to calculate the lifetime of a single exponential decay. However, in cases where the fluorescence emission of the sample contains more than one exponential decay, the phase and modulation responses need to be fit by more complicated equations involving nonlinear weighting of various lifetime components.6,15 Traditionally, equations that contain terms for only an expected number of fluorescent components are solved through the use of standard nonlinear least-squares (NLLS) methods. As an alternate approach, a large number of exponential terms (“cells”) using closely spaced lifetimes can be used to approximate any real combination or distribution of lifetimes. This approach is used by MEM in conjunction with a statistical entropy function to select a single distribution.2,3 Unlike NLLS, MEM is self-modeling in that it provides a unique solution and does not require a priori modeling of the system. It has recently been shown that MEM provides a more consistent and less biased analysis of frequency-domain data, as well as a unique view of the uncertainty in the data.1,5,6 However, there have been no systematic studies of the effects of systematic errors on distributions recovered by MEM from frequency-domain data. It is expected that the manifestation of systematic errors will be different in the self-modeling MEM from that in NLLS in which the errors must be accounted for by the values and fractional intensities of the designated number of lifetime components. In the present work, both MEM and NLLS were used in their most general forms, without expectation of error from any particular source or incorporation of mechanisms for identification and correction of such errors, in order to evaluate the effects of the errors on an “unsuspecting” algorithm. Reference Lifetime Errors. Correction of the measured responses for the reference phase and modulation, using eqs 3 and 4, is one potential source of error. If the reference is a scattering solution, then the measured phase and modulation ratio in eqs 3 and 4 reduce to those of the sample. If a reference 612
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fluorophore is used, then the reference phase shift, φr′, and reference modulation ratio, mr′, can be calculated using eqs 1 and 2. If the predetermined lifetime for the reference solution is correct, then the sample phase shift and sample modulation ratio can be obtained through eqs 3 and 4. If, however, the predetermined lifetime is incorrect, the following two equations result:
φmeas + φr′ ) φs - φr + φr′ * φs
(5)
mmeasmr′ ) (ms/mr)mr′ * ms
(6)
in which the r subscript denotes the actual reference phase shift and modulation ratio and the r′ subscript denotes the reference phase shift and modulation ratio calculated from the incorrect reference lifetime. While it has been suggested that fluorophores with similar emission and lifetime characteristics be used to avoid these inaccuracies,12,17 this is not always experimentally possible. Litwiler and co-workers have shown that reference lifetime inaccuracy greatly complicates lifetime analysis,13 causing oversimplification of distributed exponential decays as well as shifting of lifetime peaks and the appearance of anomalous lifetime components. Correction of reference lifetime errors, through inclusion of an extra lifetime component in a standard NLLS decay model, was demonstrated for several simulations in the absence of random noise. The results described here for both simulations and real data show that MEM can be used to indicate, diagnose, and correct reference lifetime errors. The results also demonstrate the effects of random noise on the diagnosis of a reference lifetime error in either MEM or NLLS analysis, leading to conclusions that differ from those described13 for noise-free data simulations. Timing and Mismatched Intensity Errors. Because of the dynamic nature of the frequency-domain experiment, accurate measurement of the phase and modulation responses requires photovoltaic detectors that are sensitive to intensity changes at low light levels. Although different detectors will have different response characteristics, two dynamic anomalies are often seen for commonly used detectors as a result of their sensitivity: timing and mismatched intensity errors.7-11,18 Timing errors result when one of several events causes a change in the time it takes for an electronic response to be measured following the impact of a photon on the detector. Several physical characteristics of the detected light have an effect on the detector delay time, or “transit time”, including wavelength/ energy of the photons and spatial positioning of incident light on the detector face.7-9,18 If the timing delay is different for the reference and sample measurements, the difference will introduce an error into the measured phase shift such that
φmeas ) φs - φr + ω∆t
(7)
where ∆t is a constant describing the difference in transit time observed for the sample and reference conditions.11 Although avoidance of the error is preferable, determination and correction of timing errors has also been shown.11,17 Empirical determination of ∆t requires a reference fluorophore of “known” lifetime which excites and emits at the same wavelengths as the (17) Lakowicz, J. R.; Cherek, H.; Balter, A. J Biochem. Biophys. Methods 1981, 5, 131-146. (18) Weber, G.; Jameson, D. M. J. Phys. Chem. 1981, 85, 953-958.
sample. The correction factor is determined for specific experimental conditions (e.g., detector voltage and wavelength) and will vary as the conditions vary. Additionally, although many experiments permit timing error correction, data collected with broad wavelength filters (band-pass > 10 nm)5 may not be readily correctable. If the error is not corrected prior to lifetime determination, problems will arise because the linear error is not easily accounted for by the nonlinear equations used in the lifetime determination algorithms. Also, because the error is only in phase, different fits are needed for the phase and modulation data, which can cause convergence problems in the fitting algorithms. Similarly, the effect of mismatched sample and reference intensities can complicate lifetime analysis.15,19 If the sample and reference solutions are not matched with respect to total intensity incident on the detector, the responses of the detector to the modulated portions of the fluorescence of the two solutions will be different. As a result, modulation ratios, which are measured as a ratio of modulated intensity to nonmodulated intensity (ac/ dc), will have an additional factor complicating their response. The ability of MEM to deal with unexpected components through self-modeling may allow for improved lifetime analysis in the presence of both timing and mismatched intensity errors, as well as improved indication of their presence. In this paper we will show examples of both of these detector response errors and their effects on MEM results. EXPERIMENTAL SECTION Data Collection and Simulation. Several different ethanolic solutions (100%, AAper) were made from benzo[k]fluoranthene (B(k)F, AccuStandard), benzo[a]pyrene (B(a)P, AccuStandard), chrysene (AccuStandard), and 1,4-bis(5-phenyl-2-oxazolyl)benzene (POPOP, Packard, scintillation grade). Measurements were made using a commercial, multifrequency phase-modulation spectrofluorometer (Model 48000 MHF, SLM Instruments, Rochester, NY), which uses the multiharmonic Fourier transform technology described elsewhere.6,16 The 325 nm line of a continuous-wave helium-cadmium laser was used for excitation, and the fluorescence emission was passed through one or more optical filters to eliminate both Rayleigh and Raman scatter in the signal. The instrument makes use of a standard photomultiplier tube (PMT, Model R928, Hamamatsu Corp., Bridgewater, NJ) for detection of the fluorescence. Unless otherwise specified, solutions were measured versus a scatter reference solution, exhibiting an effective lifetime of zero, which was matched in intensity to the sample. Multifrequency phase and modulation data were collected from 4.1 to 205 MHz at 4.1 MHz intervals. Phase and modulation data were simulated for several systems, detailed in the Results and Discussion section, using the MATLAB (Mathworks, Natick, MA) software package. The equations outlined in Part 1 of this series6 were used to calculate phase and modulation values at the same frequencies measured for the real systems. Simulated noise in the phase and modulation data, based on a normal distribution, was added either as constant (linear, or frequency-independent) absolute standard deviations for phase and modulation or as nonlinear standard deviations, for which an empirically determined relationship between frequency and standard deviation was used as previously described in Part 1 of this series.6 For all experiments, 15 replicates were simulated to (19) Weiner, E. R.; Goldberg, M. C. Am. Lab. 1986, 18 (11), 138-139.
minimize any artifacts arising from artificial generation of the noise. Specific sample compositions and frequency ranges are given with the results. Lifetime Distribution Recovery and Analysis. The MEM algorithm used in these studies was purchased from Maximum Entropy Data Consultant Ltd. (Cambridge, UK) and was run on a SPARCstation 2 (Sun Microsystems, Mountain View, CA). The algorithm has been described in detail elsewhere.1,2 Except where noted, lifetime windows contained 200 discrete, exponentially spaced cells from 10 ps to 50 ns. The recovered lifetime distributions were further analyzed using various in-house routines written in MATLAB. Peak lifetime and peak width (full width at half maximum) were determined after 8-to-1 spline interpolation of relevant regions of the distribution. A routine packaged with MATLAB was used for the spline calculation, using the cell maxima as spline anchors. Peak area, which is proportional to fractional intensity for a single (homogeneous) lifetime component, was measured by numerical integration of all cells under the peak. Standard NLLS results were obtained from a commercial software package (Globals Unlimited, Urbana, IL). For these analyses, several models were investigated, and, except where noted, only the answers provided by the model with the lowest χ2 are reported. RESULTS AND DISCUSSION Reference Lifetime Errors. It is clear from the work of Litwiler and co-workers that an incorrect reference lifetime in the absence of random noise has a significant effect on frequencydomain lifetime analysis.13 It is important, however, to consider how these errors manifest themselves in the presence of random noise. To ascertain their effect on both MEM and NLLS analyses of real data, reference lifetime errors were investigated for a 3 µM solution of B(a)P (τ ≈ 14.8 ns) using an intensity-matched B(k)F solution (τ ≈ 7.8 ns) as a reference. Lifetimes for the B(k)F reference were varied between 7.0 and 8.8 ns, in 0.2 ns intervals. The results for both one- and two-component NLLS fits are shown in Table 1 for the various assignments of reference lifetime. When the assumed reference lifetime (τr′) is below 8.2 ns, two lifetime components are required to describe the data. One lifetime, attributed to B(a)P, is found at 14.4 ns. However, a second, anomalous lifetime is also found that varies from 0 to 5 ns and accounts for 0-10% of the intensity. This component is attributed to the noncancellation error shown in eqs 5 and 6 and is similar to noise-free simulations.13 The anomalous component decreases in lifetime and fractional intensity as τr′ approaches 8.0 ns. When τr′ ) 8.2 ns, both the one- and two-component models give the same answer; that is, a single lifetime of 14.8 ns is recovered. The χ2 values for both one- and two-component models at τr′ ) 8.2 ns are also nearly identical, indicating the cancellation of φr and mr in eqs 5 and 6. It can therefore be ascertained that the true reference lifetime for these experimental conditions is in the range of 8.0-8.2 ns. This conclusion is further supported by the increase in χ2 above a τr′ of 8.2 ns and the reappearance of the anomalous peak above 8.4 ns. The lifetimes observed for the anomalous peak are, however, in disagreement with the results shown by Litwiler and coworkers.13 In their simulations, the anomalous peak was observed at the same lifetime as τr′. This discrepancy can be traced to the presence of random noise in the present work. When a noisefree simulation of a 15 ns sample and a 7.8 ns reference is analyzed Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
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Table 1. Lifetimes (τi, ns) and Fractional Intensities (fi) Recovered by NLLS Analysis of B(a)P for Various Assumed Lifetimes (τr′) for the B(k)F Reference one-component
two-component
τr′
τ1
χ2
τ1 (f1)
τ2 (f2)
χ2
7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8
13.0 13.3 13.6 13.9 14.2 14.5 14.8 15.1 15.3 15.6
>300 >200 >100 88.9 44.2 16.9 5.88 9.92 28.0 59.2
5.7 (0.10) 5.6 (0.08) 5.3 (0.05) 4.8 (0.03) 3.7 (0.01) 1.8 (4 × 10-3) 4 × 10-3 (4 × 10-4) 23 (4 × 10-5) 11.5 (-0.31) 11.1 (-0.33)
14.4 (0.90) 14.4 (0.92) 14.4 (0.95) 14.4 (0.97) 14.5 (0.99) 14.6 (1.00) 14.8 (1.00) 15.0 (1.00) 14.3 (1.31) 14.3 (1.33)
1.96 1.91 1.81 1.62 1.24 0.56 4.49 10.3 4.11 3.89
as described above, the lifetime of the second component always matches that of the assumed reference lifetime, in agreement with the previous findings.13 However, when random noise is added to the simulation, the results agree with those found here for the real system in that the lifetime of the anomalous peak is not equal to τr′. Because of this, care should be taken not to misinterpret this anomalous peak as an additional component in the sample. In theory, the technique for reference lifetime correction previously proposed,13 which involves fixing the lifetime of an extra component to τr′ and allowing only the fractional intensity to vary, should still work in the presence of random noise, although minimization of χ2 may be more difficult. The results obtained by MEM for the same range of B(k)F reference lifetimes are comparable to those of the NLLS analysis but are generally more consistent. When τr′ e 8.0 ns (Figure 1A), an anomalous peak is observed between 4 and 7 ns, with decreasing lifetime and fractional intensity as τr′ approaches 8.0 ns. All distributions indicated a lifetime of 14.5 or 14.6 ns for the B(a)P. At τr′ g 8.0 ns (Figure 1B), only a single peak is recovered. The peak is significantly narrowed and begins to shift toward longer lifetimes as the reference lifetime is increased above 8.0 ns (Figure 1C). The significance of the peak narrowing will be addressed later. From the disappearance of the anomalous peak, and the subsequent overnarrowing of peaks, we infer that the true reference lifetime (τr) is between 8.0 and 8.2 ns, in agreement with the NLLS results. Similar trends were observed for MEM analysis of several multicomponent systems. Phase shift and modulation ratio were measured in triplicate for the following three solutions using scattered light as a reference: a binary mixture of 21 nM B(k)F and 120 nM POPOP, a binary mixture of 32 nM B(k)F and 1.8 µM chrysene, and a ternary mixture of 28 nM B(k)F, 40 nM POPOP, and 1.2 µM chrysene. An unexpected, short-lifetime peak was observed in distributions recovered for all three samples when a reference lifetime of zero was assumed for the scattering solution. Based on the assumption that this was an anomaly due to an incorrect reference lifetime, additional distributions were recovered for each sample using τr′ values between 5 and 100 ps, in 5 ps steps. Three of the distributions for the binary system of B(k)F and POPOP are shown in Figure 1D-F. As τr′ was increased, the lifetime of the anomalous peak increased until it coalesced into the next longer lifetime peak. No other peaks appear to shift as a result of the reference lifetime change up to this point. Beyond the τr′ at which the anomalous peak disappeared, which is assumed to be the true reference lifetime (τr), 614 Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
there was a sudden overnarrowing of the remaining peaks, similar to what was observed for the B(a)P solution, and all peaks began to shift to longer lifetimes. To better interpret changes in the distributions recovered from multicomponent samples, a simulation of the B(k)F and POPOP mixture was created using a true reference lifetime of 43 ps and a constant, random noise of (0.75% of full scale for phase (i.e., (0.675°) and (0.25% of full scale for modulation (i.e., (0.0025). Lifetime distributions were recovered from the simulated data for all τr′ between 0 and 100 ps, in 5 ps steps. Although no anomalous peaks were observed in the distributions recovered from the simulated system for any τr′, the characteristic narrowing and shifting were observed. The absence of anomalous peaks is attributed to imperfect noise simulation. Recovered distributions were analyzed for peak width, lifetime, and area as a function of τr′ (Figure 2A, B, and C, respectively). To facilitate comparison of the relative B(k)F and POPOP peak changes, the peak widths and peak lifetimes were scaled by the lifetime and median lifetime (for all τr′ values) of the component, respectively. Additionally, to clarify trends, peak area is given as the change in peak area with change in τr′ (δarea/δτr′). In each figure, the broken vertical line represents the true reference lifetime of 43 ps. All three distribution characteristics show distinct trends as τr′ increases. The peak widths remain fairly constant when τr′ < τr, decrease sharply near τr′ ) τr, and then remain fairly constant when τr′ > τr. Interestingly, the width of the POPOP peak is reduced by 50% and the B(k)F peak starts to narrow before the reference lifetime is reached. When τr′ ) τr, the peak widths are not altered by reference lifetime errors and, thus, reflect the true measurement uncertainty.6 In a similar indication of the transition through the true reference lifetime, the peak lifetimes remain constant up to τr′ ) 45 ps and then slowly increase. The most interesting trend, and the best indicator of correct reference lifetime, is seen, however, in peak area. The peak area of B(k)F increases suddenly just before τr′ ) τr and then decreases as τr′ is further increased. Simultaneously, the POPOP peak undergoes a sharp decrease in area, followed by a return to a slow, steady increase. The trends in peak width, lifetime, and area for the real B(k)F and POPOP data are similar to those for the simulated data. Initially, the peak widths observed for the real data (Figure 2D) increase as τr′ approaches 30 ps. The POPOP peak then rapidly narrows between 35 and 50 ps, followed by the B(k)F peak as τr′ further increases. The initial increase, which was not seen for the simulated data and may be due to the coalescing of the
Figure 1. Distributions recovered by MEM for various assumed reference lifetimes (indicated in plot) for (A-C) B(a)P measured using B(k)F reference and (D-F) a mixture of B(k)F and POPOP measured using scattered light reference. Distribution windows included lifetimes from 10 to 50 ps, although only a subset is shown.
anomalous peak and the POPOP peak, does not hamper identification of the true reference lifetime, which is still indicated by the sudden decrease in peak width. The observed lifetime of the POPOP peak is also affected by the coalescing anomalous peak, as evidenced by its rapid decrease around τr′ ) 30 ps. Its slow rise when τr′ > 35 ps is, however, similar to the trend in the simulation at τr′ g τr (Figure 2E). Additionally, the B(k)F peak lifetime shows a smooth trend with only a slight decrease before the slow increase in lifetime. These observations of peak width and peak lifetime indicate that the true reference lifetime is between 35 and 40 ps. Again, however, the most significant and useful trends are seen in the peak area changes (Figure 2F). The area of the longer lived B(k)F peak shows a trend identical to that seen in the simulation, in which there is a sharp increase in peak area (at 35 ps in the real data), followed by a slow decrease at longer τr′. The change in the POPOP peak area is also comparable to the simulations in that a sudden decrease occurs in conjunction with the B(k)F area change at 35 ps. There is, however, an increase in POPOP peak area at 20 ps as the peak is adjusted for the inclusion of the anomalous peak. Despite this dissimilarity in
simulation and real data, the changes in area strongly indicate that a reference lifetime of 35-40 ps is appropriate for this sample. Analysis of the two additional replicates produced similar results, although reference lifetimes varied by nearly 10 ps (30%) because of poor instrumental precision for such short lifetimes. Additionally, analysis of only the modulation data produced results nearly identical to those just described, indicating that color effect (see below), if present under these experimental conditions, did not significantly affect the results. Although the change in peak area is useful as an indicator of the true reference lifetime, its absolute magnitude is not greatly affected by reference lifetime errors. Both the POPOP and B(k)F fractional intensities changed by only about 5% across the entire 100 ps range of τr′, and changes of only 1.4% for B(k)F and 0.6% for POPOP were observed between 20 and 55 ps. Similarly, peak lifetimes of POPOP and B(k)F changed by only about 5% and 2%, respectively, over the 20-55 ps range. In contrast, peak width is much more dependent on τr′. Although peak width has been shown to be strongly correlated to measurement precision,6,20 peak narrowing above the true reference lifetime is due not to improved precision but to Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
615
Figure 2. Recovered values for relative peak width, relative peak lifetime, and change in peak area as a function of assumed reference lifetime for a binary mixture of B(k)F and POPOP from (A-C) simulated data and (D-F) real data. The broken vertical line in A-C indicates the simulated true reference lifetime of 43 ps, and the shaded bar in D-F indicates the probable true reference lifetime for the real system. Peak widths and lifetimes are scaled to the peak lifetime and mean lifetime, respectively. All vertical axes are arbitrarily scaled.
systematic inclusion of the φr - φr′ and mr′/mr terms from eqs 5 and 6. The systematic overnarrowing can be clearly seen in the width recovered from the simulation of a distributed component (Figure 3). A component with a Gaussian distribution of lifetimes, centered at 34 ns with a width of 21.7 ns at 4% peak height, was simulated with a true reference of 7.8 ns and a nonlinear, random error, as previously described.6 This simulation was then performed for τr′ between 6.3 and 9.3 ns, in 0.1 ns steps. Notice that the recovered width only matches the true width (within about 5% error) for τr′ in the range of 7.5-7.8 ns and that the recovered width is nearly 25% of the true width when τr′ exceeds τr by only 0.1 ns (τr′ ) 7.9 ns). The same trend is seen for the ternary mixture of POPOP, B(k)F, and chrysene. When τr′ is near the true value of ∼30 ps, a single broad peak centered near 9 ns is observed for B(k)F and chrysene (Figure 4A). Measurement imprecision precludes the (20) Brochon, J. C. In Numerical Computer Methods; Johnson, M. L., Ed.; Methods in Enzymology; Academic Press: San Deigo, CA, 1994; pp 262-311.
616 Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
resolution of these two components.6 However, in the distribution recovered when τr′ . τr (Figure 4B), the peak associated with B(k)F and chrysene is narrowed, reducing the long-lifetime edge of the peak to a shoulder. For all τr′ values examined, all peaks were narrowed, but no peaks attributed to real components were ever lost. Thus, artificially increasing τr′ may help to distinguish between anomalous peaks and real peaks. By such an analysis, a peak observed in the B(k)F/chrysene binary mixture that was thought to be caused by a reference lifetime error was instead identified as trace POPOP contamination (Figure 5). Varying τr′ is not always successful for identification of the true reference lifetime. When this technique was used on real data for several different complex samples containing a significant number of components, or a number of broadly distributed components (e.g., see ref 5), no change in distribution was observed with large changes in τr′. It appears that the complex signal of these samples is not greatly affected, even by severe reference lifetime errors (7.8 ns ( 1.0 ns), nor does MEM
Figure 3. Peak width (full width, 4% peak height) recovered by MEM from the simulation of a distributed component as a function of assumed reference lifetime. The broken horizontal line indicates the true width of 21.7 ns, and the broken vertical line indicates the simulated true reference lifetime of 7.8 ns.
Figure 5. Distributions recovered by MEM for a mixture believed to contain only B(k)F and chrysene both below the empirically determined, true reference lifetime, τr, and above τr. The arrow indicates a peak attributed to an impurity of POPOP present in the solution. The nonartifactual nature of the peak is suggested by its persistence after the assumed reference lifetime was intentionally raised above the true reference lifetime.
Figure 4. Distributions recovered by MEM for a ternary mixture of POPOP, B(k)F, and chrysene at and above the empirically determined, true reference lifetime, τr. The arrow indicates the artifactual narrowing of the joint B(k)F/chrysene peak above τr.
artificially simplify the distribution in response to changes in the assumed reference lifetime. In these cases, distributional information is limited by data precision and quantity instead of systematic error. Timing Errors. To quantitate timing and mismatched intensity (MMI) errors for the MHF instrument and to verify that the characteristics of these errors are similar to results previously reported for other instruments,7-9,11 several samples were run under conditions that would cause the errors. To quantitate
timing delays, a sample of 6.3 µM B(k)F was measured using an identical B(k)F solution as a reference. Because the sample itself is used as the reference, measured phase shifts of 0° are expected (φr ) φs in eq 3). However, by the use of different emission channel filters for the sample and reference measurements, a wavelength-related timing delay was introduced into the phase measurement. By using a 600 nm short pass filter (transmission range of 380600 nm) for the reference measurements, and various band-pass filters for the sample measurements, a range of timing delays were measured. Band-pass filters of 400, 420, 440, and 470 nm, when used for the sample measurement, provided timing errors of -1.3, -1.4, -0.3, and 0.0 ns, respectively. All of the measured phase shifts were negative, indicating that the average transit time of the sample signal was less then that of the reference signal and, thus, φs < φr. The lower transit time is the result of shorter wavelength (i.e., higher energy) photons being observed for the sample measurements. If the sample and reference filters are switched, then positive phase shifts and, consequently, positive timing errors are observed. The magnitudes of the timing errors are somewhat large, possibly due to the use of the broad (>200 nm) emission window for the reference fluorophore and narrow band-pass filters (10 nm band-pass) for the samples. To determine the effect of timing errors on lifetime recovery, a two-component sample containing POPOP and B(k)F was simulated using lifetimes of 1.32 and 7.8 ns, respectively, with a fractional intensity ratio of 7:3. Noise and timing errors were then introduced into the simulation by two separate methods. First, the real noise and timing errors observed for the four band-pass Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
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Table 2. Lifetimes (τi, ns) and Fractional Intensities (fi) Recovered by MEM and NLLS Analyses of Simulated Phase and Modulation Data Containing Real Noise and Timing Error Measured for Four Different Filter Combinations NLLS
c
filtera
∆tb
POPOP τ1 (f1)
400 420 440 470
1.3 1.4 0.3 0.0
1.28 (0.69) 1.27 (0.69) 1.31 (0.70) 1.32 (0.70)
MEM
B(k)F τ2 (f2)
otherc τ3 (f3)
7.5 (0.29) 7.4 (0.29) 7.8 (0.30) 7.8 (0.30)
11.7 (0.01) 11.3 (0.03) 25 (1 × 10-3)
χ2
POPOP τ1 (f1)
B(k)F τ2 (f2)
0.89 0.98 0.42 1.61
1.36 (0.70) 1.31 (0.69) 1.31 (0.70) 1.31 (0.70)
7.7 (0.30) 7.7 (0.30) 7.7 (0.30) 7.7 (0.30)
a Wavelength (nm) of band-pass used to collect noise and timing delay error, as described in text. b Observed timing delay error in nanoseconds. Third component in three-component fits.
Table 3. Lifetimes (τi, ns) and Fractional Intensities (fi) Recovered by MEM and NLLS Analyses of Simulated Phase and Modulation Data Containing Simulated Noise and Timing Error (∆t)a NLLS
a
B(k)F τ2 (f2)
B(k)F τ2 (f2)
∆t(ns)
-1.2 -0.7 -0.2 0.0 +0.2 +0.7 +1.2
1.26 (0.68) 1.28 (0.69) 1.31 (0.70) 1.32 (0.70) 1.32 (0.70) 1.36 (0.71) 1.38 (0.72)
Constant Standard Deviation 7.4 (0.32) 2.92 7.5 (0.31) 1.59 7.7 (0.30) 0.97 7.8 (0.30) 0.93 7.9 (0.30) 1.01 8.1 (0.29) 1.77 8.2 (0.28) 3.34
1.32 (0.69) 1.32 (0.69) 1.32 (0.70) 1.32 (0.70) 1.32 (0.70) 1.37 (0.71) 1.37 (0.72)
7.9 (0.30) 7.9 (0.30) 7.9 (0.30) 7.9 (0.30) 7.9 (0.30) 8.3 (0.29) 8.6 (0.28)
-1.2 -0.7 -0.2 0.0 +0.2 +0.7 +1.2
1.28 (0.69) 1.29 (0.70) 1.31 (0.70) 1.32 (0.70) 1.33 (0.70) 1.34 (0.70) 1.36 (0.71)
Nonlinear Standard Deviation 7.7 (0.31) 3.99 7.7 (0.30) 1.86 7.8 (0.30) 0.94 7.8 (0.30) 0.93 7.8 (0.30) 1.12 7.9 (0.30) 2.48 7.9 (0.29) 5.15
1.32 (0.69) 1.32 (0.69) 1.32 (0.70) 1.32 (0.70) 1.32 (0.70) 1.37 (0.70) 1.37 (0.71)
7.9 (0.30) 7.9 (0.30) 7.9 (0.30) 7.9 (0.30) 7.9 (0.30) 7.9 (0.30) 7.9 (0.29)
χ2
Noise was simulated with both frequency-independent (constant) and frequency-dependent (nonlinear) standard deviations.
filters were added to the simulated data to give four data sets with the timing errors listed above. Second, simulated noise and timing errors were added. For these simulations, positive and negative timing errors of 1.2, 0.7, and 0.2 ns were calculated and introduced through eq 7. Data without a timing error were also simulated. Random noise was added with one of two different profiles: either a constant standard deviation of (0.2% full scale (i.e., (0.18° in phase and (0.002 in modulation) for all frequencies or a nonlinear, frequency-dependent standard deviation based on that observed for real data.6 The same series of normally distributed random numbers, scaled by the appropriate standard deviation, was used for all simulated data to facilitate comparison of recovered distributions. While the lifetimes and fractional intensities recovered by both MEM and NLLS for the case of real timing errors are very similar for all four levels of error (Table 2), the MEM results showed better internal consistency. Except for the lifetime of POPOP recovered from the data containing the 1.3 ns timing error, identical distributions were recovered by MEM for all four levels of error. Increased peak widths were observed for the data containing the highest ∆t errors, which were those collected with the 400 and 420 nm band-pass filters. When the same data were analyzed by NLLS, three-component models provided the best fits for all error levels except for the 470 nm band-pass filter, which required only two components. Notice the significant variation of both POPOP and B(k)F lifetimes with the different errors. If 618
MEM POPOP τ1 (f1)
POPOP τ1 (f1)
Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
the χ2 criterion is ignored and all of the data are fit using twocomponent models (not shown), the NLLS results are a little more consistent. In practice, however, selection of the simpler model might be difficult to justify without a priori knowledge of the system. The ability of MEM to provide more consistent lifetime recovery than NLLS is further indicated by the results for simulated data containing simulated error and noise. Results of MEM and NLLS analyses are shown in Table 3 for both constant and frequency-dependent noise. Although two-component models provided very reasonable χ2 values for all NLLS fits, the results for the constant noise simulations show lifetimes varying from 1.26 to 1.38 ns for POPOP and from 7.4 to 8.2 ns for B(k)F. The recovered fractional intensities are also mildly affected by the different timing errors. Although the variations are less in the frequency-dependent noise simulations, they are still significant relative to the consistent MEM results. Similar to what was seen in the presence of real error, the MEM results are impressively consistent for all simulated timing errors below +0.2 ns. Some of this consistency can be attributed to the inclusion of a very small amount (1% or less, depending on the ∆t) of the shortest lifetime cell in the window (10 ps), which helps to account for some of the systematic error. Apparently, this adjustment is not possible at the two highest timing errors since, at timing delays of +1.2 and +0.7 ns, the B(k)F and POPOP peaks become narrow and are shifted. The shifts are, however,
Table 4. Lifetimes (τi, ns) and Fractional Intensities (fi) Recovered by MEM and NLLS Analyses of B(k)F Using a POPOP Reference with Various Induced Mismatched Intensity Errors NLLS two-compc
one-comp ratioa
∆Ib
1:1 2:1 3:1 4:1 5:1
0.99 0.96 0.98 0.87 0.87
τ1
χ2
7.6 7.8 7.6 8.3 8.2
11.7 94.4 11.6 >600 >250
MEM
τ1 (f1)
χ2
τ1 (f1)
τ2 (f2)
7.5 (0.98) 7.5 (0.95) 7.7 (0.98) 8.6 (0.98) 8.5 (0.98)
1.37 1.50 4.75 >500 >200
7.6 (1.00) 7.6 (0.99) 7.6 (0.99) 7.6 (0.99) 7.6 (0.99)
1.7 (0.01) 2.1 (0.01) 1.7 (0.01) 2.4 (0.01)
a Intensity ratio of B(k)F to POPOP. b Mismatch constant (see eq 8), empirically determined from fit to phase data. c The τ values were 4 × 102, 2 9 × 102, 1.88, 3 × 10-7, and 1 × 10-7 ns, respectively.
much smaller when nonlinear noise is used, and lifetime and fractional intensity recoveries are nearly perfect. Mismatched Intensity Errors. Because little prior work could be found about the manifestation of MMI errors in frequency-domain data, in initial studies this error was induced, and subsequently quantitated, in the measurement of a 6.3 µM solution of B(k)F. Using an intensity-matched POPOP solution as the reference, approximate intensity ratios (B(k)F:POPOP) of 1:1, 2:1, 3:1, 4:1, and 5:1 were induced by placing neutral density filters in the emission channel for the reference measurements. Filtering of scattered light was performed for both samples by placing a 400 nm band-pass filter in the light path after the neutral density filters. Phase and modulation responses were measured for each intensity combination, and, although all of the phase responses were identical within measurement precision, the modulation responses varied with the intensity ratio. Using a reference lifetime of 1.32 ns for POPOP, the lifetime of B(k)F was determined from the phase data only, using standard NLLS analysis. The B(k)F lifetime was then used to calculate the expected modulation (mtrue), and this was compared to the observed modulation (mobs) for each intensity ratio. The relationship between mobs and mtrue can be approximated by
mobs ) ∆Imtrue
(8)
where ∆I is an empirically determined constant describing the extent of MMI error. A constant of 1.00 indicates no error. We will refer to MMI errors with ∆I < 1.00 and ∆I > 1.00 as negative and positive errors, respectively. NLLS and MEM analyses of the combined phase and modulation data provided the lifetime results shown in Table 4. Also shown are the five ∆I values determined from the phase-only NLLS analysis. It is clear from the NLLS results that, although a twocomponent fit is adequate for the intensity-matched solution, the most mismatched solutions cannot be fit by such models at all. The lifetimes recovered for B(k)F are badly shifted, and the χ2 values for the fits are high. In contrast, MEM was able to recover a well-resolved peak at 7.6 ns for all five intensity ratios without significant appearance of anomalous peaks. To further verify the effects of MMI errors on MEM and NLLS analyses of real data, simulations of data, noise, and MMI error were combined and analyzed. The data were simulated to contain POPOP (τ ) 1.32 ns) and B(k)F (τ ) 7.8 ns) in a 7:3 fractional intensity ratio. Error was introduced into the modulation through
eq 8 using ∆I values of 0.85, 0.90, 0.95, 1.00, 1.05, 1.10, and 1.15. Noise was simulated using both constant and nonlinear standard deviations, as in the color effect simulations. The results of MEM and NLLS analyses are shown in Table 5. As for the NLLS analysis of the real MMI data, an extra component was required for the best fit of all data simulated with ∆I values less than 1 (negative errors). For both nonlinear and constant noise, the lifetime recovered for POPOP was relatively constant across the range of negative errors, but the B(k)F lifetime changed considerably. For all positive errors, both lifetimes shifted, and the χ2 values increased. The presence of the third component decreases the fractional intensities recovered for POPOP and B(k)F at negative errors. The ratio of these fractional intensities is, however, always 7:3, as expected. In contrast to the inconsistent NLLS results, MEM consistently recovered correct lifetimes and fractional intensities for both components for all MMI errors. Peak widths also remained essentially constant over the entire ∆I range. In fact, the only effect of the MMI error was a failure to normalize the distribution (i.e., to constrain the sum of fractional intensities to equal unity). This easily identifiable and correctable effect is the response of MEM to a particular behavior of the phase and modulation equations.6 While the calculated phase shift does not change if the fractional intensities are not normalized, the modulation does. The calculated modulation ratio is scaled by exactly the sum of the nonnormalized fractional intensities. Since this is equivalent to the scaling introduced by mismatched intensities (eq 8), it provides a convenient route for their correction. The lack of the normalization constraint in MEM is clearly useful in this situation and could also be used in NLLS analysis. Unlike the real data, no anomalous peaks were recovered in the distributions for the simulated data. These peaks, as well as those observed for reference lifetime errors, are attributed to some synergistic mix of the systematic error and random noise. The imperfect simulation of noise due, for example, to imperfect random number generation could be the cause of the discrepancy between the real and simulated systems. It should be pointed out that, for some unrelated real data containing MMI errors as small as 3%, MEM completely failed to recover any distribution. The failure, evidenced by a recovery of zero fractional intensity for all cells, seems to be related to the number of components and their lifetimes as well as noise. The exact conditions causing the failure are, however, unclear, including whether the presence of MMI error is the cause or simply concomitant with the actual cause. In general, despite these Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
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Table 5. Lifetimes (τi, ns) and Fractional Intensities (fi) Recovered by MEM and NLLS Analyses of Simulated Phase and Modulation Data Containing Simulated Noise and Mismatched Intensity Errora NLLS
MEM
∆Ib
POPOP τ1 (f1)
B(k)F τ2 (f2)
0.85 0.90 0.95 1.00 1.05 1.10 1.15
1.33 (0.60) 1.33 (0.64) 1.32 (0.67) 1.32 (0.70) 1.29 (0.71) 1.27 (0.70) 1.26 (0.69)
Constant Standard Deviation 8.2 (0.25)c 0.97 8.1 (0.27)c 0.94 8.0 (0.28)c 0.92 7.8 (0.30) 0.93 5.8 (0.29) 16.0 4.3 (0.30) 67.6 3.2 (0.31) 172
1.32 (0.69) 1.32 (0.70) 1.32 (0.69) 1.32 (0.70) 1.32 (0.70) 1.32 (0.70) 1.32 (0.70)
7.8 (0.31) 7.8 (0.30) 7.8 (0.31) 7.8 (0.30) 7.8 (0.30) 7.8 (0.30) 7.8 (0.30)
0.85 0.90 0.95 1.00 1.05 1.10 1.15
1.32 (0.67) 1.32 (0.68) 1.31 (0.68) 1.32 (0.70) 1.33 (0.72) 1.36 (0.75) 1.40 (0.79)
Nonlinear Standard Deviation 8.6 (0.27)c 2874 8.3 (0.28)c 1137 8.0 (0.29)c 202 7.8 (0.30) 0.93 6.8 (0.28) 335 5.8 (0.25) 1330 5.0 (0.21) 2988
1.32 (0.69) 1.32 (0.70) 1.32 (0.70) 1.32 (0.70) 1.32 (0.70) 1.32 (0.70) 1.32 (0.70)
7.8 (0.31) 7.8 (0.30) 7.8 (0.30) 7.8 (0.30) 7.8 (0.30) 7.8 (0.30) 7.8 (0.30)
χ2
POPOP τ1 (f1)
B(k)F τ2 (f2)
a Noise was simulated with both frequency-independent (constant) and frequency-dependent (nonlinear) standard deviations. b Mismatch constant (see eq 8). c Three-component fit used; τ3 > 109 ns.
isolated situations, MEM seems to handle MMI error well, with only minor perturbation of the recovered distributions. CONCLUSIONS The investigation of the effects of timing, mismatched intensity, and reference lifetime errors on lifetime recovery by MEM and NLLS analyses has identified some similarities and differences in the ability of the two methods to deal with the errors in frequencydomain lifetime data. Further, the characterization of these effects suggests ways to diagnose and correct for the presence of these errors. Our study of reference lifetime errors has, while not discounting findings for simulated data in the absence of noise,13 shown that the presence of random noise complicates the correction of this error in NLLS analysis as well as in MEM. It is observed, however, that while NLLS results become somewhat consistent when the assumed reference lifetime is near the true reference lifetime, MEM provides better, more consistent results for small errors. Additionally, changes in peak area, lifetime, and width with changes in assumed reference lifetime were identified as useful clues for the identification and correction of reference lifetime errors in MEM analysis. When combined with NLLS results, a very accurate determination of the true reference lifetime should be possible in many cases. We have also shown that MEM analysis provides greater consistency than NLLS in the presence of both mismatched
620 Analytical Chemistry, Vol. 68, No. 4, February 15, 1996
intensity and timing errors. The recovered distributions from both real and simulated data contain few, if any, anomalies over a large range of errors. The lack of anomalies indicates that the consistency is probably afforded by the entropy constraint, which will tend to discount data such as those introduced by the systematic errors that are poor evidence for real lifetime features. Although the specific effects of systematic error may depend upon the system being studied as well as the individual instrumental characteristics, the general nature of these studies provides compelling evidence that MEM improves the consistency of lifetime analysis in the presence of systematic errors and, especially in the presence of reference lifetime error, provides valuable information to aid in the interpretation of lifetime results and correction of the errors. ACKNOWLEDGMENT This work was supported by the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U.S. Department of Energy (Grant No. DE-FG0588ER13931) and the National Institutes of Health (Grant No. 1RO1-HG-01161). Received for review August 16, 1995. Accepted November 27, 1995.X AC950833L X
Abstract published in Advance ACS Abstracts, January 15, 1996.
